The limitations of staggered grid finite differences in plasticity problems

NASA Astrophysics Data System (ADS)

Pranger, Casper; Herrendörfer, Robert; Le Pourhiet, Laetitia

2017-04-01

Most crustal-scale applications operate at grid sizes much larger than those at which plasticity occurs in nature. As a consequence, plastic shear bands often localize to the scale of one grid cell, and numerical ploys — like introducing an artificial length scale — are needed to counter this. If for whatever reasons (good or bad) this is not done, we find that problems may arise due to the fact that in the staggered grid finite difference discretization, unknowns like components of the stress tensor and velocity vector are located in physically different positions. This incurs frequent interpolation, reducing the accuracy of the discretization. For purely stress-dependent plasticity problems the adverse effects might be contained because the magnitude of the stress discontinuity across a plastic shear band is limited. However, we find that when rate-dependence of friction is added in the mix, things become ugly really fast and the already hard-to-solve and highly nonlinear problem of plasticity incurs an extra penalty.

Nonlinear Conservation Laws and Finite Volume Methods

NASA Astrophysics Data System (ADS)

Leveque, Randall J.

Introduction Software Notation Classification of Differential Equations Derivation of Conservation Laws The Euler Equations of Gas Dynamics Dissipative Fluxes Source Terms Radiative Transfer and Isothermal Equations Multi-dimensional Conservation Laws The Shock Tube Problem Mathematical Theory of Hyperbolic Systems Scalar Equations Linear Hyperbolic Systems Nonlinear Systems The Riemann Problem for the Euler Equations Numerical Methods in One Dimension Finite Difference Theory Finite Volume Methods Importance of Conservation Form - Incorrect Shock Speeds Numerical Flux Functions Godunov's Method Approximate Riemann Solvers High-Resolution Methods Other Approaches Boundary Conditions Source Terms and Fractional Steps Unsplit Methods Fractional Step Methods General Formulation of Fractional Step Methods Stiff Source Terms Quasi-stationary Flow and Gravity Multi-dimensional Problems Dimensional Splitting Multi-dimensional Finite Volume Methods Grids and Adaptive Refinement Computational Difficulties Low-Density Flows Discrete Shocks and Viscous Profiles Start-Up Errors Wall Heating Slow-Moving Shocks Grid Orientation Effects Grid-Aligned Shocks Magnetohydrodynamics The MHD Equations One-Dimensional MHD Solving the Riemann Problem Nonstrict Hyperbolicity Stiffness The Divergence of B Riemann Problems in Multi-dimensional MHD Staggered Grids The 8-Wave Riemann Solver Relativistic Hydrodynamics Conservation Laws in Spacetime The Continuity Equation The 4-Momentum of a Particle The Stress-Energy Tensor Finite Volume Methods Multi-dimensional Relativistic Flow Gravitation and General Relativity References

A Non-Dissipative Staggered Fourth-Order Accurate Explicit Finite Difference Scheme for the Time-Domain Maxwell's Equations

NASA Technical Reports Server (NTRS)

Yefet, Amir; Petropoulos, Peter G.

1999-01-01

We consider a divergence-free non-dissipative fourth-order explicit staggered finite difference scheme for the hyperbolic Maxwell's equations. Special one-sided difference operators are derived in order to implement the scheme near metal boundaries and dielectric interfaces. Numerical results show the scheme is long-time stable, and is fourth-order convergent over complex domains that include dielectric interfaces and perfectly conducting surfaces. We also examine the scheme's behavior near metal surfaces that are not aligned with the grid axes, and compare its accuracy to that obtained by the Yee scheme.

A multi-resolution approach to electromagnetic modelling

NASA Astrophysics Data System (ADS)

Cherevatova, M.; Egbert, G. D.; Smirnov, M. Yu

2018-07-01

We present a multi-resolution approach for 3-D magnetotelluric forward modelling. Our approach is motivated by the fact that fine-grid resolution is typically required at shallow levels to adequately represent near surface inhomogeneities, topography and bathymetry, while a much coarser grid may be adequate at depth where the diffusively propagating electromagnetic fields are much smoother. With a conventional structured finite difference grid, the fine discretization required to adequately represent rapid variations near the surface is continued to all depths, resulting in higher computational costs. Increasing the computational efficiency of the forward modelling is especially important for solving regularized inversion problems. We implement a multi-resolution finite difference scheme that allows us to decrease the horizontal grid resolution with depth, as is done with vertical discretization. In our implementation, the multi-resolution grid is represented as a vertical stack of subgrids, with each subgrid being a standard Cartesian tensor product staggered grid. Thus, our approach is similar to the octree discretization previously used for electromagnetic modelling, but simpler in that we allow refinement only with depth. The major difficulty arose in deriving the forward modelling operators on interfaces between adjacent subgrids. We considered three ways of handling the interface layers and suggest a preferable one, which results in similar accuracy as the staggered grid solution, while retaining the symmetry of coefficient matrix. A comparison between multi-resolution and staggered solvers for various models shows that multi-resolution approach improves on computational efficiency without compromising the accuracy of the solution.

A 3D finite element ALE method using an approximate Riemann solution

DOE PAGES

Chiravalle, V. P.; Morgan, N. R.

2016-08-09

Arbitrary Lagrangian–Eulerian finite volume methods that solve a multidimensional Riemann-like problem at the cell center in a staggered grid hydrodynamic (SGH) arrangement have been proposed. This research proposes a new 3D finite element arbitrary Lagrangian–Eulerian SGH method that incorporates a multidimensional Riemann-like problem. Here, two different Riemann jump relations are investigated. A new limiting method that greatly improves the accuracy of the SGH method on isentropic flows is investigated. A remap method that improves upon a well-known mesh relaxation and remapping technique in order to ensure total energy conservation during the remap is also presented. Numerical details and test problemmore » results are presented.« less

A 3D finite element ALE method using an approximate Riemann solution

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chiravalle, V. P.; Morgan, N. R.

Arbitrary Lagrangian–Eulerian finite volume methods that solve a multidimensional Riemann-like problem at the cell center in a staggered grid hydrodynamic (SGH) arrangement have been proposed. This research proposes a new 3D finite element arbitrary Lagrangian–Eulerian SGH method that incorporates a multidimensional Riemann-like problem. Here, two different Riemann jump relations are investigated. A new limiting method that greatly improves the accuracy of the SGH method on isentropic flows is investigated. A remap method that improves upon a well-known mesh relaxation and remapping technique in order to ensure total energy conservation during the remap is also presented. Numerical details and test problemmore » results are presented.« less

A multi-resolution approach to electromagnetic modeling.

NASA Astrophysics Data System (ADS)

Cherevatova, M.; Egbert, G. D.; Smirnov, M. Yu

2018-04-01

We present a multi-resolution approach for three-dimensional magnetotelluric forward modeling. Our approach is motivated by the fact that fine grid resolution is typically required at shallow levels to adequately represent near surface inhomogeneities, topography, and bathymetry, while a much coarser grid may be adequate at depth where the diffusively propagating electromagnetic fields are much smoother. This is especially true for forward modeling required in regularized inversion, where conductivity variations at depth are generally very smooth. With a conventional structured finite-difference grid the fine discretization required to adequately represent rapid variations near the surface are continued to all depths, resulting in higher computational costs. Increasing the computational efficiency of the forward modeling is especially important for solving regularized inversion problems. We implement a multi-resolution finite-difference scheme that allows us to decrease the horizontal grid resolution with depth, as is done with vertical discretization. In our implementation, the multi-resolution grid is represented as a vertical stack of sub-grids, with each sub-grid being a standard Cartesian tensor product staggered grid. Thus, our approach is similar to the octree discretization previously used for electromagnetic modeling, but simpler in that we allow refinement only with depth. The major difficulty arose in deriving the forward modeling operators on interfaces between adjacent sub-grids. We considered three ways of handling the interface layers and suggest a preferable one, which results in similar accuracy as the staggered grid solution, while retaining the symmetry of coefficient matrix. A comparison between multi-resolution and staggered solvers for various models show that multi-resolution approach improves on computational efficiency without compromising the accuracy of the solution.

Compatible diagonal-norm staggered and upwind SBP operators

NASA Astrophysics Data System (ADS)

Mattsson, Ken; O'Reilly, Ossian

2018-01-01

The main motivation with the present study is to achieve a provably stable high-order accurate finite difference discretisation of linear first-order hyperbolic problems on a staggered grid. The use of a staggered grid makes it non-trivial to discretise advective terms. To overcome this difficulty we discretise the advective terms using upwind Summation-By-Parts (SBP) operators, while the remaining terms are discretised using staggered SBP operators. The upwind and staggered SBP operators (for each order of accuracy) are compatible, here meaning that they are based on the same diagonal norms, allowing for energy estimates to be formulated. The boundary conditions are imposed using a penalty (SAT) technique, to guarantee linear stability. The resulting SBP-SAT approximations lead to fully explicit ODE systems. The accuracy and stability properties are demonstrated for linear hyperbolic problems in 1D, and for the 2D linearised Euler equations with constant background flow. The newly derived upwind and staggered SBP operators lead to significantly more accurate numerical approximations, compared with the exclusive usage of (previously derived) central-difference first derivative SBP operators.

JIGSAW-GEO (1.0): Locally Orthogonal Staggered Unstructured Grid Generation for General Circulation Modelling on the Sphere

NASA Technical Reports Server (NTRS)

Engwirda, Darren

2017-01-01

An algorithm for the generation of non-uniform, locally orthogonal staggered unstructured spheroidal grids is described. This technique is designed to generate very high-quality staggered VoronoiDelaunay meshes appropriate for general circulation modelling on the sphere, including applications to atmospheric simulation, ocean-modelling and numerical weather prediction. Using a recently developed Frontal-Delaunay refinement technique, a method for the construction of high-quality unstructured spheroidal Delaunay triangulations is introduced. A locally orthogonal polygonal grid, derived from the associated Voronoi diagram, is computed as the staggered dual. It is shown that use of the Frontal-Delaunay refinement technique allows for the generation of very high-quality unstructured triangulations, satisfying a priori bounds on element size and shape. Grid quality is further improved through the application of hill-climbing-type optimisation techniques. Overall, the algorithm is shown to produce grids with very high element quality and smooth grading characteristics, while imposing relatively low computational expense. A selection of uniform and non-uniform spheroidal grids appropriate for high-resolution, multi-scale general circulation modelling are presented. These grids are shown to satisfy the geometric constraints associated with contemporary unstructured C-grid-type finite-volume models, including the Model for Prediction Across Scales (MPAS-O). The use of user-defined mesh-spacing functions to generate smoothly graded, non-uniform grids for multi-resolution-type studies is discussed in detail.

JIGSAW-GEO (1.0): locally orthogonal staggered unstructured grid generation for general circulation modelling on the sphere

NASA Astrophysics Data System (ADS)

Engwirda, Darren

2017-06-01

An algorithm for the generation of non-uniform, locally orthogonal staggered unstructured spheroidal grids is described. This technique is designed to generate very high-quality staggered Voronoi-Delaunay meshes appropriate for general circulation modelling on the sphere, including applications to atmospheric simulation, ocean-modelling and numerical weather prediction. Using a recently developed Frontal-Delaunay refinement technique, a method for the construction of high-quality unstructured spheroidal Delaunay triangulations is introduced. A locally orthogonal polygonal grid, derived from the associated Voronoi diagram, is computed as the staggered dual. It is shown that use of the Frontal-Delaunay refinement technique allows for the generation of very high-quality unstructured triangulations, satisfying a priori bounds on element size and shape. Grid quality is further improved through the application of hill-climbing-type optimisation techniques. Overall, the algorithm is shown to produce grids with very high element quality and smooth grading characteristics, while imposing relatively low computational expense. A selection of uniform and non-uniform spheroidal grids appropriate for high-resolution, multi-scale general circulation modelling are presented. These grids are shown to satisfy the geometric constraints associated with contemporary unstructured C-grid-type finite-volume models, including the Model for Prediction Across Scales (MPAS-O). The use of user-defined mesh-spacing functions to generate smoothly graded, non-uniform grids for multi-resolution-type studies is discussed in detail.

Calculation of flow about two-dimensional bodies by means of the velocity-vorticity formulation on a staggered grid

NASA Technical Reports Server (NTRS)

Stremel, Paul M.

1991-01-01

A method for calculating the incompressible viscous flow about two-dimensional bodies, utilizing the velocity-vorticity form of the Navier-Stokes equations using a staggered-grid formulation is presented. The solution is obtained by employing an alternative-direction implicit method for the solution of the block tridiagonal matrix resulting from the finite-difference representation of the governing equations. The boundary vorticity and the conservation of mass are calculated implicitly as a part of the solution. The mass conservation is calculated to machine zero for the duration of the computation. Calculations for the flow about a circular cylinder, a 2-pct thick flat plate at 90-deg incidence, an elliptic cylinder at 45-deg incidence, and a NACA 0012, with and without a deflected flap, at - 90-deg incidence are performed and compared with the results of other numerical investigations.

An optimal implicit staggered-grid finite-difference scheme based on the modified Taylor-series expansion with minimax approximation method for elastic modeling

NASA Astrophysics Data System (ADS)

Yang, Lei; Yan, Hongyong; Liu, Hong

2017-03-01

Implicit staggered-grid finite-difference (ISFD) scheme is competitive for its great accuracy and stability, whereas its coefficients are conventionally determined by the Taylor-series expansion (TE) method, leading to a loss in numerical precision. In this paper, we modify the TE method using the minimax approximation (MA), and propose a new optimal ISFD scheme based on the modified TE (MTE) with MA method. The new ISFD scheme takes the advantage of the TE method that guarantees great accuracy at small wavenumbers, and keeps the property of the MA method that keeps the numerical errors within a limited bound at the same time. Thus, it leads to great accuracy for numerical solution of the wave equations. We derive the optimal ISFD coefficients by applying the new method to the construction of the objective function, and using a Remez algorithm to minimize its maximum. Numerical analysis is made in comparison with the conventional TE-based ISFD scheme, indicating that the MTE-based ISFD scheme with appropriate parameters can widen the wavenumber range with high accuracy, and achieve greater precision than the conventional ISFD scheme. The numerical modeling results also demonstrate that the MTE-based ISFD scheme performs well in elastic wave simulation, and is more efficient than the conventional ISFD scheme for elastic modeling.

Flow Applications of the Least Squares Finite Element Method

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan

1998-01-01

The main thrust of the effort has been towards the development, analysis and implementation of the least-squares finite element method (LSFEM) for fluid dynamics and electromagnetics applications. In the past year, there were four major accomplishments: 1) special treatments in computational fluid dynamics and computational electromagnetics, such as upwinding, numerical dissipation, staggered grid, non-equal order elements, operator splitting and preconditioning, edge elements, and vector potential are unnecessary; 2) the analysis of the LSFEM for most partial differential equations can be based on the bounded inverse theorem; 3) the finite difference and finite volume algorithms solve only two Maxwell equations and ignore the divergence equations; and 4) the first numerical simulation of three-dimensional Marangoni-Benard convection was performed using the LSFEM.

The least-squares finite element method for low-mach-number compressible viscous flows

NASA Technical Reports Server (NTRS)

Yu, Sheng-Tao

1994-01-01

The present paper reports the development of the Least-Squares Finite Element Method (LSFEM) for simulating compressible viscous flows at low Mach numbers in which the incompressible flows pose as an extreme. Conventional approach requires special treatments for low-speed flows calculations: finite difference and finite volume methods are based on the use of the staggered grid or the preconditioning technique; and, finite element methods rely on the mixed method and the operator-splitting method. In this paper, however, we show that such difficulty does not exist for the LSFEM and no special treatment is needed. The LSFEM always leads to a symmetric, positive-definite matrix through which the compressible flow equations can be effectively solved. Two numerical examples are included to demonstrate the method: first, driven cavity flows at various Reynolds numbers; and, buoyancy-driven flows with significant density variation. Both examples are calculated by using full compressible flow equations.

Optimal rotated staggered-grid finite-difference schemes for elastic wave modeling in TTI media

NASA Astrophysics Data System (ADS)

Yang, Lei; Yan, Hongyong; Liu, Hong

2015-11-01

The rotated staggered-grid finite-difference (RSFD) is an effective approach for numerical modeling to study the wavefield characteristics in tilted transversely isotropic (TTI) media. But it surfaces from serious numerical dispersion, which directly affects the modeling accuracy. In this paper, we propose two different optimal RSFD schemes based on the sampling approximation (SA) method and the least-squares (LS) method respectively to overcome this problem. We first briefly introduce the RSFD theory, based on which we respectively derive the SA-based RSFD scheme and the LS-based RSFD scheme. Then different forms of analysis are used to compare the SA-based RSFD scheme and the LS-based RSFD scheme with the conventional RSFD scheme, which is based on the Taylor-series expansion (TE) method. The contrast in numerical accuracy analysis verifies the greater accuracy of the two proposed optimal schemes, and indicates that these schemes can effectively widen the wavenumber range with great accuracy compared with the TE-based RSFD scheme. Further comparisons between these two optimal schemes show that at small wavenumbers, the SA-based RSFD scheme performs better, while at large wavenumbers, the LS-based RSFD scheme leads to a smaller error. Finally, the modeling results demonstrate that for the same operator length, the SA-based RSFD scheme and the LS-based RSFD scheme can achieve greater accuracy than the TE-based RSFD scheme, while for the same accuracy, the optimal schemes can adopt shorter difference operators to save computing time.

Dispersion analysis of the Pn -Pn-1DG mixed finite element pair for atmospheric modelling

NASA Astrophysics Data System (ADS)

Melvin, Thomas

2018-02-01

Mixed finite element methods provide a generalisation of staggered grid finite difference methods with a framework to extend the method to high orders. The ability to generate a high order method is appealing for applications on the kind of quasi-uniform grids that are popular for atmospheric modelling, so that the method retains an acceptable level of accuracy even around special points in the grid. The dispersion properties of such schemes are important to study as they provide insight into the numerical adjustment to imbalance that is an important component in atmospheric modelling. This paper extends the recent analysis of the P2 - P1DG pair, that is a quadratic continuous and linear discontinuous finite element pair, to higher polynomial orders and also spectral element type pairs. In common with the previously studied element pair, and also with other schemes such as the spectral element and discontinuous Galerkin methods, increasing the polynomial order is found to provide a more accurate dispersion relation for the well resolved part of the spectrum but at the cost of a number of unphysical spectral gaps. The effects of these spectral gaps are investigated and shown to have a varying impact depending upon the width of the gap. Finally, the tensor product nature of the finite element spaces is exploited to extend the dispersion analysis into two-dimensions.

Finite-difference modeling and dispersion analysis of high-frequency love waves for near-surface applications

USGS Publications Warehouse

Luo, Y.; Xia, J.; Xu, Y.; Zeng, C.; Liu, J.

2010-01-01

Love-wave propagation has been a topic of interest to crustal, earthquake, and engineering seismologists for many years because it is independent of Poisson's ratio and more sensitive to shear (S)-wave velocity changes and layer thickness changes than are Rayleigh waves. It is well known that Love-wave generation requires the existence of a low S-wave velocity layer in a multilayered earth model. In order to study numerically the propagation of Love waves in a layered earth model and dispersion characteristics for near-surface applications, we simulate high-frequency (>5 Hz) Love waves by the staggered-grid finite-difference (FD) method. The air-earth boundary (the shear stress above the free surface) is treated using the stress-imaging technique. We use a two-layer model to demonstrate the accuracy of the staggered-grid modeling scheme. We also simulate four-layer models including a low-velocity layer (LVL) or a high-velocity layer (HVL) to analyze dispersive energy characteristics for near-surface applications. Results demonstrate that: (1) the staggered-grid FD code and stress-imaging technique are suitable for treating the free-surface boundary conditions for Love-wave modeling, (2) Love-wave inversion should be treated with extra care when a LVL exists because of a lack of LVL information in dispersions aggravating uncertainties in the inversion procedure, and (3) energy of high modes in a low-frequency range is very weak, so that it is difficult to estimate the cutoff frequency accurately, and "mode-crossing" occurs between the second higher and third higher modes when a HVL exists. ?? 2010 Birkh??user / Springer Basel AG.

Hybrid multicore/vectorisation technique applied to the elastic wave equation on a staggered grid

NASA Astrophysics Data System (ADS)

Titarenko, Sofya; Hildyard, Mark

2017-07-01

In modern physics it has become common to find the solution of a problem by solving numerically a set of PDEs. Whether solving them on a finite difference grid or by a finite element approach, the main calculations are often applied to a stencil structure. In the last decade it has become usual to work with so called big data problems where calculations are very heavy and accelerators and modern architectures are widely used. Although CPU and GPU clusters are often used to solve such problems, parallelisation of any calculation ideally starts from a single processor optimisation. Unfortunately, it is impossible to vectorise a stencil structured loop with high level instructions. In this paper we suggest a new approach to rearranging the data structure which makes it possible to apply high level vectorisation instructions to a stencil loop and which results in significant acceleration. The suggested method allows further acceleration if shared memory APIs are used. We show the effectiveness of the method by applying it to an elastic wave propagation problem on a finite difference grid. We have chosen Intel architecture for the test problem and OpenMP (Open Multi-Processing) since they are extensively used in many applications.

Staggered-grid finite-difference acoustic modeling with the Time-Domain Atmospheric Acoustic Propagation Suite (TDAAPS).

DOE Office of Scientific and Technical Information (OSTI.GOV)

Aldridge, David Franklin; Collier, Sandra L.; Marlin, David H.

2005-05-01

This document is intended to serve as a users guide for the time-domain atmospheric acoustic propagation suite (TDAAPS) program developed as part of the Department of Defense High-Performance Modernization Office (HPCMP) Common High-Performance Computing Scalable Software Initiative (CHSSI). TDAAPS performs staggered-grid finite-difference modeling of the acoustic velocity-pressure system with the incorporation of spatially inhomogeneous winds. Wherever practical the control structure of the codes are written in C++ using an object oriented design. Sections of code where a large number of calculations are required are written in C or F77 in order to enable better compiler optimization of these sections. Themore » TDAAPS program conforms to a UNIX style calling interface. Most of the actions of the codes are controlled by adding flags to the invoking command line. This document presents a large number of examples and provides new users with the necessary background to perform acoustic modeling with TDAAPS.« less

Microseismic response characteristics modeling and locating of underground water supply pipe leak

NASA Astrophysics Data System (ADS)

Wang, J.; Liu, J.

2015-12-01

In traditional methods of pipeline leak location, geophones must be located on the pipe wall. If the exact location of the pipeline is unknown, the leaks cannot be identified accurately. To solve this problem, taking into account the characteristics of the pipeline leak, we propose a continuous random seismic source model and construct geological models to investigate the proposed method for locating underground pipeline leaks. Based on two dimensional (2D) viscoacoustic equations and the staggered grid finite-difference (FD) algorithm, the microseismic wave field generated by a leaking pipe is modeled. Cross-correlation analysis and the simulated annealing (SA) algorithm were utilized to obtain the time difference and the leak location. We also analyze and discuss the effect of the number of recorded traces, the survey layout, and the offset and interval of the traces on the accuracy of the estimated location. The preliminary results of the simulation and data field experiment indicate that (1) a continuous random source can realistically represent the leak microseismic wave field in a simulation using 2D visco-acoustic equations and a staggered grid FD algorithm. (2) The cross-correlation method is effective for calculating the time difference of the direct wave relative to the reference trace. However, outside the refraction blind zone, the accuracy of the time difference is reduced by the effects of the refracted wave. (3) The acquisition method of time difference based on the microseismic theory and SA algorithm has a great potential for locating leaks from underground pipelines from an array located on the ground surface. Keywords: Viscoacoustic finite-difference simulation; continuous random source; simulated annealing algorithm; pipeline leak location

Numerical analysis of laser ablation using the axisymmetric two-temperature model

NASA Astrophysics Data System (ADS)

Dziatkiewicz, Jolanta; Majchrzak, Ewa

2018-01-01

Laser ablation of the axisymmetric micro-domain is analyzed. To describe the thermal processes occurring in the micro-domain the two-temperature hyperbolic model supplemented by the boundary and initial conditions is used. This model takes into account the phase changes of material (solid-liquid and liquid-vapour) and the ablation process. At the stage of numerical computations the finite difference method with staggered grid is used. In the final part the results of computations are shown.

Toward the Reliability of Fault Representation Methods in Finite Difference Schemes for Simulation of Shear Rupture Propagation

NASA Astrophysics Data System (ADS)

Dalguer, L. A.; Day, S. M.

2006-12-01

Accuracy in finite difference (FD) solutions to spontaneous rupture problems is controlled principally by the scheme used to represent the fault discontinuity, and not by the grid geometry used to represent the continuum. We have numerically tested three fault representation methods, the Thick Fault (TF) proposed by Madariaga et al (1998), the Stress Glut (SG) described by Andrews (1999), and the Staggered-Grid Split-Node (SGSN) methods proposed by Dalguer and Day (2006), each implemented in a the fourth-order velocity-stress staggered-grid (VSSG) FD scheme. The TF and the SG methods approximate the discontinuity through inelastic increments to stress components ("inelastic-zone" schemes) at a set of stress grid points taken to lie on the fault plane. With this type of scheme, the fault surface is indistinguishable from an inelastic zone with a thickness given by a spatial step dx for the SG, and 2dx for the TF model. The SGSN method uses the traction-at-split-node (TSN) approach adapted to the VSSG FD. This method represents the fault discontinuity by explicitly incorporating discontinuity terms at velocity nodes in the grid, with interactions between the "split nodes" occurring exclusively through the tractions (frictional resistance) acting between them. These tractions in turn are controlled by the jump conditions and a friction law. Our 3D tests problem solutions show that the inelastic-zone TF and SG methods show much poorer performance than does the SGSN formulation. The SG inelastic-zone method achieved solutions that are qualitatively meaningful and quantitatively reliable to within a few percent. The TF inelastic-zone method did not achieve qualitatively agreement with the reference solutions to the 3D test problem, and proved to be sufficiently computationally inefficient that it was not feasible to explore convergence quantitatively. The SGSN method gives very accurate solutions, and is also very efficient. Reliable solution of the rupture time is reached with a median resolution of the cohesive zone of only ~2 grid points, and efficiency is competitive with the Boundary Integral (BI) method. The results presented here demonstrate that appropriate fault representation in a numerical scheme is crucial to reduce uncertainties in numerical simulations of earthquake source dynamics and ground motion, and therefore important to improving our understanding of earthquake physics in general.

Numerically stable finite difference simulation for ultrasonic NDE in anisotropic composites

NASA Astrophysics Data System (ADS)

Leckey, Cara A. C.; Quintanilla, Francisco Hernando; Cole, Christina M.

2018-04-01

Simulation tools can enable optimized inspection of advanced materials and complex geometry structures. Recent work at NASA Langley is focused on the development of custom simulation tools for modeling ultrasonic wave behavior in composite materials. Prior work focused on the use of a standard staggered grid finite difference type of mathematical approach, by implementing a three-dimensional (3D) anisotropic Elastodynamic Finite Integration Technique (EFIT) code. However, observations showed that the anisotropic EFIT method displays numerically unstable behavior at the locations of stress-free boundaries for some cases of anisotropic materials. This paper gives examples of the numerical instabilities observed for EFIT and discusses the source of instability. As an alternative to EFIT, the 3D Lebedev Finite Difference (LFD) method has been implemented. The paper briefly describes the LFD approach and shows examples of stable behavior in the presence of stress-free boundaries for a monoclinic anisotropy case. The LFD results are also compared to experimental results and dispersion curves.

Elastic modelling in tilted transversely isotropic media with convolutional perfectly matched layer boundary conditions

NASA Astrophysics Data System (ADS)

Han, Byeongho; Seol, Soon Jee; Byun, Joongmoo

2012-04-01

To simulate wave propagation in a tilted transversely isotropic (TTI) medium with a tilting symmetry-axis of anisotropy, we develop a 2D elastic forward modelling algorithm. In this algorithm, we use the staggered-grid finite-difference method which has fourth-order accuracy in space and second-order accuracy in time. Since velocity-stress formulations are defined for staggered grids, we include auxiliary grid points in the z-direction to meet the free surface boundary conditions for shear stress. Through comparisons of displacements obtained from our algorithm, not only with analytical solutions but also with finite element solutions, we are able to validate that the free surface conditions operate appropriately and elastic waves propagate correctly. In order to handle the artificial boundary reflections efficiently, we also implement convolutional perfectly matched layer (CPML) absorbing boundaries in our algorithm. The CPML sufficiently attenuates energy at the grazing incidence by modifying the damping profile of the PML boundary. Numerical experiments indicate that the algorithm accurately expresses elastic wave propagation in the TTI medium. At the free surface, the numerical results show good agreement with analytical solutions not only for body waves but also for the Rayleigh wave which has strong amplitude along the surface. In addition, we demonstrate the efficiency of CPML for a homogeneous TI medium and a dipping layered model. Only using 10 grid points to the CPML regions, the artificial reflections are successfully suppressed and the energy of the boundary reflection back into the effective modelling area is significantly decayed.

3D Staggered-Grid Finite-Difference Simulation of Acoustic Waves in Turbulent Moving Media

NASA Astrophysics Data System (ADS)

Symons, N. P.; Aldridge, D. F.; Marlin, D.; Wilson, D. K.; Sullivan, P.; Ostashev, V.

2003-12-01

Acoustic wave propagation in a three-dimensional heterogeneous moving atmosphere is accurately simulated with a numerical algorithm recently developed under the DOD Common High Performance Computing Software Support Initiative (CHSSI). Sound waves within such a dynamic environment are mathematically described by a set of four, coupled, first-order partial differential equations governing small-amplitude fluctuations in pressure and particle velocity. The system is rigorously derived from fundamental principles of continuum mechanics, ideal-fluid constitutive relations, and reasonable assumptions that the ambient atmospheric motion is adiabatic and divergence-free. An explicit, time-domain, finite-difference (FD) numerical scheme is used to solve the system for both pressure and particle velocity wavefields. The atmosphere is characterized by 3D gridded models of sound speed, mass density, and the three components of the wind velocity vector. Dependent variables are stored on staggered spatial and temporal grids, and centered FD operators possess 2nd-order and 4th-order space/time accuracy. Accurate sound wave simulation is achieved provided grid intervals are chosen appropriately. The gridding must be fine enough to reduce numerical dispersion artifacts to an acceptable level and maintain stability. The algorithm is designed to execute on parallel computational platforms by utilizing a spatial domain-decomposition strategy. Currently, the algorithm has been validated on four different computational platforms, and parallel scalability of approximately 85% has been demonstrated. Comparisons with analytic solutions for uniform and vertically stratified wind models indicate that the FD algorithm generates accurate results with either a vanishing pressure or vanishing vertical-particle velocity boundary condition. Simulations are performed using a kinematic turbulence wind profile developed with the quasi-wavelet method. In addition, preliminary results are presented using high-resolution 3D dynamic turbulent flowfields generated by a large-eddy simulation model of a stably stratified planetary boundary layer. Sandia National Laboratories is a operated by Sandia Corporation, a Lockheed Martin Company, for the USDOE under contract 94-AL85000.

3D frequency-domain finite-difference modeling of acoustic wave propagation

NASA Astrophysics Data System (ADS)

Operto, S.; Virieux, J.

2006-12-01

We present a 3D frequency-domain finite-difference method for acoustic wave propagation modeling. This method is developed as a tool to perform 3D frequency-domain full-waveform inversion of wide-angle seismic data. For wide-angle data, frequency-domain full-waveform inversion can be applied only to few discrete frequencies to develop reliable velocity model. Frequency-domain finite-difference (FD) modeling of wave propagation requires resolution of a huge sparse system of linear equations. If this system can be solved with a direct method, solutions for multiple sources can be computed efficiently once the underlying matrix has been factorized. The drawback of the direct method is the memory requirement resulting from the fill-in of the matrix during factorization. We assess in this study whether representative problems can be addressed in 3D geometry with such approach. We start from the velocity-stress formulation of the 3D acoustic wave equation. The spatial derivatives are discretized with second-order accurate staggered-grid stencil on different coordinate systems such that the axis span over as many directions as possible. Once the discrete equations were developed on each coordinate system, the particle velocity fields are eliminated from the first-order hyperbolic system (following the so-called parsimonious staggered-grid method) leading to second-order elliptic wave equations in pressure. The second-order wave equations discretized on each coordinate system are combined linearly to mitigate the numerical anisotropy. Secondly, grid dispersion is minimized by replacing the mass term at the collocation point by its weighted averaging over all the grid points of the stencil. Use of second-order accurate staggered- grid stencil allows to reduce the bandwidth of the matrix to be factorized. The final stencil incorporates 27 points. Absorbing conditions are PML. The system is solved using the parallel direct solver MUMPS developed for distributed-memory computers. The MUMPS solver is based on a multifrontal method for LU factorization. We used the METIS algorithm to perform re-ordering of the matrix coefficients before factorization. Four grid points per minimum wavelength is used for discretization. We applied our algorithm to the 3D SEG/EAGE synthetic onshore OVERTHRUST model of dimensions 20 x 20 x 4.65 km. The velocities range between 2 and 6 km/s. We performed the simulations using 192 processors with 2 Gbytes of RAM memory per processor. We performed simulations for the 5 Hz, 7 Hz and 10 Hz frequencies in some fractions of the OVERTHRUST model. The grid interval was 100 m, 75 m and 50 m respectively. The grid dimensions were 207x207x53, 275x218x71 and 409x109x102 respectively corresponding to 100, 80 and 25 percents of the model respectively. The time for factorization is 20 mn, 108 mn and 163 mn respectively. The time for resolution was 3.8, 9.3 and 10.3 s per source. The total memory used during factorization is 143, 384 and 449 Gbytes respectively. One can note the huge memory requirement for factorization and the efficiency of the direct method to compute solutions for a large number of sources. This highlights the respective drawback and merit of the frequency-domain approach with respect to the time- domain counterpart. These results show that 3D acoustic frequency-domain wave propagation modeling can be performed at low frequencies using direct solver on large clusters of Pcs. This forward modeling algorithm may be used in the future as a tool to image the first kilometers of the crust by frequency-domain full-waveform inversion. For larger problems, we will use the out-of-core memory during factorization that has been implemented by the authors of MUMPS.

Developments in the Gung Ho dynamical core

NASA Astrophysics Data System (ADS)

Melvin, Thomas

2017-04-01

Gung Ho is the new dynamical core being developed for the next generation Met Office weather and climate model, suitable for meeting the exascale challenge on emerging computer architectures. It builds upon the earlier collaborative project between the Met Office, NERC and STFC Daresbury of the same name to investigate suitable numerical methods for dynamical cores. A mixed-finite element approach is used, where different finite element spaces are used to represent various fields. This method provides a number of beneficial improvements over the current model, such a compatibility and inherent conservation on quasi-uniform unstructured meshes, whilst maintaining the accuracy and good dispersion properties of the staggered grid currently used. Furthermore, the mixed finite element approach allows a large degree of flexibility in the type of mesh, order of approximation and discretisation, providing a simple way to test alternative options to obtain the best model possible.

FAS multigrid calculations of three dimensional flow using non-staggered grids

NASA Technical Reports Server (NTRS)

Matovic, D.; Pollard, A.; Becker, H. A.; Grandmaison, E. W.

1993-01-01

Grid staggering is a well known remedy for the problem of velocity/pressure coupling in incompressible flow calculations. Numerous inconveniences occur, however, when staggered grids are implemented, particularly when a general-purpose code, capable of handling irregular three-dimensional domains, is sought. In several non-staggered grid numerical procedures proposed in the literature, the velocity/pressure coupling is achieved by either pressure or velocity (momentum) averaging. This approach is not convenient for simultaneous (block) solvers that are preferred when using multigrid methods. A new method is introduced in this paper that is based upon non-staggered grid formulation with a set of virtual cell face velocities used for pressure/velocity coupling. Instead of pressure or velocity averaging, a momentum balance at the cell face is used as a link between the momentum and mass balance constraints. The numerical stencil is limited to 9 nodes (in 2D) or 27 nodes (in 3D), both during the smoothing and inter-grid transfer, which is a convenient feature when a block point solver is applied. The results for a lid-driven cavity and a cube in a lid-driven cavity are presented and compared to staggered grid calculations using the same multigrid algorithm. The method is shown to be stable and produce a smooth (wiggle-free) pressure field.

Analytical Computation of Effective Grid Parameters for the Finite-Difference Seismic Waveform Modeling With the PREM, IASP91, SP6, and AK135

NASA Astrophysics Data System (ADS)

Toyokuni, G.; Takenaka, H.

2007-12-01

We propose a method to obtain effective grid parameters for the finite-difference (FD) method with standard Earth models using analytical ways. In spite of the broad use of the heterogeneous FD formulation for seismic waveform modeling, accurate treatment of material discontinuities inside the grid cells has been a serious problem for many years. One possible way to solve this problem is to introduce effective grid elastic moduli and densities (effective parameters) calculated by the volume harmonic averaging of elastic moduli and volume arithmetic averaging of density in grid cells. This scheme enables us to put a material discontinuity into an arbitrary position in the spatial grids. Most of the methods used for synthetic seismogram calculation today receives the blessing of the standard Earth models, such as the PREM, IASP91, SP6, and AK135, represented as functions of normalized radius. For the FD computation of seismic waveform with such models, we first need accurate treatment of material discontinuities in radius. This study provides a numerical scheme for analytical calculations of the effective parameters for an arbitrary spatial grids in radial direction as to these major four standard Earth models making the best use of their functional features. This scheme can analytically obtain the integral volume averages through partial fraction decompositions (PFDs) and integral formulae. We have developed a FORTRAN subroutine to perform the computations, which is opened to utilization in a large variety of FD schemes ranging from 1-D to 3-D, with conventional- and staggered-grids. In the presentation, we show some numerical examples displaying the accuracy of the FD synthetics simulated with the analytical effective parameters.

A staggered-grid finite-difference scheme optimized in the time–space domain for modeling scalar-wave propagation in geophysical problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tan, Sirui, E-mail: siruitan@hotmail.com; Huang, Lianjie, E-mail: ljh@lanl.gov

For modeling scalar-wave propagation in geophysical problems using finite-difference schemes, optimizing the coefficients of the finite-difference operators can reduce numerical dispersion. Most optimized finite-difference schemes for modeling seismic-wave propagation suppress only spatial but not temporal dispersion errors. We develop a novel optimized finite-difference scheme for numerical scalar-wave modeling to control dispersion errors not only in space but also in time. Our optimized scheme is based on a new stencil that contains a few more grid points than the standard stencil. We design an objective function for minimizing relative errors of phase velocities of waves propagating in all directions within amore » given range of wavenumbers. Dispersion analysis and numerical examples demonstrate that our optimized finite-difference scheme is computationally up to 2.5 times faster than the optimized schemes using the standard stencil to achieve the similar modeling accuracy for a given 2D or 3D problem. Compared with the high-order finite-difference scheme using the same new stencil, our optimized scheme reduces 50 percent of the computational cost to achieve the similar modeling accuracy. This new optimized finite-difference scheme is particularly useful for large-scale 3D scalar-wave modeling and inversion.« less

Numerical simulation using vorticity-vector potential formulation

NASA Technical Reports Server (NTRS)

Tokunaga, Hiroshi

1993-01-01

An accurate and efficient computational method is needed for three-dimensional incompressible viscous flows in engineering applications. On solving the turbulent shear flows directly or using the subgrid scale model, it is indispensable to resolve the small scale fluid motions as well as the large scale motions. From this point of view, the pseudo-spectral method is used so far as the computational method. However, the finite difference or the finite element methods are widely applied for computing the flow with practical importance since these methods are easily applied to the flows with complex geometric configurations. However, there exist several problems in applying the finite difference method to direct and large eddy simulations. Accuracy is one of most important problems. This point was already addressed by the present author on the direct simulations on the instability of the plane Poiseuille flow and also on the transition to turbulence. In order to obtain high efficiency, the multi-grid Poisson solver is combined with the higher-order, accurate finite difference method. The formulation method is also one of the most important problems in applying the finite difference method to the incompressible turbulent flows. The three-dimensional Navier-Stokes equations have been solved so far in the primitive variables formulation. One of the major difficulties of this method is the rigorous satisfaction of the equation of continuity. In general, the staggered grid is used for the satisfaction of the solenoidal condition for the velocity field at the wall boundary. However, the velocity field satisfies the equation of continuity automatically in the vorticity-vector potential formulation. From this point of view, the vorticity-vector potential method was extended to the generalized coordinate system. In the present article, we adopt the vorticity-vector potential formulation, the generalized coordinate system, and the 4th-order accurate difference method as the computational method. We present the computational method and apply the present method to computations of flows in a square cavity at large Reynolds number in order to investigate its effectiveness.

The Finite-Surface Method for incompressible flow: a step beyond staggered grid

NASA Astrophysics Data System (ADS)

Hokpunna, Arpiruk; Misaka, Takashi; Obayashi, Shigeru

2017-11-01

We present a newly developed higher-order finite surface method for the incompressible Navier-Stokes equations (NSE). This method defines the velocities as a surface-averaged value on the surfaces of the pressure cells. Consequently, the mass conservation on the pressure cells becomes an exact equation. The only things left to approximate is the momentum equation and the pressure at the new time step. At certain conditions, the exact mass conservation enables the explicit n-th order accurate NSE solver to be used with the pressure treatment that is two or four order less accurate without loosing the apparent convergence rate. This feature was not possible with finite volume of finite difference methods. We use Fourier analysis with a model spectrum to determine the condition and found that the range covers standard boundary layer flows. The formal convergence and the performance of the proposed scheme is compared with a sixth-order finite volume method. Finally, the accuracy and performance of the method is evaluated in turbulent channel flows. This work is partially funded by a research colloaboration from IFS, Tohoku university and ASEAN+3 funding scheme from CMUIC, Chiang Mai University.

A Least-Squares Finite Element Method for Electromagnetic Scattering Problems

NASA Technical Reports Server (NTRS)

Wu, Jie; Jiang, Bo-nan

1996-01-01

The least-squares finite element method (LSFEM) is applied to electromagnetic scattering and radar cross section (RCS) calculations. In contrast to most existing numerical approaches, in which divergence-free constraints are omitted, the LSFF-M directly incorporates two divergence equations in the discretization process. The importance of including the divergence equations is demonstrated by showing that otherwise spurious solutions with large divergence occur near the scatterers. The LSFEM is based on unstructured grids and possesses full flexibility in handling complex geometry and local refinement Moreover, the LSFEM does not require any special handling, such as upwinding, staggered grids, artificial dissipation, flux-differencing, etc. Implicit time discretization is used and the scheme is unconditionally stable. By using a matrix-free iterative method, the computational cost and memory requirement for the present scheme is competitive with other approaches. The accuracy of the LSFEM is verified by several benchmark test problems.

A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equations at all Mach numbers

NASA Astrophysics Data System (ADS)

Tavelli, Maurizio; Dumbser, Michael

2017-07-01

We propose a new arbitrary high order accurate semi-implicit space-time discontinuous Galerkin (DG) method for the solution of the two and three dimensional compressible Euler and Navier-Stokes equations on staggered unstructured curved meshes. The method is pressure-based and semi-implicit and is able to deal with all Mach number flows. The new DG scheme extends the seminal ideas outlined in [1], where a second order semi-implicit finite volume method for the solution of the compressible Navier-Stokes equations with a general equation of state was introduced on staggered Cartesian grids. Regarding the high order extension we follow [2], where a staggered space-time DG scheme for the incompressible Navier-Stokes equations was presented. In our scheme, the discrete pressure is defined on the primal grid, while the discrete velocity field and the density are defined on a face-based staggered dual grid. Then, the mass conservation equation, as well as the nonlinear convective terms in the momentum equation and the transport of kinetic energy in the energy equation are discretized explicitly, while the pressure terms appearing in the momentum and energy equation are discretized implicitly. Formal substitution of the discrete momentum equation into the total energy conservation equation yields a linear system for only one unknown, namely the scalar pressure. Here the equation of state is assumed linear with respect to the pressure. The enthalpy and the kinetic energy are taken explicitly and are then updated using a simple Picard procedure. Thanks to the use of a staggered grid, the final pressure system is a very sparse block five-point system for three dimensional problems and it is a block four-point system in the two dimensional case. Furthermore, for high order in space and piecewise constant polynomials in time, the system is observed to be symmetric and positive definite. This allows to use fast linear solvers such as the conjugate gradient (CG) method. In addition, all the volume and surface integrals needed by the scheme depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessing stage. This leads to significant savings in terms of computational effort for the time evolution part. In this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step. The viscous terms and the heat flux are also discretized making use of the staggered grid by defining the viscous stress tensor and the heat flux vector on the dual grid, which corresponds to the use of a lifting operator, but on the dual grid. The time step of our new numerical method is limited by a CFL condition based only on the fluid velocity and not on the sound speed. This makes the method particularly interesting for low Mach number flows. Finally, a very simple combination of artificial viscosity and the a posteriori MOOD technique allows to deal with shock waves and thus permits also to simulate high Mach number flows. We show computational results for a large set of two and three-dimensional benchmark problems, including both low and high Mach number flows and using polynomial approximation degrees up to p = 4.

Vector Potential Generation for Numerical Relativity Simulations

NASA Astrophysics Data System (ADS)

Silberman, Zachary; Faber, Joshua; Adams, Thomas; Etienne, Zachariah; Ruchlin, Ian

2017-01-01

Many different numerical codes are employed in studies of highly relativistic magnetized accretion flows around black holes. Based on the formalisms each uses, some codes evolve the magnetic field vector B, while others evolve the magnetic vector potential A, the two being related by the curl: B=curl(A). Here, we discuss how to generate vector potentials corresponding to specified magnetic fields on staggered grids, a surprisingly difficult task on finite cubic domains. The code we have developed solves this problem in two ways: a brute-force method, whose scaling is nearly linear in the number of grid cells, and a direct linear algebra approach. We discuss the success both algorithms have in generating smooth vector potential configurations and how both may be extended to more complicated cases involving multiple mesh-refinement levels. NSF ACI-1550436

An in-depth stability analysis of nonuniform FDTD combined with novel local implicitization techniques

NASA Astrophysics Data System (ADS)

Van Londersele, Arne; De Zutter, Daniël; Vande Ginste, Dries

2017-08-01

This work focuses on efficient full-wave solutions of multiscale electromagnetic problems in the time domain. Three local implicitization techniques are proposed and carefully analyzed in order to relax the traditional time step limit of the Finite-Difference Time-Domain (FDTD) method on a nonuniform, staggered, tensor product grid: Newmark, Crank-Nicolson (CN) and Alternating-Direction-Implicit (ADI) implicitization. All of them are applied in preferable directions, alike Hybrid Implicit-Explicit (HIE) methods, as to limit the rank of the sparse linear systems. Both exponential and linear stability are rigorously investigated for arbitrary grid spacings and arbitrary inhomogeneous, possibly lossy, isotropic media. Numerical examples confirm the conservation of energy inside a cavity for a million iterations if the time step is chosen below the proposed, relaxed limit. Apart from the theoretical contributions, new accomplishments such as the development of the leapfrog Alternating-Direction-Hybrid-Implicit-Explicit (ADHIE) FDTD method and a less stringent Courant-like time step limit for the conventional, fully explicit FDTD method on a nonuniform grid, have immediate practical applications.

On long-time instabilities in staggered finite difference simulations of the seismic acoustic wave equations on discontinuous grids

NASA Astrophysics Data System (ADS)

Gao, Longfei; Ketcheson, David; Keyes, David

2018-02-01

We consider the long-time instability issue associated with finite difference simulation of seismic acoustic wave equations on discontinuous grids. This issue is exhibited by a prototype algebraic problem abstracted from practical application settings. Analysis of this algebraic problem leads to better understanding of the cause of the instability and provides guidance for its treatment. Specifically, we use the concept of discrete energy to derive the proper solution transfer operators and design an effective way to damp the unstable solution modes. Our investigation shows that the interpolation operators need to be matched with their companion restriction operators in order to properly couple the coarse and fine grids. Moreover, to provide effective damping, specially designed diffusive terms are introduced to the equations at designated locations and discretized with specially designed schemes. These techniques are applied to simulations in practical settings and are shown to lead to superior results in terms of both stability and accuracy.

A Highly Accurate Technique for the Treatment of Flow Equations at the Polar Axis in Cylindrical Coordinates using Series Expansions. Appendix A

NASA Technical Reports Server (NTRS)

Constantinescu, George S.; Lele, S. K.

2001-01-01

Numerical methods for solving the flow equations in cylindrical or spherical coordinates should be able to capture the behavior of the exact solution near the regions where the particular form of the governing equations is singular. In this work we focus on the treatment of these numerical singularities for finite-differences methods by reinterpreting the regularity conditions developed in the context of pseudo-spectral methods. A generally applicable numerical method for treating the singularities present at the polar axis, when nonaxisymmetric flows are solved in cylindrical, coordinates using highly accurate finite differences schemes (e.g., Pade schemes) on non-staggered grids, is presented. Governing equations for the flow at the polar axis are derived using series expansions near r=0. The only information needed to calculate the coefficients in these equations are the values of the flow variables and their radial derivatives at the previous iteration (or time) level. These derivatives, which are multi-valued at the polar axis, are calculated without dropping the accuracy of the numerical method using a mapping of the flow domain from (0,R)*(0,2pi) to (-R,R)*(0,pi), where R is the radius of the computational domain. This allows the radial derivatives to be evaluated using high-order differencing schemes (e.g., compact schemes) at points located on the polar axis. The proposed technique is illustrated by results from simulations of laminar-forced jets and turbulent compressible jets using large eddy simulation (LES) methods. In term of the general robustness of the numerical method and smoothness of the solution close to the polar axis, the present results compare very favorably to similar calculations in which the equations are solved in Cartesian coordinates at the polar axis, or in which the singularity is removed by employing a staggered mesh in the radial direction without a mesh point at r=0, following the method proposed recently by Mohseni and Colonius (1). Extension of the method described here for incompressible flows or for any other set of equations that are solved on a non-staggered mesh in cylindrical or spherical coordinates with finite-differences schemes of various level of accuracy is immediate.

Arbitrary Lagrangian-Eulerian Method with Local Structured Adaptive Mesh Refinement for Modeling Shock Hydrodynamics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Anderson, R W; Pember, R B; Elliott, N S

2001-10-22

A new method that combines staggered grid Arbitrary Lagrangian-Eulerian (ALE) techniques with structured local adaptive mesh refinement (AMR) has been developed for solution of the Euler equations. This method facilitates the solution of problems currently at and beyond the boundary of soluble problems by traditional ALE methods by focusing computational resources where they are required through dynamic adaption. Many of the core issues involved in the development of the combined ALEAMR method hinge upon the integration of AMR with a staggered grid Lagrangian integration method. The novel components of the method are mainly driven by the need to reconcile traditionalmore » AMR techniques, which are typically employed on stationary meshes with cell-centered quantities, with the staggered grids and grid motion employed by Lagrangian methods. Numerical examples are presented which demonstrate the accuracy and efficiency of the method.« less

Multigrid solutions to quasi-elliptic schemes

NASA Technical Reports Server (NTRS)

Brandt, A.; Taasan, S.

1985-01-01

Quasi-elliptic schemes arise from central differencing or finite element discretization of elliptic systems with odd order derivatives on non-staggered grids. They are somewhat unstable and less accurate then corresponding staggered-grid schemes. When usual multigrid solvers are applied to them, the asymptotic algebraic convergence is necessarily slow. Nevertheless, it is shown by mode analyses and numerical experiments that the usual FMG algorithm is very efficient in solving quasi-elliptic equations to the level of truncation errors. Also, a new type of multigrid algorithm is presented, mode analyzed and tested, for which even the asymptotic algebraic convergence is fast. The essence of that algorithm is applicable to other kinds of problems, including highly indefinite ones.

Multigrid solutions to quasi-elliptic schemes

NASA Technical Reports Server (NTRS)

Brandt, A.; Taasan, S.

1985-01-01

Quasi-elliptic schemes arise from central differencing or finite element discretization of elliptic systems with odd order derivatives on non-staggered grids. They are somewhat unstable and less accurate than corresponding staggered-grid schemes. When usual multigrid solvers are applied to them, the asymptotic algebraic convergence is necessarily slow. Nevertheless, it is shown by mode analyses and numerical experiments that the usual FMG algorithm is very efficient in solving quasi-elliptic equations to the level of truncation errors. Also, a new type of multigrid algorithm is presented, mode analyzed and tested, for which even the asymptotic algebraic convergence is fast. The essence of that algorithm is applicable to other kinds of problems, including highly indefinite ones.

A high-order staggered meshless method for elliptic problems

DOE PAGES

Trask, Nathaniel; Perego, Mauro; Bochev, Pavel Blagoveston

2017-03-21

Here, we present a new meshless method for scalar diffusion equations, which is motivated by their compatible discretizations on primal-dual grids. Unlike the latter though, our approach is truly meshless because it only requires the graph of nearby neighbor connectivity of the discretization points. This graph defines a local primal-dual grid complex with a virtual dual grid, in the sense that specification of the dual metric attributes is implicit in the method's construction. Our method combines a topological gradient operator on the local primal grid with a generalized moving least squares approximation of the divergence on the local dual grid. We show that the resulting approximation of the div-grad operator maintains polynomial reproduction to arbitrary orders and yields a meshless method, which attainsmore » $$O(h^{m})$$ convergence in both $L^2$- and $H^1$-norms, similar to mixed finite element methods. We demonstrate this convergence on curvilinear domains using manufactured solutions in two and three dimensions. Application of the new method to problems with discontinuous coefficients reveals solutions that are qualitatively similar to those of compatible mesh-based discretizations.« less

Reynolds-averaged Navier-Stokes computation on tip clearance flow in a compressor cascade using an unstructured grid

NASA Astrophysics Data System (ADS)

Shin, Sangmook

2001-07-01

A three-dimensional unstructured incompressible RANS code has been developed using artificial compressibility and Spalart-Allmaras eddy viscosity model. A node-based finite volume method is used in which all flow variables are defined at the vertices of tetrahedrons in an unstructured grid. The inviscid fluxes are computed by using the Roe's flux difference splitting method, and higher order accuracy is attained by data reconstruction based on Taylor series expansion. Gauss theorem is used to formulate necessary gradients. For time integration, an implicit scheme based on linearized Euler backward method is used. A tetrahedral unstructured grid generation code has been also developed and applied to the tip clearance flow in a highly staggered cascade. Surface grids are first generated in the flow passage and blade tip by using several triangulation methods including Delaunay triangulation, advancing front method and advancing layer method. Then the whole computational domain including tip gap region is filled with prisms using the surface grids. The code has been validated by comparisons with available computational and experimental results for several test cases: inviscid flow around NACA section, laminar and turbulent flow over a flat plate, turbulent flow through double-circular arc cascade and laminar flow through a square duct with 90° bend. Finally the code is applied to a linear cascade that has GE rotor B section with tip clearance and a high stagger angle of 56.9°. The overall structure of the tip clearance flow is well predicted. Loss of loading due to tip leakage flow and reloading due to tip leakage vortex are presented. On the end wall, separation line of the tip leakage vortex and reattachment line of passage vortex are identified. Prediction of such an interaction presents a challenge to RANS computations. The effects of blade span on the flow structure have been also investigated. Two cascades with blades of aspect ratios of 0.5 and 1.0 are considered. By comparing pressure distributions on the blade, it is shown that the aspect ratio has strong effects on loading distribution on the blade although the tip gap height is very small (0.016 chord). Grid convergence study has been carried out with three different grids for pressure distributions and limiting streamlines on the end wall. (Abstract shortened by UMI.)

A finite difference method for a coupled model of wave propagation in poroelastic materials.

PubMed

Zhang, Yang; Song, Limin; Deffenbaugh, Max; Toksöz, M Nafi

2010-05-01

A computational method for time-domain multi-physics simulation of wave propagation in a poroelastic medium is presented. The medium is composed of an elastic matrix saturated with a Newtonian fluid, and the method operates on a digital representation of the medium where a distinct material phase and properties are specified at each volume cell. The dynamic response to an acoustic excitation is modeled mathematically with a coupled system of equations: elastic wave equation in the solid matrix and linearized Navier-Stokes equation in the fluid. Implementation of the solution is simplified by introducing a common numerical form for both solid and fluid cells and using a rotated-staggered-grid which allows stable solutions without explicitly handling the fluid-solid boundary conditions. A stability analysis is presented which can be used to select gridding and time step size as a function of material properties. The numerical results are shown to agree with the analytical solution for an idealized porous medium of periodically alternating solid and fluid layers.

Monolithic multigrid method for the coupled Stokes flow and deformable porous medium system

NASA Astrophysics Data System (ADS)

Luo, P.; Rodrigo, C.; Gaspar, F. J.; Oosterlee, C. W.

2018-01-01

The interaction between fluid flow and a deformable porous medium is a complicated multi-physics problem, which can be described by a coupled model based on the Stokes and poroelastic equations. A monolithic multigrid method together with either a coupled Vanka smoother or a decoupled Uzawa smoother is employed as an efficient numerical technique for the linear discrete system obtained by finite volumes on staggered grids. A specialty in our modeling approach is that at the interface of the fluid and poroelastic medium, two unknowns from the different subsystems are defined at the same grid point. We propose a special discretization at and near the points on the interface, which combines the approximation of the governing equations and the considered interface conditions. In the decoupled Uzawa smoother, Local Fourier Analysis (LFA) helps us to select optimal values of the relaxation parameter appearing. To implement the monolithic multigrid method, grid partitioning is used to deal with the interface updates when communication is required between two subdomains. Numerical experiments show that the proposed numerical method has an excellent convergence rate. The efficiency and robustness of the method are confirmed in numerical experiments with typically small realistic values of the physical coefficients.

Forward modeling and inversion of tensor CSAMT in 3D anisotropic media

NASA Astrophysics Data System (ADS)

Wang, Tao; Wang, Kun-Peng; Tan, Han-Dong

2017-12-01

Tensor controlled-source audio-frequency magnetotellurics (CSAMT) can yield information about electric and magnetic fields owing to its multi-transmitter configuration compared with the common scalar CSAMT. The most current theories, numerical simulations, and inversion of tensor CSAMT are based on far-field measurements and the assumption that underground media have isotropic resistivity. We adopt a three-dimensional (3D) staggered-grid finite difference numerical simulation method to analyze the resistivity in axial anisotropic and isotropic media. We further adopt the limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) method to perform 3D tensor CSAMT axial anisotropic inversion. The inversion results suggest that when the underground structure is anisotropic, the isotropic inversion will introduce errors to the interpretation.

Numerical Simulations of Buoyancy Effects in low Density Gas Jets

NASA Technical Reports Server (NTRS)

Satti, R. P.; Pasumarthi, K. S.; Agrawal, A. K.

2004-01-01

This paper deals with the computational analysis of buoyancy effects in the near field of an isothermal helium jet injected into quiescent ambient air environment. The transport equations of helium mass fraction coupled with the conservation equations of mixture mass and momentum were solved using a staggered grid finite volume method. Laminar, axisymmetric, unsteady flow conditions were considered for the analysis. An orthogonal system with non-uniform grids was used to capture the instability phenomena. Computations were performed for Earth gravity and during transition from Earth to different gravitational levels. The flow physics was described by simultaneous visualizations of velocity and concentration fields at Earth and microgravity conditions. Computed results were validated by comparing with experimental data substantiating that buoyancy induced global flow oscillations present in Earth gravity are absent in microgravity. The dependence of oscillation frequency and amplitude on gravitational forcing was presented to further quantify the buoyancy effects.

Finite volume multigrid method of the planar contraction flow of a viscoelastic fluid

NASA Astrophysics Data System (ADS)

Moatssime, H. Al; Esselaoui, D.; Hakim, A.; Raghay, S.

2001-08-01

This paper reports on a numerical algorithm for the steady flow of viscoelastic fluid. The conservative and constitutive equations are solved using the finite volume method (FVM) with a hybrid scheme for the velocities and first-order upwind approximation for the viscoelastic stress. A non-uniform staggered grid system is used. The iterative SIMPLE algorithm is employed to relax the coupled momentum and continuity equations. The non-linear algebraic equations over the flow domain are solved iteratively by the symmetrical coupled Gauss-Seidel (SCGS) method. In both, the full approximation storage (FAS) multigrid algorithm is used. An Oldroyd-B fluid model was selected for the calculation. Results are reported for planar 4:1 abrupt contraction at various Weissenberg numbers. The solutions are found to be stable and smooth. The solutions show that at high Weissenberg number the domain must be long enough. The convergence of the method has been verified with grid refinement. All the calculations have been performed on a PC equipped with a Pentium III processor at 550 MHz. Copyright

Implementing a Matrix-free Analytical Jacobian to Handle Nonlinearities in Models of 3D Lithospheric Deformation

NASA Astrophysics Data System (ADS)

Kaus, B.; Popov, A.

2015-12-01

The analytical expression for the Jacobian is a key component to achieve fast and robust convergence of the nonlinear Newton-Raphson iterative solver. Accomplishing this task in practice often requires a significant algebraic effort. Therefore it is quite common to use a cheap alternative instead, for example by approximating the Jacobian with a finite difference estimation. Despite its simplicity it is a relatively fragile and unreliable technique that is sensitive to the scaling of the residual and unknowns, as well as to the perturbation parameter selection. Unfortunately no universal rule can be applied to provide both a robust scaling and a perturbation. The approach we use here is to derive the analytical Jacobian for the coupled set of momentum, mass, and energy conservation equations together with the elasto-visco-plastic rheology and a marker in cell/staggered finite difference method. The software project LaMEM (Lithosphere and Mantle Evolution Model) is primarily developed for the thermo-mechanically coupled modeling of the 3D lithospheric deformation. The code is based on a staggered grid finite difference discretization in space, and uses customized scalable solvers form PETSc library to efficiently run on the massively parallel machines (such as IBM Blue Gene/Q). Currently LaMEM relies on the Jacobian-Free Newton-Krylov (JFNK) nonlinear solver, which approximates the Jacobian-vector product using a simple finite difference formula. This approach never requires an assembled Jacobian matrix and uses only the residual computation routine. We use an approximate Jacobian (Picard) matrix to precondition the Krylov solver with the Galerkin geometric multigrid. Because of the inherent problems of the finite difference Jacobian estimation, this approach doesn't always result in stable convergence. In this work we present and discuss a matrix-free technique in which the Jacobian-vector product is replaced by analytically-derived expressions and compare results with those obtained with a finite difference approximation of the Jacobian. This project is funded by ERC Starting Grant 258830 and computer facilities were provided by Jülich supercomputer center (Germany).

SPIREs: A Finite-Difference Frequency-Domain electromagnetic solver for inhomogeneous magnetized plasma cylinders

NASA Astrophysics Data System (ADS)

Melazzi, D.; Curreli, D.; Manente, M.; Carlsson, J.; Pavarin, D.

2012-06-01

We present SPIREs (plaSma Padova Inhomogeneous Radial Electromagnetic solver), a Finite-Difference Frequency-Domain (FDFD) electromagnetic solver in one dimension for the rapid calculation of the electromagnetic fields and the deposited power of a large variety of cylindrical plasma problems. The two Maxwell wave equations have been discretized using a staggered Yee mesh along the radial direction of the cylinder, and Fourier transformed along the other two dimensions and in time. By means of this kind of discretization, we have found that mode-coupling of fast and slow branches can be fully resolved without singularity issues that flawed other well-established methods in the past. Fields are forced by an antenna placed at a given distance from the plasma. The plasma can be inhomogeneous, finite-temperature, collisional, magnetized and multi-species. Finite-temperature Maxwellian effects, comprising Landau and cyclotron damping, have been included by means of the plasma Z dispersion function. Finite Larmor radius effects have been neglected. Radial variations of the plasma parameters are taken into account, thus extending the range of applications to a large variety of inhomogeneous plasma systems. The method proved to be fast and reliable, with accuracy depending on the spatial grid size. Two physical examples are reported: fields in a forced vacuum waveguide with the antenna inside, and forced plasma oscillations in the helicon radiofrequency range.

Single-cone finite-difference schemes for the (2+1)-dimensional Dirac equation in general electromagnetic textures

NASA Astrophysics Data System (ADS)

Pötz, Walter

2017-11-01

A single-cone finite-difference lattice scheme is developed for the (2+1)-dimensional Dirac equation in presence of general electromagnetic textures. The latter is represented on a (2+1)-dimensional staggered grid using a second-order-accurate finite difference scheme. A Peierls-Schwinger substitution to the wave function is used to introduce the electromagnetic (vector) potential into the Dirac equation. Thereby, the single-cone energy dispersion and gauge invariance are carried over from the continuum to the lattice formulation. Conservation laws and stability properties of the formal scheme are identified by comparison with the scheme for zero vector potential. The placement of magnetization terms is inferred from consistency with the one for the vector potential. Based on this formal scheme, several numerical schemes are proposed and tested. Elementary examples for single-fermion transport in the presence of in-plane magnetization are given, using material parameters typical for topological insulator surfaces.

3D Tensorial Elastodynamics for Isotropic Media on Vertically Deformed Meshes

NASA Astrophysics Data System (ADS)

Shragge, J. C.

2017-12-01

Solutions of the 3D elastodynamic wave equation are sometimes required in industrial and academic applications of elastic reverse-time migration (E-RTM) and full waveform inversion (E-FWI) that involve vertically deformed meshes. Examples include incorporating irregular free-surface topography and handling internal boundaries (e.g., water bottom) directly into the computational meshes. In 3D E-RTM and E-FWI applications, the number of forward modeling simulations can number in the tens of thousands (per iteration), which necessitates the development of stable, accurate and efficient 3D elastodynamics solvers. For topographic scenarios, most finite-difference solution approaches use a change-of-variable strategy that has a number of associated computational challenges, including difficulties in handling of the free-surface boundary condition. In this study, I follow a tensorial approach and use a generalized family of analytic transforms to develop a set of analytic equations for 3D elastodynamics that directly incorporates vertical grid deformations. Importantly, this analytic approach allows for the specification of an analytic free-surface boundary condition appropriate for vertically deformed meshes. These equations are both straightforward and efficient to solve using a velocity-stress formulation with finite-difference (MFD) operators implemented on a fully staggered grid. Moreover, I demonstrate that the use of mimetic finite difference (MFD) methods allows stable, accurate, and efficient numerical solutions to be simulated for typical topographic scenarios. Examples demonstrate that high-quality elastic wavefields can be generated for topographic surfaces exhibiting significant topographic relief.

An algorithm for fast elastic wave simulation using a vectorized finite difference operator

NASA Astrophysics Data System (ADS)

Malkoti, Ajay; Vedanti, Nimisha; Tiwari, Ram Krishna

2018-07-01

Modern geophysical imaging techniques exploit the full wavefield information which can be simulated numerically. These numerical simulations are computationally expensive due to several factors, such as a large number of time steps and nodes, big size of the derivative stencil and huge model size. Besides these constraints, it is also important to reformulate the numerical derivative operator for improved efficiency. In this paper, we have introduced a vectorized derivative operator over the staggered grid with shifted coordinate systems. The operator increases the efficiency of simulation by exploiting the fact that each variable can be represented in the form of a matrix. This operator allows updating all nodes of a variable defined on the staggered grid, in a manner similar to the collocated grid scheme and thereby reducing the computational run-time considerably. Here we demonstrate an application of this operator to simulate the seismic wave propagation in elastic media (Marmousi model), by discretizing the equations on a staggered grid. We have compared the performance of this operator on three programming languages, which reveals that it can increase the execution speed by a factor of at least 2-3 times for FORTRAN and MATLAB; and nearly 100 times for Python. We have further carried out various tests in MATLAB to analyze the effect of model size and the number of time steps on total simulation run-time. We find that there is an additional, though small, computational overhead for each step and it depends on total number of time steps used in the simulation. A MATLAB code package, 'FDwave', for the proposed simulation scheme is available upon request.

A time-space domain stereo finite difference method for 3D scalar wave propagation

NASA Astrophysics Data System (ADS)

Chen, Yushu; Yang, Guangwen; Ma, Xiao; He, Conghui; Song, Guojie

2016-11-01

The time-space domain finite difference methods reduce numerical dispersion effectively by minimizing the error in the joint time-space domain. However, their interpolating coefficients are related with the Courant numbers, leading to significantly extra time costs for loading the coefficients consecutively according to velocity in heterogeneous models. In the present study, we develop a time-space domain stereo finite difference (TSSFD) method for 3D scalar wave equation. The method propagates both the displacements and their gradients simultaneously to keep more information of the wavefields, and minimizes the maximum phase velocity error directly using constant interpolation coefficients for different Courant numbers. We obtain the optimal constant coefficients by combining the truncated Taylor series approximation and the time-space domain optimization, and adjust the coefficients to improve the stability condition. Subsequent investigation shows that the TSSFD can suppress numerical dispersion effectively with high computational efficiency. The maximum phase velocity error of the TSSFD is just 3.09% even with only 2 sampling points per minimum wavelength when the Courant number is 0.4. Numerical experiments show that to generate wavefields with no visible numerical dispersion, the computational efficiency of the TSSFD is 576.9%, 193.5%, 699.0%, and 191.6% of those of the 4th-order and 8th-order Lax-Wendroff correction (LWC) method, the 4th-order staggered grid method (SG), and the 8th-order optimal finite difference method (OFD), respectively. Meanwhile, the TSSFD is compatible to the unsplit convolutional perfectly matched layer (CPML) boundary condition for absorbing artificial boundaries. The efficiency and capability to handle complex velocity models make it an attractive tool in imaging methods such as acoustic reverse time migration (RTM).

A conservative staggered-grid Chebyshev multidomain method for compressible flows

NASA Technical Reports Server (NTRS)

Kopriva, David A.; Kolias, John H.

1995-01-01

We present a new multidomain spectral collocation method that uses staggered grids for the solution of compressible flow problems. The solution unknowns are defined at the nodes of a Gauss quadrature rule. The fluxes are evaluated at the nodes of a Gauss-Lobatto rule. The method is conservative, free-stream preserving, and exponentially accurate. A significant advantage of the method is that subdomain corners are not included in the approximation, making solutions in complex geometries easier to compute.

Parallel Cartesian grid refinement for 3D complex flow simulations

NASA Astrophysics Data System (ADS)

Angelidis, Dionysios; Sotiropoulos, Fotis

2013-11-01

A second order accurate method for discretizing the Navier-Stokes equations on 3D unstructured Cartesian grids is presented. Although the grid generator is based on the oct-tree hierarchical method, fully unstructured data-structure is adopted enabling robust calculations for incompressible flows, avoiding both the need of synchronization of the solution between different levels of refinement and usage of prolongation/restriction operators. The current solver implements a hybrid staggered/non-staggered grid layout, employing the implicit fractional step method to satisfy the continuity equation. The pressure-Poisson equation is discretized by using a novel second order fully implicit scheme for unstructured Cartesian grids and solved using an efficient Krylov subspace solver. The momentum equation is also discretized with second order accuracy and the high performance Newton-Krylov method is used for integrating them in time. Neumann and Dirichlet conditions are used to validate the Poisson solver against analytical functions and grid refinement results to a significant reduction of the solution error. The effectiveness of the fractional step method results in the stability of the overall algorithm and enables the performance of accurate multi-resolution real life simulations. This material is based upon work supported by the Department of Energy under Award Number DE-EE0005482.

Laser-plasma interactions with a Fourier-Bessel particle-in-cell method

DOE Office of Scientific and Technical Information (OSTI.GOV)

Andriyash, Igor A., E-mail: igor.andriyash@gmail.com; LOA, ENSTA ParisTech, CNRS, Ecole polytechnique, Université Paris-Saclay, 828 bd des Maréchaux, 91762 Palaiseau cedex; Lehe, Remi

A new spectral particle-in-cell (PIC) method for plasma modeling is presented and discussed. In the proposed scheme, the Fourier-Bessel transform is used to translate the Maxwell equations to the quasi-cylindrical spectral domain. In this domain, the equations are solved analytically in time, and the spatial derivatives are approximated with high accuracy. In contrast to the finite-difference time domain (FDTD) methods, that are used commonly in PIC, the developed method does not produce numerical dispersion and does not involve grid staggering for the electric and magnetic fields. These features are especially valuable in modeling the wakefield acceleration of particles in plasmas.more » The proposed algorithm is implemented in the code PLARES-PIC, and the test simulations of laser plasma interactions are compared to the ones done with the quasi-cylindrical FDTD PIC code CALDER-CIRC.« less

Compatible Spatial Discretizations for Partial Differential Equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Arnold, Douglas, N, ed.

From May 11--15, 2004, the Institute for Mathematics and its Applications held a hot topics workshop on Compatible Spatial Discretizations for Partial Differential Equations. The numerical solution of partial differential equations (PDE) is a fundamental task in science and engineering. The goal of the workshop was to bring together a spectrum of scientists at the forefront of the research in the numerical solution of PDEs to discuss compatible spatial discretizations. We define compatible spatial discretizations as those that inherit or mimic fundamental properties of the PDE such as topology, conservation, symmetries, and positivity structures and maximum principles. A wide varietymore » of discretization methods applied across a wide range of scientific and engineering applications have been designed to or found to inherit or mimic intrinsic spatial structure and reproduce fundamental properties of the solution of the continuous PDE model at the finite dimensional level. A profusion of such methods and concepts relevant to understanding them have been developed and explored: mixed finite element methods, mimetic finite differences, support operator methods, control volume methods, discrete differential forms, Whitney forms, conservative differencing, discrete Hodge operators, discrete Helmholtz decomposition, finite integration techniques, staggered grid and dual grid methods, etc. This workshop seeks to foster communication among the diverse groups of researchers designing, applying, and studying such methods as well as researchers involved in practical solution of large scale problems that may benefit from advancements in such discretizations; to help elucidate the relations between the different methods and concepts; and to generally advance our understanding in the area of compatible spatial discretization methods for PDE. Particular points of emphasis included: + Identification of intrinsic properties of PDE models that are critical for the fidelity of numerical simulations. + Identification and design of compatible spatial discretizations of PDEs, their classification, analysis, and relations. + Relationships between different compatible spatial discretization methods and concepts which have been developed; + Impact of compatible spatial discretizations upon physical fidelity, verification and validation of simulations, especially in large-scale, multiphysics settings. + How solvers address the demands placed upon them by compatible spatial discretizations. This report provides information about the program and abstracts of all the presentations.« less

Nonuniform grid implicit spatial finite difference method for acoustic wave modeling in tilted transversely isotropic media

NASA Astrophysics Data System (ADS)

Chu, Chunlei; Stoffa, Paul L.

2012-01-01

Discrete earth models are commonly represented by uniform structured grids. In order to ensure accurate numerical description of all wave components propagating through these uniform grids, the grid size must be determined by the slowest velocity of the entire model. Consequently, high velocity areas are always oversampled, which inevitably increases the computational cost. A practical solution to this problem is to use nonuniform grids. We propose a nonuniform grid implicit spatial finite difference method which utilizes nonuniform grids to obtain high efficiency and relies on implicit operators to achieve high accuracy. We present a simple way of deriving implicit finite difference operators of arbitrary stencil widths on general nonuniform grids for the first and second derivatives and, as a demonstration example, apply these operators to the pseudo-acoustic wave equation in tilted transversely isotropic (TTI) media. We propose an efficient gridding algorithm that can be used to convert uniformly sampled models onto vertically nonuniform grids. We use a 2D TTI salt model to demonstrate its effectiveness and show that the nonuniform grid implicit spatial finite difference method can produce highly accurate seismic modeling results with enhanced efficiency, compared to uniform grid explicit finite difference implementations.

Three-dimensional magnetotelluric axial anisotropic forward modeling and inversion

NASA Astrophysics Data System (ADS)

Cao, Hui; Wang, Kunpeng; Wang, Tao; Hua, Boguang

2018-06-01

Magnetotelluric (MT) data has been widely used to image underground electrical structural. However, when the significant axial resistivity anisotropy presents, how this influences three-dimensional MT data has not been resolved clearly yet. We here propose a scheme for three-dimensional modeling of MT data in presence of axial anisotropic resistivity, where the electromagnetic fields are decomposed into primary and secondary components. A 3D staggered-grid finite difference method is then used to resolve the resulting 3D governing equations. Numerical tests have completed to validate the correctness and accuracy of the present algorithm. A limited-memory Broyden-Fletcher-Goldfarb-Shanno method is then utilized to realize the 3D MT axial anisotropic inversion. The testing results show that, compared to the results of isotropic resistivity inversion, taking account the axial anisotropy can much improve the inverted results.

Generalization of von Neumann analysis for a model of two discrete half-spaces: The acoustic case

USGS Publications Warehouse

Haney, M.M.

2007-01-01

Evaluating the performance of finite-difference algorithms typically uses a technique known as von Neumann analysis. For a given algorithm, application of the technique yields both a dispersion relation valid for the discrete time-space grid and a mathematical condition for stability. In practice, a major shortcoming of conventional von Neumann analysis is that it can be applied only to an idealized numerical model - that of an infinite, homogeneous whole space. Experience has shown that numerical instabilities often arise in finite-difference simulations of wave propagation at interfaces with strong material contrasts. These interface instabilities occur even though the conventional von Neumann stability criterion may be satisfied at each point of the numerical model. To address this issue, I generalize von Neumann analysis for a model of two half-spaces. I perform the analysis for the case of acoustic wave propagation using a standard staggered-grid finite-difference numerical scheme. By deriving expressions for the discrete reflection and transmission coefficients, I study under what conditions the discrete reflection and transmission coefficients become unbounded. I find that instabilities encountered in numerical modeling near interfaces with strong material contrasts are linked to these cases and develop a modified stability criterion that takes into account the resulting instabilities. I test and verify the stability criterion by executing a finite-difference algorithm under conditions predicted to be stable and unstable. ?? 2007 Society of Exploration Geophysicists.

Momentum Advection on a Staggered Mesh

NASA Astrophysics Data System (ADS)

Benson, David J.

1992-05-01

Eulerian and ALE (arbitrary Lagrangian-Eulerian) hydrodynamics programs usually split a timestep into two parts. The first part is a Lagrangian step, which calculates the incremental motion of the material. The second part is referred to as the Eulerian step, the advection step, or the remap step, and it accounts for the transport of material between cells. In most finite difference and finite element formulations, all the solution variables except the velocities are cell-centered while the velocities are edge- or vertex-centered. As a result, the advection algorithm for the momentum is, by necessity, different than the algorithm used for the other variables. This paper reviews three momentum advection methods and proposes a new one. One method, pioneered in YAQUI, creates a new staggered mesh, while the other two, used in SALE and SHALE, are cell-centered. The new method is cell-centered and its relationship to the other methods is discussed. Both pure advection and strong shock calculations are presented to substantiate the mathematical analysis. From the standpoint of numerical accuracy, both the staggered mesh and the cell-centered algorithms can give good results, while the computational costs are highly dependent on the overall architecture of a code.

FDTD simulation of EM wave propagation in 3-D media

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wang, T.; Tripp, A.C.

1996-01-01

A finite-difference, time-domain solution to Maxwell`s equations has been developed for simulating electromagnetic wave propagation in 3-D media. The algorithm allows arbitrary electrical conductivity and permittivity variations within a model. The staggered grid technique of Yee is used to sample the fields. A new optimized second-order difference scheme is designed to approximate the spatial derivatives. Like the conventional fourth-order difference scheme, the optimized second-order scheme needs four discrete values to calculate a single derivative. However, the optimized scheme is accurate over a wider wavenumber range. Compared to the fourth-order scheme, the optimized scheme imposes stricter limitations on the time stepmore » sizes but allows coarser grids. The net effect is that the optimized scheme is more efficient in terms of computation time and memory requirement than the fourth-order scheme. The temporal derivatives are approximated by second-order central differences throughout. The Liao transmitting boundary conditions are used to truncate an open problem. A reflection coefficient analysis shows that this transmitting boundary condition works very well. However, it is subject to instability. A method that can be easily implemented is proposed to stabilize the boundary condition. The finite-difference solution is compared to closed-form solutions for conducting and nonconducting whole spaces and to an integral-equation solution for a 3-D body in a homogeneous half-space. In all cases, the finite-difference solutions are in good agreement with the other solutions. Finally, the use of the algorithm is demonstrated with a 3-D model. Numerical results show that both the magnetic field response and electric field response can be useful for shallow-depth and small-scale investigations.« less

Parallel Computing of Upwelling in a Rotating Stratified Flow

NASA Astrophysics Data System (ADS)

Cui, A.; Street, R. L.

1997-11-01

A code for the three-dimensional, unsteady, incompressible, and turbulent flow has been implemented on the IBM SP2, using message passing. The effects of rotation and variable density are included. A finite volume method is used to discretize the Navier-Stokes equations in general curvilinear coordinates on a non-staggered grid. All the spatial derivatives are approximated using second-order central differences with the exception of the convection terms, which are handled with special upwind-difference schemes. The semi-implicit, second-order accurate, time-advancement scheme employs the Adams-Bashforth method for the explicit terms and Crank-Nicolson for the implicit terms. A multigrid method, with the four-color ZEBRA as smoother, is used to solve the Poisson equation for pressure, while the momentum equations are solved with an approximate factorization technique. The code was successfully validated for a variety test cases. Simulations of a laboratory model of coastal upwelling in a rotating annulus are in progress and will be presented.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wiley, J.C.

The author describes a general `hp` finite element method with adaptive grids. The code was based on the work of Oden, et al. The term `hp` refers to the method of spatial refinement (h), in conjunction with the order of polynomials used as a part of the finite element discretization (p). This finite element code seems to handle well the different mesh grid sizes occuring between abuted grids with different resolutions.

Technical report series on global modeling and data assimilation. Volume 5: Documentation of the AIRES/GEOS dynamical core, version 2

NASA Technical Reports Server (NTRS)

Suarez, Max J. (Editor); Takacs, Lawrence L.

1995-01-01

A detailed description of the numerical formulation of Version 2 of the ARIES/GEOS 'dynamical core' is presented. This code is a nearly 'plug-compatible' dynamics for use in atmospheric general circulation models (GCMs). It is a finite difference model on a staggered latitude-longitude C-grid. It uses second-order differences for all terms except the advection of vorticity by the rotation part of the flow, which is done at fourth-order accuracy. This dynamical core is currently being used in the climate (ARIES) and data assimilation (GEOS) GCMs at Goddard.

A compatible high-order meshless method for the Stokes equations with applications to suspension flows

NASA Astrophysics Data System (ADS)

Trask, Nathaniel; Maxey, Martin; Hu, Xiaozhe

2018-02-01

A stable numerical solution of the steady Stokes problem requires compatibility between the choice of velocity and pressure approximation that has traditionally proven problematic for meshless methods. In this work, we present a discretization that couples a staggered scheme for pressure approximation with a divergence-free velocity reconstruction to obtain an adaptive, high-order, finite difference-like discretization that can be efficiently solved with conventional algebraic multigrid techniques. We use analytic benchmarks to demonstrate equal-order convergence for both velocity and pressure when solving problems with curvilinear geometries. In order to study problems in dense suspensions, we couple the solution for the flow to the equations of motion for freely suspended particles in an implicit monolithic scheme. The combination of high-order accuracy with fully-implicit schemes allows the accurate resolution of stiff lubrication forces directly from the solution of the Stokes problem without the need to introduce sub-grid lubrication models.

A numerical homogenization method for heterogeneous, anisotropic elastic media based on multiscale theory

DOE PAGES

Gao, Kai; Chung, Eric T.; Gibson, Richard L.; ...

2015-06-05

The development of reliable methods for upscaling fine scale models of elastic media has long been an important topic for rock physics and applied seismology. Several effective medium theories have been developed to provide elastic parameters for materials such as finely layered media or randomly oriented or aligned fractures. In such cases, the analytic solutions for upscaled properties can be used for accurate prediction of wave propagation. However, such theories cannot be applied directly to homogenize elastic media with more complex, arbitrary spatial heterogeneity. We therefore propose a numerical homogenization algorithm based on multiscale finite element methods for simulating elasticmore » wave propagation in heterogeneous, anisotropic elastic media. Specifically, our method used multiscale basis functions obtained from a local linear elasticity problem with appropriately defined boundary conditions. Homogenized, effective medium parameters were then computed using these basis functions, and the approach applied a numerical discretization that is similar to the rotated staggered-grid finite difference scheme. Comparisons of the results from our method and from conventional, analytical approaches for finely layered media showed that the homogenization reliably estimated elastic parameters for this simple geometry. Additional tests examined anisotropic models with arbitrary spatial heterogeneity where the average size of the heterogeneities ranged from several centimeters to several meters, and the ratio between the dominant wavelength and the average size of the arbitrary heterogeneities ranged from 10 to 100. Comparisons to finite-difference simulations proved that the numerical homogenization was equally accurate for these complex cases.« less

An overlapped grid method for multigrid, finite volume/difference flow solvers: MaGGiE

NASA Technical Reports Server (NTRS)

Baysal, Oktay; Lessard, Victor R.

1990-01-01

The objective is to develop a domain decomposition method via overlapping/embedding the component grids, which is to be used by upwind, multi-grid, finite volume solution algorithms. A computer code, given the name MaGGiE (Multi-Geometry Grid Embedder) is developed to meet this objective. MaGGiE takes independently generated component grids as input, and automatically constructs the composite mesh and interpolation data, which can be used by the finite volume solution methods with or without multigrid convergence acceleration. Six demonstrative examples showing various aspects of the overlap technique are presented and discussed. These cases are used for developing the procedure for overlapping grids of different topologies, and to evaluate the grid connection and interpolation data for finite volume calculations on a composite mesh. Time fluxes are transferred between mesh interfaces using a trilinear interpolation procedure. Conservation losses are minimal at the interfaces using this method. The multi-grid solution algorithm, using the coaser grid connections, improves the convergence time history as compared to the solution on composite mesh without multi-gridding.

The Stagger-grid: A grid of 3D stellar atmosphere models. II. Horizontal and temporal averaging and spectral line formation

NASA Astrophysics Data System (ADS)

Magic, Z.; Collet, R.; Hayek, W.; Asplund, M.

2013-12-01

Aims: We study the implications of averaging methods with different reference depth scales for 3D hydrodynamical model atmospheres computed with the Stagger-code. The temporally and spatially averaged (hereafter denoted as ⟨3D⟩) models are explored in the light of local thermodynamic equilibrium (LTE) spectral line formation by comparing spectrum calculations using full 3D atmosphere structures with those from ⟨3D⟩ averages. Methods: We explored methods for computing mean ⟨3D⟩ stratifications from the Stagger-grid time-dependent 3D radiative hydrodynamical atmosphere models by considering four different reference depth scales (geometrical depth, column-mass density, and two optical depth scales). Furthermore, we investigated the influence of alternative averages (logarithmic, enforced hydrostatic equilibrium, flux-weighted temperatures). For the line formation we computed curves of growth for Fe i and Fe ii lines in LTE. Results: The resulting ⟨3D⟩ stratifications for the four reference depth scales can be very different. We typically find that in the upper atmosphere and in the superadiabatic region just below the optical surface, where the temperature and density fluctuations are highest, the differences become considerable and increase for higher Teff, lower log g, and lower [Fe / H]. The differential comparison of spectral line formation shows distinctive differences depending on which ⟨3D⟩ model is applied. The averages over layers of constant column-mass density yield the best mean ⟨3D⟩ representation of the full 3D models for LTE line formation, while the averages on layers at constant geometrical height are the least appropriate. Unexpectedly, the usually preferred averages over layers of constant optical depth are prone to increasing interference by reversed granulation towards higher effective temperature, in particular at low metallicity. Appendix A is available in electronic form at http://www.aanda.orgMean ⟨3D⟩ models are available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/560/A8 as well as at http://www.stagger-stars.net

Finite-difference time-domain simulation of GPR data

NASA Astrophysics Data System (ADS)

Chen, How-Wei; Huang, Tai-Min

1998-10-01

Simulation of digital ground penetrating radar (GPR) wave propagation in two-dimensional (2-D) media is developed, tested, implemented, and applied using a time-domain staggered-grid finite-difference (FD) numerical method. Three types of numerical algorithms for constructing synthetic common-shot, constant-offset radar profiles based on an actual transmitter-to-receiver configuration and based on the exploding reflector concept are demonstrated to mimic different types of radar survey geometries. Frequency-dependent attenuation is also incorporated to account for amplitude decay and time shift in the recorded responses. The algorithms are based on an explicit FD solution to Maxwell's curl equations. In addition, the first-order TE mode responses of wave propagation phenomena are considered due to the operating frequency of current GPR instruments. The staggered-grid technique is used to sample the fields and approximate the spatial derivatives with fourth-order FDs. The temporal derivatives are approximated by an explicit second-order difference time-marching scheme. By combining paraxial approximation of the one-way wave equation ( A2) and the damping mechanisms (sponge filter), we propose a new composite absorbing boundary conditions (ABC) algorithm that effectively absorb both incoming and outgoing waves. To overcome the angle- and frequency-dependent characteristic of the absorbing behaviors, each ABC has two types of absorption mechanism. The first ABC uses a modified Clayton and Enquist's A2 condition. Moreover, a fixed and a floating A2 ABC that operates at one grid point is proposed. The second ABC uses a damping mechanism. By superimposing artificial damping and by alternating the physical attenuation properties and impedance contrast of the media within the absorbing region, those waves impinging on the boundary can be effectively attenuated and can prevent waves from reflecting back into the grid. The frequency-dependent characteristic of the damping mechanism can be used to adjust the width of the absorbing zone around the computational domain. By applying any combination of absorbing mechanism, non-physical reflections from the computation domain boundary can be effectively minimized. The algorithm enables us to use very thin absorbing boundaries. The model can be parameterized through velocity, relative electrical permittivity (dielectric constants), electrical conductivity, magnetic permeability, loss tangent, Q values, and attenuation. According to this scheme, widely varying electrical properties of near-surface earth materials can be modeled. The capability of simulating common-source, constant-offset and zero-offset gathers is also demonstrated through various synthetic examples. The synthetic cases for typical GPR applications include buried objects such as pipes of different materials, AVO analysis for ground water exploration, archaeological site investigation, and stratigraphy studies. The algorithms are also applied to iterative modeling of GPR data acquired over a gymnasium construction site on the NCCU campus.

Subsurface Void Characterization with 3-D Time Domain Full Waveform Tomography.

NASA Astrophysics Data System (ADS)

Nguyen, T. D.

2017-12-01

A new three dimensional full waveform inversion (3-D FWI) method is presented for subsurface site characterization at engineering scales (less than 30 m in depth). The method is based on a solution of 3-D elastic wave equations for forward modeling, and a cross-adjoint gradient approach for model updating. The staggered-grid finite-difference technique is used to solve the wave equations, together with implementation of the perfectly matched layer condition for boundary truncation. The gradient is calculated from the forward and backward wavefields. Reversed-in-time displacement residuals are induced as multiple sources at all receiver locations for the backward wavefield. The capability of the presented FWI method is tested on both synthetic and field experimental datasets. The test configuration uses 96 receivers and 117 shots at equal spacing (Fig 1). The inversion results from synthetic data show the ability of characterizing variable low- and high-velocity layers with embedded void (Figs 2-3). The synthetic study shows good potential for detection of voids and abnormalities in the field.

Three-dimensional local grid refinement for block-centered finite-difference groundwater models using iteratively coupled shared nodes: A new method of interpolation and analysis of errors

USGS Publications Warehouse

Mehl, S.; Hill, M.C.

2004-01-01

This paper describes work that extends to three dimensions the two-dimensional local-grid refinement method for block-centered finite-difference groundwater models of Mehl and Hill [Development and evaluation of a local grid refinement method for block-centered finite-difference groundwater models using shared nodes. Adv Water Resour 2002;25(5):497-511]. In this approach, the (parent) finite-difference grid is discretized more finely within a (child) sub-region. The grid refinement method sequentially solves each grid and uses specified flux (parent) and specified head (child) boundary conditions to couple the grids. Iteration achieves convergence between heads and fluxes of both grids. Of most concern is how to interpolate heads onto the boundary of the child grid such that the physics of the parent-grid flow is retained in three dimensions. We develop a new two-step, "cage-shell" interpolation method based on the solution of the flow equation on the boundary of the child between nodes shared with the parent grid. Error analysis using a test case indicates that the shared-node local grid refinement method with cage-shell boundary head interpolation is accurate and robust, and the resulting code is used to investigate three-dimensional local grid refinement of stream-aquifer interactions. Results reveal that (1) the parent and child grids interact to shift the true head and flux solution to a different solution where the heads and fluxes of both grids are in equilibrium, (2) the locally refined model provided a solution for both heads and fluxes in the region of the refinement that was more accurate than a model without refinement only if iterations are performed so that both heads and fluxes are in equilibrium, and (3) the accuracy of the coupling is limited by the parent-grid size - A coarse parent grid limits correct representation of the hydraulics in the feedback from the child grid.

Stability of finite difference numerical simulations of acoustic logging-while-drilling with different perfectly matched layer schemes

NASA Astrophysics Data System (ADS)

Wang, Hua; Tao, Guo; Shang, Xue-Feng; Fang, Xin-Ding; Burns, Daniel R.

2013-12-01

In acoustic logging-while-drilling (ALWD) finite difference in time domain (FDTD) simulations, large drill collar occupies, most of the fluid-filled borehole and divides the borehole fluid into two thin fluid columns (radius ˜27 mm). Fine grids and large computational models are required to model the thin fluid region between the tool and the formation. As a result, small time step and more iterations are needed, which increases the cumulative numerical error. Furthermore, due to high impedance contrast between the drill collar and fluid in the borehole (the difference is >30 times), the stability and efficiency of the perfectly matched layer (PML) scheme is critical to simulate complicated wave modes accurately. In this paper, we compared four different PML implementations in a staggered grid finite difference in time domain (FDTD) in the ALWD simulation, including field-splitting PML (SPML), multiaxial PML(MPML), non-splitting PML (NPML), and complex frequency-shifted PML (CFS-PML). The comparison indicated that NPML and CFS-PML can absorb the guided wave reflection from the computational boundaries more efficiently than SPML and M-PML. For large simulation time, SPML, M-PML, and NPML are numerically unstable. However, the stability of M-PML can be improved further to some extent. Based on the analysis, we proposed that the CFS-PML method is used in FDTD to eliminate the numerical instability and to improve the efficiency of absorption in the PML layers for LWD modeling. The optimal values of CFS-PML parameters in the LWD simulation were investigated based on thousands of 3D simulations. For typical LWD cases, the best maximum value of the quadratic damping profile was obtained using one d 0. The optimal parameter space for the maximum value of the linear frequency-shifted factor ( α 0) and the scaling factor ( β 0) depended on the thickness of the PML layer. For typical formations, if the PML thickness is 10 grid points, the global error can be reduced to <1% using the optimal PML parameters, and the error will decrease as the PML thickness increases.

Calculation of three-dimensional (3-D) internal flow by means of the velocity-vorticity formulation on a staggered grid

NASA Technical Reports Server (NTRS)

Stremel, Paul M.

1995-01-01

A method has been developed to accurately compute the viscous flow in three-dimensional (3-D) enclosures. This method is the 3-D extension of a two-dimensional (2-D) method developed for the calculation of flow over airfoils. The 2-D method has been tested extensively and has been shown to accurately reproduce experimental results. As in the 2-D method, the 3-D method provides for the non-iterative solution of the incompressible Navier-Stokes equations by means of a fully coupled implicit technique. The solution is calculated on a body fitted computational mesh incorporating a staggered grid methodology. In the staggered grid method, the three components of vorticity are defined at the centers of the computational cell sides, while the velocity components are defined as normal vectors at the centers of the computational cell faces. The staggered grid orientation provides for the accurate definition of the vorticity components at the vorticity locations, the divergence of vorticity at the mesh cell nodes and the conservation of mass at the mesh cell centers. The solution is obtained by utilizing a fractional step solution technique in the three coordinate directions. The boundary conditions for the vorticity and velocity are calculated implicitly as part of the solution. The method provides for the non-iterative solution of the flow field and satisfies the conservation of mass and divergence of vorticity to machine zero at each time step. To test the method, the calculation of simple driven cavity flows have been computed. The driven cavity flow is defined as the flow in an enclosure driven by a moving upper plate at the top of the enclosure. To demonstrate the ability of the method to predict the flow in arbitrary cavities, results will he shown for both cubic and curved cavities.

Modeling 3D Dynamic Rupture on Arbitrarily-Shaped faults by Boundary-Conforming Finite Difference Method

NASA Astrophysics Data System (ADS)

Zhu, D.; Zhu, H.; Luo, Y.; Chen, X.

2008-12-01

We use a new finite difference method (FDM) and the slip-weakening law to model the rupture dynamics of a non-planar fault embedded in a 3-D elastic media with free surface. The new FDM, based on boundary- conforming grid, sets up the mapping equations between the curvilinear coordinate and the Cartesian coordinate and transforms irregular physical space to regular computational space; it also employs a higher- order non-staggered DRP/opt MacCormack scheme which is of low dispersion and low dissipation so that the high accuracy and stability of our rupture modeling are guaranteed. Compared with the previous methods, not only we can compute the spontaneous rupture of an arbitrarily shaped fault, but also can model the influence of the surface topography on the rupture process of earthquake. In order to verify the feasibility of this method, we compared our results and other previous results, and found out they matched perfectly. Thanks to the boundary-conforming FDM, problems such as dynamic rupture with arbitrary dip, strike and rake over an arbitrary curved plane can be handled; and supershear or subshear rupture can be simulated with different parameters such as the initial stresses and the critical slip displacement Dc. Besides, our rupture modeling is economical to be implemented owing to its high efficiency and does not suffer from displacement leakage. With the help of inversion data of rupture by field observations, this method is convenient to model rupture processes and seismograms of natural earthquakes.

Conjugate Heat Transfer in Rayleigh-Bénard Convection in a Square Enclosure

PubMed Central

Hashim, Ishak

2014-01-01

Conjugate natural convection-conduction heat transfer in a square enclosure with a finite wall thickness is studied numerically in the present paper. The governing parameters considered are the Rayleigh number (5 × 103 ≤ Ra ≤ 106), the wall-to-fluid thermal conductivity ratio (0.5 ≤ Kr ≤ 10), and the ratio of wall thickness to its height (0.2 ≤ D ≤ 0.4). The staggered grid arrangement together with MAC method was employed to solve the governing equations. It is found that the fluid flow and the heat transfer can be controlled by the thickness of the bottom wall, the thermal conductivity ratio, and the Rayleigh number. PMID:24971390

Development of a fractional-step method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems

NASA Technical Reports Server (NTRS)

Rosenfeld, Moshe; Kwak, Dochan; Vinokur, Marcel

1992-01-01

A fractional step method is developed for solving the time-dependent three-dimensional incompressible Navier-Stokes equations in generalized coordinate systems. The primitive variable formulation uses the pressure, defined at the center of the computational cell, and the volume fluxes across the faces of the cells as the dependent variables, instead of the Cartesian components of the velocity. This choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell. The governing equations are discretized by finite volumes using a staggered mesh system. The solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a Poisson equation. This procedure, combined with the consistent approximations of the geometric quantities, is done to satisfy the discretized mass conservation equation to machine accuracy, as well as to gain the favorable convergence properties of the Poisson solver. The momentum equations are solved by an approximate factorization method, and a novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two- and three-dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method. Good agreement is obtained in all cases.

2-D Model for Normal and Sickle Cell Blood Microcirculation

NASA Astrophysics Data System (ADS)

Tekleab, Yonatan; Harris, Wesley

2011-11-01

Sickle cell disease (SCD) is a genetic disorder that alters the red blood cell (RBC) structure and function such that hemoglobin (Hb) cannot effectively bind and release oxygen. Previous computational models have been designed to study the microcirculation for insight into blood disorders such as SCD. Our novel 2-D computational model represents a fast, time efficient method developed to analyze flow dynamics, O2 diffusion, and cell deformation in the microcirculation. The model uses a finite difference, Crank-Nicholson scheme to compute the flow and O2 concentration, and the level set computational method to advect the RBC membrane on a staggered grid. Several sets of initial and boundary conditions were tested. Simulation data indicate a few parameters to be significant in the perturbation of the blood flow and O2 concentration profiles. Specifically, the Hill coefficient, arterial O2 partial pressure, O2 partial pressure at 50% Hb saturation, and cell membrane stiffness are significant factors. Results were found to be consistent with those of Le Floch [2010] and Secomb [2006].

A Calculation Method for Convective Heat and Mass Transfer in Multiply-Slotted Film-Cooling Applications.

DTIC Science & Technology

1980-01-01

Transport of Heat ..... .......... 8 3. THE SOLUTION PROCEDURE ..... .. ................. 8 3.1 The Finite-Difference Grid Network ... .......... 8 3.2...The Finite-Difference Grid Network. Figure 4: The Iterative Solution Procedure used at each Streamwise Station. Figure 5: Velocity Profiles in the...the finite-difference grid in the y-direction. I is the mixing length. L is the distance in the x-direction from the injection slot entrance to the

Implicit finite difference methods on composite grids

NASA Technical Reports Server (NTRS)

Mastin, C. Wayne

1987-01-01

Techniques for eliminating time lags in the implicit finite-difference solution of partial differential equations are investigated analytically, with a focus on transient fluid dynamics problems on overlapping multicomponent grids. The fundamental principles of the approach are explained, and the method is shown to be applicable to both rectangular and curvilinear grids. Numerical results for sample problems are compared with exact solutions in graphs, and good agreement is demonstrated.

On the wavelet optimized finite difference method

NASA Technical Reports Server (NTRS)

Jameson, Leland

1994-01-01

When one considers the effect in the physical space, Daubechies-based wavelet methods are equivalent to finite difference methods with grid refinement in regions of the domain where small scale structure exists. Adding a wavelet basis function at a given scale and location where one has a correspondingly large wavelet coefficient is, essentially, equivalent to adding a grid point, or two, at the same location and at a grid density which corresponds to the wavelet scale. This paper introduces a wavelet optimized finite difference method which is equivalent to a wavelet method in its multiresolution approach but which does not suffer from difficulties with nonlinear terms and boundary conditions, since all calculations are done in the physical space. With this method one can obtain an arbitrarily good approximation to a conservative difference method for solving nonlinear conservation laws.

Analytical and numerical solution for wave reflection from a porous wave absorber

NASA Astrophysics Data System (ADS)

Magdalena, Ikha; Roque, Marian P.

2018-03-01

In this paper, wave reflection from a porous wave absorber is investigated theoretically and numerically. The equations that we used are based on shallow water type model. Modification of motion inside the absorber is by including linearized friction term in momentum equation and introducing a filtered velocity. Here, an analytical solution for wave reflection coefficient from a porous wave absorber over a flat bottom is derived. Numerically, we solve the equations using the finite volume method on a staggered grid. To validate our numerical model, comparison of the numerical reflection coefficient is made against the analytical solution. Further, we implement our numerical scheme to study the evolution of surface waves pass through a porous absorber over varied bottom topography.

Computational Analysis of Gravitational Effects in Low-Density Gas Jets

NASA Technical Reports Server (NTRS)

Satti, Rajani P.; Agrawal, Ajay K.

2004-01-01

This study deals with the computational analysis of buoyancy-induced instability in the nearfield of an isothermal helium jet injected into quiescent ambient air environment. Laminar, axisymmetric, unsteady flow conditions were considered for the analysis. The transport equations of helium mass fraction coupled with the conservation equations of mixture mass and momentum were solved using a staggered grid finite volume method. The jet Richardson numbers of 1.5 and 0.018 were considered to encompass both buoyant and inertial jet flow regimes. Buoyancy effects were isolated by initiating computations in Earth gravity and subsequently, reducing gravity to simulate the microgravity conditions. Computed results concur with experimental observations that the periodic flow oscillations observed in Earth gravity subside in microgravity.

Calculations of separated 3-D flows with a pressure-staggered Navier-Stokes equations solver

NASA Technical Reports Server (NTRS)

Kim, S.-W.

1991-01-01

A Navier-Stokes equations solver based on a pressure correction method with a pressure-staggered mesh and calculations of separated three-dimensional flows are presented. It is shown that the velocity pressure decoupling, which occurs when various pressure correction algorithms are used for pressure-staggered meshes, is caused by the ill-conditioned discrete pressure correction equation. The use of a partial differential equation for the incremental pressure eliminates the velocity pressure decoupling mechanism by itself and yields accurate numerical results. Example flows considered are a three-dimensional lid driven cavity flow and a laminar flow through a 90 degree bend square duct. For the lid driven cavity flow, the present numerical results compare more favorably with the measured data than those obtained using a formally third order accurate quadratic upwind interpolation scheme. For the curved duct flow, the present numerical method yields a grid independent solution with a very small number of grid points. The calculated velocity profiles are in good agreement with the measured data.

Numerical investigation of implementation of air-earth boundary by acoustic-elastic boundary approach

USGS Publications Warehouse

Xu, Y.; Xia, J.; Miller, R.D.

2007-01-01

The need for incorporating the traction-free condition at the air-earth boundary for finite-difference modeling of seismic wave propagation has been discussed widely. A new implementation has been developed for simulating elastic wave propagation in which the free-surface condition is replaced by an explicit acoustic-elastic boundary. Detailed comparisons of seismograms with different implementations for the air-earth boundary were undertaken using the (2,2) (the finite-difference operators are second order in time and space) and the (2,6) (second order in time and sixth order in space) standard staggered-grid (SSG) schemes. Methods used in these comparisons to define the air-earth boundary included the stress image method (SIM), the heterogeneous approach, the scheme of modifying material properties based on transversely isotropic medium approach, the acoustic-elastic boundary approach, and an analytical approach. The method proposed achieves the same or higher accuracy of modeled body waves relative to the SIM. Rayleigh waves calculated using the explicit acoustic-elastic boundary approach differ slightly from those calculated using the SIM. Numerical results indicate that when using the (2,2) SSG scheme for SIM and our new method, a spatial step of 16 points per minimum wavelength is sufficient to achieve 90% accuracy; 32 points per minimum wavelength achieves 95% accuracy in modeled Rayleigh waves. When using the (2,6) SSG scheme for the two methods, a spatial step of eight points per minimum wavelength achieves 95% accuracy in modeled Rayleigh waves. Our proposed method is physically reasonable and, based on dispersive analysis of simulated seismographs from a layered half-space model, is highly accurate. As a bonus, our proposed method is easy to program and slightly faster than the SIM. ?? 2007 Society of Exploration Geophysicists.

Multigrid direct numerical simulation of the whole process of flow transition in 3-D boundary layers

NASA Technical Reports Server (NTRS)

Liu, Chaoqun; Liu, Zhining

1993-01-01

A new technology was developed in this study which provides a successful numerical simulation of the whole process of flow transition in 3-D boundary layers, including linear growth, secondary instability, breakdown, and transition at relatively low CPU cost. Most other spatial numerical simulations require high CPU cost and blow up at the stage of flow breakdown. A fourth-order finite difference scheme on stretched and staggered grids, a fully implicit time marching technique, a semi-coarsening multigrid based on the so-called approximate line-box relaxation, and a buffer domain for the outflow boundary conditions were all used for high-order accuracy, good stability, and fast convergence. A new fine-coarse-fine grid mapping technique was developed to keep the code running after the laminar flow breaks down. The computational results are in good agreement with linear stability theory, secondary instability theory, and some experiments. The cost for a typical case with 162 x 34 x 34 grid is around 2 CRAY-YMP CPU hours for 10 T-S periods.

Numerical Modeling of Liquid-Vapor Phase Change

NASA Technical Reports Server (NTRS)

Esmaeeli, Asghar; Arpaci, Vedat S.

2001-01-01

We implemented a two- and three-dimensional finite difference/front tracking technique to solve liquid-vapor phase change problems. The mathematical and the numerical features of the method were explained in great detail in our previous reports, Briefly, we used a single formula representation which incorporated jump conditions into the governing equations. The interfacial terms were distributed as singular terms using delta functions so that the governing equations would be the same as conventional conservation equations away from the interface and in the vicinity of the interface they would provide correct jump conditions. We used a fixed staggered grid to discretize these equations and an unstructured grid to explicitly track the front. While in two dimensions the front was simply a connection of small line segments, in three dimensions it was represented by a connection of small triangular elements. The equations were written in conservative forms and during the course of computations we used regriding to control the size of the elements of the unstructured grid. Moreover, we implemented a coalescence in two dimensions which allowed the merging of different fronts or two segments of the same front when they were sufficiently close. We used our code to study thermocapillary migration of bubbles, burst of bubbles at a free surface, buoyancy-driven interactions of bubbles, evaporation of drops, rapid evaporation of an interface, planar solidification of an undercooled melt, dendritic solidification, and a host of other problems cited in the reference.

Analysis of vegetation effect on waves using a vertical 2-D RANS model

USDA-ARS?s Scientific Manuscript database

A vertical two-dimensional (2-D) model has been applied in the simulation of wave propagation through vegetated water bodies. The model is based on an existing model SOLA-VOF which solves the Reynolds-Averaged Navier-Stokes (RANS) equations with the finite difference method on a staggered rectangula...

A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions

NASA Astrophysics Data System (ADS)

Liska, Sebastian; Colonius, Tim

2017-02-01

A new parallel, computationally efficient immersed boundary method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. Immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original (Peskin's) immersed boundary method. Unlike Peskin's method, boundary forces are regarded as Lagrange multipliers that are used to satisfy the no-slip condition. The incompressible Navier-Stokes equations are discretized on an unbounded staggered Cartesian grid and are solved in a finite number of operations using lattice Green's function techniques. These techniques are used to automatically enforce the natural free-space boundary conditions and to implement a novel block-wise adaptive grid that significantly reduces the run-time cost of solutions by limiting operations to grid cells in the immediate vicinity and near-wake region of the immersed surface. These techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half-explicit Runge-Kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation, incompressibility constraint, and no-slip constraint. Linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation-free nested projection technique. The algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems, e.g. discrete Poisson problems, that are compatible with recent parallel fast multipole methods for difference equations. Numerical experiments on low-aspect-ratio flat plates and spheres at Reynolds numbers up to 3700 are used to verify the accuracy and physical fidelity of the formulation.

Site-specific strong ground motion prediction using 2.5-D modelling

NASA Astrophysics Data System (ADS)

Narayan, J. P.

2001-08-01

An algorithm was developed using the 2.5-D elastodynamic wave equation, based on the displacement-stress relation. One of the most significant advantages of the 2.5-D simulation is that the 3-D radiation pattern can be generated using double-couple point shear-dislocation sources in the 2-D numerical grid. A parsimonious staggered grid scheme was adopted instead of the standard staggered grid scheme, since this is the only scheme suitable for computing the dislocation. This new 2.5-D numerical modelling avoids the extensive computational cost of 3-D modelling. The significance of this exercise is that it makes it possible to simulate the strong ground motion (SGM), taking into account the energy released, 3-D radiation pattern, path effects and local site conditions at any location around the epicentre. The slowness vector (py) was used in the supersonic region for each layer, so that all the components of the inertia coefficient are positive. The double-couple point shear-dislocation source was implemented in the numerical grid using the moment tensor components as the body-force couples. The moment per unit volume was used in both the 3-D and 2.5-D modelling. A good agreement in the 3-D and 2.5-D responses for different grid sizes was obtained when the moment per unit volume was further reduced by a factor equal to the finite-difference grid size in the case of the 2.5-D modelling. The components of the radiation pattern were computed in the xz-plane using 3-D and 2.5-D algorithms for various focal mechanisms, and the results were in good agreement. A comparative study of the amplitude behaviour of the 3-D and 2.5-D wavefronts in a layered medium reveals the spatial and temporal damped nature of the 2.5-D elastodynamic wave equation. 3-D and 2.5-D simulated responses at a site using a different strike direction reveal that strong ground motion (SGM) can be predicted just by rotating the strike of the fault counter-clockwise by the same amount as the azimuth of the site with respect to the epicentre. This adjustment is necessary since the response is computed keeping the epicentre, focus and the desired site in the same xz-plane, with the x-axis pointing in the north direction.

A Second Order Semi-Discrete Cosserat Rod Model Suitable for Dynamic Simulations in Real Time

NASA Astrophysics Data System (ADS)

Lang, Holger; Linn, Joachim

2009-09-01

We present an alternative approach for a semi-discrete viscoelastic Cosserat rod model that allows both fast dynamic computations within milliseconds and accurate results compared to detailed finite element solutions. The model is able to represent extension, shearing, bending and torsion. For inner dissipation, a consistent damping potential from Antman is chosen. The continuous equations of motion, which consist a system of nonlinear hyperbolic partial differential algebraic equations, are derived from a two dimensional variational principle. The semi-discrete balance equations are obtained by spatial finite difference schemes on a staggered grid and standard index reduction techniques. The right-hand side of the model and its Jacobian can be chosen free of higher algebraic (e.g. root) or transcendent (e.g. trigonometric or exponential) functions and is therefore extremely cheap to evaluate numerically. For the time integration of the system, we use well established stiff solvers. As our model yields computational times within milliseconds, it is suitable for interactive manipulation. It reflects structural mechanics solutions sufficiently correct, as comparison with detailed finite element results shows.

Comparison of three explicit multigrid methods for the Euler and Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Chima, Rodrick V.; Turkel, Eli; Schaffer, Steve

1987-01-01

Three explicit multigrid methods, Ni's method, Jameson's finite-volume method, and a finite-difference method based on Brandt's work, are described and compared for two model problems. All three methods use an explicit multistage Runge-Kutta scheme on the fine grid, and this scheme is also described. Convergence histories for inviscid flow over a bump in a channel for the fine-grid scheme alone show that convergence rate is proportional to Courant number and that implicit residual smoothing can significantly accelerate the scheme. Ni's method was slightly slower than the implicitly-smoothed scheme alone. Brandt's and Jameson's methods are shown to be equivalent in form but differ in their node versus cell-centered implementations. They are about 8.5 times faster than Ni's method in terms of CPU time. Results for an oblique shock/boundary layer interaction problem verify the accuracy of the finite-difference code. All methods slowed considerably on the stretched viscous grid but Brandt's method was still 2.1 times faster than Ni's method.

A chimera grid scheme. [multiple overset body-conforming mesh system for finite difference adaptation to complex aircraft configurations

NASA Technical Reports Server (NTRS)

Steger, J. L.; Dougherty, F. C.; Benek, J. A.

1983-01-01

A mesh system composed of multiple overset body-conforming grids is described for adapting finite-difference procedures to complex aircraft configurations. In this so-called 'chimera mesh,' a major grid is generated about a main component of the configuration and overset minor grids are used to resolve all other features. Methods for connecting overset multiple grids and modifications of flow-simulation algorithms are discussed. Computational tests in two dimensions indicate that the use of multiple overset grids can simplify the task of grid generation without an adverse effect on flow-field algorithms and computer code complexity.

Development of a pressure based multigrid solution method for complex fluid flows

NASA Technical Reports Server (NTRS)

Shyy, Wei

1991-01-01

In order to reduce the computational difficulty associated with a single grid (SG) solution procedure, the multigrid (MG) technique was identified as a useful means for improving the convergence rate of iterative methods. A full MG full approximation storage (FMG/FAS) algorithm is used to solve the incompressible recirculating flow problems in complex geometries. The algorithm is implemented in conjunction with a pressure correction staggered grid type of technique using the curvilinear coordinates. In order to show the performance of the method, two flow configurations, one a square cavity and the other a channel, are used as test problems. Comparisons are made between the iterations, equivalent work units, and CPU time. Besides showing that the MG method can yield substantial speed-up with wide variations in Reynolds number, grid distributions, and geometry, issues such as the convergence characteristics of different grid levels, the choice of convection schemes, and the effectiveness of the basic iteration smoothers are studied. An adaptive grid scheme is also combined with the MG procedure to explore the effects of grid resolution on the MG convergence rate as well as the numerical accuracy.

Slat Noise Predictions Using Higher-Order Finite-Difference Methods on Overset Grids

NASA Technical Reports Server (NTRS)

Housman, Jeffrey A.; Kiris, Cetin

2016-01-01

Computational aeroacoustic simulations using the structured overset grid approach and higher-order finite difference methods within the Launch Ascent and Vehicle Aerodynamics (LAVA) solver framework are presented for slat noise predictions. The simulations are part of a collaborative study comparing noise generation mechanisms between a conventional slat and a Krueger leading edge flap. Simulation results are compared with experimental data acquired during an aeroacoustic test in the NASA Langley Quiet Flow Facility. Details of the structured overset grid, numerical discretization, and turbulence model are provided.

Fully anisotropic 3-D EM modelling on a Lebedev grid with a multigrid pre-conditioner

NASA Astrophysics Data System (ADS)

Jaysaval, Piyoosh; Shantsev, Daniil V.; de la Kethulle de Ryhove, Sébastien; Bratteland, Tarjei

2016-12-01

We present a numerical algorithm for 3-D electromagnetic (EM) simulations in conducting media with general electric anisotropy. The algorithm is based on the finite-difference discretization of frequency-domain Maxwell's equations on a Lebedev grid, in which all components of the electric field are collocated but half a spatial step staggered with respect to the magnetic field components, which also are collocated. This leads to a system of linear equations that is solved using a stabilized biconjugate gradient method with a multigrid preconditioner. We validate the accuracy of the numerical results for layered and 3-D tilted transverse isotropic (TTI) earth models representing typical scenarios used in the marine controlled-source EM method. It is then demonstrated that not taking into account the full anisotropy of the conductivity tensor can lead to misleading inversion results. For synthetic data corresponding to a 3-D model with a TTI anticlinal structure, a standard vertical transverse isotropic (VTI) inversion is not able to image a resistor, while for a 3-D model with a TTI synclinal structure it produces a false resistive anomaly. However, if the VTI forward solver used in the inversion is replaced by the proposed TTI solver with perfect knowledge of the strike and dip of the dipping structures, the resulting resistivity images become consistent with the true models.

Multigrid methods for flow transition in three-dimensional boundary layers with surface roughness

NASA Technical Reports Server (NTRS)

Liu, Chaoqun; Liu, Zhining; Mccormick, Steve

1993-01-01

The efficient multilevel adaptive method has been successfully applied to perform direct numerical simulations (DNS) of flow transition in 3-D channels and 3-D boundary layers with 2-D and 3-D isolated and distributed roughness in a curvilinear coordinate system. A fourth-order finite difference technique on stretched and staggered grids, a fully-implicit time marching scheme, a semi-coarsening multigrid method associated with line distributive relaxation scheme, and an improved outflow boundary-condition treatment, which needs only a very short buffer domain to damp all order-one wave reflections, are developed. These approaches make the multigrid DNS code very accurate and efficient. This allows us not only to be able to do spatial DNS for the 3-D channel and flat plate at low computational costs, but also to do spatial DNS for transition in the 3-D boundary layer with 3-D single and multiple roughness elements, which would have extremely high computational costs with conventional methods. Numerical results show good agreement with the linear stability theory, the secondary instability theory, and a number of laboratory experiments. The contribution of isolated and distributed roughness to transition is analyzed.

Entropy Stable Staggered Grid Spectral Collocation for the Burgers' and Compressible Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Carpenter, Mark H.; Parsani, Matteo; Fisher, Travis C.; Nielsen, Eric J.

2015-01-01

Staggered grid, entropy stable discontinuous spectral collocation operators of any order are developed for Burgers' and the compressible Navier-Stokes equations on unstructured hexahedral elements. This generalization of previous entropy stable spectral collocation work [1, 2], extends the applicable set of points from tensor product, Legendre-Gauss-Lobatto (LGL) to a combination of tensor product Legendre-Gauss (LG) and LGL points. The new semi-discrete operators discretely conserve mass, momentum, energy and satisfy a mathematical entropy inequality for both Burgers' and the compressible Navier-Stokes equations in three spatial dimensions. They are valid for smooth as well as discontinuous flows. The staggered LG and conventional LGL point formulations are compared on several challenging test problems. The staggered LG operators are significantly more accurate, although more costly to implement. The LG and LGL operators exhibit similar robustness, as is demonstrated using test problems known to be problematic for operators that lack a nonlinearly stability proof for the compressible Navier-Stokes equations (e.g., discontinuous Galerkin, spectral difference, or flux reconstruction operators).

An analysis of finite-difference and finite-volume formulations of conservation laws

NASA Technical Reports Server (NTRS)

Vinokur, Marcel

1986-01-01

Finite-difference and finite-volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws. A new coordinate-free formulation of systems of conservation laws is developed, which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system. The analysis considers general types of equations--potential, Euler, and Navier-Stokes. Three-dimensional unsteady flows with time-varying grids are described using a single, consistent nomeclature for both formulations. Grid motion due to a non-inertial reference frame as well as flow adaptation is covered. In comparing the two formulations, it is found useful to distinguish between differences in numerical methods and differences in grid definition. The former plays a role for non-Cartesian grids, and results in only cosmetic differences in the manner in which geometric terms are handled. The differences in grid definition for the two formulations is found to be more important, since it affects the manner in which boundary conditions, zonal procedures, and grid singularities are handled at computational boundaries. The proper interpretation of strong and weak conservation-law forms for quasi-one-dimensional and axisymmetric flows is brought out.

An analysis of finite-difference and finite-volume formulations of conservation laws

NASA Technical Reports Server (NTRS)

Vinokur, Marcel

1989-01-01

Finite-difference and finite-volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws. A new coordinate-free formulation of systems of conservation laws is developed, which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system. The analysis considers general types of equations: potential, Euler, and Navier-Stokes. Three-dimensional unsteady flows with time-varying grids are described using a single, consistent nomenclature for both formulations. Grid motion due to a non-inertial reference frame as well as flow adaptation is covered. In comparing the two formulations, it is found useful to distinguish between differences in numerical methods and differences in grid definition. The former plays a role for non-Cartesian grids, and results in only cosmetic differences in the manner in which geometric terms are handled. The differences in grid definition for the two formulations is found to be more important, since it affects the manner in which boundary conditions, zonal procedures, and grid singularities are handled at computational boundaries. The proper interpretation of strong and weak conservation-law forms for quasi-one-dimensional and axisymmetric flows is brought out.

An immersed boundary method for fluid-structure interaction with compressible multiphase flows

NASA Astrophysics Data System (ADS)

Wang, Li; Currao, Gaetano M. D.; Han, Feng; Neely, Andrew J.; Young, John; Tian, Fang-Bao

2017-10-01

This paper presents a two-dimensional immersed boundary method for fluid-structure interaction with compressible multiphase flows involving large structure deformations. This method involves three important parts: flow solver, structure solver and fluid-structure interaction coupling. In the flow solver, the compressible multiphase Navier-Stokes equations for ideal gases are solved by a finite difference method based on a staggered Cartesian mesh, where a fifth-order accuracy Weighted Essentially Non-Oscillation (WENO) scheme is used to handle spatial discretization of the convective term, a fourth-order central difference scheme is employed to discretize the viscous term, the third-order TVD Runge-Kutta scheme is used to discretize the temporal term, and the level-set method is adopted to capture the multi-material interface. In this work, the structure considered is a geometrically non-linear beam which is solved by using a finite element method based on the absolute nodal coordinate formulation (ANCF). The fluid dynamics and the structure motion are coupled in a partitioned iterative manner with a feedback penalty immersed boundary method where the flow dynamics is defined on a fixed Lagrangian grid and the structure dynamics is described on a global coordinate. We perform several validation cases (including fluid over a cylinder, structure dynamics, flow induced vibration of a flexible plate, deformation of a flexible panel induced by shock waves in a shock tube, an inclined flexible plate in a hypersonic flow, and shock-induced collapse of a cylindrical helium cavity in the air), and compare the results with experimental and other numerical data. The present results agree well with the published data and the current experiment. Finally, we further demonstrate the versatility of the present method by applying it to a flexible plate interacting with multiphase flows.

Wave propagation in anisotropic elastic materials and curvilinear coordinates using a summation-by-parts finite difference method

DOE PAGES

Petersson, N. Anders; Sjogreen, Bjorn

2015-07-20

We develop a fourth order accurate finite difference method for solving the three-dimensional elastic wave equation in general heterogeneous anisotropic materials on curvilinear grids. The proposed method is an extension of the method for isotropic materials, previously described in the paper by Sjögreen and Petersson (2012) [11]. The method we proposed discretizes the anisotropic elastic wave equation in second order formulation, using a node centered finite difference method that satisfies the principle of summation by parts. The summation by parts technique results in a provably stable numerical method that is energy conserving. Also, we generalize and evaluate the super-grid far-fieldmore » technique for truncating unbounded domains. Unlike the commonly used perfectly matched layers (PML), the super-grid technique is stable for general anisotropic material, because it is based on a coordinate stretching combined with an artificial dissipation. Moreover, the discretization satisfies an energy estimate, proving that the numerical approximation is stable. We demonstrate by numerical experiments that sufficiently wide super-grid layers result in very small artificial reflections. Applications of the proposed method are demonstrated by three-dimensional simulations of anisotropic wave propagation in crystals.« less

Implicit solution of Navier-Stokes equations on staggered curvilinear grids using a Newton-Krylov method with a novel analytical Jacobian.

NASA Astrophysics Data System (ADS)

Borazjani, Iman; Asgharzadeh, Hafez

2015-11-01

Flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates with explicit and semi-implicit schemes. Implicit schemes can be used to overcome these restrictions. However, implementing implicit solver for nonlinear equations including Navier-Stokes is not straightforward. Newton-Krylov subspace methods (NKMs) are one of the most advanced iterative methods to solve non-linear equations such as implicit descritization of the Navier-Stokes equation. The efficiency of NKMs massively depends on the Jacobian formation method, e.g., automatic differentiation is very expensive, and matrix-free methods slow down as the mesh is refined. Analytical Jacobian is inexpensive method, but derivation of analytical Jacobian for Navier-Stokes equation on staggered grid is challenging. The NKM with a novel analytical Jacobian was developed and validated against Taylor-Green vortex and pulsatile flow in a 90 degree bend. The developed method successfully handled the complex geometries such as an intracranial aneurysm with multiple overset grids, and immersed boundaries. It is shown that the NKM with an analytical Jacobian is 3 to 25 times faster than the fixed-point implicit Runge-Kutta method, and more than 100 times faster than automatic differentiation depending on the grid (size) and the flow problem. The developed methods are fully parallelized with parallel efficiency of 80-90% on the problems tested.

Computer-Aided Engineering of Semiconductor Integrated Circuits

DTIC Science & Technology

1979-07-01

equation using a five point finite difference approximation. Section 4.3.6 describes the numerical techniques and iterative algorithms which are used...neighbor points. This is generally referred to as a five point finite difference scheme on a rectangular grid, as described below. The finite difference ...problems in steady state have been analyzed by the finite difference method [4. 16 ] [4.17 3 or finite element method [4. 18 3, [4. 19 3 as reported last

Optimal implicit 2-D finite differences to model wave propagation in poroelastic media

NASA Astrophysics Data System (ADS)

Itzá, Reymundo; Iturrarán-Viveros, Ursula; Parra, Jorge O.

2016-08-01

Numerical modeling of seismic waves in heterogeneous porous reservoir rocks is an important tool for the interpretation of seismic surveys in reservoir engineering. We apply globally optimal implicit staggered-grid finite differences (FD) to model 2-D wave propagation in heterogeneous poroelastic media at a low-frequency range (<10 kHz). We validate the numerical solution by comparing it to an analytical-transient solution obtaining clear seismic wavefields including fast P and slow P and S waves (for a porous media saturated with fluid). The numerical dispersion and stability conditions are derived using von Neumann analysis, showing that over a wide range of porous materials the Courant condition governs the stability and this optimal implicit scheme improves the stability of explicit schemes. High-order explicit FD can be replaced by some lower order optimal implicit FD so computational cost will not be as expensive while maintaining the accuracy. Here, we compute weights for the optimal implicit FD scheme to attain an accuracy of γ = 10-8. The implicit spatial differentiation involves solving tridiagonal linear systems of equations through Thomas' algorithm.

Numerical Simulations of STOVL Hot Gas Ingestion in Ground Proximity Using a Multigrid Solution Procedure

NASA Technical Reports Server (NTRS)

Wang, Gang

2003-01-01

A multi grid solution procedure for the numerical simulation of turbulent flows in complex geometries has been developed. A Full Multigrid-Full Approximation Scheme (FMG-FAS) is incorporated into the continuity and momentum equations, while the scalars are decoupled from the multi grid V-cycle. A standard kappa-Epsilon turbulence model with wall functions has been used to close the governing equations. The numerical solution is accomplished by solving for the Cartesian velocity components either with a traditional grid staggering arrangement or with a multiple velocity grid staggering arrangement. The two solution methodologies are evaluated for relative computational efficiency. The solution procedure with traditional staggering arrangement is subsequently applied to calculate the flow and temperature fields around a model Short Take-off and Vertical Landing (STOVL) aircraft hovering in ground proximity.

Research on the forward modeling of controlled-source audio-frequency magnetotellurics in three-dimensional axial anisotropic media

NASA Astrophysics Data System (ADS)

Wang, Kunpeng; Tan, Handong

2017-11-01

Controlled-source audio-frequency magnetotellurics (CSAMT) has developed rapidly in recent years and are widely used in the area of mineral and oil resource exploration as well as other fields. The current theory, numerical simulation, and inversion research are based on the assumption that the underground media have resistivity isotropy. However a large number of rock and mineral physical property tests show the resistivity of underground media is generally anisotropic. With the increasing application of CSAMT, the demand for probe accuracy of practical exploration to complex targets continues to increase. The question of how to evaluate the influence of anisotropic resistivity to CSAMT response is becoming important. To meet the demand for CSAMT response research of resistivity anisotropic media, this paper examines the CSAMT electric equations, derives and realizes a three-dimensional (3D) staggered-grid finite difference numerical simulation method of CSAMT resistivity axial anisotropy. Through building a two-dimensional (2D) resistivity anisotropy geoelectric model, we validate the 3D computation result by comparing it to the result of controlled-source electromagnetic method (CSEM) resistivity anisotropy 2D finite element program. Through simulating a 3D resistivity axial anisotropy geoelectric model, we compare and analyze the responses of equatorial configuration, axial configuration, two oblique sources and tensor source. The research shows that the tensor source is suitable for CSAMT to recognize the anisotropic effect of underground structure.

An efficient semi-implicit method for three-dimensional non-hydrostatic flows in compliant arterial vessels.

PubMed

Fambri, Francesco; Dumbser, Michael; Casulli, Vincenzo

2014-11-01

Blood flow in arterial systems can be described by the three-dimensional Navier-Stokes equations within a time-dependent spatial domain that accounts for the elasticity of the arterial walls. In this article, blood is treated as an incompressible Newtonian fluid that flows through compliant vessels of general cross section. A three-dimensional semi-implicit finite difference and finite volume model is derived so that numerical stability is obtained at a low computational cost on a staggered grid. The key idea of the method consists in a splitting of the pressure into a hydrostatic and a non-hydrostatic part, where first a small quasi-one-dimensional nonlinear system is solved for the hydrostatic pressure and only in a second step the fully three-dimensional non-hydrostatic pressure is computed from a three-dimensional nonlinear system as a correction to the hydrostatic one. The resulting algorithm is robust, efficient, locally and globally mass conservative, and applies to hydrostatic and non-hydrostatic flows in one, two and three space dimensions. These features are illustrated on nontrivial test cases for flows in tubes with circular or elliptical cross section where the exact analytical solution is known. Test cases of steady and pulsatile flows in uniformly curved rigid and elastic tubes are presented. Wherever possible, axial velocity development and secondary flows are shown and compared with previously published results. Copyright © 2014 John Wiley & Sons, Ltd.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Balsa Terzic, Gabriele Bassi

In this paper we discuss representations of charge particle densities in particle-in-cell (PIC) simulations, analyze the sources and profiles of the intrinsic numerical noise, and present efficient methods for their removal. We devise two alternative estimation methods for charged particle distribution which represent significant improvement over the Monte Carlo cosine expansion used in the 2d code of Bassi, designed to simulate coherent synchrotron radiation (CSR) in charged particle beams. The improvement is achieved by employing an alternative beam density estimation to the Monte Carlo cosine expansion. The representation is first binned onto a finite grid, after which two grid-based methodsmore » are employed to approximate particle distributions: (i) truncated fast cosine transform (TFCT); and (ii) thresholded wavelet transform (TWT). We demonstrate that these alternative methods represent a staggering upgrade over the original Monte Carlo cosine expansion in terms of efficiency, while the TWT approximation also provides an appreciable improvement in accuracy. The improvement in accuracy comes from a judicious removal of the numerical noise enabled by the wavelet formulation. The TWT method is then integrated into Bassi's CSR code, and benchmarked against the original version. We show that the new density estimation method provides a superior performance in terms of efficiency and spatial resolution, thus enabling high-fidelity simulations of CSR effects, including microbunching instability.« less

Accurate finite difference methods for time-harmonic wave propagation

NASA Technical Reports Server (NTRS)

Harari, Isaac; Turkel, Eli

1994-01-01

Finite difference methods for solving problems of time-harmonic acoustics are developed and analyzed. Multidimensional inhomogeneous problems with variable, possibly discontinuous, coefficients are considered, accounting for the effects of employing nonuniform grids. A weighted-average representation is less sensitive to transition in wave resolution (due to variable wave numbers or nonuniform grids) than the standard pointwise representation. Further enhancement in method performance is obtained by basing the stencils on generalizations of Pade approximation, or generalized definitions of the derivative, reducing spurious dispersion, anisotropy and reflection, and by improving the representation of source terms. The resulting schemes have fourth-order accurate local truncation error on uniform grids and third order in the nonuniform case. Guidelines for discretization pertaining to grid orientation and resolution are presented.

Grid-size dependence of Cauchy boundary conditions used to simulate stream-aquifer interactions

USGS Publications Warehouse

Mehl, S.; Hill, M.C.

2010-01-01

This work examines the simulation of stream–aquifer interactions as grids are refined vertically and horizontally and suggests that traditional methods for calculating conductance can produce inappropriate values when the grid size is changed. Instead, different grid resolutions require different estimated values. Grid refinement strategies considered include global refinement of the entire model and local refinement of part of the stream. Three methods of calculating the conductance of the Cauchy boundary conditions are investigated. Single- and multi-layer models with narrow and wide streams produced stream leakages that differ by as much as 122% as the grid is refined. Similar results occur for globally and locally refined grids, but the latter required as little as one-quarter the computer execution time and memory and thus are useful for addressing some scale issues of stream–aquifer interactions. Results suggest that existing grid-size criteria for simulating stream–aquifer interactions are useful for one-layer models, but inadequate for three-dimensional models. The grid dependence of the conductance terms suggests that values for refined models using, for example, finite difference or finite-element methods, cannot be determined from previous coarse-grid models or field measurements. Our examples demonstrate the need for a method of obtaining conductances that can be translated to different grid resolutions and provide definitive test cases for investigating alternative conductance formulations.

A staggered conservative scheme for every Froude number in rapidly varied shallow water flows

NASA Astrophysics Data System (ADS)

Stelling, G. S.; Duinmeijer, S. P. A.

2003-12-01

This paper proposes a numerical technique that in essence is based upon the classical staggered grids and implicit numerical integration schemes, but that can be applied to problems that include rapidly varied flows as well. Rapidly varied flows occur, for instance, in hydraulic jumps and bores. Inundation of dry land implies sudden flow transitions due to obstacles such as road banks. Near such transitions the grid resolution is often low compared to the gradients of the bathymetry. In combination with the local invalidity of the hydrostatic pressure assumption, conservation properties become crucial. The scheme described here, combines the efficiency of staggered grids with conservation properties so as to ensure accurate results for rapidly varied flows, as well as in expansions as in contractions. In flow expansions, a numerical approximation is applied that is consistent with the momentum principle. In flow contractions, a numerical approximation is applied that is consistent with the Bernoulli equation. Both approximations are consistent with the shallow water equations, so under sufficiently smooth conditions they converge to the same solution. The resulting method is very efficient for the simulation of large-scale inundations.

A new multigrid formulation for high order finite difference methods on summation-by-parts form

NASA Astrophysics Data System (ADS)

Ruggiu, Andrea A.; Weinerfelt, Per; Nordström, Jan

2018-04-01

Multigrid schemes for high order finite difference methods on summation-by-parts form are studied by comparing the effect of different interpolation operators. By using the standard linear prolongation and restriction operators, the Galerkin condition leads to inaccurate coarse grid discretizations. In this paper, an alternative class of interpolation operators that bypass this issue and preserve the summation-by-parts property on each grid level is considered. Clear improvements of the convergence rate for relevant model problems are achieved.

An unconditionally stable staggered algorithm for transient finite element analysis of coupled thermoelastic problems

NASA Technical Reports Server (NTRS)

Farhat, C.; Park, K. C.; Dubois-Pelerin, Y.

1991-01-01

An unconditionally stable second order accurate implicit-implicit staggered procedure for the finite element solution of fully coupled thermoelasticity transient problems is proposed. The procedure is stabilized with a semi-algebraic augmentation technique. A comparative cost analysis reveals the superiority of the proposed computational strategy to other conventional staggered procedures. Numerical examples of one and two-dimensional thermomechanical coupled problems demonstrate the accuracy of the proposed numerical solution algorithm.

A Solution Adaptive Structured/Unstructured Overset Grid Flow Solver with Applications to Helicopter Rotor Flows

NASA Technical Reports Server (NTRS)

Duque, Earl P. N.; Biswas, Rupak; Strawn, Roger C.

1995-01-01

This paper summarizes a method that solves both the three dimensional thin-layer Navier-Stokes equations and the Euler equations using overset structured and solution adaptive unstructured grids with applications to helicopter rotor flowfields. The overset structured grids use an implicit finite-difference method to solve the thin-layer Navier-Stokes/Euler equations while the unstructured grid uses an explicit finite-volume method to solve the Euler equations. Solutions on a helicopter rotor in hover show the ability to accurately convect the rotor wake. However, isotropic subdivision of the tetrahedral mesh rapidly increases the overall problem size.

A Comparison of Spectral Element and Finite Difference Methods Using Statically Refined Nonconforming Grids for the MHD Island Coalescence Instability Problem

NASA Astrophysics Data System (ADS)

Ng, C. S.; Rosenberg, D.; Pouquet, A.; Germaschewski, K.; Bhattacharjee, A.

2009-04-01

A recently developed spectral-element adaptive refinement incompressible magnetohydrodynamic (MHD) code [Rosenberg, Fournier, Fischer, Pouquet, J. Comp. Phys. 215, 59-80 (2006)] is applied to simulate the problem of MHD island coalescence instability (\\ci) in two dimensions. \\ci is a fundamental MHD process that can produce sharp current layers and subsequent reconnection and heating in a high-Lundquist number plasma such as the solar corona [Ng and Bhattacharjee, Phys. Plasmas, 5, 4028 (1998)]. Due to the formation of thin current layers, it is highly desirable to use adaptively or statically refined grids to resolve them, and to maintain accuracy at the same time. The output of the spectral-element static adaptive refinement simulations are compared with simulations using a finite difference method on the same refinement grids, and both methods are compared to pseudo-spectral simulations with uniform grids as baselines. It is shown that with the statically refined grids roughly scaling linearly with effective resolution, spectral element runs can maintain accuracy significantly higher than that of the finite difference runs, in some cases achieving close to full spectral accuracy.

Distributed Relaxation for Conservative Discretizations

NASA Technical Reports Server (NTRS)

Diskin, Boris; Thomas, James L.

2001-01-01

A multigrid method is defined as having textbook multigrid efficiency (TME) if the solutions to the governing system of equations are attained in a computational work that is a small (less than 10) multiple of the operation count in one target-grid residual evaluation. The way to achieve this efficiency is the distributed relaxation approach. TME solvers employing distributed relaxation have already been demonstrated for nonconservative formulations of high-Reynolds-number viscous incompressible and subsonic compressible flow regimes. The purpose of this paper is to provide foundations for applications of distributed relaxation to conservative discretizations. A direct correspondence between the primitive variable interpolations for calculating fluxes in conservative finite-volume discretizations and stencils of the discretized derivatives in the nonconservative formulation has been established. Based on this correspondence, one can arrive at a conservative discretization which is very efficiently solved with a nonconservative relaxation scheme and this is demonstrated for conservative discretization of the quasi one-dimensional Euler equations. Formulations for both staggered and collocated grid arrangements are considered and extensions of the general procedure to multiple dimensions are discussed.

Development and evaluation of a local grid refinement method for block-centered finite-difference groundwater models using shared nodes

USGS Publications Warehouse

Mehl, S.; Hill, M.C.

2002-01-01

A new method of local grid refinement for two-dimensional block-centered finite-difference meshes is presented in the context of steady-state groundwater-flow modeling. The method uses an iteration-based feedback with shared nodes to couple two separate grids. The new method is evaluated by comparison with results using a uniform fine mesh, a variably spaced mesh, and a traditional method of local grid refinement without a feedback. Results indicate: (1) The new method exhibits quadratic convergence for homogeneous systems and convergence equivalent to uniform-grid refinement for heterogeneous systems. (2) Coupling the coarse grid with the refined grid in a numerically rigorous way allowed for improvement in the coarse-grid results. (3) For heterogeneous systems, commonly used linear interpolation of heads from the large model onto the boundary of the refined model produced heads that are inconsistent with the physics of the flow field. (4) The traditional method works well in situations where the better resolution of the locally refined grid has little influence on the overall flow-system dynamics, but if this is not true, lack of a feedback mechanism produced errors in head up to 3.6% and errors in cell-to-cell flows up to 25%. ?? 2002 Elsevier Science Ltd. All rights reserved.

Large-eddy simulation of wind turbine wake interactions on locally refined Cartesian grids

NASA Astrophysics Data System (ADS)

Angelidis, Dionysios; Sotiropoulos, Fotis

2014-11-01

Performing high-fidelity numerical simulations of turbulent flow in wind farms remains a challenging issue mainly because of the large computational resources required to accurately simulate the turbine wakes and turbine/turbine interactions. The discretization of the governing equations on structured grids for mesoscale calculations may not be the most efficient approach for resolving the large disparity of spatial scales. A 3D Cartesian grid refinement method enabling the efficient coupling of the Actuator Line Model (ALM) with locally refined unstructured Cartesian grids adapted to accurately resolve tip vortices and multi-turbine interactions, is presented. Second order schemes are employed for the discretization of the incompressible Navier-Stokes equations in a hybrid staggered/non-staggered formulation coupled with a fractional step method that ensures the satisfaction of local mass conservation to machine zero. The current approach enables multi-resolution LES of turbulent flow in multi-turbine wind farms. The numerical simulations are in good agreement with experimental measurements and are able to resolve the rich dynamics of turbine wakes on grids containing only a small fraction of the grid nodes that would be required in simulations without local mesh refinement. This material is based upon work supported by the Department of Energy under Award Number DE-EE0005482 and the National Science Foundation under Award number NSF PFI:BIC 1318201.

StagBL : A Scalable, Portable, High-Performance Discretization and Solver Layer for Geodynamic Simulation

NASA Astrophysics Data System (ADS)

Sanan, P.; Tackley, P. J.; Gerya, T.; Kaus, B. J. P.; May, D.

2017-12-01

StagBL is an open-source parallel solver and discretization library for geodynamic simulation,encapsulating and optimizing operations essential to staggered-grid finite volume Stokes flow solvers.It provides a parallel staggered-grid abstraction with a high-level interface in C and Fortran.On top of this abstraction, tools are available to define boundary conditions and interact with particle systems.Tools and examples to efficiently solve Stokes systems defined on the grid are provided in small (direct solver), medium (simple preconditioners), and large (block factorization and multigrid) model regimes.By working directly with leading application codes (StagYY, I3ELVIS, and LaMEM) and providing an API and examples to integrate with others, StagBL aims to become a community tool supplying scalable, portable, reproducible performance toward novel science in regional- and planet-scale geodynamics and planetary science.By implementing kernels used by many research groups beneath a uniform abstraction layer, the library will enable optimization for modern hardware, thus reducing community barriers to large- or extreme-scale parallel simulation on modern architectures. In particular, the library will include CPU-, Manycore-, and GPU-optimized variants of matrix-free operators and multigrid components.The common layer provides a framework upon which to introduce innovative new tools.StagBL will leverage p4est to provide distributed adaptive meshes, and incorporate a multigrid convergence analysis tool.These options, in addition to a wealth of solver options provided by an interface to PETSc, will make the most modern solution techniques available from a common interface. StagBL in turn provides a PETSc interface, DMStag, to its central staggered grid abstraction.We present public version 0.5 of StagBL, including preliminary integration with application codes and demonstrations with its own demonstration application, StagBLDemo. Central to StagBL is the notion of an uninterrupted pipeline from toy/teaching codes to high-performance, extreme-scale solves. StagBLDemo replicates the functionality of an advanced MATLAB-style regional geodynamics code, thus providing users with a concrete procedure to exceed the performance and scalability limitations of smaller-scale tools.

Simulating incompressible flow on moving meshfree grids using General Finite Differences (GFD)

NASA Astrophysics Data System (ADS)

Vasyliv, Yaroslav; Alexeev, Alexander

2016-11-01

We simulate incompressible flow around an oscillating cylinder at different Reynolds numbers using General Finite Differences (GFD) on a meshfree grid. We evolve the meshfree grid by treating each grid node as a particle. To compute velocities and accelerations, we consider the particles at a particular instance as Eulerian observation points. The incompressible Navier-Stokes equations are directly discretized using GFD with boundary conditions enforced using a sharp interface treatment. Cloud sizes are set such that the local approximations use only 16 neighbors. To enforce incompressibility, we apply a semi-implicit approximate projection method. To prevent overlapping particles and formation of voids in the grid, we propose a particle regularization scheme based on a local minimization principle. We validate the GFD results for an oscillating cylinder against the lattice Boltzmann method and find good agreement. Financial support provided by National Science Foundation (NSF) Graduate Research Fellowship, Grant No. DGE-1148903.

Influence of model parameters on synthesized high-frequency strong-motion waveforms

NASA Astrophysics Data System (ADS)

Zadonina, Ekaterina; Caldeira, Bento; Bezzeghoud, Mourad; Borges, José F.

2010-05-01

Waveform modeling is an important and helpful instrument of modern seismology that may provide valuable information. However, synthesizing seismograms requires to define many parameters, which differently affect the final result. Such parameters may be: the design of the grid, the structure model, the source time functions, the source mechanism, the rupture velocity. Variations in parameters may produce significantly different seismograms. We synthesize seismograms from a hypothetical earthquake and numerically estimate the influence of some of the used parameters. Firstly, we present the results for high-frequency near-fault waveforms obtained from defined model by changing tested parameters. Secondly, we present the results of a quantitative comparison of contributions from certain parameters on synthetic waveforms by using misfit criteria. For the synthesis of waveforms we used 2D/3D elastic finite-difference wave propagation code E3D [1] based on the elastodynamic formulation of the wave equation on a staggered grid. This code gave us the opportunity to perform all needed manipulations using a computer cluster. To assess the obtained results, we use misfit criteria [2] where seismograms are compared in time-frequency and phase by applying a continuous wavelet transform to the seismic signal. [1] - Larsen, S. and C.A. Schultz (1995). ELAS3D: 2D/3D elastic finite-difference wave propagation code, Technical Report No. UCRL-MA-121792, 19 pp. [2] - Kristekova, M., Kristek, J., Moczo, P., Day, S.M., 2006. Misfit criteria for quantitative comparison of seismograms. Bul. of Seis. Soc. of Am. 96(5), 1836-1850.

Mathematical Aspects of Finite Element Methods for Incompressible Viscous Flows.

DTIC Science & Technology

1986-09-01

respectively. Here h is a parameter which is usually related to the size of the grid associated with the finite element partitioning of Q. Then one... grid and of not at least performing serious mesh refinement studies. It also points out the usefulness of rigorous results concerning the stability...overconstrained the .1% approximate velocity field. However, by employing different grids for the ’z pressure and velocity fields, the linear-constant

Numerical implementation, verification and validation of two-phase flow four-equation drift flux model with Jacobian-free Newton–Krylov method

DOE PAGES

Zou, Ling; Zhao, Haihua; Zhang, Hongbin

2016-08-24

This study presents a numerical investigation on using the Jacobian-free Newton–Krylov (JFNK) method to solve the two-phase flow four-equation drift flux model with realistic constitutive correlations (‘closure models’). The drift flux model is based on Isshi and his collaborators’ work. Additional constitutive correlations for vertical channel flow, such as two-phase flow pressure drop, flow regime map, wall boiling and interfacial heat transfer models, were taken from the RELAP5-3D Code Manual and included to complete the model. The staggered grid finite volume method and fully implicit backward Euler method was used for the spatial discretization and time integration schemes, respectively. Themore » Jacobian-free Newton–Krylov method shows no difficulty in solving the two-phase flow drift flux model with a discrete flow regime map. In addition to the Jacobian-free approach, the preconditioning matrix is obtained by using the default finite differencing method provided in the PETSc package, and consequently the labor-intensive implementation of complex analytical Jacobian matrix is avoided. Extensive and successful numerical verification and validation have been performed to prove the correct implementation of the models and methods. Code-to-code comparison with RELAP5-3D has further demonstrated the successful implementation of the drift flux model.« less

A Computer Code for Swirling Turbulent Axisymmetric Recirculating Flows in Practical Isothermal Combustor Geometries

NASA Technical Reports Server (NTRS)

Lilley, D. G.; Rhode, D. L.

1982-01-01

A primitive pressure-velocity variable finite difference computer code was developed to predict swirling recirculating inert turbulent flows in axisymmetric combustors in general, and for application to a specific idealized combustion chamber with sudden or gradual expansion. The technique involves a staggered grid system for axial and radial velocities, a line relaxation procedure for efficient solution of the equations, a two-equation k-epsilon turbulence model, a stairstep boundary representation of the expansion flow, and realistic accommodation of swirl effects. A user's manual, dealing with the computational problem, showing how the mathematical basis and computational scheme may be translated into a computer program is presented. A flow chart, FORTRAN IV listing, notes about various subroutines and a user's guide are supplied as an aid to prospective users of the code.

A Virtual World of Visualization

NASA Technical Reports Server (NTRS)

1998-01-01

In 1990, Sterling Software, Inc., developed the Flow Analysis Software Toolkit (FAST) for NASA Ames on contract. FAST is a workstation based modular analysis and visualization tool. It is used to visualize and animate grids and grid oriented data, typically generated by finite difference, finite element and other analytical methods. FAST is now available through COSMIC, NASA's software storehouse.

Moving and adaptive grid methods for compressible flows

NASA Technical Reports Server (NTRS)

Trepanier, Jean-Yves; Camarero, Ricardo

1995-01-01

This paper describes adaptive grid methods developed specifically for compressible flow computations. The basic flow solver is a finite-volume implementation of Roe's flux difference splitting scheme or arbitrarily moving unstructured triangular meshes. The grid adaptation is performed according to geometric and flow requirements. Some results are included to illustrate the potential of the methodology.

Analysis of the influence of the interlayer staggered zone in the basalt of Jinsha River Basin on the main buildings

NASA Astrophysics Data System (ADS)

Guo, Qiaona; Huang, Jiangwei

2018-02-01

In this paper, the finite element software FEFLOW is used to simulate the seepage field of the interlayer staggered zone C2 in the basalt of Jinsha River Basin. The influence of the interlayer staggered zone C2 on the building is analyzed. Combined with the waterproof effect of current design scheme of anti-seepage curtain, the seepage field in the interlayer staggered zone C2 is discussed under different design schemes. The optimal design scheme of anti-seepage curtain is put forward. The results showed that the case four can effectively reduce the head and hydraulic gradient of underground powerhouse area, and improve the groundwater seepage field in the plant area.

Waveform Modeling Reveals Important Features of the Subduction Zone Seismic Structure Beneath the Tyrrhenian Sea, Italy

NASA Astrophysics Data System (ADS)

Di Luccio, F.; Persaud, P.; Pino, N. A.; Clayton, R. W.; Helmberger, D. V.; Li, D.

2016-12-01

Seismic images of the slab in southern Italy indicate a complex geodynamic system, although these images are strongly affected by limitations due to instrumental coverage, in terms of depth resolution and lateral extent. To help improve our knowledge of the structure of the Calabrian subduction zone, we analyze waveforms of regional events that occurred between 2001 and 2015 beneath the Tyrrhenian sea in the western Mediterranean. The selected events are deeper than 200 km and they were recorded at the Italian seismic network managed by Istituto Nazionale di Geofisica e Vulcanologia in Italy. We have also included recordings at ocean bottom seismometers and hydrophones, which were installed for a few months in 2000-2001, 2004-2005 and 2007-2008. Accurate selection of the source-to receiver raypaths can reveal significant differences at receivers, which are perpendicular to the trench with respect to other stations. P-wave complexity, converted phases and frequency content are some of the features we have observed for selected events. To investigate the slab structure, we model the waveforms using the 2D staggered grid Finite Difference method on graphics processing units developed by Li et al. (Geophys. J. Int., 2014).

A multi-scale residual-based anti-hourglass control for compatible staggered Lagrangian hydrodynamics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kucharik, M.; Scovazzi, Guglielmo; Shashkov, Mikhail Jurievich

Hourglassing is a well-known pathological numerical artifact affecting the robustness and accuracy of Lagrangian methods. There exist a large number of hourglass control/suppression strategies. In the community of the staggered compatible Lagrangian methods, the approach of sub-zonal pressure forces is among the most widely used. However, this approach is known to add numerical strength to the solution, which can cause potential problems in certain types of simulations, for instance in simulations of various instabilities. To avoid this complication, we have adapted the multi-scale residual-based stabilization typically used in the finite element approach for staggered compatible framework. In this study, wemore » describe two discretizations of the new approach and demonstrate their properties and compare with the method of sub-zonal pressure forces on selected numerical problems.« less

A multi-scale residual-based anti-hourglass control for compatible staggered Lagrangian hydrodynamics

DOE PAGES

Kucharik, M.; Scovazzi, Guglielmo; Shashkov, Mikhail Jurievich; ...

2017-10-28

Hourglassing is a well-known pathological numerical artifact affecting the robustness and accuracy of Lagrangian methods. There exist a large number of hourglass control/suppression strategies. In the community of the staggered compatible Lagrangian methods, the approach of sub-zonal pressure forces is among the most widely used. However, this approach is known to add numerical strength to the solution, which can cause potential problems in certain types of simulations, for instance in simulations of various instabilities. To avoid this complication, we have adapted the multi-scale residual-based stabilization typically used in the finite element approach for staggered compatible framework. In this study, wemore » describe two discretizations of the new approach and demonstrate their properties and compare with the method of sub-zonal pressure forces on selected numerical problems.« less

Stable Artificial Dissipation Operators for Finite Volume Schemes on Unstructured Grids

NASA Technical Reports Server (NTRS)

Svard, Magnus; Gong, Jing; Nordstrom, Jan

2006-01-01

Our objective is to derive stable first-, second- and fourth-order artificial dissipation operators for node based finite volume schemes. Of particular interest are general unstructured grids where the strength of the finite volume method is fully utilized. A commonly used finite volume approximation of the Laplacian will be the basis in the construction of the artificial dissipation. Both a homogeneous dissipation acting in all directions with equal strength and a modification that allows different amount of dissipation in different directions are derived. Stability and accuracy of the new operators are proved and the theoretical results are supported by numerical computations.

Multigrid finite element method in stress analysis of three-dimensional elastic bodies of heterogeneous structure

NASA Astrophysics Data System (ADS)

Matveev, A. D.

2016-11-01

To calculate the three-dimensional elastic body of heterogeneous structure under static loading, a method of multigrid finite element is provided, when implemented on the basis of algorithms of finite element method (FEM), using homogeneous and composite threedimensional multigrid finite elements (MFE). Peculiarities and differences of MFE from the currently available finite elements (FE) are to develop composite MFE (without increasing their dimensions), arbitrarily small basic partition of composite solids consisting of single-grid homogeneous FE of the first order can be used, i.e. in fact, to use micro approach in finite element form. These small partitions allow one to take into account in MFE, i.e. in the basic discrete models of composite solids, complex heterogeneous and microscopically inhomogeneous structure, shape, the complex nature of the loading and fixation and describe arbitrarily closely the stress and stain state by the equations of three-dimensional elastic theory without any additional simplifying hypotheses. When building the m grid FE, m of nested grids is used. The fine grid is generated by a basic partition of MFE, the other m —1 large grids are applied to reduce MFE dimensionality, when m is increased, MFE dimensionality becomes smaller. The procedures of developing MFE of rectangular parallelepiped, irregular shape, plate and beam types are given. MFE generate the small dimensional discrete models and numerical solutions with a high accuracy. An example of calculating the laminated plate, using three-dimensional 3-grid FE and the reference discrete model is given, with that having 2.2 milliards of FEM nodal unknowns.

On a multigrid method for the coupled Stokes and porous media flow problem

NASA Astrophysics Data System (ADS)

Luo, P.; Rodrigo, C.; Gaspar, F. J.; Oosterlee, C. W.

2017-07-01

The multigrid solution of coupled porous media and Stokes flow problems is considered. The Darcy equation as the saturated porous medium model is coupled to the Stokes equations by means of appropriate interface conditions. We focus on an efficient multigrid solution technique for the coupled problem, which is discretized by finite volumes on staggered grids, giving rise to a saddle point linear system. Special treatment is required regarding the discretization at the interface. An Uzawa smoother is employed in multigrid, which is a decoupled procedure based on symmetric Gauss-Seidel smoothing for velocity components and a simple Richardson iteration for the pressure field. Since a relaxation parameter is part of a Richardson iteration, Local Fourier Analysis (LFA) is applied to determine the optimal parameters. Highly satisfactory multigrid convergence is reported, and, moreover, the algorithm performs well for small values of the hydraulic conductivity and fluid viscosity, that are relevant for applications.

Turbulent heat transfer performance of single stage turbine

DOE Office of Scientific and Technical Information (OSTI.GOV)

Amano, R.S.; Song, B.

1999-07-01

To increase the efficiency and the power of modern power plant gas turbines, designers are continually trying to raise the maximum turbine inlet temperature. Here, a numerical study based on the Navier-Stokes equations on a three-dimensional turbulent flow in a single stage turbine stator/rotor passage has been conducted and reported in this paper. The full Reynolds-stress closure model (RSM) was used for the computations and the results were also compared with the computations made by using the Launder-Sharma low-Reynolds-number {kappa}-{epsilon} model. The computational results obtained using these models were compared in order to investigate the turbulence effect in the near-wallmore » region. The set of the governing equations in a generalized curvilinear coordinate system was discretized by using the finite volume method with non-staggered grids. The numerical modeling was performed to interact between the stator and rotor blades.« less

Mapping PetaSHA Applications to TeraGrid Architectures

NASA Astrophysics Data System (ADS)

Cui, Y.; Moore, R.; Olsen, K.; Zhu, J.; Dalguer, L. A.; Day, S.; Cruz-Atienza, V.; Maechling, P.; Jordan, T.

2007-12-01

The Southern California Earthquake Center (SCEC) has a science program in developing an integrated cyberfacility - PetaSHA - for executing physics-based seismic hazard analysis (SHA) computations. The NSF has awarded PetaSHA 15 million allocation service units this year on the fastest supercomputers available within the NSF TeraGrid. However, one size does not fit all, a range of systems are needed to support this effort at different stages of the simulations. Enabling PetaSHA simulations on those TeraGrid architectures to solve both dynamic rupture and seismic wave propagation have been a challenge from both hardware and software levels. This is an adaptation procedure to meet specific requirements of each architecture. It is important to determine how fundamental system attributes affect application performance. We present an adaptive approach in our PetaSHA application that enables the simultaneous optimization of both computation and communication at run-time using flexible settings. These techniques optimize initialization, source/media partition and MPI-IO output in different ways to achieve optimal performance on the target machines. The resulting code is a factor of four faster than the orignial version. New MPI-I/O capabilities have been added for the accurate Staggered-Grid Split-Node (SGSN) method for dynamic rupture propagation in the velocity-stress staggered-grid finite difference scheme (Dalguer and Day, JGR, 2007), We use execution workflow across TeraGrid sites for managing the resulting data volumes. Our lessons learned indicate that minimizing time to solution is most critical, in particular when scheduling large scale simulations across supercomputer sites. The TeraShake platform has been ported to multiple architectures including TACC Dell lonestar and Abe, Cray XT3 Bigben and Blue Gene/L. Parallel efficiency of 96% with the PetaSHA application Olsen-AWM has been demonstrated on 40,960 Blue Gene/L processors at IBM TJ Watson Center. Notable accomplishments using the optimized code include the M7.8 ShakeOut rupture scenario, as part of the southern San Andreas Fault evaluation SoSAFE. The ShakeOut simulation domain is the same as used for the SCEC TeraShake simulations (600 km by 300 km by 80 km). However, the higher resolution of 100 m with frequency content up to 1 Hz required 14.4 billion grid points, eight times more than the TeraShake scenarios. The simulation used 2000 TACC Dell linux Lonestar processors and took 56 hours to compute 240 seconds of wave propagation. The pre-processing input partition, as well as post-processing analysis has been performed on the SDSC IBM Datastar p655 and p690. In addition, as part of the SCEC DynaShake computational platform, the SGSN capability was used to model dynamic rupture propagation for the ShakeOut scenario that match the proposed surface slip and size of the event. Mapping applications to different architectures require coordination of many areas of expertise in hardware and application level, an outstanding challenge faced on the current petascale computing effort. We believe our techniques as well as distributed data management through data grids have provided a practical example of how to effectively use multiple compute resources, and our results will benefit other geoscience disciplines as well.

Seismic wavefield modeling based on time-domain symplectic and Fourier finite-difference method

NASA Astrophysics Data System (ADS)

Fang, Gang; Ba, Jing; Liu, Xin-xin; Zhu, Kun; Liu, Guo-Chang

2017-06-01

Seismic wavefield modeling is important for improving seismic data processing and interpretation. Calculations of wavefield propagation are sometimes not stable when forward modeling of seismic wave uses large time steps for long times. Based on the Hamiltonian expression of the acoustic wave equation, we propose a structure-preserving method for seismic wavefield modeling by applying the symplectic finite-difference method on time grids and the Fourier finite-difference method on space grids to solve the acoustic wave equation. The proposed method is called the symplectic Fourier finite-difference (symplectic FFD) method, and offers high computational accuracy and improves the computational stability. Using acoustic approximation, we extend the method to anisotropic media. We discuss the calculations in the symplectic FFD method for seismic wavefield modeling of isotropic and anisotropic media, and use the BP salt model and BP TTI model to test the proposed method. The numerical examples suggest that the proposed method can be used in seismic modeling of strongly variable velocities, offering high computational accuracy and low numerical dispersion. The symplectic FFD method overcomes the residual qSV wave of seismic modeling in anisotropic media and maintains the stability of the wavefield propagation for large time steps.

Generalized energy and potential enstrophy conserving finite difference schemes for the shallow water equations

NASA Technical Reports Server (NTRS)

Abramopoulos, Frank

1988-01-01

The conditions under which finite difference schemes for the shallow water equations can conserve both total energy and potential enstrophy are considered. A method of deriving such schemes using operator formalism is developed. Several such schemes are derived for the A-, B- and C-grids. The derived schemes include second-order schemes and pseudo-fourth-order schemes. The simplest B-grid pseudo-fourth-order schemes are presented.

Rheological Models in the Time-Domain Modeling of Seismic Motion

NASA Astrophysics Data System (ADS)

Moczo, P.; Kristek, J.

2004-12-01

The time-domain stress-strain relation in a viscoelastic medium has a form of the convolutory integral which is numerically intractable. This was the reason for the oversimplified models of attenuation in the time-domain seismic wave propagation and earthquake motion modeling. In their pioneering work, Day and Minster (1984) showed the way how to convert the integral into numerically tractable differential form in the case of a general viscoelastic modulus. In response to the work by Day and Minster, Emmerich and Korn (1987) suggested using the rheology of their generalized Maxwell body (GMB) while Carcione et al. (1988) suggested using the generalized Zener body (GZB). The viscoelastic moduli of both rheological models have a form of the rational function and thus the differential form of the stress-strain relation is rather easy to obtain. After the papers by Emmerich and Korn and Carcione et al. numerical modelers decided either for the GMB or GZB rheology and developed 'non-communicating' algorithms. In the many following papers the authors using the GMB never commented the GZB rheology and the corresponding algorithms, and the authors using the GZB never related their methods to the GMB rheology and algorithms. We analyze and compare both rheologies and the corresponding incorporations of the realistic attenuation into the time-domain computations. We then focus on the most recent staggered-grid finite-difference modeling, mainly on accounting for the material heterogeneity in the viscoelastic media, and the computational efficiency of the finite-difference algorithms.

Linear-stability theory of thermocapillary convection in a model of float-zone crystal growth

NASA Technical Reports Server (NTRS)

Neitzel, G. P.; Chang, K.-T.; Jankowski, D. F.; Mittelmann, H. D.

1992-01-01

Linear-stability theory has been applied to a basic state of thermocapillary convection in a model half-zone to determine values of the Marangoni number above which instability is guaranteed. The basic state must be determined numerically since the half-zone is of finite, O(1) aspect ratio with two-dimensional flow and temperature fields. This, in turn, means that the governing equations for disturbance quantities will remain partial differential equations. The disturbance equations are treated by a staggered-grid discretization scheme. Results are presented for a variety of parameters of interest in the problem, including both terrestrial and microgravity cases.

Demonstration Of Ultra HI-FI (UHF) Methods

NASA Technical Reports Server (NTRS)

Dyson, Rodger W.

2004-01-01

Computational aero-acoustics (CAA) requires efficient, high-resolution simulation tools. Most current techniques utilize finite-difference approaches because high order accuracy is considered too difficult or expensive to achieve with finite volume or finite element methods. However, a novel finite volume approach (Ultra HI-FI or UHF) which utilizes Hermite fluxes is presented which can achieve both arbitrary accuracy and fidelity in space and time. The technique can be applied to unstructured grids with some loss of fidelity or with multi-block structured grids for maximum efficiency and resolution. In either paradigm, it is possible to resolve ultra-short waves (less than 2 PPW). This is demonstrated here by solving the 4th CAA workshop Category 1 Problem 1.

Pathloss Calculation Using the Transmission Line Matrix and Finite Difference Time Domain Methods With Coarse Grids

DOE PAGES

Nutaro, James; Kuruganti, Teja

2017-02-24

Numerical simulations of the wave equation that are intended to provide accurate time domain solutions require a computational mesh with grid points separated by a distance less than the wavelength of the source term and initial data. However, calculations of radio signal pathloss generally do not require accurate time domain solutions. This paper describes an approach for calculating pathloss by using the finite difference time domain and transmission line matrix models of wave propagation on a grid with points separated by distances much greater than the signal wavelength. The calculated pathloss can be kept close to the true value formore » freespace propagation with an appropriate selection of initial conditions. This method can also simulate diffraction with an error governed by the ratio of the signal wavelength to the grid spacing.« less

Relative and Absolute Error Control in a Finite-Difference Method Solution of Poisson's Equation

ERIC Educational Resources Information Center

Prentice, J. S. C.

2012-01-01

An algorithm for error control (absolute and relative) in the five-point finite-difference method applied to Poisson's equation is described. The algorithm is based on discretization of the domain of the problem by means of three rectilinear grids, each of different resolution. We discuss some hardware limitations associated with the algorithm,…

Direct measurement of the breakdown slip from near-fault strong motion data

NASA Astrophysics Data System (ADS)

Cruz-Atienza, V. M.; Olsen, K. B.; Dalguer, L. A.

2007-12-01

Obtaining reliable estimates of the frictional behaviour on earthquake faults is a fundamental task, particularly the breakdown slip Dc, which has an important role on rupture propagation through the earthquake energy budget. Several studies have attempted to estimate Dc indirectly from kinematical analysis of fault ruptures (e.g., Ide and Takeo, JGR, 1997). However, such estimates are complicated because of both the limited band-width of the observed seismograms used to image the rupture process and the rapid decay of high frequencies with distance from the fault. Mikumo et al. (BSSA, 2003) proposed a method to estimate Dc on the fault plane as the slip at the time of the peak sliprate function (Dc'). Fukuyama and Mikumo (GRL, 2007) proposed to extend this method beyond the fault plane, by estimating Dc as twice the rake-parallel particle displacement at the time of the peak particle velocity. The factor of two arises from an equal amount of opposite displacement on either side of the fault. They concluded that such method allows reliable Dc' estimates with negligible dependence on the perpendicular distance from the fault, and used it to obtain Dc' estimates for the 2000 M6.6 Tottori (0.3 m) and the 2002 M7.9 Denali (2.5 m) earthquakes. The study by Fukuyama and Mikumo was based on simple two-dimensional Green's functions in a homogeneous full space for an anti-plane kinematic crack, and suffers from three fundamental omissions: 1) the free surface and heterogeneous structure, 2) the finiteness of the rupture surface and 3) the dynamic rupture complexity of real 3D earthquakes. Here, we re-examine the methodology proposed by Fukuyama and Mikumo by means of a more realistic approach. We use spontaneous rupture propagation simulated by a recently developed and highly accurate approach, namely the staggered-grid split-node (SGSN) method in a fourth-order staggered- grid finite difference method (Dalguer and Day, JGR, 2007). We assume a vertical strike-slip fault governed by both linear and non-linear slip-weakening friction laws. Our results show that both the free surface and the stopping phases strongly affect Dc estimates. The particle motion recorded by surface instruments is amplified roughly by a factor of two due to the presence of the free surface. As a consequence, the method by Fukuyama and Mikumo over-estimates Dc when applied to strong motion data recorded on the earth's surface. Moreover, contrary to the results by Fukuyama and Mikumo, we observe a strong distance-dependence of the Dc estimates perpendicular to the fault. This variation includes a minimum near the fault, increasing up to about 140% of the target Dc value at a distance 2-3 km from the fault. At further distances from the fault the Dc estimate decreases to about 60% of the target value 10 km away. This distance dependence of the Dc estimate is presumably caused mainly by stopping phases propagating from the fault boundaries. Simulations in heterogeneous media including a low-velocity layer, intrinsic attenuation (Q) and stochastic initial stress conditions allow us to asses the reliability and uncertainty involved in the method proposed by Fukuyama and Mikumo. Dc estimates under these realistic conditions are important but remain below a factor of two in most of the cases we have analyzed. In summary, the accuracy of the method is strongly affected by the presence of the free surface, finite fault extent, and likely by complexity in the velocity structure and rupture propagation.

The effect of Reynolds number and turbulence on airfoil aerodynamics at -90 degrees incidence

NASA Technical Reports Server (NTRS)

Stremel, Paul M.

1993-01-01

A method has been developed for calculating the viscous flow about airfoils in with and without deflected flaps at -90 deg incidence. This method provides for the solution of the unsteady incompressible Navier-Stokes equations by means of an implicit technique. The solution is calculated on a body-fitted computational mesh using a staggered grid method. The vorticity is defined at the node points, and the velocity components are defined at the mesh-cell sides. The staggered-grid orientation provides for accurate representation of vorticity at the node points and the continuity equation at the mesh-cell centers. The method provides for the direct solution of the flow field and satisfies the continuity equation to machine zero at each time-step. The method is evaluated in terms of its ability to predict two-dimensional flow about an airfoil at -90 degrees incidence for varying Reynolds number and different boundary layer models. A laminar and a turbulent boundary layer models. A laminar and a turbulent boundary layer model are considered in the evaluation of the method. The variation of the average loading and surface pressure distribution due to flap deflection, Reynolds number, and laminar or turbulent flow are presented and compared with experimental results. The comparisons indicate that the calculated drag and drag reduction caused by flap deflection and the calculated average surface pressure are in excellent agreement with the measured results at a similar Reynolds number.

Solving phase appearance/disappearance two-phase flow problems with high resolution staggered grid and fully implicit schemes by the Jacobian-free Newton–Krylov Method

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zou, Ling; Zhao, Haihua; Zhang, Hongbin

2016-04-01

The phase appearance/disappearance issue presents serious numerical challenges in two-phase flow simulations. Many existing reactor safety analysis codes use different kinds of treatments for the phase appearance/disappearance problem. However, to our best knowledge, there are no fully satisfactory solutions. Additionally, the majority of the existing reactor system analysis codes were developed using low-order numerical schemes in both space and time. In many situations, it is desirable to use high-resolution spatial discretization and fully implicit time integration schemes to reduce numerical errors. In this work, we adapted a high-resolution spatial discretization scheme on staggered grid mesh and fully implicit time integrationmore » methods (such as BDF1 and BDF2) to solve the two-phase flow problems. The discretized nonlinear system was solved by the Jacobian-free Newton Krylov (JFNK) method, which does not require the derivation and implementation of analytical Jacobian matrix. These methods were tested with a few two-phase flow problems with phase appearance/disappearance phenomena considered, such as a linear advection problem, an oscillating manometer problem, and a sedimentation problem. The JFNK method demonstrated extremely robust and stable behaviors in solving the two-phase flow problems with phase appearance/disappearance. No special treatments such as water level tracking or void fraction limiting were used. High-resolution spatial discretization and second- order fully implicit method also demonstrated their capabilities in significantly reducing numerical errors.« less

Comparative Study of Advanced Turbulence Models for Turbomachinery

NASA Technical Reports Server (NTRS)

Hadid, Ali H.; Sindir, Munir M.

1996-01-01

A computational study has been undertaken to study the performance of advanced phenomenological turbulence models coded in a modular form to describe incompressible turbulent flow behavior in two dimensional/axisymmetric and three dimensional complex geometry. The models include a variety of two equation models (single and multi-scale k-epsilon models with different near wall treatments) and second moment algebraic and full Reynolds stress closure models. These models were systematically assessed to evaluate their performance in complex flows with rotation, curvature and separation. The models are coded as self contained modules that can be interfaced with a number of flow solvers. These modules are stand alone satellite programs that come with their own formulation, finite-volume discretization scheme, solver and boundary condition implementation. They will take as input (from any generic Navier-Stokes solver) the velocity field, grid (structured H-type grid) and computational domain specification (boundary conditions), and will deliver, depending on the model used, turbulent viscosity, or the components of the Reynolds stress tensor. There are separate 2D/axisymmetric and/or 3D decks for each module considered. The modules are tested using Rocketdyn's proprietary code REACT. The code utilizes an efficient solution procedure to solve Navier-Stokes equations in a non-orthogonal body-fitted coordinate system. The differential equations are discretized over a finite-volume grid using a non-staggered variable arrangement and an efficient solution procedure based on the SIMPLE algorithm for the velocity-pressure coupling is used. The modules developed have been interfaced and tested using finite-volume, pressure-correction CFD solvers which are widely used in the CFD community. Other solvers can also be used to test these modules since they are independently structured with their own discretization scheme and solver methodology. Many of these modules have been independently tested by Professor C.P. Chen and his group at the University of Alabama at Huntsville (UAH) by interfacing them with own flow solver (MAST).

A systematic approach to numerical dispersion in Maxwell solvers

NASA Astrophysics Data System (ADS)

Blinne, Alexander; Schinkel, David; Kuschel, Stephan; Elkina, Nina; Rykovanov, Sergey G.; Zepf, Matt

2018-03-01

The finite-difference time-domain (FDTD) method is a well established method for solving the time evolution of Maxwell's equations. Unfortunately the scheme introduces numerical dispersion and therefore phase and group velocities which deviate from the correct values. The solution to Maxwell's equations in more than one dimension results in non-physical predictions such as numerical dispersion or numerical Cherenkov radiation emitted by a relativistic electron beam propagating in vacuum. Improved solvers, which keep the staggered Yee-type grid for electric and magnetic fields, generally modify the spatial derivative operator in the Maxwell-Faraday equation by increasing the computational stencil. These modified solvers can be characterized by different sets of coefficients, leading to different dispersion properties. In this work we introduce a norm function to rewrite the choice of coefficients into a minimization problem. We solve this problem numerically and show that the minimization procedure leads to phase and group velocities that are considerably closer to c as compared to schemes with manually set coefficients available in the literature. Depending on a specific problem at hand (e.g. electron beam propagation in plasma, high-order harmonic generation from plasma surfaces, etc.), the norm function can be chosen accordingly, for example, to minimize the numerical dispersion in a certain given propagation direction. Particle-in-cell simulations of an electron beam propagating in vacuum using our solver are provided.

Role of deformation in odd-even staggering in reaction cross sections for 30,31,32Ne and 36,37,38Mg isotopes

NASA Astrophysics Data System (ADS)

Urata, Y.; Hagino, K.; Sagawa, H.

2017-12-01

We discuss the role of pairing antihalo effect in the observed odd-even staggering in reaction cross sections for 30,31,32Ne and 36,37,38Mg isotopes by taking into account the ground-state deformation of these nuclei. To this end, we construct the ground-state density for the Ne,3130 and Mg,3736 nuclei based on a deformed Woods-Saxon potential, while for the 32Ne and 38Mg nuclei we also take into account the pairing correlation using the Hartree-Fock-Bogoliubov method. We demonstrate that, when the one-neutron separation energy is small for the odd-mass nuclei, a significant odd-even staggering still appears even with finite deformation, although the degree of staggering is somewhat reduced compared to the spherical case. This implies that the pairing antihalo effect in general plays an important role in generating the odd-even staggering in reaction cross sections for weakly bound nuclei.

Problems with heterogeneous and non-isotropic media or distorted grids

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hyman, J.; Shashkov, M.; Steinberg, S.

1996-08-01

This paper defines discretizations of the divergence and flux operators that produce symmetric, positive-definite, and accurate approximations to steady-state diffusion problems. Because discontinuous material properties and highly distorted grids are allowed, the flux operator, rather than the gradient, is used as a fundamental operator to be discretized. Resulting finite-difference scheme is similar to those obtained from the mixed finite-element method.

New ghost-node method for linking different models with varied grid refinement

USGS Publications Warehouse

James, S.C.; Dickinson, J.E.; Mehl, S.W.; Hill, M.C.; Leake, S.A.; Zyvoloski, G.A.; Eddebbarh, A.-A.

2006-01-01

A flexible, robust method for linking grids of locally refined ground-water flow models constructed with different numerical methods is needed to address a variety of hydrologic problems. This work outlines and tests a new ghost-node model-linking method for a refined "child" model that is contained within a larger and coarser "parent" model that is based on the iterative method of Steffen W. Mehl and Mary C. Hill (2002, Advances in Water Res., 25, p. 497-511; 2004, Advances in Water Res., 27, p. 899-912). The method is applicable to steady-state solutions for ground-water flow. Tests are presented for a homogeneous two-dimensional system that has matching grids (parent cells border an integer number of child cells) or nonmatching grids. The coupled grids are simulated by using the finite-difference and finite-element models MODFLOW and FEHM, respectively. The simulations require no alteration of the MODFLOW or FEHM models and are executed using a batch file on Windows operating systems. Results indicate that when the grids are matched spatially so that nodes and child-cell boundaries are aligned, the new coupling technique has error nearly equal to that when coupling two MODFLOW models. When the grids are nonmatching, model accuracy is slightly increased compared to that for matching-grid cases. Overall, results indicate that the ghost-node technique is a viable means to couple distinct models because the overall head and flow errors relative to the analytical solution are less than if only the regional coarse-grid model was used to simulate flow in the child model's domain.

Improving sub-grid scale accuracy of boundary features in regional finite-difference models

USGS Publications Warehouse

Panday, Sorab; Langevin, Christian D.

2012-01-01

As an alternative to grid refinement, the concept of a ghost node, which was developed for nested grid applications, has been extended towards improving sub-grid scale accuracy of flow to conduits, wells, rivers or other boundary features that interact with a finite-difference groundwater flow model. The formulation is presented for correcting the regular finite-difference groundwater flow equations for confined and unconfined cases, with or without Newton Raphson linearization of the nonlinearities, to include the Ghost Node Correction (GNC) for location displacement. The correction may be applied on the right-hand side vector for a symmetric finite-difference Picard implementation, or on the left-hand side matrix for an implicit but asymmetric implementation. The finite-difference matrix connectivity structure may be maintained for an implicit implementation by only selecting contributing nodes that are a part of the finite-difference connectivity. Proof of concept example problems are provided to demonstrate the improved accuracy that may be achieved through sub-grid scale corrections using the GNC schemes.

Three-dimensional wave field modeling by a collocated-grid finite-difference method in the anelastic model with surface topography

NASA Astrophysics Data System (ADS)

Wang, N.; Li, J.; Borisov, D.; Gharti, H. N.; Shen, Y.; Zhang, W.; Savage, B. K.

2016-12-01

We incorporate 3D anelastic attenuation into the collocated-grid finite-difference method on curvilinear grids (Zhang et al., 2012), using the rheological model of the generalized Maxwell body (Emmerich and Korn, 1987; Moczo and Kristek, 2005; Käser et al., 2007). We follow a conventional procedure to calculate the anelastic coefficients (Emmerich and Korn, 1987) determined by the Q(ω)-law, with a modification in the choice of frequency band and thus the relaxation frequencies that equidistantly cover the logarithmic frequency range. We show that such an optimization of anelastic coefficients is more accurate when using a fixed number of relaxation mechanisms to fit the frequency independent Q-factors. We use curvilinear grids to represent the surface topography. The velocity-stress form of the 3D isotropic anelastic wave equation is solved with a collocated-grid finite-difference method. Compared with the elastic case, we need to solve additional material-independent anelastic functions (Kristek and Moczo, 2003) for the mechanisms at each relaxation frequency. Based on the stress-strain relation, we calculate the spatial partial derivatives of the anelastic functions indirectly thereby saving computational storage and improving computational efficiency. The complex-frequency-shifted perfectly matched layer (CFS-PML) is used for the absorbing boundary condition based on the auxiliary difference equation (Zhang and Shen, 2010). The traction image method (Zhang and Chen, 2006) is employed for the free-surface boundary condition. We perform several numerical experiments including homogeneous full-space models and layered half-space models, considering both flat and 3D Gaussian-shape hill surfaces. The results match very well with those of the spectral-element method (Komatitisch and Tromp, 2002; Savage et al., 2010), verifying the simulations by our method in the anelastic model with surface topography.

Domain-adaptive finite difference methods for collapsing annular liquid jets

NASA Astrophysics Data System (ADS)

Ramos, J. I.

1993-01-01

A domain-adaptive technique which maps a time-dependent, curvilinear geometry into a unit square is used to determine the steady state mass absorption rate and the collapse of annular liquid jets. A method of lines is used to solve the one-dimensional fluid dynamics equations written in weak conservation-law form, and upwind differences are employed to evaluate the axial convective fluxes. The unknown, time-dependent, axial location of the downstream boundary is determined from the solution of an ordinary differential equation which is nonlinearly coupled to the fluid dynamics and gas concentration equations. The equation for the gas concentration in the annular liquid jet is written in strong conservation-law form and solved by means of a method of lines at high Peclet numbers and a line Gauss-Seidel method at low Peclet numbers. The effects of the number of grid points along and across the annular jet, time step, and discretization of the radial convective fluxes on both the steady state mass absorption rate and the jet's collapse rate have been analyzed on staggered and non-staggered grids. The steady state mass absorption rate and the collapse of annular liquid jets are determined as a function of the Froude, Peclet and Weber numbers, annular jet's thickness-to-radius ratio at the nozzle exit, initial pressure difference across the annular jet, nozzle exit angle, temperature of the gas enclosed by the annular jet, pressure of the gas surrounding the jet, solubilities at the inner and outer interfaces of the annular jet, and gas concentration at the nozzle exit. It is shown that the steady state mass absorption rate is proportional to the inverse square root of the Peclet number except for low values of this parameter, and that the possible mathematical incompatibilities in the concentration field at the nozzle exit exert a great influence on the steady state mass absorption rate and on the jet collapse. It is also shown that the steady state mass absorption rate increases as the Weber number, nozzle exit angle, gas concentration at the nozzle exit, and temperature of the gases enclosed by the annular liquid jet are increased, but it decreases as the Froude and Peclet numbers, and annular liquid jet's thickness-to-radius ratio at the nozzle exit are increased. It is also shown that the annular liquid jet's collapse rate increases as the Weber number, nozzle exit angle, temperature of the gases enclosed by the annular liquid jet, and pressure of the gases which surround the jet are increased, but decreases as the Froude and Peclet numbers, and annular liquid jet's thickness-toradius ratio at the nozzle exit are increased. It is also shown that both the ratio of the initial pressure of the gas enclosed by the jet to the pressure of the gas surrounding the jet and the ratio of solubilities at the annular liquid jet's inner and outer interfaces play an important role on both the steady state mass absorption rate and the jet collapse. If the product of these ratios is greater or less than one, both the pressure and the mass of the gas enclosed by the annular liquid jet decrease or increase, respectively, with time. It is also shown that the numerical results obtained with the conservative, domain-adaptive method of lines technique presented in this paper are in excellent agreement with those of a domain-adaptive, iterative, non-conservative, block-bidiagonal, finite difference method which uncouples the solution of the fluid dynamics equations from that of the convergence length.

Semianalytical computation of path lines for finite-difference models

USGS Publications Warehouse

Pollock, D.W.

1988-01-01

A semianalytical particle tracking method was developed for use with velocities generated from block-centered finite-difference ground-water flow models. Based on the assumption that each directional velocity component varies linearly within a grid cell in its own coordinate directions, the method allows an analytical expression to be obtained describing the flow path within an individual grid cell. Given the intitial position of a particle anywhere in a cell, the coordinates of any other point along its path line within the cell, and the time of travel between them, can be computed directly. For steady-state systems, the exit point for a particle entering a cell at any arbitrary location can be computed in a single step. By following the particle as it moves from cell to cell, this method can be used to trace the path of a particle through any multidimensional flow field generated from a block-centered finite-difference flow model. -Author

The Development of a Finite Volume Method for Modeling Sound in Coastal Ocean Environment

DOE Office of Scientific and Technical Information (OSTI.GOV)

Long, Wen; Yang, Zhaoqing; Copping, Andrea E.

: As the rapid growth of marine renewable energy and off-shore wind energy, there have been concerns that the noises generated from construction and operation of the devices may interfere marine animals’ communication. In this research, a underwater sound model is developed to simulate sound prorogation generated by marine-hydrokinetic energy (MHK) devices or offshore wind (OSW) energy platforms. Finite volume and finite difference methods are developed to solve the 3D Helmholtz equation of sound propagation in the coastal environment. For finite volume method, the grid system consists of triangular grids in horizontal plane and sigma-layers in vertical dimension. A 3Dmore » sparse matrix solver with complex coefficients is formed for solving the resulting acoustic pressure field. The Complex Shifted Laplacian Preconditioner (CSLP) method is applied to efficiently solve the matrix system iteratively with MPI parallelization using a high performance cluster. The sound model is then coupled with the Finite Volume Community Ocean Model (FVCOM) for simulating sound propagation generated by human activities in a range-dependent setting, such as offshore wind energy platform constructions and tidal stream turbines. As a proof of concept, initial validation of the finite difference solver is presented for two coastal wedge problems. Validation of finite volume method will be reported separately.« less

Curvilinear grids for WENO methods in astrophysical simulations

NASA Astrophysics Data System (ADS)

Grimm-Strele, H.; Kupka, F.; Muthsam, H. J.

2014-03-01

We investigate the applicability of curvilinear grids in the context of astrophysical simulations and WENO schemes. With the non-smooth mapping functions from Calhoun et al. (2008), we can tackle many astrophysical problems which were out of scope with the standard grids in numerical astrophysics. We describe the difficulties occurring when implementing curvilinear coordinates into our WENO code, and how we overcome them. We illustrate the theoretical results with numerical data. The WENO finite difference scheme works only for high Mach number flows and smooth mapping functions, whereas the finite volume scheme gives accurate results even for low Mach number flows and on non-smooth grids.

Seismic wavefield propagation in 2D anisotropic media: Ray theory versus wave-equation simulation

NASA Astrophysics Data System (ADS)

Bai, Chao-ying; Hu, Guang-yi; Zhang, Yan-teng; Li, Zhong-sheng

2014-05-01

Despite the ray theory that is based on the high frequency assumption of the elastic wave-equation, the ray theory and the wave-equation simulation methods should be mutually proof of each other and hence jointly developed, but in fact parallel independent progressively. For this reason, in this paper we try an alternative way to mutually verify and test the computational accuracy and the solution correctness of both the ray theory (the multistage irregular shortest-path method) and the wave-equation simulation method (both the staggered finite difference method and the pseudo-spectral method) in anisotropic VTI and TTI media. Through the analysis and comparison of wavefield snapshot, common source gather profile and synthetic seismogram, it is able not only to verify the accuracy and correctness of each of the methods at least for kinematic features, but also to thoroughly understand the kinematic and dynamic features of the wave propagation in anisotropic media. The results show that both the staggered finite difference method and the pseudo-spectral method are able to yield the same results even for complex anisotropic media (such as a fault model); the multistage irregular shortest-path method is capable of predicting similar kinematic features as the wave-equation simulation method does, which can be used to mutually test each other for methodology accuracy and solution correctness. In addition, with the aid of the ray tracing results, it is easy to identify the multi-phases (or multiples) in the wavefield snapshot, common source point gather seismic section and synthetic seismogram predicted by the wave-equation simulation method, which is a key issue for later seismic application.

Calculation of unsteady airfoil loads with and without flap deflection at -90 degrees incidence

NASA Technical Reports Server (NTRS)

Stremel, Paul M.

1991-01-01

A method has been developed for calculating the viscous flow about airfoils with and without deflected flaps at -90 deg incidence. This unique method provides for the direct solution of the incompressible Navier-Stokes equations by means of a fully coupled implicit technique. The solution is calculated on a body-fitted computational mesh incorporating a staggered grid method. The vorticity is determined at the node points, and the velocity components are defined at the mesh-cell sides. The staggered-grid orientation provides for accurate representation of vorticity at the node points and for the conservation of mass at the mesh-cell centers. The method provides for the direct solution of the flow field and satisfies the conservation of mass to machine zero at each time-step. The results of the present analysis and experimental results obtained for a XV-15 airfoil are compared. The comparisons indicate that the calculated drag reduction caused by flap deflection and the calculated average surface pressure are in excellent agreement with the measured results. Comparisons of the numerical results of the present method for several airfoils demonstrate the significant influence of airfoil curvature and flap deflection on the predicted download.

Interior Fluid Dynamics of Liquid-Filled Projectiles

DTIC Science & Technology

1989-12-01

the Sandia code. The previous codes are primarily based on finite-difference approximations with relatively coarse grid and were designed without...exploits Chorin’s method of artificial compressibility. The steady solution at 11 X 24 X 21 grid points in r, 0, z-direction is obtained by integrating...differences in radial and axial direction and pseudoepectral differencing in the azimuthal direction. Nonuniform grids are introduced for increased

3D electromagnetic modelling of a TTI medium and TTI effects in inversion

NASA Astrophysics Data System (ADS)

Jaysaval, Piyoosh; Shantsev, Daniil; de la Kethulle de Ryhove, Sébastien

2016-04-01

We present a numerical algorithm for 3D electromagnetic (EM) forward modelling in conducting media with general electric anisotropy. The algorithm is based on the finite-difference discretization of frequency-domain Maxwell's equations on a Lebedev grid, in which all components of the electric field are collocated but half a spatial step staggered with respect to the magnetic field components, which also are collocated. This leads to a system of linear equations that is solved using a stabilized biconjugate gradient method with a multigrid preconditioner. We validate the accuracy of the numerical results for layered and 3D tilted transverse isotropic (TTI) earth models representing typical scenarios used in the marine controlled-source EM method. It is then demonstrated that not taking into account the full anisotropy of the conductivity tensor can lead to misleading inversion results. For simulation data corresponding to a 3D model with a TTI anticlinal structure, a standard vertical transverse isotropic inversion is not able to image a resistor, while for a 3D model with a TTI synclinal structure the inversion produces a false resistive anomaly. If inversion uses the proposed forward solver that can handle TTI anisotropy, it produces resistivity images consistent with the true models.

Numerical simulation of unsteady generalized Newtonian blood flow through differently shaped distensible arterial stenoses.

PubMed

Sarifuddin; Chakravarty, S; Mandal, P K; Layek, G C

2008-01-01

An updated numerical simulation of unsteady generalized Newtonian blood flow through differently shaped distensible arterial stenoses is developed. A shear-thinning fluid modelling the deformation dependent viscosity of blood is considered for the characterization of generalized Newtonian behaviour of blood. The arterial model is treated as two-dimensional and axisymmetric with an outline of the stenosis obtained from a three-dimensional casting of a mildly stenosed artery. The full Navier-Stokes equations governing blood flow are written in the dimensionless form and the solution is accomplished by finite time-step advancement through their finite difference staggered grid representations. The marker and cell (MAC) method comprising the use of a set of marker particles moving with the fluid is used for the purpose. Results are obtained for three differently shaped stenoses - irregular, smooth and cosine curve representations. The present results do agree well with those of existing investigations in the steady state, but contrary to their conclusions the present findings demonstrate that the excess pressure drop across the cosine and the smooth stenoses is caused by neither their smoothness nor their higher degree of symmetry relative to the irregular stenosis, but is rather an effect of area cover with respect to the irregular stenosis. This effect clearly prevails throughout the entire physiological range of Reynolds numbers. Further the in-depth study in flow patterns reveals the development of flow separation zones in the diverging part of the stenosis towards the arterial wall, and they are influenced by non-Newtonian blood rheology, distensibility of the wall and flow unsteadiness in order to validate the applicability of the present model.

Hybrid finite difference/finite element immersed boundary method.

PubMed

E Griffith, Boyce; Luo, Xiaoyu

2017-12-01

The immersed boundary method is an approach to fluid-structure interaction that uses a Lagrangian description of the structural deformations, stresses, and forces along with an Eulerian description of the momentum, viscosity, and incompressibility of the fluid-structure system. The original immersed boundary methods described immersed elastic structures using systems of flexible fibers, and even now, most immersed boundary methods still require Lagrangian meshes that are finer than the Eulerian grid. This work introduces a coupling scheme for the immersed boundary method to link the Lagrangian and Eulerian variables that facilitates independent spatial discretizations for the structure and background grid. This approach uses a finite element discretization of the structure while retaining a finite difference scheme for the Eulerian variables. We apply this method to benchmark problems involving elastic, rigid, and actively contracting structures, including an idealized model of the left ventricle of the heart. Our tests include cases in which, for a fixed Eulerian grid spacing, coarser Lagrangian structural meshes yield discretization errors that are as much as several orders of magnitude smaller than errors obtained using finer structural meshes. The Lagrangian-Eulerian coupling approach developed in this work enables the effective use of these coarse structural meshes with the immersed boundary method. This work also contrasts two different weak forms of the equations, one of which is demonstrated to be more effective for the coarse structural discretizations facilitated by our coupling approach. © 2017 The Authors International  Journal  for  Numerical  Methods  in  Biomedical  Engineering Published by John Wiley & Sons Ltd.

First International Conference on Numerical Ship Hydrodynamics Held in Gaithersburg, Maryland on 20-22 October 1975.

DTIC Science & Technology

1975-01-01

instance, Harlow and Amsden, 1971 ). In this method a staggered finite difference mesh as shown in Figure 2 is used. 4X 1X 41X 0X 4 - -* - 4 - * - 4 - -X I...in Harlow and Amsden ( 1971 ). It suffices to state here that spatial derivatives are approximated by central differences throughout giving 0(h 2...flow divergence.) The boundary conditions are not elaborated here (but see Harlow and Amsden, 1971 ) except to note that the free surface is advanced

Direct Coupling Method for Time-Accurate Solution of Incompressible Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Soh, Woo Y.

1992-01-01

A noniterative finite difference numerical method is presented for the solution of the incompressible Navier-Stokes equations with second order accuracy in time and space. Explicit treatment of convection and diffusion terms and implicit treatment of the pressure gradient give a single pressure Poisson equation when the discretized momentum and continuity equations are combined. A pressure boundary condition is not needed on solid boundaries in the staggered mesh system. The solution of the pressure Poisson equation is obtained directly by Gaussian elimination. This method is tested on flow problems in a driven cavity and a curved duct.

Three-Dimensional High-Order Spectral Finite Volume Method for Unstructured Grids

NASA Technical Reports Server (NTRS)

Liu, Yen; Vinokur, Marcel; Wang, Z. J.; Kwak, Dochan (Technical Monitor)

2002-01-01

Many areas require a very high-order accurate numerical solution of conservation laws for complex shapes. This paper deals with the extension to three dimensions of the Spectral Finite Volume (SV) method for unstructured grids, which was developed to solve such problems. We first summarize the limitations of traditional methods such as finite-difference, and finite-volume for both structured and unstructured grids. We then describe the basic formulation of the spectral finite volume method. What distinguishes the SV method from conventional high-order finite-volume methods for unstructured triangular or tetrahedral grids is the data reconstruction. Instead of using a large stencil of neighboring cells to perform a high-order reconstruction, the stencil is constructed by partitioning each grid cell, called a spectral volume (SV), into 'structured' sub-cells, called control volumes (CVs). One can show that if all the SV cells are partitioned into polygonal or polyhedral CV sub-cells in a geometrically similar manner, the reconstructions for all the SVs become universal, irrespective of their shapes, sizes, orientations, or locations. It follows that the reconstruction is reduced to a weighted sum of unknowns involving just a few simple adds and multiplies, and those weights are universal and can be pre-determined once for all. The method is thus very efficient, accurate, and yet geometrically flexible. The most critical part of the SV method is the partitioning of the SV into CVs. In this paper we present the partitioning of a tetrahedral SV into polyhedral CVs with one free parameter for polynomial reconstructions up to degree of precision five. (Note that the order of accuracy of the method is one order higher than the reconstruction degree of precision.) The free parameter will be determined by minimizing the Lebesgue constant of the reconstruction matrix or similar criteria to obtain optimized partitions. The details of an efficient, parallelizable code to solve three-dimensional problems for any order of accuracy are then presented. Important aspects of the data structure are discussed. Comparisons with the Discontinuous Galerkin (DG) method are made. Numerical examples for wave propagation problems are presented.

A finite-difference method for the variable coefficient Poisson equation on hierarchical Cartesian meshes

NASA Astrophysics Data System (ADS)

Raeli, Alice; Bergmann, Michel; Iollo, Angelo

2018-02-01

We consider problems governed by a linear elliptic equation with varying coefficients across internal interfaces. The solution and its normal derivative can undergo significant variations through these internal boundaries. We present a compact finite-difference scheme on a tree-based adaptive grid that can be efficiently solved using a natively parallel data structure. The main idea is to optimize the truncation error of the discretization scheme as a function of the local grid configuration to achieve second-order accuracy. Numerical illustrations are presented in two and three-dimensional configurations.

Subresolution Displacements in Finite Difference Simulations of Ultrasound Propagation and Imaging.

PubMed

Pinton, Gianmarco F

2017-03-01

Time domain finite difference simulations are used extensively to simulate wave propagation. They approximate the wave field on a discrete domain with a grid spacing that is typically on the order of a tenth of a wavelength. The smallest displacements that can be modeled by this type of simulation are thus limited to discrete values that are integer multiples of the grid spacing. This paper presents a method to represent continuous and subresolution displacements by varying the impedance of individual elements in a multielement scatterer. It is demonstrated that this method removes the limitations imposed by the discrete grid spacing by generating a continuum of displacements as measured by the backscattered signal. The method is first validated on an ideal perfect correlation case with a single scatterer. It is subsequently applied to a more complex case with a field of scatterers that model an acoustic radiation force-induced displacement used in ultrasound elasticity imaging. A custom finite difference simulation tool is used to simulate propagation from ultrasound imaging pulses in the scatterer field. These simulated transmit-receive events are then beamformed into images, which are tracked with a correlation-based algorithm to determine the displacement. A linear predictive model is developed to analytically describe the relationship between element impedance and backscattered phase shift. The error between model and simulation is λ/ 1364 , where λ is the acoustical wavelength. An iterative method is also presented that reduces the simulation error to λ/ 5556 over one iteration. The proposed technique therefore offers a computationally efficient method to model continuous subresolution displacements of a scattering medium in ultrasound imaging. This method has applications that include ultrasound elastography, blood flow, and motion tracking. This method also extends generally to finite difference simulations of wave propagation, such as electromagnetic or seismic waves.

Three-dimensional long-period groundmotion simulations in the upper Mississippi embayment

USGS Publications Warehouse

Macpherson, K.A.; Woolery, E.W.; Wang, Z.; Liu, P.

2010-01-01

We employed a 3D velocity model and 3D wave propagation code to simulate long-period ground motions in the upper Mississippi embayment. This region is at risk from large earthquakes in the New Madrid seismic zone (NMSZ) and observational data are sparse, making simulation a valuable tool for predicting the effects of large events. We undertook these simulations to estimate the magnitude of shaking likely to occur and to investigate the influence of the 3D embayment structure and finite-fault mechanics on ground motions. There exist three primary fault zones in the NMSZ, each of which was likely associated with one of the main shocks of the 1811-12 earthquake triplet. For this study, three simulations have been conducted on each major segment, exploring the impact of different epicentral locations and rupture directions on ground motions. The full wave field up to a frequency of 0.5 Hz is computed on a 200 ?? 200 ?? 50-km 3 volume using a staggered-grid finite-difference code. Peak horizontal velocity and bracketed durations were calculated at the free surface. The NMSZ simulations indicate that for the considered bandwidth, finite-fault mechanics such as fault proximity, directivity effect, and slip distribution exert the most control on ground motions. The 3D geologic structure of the upper Mississippi embayment also influences ground motion with indications that amplification is induced by the sharp velocity contrast at the basin edge.

Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids II: Extension to Two Dimensional Scalar Equation

NASA Technical Reports Server (NTRS)

Wang, Z. J.; Liu, Yen; Kwak, Dochan (Technical Monitor)

2002-01-01

The framework for constructing a high-order, conservative Spectral (Finite) Volume (SV) method is presented for two-dimensional scalar hyperbolic conservation laws on unstructured triangular grids. Each triangular grid cell forms a spectral volume (SV), and the SV is further subdivided into polygonal control volumes (CVs) to supported high-order data reconstructions. Cell-averaged solutions from these CVs are used to reconstruct a high order polynomial approximation in the SV. Each CV is then updated independently with a Godunov-type finite volume method and a high-order Runge-Kutta time integration scheme. A universal reconstruction is obtained by partitioning all SVs in a geometrically similar manner. The convergence of the SV method is shown to depend on how a SV is partitioned. A criterion based on the Lebesgue constant has been developed and used successfully to determine the quality of various partitions. Symmetric, stable, and convergent linear, quadratic, and cubic SVs have been obtained, and many different types of partitions have been evaluated. The SV method is tested for both linear and non-linear model problems with and without discontinuities.

Numerical simulation and experiment on multilayer stagger-split die.

PubMed

Liu, Zhiwei; Li, Mingzhe; Han, Qigang; Yang, Yunfei; Wang, Bolong; Sui, Zhou

2013-05-01

A novel ultra-high pressure device, multilayer stagger-split die, has been constructed based on the principle of "dividing dies before cracking." Multilayer stagger-split die includes an encircling ring and multilayer assemblages, and the mating surfaces of the multilayer assemblages are mutually staggered between adjacent layers. In this paper, we investigated the stressing features of this structure through finite element techniques, and the results were compared with those of the belt type die and single split die. The contrast experiments were also carried out to test the bearing pressure performance of multilayer stagger-split die. It is concluded that the stress distributions are reasonable and the materials are utilized effectively for multilayer stagger-split die. And experiments indicate that the multilayer stagger-split die can bear the greatest pressure.

Multigrid for Staggered Lattice Fermions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Brower, Richard C.; Clark, M. A.; Strelchenko, Alexei

Critical slowing down in Krylov methods for the Dirac operator presents a major obstacle to further advances in lattice field theory as it approaches the continuum solution. Here we formulate a multi-grid algorithm for the Kogut-Susskind (or staggered) fermion discretization which has proven difficult relative to Wilson multigrid due to its first-order anti-Hermitian structure. The solution is to introduce a novel spectral transformation by the K\\"ahler-Dirac spin structure prior to the Galerkin projection. We present numerical results for the two-dimensional, two-flavor Schwinger model, however, the general formalism is agnostic to dimension and is directly applicable to four-dimensional lattice QCD.

Multi-grid finite element method used for enhancing the reconstruction accuracy in Cerenkov luminescence tomography

NASA Astrophysics Data System (ADS)

Guo, Hongbo; He, Xiaowei; Liu, Muhan; Zhang, Zeyu; Hu, Zhenhua; Tian, Jie

2017-03-01

Cerenkov luminescence tomography (CLT), as a promising optical molecular imaging modality, can be applied to cancer diagnostic and therapeutic. Most researches about CLT reconstruction are based on the finite element method (FEM) framework. However, the quality of FEM mesh grid is still a vital factor to restrict the accuracy of the CLT reconstruction result. In this paper, we proposed a multi-grid finite element method framework, which was able to improve the accuracy of reconstruction. Meanwhile, the multilevel scheme adaptive algebraic reconstruction technique (MLS-AART) based on a modified iterative algorithm was applied to improve the reconstruction accuracy. In numerical simulation experiments, the feasibility of our proposed method were evaluated. Results showed that the multi-grid strategy could obtain 3D spatial information of Cerenkov source more accurately compared with the traditional single-grid FEM.

Radiation boundary condition and anisotropy correction for finite difference solutions of the Helmholtz equation

NASA Technical Reports Server (NTRS)

Tam, Christopher K. W.; Webb, Jay C.

1994-01-01

In this paper finite-difference solutions of the Helmholtz equation in an open domain are considered. By using a second-order central difference scheme and the Bayliss-Turkel radiation boundary condition, reasonably accurate solutions can be obtained when the number of grid points per acoustic wavelength used is large. However, when a smaller number of grid points per wavelength is used excessive reflections occur which tend to overwhelm the computed solutions. Excessive reflections are due to the incompability between the governing finite difference equation and the Bayliss-Turkel radiation boundary condition. The Bayliss-Turkel radiation boundary condition was developed from the asymptotic solution of the partial differential equation. To obtain compatibility, the radiation boundary condition should be constructed from the asymptotic solution of the finite difference equation instead. Examples are provided using the improved radiation boundary condition based on the asymptotic solution of the governing finite difference equation. The computed results are free of reflections even when only five grid points per wavelength are used. The improved radiation boundary condition has also been tested for problems with complex acoustic sources and sources embedded in a uniform mean flow. The present method of developing a radiation boundary condition is also applicable to higher order finite difference schemes. In all these cases no reflected waves could be detected. The use of finite difference approximation inevita bly introduces anisotropy into the governing field equation. The effect of anisotropy is to distort the directional distribution of the amplitude and phase of the computed solution. It can be quite large when the number of grid points per wavelength used in the computation is small. A way to correct this effect is proposed. The correction factor developed from the asymptotic solutions is source independent and, hence, can be determined once and for all. The effectiveness of the correction factor in providing improvements to the computed solution is demonstrated in this paper.

Study of grid independence of finite element method on MHD free convective casson fluid flow with slip effect

NASA Astrophysics Data System (ADS)

Raju, R. Srinivasa; Ramesh, K.

2018-05-01

The purpose of this work is to study the grid independence of finite element method on MHD Casson fluid flow past a vertically inclined plate filled in a porous medium in presence of chemical reaction, heat absorption, an external magnetic field and slip effect has been investigated. For this study of grid independence, a mathematical model is developed and analyzed by using appropriate mathematical technique, called finite element method. Grid study discussed with the help of numerical values of velocity, temperature and concentration profiles in tabular form. avourable comparisons with previously published work on various special cases of the problem are obtained.

Critical line of 2+1 flavor QCD

NASA Astrophysics Data System (ADS)

Cea, Paolo; Cosmai, Leonardo; Papa, Alessandro

2014-04-01

We determine the curvature of the (pseudo)critical line of QCD with nf = 2 + 1 staggered fermions at nonzero temperature and quark density by analytic continuation from imaginary chemical potentials. Monte Carlo simulations are performed by adopting the highly improved staggered quarks /tree action discretization, as implemented in the code by the MILC Collaboration, suitably modified to include a nonzero imaginary baryon chemical potential. We work on a line of constant physics, as determined in Ref. [1], adjusting the couplings so as to keep the strange quark mass ms fixed at its physical value, with a light to strange mass ratio of ml/ms=1/20. In the present investigation, we set the chemical potential at the same value for the three quark species, μl=μs≡μ. We explore lattices of different spatial extensions, 163×6 and 243×6, to check for finite size effects, and present results on a 323×8 lattice, to check for finite cutoff effects. We discuss our results for the curvature κ of the (pseudo)critical line at μ =0, which indicate κ=0.018(4), and compare them with previous lattice determinations by alternative methods and with experimental determinations of the freeze-out curve.

TranAir: A full-potential, solution-adaptive, rectangular grid code for predicting subsonic, transonic, and supersonic flows about arbitrary configurations. Theory document

NASA Technical Reports Server (NTRS)

Johnson, F. T.; Samant, S. S.; Bieterman, M. B.; Melvin, R. G.; Young, D. P.; Bussoletti, J. E.; Hilmes, C. L.

1992-01-01

A new computer program, called TranAir, for analyzing complex configurations in transonic flow (with subsonic or supersonic freestream) was developed. This program provides accurate and efficient simulations of nonlinear aerodynamic flows about arbitrary geometries with the ease and flexibility of a typical panel method program. The numerical method implemented in TranAir is described. The method solves the full potential equation subject to a set of general boundary conditions and can handle regions with differing total pressure and temperature. The boundary value problem is discretized using the finite element method on a locally refined rectangular grid. The grid is automatically constructed by the code and is superimposed on the boundary described by networks of panels; thus no surface fitted grid generation is required. The nonlinear discrete system arising from the finite element method is solved using a preconditioned Krylov subspace method embedded in an inexact Newton method. The solution is obtained on a sequence of successively refined grids which are either constructed adaptively based on estimated solution errors or are predetermined based on user inputs. Many results obtained by using TranAir to analyze aerodynamic configurations are presented.

Combination of the discontinuous Galerkin method with finite differences for simulation of seismic wave propagation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lisitsa, Vadim, E-mail: lisitsavv@ipgg.sbras.ru; Novosibirsk State University, Novosibirsk; Tcheverda, Vladimir

We present an algorithm for the numerical simulation of seismic wave propagation in models with a complex near surface part and free surface topography. The approach is based on the combination of finite differences with the discontinuous Galerkin method. The discontinuous Galerkin method can be used on polyhedral meshes; thus, it is easy to handle the complex surfaces in the models. However, this approach is computationally intense in comparison with finite differences. Finite differences are computationally efficient, but in general, they require rectangular grids, leading to the stair-step approximation of the interfaces, which causes strong diffraction of the wavefield. Inmore » this research we present a hybrid algorithm where the discontinuous Galerkin method is used in a relatively small upper part of the model and finite differences are applied to the main part of the model.« less

Discontinuous Spectral Difference Method for Conservation Laws on Unstructured Grids

NASA Technical Reports Server (NTRS)

Liu, Yen; Vinokur, Marcel

2004-01-01

A new, high-order, conservative, and efficient discontinuous spectral finite difference (SD) method for conservation laws on unstructured grids is developed. The concept of discontinuous and high-order local representations to achieve conservation and high accuracy is utilized in a manner similar to the Discontinuous Galerkin (DG) and the Spectral Volume (SV) methods, but while these methods are based on the integrated forms of the equations, the new method is based on the differential form to attain a simpler formulation and higher efficiency. Conventional unstructured finite-difference and finite-volume methods require data reconstruction based on the least-squares formulation using neighboring point or cell data. Since each unknown employs a different stencil, one must repeat the least-squares inversion for every point or cell at each time step, or to store the inversion coefficients. In a high-order, three-dimensional computation, the former would involve impractically large CPU time, while for the latter the memory requirement becomes prohibitive. In addition, the finite-difference method does not satisfy the integral conservation in general. By contrast, the DG and SV methods employ a local, universal reconstruction of a given order of accuracy in each cell in terms of internally defined conservative unknowns. Since the solution is discontinuous across cell boundaries, a Riemann solver is necessary to evaluate boundary flux terms and maintain conservation. In the DG method, a Galerkin finite-element method is employed to update the nodal unknowns within each cell. This requires the inversion of a mass matrix, and the use of quadratures of twice the order of accuracy of the reconstruction to evaluate the surface integrals and additional volume integrals for nonlinear flux functions. In the SV method, the integral conservation law is used to update volume averages over subcells defined by a geometrically similar partition of each grid cell. As the order of accuracy increases, the partitioning for 3D requires the introduction of a large number of parameters, whose optimization to achieve convergence becomes increasingly more difficult. Also, the number of interior facets required to subdivide non-planar faces, and the additional increase in the number of quadrature points for each facet, increases the computational cost greatly.

An RBF-FD closest point method for solving PDEs on surfaces

NASA Astrophysics Data System (ADS)

Petras, A.; Ling, L.; Ruuth, S. J.

2018-10-01

Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman (2008) [17]) is an embedding method for solving PDEs on surfaces using standard finite difference schemes. In this paper, we formulate an explicit closest point method using finite difference schemes derived from radial basis functions (RBF-FD). Unlike the orthogonal gradients method (Piret (2012) [22]), our proposed method uses RBF centers on regular grid nodes. This formulation not only reduces the computational cost but also avoids the ill-conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm (Leung and Zhao (2009) [26]). When compared to the standard finite difference discretization of the closest point method, the proposed method requires a smaller computational domain surrounding the surface, resulting in a decrease in the number of sampling points on the surface. In addition, higher-order schemes can easily be constructed by increasing the number of points in the RBF-FD stencil. Applications to a variety of examples are provided to illustrate the numerical convergence of the method.

A new ghost-node method for linking different models and initial investigations of heterogeneity and nonmatching grids

USGS Publications Warehouse

Dickinson, J.E.; James, S.C.; Mehl, S.; Hill, M.C.; Leake, S.A.; Zyvoloski, G.A.; Faunt, C.C.; Eddebbarh, A.-A.

2007-01-01

A flexible, robust method for linking parent (regional-scale) and child (local-scale) grids of locally refined models that use different numerical methods is developed based on a new, iterative ghost-node method. Tests are presented for two-dimensional and three-dimensional pumped systems that are homogeneous or that have simple heterogeneity. The parent and child grids are simulated using the block-centered finite-difference MODFLOW and control-volume finite-element FEHM models, respectively. The models are solved iteratively through head-dependent (child model) and specified-flow (parent model) boundary conditions. Boundary conditions for models with nonmatching grids or zones of different hydraulic conductivity are derived and tested against heads and flows from analytical or globally-refined models. Results indicate that for homogeneous two- and three-dimensional models with matched grids (integer number of child cells per parent cell), the new method is nearly as accurate as the coupling of two MODFLOW models using the shared-node method and, surprisingly, errors are slightly lower for nonmatching grids (noninteger number of child cells per parent cell). For heterogeneous three-dimensional systems, this paper compares two methods for each of the two sets of boundary conditions: external heads at head-dependent boundary conditions for the child model are calculated using bilinear interpolation or a Darcy-weighted interpolation; specified-flow boundary conditions for the parent model are calculated using model-grid or hydrogeologic-unit hydraulic conductivities. Results suggest that significantly more accurate heads and flows are produced when both Darcy-weighted interpolation and hydrogeologic-unit hydraulic conductivities are used, while the other methods produce larger errors at the boundary between the regional and local models. The tests suggest that, if posed correctly, the ghost-node method performs well. Additional testing is needed for highly heterogeneous systems. ?? 2007 Elsevier Ltd. All rights reserved.

The multidimensional Self-Adaptive Grid code, SAGE, version 2

NASA Technical Reports Server (NTRS)

Davies, Carol B.; Venkatapathy, Ethiraj

1995-01-01

This new report on Version 2 of the SAGE code includes all the information in the original publication plus all upgrades and changes to the SAGE code since that time. The two most significant upgrades are the inclusion of a finite-volume option and the ability to adapt and manipulate zonal-matching multiple-grid files. In addition, the original SAGE code has been upgraded to Version 1.1 and includes all options mentioned in this report, with the exception of the multiple grid option and its associated features. Since Version 2 is a larger and more complex code, it is suggested (but not required) that Version 1.1 be used for single-grid applications. This document contains all the information required to run both versions of SAGE. The formulation of the adaption method is described in the first section of this document. The second section is presented in the form of a user guide that explains the input and execution of the code. The third section provides many examples. Successful application of the SAGE code in both two and three dimensions for the solution of various flow problems has proven the code to be robust, portable, and simple to use. Although the basic formulation follows the method of Nakahashi and Deiwert, many modifications have been made to facilitate the use of the self-adaptive grid method for complex grid structures. Modifications to the method and the simple but extensive input options make this a flexible and user-friendly code. The SAGE code can accommodate two-dimensional and three-dimensional, finite-difference and finite-volume, single grid, and zonal-matching multiple grid flow problems.

Effect of Reynolds number and turbulence on airfoil aerodynamics at -90-degree incidence

NASA Technical Reports Server (NTRS)

Stremel, Paul M.

1994-01-01

A method has been developed for calculating the viscous flow about airfoils with and without deflected flaps at -90 deg incidence. This method provides for the solution of the unsteady incompressible Navier-Stokes equations by means of an implicit technique. The solution is calculated on a body-fitted computational mesh using a staggered-grid method. The vorticity is defined at the node points, and the velocity components are defined at the mesh-cell sides. The staggered-grid orientation provides for accurate representation of vorticity at the node points and the continuity equation at the mesh-cell centers. The method provides for the noniterative solution of the flowfield and satisfies the continuity equation to machine zero at each time step. The method is evaluated in terms of its stability to predict two-dimensional flow about an airfoil at -90-deg incidence for varying Reynolds number and laminar/turbulent models. The variations of the average loading and surface pressure distribution due to flap deflection, Reynolds number, and laminar or turbulent flow are presented and compared with experimental results. The comparisom indicate that the calculated drag and drag reduction caused by flap deflection and the calculated average surface pressure are in excellent agreement with the measured results at a similar Reynolds number.

A Note on Multigrid Theory for Non-nested Grids and/or Quadrature

NASA Technical Reports Server (NTRS)

Douglas, C. C.; Douglas, J., Jr.; Fyfe, D. E.

1996-01-01

We provide a unified theory for multilevel and multigrid methods when the usual assumptions are not present. For example, we do not assume that the solution spaces or the grids are nested. Further, we do not assume that there is an algebraic relationship between the linear algebra problems on different levels. What we provide is a computationally useful theory for adaptively changing levels. Theory is provided for multilevel correction schemes, nested iteration schemes, and one way (i.e., coarse to fine grid with no correction iterations) schemes. We include examples showing the applicability of this theory: finite element examples using quadrature in the matrix assembly and finite volume examples with non-nested grids. Our theory applies directly to other discretizations as well.

Implicit method for the computation of unsteady flows on unstructured grids

NASA Technical Reports Server (NTRS)

Venkatakrishnan, V.; Mavriplis, D. J.

1995-01-01

An implicit method for the computation of unsteady flows on unstructured grids is presented. Following a finite difference approximation for the time derivative, the resulting nonlinear system of equations is solved at each time step by using an agglomeration multigrid procedure. The method allows for arbitrarily large time steps and is efficient in terms of computational effort and storage. Inviscid and viscous unsteady flows are computed to validate the procedure. The issue of the mass matrix which arises with vertex-centered finite volume schemes is addressed. The present formulation allows the mass matrix to be inverted indirectly. A mesh point movement and reconnection procedure is described that allows the grids to evolve with the motion of bodies. As an example of flow over bodies in relative motion, flow over a multi-element airfoil system undergoing deployment is computed.

Test problems for inviscid transonic flow

NASA Technical Reports Server (NTRS)

Carlson, L. A.

1979-01-01

Solving of test problems with the TRANDES program is discussed. This method utilizes the full, inviscid, perturbation potential flow equation in a Cartesian grid system that is stretched to infinity. This equation is represented by a nonconservative system of finite difference equations that includes at supersonic points a rotated difference scheme and is solved by column relaxation. The solution usually starts from a zero perturbation potential on a very coarse grid (typically 13 by 7) followed by several grid halvings until a final solution is obtained on a fine grid (97 by 49).

Parallel Semi-Implicit Spectral Element Atmospheric Model

NASA Astrophysics Data System (ADS)

Fournier, A.; Thomas, S.; Loft, R.

2001-05-01

The shallow-water equations (SWE) have long been used to test atmospheric-modeling numerical methods. The SWE contain essential wave-propagation and nonlinear effects of more complete models. We present a semi-implicit (SI) improvement of the Spectral Element Atmospheric Model to solve the SWE (SEAM, Taylor et al. 1997, Fournier et al. 2000, Thomas & Loft 2000). SE methods are h-p finite element methods combining the geometric flexibility of size-h finite elements with the accuracy of degree-p spectral methods. Our work suggests that exceptional parallel-computation performance is achievable by a General-Circulation-Model (GCM) dynamical core, even at modest climate-simulation resolutions (>1o). The code derivation involves weak variational formulation of the SWE, Gauss(-Lobatto) quadrature over the collocation points, and Legendre cardinal interpolators. Appropriate weak variation yields a symmetric positive-definite Helmholtz operator. To meet the Ladyzhenskaya-Babuska-Brezzi inf-sup condition and avoid spurious modes, we use a staggered grid. The SI scheme combines leapfrog and Crank-Nicholson schemes for the nonlinear and linear terms respectively. The localization of operations to elements ideally fits the method to cache-based microprocessor computer architectures --derivatives are computed as collections of small (8x8), naturally cache-blocked matrix-vector products. SEAM also has desirable boundary-exchange communication, like finite-difference models. Timings on on the IBM SP and Compaq ES40 supercomputers indicate that the SI code (20-min timestep) requires 1/3 the CPU time of the explicit code (2-min timestep) for T42 resolutions. Both codes scale nearly linearly out to 400 processors. We achieved single-processor performance up to 30% of peak for both codes on the 375-MHz IBM Power-3 processors. Fast computation and linear scaling lead to a useful climate-simulation dycore only if enough model time is computed per unit wall-clock time. An efficient SI solver is essential to substantially increase this rate. Parallel preconditioning for an iterative conjugate-gradient elliptic solver is described. We are building a GCM dycore capable of 200 GF% lOPS sustained performance on clustered RISC/cache architectures using hybrid MPI/OpenMP programming.

Full Wave Analysis of Passive Microwave Monolithic Integrated Circuit Devices Using a Generalized Finite Difference Time Domain (GFDTD) Algorithm

NASA Technical Reports Server (NTRS)

Lansing, Faiza S.; Rascoe, Daniel L.

1993-01-01

This paper presents a modified Finite-Difference Time-Domain (FDTD) technique using a generalized conformed orthogonal grid. The use of the Conformed Orthogonal Grid, Finite Difference Time Domain (GFDTD) enables the designer to match all the circuit dimensions, hence eliminating a major source o error in the analysis.

Efficient discretization in finite difference method

NASA Astrophysics Data System (ADS)

Rozos, Evangelos; Koussis, Antonis; Koutsoyiannis, Demetris

2015-04-01

Finite difference method (FDM) is a plausible and simple method for solving partial differential equations. The standard practice is to use an orthogonal discretization to form algebraic approximate formulations of the derivatives of the unknown function and a grid, much like raster maps, to represent the properties of the function domain. For example, for the solution of the groundwater flow equation, a raster map is required for the characterization of the discretization cells (flow cell, no-flow cell, boundary cell, etc.), and two raster maps are required for the hydraulic conductivity and the storage coefficient. Unfortunately, this simple approach to describe the topology comes along with the known disadvantages of the FDM (rough representation of the geometry of the boundaries, wasted computational resources in the unavoidable expansion of the grid refinement in all cells of the same column and row, etc.). To overcome these disadvantages, Hunt has suggested an alternative approach to describe the topology, the use of an array of neighbours. This limits the need for discretization nodes only for the representation of the boundary conditions and the flow domain. Furthermore, the geometry of the boundaries is described more accurately using a vector representation. Most importantly, graded meshes can be employed, which are capable of restricting grid refinement only in the areas of interest (e.g. regions where hydraulic head varies rapidly, locations of pumping wells, etc.). In this study, we test the Hunt approach against MODFLOW, a well established finite difference model, and the Finite Volume Method with Simplified Integration (FVMSI). The results of this comparison are examined and critically discussed.

Comparison of Cartesian grid configurations for application of the finite-difference time-domain method to electromagnetic scattering by dielectric particles.

PubMed

Yang, Ping; Kattawar, George W; Liou, Kuo-Nan; Lu, Jun Q

2004-08-10

Two grid configurations can be employed to implement the finite-difference time-domain (FDTD) technique in a Cartesian system. One configuration defines the electric and magnetic field components at the cell edges and cell-face centers, respectively, whereas the other reverses these definitions. These two grid configurations differ in terms of implication on the electromagnetic boundary conditions if the scatterer in the FDTD computation is a dielectric particle. The permittivity has an abrupt transition at the cell interface if the dielectric properties of two adjacent cells are not identical. Similarly, the discontinuity of permittivity is also observed at the edges of neighboring cells that are different in terms of their dielectric constants. We present two FDTD schemes for light scattering by dielectric particles to overcome the above-mentioned discontinuity on the basis of the electromagnetic boundary conditions for the two Cartesian grid configurations. We also present an empirical approach to accelerate the convergence of the discrete Fourier transform to obtain the field values in the frequency domain. As a new application of the FDTD method, we investigate the scattering properties of multibranched bullet-rosette ice crystals at both visible and thermal infrared wavelengths.

Finite-Difference Numerical Simulation of Seismic Gradiometry

NASA Astrophysics Data System (ADS)

Aldridge, D. F.; Symons, N. P.; Haney, M. M.

2006-12-01

We use the phrase seismic gradiometry to refer to the developing research area involving measurement, modeling, analysis, and interpretation of spatial derivatives (or differences) of a seismic wavefield. In analogy with gradiometric methods used in gravity and magnetic exploration, seismic gradiometry offers the potential for enhancing resolution, and revealing new (or hitherto obscure) information about the subsurface. For example, measurement of pressure and rotation enables the decomposition of recorded seismic data into compressional (P) and shear (S) components. Additionally, a complete observation of the total seismic wavefield at a single receiver (including both rectilinear and rotational motions) offers the possibility of inferring the type, speed, and direction of an incident seismic wave. Spatially extended receiver arrays, conventionally used for such directional and phase speed determinations, may be dispensed with. Seismic wave propagation algorithms based on the explicit, time-domain, finite-difference (FD) numerical method are well-suited for investigating gradiometric effects. We have implemented in our acoustic, elastic, and poroelastic algorithms a point receiver that records the 9 components of the particle velocity gradient tensor. Pressure and particle rotation are obtained by forming particular linear combinations of these tensor components, and integrating with respect to time. All algorithms entail 3D O(2,4) FD solutions of coupled, first- order systems of partial differential equations on uniformly-spaced staggered spatial and temporal grids. Numerical tests with a 1D model composed of homogeneous and isotropic elastic layers show isolation of P, SV, and SH phases recorded in a multiple borehole configuration, even in the case of interfering events. Synthetic traces recorded by geophones and rotation receivers in a shallow crosswell geometry with randomly heterogeneous poroelastic models also illustrate clear P (fast and slow) and S separation. Finally, numerical tests of the "point seismic array" concept are oriented toward understanding its potential and limitations. Sandia National Laboratories is a multiprogram science and engineering facility operated by Sandia Corporation, a Lockheed-Martin company, for the United States Department of Energy under contract DE- AC04-94AL85000.

Phase diagrams and Hofstadter butterflies in the strongly correlated bosonic systems on the lattices with Dirac points

NASA Astrophysics Data System (ADS)

Sajna, A. S.; Polak, T. P.

2018-06-01

Gauge potentials with different configurations have been recently realized in the optical lattice experiments. It is remarkable that one of the simplest gauge potential can generate particle energy spectrum with the self-similar structure known as a Hofstadter butterfly. We investigate theoretically the impact of strong on-site interaction on such a spectrum in the bosonic Mott insulator within Bose-Hubbard model. In particular, it is shown that the fractal structure is encoded in the quasi-particle and hole bosonic branches for different lattice backgrounds. For example a square lattice and other structures (brick-wall and staggered magnetic flux lattice) which contain Dirac points in energy dispersions are considered. This shows that single-particle physics is still present even in the strong interaction limit for whole Hofstadter spectrum. Additionally we observe, that although in brick-wall and staggered flux lattices the quasi-particle densities of states look qualitatively similar, the corresponding Hofstadter butterfly assumes different forms. In particular, we use a superposition of two different synthetic gauge fields which appears to be a generator of non-trivial phenomena in the optical lattice systems. We also discuss the consequences of these phenomena on the phase diagrams between bosonic Mott insulator and superfluid phase. The analysis is carried out within the strong coupling expansion method on the finite size lattices and also at finite temperatures which are relevant for the currently made experiments.

A multigrid method for steady Euler equations on unstructured adaptive grids

NASA Technical Reports Server (NTRS)

Riemslagh, Kris; Dick, Erik

1993-01-01

A flux-difference splitting type algorithm is formulated for the steady Euler equations on unstructured grids. The polynomial flux-difference splitting technique is used. A vertex-centered finite volume method is employed on a triangular mesh. The multigrid method is in defect-correction form. A relaxation procedure with a first order accurate inner iteration and a second-order correction performed only on the finest grid, is used. A multi-stage Jacobi relaxation method is employed as a smoother. Since the grid is unstructured a Jacobi type is chosen. The multi-staging is necessary to provide sufficient smoothing properties. The domain is discretized using a Delaunay triangular mesh generator. Three grids with more or less uniform distribution of nodes but with different resolution are generated by successive refinement of the coarsest grid. Nodes of coarser grids appear in the finer grids. The multigrid method is started on these grids. As soon as the residual drops below a threshold value, an adaptive refinement is started. The solution on the adaptively refined grid is accelerated by a multigrid procedure. The coarser multigrid grids are generated by successive coarsening through point removement. The adaption cycle is repeated a few times. Results are given for the transonic flow over a NACA-0012 airfoil.

Applying time-reverse-imaging techniques to locate individual low-frequency earthquakes on the San Andreas fault near Cholame, California

NASA Astrophysics Data System (ADS)

Horstmann, T.; Harrington, R. M.; Cochran, E.; Shelly, D. R.

2013-12-01

Observations of non-volcanic tremor have become ubiquitous in recent years. In spite of the abundance of observations, locating tremor remains a difficult task because of the lack of distinctive phase arrivals. Here we use time-reverse-imaging techniques that do not require identifying phase arrivals to locate individual low-frequency-earthquakes (LFEs) within tremor episodes on the San Andreas fault near Cholame, California. Time windows of 1.5-second duration containing LFEs are selected from continuously recorded waveforms of the local seismic network filtered between 1-5 Hz. We propagate the time-reversed seismic signal back through the subsurface using a staggered-grid finite-difference code. Assuming all rebroadcasted waveforms result from similar wave fields at the source origin, we search for wave field coherence in time and space to obtain the source location and origin time where the constructive interference is a maximum. We use an interpolated velocity model with a grid spacing of 100 m and a 5 ms time step to calculate the relative curl field energy amplitudes for each rebroadcasted seismogram every 50 ms for each grid point in the model. Finally, we perform a grid search for coherency in the curl field using a sliding time window, and taking the absolute value of the correlation coefficient to account for differences in radiation pattern. The highest median cross-correlation coefficient value over at a given grid point indicates the source location for the rebroadcasted event. Horizontal location errors based on the spatial extent of the highest 10% cross-correlation coefficient are on the order of 4 km, and vertical errors on the order of 3 km. Furthermore, a test of the method using earthquake data shows that the method produces an identical hypocentral location (within errors) as that obtained by standard ray-tracing methods. We also compare the event locations to a LFE catalog that locates the LFEs from stacked waveforms of repeated LFEs identified by cross-correlation techniques [Shelly and Hardebeck, 2010]. The LFE catalog uses stacks of at least several hundred templates to identify phase arrivals used to estimate the location. We find epicentral locations for individual LFEs based on the time-reverse-imaging technique are within ~4 km relative to the LFE catalog [Shelly and Hardebeck, 2010]. LFEs locate between 15-25 km depth, and have similar focal depths found in previous studies of the region. Overall, the method can provide robust locations of individual LFEs without identifying and stacking hundreds of LFE templates; the locations are also more accurate than envelope location methods, which have errors on the order of tens of km [Horstmann et al., 2013].

Unstructured Cartesian refinement with sharp interface immersed boundary method for 3D unsteady incompressible flows

NASA Astrophysics Data System (ADS)

Angelidis, Dionysios; Chawdhary, Saurabh; Sotiropoulos, Fotis

2016-11-01

A novel numerical method is developed for solving the 3D, unsteady, incompressible Navier-Stokes equations on locally refined fully unstructured Cartesian grids in domains with arbitrarily complex immersed boundaries. Owing to the utilization of the fractional step method on an unstructured Cartesian hybrid staggered/non-staggered grid layout, flux mismatch and pressure discontinuity issues are avoided and the divergence free constraint is inherently satisfied to machine zero. Auxiliary/hanging nodes are used to facilitate the discretization of the governing equations. The second-order accuracy of the solver is ensured by using multi-dimension Lagrange interpolation operators and appropriate differencing schemes at the interface of regions with different levels of refinement. The sharp interface immersed boundary method is augmented with local near-boundary refinement to handle arbitrarily complex boundaries. The discrete momentum equation is solved with the matrix free Newton-Krylov method and the Krylov-subspace method is employed to solve the Poisson equation. The second-order accuracy of the proposed method on unstructured Cartesian grids is demonstrated by solving the Poisson equation with a known analytical solution. A number of three-dimensional laminar flow simulations of increasing complexity illustrate the ability of the method to handle flows across a range of Reynolds numbers and flow regimes. Laminar steady and unsteady flows past a sphere and the oblique vortex shedding from a circular cylinder mounted between two end walls demonstrate the accuracy, the efficiency and the smooth transition of scales and coherent structures across refinement levels. Large-eddy simulation (LES) past a miniature wind turbine rotor, parameterized using the actuator line approach, indicates the ability of the fully unstructured solver to simulate complex turbulent flows. Finally, a geometry resolving LES of turbulent flow past a complete hydrokinetic turbine illustrates the potential of the method to simulate turbulent flows past geometrically complex bodies on locally refined meshes. In all the cases, the results are found to be in very good agreement with published data and savings in computational resources are achieved.

Noniterative three-dimensional grid generation using parabolic partial differential equations

NASA Technical Reports Server (NTRS)

Edwards, T. A.

1985-01-01

A new algorithm for generating three-dimensional grids has been developed and implemented which numerically solves a parabolic partial differential equation (PDE). The solution procedure marches outward in two coordinate directions, and requires inversion of a scalar tridiagonal system in the third. Source terms have been introduced to control the spacing and angle of grid lines near the grid boundaries, and to control the outer boundary point distribution. The method has been found to generate grids about 100 times faster than comparable grids generated via solution of elliptic PDEs, and produces smooth grids for finite-difference flow calculations.

A 3-D enlarged cell technique (ECT) for elastic wave modelling of a curved free surface

NASA Astrophysics Data System (ADS)

Wei, Songlin; Zhou, Jianyang; Zhuang, Mingwei; Liu, Qing Huo

2016-09-01

The conventional finite-difference time-domain (FDTD) method for elastic waves suffers from the staircasing error when applied to model a curved free surface because of its structured grid. In this work, an improved, stable and accurate 3-D FDTD method for elastic wave modelling on a curved free surface is developed based on the finite volume method and enlarged cell technique (ECT). To achieve a sufficiently accurate implementation, a finite volume scheme is applied to the curved free surface to remove the staircasing error; in the mean time, to achieve the same stability as the FDTD method without reducing the time step increment, the ECT is introduced to preserve the solution stability by enlarging small irregular cells into adjacent cells under the condition of conservation of force. This method is verified by several 3-D numerical examples. Results show that the method is stable at the Courant stability limit for a regular FDTD grid, and has much higher accuracy than the conventional FDTD method.

Mixed finite-difference scheme for free vibration analysis of noncircular cylinders

NASA Technical Reports Server (NTRS)

Noor, A. K.; Stephens, W. B.

1973-01-01

A mixed finite-difference scheme is presented for the free-vibration analysis of simply supported closed noncircular cylindrical shells. The problem is formulated in terms of eight first-order differential equations in the circumferential coordinate which possess a symmetric coefficient matrix and are free of the derivatives of the elastic and geometric characteristics of the shell. In the finite-difference discretization, two interlacing grids are used for the different fundamental unknowns in such a way as to avoid averaging in the difference-quotient expressions used for the first derivative. The resulting finite-difference equations are symmetric. The inverse-power method is used for obtaining the eigenvalues and eigenvectors.

Semi-implicit integration factor methods on sparse grids for high-dimensional systems

NASA Astrophysics Data System (ADS)

Wang, Dongyong; Chen, Weitao; Nie, Qing

2015-07-01

Numerical methods for partial differential equations in high-dimensional spaces are often limited by the curse of dimensionality. Though the sparse grid technique, based on a one-dimensional hierarchical basis through tensor products, is popular for handling challenges such as those associated with spatial discretization, the stability conditions on time step size due to temporal discretization, such as those associated with high-order derivatives in space and stiff reactions, remain. Here, we incorporate the sparse grids with the implicit integration factor method (IIF) that is advantageous in terms of stability conditions for systems containing stiff reactions and diffusions. We combine IIF, in which the reaction is treated implicitly and the diffusion is treated explicitly and exactly, with various sparse grid techniques based on the finite element and finite difference methods and a multi-level combination approach. The overall method is found to be efficient in terms of both storage and computational time for solving a wide range of PDEs in high dimensions. In particular, the IIF with the sparse grid combination technique is flexible and effective in solving systems that may include cross-derivatives and non-constant diffusion coefficients. Extensive numerical simulations in both linear and nonlinear systems in high dimensions, along with applications of diffusive logistic equations and Fokker-Planck equations, demonstrate the accuracy, efficiency, and robustness of the new methods, indicating potential broad applications of the sparse grid-based integration factor method.

Grid orthogonality effects on predicted turbine midspan heat transfer and performance

NASA Technical Reports Server (NTRS)

Boyle, R. J.; Ameri, A. A.

1995-01-01

The effect of five different C type grid geometries on the predicted heat transfer and aerodynamic performance of a turbine stator is examined. Predictions were obtained using two flow analysis codes. One was a finite difference analysis, and the other was a finite volume analysis. Differences among the grids in terms of heat transfer and overall performance were small. The most significant difference among the five grids occurred in the prediction of pitchwise variation in total pressure. There was consistency between results obtained with each of the flow analysis codes when the same grid was used. A grid generating procedure in which the viscous grid is embedded within an inviscid type grid resulted in the best overall performance.

Computation of the acoustic radiation force using the finite-difference time-domain method.

PubMed

Cai, Feiyan; Meng, Long; Jiang, Chunxiang; Pan, Yu; Zheng, Hairong

2010-10-01

The computational details related to calculating the acoustic radiation force on an object using a 2-D grid finite-difference time-domain method (FDTD) are presented. The method is based on propagating the stress and velocity fields through the grid and determining the energy flow with and without the object. The axial and radial acoustic radiation forces predicted by FDTD method are in excellent agreement with the results obtained by analytical evaluation of the scattering method. In particular, the results indicate that it is possible to trap the steel cylinder in the radial direction by optimizing the width of Gaussian source and the operation frequency. As the sizes of the relating objects are smaller than or comparable to wavelength, the algorithm presented here can be easily extended to 3-D and include torque computation algorithms, thus providing a highly flexible and universally usable computation engine.

New density estimation methods for charged particle beams with applications to microbunching instability

NASA Astrophysics Data System (ADS)

Terzić, Balša; Bassi, Gabriele

2011-07-01

In this paper we discuss representations of charge particle densities in particle-in-cell simulations, analyze the sources and profiles of the intrinsic numerical noise, and present efficient methods for their removal. We devise two alternative estimation methods for charged particle distribution which represent significant improvement over the Monte Carlo cosine expansion used in the 2D code of Bassi et al. [G. Bassi, J. A. Ellison, K. Heinemann, and R. Warnock, Phys. Rev. ST Accel. Beams 12, 080704 (2009); PRABFM1098-440210.1103/PhysRevSTAB.12.080704G. Bassi and B. Terzić, in Proceedings of the 23rd Particle Accelerator Conference, Vancouver, Canada, 2009 (IEEE, Piscataway, NJ, 2009), TH5PFP043], designed to simulate coherent synchrotron radiation (CSR) in charged particle beams. The improvement is achieved by employing an alternative beam density estimation to the Monte Carlo cosine expansion. The representation is first binned onto a finite grid, after which two grid-based methods are employed to approximate particle distributions: (i) truncated fast cosine transform; and (ii) thresholded wavelet transform (TWT). We demonstrate that these alternative methods represent a staggering upgrade over the original Monte Carlo cosine expansion in terms of efficiency, while the TWT approximation also provides an appreciable improvement in accuracy. The improvement in accuracy comes from a judicious removal of the numerical noise enabled by the wavelet formulation. The TWT method is then integrated into the CSR code [G. Bassi, J. A. Ellison, K. Heinemann, and R. Warnock, Phys. Rev. ST Accel. Beams 12, 080704 (2009)PRABFM1098-440210.1103/PhysRevSTAB.12.080704], and benchmarked against the original version. We show that the new density estimation method provides a superior performance in terms of efficiency and spatial resolution, thus enabling high-fidelity simulations of CSR effects, including microbunching instability.

Grid sensitivity capability for large scale structures

NASA Technical Reports Server (NTRS)

Nagendra, Gopal K.; Wallerstein, David V.

1989-01-01

The considerations and the resultant approach used to implement design sensitivity capability for grids into a large scale, general purpose finite element system (MSC/NASTRAN) are presented. The design variables are grid perturbations with a rather general linking capability. Moreover, shape and sizing variables may be linked together. The design is general enough to facilitate geometric modeling techniques for generating design variable linking schemes in an easy and straightforward manner. Test cases have been run and validated by comparison with the overall finite difference method. The linking of a design sensitivity capability for shape variables in MSC/NASTRAN with an optimizer would give designers a powerful, automated tool to carry out practical optimization design of real life, complicated structures.

Effect of surface-related Rayleigh and multiple waves on velocity reconstruction with time-domain elastic FWI

NASA Astrophysics Data System (ADS)

Fang, Jinwei; Zhou, Hui; Zhang, Qingchen; Chen, Hanming; Wang, Ning; Sun, Pengyuan; Wang, Shucheng

2018-01-01

It is critically important to assess the effectiveness of elastic full waveform inversion (FWI) algorithms when FWI is applied to real land seismic data including strong surface and multiple waves related to the air-earth boundary. In this paper, we review the realization of the free surface boundary condition in staggered-grid finite-difference (FD) discretization of elastic wave equation, and analyze the impact of the free surface on FWI results. To reduce inputs/outputs (I/O) operations in gradient calculation, we adopt the boundary value reconstruction method to rebuild the source wavefields during the backward propagation of the residual data. A time-domain multiscale inversion strategy is conducted by using a convolutional objective function, and a multi-GPU parallel programming technique is used to accelerate our elastic FWI further. Forward simulation and elastic FWI examples without and with considering the free surface are shown and analyzed, respectively. Numerical results indicate that no free surface incorporated elastic FWI fails to recover a good inversion result from the Rayleigh wave contaminated observed data. By contrast, when the free surface is incorporated into FWI, the inversion results become better. We also discuss the dependency of the Rayleigh waveform incorporated FWI on the accuracy of initial models, especially the accuracy of the shallow part of the initial models.

Definition of NASTRAN sets by use of parametric geometry

NASA Technical Reports Server (NTRS)

Baughn, Terry V.; Tiv, Mehran

1989-01-01

Many finite element preprocessors describe finite element model geometry with points, lines, surfaces and volumes. One method for describing these basic geometric entities is by use of parametric cubics which are useful for representing complex shapes. The lines, surfaces and volumes may be discretized for follow on finite element analysis. The ability to limit or selectively recover results from the finite element model is extremely important to the analyst. Equally important is the ability to easily apply boundary conditions. Although graphical preprocessors have made these tasks easier, model complexity may not lend itself to easily identify a group of grid points desired for data recovery or application of constraints. A methodology is presented which makes use of the assignment of grid point locations in parametric coordinates. The parametric coordinates provide a convenient ordering of the grid point locations and a method for retrieving the grid point ID's from the parent geometry. The selected grid points may then be used for the generation of the appropriate set and constraint cards.

Collocated electrodynamic FDTD schemes using overlapping Yee grids and higher-order Hodge duals

NASA Astrophysics Data System (ADS)

Deimert, C.; Potter, M. E.; Okoniewski, M.

2016-12-01

The collocated Lebedev grid has previously been proposed as an alternative to the Yee grid for electromagnetic finite-difference time-domain (FDTD) simulations. While it performs better in anisotropic media, it performs poorly in isotropic media because it is equivalent to four overlapping, uncoupled Yee grids. We propose to couple the four Yee grids and fix the Lebedev method using discrete exterior calculus (DEC) with higher-order Hodge duals. We find that higher-order Hodge duals do improve the performance of the Lebedev grid, but they also improve the Yee grid by a similar amount. The effectiveness of coupling overlapping Yee grids with a higher-order Hodge dual is thus questionable. However, the theoretical foundations developed to derive these methods may be of interest in other problems.

Multiscale Approach For Simulating Nonlinear Wave Propagation In Materials with Localized Microdamage

NASA Astrophysics Data System (ADS)

Vanaverbeke, Sigfried; Van Den Abeele, Koen

2006-05-01

A multiscale model for the simulation of two-dimensional nonlinear wave propagation in microcracked materials exhibiting hysteretic nonlinearity is presented. We use trigger-like elements with a two state nonlinear stress-strain relation to simulate microcracks at the microlevel. A generalized Preisach space approach, based on the eigenstress-eigenstrain formulation, upscales the microscopic state relation to the mesoscopic level. The macroscopic response of the sample to an arbitrary excitation signal is then predicted using a staggered grid Elastodynamic Finite Integration Technique (EFIT) formalism. We apply the model to investigate spectral changes of a pulsed signal traversing a localized microdamaged region with hysteretic nonlinearity in a plate, and to study the influence of a superficial region with hysteretic nonlinearity on the nonlinear Rayleigh wave propagation.

Deformation of two-phase aggregates using standard numerical methods

NASA Astrophysics Data System (ADS)

Duretz, Thibault; Yamato, Philippe; Schmalholz, Stefan M.

2013-04-01

Geodynamic problems often involve the large deformation of material encompassing material boundaries. In geophysical fluids, such boundaries often coincide with a discontinuity in the viscosity (or effective viscosity) field and subsequently in the pressure field. Here, we employ popular implementations of the finite difference and finite element methods for solving viscous flow problems. On one hand, we implemented finite difference method coupled with a Lagrangian marker-in-cell technique to represent the deforming fluid. Thanks to it Eulerian nature, this method has a limited geometric flexibility but is characterized by a light and stable discretization. On the other hand, we employ the Lagrangian finite element method which offers full geometric flexibility at the cost of relatively heavier discretization. In order to test the accuracy of the finite difference scheme, we ran large strain simple shear deformation of aggregates containing either weak of strong circular inclusion (1e6 viscosity ratio). The results, obtained for different grid resolutions, are compared to Lagrangian finite element results which are considered as reference solution. The comparison is then used to establish up to which strain can finite difference simulations be run given the nature of the inclusions (dimensions, viscosity) and the resolution of the Eulerian mesh.

Sharp Boundary Inversion of 2D Magnetotelluric Data using Bayesian Method.

NASA Astrophysics Data System (ADS)

Zhou, S.; Huang, Q.

2017-12-01

Normally magnetotelluric(MT) inversion method cannot show the distribution of underground resistivity with clear boundary, even if there are obviously different blocks. Aiming to solve this problem, we develop a Bayesian structure to inverse 2D MT sharp boundary data, using boundary location and inside resistivity as the random variables. Firstly, we use other MT inversion results, like ModEM, to analyze the resistivity distribution roughly. Then, we select the suitable random variables and change its data format to traditional staggered grid parameters, which can be used to do finite difference forward part. Finally, we can shape the posterior probability density(PPD), which contains all the prior information and model-data correlation, by Markov Chain Monte Carlo(MCMC) sampling from prior distribution. The depth, resistivity and their uncertainty can be valued. It also works for sensibility estimation. We applied the method to a synthetic case, which composes two large abnormal blocks in a trivial background. We consider the boundary smooth and the near true model weight constrains that mimic joint inversion or constrained inversion, then we find that the model results a more precise and focused depth distribution. And we also test the inversion without constrains and find that the boundary could also be figured, though not as well. Both inversions have a good valuation of resistivity. The constrained result has a lower root mean square than ModEM inversion result. The data sensibility obtained via PPD shows that the resistivity is the most sensible, center depth comes second and both sides are the worst.

Finite-Difference Algorithm for Simulating 3D Electromagnetic Wavefields in Conductive Media

NASA Astrophysics Data System (ADS)

Aldridge, D. F.; Bartel, L. C.; Knox, H. A.

2013-12-01

Electromagnetic (EM) wavefields are routinely used in geophysical exploration for detection and characterization of subsurface geological formations of economic interest. Recorded EM signals depend strongly on the current conductivity of geologic media. Hence, they are particularly useful for inferring fluid content of saturated porous bodies. In order to enhance understanding of field-recorded data, we are developing a numerical algorithm for simulating three-dimensional (3D) EM wave propagation and diffusion in heterogeneous conductive materials. Maxwell's equations are combined with isotropic constitutive relations to obtain a set of six, coupled, first-order partial differential equations governing the electric and magnetic vectors. An advantage of this system is that it does not contain spatial derivatives of the three medium parameters electric permittivity, magnetic permeability, and current conductivity. Numerical solution methodology consists of explicit, time-domain finite-differencing on a 3D staggered rectangular grid. Temporal and spatial FD operators have order 2 and N, where N is user-selectable. We use an artificially-large electric permittivity to maximize the FD timestep, and thus reduce execution time. For the low frequencies typically used in geophysical exploration, accuracy is not unduly compromised. Grid boundary reflections are mitigated via convolutional perfectly matched layers (C-PMLs) imposed at the six grid flanks. A shared-memory-parallel code implementation via OpenMP directives enables rapid algorithm execution on a multi-thread computational platform. Good agreement is obtained in comparisons of numerically-generated data with reference solutions. EM wavefields are sourced via point current density and magnetic dipole vectors. Spatially-extended inductive sources (current carrying wire loops) are under development. We are particularly interested in accurate representation of high-conductivity sub-grid-scale features that are common in industrial environments (borehole casing, pipes, railroad tracks). Present efforts are oriented toward calculating the EM responses of these objects via a First Born Approximation approach. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the US Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

Finite-volume effects and the electromagnetic contributions to kaon and pion masses

DOE Office of Scientific and Technical Information (OSTI.GOV)

Basak, Subhasish; Bazavov, Alexei; Bernard, Claude

2014-09-25

We report on the MILC Collaboration calculation of electromagnetic effects on light pseudoscalar mesons. The simulations employ asqtad staggered dynamical quarks in QCD plus quenched photons, with lattice spacings varying from 0.12 to 0.06 fm. Finite volume corrections for the MILC realization of lattice electrodynamics have been calculated in chiral perturbation theory and applied to the lattice data. These corrections differ from those calculated by Hayakawa and Uno because our treatment of zero modes differs from theirs. Updated results for the corrections to "Dashen's theorem" are presented.

An Immersed Boundary - Adaptive Mesh Refinement solver (IB-AMR) for high fidelity fully resolved wind turbine simulations

NASA Astrophysics Data System (ADS)

Angelidis, Dionysios; Sotiropoulos, Fotis

2015-11-01

The geometrical details of wind turbines determine the structure of the turbulence in the near and far wake and should be taken in account when performing high fidelity calculations. Multi-resolution simulations coupled with an immersed boundary method constitutes a powerful framework for high-fidelity calculations past wind farms located over complex terrains. We develop a 3D Immersed-Boundary Adaptive Mesh Refinement flow solver (IB-AMR) which enables turbine-resolving LES of wind turbines. The idea of using a hybrid staggered/non-staggered grid layout adopted in the Curvilinear Immersed Boundary Method (CURVIB) has been successfully incorporated on unstructured meshes and the fractional step method has been employed. The overall performance and robustness of the second order accurate, parallel, unstructured solver is evaluated by comparing the numerical simulations against conforming grid calculations and experimental measurements of laminar and turbulent flows over complex geometries. We also present turbine-resolving multi-scale LES considering all the details affecting the induced flow field; including the geometry of the tower, the nacelle and especially the rotor blades of a wind tunnel scale turbine. This material is based upon work supported by the Department of Energy under Award Number DE-EE0005482 and the Sandia National Laboratories.

Sub-basalt Imaging of Hydrocarbon-Bearing Mesozoic Sediments Using Ray-Trace Inversion of First-Arrival Seismic Data and Elastic Finite-Difference Full-Wave Modeling Along Sinor-Valod Profile of Deccan Syneclise, India

NASA Astrophysics Data System (ADS)

Talukdar, Karabi; Behera, Laxmidhar

2018-03-01

Imaging below the basalt for hydrocarbon exploration is a global problem because of poor penetration and significant loss of seismic energy due to scattering, attenuation, absorption and mode-conversion when the seismic waves encounter a highly heterogeneous and rugose basalt layer. The conventional (short offset) seismic data acquisition, processing and modeling techniques adopted by the oil industry generally fails to image hydrocarbon-bearing sub-trappean Mesozoic sediments hidden below the basalt and is considered as a serious problem for hydrocarbon exploration in the world. To overcome this difficulty of sub-basalt imaging, we have generated dense synthetic seismic data with the help of elastic finite-difference full-wave modeling using staggered-grid scheme for the model derived from ray-trace inversion using sparse wide-angle seismic data acquired along Sinor-Valod profile in the Deccan Volcanic Province of India. The full-wave synthetic seismic data generated have been processed and imaged using conventional seismic data processing technique with Kirchhoff pre-stack time and depth migrations. The seismic image obtained correlates with all the structural features of the model obtained through ray-trace inversion of wide-angle seismic data, validating the effectiveness of robust elastic finite-difference full-wave modeling approach for imaging below thick basalts. Using the full-wave modeling also allows us to decipher small-scale heterogeneities imposed in the model as a measure of the rugose basalt interfaces, which could not be dealt with ray-trace inversion. Furthermore, we were able to accurately image thin low-velocity hydrocarbon-bearing Mesozoic sediments sandwiched between and hidden below two thick sequences of high-velocity basalt layers lying above the basement.

Finite difference methods for transient signal propagation in stratified dispersive media

NASA Technical Reports Server (NTRS)

Lam, D. H.

1975-01-01

Explicit difference equations are presented for the solution of a signal of arbitrary waveform propagating in an ohmic dielectric, a cold plasma, a Debye model dielectric, and a Lorentz model dielectric. These difference equations are derived from the governing time-dependent integro-differential equations for the electric fields by a finite difference method. A special difference equation is derived for the grid point at the boundary of two different media. Employing this difference equation, transient signal propagation in an inhomogeneous media can be solved provided that the medium is approximated in a step-wise fashion. The solutions are generated simply by marching on in time. It is concluded that while the classical transform methods will remain useful in certain cases, with the development of the finite difference methods described, an extensive class of problems of transient signal propagating in stratified dispersive media can be effectively solved by numerical methods.

Solution of Poisson equations for 3-dimensional grid generations. [computations of a flow field over a thin delta wing

NASA Technical Reports Server (NTRS)

Fujii, K.

1983-01-01

A method for generating three dimensional, finite difference grids about complicated geometries by using Poisson equations is developed. The inhomogenous terms are automatically chosen such that orthogonality and spacing restrictions at the body surface are satisfied. Spherical variables are used to avoid the axis singularity, and an alternating-direction-implicit (ADI) solution scheme is used to accelerate the computations. Computed results are presented that show the capability of the method. Since most of the results presented have been used as grids for flow-field computations, this is indicative that the method is a useful tool for generating three-dimensional grids about complicated geometries.

Coupling of Peridynamics and Finite Element Formulation for Multiscale Simulations

DTIC Science & Technology

2012-10-16

unidirectional fiber - reinforced composites, Computer Methods in Applied Mechanics and Engineering 217 (2012) 247-261. [44] S. A. Silling, M. Epton...numerical testing for different grid widths to horizon ratios , (4) development of an approach to add another material variable in the given approach...partition of unity principle, (3) numerical testing for different grid widths to horizon ratios , (4) development of an approach to add another

OpenMP performance for benchmark 2D shallow water equations using LBM

NASA Astrophysics Data System (ADS)

Sabri, Khairul; Rabbani, Hasbi; Gunawan, Putu Harry

2018-03-01

Shallow water equations or commonly referred as Saint-Venant equations are used to model fluid phenomena. These equations can be solved numerically using several methods, like Lattice Boltzmann method (LBM), SIMPLE-like Method, Finite Difference Method, Godunov-type Method, and Finite Volume Method. In this paper, the shallow water equation will be approximated using LBM or known as LABSWE and will be simulated in performance of parallel programming using OpenMP. To evaluate the performance between 2 and 4 threads parallel algorithm, ten various number of grids Lx and Ly are elaborated. The results show that using OpenMP platform, the computational time for solving LABSWE can be decreased. For instance using grid sizes 1000 × 500, the speedup of 2 and 4 threads is observed 93.54 s and 333.243 s respectively.

Vibration response comparison of twisted shrouded blades using different impact models

NASA Astrophysics Data System (ADS)

Xie, Fangtao; Ma, Hui; Cui, Can; Wen, Bangchun

2017-06-01

On the basis of our previous work (Ma et al., 2016, Journal of Sound and Vibration, 378, 92-108) [36], an improved analytical model (IAM) of a rotating twisted shrouded blade with stagger angle simulated by flexible beam with a tip-mass is established based on Timoshenko beam theory, whose effectiveness is verified using finite element (FE) method. The effects of different parameters such as shroud gaps, contact stiffness, stagger angles and twist angels on the vibration responses of the shrouded blades are analyzed using two different impact models where the adjacent two shrouded blades are simulated by massless springs in impact model 1 (IM1) and those are simulated by Timoshenko beam in impact model 2 (IM2). The results indicate that two impact models agree well under some cases such as big shroud gaps and small contact stiffness due to the small vibration effects of adjacent blades, but not vice versa under the condition of small shroud gaps and big contact stiffness. As for IM2, the resonance appears because the limitation of the adjacent blades is weakened due to their inertia effects, however, the resonance does not appear because of the strong limitation of the springs used to simulate adjacent blades for IM1. With the increase of stagger angles and twist angles, the first-order resonance rotational speed increases due to the increase of the dynamic stiffness under no-impact condition, and the rotational speeds of starting impact and ending impact rise under the impact condition.

A diffuse-interface method for two-phase flows with soluble surfactants

PubMed Central

Teigen, Knut Erik; Song, Peng; Lowengrub, John; Voigt, Axel

2010-01-01

A method is presented to solve two-phase problems involving soluble surfactants. The incompressible Navier–Stokes equations are solved along with equations for the bulk and interfacial surfactant concentrations. A non-linear equation of state is used to relate the surface tension to the interfacial surfactant concentration. The method is based on the use of a diffuse interface, which allows a simple implementation using standard finite difference or finite element techniques. Here, finite difference methods on a block-structured adaptive grid are used, and the resulting equations are solved using a non-linear multigrid method. Results are presented for a drop in shear flow in both 2D and 3D, and the effect of solubility is discussed. PMID:21218125

Adaptively-refined overlapping grids for the numerical solution of systems of hyperbolic conservation laws

NASA Technical Reports Server (NTRS)

Brislawn, Kristi D.; Brown, David L.; Chesshire, Geoffrey S.; Saltzman, Jeffrey S.

1995-01-01

Adaptive mesh refinement (AMR) in conjunction with higher-order upwind finite-difference methods have been used effectively on a variety of problems in two and three dimensions. In this paper we introduce an approach for resolving problems that involve complex geometries in which resolution of boundary geometry is important. The complex geometry is represented by using the method of overlapping grids, while local resolution is obtained by refining each component grid with the AMR algorithm, appropriately generalized for this situation. The CMPGRD algorithm introduced by Chesshire and Henshaw is used to automatically generate the overlapping grid structure for the underlying mesh.

Three-dimensional zonal grids about arbitrary shapes by Poisson's equation

NASA Technical Reports Server (NTRS)

Sorenson, Reese L.

1988-01-01

A method for generating 3-D finite difference grids about or within arbitrary shapes is presented. The 3-D Poisson equations are solved numerically, with values for the inhomogeneous terms found automatically by the algorithm. Those inhomogeneous terms have the effect near boundaries of reducing cell skewness and imposing arbitrary cell height. The method allows the region of interest to be divided into zones (blocks), allowing the method to be applicable to almost any physical domain. A FORTRAN program called 3DGRAPE has been written to implement the algorithm. Lastly, a method for redistributing grid points along lines normal to boundaries will be described.

Prediction of the Thrust Performance and the Flowfield of Liquid Rocket Engines

NASA Technical Reports Server (NTRS)

Wang, T.-S.

1990-01-01

In an effort to improve the current solutions in the design and analysis of liquid propulsive engines, a computational fluid dynamics (CFD) model capable of calculating the reacting flows from the combustion chamber, through the nozzle to the external plume, was developed. The Space Shuttle Main Engine (SSME) fired at sea level, was investigated as a sample case. The CFD model, FDNS, is a pressure based, non-staggered grid, viscous/inviscid, ideal gas/real gas, reactive code. An adaptive upwinding differencing scheme is employed for the spatial discretization. The upwind scheme is based on fourth order central differencing with fourth order damping for smooth regions, and second order central differencing with second order damping for shock capturing. It is equipped with a CHMQGM equilibrium chemistry algorithm and a PARASOL finite rate chemistry algorithm using the point implicit method. The computed flow results and performance compared well with those of other standard codes and engine hot fire test data. In addition, the transient nozzle flowfield calculation was also performed to demonstrate the ability of FDNS in capturing the flow separation during the startup process.

A comparison of two central difference schemes for solving the Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Maksymiuk, C. M.; Swanson, R. C.; Pulliam, T. H.

1990-01-01

Five viscous transonic airfoil cases were computed by two significantly different computational fluid dynamics codes: An explicit finite-volume algorithm with multigrid, and an implicit finite-difference approximate-factorization method with Eigenvector diagonalization. Both methods are described in detail, and their performance on the test cases is compared. The codes utilized the same grids, turbulence model, and computer to provide the truest test of the algorithms. The two approaches produce very similar results, which, for attached flows, also agree well with experimental results; however, the explicit code is considerably faster.

SOME NEW FINITE DIFFERENCE METHODS FOR HELMHOLTZ EQUATIONS ON IRREGULAR DOMAINS OR WITH INTERFACES

PubMed Central

Wan, Xiaohai; Li, Zhilin

2012-01-01

Solving a Helmholtz equation Δu + λu = f efficiently is a challenge for many applications. For example, the core part of many efficient solvers for the incompressible Navier-Stokes equations is to solve one or several Helmholtz equations. In this paper, two new finite difference methods are proposed for solving Helmholtz equations on irregular domains, or with interfaces. For Helmholtz equations on irregular domains, the accuracy of the numerical solution obtained using the existing augmented immersed interface method (AIIM) may deteriorate when the magnitude of λ is large. In our new method, we use a level set function to extend the source term and the PDE to a larger domain before we apply the AIIM. For Helmholtz equations with interfaces, a new maximum principle preserving finite difference method is developed. The new method still uses the standard five-point stencil with modifications of the finite difference scheme at irregular grid points. The resulting coefficient matrix of the linear system of finite difference equations satisfies the sign property of the discrete maximum principle and can be solved efficiently using a multigrid solver. The finite difference method is also extended to handle temporal discretized equations where the solution coefficient λ is inversely proportional to the mesh size. PMID:22701346

SOME NEW FINITE DIFFERENCE METHODS FOR HELMHOLTZ EQUATIONS ON IRREGULAR DOMAINS OR WITH INTERFACES.

PubMed

Wan, Xiaohai; Li, Zhilin

2012-06-01

Solving a Helmholtz equation Δu + λu = f efficiently is a challenge for many applications. For example, the core part of many efficient solvers for the incompressible Navier-Stokes equations is to solve one or several Helmholtz equations. In this paper, two new finite difference methods are proposed for solving Helmholtz equations on irregular domains, or with interfaces. For Helmholtz equations on irregular domains, the accuracy of the numerical solution obtained using the existing augmented immersed interface method (AIIM) may deteriorate when the magnitude of λ is large. In our new method, we use a level set function to extend the source term and the PDE to a larger domain before we apply the AIIM. For Helmholtz equations with interfaces, a new maximum principle preserving finite difference method is developed. The new method still uses the standard five-point stencil with modifications of the finite difference scheme at irregular grid points. The resulting coefficient matrix of the linear system of finite difference equations satisfies the sign property of the discrete maximum principle and can be solved efficiently using a multigrid solver. The finite difference method is also extended to handle temporal discretized equations where the solution coefficient λ is inversely proportional to the mesh size.

Efficient parallel seismic simulations including topography and 3-D material heterogeneities on locally refined composite grids

NASA Astrophysics Data System (ADS)

Petersson, Anders; Rodgers, Arthur

2010-05-01

The finite difference method on a uniform Cartesian grid is a highly efficient and easy to implement technique for solving the elastic wave equation in seismic applications. However, the spacing in a uniform Cartesian grid is fixed throughout the computational domain, whereas the resolution requirements in realistic seismic simulations usually are higher near the surface than at depth. This can be seen from the well-known formula h ≤ L-P which relates the grid spacing h to the wave length L, and the required number of grid points per wavelength P for obtaining an accurate solution. The compressional and shear wave lengths in the earth generally increase with depth and are often a factor of ten larger below the Moho discontinuity (at about 30 km depth), than in sedimentary basins near the surface. A uniform grid must have a grid spacing based on the small wave lengths near the surface, which results in over-resolving the solution at depth. As a result, the number of points in a uniform grid is unnecessarily large. In the wave propagation project (WPP) code, we address the over-resolution-at-depth issue by generalizing our previously developed single grid finite difference scheme to work on a composite grid consisting of a set of structured rectangular grids of different spacings, with hanging nodes on the grid refinement interfaces. The computational domain in a regional seismic simulation often extends to depth 40-50 km. Hence, using a refinement ratio of two, we need about three grid refinements from the bottom of the computational domain to the surface, to keep the local grid size in approximate parity with the local wave lengths. The challenge of the composite grid approach is to find a stable and accurate method for coupling the solution across the grid refinement interface. Of particular importance is the treatment of the solution at the hanging nodes, i.e., the fine grid points which are located in between coarse grid points. WPP implements a new, energy conserving, coupling procedure for the elastic wave equation at grid refinement interfaces. When used together with our single grid finite difference scheme, it results in a method which is provably stable, without artificial dissipation, for arbitrary heterogeneous isotropic elastic materials. The new coupling procedure is based on satisfying the summation-by-parts principle across refinement interfaces. From a practical standpoint, an important advantage of the proposed method is the absence of tunable numerical parameters, which seldom are appreciated by application experts. In WPP, the composite grid discretization is combined with a curvilinear grid approach that enables accurate modeling of free surfaces on realistic (non-planar) topography. The overall method satisfies the summation-by-parts principle and is stable under a CFL time step restriction. A feature of great practical importance is that WPP automatically generates the composite grid based on the user provided topography and the depths of the grid refinement interfaces. The WPP code has been verified extensively, for example using the method of manufactured solutions, by solving Lamb's problem, by solving various layer over half- space problems and comparing to semi-analytic (FK) results, and by simulating scenario earthquakes where results from other seismic simulation codes are available. WPP has also been validated against seismographic recordings of moderate earthquakes. WPP performs well on large parallel computers and has been run on up to 32,768 processors using about 26 Billion grid points (78 Billion DOF) and 41,000 time steps. WPP is an open source code that is available under the Gnu general public license.

Numerical stability of an explicit finite difference scheme for the solution of transient conduction in composite media

NASA Technical Reports Server (NTRS)

Campbell, W.

1981-01-01

A theoretical evaluation of the stability of an explicit finite difference solution of the transient temperature field in a composite medium is presented. The grid points of the field are assumed uniformly spaced, and media interfaces are either vertical or horizontal and pass through grid points. In addition, perfect contact between different media (infinite interfacial conductance) is assumed. A finite difference form of the conduction equation is not valid at media interfaces; therefore, heat balance forms are derived. These equations were subjected to stability analysis, and a computer graphics code was developed that permitted determination of a maximum time step for a given grid spacing.

An arbitrary-order staggered time integrator for the linear acoustic wave equation

NASA Astrophysics Data System (ADS)

Lee, Jaejoon; Park, Hyunseo; Park, Yoonseo; Shin, Changsoo

2018-02-01

We suggest a staggered time integrator whose order of accuracy can arbitrarily be extended to solve the linear acoustic wave equation. A strategy to select the appropriate order of accuracy is also proposed based on the error analysis that quantitatively predicts the truncation error of the numerical solution. This strategy not only reduces the computational cost several times, but also allows us to flexibly set the modelling parameters such as the time step length, grid interval and P-wave speed. It is demonstrated that the proposed method can almost eliminate temporal dispersive errors during long term simulations regardless of the heterogeneity of the media and time step lengths. The method can also be successfully applied to the source problem with an absorbing boundary condition, which is frequently encountered in the practical usage for the imaging algorithms or the inverse problems.

High-order central ENO finite-volume scheme for hyperbolic conservation laws on three-dimensional cubed-sphere grids

NASA Astrophysics Data System (ADS)

Ivan, L.; De Sterck, H.; Susanto, A.; Groth, C. P. T.

2015-02-01

A fourth-order accurate finite-volume scheme for hyperbolic conservation laws on three-dimensional (3D) cubed-sphere grids is described. The approach is based on a central essentially non-oscillatory (CENO) finite-volume method that was recently introduced for two-dimensional compressible flows and is extended to 3D geometries with structured hexahedral grids. Cubed-sphere grids feature hexahedral cells with nonplanar cell surfaces, which are handled with high-order accuracy using trilinear geometry representations in the proposed approach. Varying stencil sizes and slope discontinuities in grid lines occur at the boundaries and corners of the six sectors of the cubed-sphere grid where the grid topology is unstructured, and these difficulties are handled naturally with high-order accuracy by the multidimensional least-squares based 3D CENO reconstruction with overdetermined stencils. A rotation-based mechanism is introduced to automatically select appropriate smaller stencils at degenerate block boundaries, where fewer ghost cells are available and the grid topology changes, requiring stencils to be modified. Combining these building blocks results in a finite-volume discretization for conservation laws on 3D cubed-sphere grids that is uniformly high-order accurate in all three grid directions. While solution-adaptivity is natural in the multi-block setting of our code, high-order accurate adaptive refinement on cubed-sphere grids is not pursued in this paper. The 3D CENO scheme is an accurate and robust solution method for hyperbolic conservation laws on general hexahedral grids that is attractive because it is inherently multidimensional by employing a K-exact overdetermined reconstruction scheme, and it avoids the complexity of considering multiple non-central stencil configurations that characterizes traditional ENO schemes. Extensive numerical tests demonstrate fourth-order convergence for stationary and time-dependent Euler and magnetohydrodynamic flows on cubed-sphere grids, and robustness against spurious oscillations at 3D shocks. Performance tests illustrate efficiency gains that can be potentially achieved using fourth-order schemes as compared to second-order methods for the same error level. Applications on extended cubed-sphere grids incorporating a seventh root block that discretizes the interior of the inner sphere demonstrate the versatility of the spatial discretization method.

Solution of the advection-dispersion equation in two dimensions by a finite-volume Eulerian-Lagrangian localized adjoint method

USGS Publications Warehouse

Healy, R.W.; Russell, T.F.

1998-01-01

We extend the finite-volume Eulerian-Lagrangian localized adjoint method (FVELLAM) for solution of the advection-dispersion equation to two dimensions. The method can conserve mass globally and is not limited by restrictions on the size of the grid Peclet or Courant number. Therefore, it is well suited for solution of advection-dominated ground-water solute transport problems. In test problem comparisons with standard finite differences, FVELLAM is able to attain accurate solutions on much coarser space and time grids. On fine grids, the accuracy of the two methods is comparable. A critical aspect of FVELLAM (and all other ELLAMs) is evaluation of the mass storage integral from the preceding time level. In FVELLAM this may be accomplished with either a forward or backtracking approach. The forward tracking approach conserves mass globally and is the preferred approach. The backtracking approach is less computationally intensive, but not globally mass conservative. Boundary terms are systematically represented as integrals in space and time which are evaluated by a common integration scheme in conjunction with forward tracking through time. Unlike the one-dimensional case, local mass conservation cannot be guaranteed, so slight oscillations in concentration can develop, particularly in the vicinity of inflow or outflow boundaries. Published by Elsevier Science Ltd.

On the use of Schwarz-Christoffel conformal mappings to the grid generation for global ocean models

NASA Astrophysics Data System (ADS)

Xu, S.; Wang, B.; Liu, J.

2015-10-01

In this article we propose two grid generation methods for global ocean general circulation models. Contrary to conventional dipolar or tripolar grids, the proposed methods are based on Schwarz-Christoffel conformal mappings that map areas with user-prescribed, irregular boundaries to those with regular boundaries (i.e., disks, slits, etc.). The first method aims at improving existing dipolar grids. Compared with existing grids, the sample grid achieves a better trade-off between the enlargement of the latitudinal-longitudinal portion and the overall smooth grid cell size transition. The second method addresses more modern and advanced grid design requirements arising from high-resolution and multi-scale ocean modeling. The generated grids could potentially achieve the alignment of grid lines to the large-scale coastlines, enhanced spatial resolution in coastal regions, and easier computational load balance. Since the grids are orthogonal curvilinear, they can be easily utilized by the majority of ocean general circulation models that are based on finite difference and require grid orthogonality. The proposed grid generation algorithms can also be applied to the grid generation for regional ocean modeling where complex land-sea distribution is present.

Grid generation about complex three-dimensional aircraft configurations

NASA Technical Reports Server (NTRS)

Klopfer, Goetz H.

1991-01-01

The problem of obtaining three dimensional grids with sufficient resolution to resolve all the flow or other physical features of interest is addressed. The generation of a computational grid involves a series of compromises to resolve several conflicting requirements. On one hand, one would like the grid to be fine enough and not too skewed to reduce the numerical errors and to adequately resolve the pertinent physical features of the flow field about the aircraft. On the other hand, the capabilities of present or even future supercomputers are finite and the number of mesh points must be limited to a reasonable number: one which is usually much less than desired for numerical accuracy. One technique to overcome this limitation is the 'zonal' grid approach. In this method, the overall field is subdivided into smaller zones or blocks in each of which an independent grid is generated with enough grid density to resolve the flow features in that zone. The zonal boundaries or interfaces require special boundary conditions such that the conservation properties of the governing equations are observed. Much work was done in 3-D zonal approaches with nonconservative zonal interfaces. A 3-D zonal conservative interfacing method that is efficient and easy to implement was developed during the past year. During the course of the work, it became apparent that it would be much more feasible to do the conservative interfacing with cell-centered finite volume codes instead of the originally planned finite difference codes. Accordingly, the CNS code was converted to finite volume form. This new version of the code is named CNSFV. The original multi-zonal interfacing capability of the CNS code was enhanced by generalizing the procedure to allow for completely arbitrarily shaped zones with no mesh continuity between the zones. While this zoning capability works well for most flow situations, it is, however, still nonconservative. The conservative interface algorithm was also implemented but was not completely validated.

A method for modeling finite-core vortices in wake-flow calculations

NASA Technical Reports Server (NTRS)

Stremel, P. M.

1984-01-01

A numerical method for computing nonplanar vortex wakes represented by finite-core vortices is presented. The approach solves for the velocity on an Eulerian grid, using standard finite-difference techniques; the vortex wake is tracked by Lagrangian methods. In this method, the distribution of continuous vorticity in the wake is replaced by a group of discrete vortices. An axially symmetric distribution of vorticity about the center of each discrete vortex is used to represent the finite-core model. Two distributions of vorticity, or core models, are investigated: a finite distribution of vorticity represented by a third-order polynomial, and a continuous distribution of vorticity throughout the wake. The method provides for a vortex-core model that is insensitive to the mesh spacing. Results for a simplified case are presented. Computed results for the roll-up of a vortex wake generated by wings with different spanwise load distributions are presented; contour plots of the flow-field velocities are included; and comparisons are made of the computed flow-field velocities with experimentally measured velocities.

High order multi-grid methods to solve the Poisson equation

NASA Technical Reports Server (NTRS)

Schaffer, S.

1981-01-01

High order multigrid methods based on finite difference discretization of the model problem are examined. The following methods are described: (1) a fixed high order FMG-FAS multigrid algorithm; (2) the high order methods; and (3) results are presented on four problems using each method with the same underlying fixed FMG-FAS algorithm.

Equation of state and QCD transition at finite temperature

NASA Astrophysics Data System (ADS)

Bazavov, A.; Bhattacharya, T.; Cheng, M.; Christ, N. H.; Detar, C.; Ejiri, S.; Gottlieb, Steven; Gupta, R.; Heller, U. M.; Huebner, K.; Jung, C.; Karsch, F.; Laermann, E.; Levkova, L.; Miao, C.; Mawhinney, R. D.; Petreczky, P.; Schmidt, C.; Soltz, R. A.; Soeldner, W.; Sugar, R.; Toussaint, D.; Vranas, P.

2009-07-01

We calculate the equation of state in 2+1 flavor QCD at finite temperature with physical strange quark mass and almost physical light quark masses using lattices with temporal extent Nτ=8. Calculations have been performed with two different improved staggered fermion actions, the asqtad and p4 actions. Overall, we find good agreement between results obtained with these two O(a2) improved staggered fermion discretization schemes. A comparison with earlier calculations on coarser lattices is performed to quantify systematic errors in current studies of the equation of state. We also present results for observables that are sensitive to deconfining and chiral aspects of the QCD transition on Nτ=6 and 8 lattices. We find that deconfinement and chiral symmetry restoration happen in the same narrow temperature interval. In an appendix we present a simple parametrization of the equation of state that can easily be used in hydrodynamic model calculations. In this parametrization we include an estimate of current uncertainties in the lattice calculations which arise from cutoff and quark mass effects.

Finite element analysis of transonic flows in cascades: Importance of computational grids in improving accuracy and convergence

NASA Technical Reports Server (NTRS)

Ecer, A.; Akay, H. U.

1981-01-01

The finite element method is applied for the solution of transonic potential flows through a cascade of airfoils. Convergence characteristics of the solution scheme are discussed. Accuracy of the numerical solutions is investigated for various flow regions in the transonic flow configuration. The design of an efficient finite element computational grid is discussed for improving accuracy and convergence.

FV-MHMM: A Discussion on Weighting Schemes.

NASA Astrophysics Data System (ADS)

Franc, J.; Gerald, D.; Jeannin, L.; Egermann, P.; Masson, R.

2016-12-01

Upscaling or homogenization techniques consist in finding block-equivalentor equivalent upscaled properties on a coarse grid from heterogeneousproperties defined on an underlying fine grid. However, this couldbecome costly and resource consuming. Harder et al., 2013, have developeda Multiscale Hybrid-Mixed Method (MHMM) of upscaling to treat Darcytype equations on heterogeneous fields formulated using a finite elementmethod. Recently, Franc et al. 2016, has extended this method of upscalingto finite volume formulation (FV-MHMM). Although convergence refiningLagrange multipliers space has been observed, numerical artefactscan occur while trapping numerically the flow in regions of low permeability. This work will present the development of the method along with theresults obtained from its classical formulation. Then, two weightingschemes and their benefits on the FV-MHMM method will be presented insome simple random permeability cases. Next example will involve alarger heterogeneous 2D permeability field extracted from the 10thSPE test case. Eventually, multiphase flow will be addressed asan extension of this single phase flow method. An elliptic pressureequation solved on the coarse grid via FV-MHMM will be sequentiallycoupled with a hyperbolic saturation equation on the fine grid. Theimproved accuracy thanks to the weighting scheme will be measuredcompared to a finite volume fine grid solution. References: Harder, C., Paredes, D. and Valentin, F., A family of multiscalehybrid-mixed finite element methods for the Darcy equation with roughcoefficients, Journal of Computational Physics, 2013. Franc J., Debenest G., Jeannin L., Egermann P. and Masson R., FV-MHMMfor reservoir modelling ECMOR XV-15th European Conference on the Mathematicsof Oil Recovery, 2015.

Finite difference time domain grid generation from AMC helicopter models

NASA Technical Reports Server (NTRS)

Cravey, Robin L.

1992-01-01

A simple technique is presented which forms a cubic grid model of a helicopter from an Aircraft Modeling Code (AMC) input file. The AMC input file defines the helicopter fuselage as a series of polygonal cross sections. The cubic grid model is used as an input to a Finite Difference Time Domain (FDTD) code to obtain predictions of antenna performance on a generic helicopter model. The predictions compare reasonably well with measured data.

The Application of COMSOL Multiphysics Package on the Modelling of Complex 3-D Lithospheric Electrical Resistivity Structures - A Case Study from the Proterozoic Orogenic belt within the North China Craton

NASA Astrophysics Data System (ADS)

Guo, L.; Yin, Y.; Deng, M.; Guo, L.; Yan, J.

2017-12-01

At present, most magnetotelluric (MT) forward modelling and inversion codes are based on finite difference method. But its structured mesh gridding cannot be well adapted for the conditions with arbitrary topography or complex tectonic structures. By contrast, the finite element method is more accurate in calculating complex and irregular 3-D region and has lower requirement of function smoothness. However, the complexity of mesh gridding and limitation of computer capacity has been affecting its application. COMSOL Multiphysics is a cross-platform finite element analysis, solver and multiphysics full-coupling simulation software. It achieves highly accurate numerical simulations with high computational performance and outstanding multi-field bi-directional coupling analysis capability. In addition, its AC/DC and RF module can be used to easily calculate the electromagnetic responses of complex geological structures. Using the adaptive unstructured grid, the calculation is much faster. In order to improve the discretization technique of computing area, we use the combination of Matlab and COMSOL Multiphysics to establish a general procedure for calculating the MT responses for arbitrary resistivity models. The calculated responses include the surface electric and magnetic field components, impedance components, magnetic transfer functions and phase tensors. Then, the reliability of this procedure is certificated by 1-D, 2-D and 3-D and anisotropic forward modeling tests. Finally, we establish the 3-D lithospheric resistivity model for the Proterozoic Wutai-Hengshan Mts. within the North China Craton by fitting the real MT data collected there. The reliability of the model is also verified by induced vectors and phase tensors. Our model shows more details and better resolution, compared with the previously published 3-D model based on the finite difference method. In conclusion, COMSOL Multiphysics package is suitable for modeling the 3-D lithospheric resistivity structures under complex tectonic deformation backgrounds, which could be a good complement to the existing finite-difference inversion algorithms.

The Benard problem: A comparison of finite difference and spectral collocation eigen value solutions

NASA Technical Reports Server (NTRS)

Skarda, J. Raymond Lee; Mccaughan, Frances E.; Fitzmaurice, Nessan

1995-01-01

The application of spectral methods, using a Chebyshev collocation scheme, to solve hydrodynamic stability problems is demonstrated on the Benard problem. Implementation of the Chebyshev collocation formulation is described. The performance of the spectral scheme is compared with that of a 2nd order finite difference scheme. An exact solution to the Marangoni-Benard problem is used to evaluate the performance of both schemes. The error of the spectral scheme is at least seven orders of magnitude smaller than finite difference error for a grid resolution of N = 15 (number of points used). The performance of the spectral formulation far exceeded the performance of the finite difference formulation for this problem. The spectral scheme required only slightly more effort to set up than the 2nd order finite difference scheme. This suggests that the spectral scheme may actually be faster to implement than higher order finite difference schemes.

ISCFD Nagoya 1989 - International Symposium on Computational Fluid Dynamics, 3rd, Nagoya, Japan, Aug. 28-31, 1989, Technical Papers

NASA Astrophysics Data System (ADS)

Recent advances in computational fluid dynamics are discussed in reviews and reports. Topics addressed include large-scale LESs for turbulent pipe and channel flows, numerical solutions of the Euler and Navier-Stokes equations on parallel computers, multigrid methods for steady high-Reynolds-number flow past sudden expansions, finite-volume methods on unstructured grids, supersonic wake flow on a blunt body, a grid-characteristic method for multidimensional gas dynamics, and CIC numerical simulation of a wave boundary layer. Consideration is given to vortex simulations of confined two-dimensional jets, supersonic viscous shear layers, spectral methods for compressible flows, shock-wave refraction at air/water interfaces, oscillatory flow in a two-dimensional collapsible channel, the growth of randomness in a spatially developing wake, and an efficient simplex algorithm for the finite-difference and dynamic linear-programming method in optimal potential control.

a Cell Vertex Algorithm for the Incompressible Navier-Stokes Equations on Non-Orthogonal Grids

NASA Astrophysics Data System (ADS)

Jessee, J. P.; Fiveland, W. A.

1996-08-01

The steady, incompressible Navier-Stokes (N-S) equations are discretized using a cell vertex, finite volume method. Quadrilateral and hexahedral meshes are used to represent two- and three-dimensional geometries respectively. The dependent variables include the Cartesian components of velocity and pressure. Advective fluxes are calculated using bounded, high-resolution schemes with a deferred correction procedure to maintain a compact stencil. This treatment insures bounded, non-oscillatory solutions while maintaining low numerical diffusion. The mass and momentum equations are solved with the projection method on a non-staggered grid. The coupling of the pressure and velocity fields is achieved using the Rhie and Chow interpolation scheme modified to provide solutions independent of time steps or relaxation factors. An algebraic multigrid solver is used for the solution of the implicit, linearized equations.A number of test cases are anlaysed and presented. The standard benchmark cases include a lid-driven cavity, flow through a gradual expansion and laminar flow in a three-dimensional curved duct. Predictions are compared with data, results of other workers and with predictions from a structured, cell-centred, control volume algorithm whenever applicable. Sensitivity of results to the advection differencing scheme is investigated by applying a number of higher-order flux limiters: the MINMOD, MUSCL, OSHER, CLAM and SMART schemes. As expected, studies indicate that higher-order schemes largely mitigate the diffusion effects of first-order schemes but also shown no clear preference among the higher-order schemes themselves with respect to accuracy. The effect of the deferred correction procedure on global convergence is discussed.

Higher-order adaptive finite-element methods for Kohn–Sham density functional theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Motamarri, P.; Nowak, M.R.; Leiter, K.

2013-11-15

We present an efficient computational approach to perform real-space electronic structure calculations using an adaptive higher-order finite-element discretization of Kohn–Sham density-functional theory (DFT). To this end, we develop an a priori mesh-adaption technique to construct a close to optimal finite-element discretization of the problem. We further propose an efficient solution strategy for solving the discrete eigenvalue problem by using spectral finite-elements in conjunction with Gauss–Lobatto quadrature, and a Chebyshev acceleration technique for computing the occupied eigenspace. The proposed approach has been observed to provide a staggering 100–200-fold computational advantage over the solution of a generalized eigenvalue problem. Using the proposedmore » solution procedure, we investigate the computational efficiency afforded by higher-order finite-element discretizations of the Kohn–Sham DFT problem. Our studies suggest that staggering computational savings—of the order of 1000-fold—relative to linear finite-elements can be realized, for both all-electron and local pseudopotential calculations, by using higher-order finite-element discretizations. On all the benchmark systems studied, we observe diminishing returns in computational savings beyond the sixth-order for accuracies commensurate with chemical accuracy, suggesting that the hexic spectral-element may be an optimal choice for the finite-element discretization of the Kohn–Sham DFT problem. A comparative study of the computational efficiency of the proposed higher-order finite-element discretizations suggests that the performance of finite-element basis is competing with the plane-wave discretization for non-periodic local pseudopotential calculations, and compares to the Gaussian basis for all-electron calculations to within an order of magnitude. Further, we demonstrate the capability of the proposed approach to compute the electronic structure of a metallic system containing 1688 atoms using modest computational resources, and good scalability of the present implementation up to 192 processors.« less

A simple finite-difference scheme for handling topography with the first-order wave equation

NASA Astrophysics Data System (ADS)

Mulder, W. A.; Huiskes, M. J.

2017-07-01

One approach to incorporate topography in seismic finite-difference codes is a local modification of the difference operators near the free surface. An earlier paper described an approach for modelling irregular boundaries in a constant-density acoustic finite-difference code, based on the second-order formulation of the wave equation that only involves the pressure. Here, a similar method is considered for the first-order formulation in terms of pressure and particle velocity, using a staggered finite-difference discretization both in space and in time. In one space dimension, the boundary conditions consist in imposing antisymmetry for the pressure and symmetry for particle velocity components. For the pressure, this means that the solution values as well as all even derivatives up to a certain order are zero on the boundary. For the particle velocity, all odd derivatives are zero. In 2D, the 1-D assumption is used along each coordinate direction, with antisymmetry for the pressure along the coordinate and symmetry for the particle velocity component parallel to that coordinate direction. Since the symmetry or antisymmetry should hold along the direction normal to the boundary rather than along the coordinate directions, this generates an additional numerical error on top of the time stepping errors and the errors due to the interior spatial discretization. Numerical experiments in 2D and 3D nevertheless produce acceptable results.

A mass and momentum conserving unsplit semi-Lagrangian framework for simulating multiphase flows

DOE Office of Scientific and Technical Information (OSTI.GOV)

Owkes, Mark, E-mail: mark.owkes@montana.edu; Desjardins, Olivier

In this work, we present a computational methodology for convection and advection that handles discontinuities with second order accuracy and maintains conservation to machine precision. This method can transport a variety of discontinuous quantities and is used in the context of an incompressible gas–liquid flow to transport the phase interface, momentum, and scalars. The proposed method provides a modification to the three-dimensional, unsplit, second-order semi-Lagrangian flux method of Owkes & Desjardins (JCP, 2014). The modification adds a refined grid that provides consistent fluxes of mass and momentum defined on a staggered grid and discrete conservation of mass and momentum, evenmore » for flows with large density ratios. Additionally, the refined grid doubles the resolution of the interface without significantly increasing the computational cost over previous non-conservative schemes. This is possible due to a novel partitioning of the semi-Lagrangian fluxes into a small number of simplices. The proposed scheme is tested using canonical verification tests, rising bubbles, and an atomizing liquid jet.« less

Multigrid solution of the Navier-Stokes equations on highly stretched grids with defect correction

NASA Technical Reports Server (NTRS)

Sockol, Peter M.

1993-01-01

Relaxation-based multigrid solvers for the steady incompressible Navier-Stokes equations are examined to determine their computational speed and robustness. Four relaxation methods with a common discretization have been used as smoothers in a single tailored multigrid procedure. The equations are discretized on a staggered grid with first order upwind used for convection in the relaxation process on all grids and defect correction to second order central on the fine grid introduced once per multigrid cycle. A fixed W(1,1) cycle with full weighting of residuals is used in the FAS multigrid process. The resulting solvers have been applied to three 2D flow problems, over a range of Reynolds numbers, on both uniform and highly stretched grids. In all cases the L(sub 2) norm of the velocity changes is reduced to 10(exp -6) in a few 10's of fine grid sweeps. The results from this study are used to draw conclusions on the strengths and weaknesses of the individual relaxation schemes as well as those of the overall multigrid procedure when used as a solver on highly stretched grids.

Accurate Finite Difference Algorithms

NASA Technical Reports Server (NTRS)

Goodrich, John W.

1996-01-01

Two families of finite difference algorithms for computational aeroacoustics are presented and compared. All of the algorithms are single step explicit methods, they have the same order of accuracy in both space and time, with examples up to eleventh order, and they have multidimensional extensions. One of the algorithm families has spectral like high resolution. Propagation with high order and high resolution algorithms can produce accurate results after O(10(exp 6)) periods of propagation with eight grid points per wavelength.

Numerical simulation of axisymmetric turbulent flow in combustors and diffusors. Ph.D. Thesis. Final Report

NASA Technical Reports Server (NTRS)

Yung, Chain Nan

1988-01-01

A method for predicting turbulent flow in combustors and diffusers is developed. The Navier-Stokes equations, incorporating a turbulence kappa-epsilon model equation, were solved in a nonorthogonal curvilinear coordinate system. The solution applied the finite volume method to discretize the differential equations and utilized the SIMPLE algorithm iteratively to solve the differenced equations. A zonal grid method, wherein the flow field was divided into several subsections, was developed. This approach permitted different computational schemes to be used in the various zones. In addition, grid generation was made a more simple task. However, treatment of the zonal boundaries required special handling. Boundary overlap and interpolating techniques were used and an adjustment of the flow variables was required to assure conservation of mass, momentum and energy fluxes. The numerical accuracy was assessed using different finite differencing methods, i.e., hybrid, quadratic upwind and skew upwind, to represent the convection terms. Flows in different geometries of combustors and diffusers were simulated and results compared with experimental data and good agreement was obtained.

Stabilized Finite Elements in FUN3D

NASA Technical Reports Server (NTRS)

Anderson, W. Kyle; Newman, James C.; Karman, Steve L.

2017-01-01

A Streamlined Upwind Petrov-Galerkin (SUPG) stabilized finite-element discretization has been implemented as a library into the FUN3D unstructured-grid flow solver. Motivation for the selection of this methodology is given, details of the implementation are provided, and the discretization for the interior scheme is verified for linear and quadratic elements by using the method of manufactured solutions. A methodology is also described for capturing shocks, and simulation results are compared to the finite-volume formulation that is currently the primary method employed for routine engineering applications. The finite-element methodology is demonstrated to be more accurate than the finite-volume technology, particularly on tetrahedral meshes where the solutions obtained using the finite-volume scheme can suffer from adverse effects caused by bias in the grid. Although no effort has been made to date to optimize computational efficiency, the finite-element scheme is competitive with the finite-volume scheme in terms of computer time to reach convergence.

Active noise attenuation in ventilation windows.

PubMed

Huang, Huahua; Qiu, Xiaojun; Kang, Jian

2011-07-01

The feasibility of applying active noise control techniques to attenuate low frequency noise transmission through a natural ventilation window into a room is investigated analytically and experimentally. The window system is constructed by staggering the opening sashes of a spaced double glazing window to allow ventilation and natural light. An analytical model based on the modal expansion method is developed to calculate the low frequency sound field inside the window and the room and to be used in the active noise control simulations. The effectiveness of the proposed analytical model is validated by using the finite element method. The performance of the active control system for a window with different source and receiver configurations are compared, and it is found that the numerical and experimental results are in good agreement and the best result is achieved when the secondary sources are placed in the center at the bottom of the staggered window. The extra attenuation at the observation points in the optimized window system is almost equivalent to the noise reduction at the error sensor and the frequency range of effective control is up to 390 Hz in the case of a single channel active noise control system. © 2011 Acoustical Society of America

A Kernel-free Boundary Integral Method for Elliptic Boundary Value Problems ⋆

PubMed Central

Ying, Wenjun; Henriquez, Craig S.

2013-01-01

This paper presents a class of kernel-free boundary integral (KFBI) methods for general elliptic boundary value problems (BVPs). The boundary integral equations reformulated from the BVPs are solved iteratively with the GMRES method. During the iteration, the boundary and volume integrals involving Green's functions are approximated by structured grid-based numerical solutions, which avoids the need to know the analytical expressions of Green's functions. The KFBI method assumes that the larger regular domain, which embeds the original complex domain, can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast Fourier transform (FFT) based Poisson/Helmholtz solvers or those based on geometric multigrid iterations are applicable. The structured grid-based solutions are obtained with standard finite difference method (FDM) or finite element method (FEM), where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme. Numerical results demonstrating the efficiency and accuracy of the KFBI methods are presented. It is observed that the number of GM-RES iterations used by the method for solving isotropic and moderately anisotropic BVPs is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals. With the standard second-order FEMs and FDMs, the KFBI method shows a second-order convergence rate in accuracy for all of the tested Dirichlet/Neumann BVPs when the anisotropy of the diffusion tensor is not too strong. PMID:23519600

Dynamical Core in Atmospheric Model Does Matter in the Simulation of Arctic Climate

NASA Astrophysics Data System (ADS)

Jun, Sang-Yoon; Choi, Suk-Jin; Kim, Baek-Min

2018-03-01

Climate models using different dynamical cores can simulate significantly different winter Arctic climates even if equipped with virtually the same physics schemes. Current climate simulated by the global climate model using cubed-sphere grid with spectral element method (SE core) exhibited significantly warmer Arctic surface air temperature compared to that using latitude-longitude grid with finite volume method core. Compared to the finite volume method core, SE core simulated additional adiabatic warming in the Arctic lower atmosphere, and this was consistent with the eddy-forced secondary circulation. Downward longwave radiation further enhanced Arctic near-surface warming with a higher surface air temperature of about 1.9 K. Furthermore, in the atmospheric response to the reduced sea ice conditions with the same physical settings, only the SE core showed a robust cooling response over North America. We emphasize that special attention is needed in selecting the dynamical core of climate models in the simulation of the Arctic climate and associated teleconnection patterns.

A method of selecting grid size to account for Hertz deformation in finite element analysis of spur gears

NASA Technical Reports Server (NTRS)

Coy, J. J.; Chao, C. H. C.

1981-01-01

A method of selecting grid size for the finite element analysis of gear tooth deflection is presented. The method is based on a finite element study of two cylinders in line contact, where the criterion for establishing element size was that there be agreement with the classical Hertzian solution for deflection. The results are applied to calculate deflection for the gear specimen used in the NASA spur gear test rig. Comparisons are made between the present results and the results of two other methods of calculation. The results have application in design of gear tooth profile modifications to reduce noise and dynamic loads.

Comparison of Accuracy and Performance for Lattice Boltzmann and Finite Difference Simulations of Steady Viscous Flow

NASA Astrophysics Data System (ADS)

Noble, David R.; Georgiadis, John G.; Buckius, Richard O.

1996-07-01

The lattice Boltzmann method (LBM) is used to simulate flow in an infinite periodic array of octagonal cylinders. Results are compared with those obtained by a finite difference (FD) simulation solved in terms of streamfunction and vorticity using an alternating direction implicit scheme. Computed velocity profiles are compared along lines common to both the lattice Boltzmann and finite difference grids. Along all such slices, both streamwise and transverse velocity predictions agree to within 05% of the average streamwise velocity. The local shear on the surface of the cylinders also compares well, with the only deviations occurring in the vicinity of the corners of the cylinders, where the slope of the shear is discontinuous. When a constant dimensionless relaxation time is maintained, LBM exhibits the same convergence behaviour as the FD algorithm, with the time step increasing as the square of the grid size. By adjusting the relaxation time such that a constant Mach number is achieved, the time step of LBM varies linearly with the grid size. The efficiency of LBM on the CM-5 parallel computer at the National Center for Supercomputing Applications (NCSA) is evaluated by examining each part of the algorithm. Overall, a speed of 139 GFLOPS is obtained using 512 processors for a domain size of 2176×2176.

A Newton-Krylov method with an approximate analytical Jacobian for implicit solution of Navier-Stokes equations on staggered overset-curvilinear grids with immersed boundaries.

PubMed

Asgharzadeh, Hafez; Borazjani, Iman

2017-02-15

The explicit and semi-implicit schemes in flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates. Implicit schemes can be used to overcome these restrictions, but implementing them to solve the Navier-Stokes equations is not straightforward due to their non-linearity. Among the implicit schemes for nonlinear equations, Newton-based techniques are preferred over fixed-point techniques because of their high convergence rate but each Newton iteration is more expensive than a fixed-point iteration. Krylov subspace methods are one of the most advanced iterative methods that can be combined with Newton methods, i.e., Newton-Krylov Methods (NKMs) to solve non-linear systems of equations. The success of NKMs vastly depends on the scheme for forming the Jacobian, e.g., automatic differentiation is very expensive, and matrix-free methods without a preconditioner slow down as the mesh is refined. A novel, computationally inexpensive analytical Jacobian for NKM is developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered overset-curvilinear grids with immersed boundaries. Moreover, the analytical Jacobian is used to form preconditioner for matrix-free method in order to improve its performance. The NKM with the analytical Jacobian was validated and verified against Taylor-Green vortex, inline oscillations of a cylinder in a fluid initially at rest, and pulsatile flow in a 90 degree bend. The capability of the method in handling complex geometries with multiple overset grids and immersed boundaries is shown by simulating an intracranial aneurysm. It was shown that the NKM with an analytical Jacobian is 1.17 to 14.77 times faster than the fixed-point Runge-Kutta method, and 1.74 to 152.3 times (excluding an intensively stretched grid) faster than automatic differentiation depending on the grid (size) and the flow problem. In addition, it was shown that using only the diagonal of the Jacobian further improves the performance by 42 - 74% compared to the full Jacobian. The NKM with an analytical Jacobian showed better performance than the fixed point Runge-Kutta because it converged with higher time steps and in approximately 30% less iterations even when the grid was stretched and the Reynold number was increased. In fact, stretching the grid decreased the performance of all methods, but the fixed-point Runge-Kutta performance decreased 4.57 and 2.26 times more than NKM with a diagonal Jacobian when the stretching factor was increased, respectively. The NKM with a diagonal analytical Jacobian and matrix-free method with an analytical preconditioner are the fastest methods and the superiority of one to another depends on the flow problem. Furthermore, the implemented methods are fully parallelized with parallel efficiency of 80-90% on the problems tested. The NKM with the analytical Jacobian can guide building preconditioners for other techniques to improve their performance in the future.

A Newton–Krylov method with an approximate analytical Jacobian for implicit solution of Navier–Stokes equations on staggered overset-curvilinear grids with immersed boundaries

PubMed Central

Asgharzadeh, Hafez; Borazjani, Iman

2016-01-01

The explicit and semi-implicit schemes in flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates. Implicit schemes can be used to overcome these restrictions, but implementing them to solve the Navier-Stokes equations is not straightforward due to their non-linearity. Among the implicit schemes for nonlinear equations, Newton-based techniques are preferred over fixed-point techniques because of their high convergence rate but each Newton iteration is more expensive than a fixed-point iteration. Krylov subspace methods are one of the most advanced iterative methods that can be combined with Newton methods, i.e., Newton-Krylov Methods (NKMs) to solve non-linear systems of equations. The success of NKMs vastly depends on the scheme for forming the Jacobian, e.g., automatic differentiation is very expensive, and matrix-free methods without a preconditioner slow down as the mesh is refined. A novel, computationally inexpensive analytical Jacobian for NKM is developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered overset-curvilinear grids with immersed boundaries. Moreover, the analytical Jacobian is used to form preconditioner for matrix-free method in order to improve its performance. The NKM with the analytical Jacobian was validated and verified against Taylor-Green vortex, inline oscillations of a cylinder in a fluid initially at rest, and pulsatile flow in a 90 degree bend. The capability of the method in handling complex geometries with multiple overset grids and immersed boundaries is shown by simulating an intracranial aneurysm. It was shown that the NKM with an analytical Jacobian is 1.17 to 14.77 times faster than the fixed-point Runge-Kutta method, and 1.74 to 152.3 times (excluding an intensively stretched grid) faster than automatic differentiation depending on the grid (size) and the flow problem. In addition, it was shown that using only the diagonal of the Jacobian further improves the performance by 42 – 74% compared to the full Jacobian. The NKM with an analytical Jacobian showed better performance than the fixed point Runge-Kutta because it converged with higher time steps and in approximately 30% less iterations even when the grid was stretched and the Reynold number was increased. In fact, stretching the grid decreased the performance of all methods, but the fixed-point Runge-Kutta performance decreased 4.57 and 2.26 times more than NKM with a diagonal Jacobian when the stretching factor was increased, respectively. The NKM with a diagonal analytical Jacobian and matrix-free method with an analytical preconditioner are the fastest methods and the superiority of one to another depends on the flow problem. Furthermore, the implemented methods are fully parallelized with parallel efficiency of 80–90% on the problems tested. The NKM with the analytical Jacobian can guide building preconditioners for other techniques to improve their performance in the future. PMID:28042172

A Newton-Krylov method with an approximate analytical Jacobian for implicit solution of Navier-Stokes equations on staggered overset-curvilinear grids with immersed boundaries

NASA Astrophysics Data System (ADS)

Asgharzadeh, Hafez; Borazjani, Iman

2017-02-01

The explicit and semi-implicit schemes in flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates. Implicit schemes can be used to overcome these restrictions, but implementing them to solve the Navier-Stokes equations is not straightforward due to their non-linearity. Among the implicit schemes for non-linear equations, Newton-based techniques are preferred over fixed-point techniques because of their high convergence rate but each Newton iteration is more expensive than a fixed-point iteration. Krylov subspace methods are one of the most advanced iterative methods that can be combined with Newton methods, i.e., Newton-Krylov Methods (NKMs) to solve non-linear systems of equations. The success of NKMs vastly depends on the scheme for forming the Jacobian, e.g., automatic differentiation is very expensive, and matrix-free methods without a preconditioner slow down as the mesh is refined. A novel, computationally inexpensive analytical Jacobian for NKM is developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered overset-curvilinear grids with immersed boundaries. Moreover, the analytical Jacobian is used to form a preconditioner for matrix-free method in order to improve its performance. The NKM with the analytical Jacobian was validated and verified against Taylor-Green vortex, inline oscillations of a cylinder in a fluid initially at rest, and pulsatile flow in a 90 degree bend. The capability of the method in handling complex geometries with multiple overset grids and immersed boundaries is shown by simulating an intracranial aneurysm. It was shown that the NKM with an analytical Jacobian is 1.17 to 14.77 times faster than the fixed-point Runge-Kutta method, and 1.74 to 152.3 times (excluding an intensively stretched grid) faster than automatic differentiation depending on the grid (size) and the flow problem. In addition, it was shown that using only the diagonal of the Jacobian further improves the performance by 42-74% compared to the full Jacobian. The NKM with an analytical Jacobian showed better performance than the fixed point Runge-Kutta because it converged with higher time steps and in approximately 30% less iterations even when the grid was stretched and the Reynold number was increased. In fact, stretching the grid decreased the performance of all methods, but the fixed-point Runge-Kutta performance decreased 4.57 and 2.26 times more than NKM with a diagonal and full Jacobian, respectivley, when the stretching factor was increased. The NKM with a diagonal analytical Jacobian and matrix-free method with an analytical preconditioner are the fastest methods and the superiority of one to another depends on the flow problem. Furthermore, the implemented methods are fully parallelized with parallel efficiency of 80-90% on the problems tested. The NKM with the analytical Jacobian can guide building preconditioners for other techniques to improve their performance in the future.

Time-partitioning simulation models for calculation on parallel computers

NASA Technical Reports Server (NTRS)

Milner, Edward J.; Blech, Richard A.; Chima, Rodrick V.

1987-01-01

A technique allowing time-staggered solution of partial differential equations is presented in this report. Using this technique, called time-partitioning, simulation execution speedup is proportional to the number of processors used because all processors operate simultaneously, with each updating of the solution grid at a different time point. The technique is limited by neither the number of processors available nor by the dimension of the solution grid. Time-partitioning was used to obtain the flow pattern through a cascade of airfoils, modeled by the Euler partial differential equations. An execution speedup factor of 1.77 was achieved using a two processor Cray X-MP/24 computer.

Projection methods for incompressible flow problems with WENO finite difference schemes

NASA Astrophysics Data System (ADS)

de Frutos, Javier; John, Volker; Novo, Julia

2016-03-01

Weighted essentially non-oscillatory (WENO) finite difference schemes have been recommended in a competitive study of discretizations for scalar evolutionary convection-diffusion equations [20]. This paper explores the applicability of these schemes for the simulation of incompressible flows. To this end, WENO schemes are used in several non-incremental and incremental projection methods for the incompressible Navier-Stokes equations. Velocity and pressure are discretized on the same grid. A pressure stabilization Petrov-Galerkin (PSPG) type of stabilization is introduced in the incremental schemes to account for the violation of the discrete inf-sup condition. Algorithmic aspects of the proposed schemes are discussed. The schemes are studied on several examples with different features. It is shown that the WENO finite difference idea can be transferred to the simulation of incompressible flows. Some shortcomings of the methods, which are due to the splitting in projection schemes, become also obvious.

A Pseudo-Temporal Multi-Grid Relaxation Scheme for Solving the Parabolized Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

White, J. A.; Morrison, J. H.

1999-01-01

A multi-grid, flux-difference-split, finite-volume code, VULCAN, is presented for solving the elliptic and parabolized form of the equations governing three-dimensional, turbulent, calorically perfect and non-equilibrium chemically reacting flows. The space marching algorithms developed to improve convergence rate and or reduce computational cost are emphasized. The algorithms presented are extensions to the class of implicit pseudo-time iterative, upwind space-marching schemes. A full approximate storage, full multi-grid scheme is also described which is used to accelerate the convergence of a Gauss-Seidel relaxation method. The multi-grid algorithm is shown to significantly improve convergence on high aspect ratio grids.

High-order flux correction/finite difference schemes for strand grids

NASA Astrophysics Data System (ADS)

Katz, Aaron; Work, Dalon

2015-02-01

A novel high-order method combining unstructured flux correction along body surfaces and high-order finite differences normal to surfaces is formulated for unsteady viscous flows on strand grids. The flux correction algorithm is applied in each unstructured layer of the strand grid, and the layers are then coupled together via a source term containing derivatives in the strand direction. Strand-direction derivatives are approximated to high-order via summation-by-parts operators for first derivatives and second derivatives with variable coefficients. We show how this procedure allows for the proper truncation error canceling properties required for the flux correction scheme. The resulting scheme possesses third-order design accuracy, but often exhibits fourth-order accuracy when higher-order derivatives are employed in the strand direction, especially for highly viscous flows. We prove discrete conservation for the new scheme and time stability in the absence of the flux correction terms. Results in two dimensions are presented that demonstrate improvements in accuracy with minimal computational and algorithmic overhead over traditional second-order algorithms.

A novel simulation theory and model system for multi-field coupling pipe-flow system

NASA Astrophysics Data System (ADS)

Chen, Yang; Jiang, Fan; Cai, Guobiao; Xu, Xu

2017-09-01

Due to the lack of a theoretical basis for multi-field coupling in many system-level models, a novel set of system-level basic equations for flow/heat transfer/combustion coupling is put forward. Then a finite volume model of quasi-1D transient flow field for multi-species compressible variable-cross-section pipe flow is established by discretising the basic equations on spatially staggered grids. Combining with the 2D axisymmetric model for pipe-wall temperature field and specific chemical reaction mechanisms, a finite volume model system is established; a set of specific calculation methods suitable for multi-field coupling system-level research is structured for various parameters in this model; specific modularisation simulation models can be further derived in accordance with specific structures of various typical components in a liquid propulsion system. This novel system can also be used to derive two sub-systems: a flow/heat transfer two-field coupling pipe-flow model system without chemical reaction and species diffusion; and a chemical equilibrium thermodynamic calculation-based multi-field coupling system. The applicability and accuracy of two sub-systems have been verified through a series of dynamic modelling and simulations in earlier studies. The validity of this system is verified in an air-hydrogen combustion sample system. The basic equations and the model system provide a unified universal theory and numerical system for modelling and simulation and even virtual testing of various pipeline systems.

Computational time analysis of the numerical solution of 3D electrostatic Poisson's equation

NASA Astrophysics Data System (ADS)

Kamboh, Shakeel Ahmed; Labadin, Jane; Rigit, Andrew Ragai Henri; Ling, Tech Chaw; Amur, Khuda Bux; Chaudhary, Muhammad Tayyab

2015-05-01

3D Poisson's equation is solved numerically to simulate the electric potential in a prototype design of electrohydrodynamic (EHD) ion-drag micropump. Finite difference method (FDM) is employed to discretize the governing equation. The system of linear equations resulting from FDM is solved iteratively by using the sequential Jacobi (SJ) and sequential Gauss-Seidel (SGS) methods, simulation results are also compared to examine the difference between the results. The main objective was to analyze the computational time required by both the methods with respect to different grid sizes and parallelize the Jacobi method to reduce the computational time. In common, the SGS method is faster than the SJ method but the data parallelism of Jacobi method may produce good speedup over SGS method. In this study, the feasibility of using parallel Jacobi (PJ) method is attempted in relation to SGS method. MATLAB Parallel/Distributed computing environment is used and a parallel code for SJ method is implemented. It was found that for small grid size the SGS method remains dominant over SJ method and PJ method while for large grid size both the sequential methods may take nearly too much processing time to converge. Yet, the PJ method reduces computational time to some extent for large grid sizes.

3D anisotropic modeling and identification for airborne EM systems based on the spectral-element method

NASA Astrophysics Data System (ADS)

Huang, Xin; Yin, Chang-Chun; Cao, Xiao-Yue; Liu, Yun-He; Zhang, Bo; Cai, Jing

2017-09-01

The airborne electromagnetic (AEM) method has a high sampling rate and survey flexibility. However, traditional numerical modeling approaches must use high-resolution physical grids to guarantee modeling accuracy, especially for complex geological structures such as anisotropic earth. This can lead to huge computational costs. To solve this problem, we propose a spectral-element (SE) method for 3D AEM anisotropic modeling, which combines the advantages of spectral and finite-element methods. Thus, the SE method has accuracy as high as that of the spectral method and the ability to model complex geology inherited from the finite-element method. The SE method can improve the modeling accuracy within discrete grids and reduce the dependence of modeling results on the grids. This helps achieve high-accuracy anisotropic AEM modeling. We first introduced a rotating tensor of anisotropic conductivity to Maxwell's equations and described the electrical field via SE basis functions based on GLL interpolation polynomials. We used the Galerkin weighted residual method to establish the linear equation system for the SE method, and we took a vertical magnetic dipole as the transmission source for our AEM modeling. We then applied fourth-order SE calculations with coarse physical grids to check the accuracy of our modeling results against a 1D semi-analytical solution for an anisotropic half-space model and verified the high accuracy of the SE. Moreover, we conducted AEM modeling for different anisotropic 3D abnormal bodies using two physical grid scales and three orders of SE to obtain the convergence conditions for different anisotropic abnormal bodies. Finally, we studied the identification of anisotropy for single anisotropic abnormal bodies, anisotropic surrounding rock, and single anisotropic abnormal body embedded in an anisotropic surrounding rock. This approach will play a key role in the inversion and interpretation of AEM data collected in regions with anisotropic geology.

The finite element method for micro-scale modeling of ultrasound propagation in cancellous bone.

PubMed

Vafaeian, B; El-Rich, M; El-Bialy, T; Adeeb, S

2014-08-01

Quantitative ultrasound for bone assessment is based on the correlations between ultrasonic parameters and the properties (mechanical and physical) of cancellous bone. To elucidate the correlations, understanding the physics of ultrasound in cancellous bone is demanded. Micro-scale modeling of ultrasound propagation in cancellous bone using the finite-difference time-domain (FDTD) method has been so far utilized as one of the approaches in this regard. However, the FDTD method accompanies two disadvantages: staircase sampling of cancellous bone by finite difference grids leads to generation of wave artifacts at the solid-fluid interface inside the bone; additionally, this method cannot explicitly satisfy the needed perfect-slip conditions at the interface. To overcome these disadvantages, the finite element method (FEM) is proposed in this study. Three-dimensional finite element models of six water-saturated cancellous bone samples with different bone volume were created. The values of speed of sound (SOS) and broadband ultrasound attenuation (BUA) were calculated through the finite element simulations of ultrasound propagation in each sample. Comparing the results with other experimental and simulation studies demonstrated the capabilities of the FEM for micro-scale modeling of ultrasound in water-saturated cancellous bone. Copyright © 2014 Elsevier B.V. All rights reserved.

Constrained CVT meshes and a comparison of triangular mesh generators

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nguyen, Hoa; Burkardt, John; Gunzburger, Max

2009-01-01

Mesh generation in regions in Euclidean space is a central task in computational science, and especially for commonly used numerical methods for the solution of partial differential equations, e.g., finite element and finite volume methods. We focus on the uniform Delaunay triangulation of planar regions and, in particular, on how one selects the positions of the vertices of the triangulation. We discuss a recently developed method, based on the centroidal Voronoi tessellation (CVT) concept, for effecting such triangulations and present two algorithms, including one new one, for CVT-based grid generation. We also compare several methods, including CVT-based methods, for triangulatingmore » planar domains. To this end, we define several quantitative measures of the quality of uniform grids. We then generate triangulations of several planar regions, including some having complexities that are representative of what one may encounter in practice. We subject the resulting grids to visual and quantitative comparisons and conclude that all the methods considered produce high-quality uniform grids and that the CVT-based grids are at least as good as any of the others.« less

Parallelized implicit propagators for the finite-difference Schrödinger equation

NASA Astrophysics Data System (ADS)

Parker, Jonathan; Taylor, K. T.

1995-08-01

We describe the application of block Gauss-Seidel and block Jacobi iterative methods to the design of implicit propagators for finite-difference models of the time-dependent Schrödinger equation. The block-wise iterative methods discussed here are mixed direct-iterative methods for solving simultaneous equations, in the sense that direct methods (e.g. LU decomposition) are used to invert certain block sub-matrices, and iterative methods are used to complete the solution. We describe parallel variants of the basic algorithm that are well suited to the medium- to coarse-grained parallelism of work-station clusters, and MIMD supercomputers, and we show that under a wide range of conditions, fine-grained parallelism of the computation can be achieved. Numerical tests are conducted on a typical one-electron atom Hamiltonian. The methods converge robustly to machine precision (15 significant figures), in some cases in as few as 6 or 7 iterations. The rate of convergence is nearly independent of the finite-difference grid-point separations.

Comparison of AGE and Spectral Methods for the Simulation of Far-Wakes

NASA Technical Reports Server (NTRS)

Bisset, D. K.; Rogers, M. M.; Kega, Dennis (Technical Monitor)

1999-01-01

Turbulent flow simulation methods based on finite differences are attractive for their simplicity, flexibility and efficiency, but not always for accuracy or stability. This report demonstrates that a good compromise is possible with the Advected Grid Explicit (AGE) method. AGE has proven to be both efficient and accurate for simulating turbulent free-shear flows, including planar mixing layers and planar jets. Its efficiency results from its localized fully explicit finite difference formulation (Bisset 1998a,b) that is very straightforward to compute, outweighing the need for a fairly small timestep. Also, most of the successful simulations were slightly under-resolved, and therefore they were, in effect, large-eddy simulations (LES) without a sub-grid-scale (SGS) model, rather than direct numerical simulations (DNS). The principle is that the role of the smallest scales of turbulent motion (when the Reynolds number is not too low) is to dissipate turbulent energy, and therefore they do not have to be simulated when the numerical method is inherently dissipative at its resolution limits. Such simulations are termed 'auto-LES' (LES with automatic SGS modeling) in this report.

Finite-element grid improvement by minimization of stiffness matrix trace

NASA Technical Reports Server (NTRS)

Kittur, Madan G.; Huston, Ronald L.; Oswald, Fred B.

1989-01-01

A new and simple method of finite-element grid improvement is presented. The objective is to improve the accuracy of the analysis. The procedure is based on a minimization of the trace of the stiffness matrix. For a broad class of problems this minimization is seen to be equivalent to minimizing the potential energy. The method is illustrated with the classical tapered bar problem examined earlier by Prager and Masur. Identical results are obtained.

Finite-element grid improvement by minimization of stiffness matrix trace

NASA Technical Reports Server (NTRS)

Kittur, Madan G.; Huston, Ronald L.; Oswald, Fred B.

1987-01-01

A new and simple method of finite-element grid improvement is presented. The objective is to improve the accuracy of the analysis. The procedure is based on a minimization of the trace of the stiffness matrix. For a broad class of problems this minimization is seen to be equivalent to minimizing the potential energy. The method is illustrated with the classical tapered bar problem examined earlier by Prager and Masur. Identical results are obtained.

Entropy generation minimization for the sloshing phenomenon in half-full elliptical storage tanks

NASA Astrophysics Data System (ADS)

Saghi, Hassan

2018-02-01

In this paper, the entropy generation in the sloshing phenomenon was obtained in elliptical storage tanks and the optimum geometry of tank was suggested. To do this, a numerical model was developed to simulate the sloshing phenomenon by using coupled Reynolds-Averaged Navier-Stokes (RANS) solver and the Volume-of-Fluid (VOF) method. The RANS equations were discretized and solved using the staggered grid finite difference and SMAC methods, and the available data were used for the model validation. Some parameters consisting of maximum free surface displacement (MFSD), maximum horizontal force exerted on the tank perimeter (MHF), tank perimeter (TP), and total entropy generation (Sgen) were introduced as design criteria for elliptical storage tanks. The entropy generation distribution provides designers with useful information about the causes of the energy loss. In this step, horizontal periodic sway motions as X =amsin(ωt) were applied to elliptical storage tanks with different aspect ratios namely ratios of large diameter to small diameter of elliptical storage tank (AR). Then, the effect of am and ω was studied on the results. The results show that the relation between MFSD and MHF is almost linear relative to the sway motion amplitude. Moreover, the results show that an increase in the AR causes a decrease in the MFSD and MHF. The results, also, show that the relation between MFSD and MHF is nonlinear relative to the sway motion angular frequency. Furthermore, the results show that an increase in the AR causes that the relation between MFSD and MHF becomes linear relative to the sway motion angular frequency. In addition, MFSD and MHF were minimized in a sway motion with a 7 rad/s angular frequency. Finally, the results show that the elliptical storage tank with AR =1.2-1.4 is the optimum section.

Numerical simulation of aerothermal loads in hypersonic engine inlets due to shock impingement

NASA Technical Reports Server (NTRS)

Ramakrishnan, R.

1992-01-01

The effect of shock impingement on an axial corner simulating the inlet of a hypersonic vehicle engine is modeled using a finite-difference procedure. A three-dimensional dynamic grid adaptation procedure is utilized to move the grids to regions with strong flow gradients. The adaptation procedure uses a grid relocation stencil that is valid at both the interior and boundary points of the finite-difference grid. A linear combination of spatial derivatives of specific flow variables, calculated with finite-element interpolation functions, are used as adaptation measures. This computational procedure is used to study laminar and turbulent Mach 6 flows in the axial corner. The description of flow physics and qualitative measures of heat transfer distributions on cowl and strut surfaces obtained from the analysis are compared with experimental observations. Conclusions are drawn regarding the capability of the numerical scheme for enhanced modeling of high-speed compressible flows.

Accuracy of Gradient Reconstruction on Grids with High Aspect Ratio

NASA Technical Reports Server (NTRS)

Thomas, James

2008-01-01

Gradient approximation methods commonly used in unstructured-grid finite-volume schemes intended for solutions of high Reynolds number flow equations are studied comprehensively. The accuracy of gradients within cells and within faces is evaluated systematically for both node-centered and cell-centered formulations. Computational and analytical evaluations are made on a series of high-aspect-ratio grids with different primal elements, including quadrilateral, triangular, and mixed element grids, with and without random perturbations to the mesh. Both rectangular and cylindrical geometries are considered; the latter serves to study the effects of geometric curvature. The study shows that the accuracy of gradient reconstruction on high-aspect-ratio grids is determined by a combination of the grid and the solution. The contributors to the error are identified and approaches to reduce errors are given, including the addition of higher-order terms in the direction of larger mesh spacing. A parameter GAMMA characterizing accuracy on curved high-aspect-ratio grids is discussed and an approximate-mapped-least-square method using a commonly-available distance function is presented; the method provides accurate gradient reconstruction on general grids. The study is intended to be a reference guide accompanying the construction of accurate and efficient methods for high Reynolds number applications

Sampling Scattered Data Onto Rectangular Grids for Volume Visualization

DTIC Science & Technology

1989-12-01

30 4.4 Building A Rectangular Grid ..... ................ 30 4.5 Sampling Methds ...... ...................... 34 4.6...dimensional data have been developed recently. In computational fluid flow analysis, methods for constructing three dimen- sional numerical grids are...structure of rectangular grids. Because finite element analysis is useful in fields other than fluid flow analysis and the numerical grid has promising

A coarse-grid-projection acceleration method for finite-element incompressible flow computations

NASA Astrophysics Data System (ADS)

Kashefi, Ali; Staples, Anne; FiN Lab Team

2015-11-01

Coarse grid projection (CGP) methodology provides a framework for accelerating computations by performing some part of the computation on a coarsened grid. We apply the CGP to pressure projection methods for finite element-based incompressible flow simulations. Based on it, the predicted velocity field data is restricted to a coarsened grid, the pressure is determined by solving the Poisson equation on the coarse grid, and the resulting data are prolonged to the preset fine grid. The contributions of the CGP method to the pressure correction technique are twofold: first, it substantially lessens the computational cost devoted to the Poisson equation, which is the most time-consuming part of the simulation process. Second, it preserves the accuracy of the velocity field. The velocity and pressure spaces are approximated by Galerkin spectral element using piecewise linear basis functions. A restriction operator is designed so that fine data are directly injected into the coarse grid. The Laplacian and divergence matrices are driven by taking inner products of coarse grid shape functions. Linear interpolation is implemented to construct a prolongation operator. A study of the data accuracy and the CPU time for the CGP-based versus non-CGP computations is presented. Laboratory for Fluid Dynamics in Nature.

Multigrid methods for numerical simulation of laminar diffusion flames

NASA Technical Reports Server (NTRS)

Liu, C.; Liu, Z.; Mccormick, S.

1993-01-01

This paper documents the result of a computational study of multigrid methods for numerical simulation of 2D diffusion flames. The focus is on a simplified combustion model, which is assumed to be a single step, infinitely fast and irreversible chemical reaction with five species (C3H8, O2, N2, CO2 and H2O). A fully-implicit second-order hybrid scheme is developed on a staggered grid, which is stretched in the streamwise coordinate direction. A full approximation multigrid scheme (FAS) based on line distributive relaxation is developed as a fast solver for the algebraic equations arising at each time step. Convergence of the process for the simplified model problem is more than two-orders of magnitude faster than other iterative methods, and the computational results show good grid convergence, with second-order accuracy, as well as qualitatively agreement with the results of other researchers.

Numerical Methods for 2-Dimensional Modeling

DTIC Science & Technology

1980-12-01

high-order finite element methods, and a multidimensional version of the method of lines, both utilizing an optimized stiff integrator for the time...integration. The finite element methods have proved disappointing, but the method of lines has provided an unexpectedly large gain in speed. Two...diffusion problems with the same number of unknowns (a 21 x 41 grid), solved by second-order finite element methods, took over seven minutes on the Cray-i

External Boundary Conditions for Three-Dimensional Problems of Computational Aerodynamics

NASA Technical Reports Server (NTRS)

Tsynkov, Semyon V.

1997-01-01

We consider an unbounded steady-state flow of viscous fluid over a three-dimensional finite body or configuration of bodies. For the purpose of solving this flow problem numerically, we discretize the governing equations (Navier-Stokes) on a finite-difference grid. The grid obviously cannot stretch from the body up to infinity, because the number of the discrete variables in that case would not be finite. Therefore, prior to the discretization we truncate the original unbounded flow domain by introducing some artificial computational boundary at a finite distance of the body. Typically, the artificial boundary is introduced in a natural way as the external boundary of the domain covered by the grid. The flow problem formulated only on the finite computational domain rather than on the original infinite domain is clearly subdefinite unless some artificial boundary conditions (ABC's) are specified at the external computational boundary. Similarly, the discretized flow problem is subdefinite (i.e., lacks equations with respect to unknowns) unless a special closing procedure is implemented at this artificial boundary. The closing procedure in the discrete case is called the ABC's as well. In this paper, we present an innovative approach to constructing highly accurate ABC's for three-dimensional flow computations. The approach extends our previous technique developed for the two-dimensional case; it employs the finite-difference counterparts to Calderon's pseudodifferential boundary projections calculated in the framework of the difference potentials method (DPM) by Ryaben'kii. The resulting ABC's appear spatially nonlocal but particularly easy to implement along with the existing solvers. The new boundary conditions have been successfully combined with the NASA-developed production code TLNS3D and used for the analysis of wing-shaped configurations in subsonic (including incompressible limit) and transonic flow regimes. As demonstrated by the computational experiments and comparisons with the standard (local) methods, the DPM-based ABC's allow one to greatly reduce the size of the computational domain while still maintaining high accuracy of the numerical solution. Moreover, they may provide for a noticeable increase of the convergence rate of multigrid iterations.

A robust, finite element model for hydrostatic surface water flows

USGS Publications Warehouse

Walters, R.A.; Casulli, V.

1998-01-01

A finite element scheme is introduced for the 2-dimensional shallow water equations using semi-implicit methods in time. A semi-Lagrangian method is used to approximate the effects of advection. A wave equation is formed at the discrete level such that the equations decouple into an equation for surface elevation and a momentum equation for the horizontal velocity. The convergence rates and relative computational efficiency are examined with the use of three test cases representing various degrees of difficulty. A test with a polar-quadrant grid investigates the response to local grid-scale forcing and the presence of spurious modes, a channel test case establishes convergence rates, and a field-scale test case examines problems with highly irregular grids.A finite element scheme is introduced for the 2-dimensional shallow water equations using semi-implicit methods in time. A semi-Lagrangian method is used to approximate the effects of advection. A wave equation is formed at the discrete level such that the equations decouple into an equation for surface elevation and a momentum equation for the horizontal velocity. The convergence rates and relative computational efficiency are examined with the use of three test cases representing various degrees of difficulty. A test with a polar-quadrant grid investigates the response to local grid-scale forcing and the presence of spurious modes, a channel test case establishes convergence rates, and a field-scale test case examines problems with highly irregular grids.

GRILLIX: a 3D turbulence code based on the flux-coordinate independent approach

NASA Astrophysics Data System (ADS)

Stegmeir, Andreas; Coster, David; Ross, Alexander; Maj, Omar; Lackner, Karl; Poli, Emanuele

2018-03-01

The GRILLIX code is presented with which plasma turbulence/transport in various geometries can be simulated in 3D. The distinguishing feature of the code is that it is based on the flux-coordinate independent approach (FCI) (Hariri and Ottaviani 2013 Comput. Phys. Commun. 184 2419; Stegmeir et al 2016 Comput. Phys. Commun. 198 139). Cylindrical or Cartesian grids are used on which perpendicular operators are discretised via standard finite difference methods and parallel operators via a field line tracing and interpolation procedure (field line map). This offers a very high flexibility with respect to geometry, especially a separatrix with X-point(s) or a magnetic axis can be treated easily in contrast to approaches which are based on field aligned coordinates and suffer from coordinate singularities. Aiming finally for simulation of edge and scrape-off layer (SOL) turbulence, an isothermal electrostatic drift-reduced Braginskii model (Zeiler et al 1997 Phys. Plasmas 4 2134) has been implemented in GRILLIX. We present the numerical approach, which is based on a toroidally staggered formulation of the FCI, we show verification of the code with the method of manufactured solutions and show a benchmark based on a TORPEX blob experiment, previously performed by several edge/SOL codes (Riva et al 2016 Plasma Phys. Control. Fusion 58 044005). Examples for slab, circular, limiter and diverted geometry are presented. Finally, the results show that the FCI approach in general and GRILLIX in particular are viable approaches in order to tackle simulation of edge/SOL turbulence in diverted geometry.

A memory-efficient staining algorithm in 3D seismic modelling and imaging

NASA Astrophysics Data System (ADS)

Jia, Xiaofeng; Yang, Lu

2017-08-01

The staining algorithm has been proven to generate high signal-to-noise ratio (S/N) images in poorly illuminated areas in two-dimensional cases. In the staining algorithm, the stained wavefield relevant to the target area and the regular source wavefield forward propagate synchronously. Cross-correlating these two wavefields with the backward propagated receiver wavefield separately, we obtain two images: the local image of the target area and the conventional reverse time migration (RTM) image. This imaging process costs massive computer memory for wavefield storage, especially in large scale three-dimensional cases. To make the staining algorithm applicable to three-dimensional RTM, we develop a method to implement the staining algorithm in three-dimensional acoustic modelling in a standard staggered grid finite difference (FD) scheme. The implementation is adaptive to the order of spatial accuracy of the FD operator. The method can be applied to elastic, electromagnetic, and other wave equations. Taking the memory requirement into account, we adopt a random boundary condition (RBC) to backward extrapolate the receiver wavefield and reconstruct it by reverse propagation using the final wavefield snapshot only. Meanwhile, we forward simulate the stained wavefield and source wavefield simultaneously using the nearly perfectly matched layer (NPML) boundary condition. Experiments on a complex geologic model indicate that the RBC-NPML collaborative strategy not only minimizes the memory consumption but also guarantees high quality imaging results. We apply the staining algorithm to three-dimensional RTM via the proposed strategy. Numerical results show that our staining algorithm can produce high S/N images in the target areas with other structures effectively muted.

Test functions for three-dimensional control-volume mixed finite-element methods on irregular grids

USGS Publications Warehouse

Naff, R.L.; Russell, T.F.; Wilson, J.D.; ,; ,; ,; ,; ,

2000-01-01

Numerical methods based on unstructured grids, with irregular cells, usually require discrete shape functions to approximate the distribution of quantities across cells. For control-volume mixed finite-element methods, vector shape functions are used to approximate the distribution of velocities across cells and vector test functions are used to minimize the error associated with the numerical approximation scheme. For a logically cubic mesh, the lowest-order shape functions are chosen in a natural way to conserve intercell fluxes that vary linearly in logical space. Vector test functions, while somewhat restricted by the mapping into the logical reference cube, admit a wider class of possibilities. Ideally, an error minimization procedure to select the test function from an acceptable class of candidates would be the best procedure. Lacking such a procedure, we first investigate the effect of possible test functions on the pressure distribution over the control volume; specifically, we look for test functions that allow for the elimination of intermediate pressures on cell faces. From these results, we select three forms for the test function for use in a control-volume mixed method code and subject them to an error analysis for different forms of grid irregularity; errors are reported in terms of the discrete L2 norm of the velocity error. Of these three forms, one appears to produce optimal results for most forms of grid irregularity.

Stability Analysis of Algebraic Reconstruction for Immersed Boundary Methods with Application in Flow and Transport in Porous Media

NASA Astrophysics Data System (ADS)

Yousefzadeh, M.; Battiato, I.

2017-12-01

Flow and reactive transport problems in porous media often involve complex geometries with stationary or evolving boundaries due to absorption and dissolution processes. Grid based methods (e.g. finite volume, finite element, etc.) are a vital tool for studying these problems. Yet, implementing these methods requires one to answer a very first question of what type of grid is to be used. Among different possible answers, Cartesian grids are one of the most attractive options as they possess simple discretization stencil and are usually straightforward to generate at roughly no computational cost. The Immersed Boundary Method, a Cartesian based methodology, maintains most of the useful features of the structured grids while exhibiting a high-level resilience in dealing with complex geometries. These features make it increasingly more attractive to model transport in evolving porous media as the cost of grid generation reduces greatly. Yet, stability issues and severe time-step restriction due to explicit-time implementation combined with limited studies on the implementation of Neumann (constant flux) and linear and non-linear Robin (e.g. reaction) boundary conditions (BCs) have significantly limited the applicability of IBMs to transport in porous media. We have developed an implicit IBM capable of handling all types of BCs and addressed some numerical issues, including unconditional stability criteria, compactness and reduction of spurious oscillations near the immersed boundary. We tested the method for several transport and flow scenarios, including dissolution processes in porous media, and demonstrate its capabilities. Successful validation against both experimental and numerical data has been carried out.

Finite volume solution of the compressible boundary-layer equations

NASA Technical Reports Server (NTRS)

Loyd, B.; Murman, E. M.

1986-01-01

A box-type finite volume discretization is applied to the integral form of the compressible boundary layer equations. Boundary layer scaling is introduced through the grid construction: streamwise grid lines follow eta = y/h = const., where y is the normal coordinate and h(x) is a scale factor proportional to the boundary layer thickness. With this grid, similarity can be applied explicity to calculate initial conditions. The finite volume method preserves the physical transparency of the integral equations in the discrete approximation. The resulting scheme is accurate, efficient, and conceptually simple. Computations for similar and non-similar flows show excellent agreement with tabulated results, solutions computed with Keller's Box scheme, and experimental data.

Computational methods for vortex dominated compressible flows

NASA Technical Reports Server (NTRS)

Murman, Earll M.

1987-01-01

The principal objectives were to: understand the mechanisms by which Euler equation computations model leading edge vortex flows; understand the vortical and shock wave structures that may exist for different wing shapes, angles of incidence, and Mach numbers; and compare calculations with experiments in order to ascertain the limitations and advantages of Euler equation models. The initial approach utilized the cell centered finite volume Jameson scheme. The final calculation utilized a cell vertex finite volume method on an unstructured grid. Both methods used Runge-Kutta four stage schemes for integrating the equations. The principal findings are briefly summarized.

A mass-conservative adaptive FAS multigrid solver for cell-centered finite difference methods on block-structured, locally-cartesian grids

NASA Astrophysics Data System (ADS)

Feng, Wenqiang; Guo, Zhenlin; Lowengrub, John S.; Wise, Steven M.

2018-01-01

We present a mass-conservative full approximation storage (FAS) multigrid solver for cell-centered finite difference methods on block-structured, locally cartesian grids. The algorithm is essentially a standard adaptive FAS (AFAS) scheme, but with a simple modification that comes in the form of a mass-conservative correction to the coarse-level force. This correction is facilitated by the creation of a zombie variable, analogous to a ghost variable, but defined on the coarse grid and lying under the fine grid refinement patch. We show that a number of different types of fine-level ghost cell interpolation strategies could be used in our framework, including low-order linear interpolation. In our approach, the smoother, prolongation, and restriction operations need never be aware of the mass conservation conditions at the coarse-fine interface. To maintain global mass conservation, we need only modify the usual FAS algorithm by correcting the coarse-level force function at points adjacent to the coarse-fine interface. We demonstrate through simulations that the solver converges geometrically, at a rate that is h-independent, and we show the generality of the solver, applying it to several nonlinear, time-dependent, and multi-dimensional problems. In several tests, we show that second-order asymptotic (h → 0) convergence is observed for the discretizations, provided that (1) at least linear interpolation of the ghost variables is employed, and (2) the mass conservation corrections are applied to the coarse-level force term.

Generalized fourier analyses of the advection-diffusion equation - Part II: two-dimensional domains

NASA Astrophysics Data System (ADS)

Voth, Thomas E.; Martinez, Mario J.; Christon, Mark A.

2004-07-01

Part I of this work presents a detailed multi-methods comparison of the spatial errors associated with the one-dimensional finite difference, finite element and finite volume semi-discretizations of the scalar advection-diffusion equation. In Part II we extend the analysis to two-dimensional domains and also consider the effects of wave propagation direction and grid aspect ratio on the phase speed, and the discrete and artificial diffusivities. The observed dependence of dispersive and diffusive behaviour on propagation direction makes comparison of methods more difficult relative to the one-dimensional results. For this reason, integrated (over propagation direction and wave number) error and anisotropy metrics are introduced to facilitate comparison among the various methods. With respect to these metrics, the consistent mass Galerkin and consistent mass control-volume finite element methods, and their streamline upwind derivatives, exhibit comparable accuracy, and generally out-perform their lumped mass counterparts and finite-difference based schemes. While this work can only be considered a first step in a comprehensive multi-methods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common mathematical framework. Published in 2004 by John Wiley & Sons, Ltd.

A Newton method for the magnetohydrodynamic equilibrium equations

NASA Astrophysics Data System (ADS)

Oliver, Hilary James

We have developed and implemented a (J, B) space Newton method to solve the full nonlinear three dimensional magnetohydrodynamic equilibrium equations in toroidal geometry. Various cases have been run successfully, demonstrating significant improvement over Picard iteration, including a 3D stellarator equilibrium at β = 2%. The algorithm first solves the equilibrium force balance equation for the current density J, given a guess for the magnetic field B. This step is taken from the Picard-iterative PIES 3D equilibrium code. Next, we apply Newton's method to Ampere's Law by expansion of the functional J(B), which is defined by the first step. An analytic calculation in magnetic coordinates, of how the Pfirsch-Schlüter currents vary in the plasma in response to a small change in the magnetic field, yields the Newton gradient term (analogous to ∇f . δx in Newton's method for f(x) = 0). The algorithm is computationally feasible because we do this analytically, and because the gradient term is flux surface local when expressed in terms of a vector potential in an Ar=0 gauge. The equations are discretized by a hybrid spectral/offset grid finite difference technique, and leading order radial dependence is factored from Fourier coefficients to improve finite- difference accuracy near the polar-like origin. After calculating the Newton gradient term we transfer the equation from the magnetic grid to a fixed background grid, which greatly improves the code's performance.

Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface

NASA Astrophysics Data System (ADS)

Coco, Armando; Russo, Giovanni

2018-05-01

In this paper we propose a second-order accurate numerical method to solve elliptic problems with discontinuous coefficients (with general non-homogeneous jumps in the solution and its gradient) in 2D and 3D. The method consists of a finite-difference method on a Cartesian grid in which complex geometries (boundaries and interfaces) are embedded, and is second order accurate in the solution and the gradient itself. In order to avoid the drop in accuracy caused by the discontinuity of the coefficients across the interface, two numerical values are assigned on grid points that are close to the interface: a real value, that represents the numerical solution on that grid point, and a ghost value, that represents the numerical solution extrapolated from the other side of the interface, obtained by enforcing the assigned non-homogeneous jump conditions on the solution and its flux. The method is also extended to the case of matrix coefficient. The linear system arising from the discretization is solved by an efficient multigrid approach. Unlike the 1D case, grid points are not necessarily aligned with the normal derivative and therefore suitable stencils must be chosen to discretize interface conditions in order to achieve second order accuracy in the solution and its gradient. A proper treatment of the interface conditions will allow the multigrid to attain the optimal convergence factor, comparable with the one obtained by Local Fourier Analysis for rectangular domains. The method is robust enough to handle large jump in the coefficients: order of accuracy, monotonicity of the errors and good convergence factor are maintained by the scheme.

Solution of free-boundary problems using finite-element/Newton methods and locally refined grids - Application to analysis of solidification microstructure

NASA Technical Reports Server (NTRS)

Tsiveriotis, K.; Brown, R. A.

1993-01-01

A new method is presented for the solution of free-boundary problems using Lagrangian finite element approximations defined on locally refined grids. The formulation allows for direct transition from coarse to fine grids without introducing non-conforming basis functions. The calculation of elemental stiffness matrices and residual vectors are unaffected by changes in the refinement level, which are accounted for in the loading of elemental data to the global stiffness matrix and residual vector. This technique for local mesh refinement is combined with recently developed mapping methods and Newton's method to form an efficient algorithm for the solution of free-boundary problems, as demonstrated here by sample calculations of cellular interfacial microstructure during directional solidification of a binary alloy.

A numerical study of the axisymmetric Couette-Taylor problem using a fast high-resolution second-order central scheme

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kupferman, R.

The author presents a numerical study of the axisymmetric Couette-Taylor problem using a finite difference scheme. The scheme is based on a staggered version of a second-order central-differencing method combined with a discrete Hodge projection. The use of central-differencing operators obviates the need to trace the characteristic flow associated with the hyperbolic terms. The result is a simple and efficient scheme which is readily adaptable to other geometries and to more complicated flows. The scheme exhibits competitive performance in terms of accuracy, resolution, and robustness. The numerical results agree accurately with linear stability theory and with previous numerical studies.

Multigrid Methods for the Computation of Propagators in Gauge Fields

NASA Astrophysics Data System (ADS)

Kalkreuter, Thomas

Multigrid methods were invented for the solution of discretized partial differential equations in order to overcome the slowness of traditional algorithms by updates on various length scales. In the present work generalizations of multigrid methods for propagators in gauge fields are investigated. Gauge fields are incorporated in algorithms in a covariant way. The kernel C of the restriction operator which averages from one grid to the next coarser grid is defined by projection on the ground-state of a local Hamiltonian. The idea behind this definition is that the appropriate notion of smoothness depends on the dynamics. The ground-state projection choice of C can be used in arbitrary dimension and for arbitrary gauge group. We discuss proper averaging operations for bosons and for staggered fermions. The kernels C can also be used in multigrid Monte Carlo simulations, and for the definition of block spins and blocked gauge fields in Monte Carlo renormalization group studies. Actual numerical computations are performed in four-dimensional SU(2) gauge fields. We prove that our proposals for block spins are “good”, using renormalization group arguments. A central result is that the multigrid method works in arbitrarily disordered gauge fields, in principle. It is proved that computations of propagators in gauge fields without critical slowing down are possible when one uses an ideal interpolation kernel. Unfortunately, the idealized algorithm is not practical, but it was important to answer questions of principle. Practical methods are able to outperform the conjugate gradient algorithm in case of bosons. The case of staggered fermions is harder. Multigrid methods give considerable speed-ups compared to conventional relaxation algorithms, but on lattices up to 184 conjugate gradient is superior.

Seismic Wave Propagation on the Tablet Computer

NASA Astrophysics Data System (ADS)

Emoto, K.

2015-12-01

Tablet computers widely used in recent years. The performance of the tablet computer is improving year by year. Some of them have performance comparable to the personal computer of a few years ago with respect to the calculation speed and the memory size. The convenience and the intuitive operation are the advantage of the tablet computer compared to the desktop PC. I developed the iPad application of the numerical simulation of the seismic wave propagation. The numerical simulation is based on the 2D finite difference method with the staggered-grid scheme. The number of the grid points is 512 x 384 = 196,608. The grid space is 200m in both horizontal and vertical directions. That is the calculation area is 102km x 77km. The time step is 0.01s. In order to reduce the user waiting time, the image of the wave field is drawn simultaneously with the calculation rather than playing the movie after the whole calculation. P and S wave energies are plotted on the screen every 20 steps (0.2s). There is the trade-off between the smooth simulation and the resolution of the wave field image. In the current setting, it takes about 30s to calculate the 10s wave propagation (50 times image updates). The seismogram at the receiver is displayed below of the wave field updated in real time. The default medium structure consists of 3 layers. The layer boundary is defined by 10 movable points with linear interpolation. Users can intuitively change to the arbitrary boundary shape by moving the point. Also users can easily change the source and the receiver positions. The favorite structure can be saved and loaded. For the advance simulation, users can introduce the random velocity fluctuation whose spectrum can be changed to the arbitrary shape. By using this application, everyone can simulate the seismic wave propagation without the special knowledge of the elastic wave equation. So far, the Japanese version of the application is released on the App Store. Now I am preparing the English version.

Locally refined block-centred finite-difference groundwater models: Evaluation of parameter sensitivity and the consequences for inverse modelling

USGS Publications Warehouse

Mehl, S.; Hill, M.C.

2002-01-01

Models with local grid refinement, as often required in groundwater models, pose special problems for model calibration. This work investigates the calculation of sensitivities and the performance of regression methods using two existing and one new method of grid refinement. The existing local grid refinement methods considered are: (a) a variably spaced grid in which the grid spacing becomes smaller near the area of interest and larger where such detail is not needed, and (b) telescopic mesh refinement (TMR), which uses the hydraulic heads or fluxes of a regional model to provide the boundary conditions for a locally refined model. The new method has a feedback between the regional and local grids using shared nodes, and thereby, unlike the TMR methods, balances heads and fluxes at the interfacing boundary. Results for sensitivities are compared for the three methods and the effect of the accuracy of sensitivity calculations are evaluated by comparing inverse modelling results. For the cases tested, results indicate that the inaccuracies of the sensitivities calculated using the TMR approach can cause the inverse model to converge to an incorrect solution.

Locally refined block-centered finite-difference groundwater models: Evaluation of parameter sensitivity and the consequences for inverse modelling and predictions

USGS Publications Warehouse

Mehl, S.; Hill, M.C.

2002-01-01

Models with local grid refinement, as often required in groundwater models, pose special problems for model calibration. This work investigates the calculation of sensitivities and performance of regression methods using two existing and one new method of grid refinement. The existing local grid refinement methods considered are (1) a variably spaced grid in which the grid spacing becomes smaller near the area of interest and larger where such detail is not needed and (2) telescopic mesh refinement (TMR), which uses the hydraulic heads or fluxes of a regional model to provide the boundary conditions for a locally refined model. The new method has a feedback between the regional and local grids using shared nodes, and thereby, unlike the TMR methods, balances heads and fluxes at the interfacing boundary. Results for sensitivities are compared for the three methods and the effect of the accuracy of sensitivity calculations are evaluated by comparing inverse modelling results. For the cases tested, results indicate that the inaccuracies of the sensitivities calculated using the TMR approach can cause the inverse model to converge to an incorrect solution.

Tectonic slicing of subducting oceanic crust along plate interfaces: Numerical modeling

NASA Astrophysics Data System (ADS)

Ruh, J. B.; Le Pourhiet, L.; Agard, Ph.; Burov, E.; Gerya, T.

2015-10-01

Multikilometer-sized slivers of high-pressure low-temperature metamorphic oceanic crust and mantle are observed in many mountain belts. These blueschist and eclogite units were detached from the descending plate during subduction. Large-scale thermo-mechanical numerical models based on finite difference marker-in-cell staggered grid technique are implemented to investigate slicing processes that lead to the detachment of oceanic slivers and their exhumation before the onset of the continental collision phase. In particular, we investigate the role of the serpentinized subcrustal slab mantle in the mechanisms of shallow and deep crustal slicing. Results show that spatially homogeneous serpentinization of the sub-Moho slab mantle leads to complete accretion of oceanic crust within the accretionary wedge. Spatially discontinuous serpentinization of the slab mantle in form of unconnected patches can lead to shallow slicing of the oceanic crust below the accretionary wedge and to its deep slicing at mantle depths depending on the patch length, slab angle, convergence velocity and continental geothermal gradient. P-T paths obtained in this study are compared to natural examples of shallow slicing of the Crescent Terrane below Vancouver Island and deeply sliced crust of the Lago Superiore and Saas-Zermatt units in the Western Alps.

Real-time Tsunami Inundation Prediction Using High Performance Computers

NASA Astrophysics Data System (ADS)

Oishi, Y.; Imamura, F.; Sugawara, D.

2014-12-01

Recently off-shore tsunami observation stations based on cabled ocean bottom pressure gauges are actively being deployed especially in Japan. These cabled systems are designed to provide real-time tsunami data before tsunamis reach coastlines for disaster mitigation purposes. To receive real benefits of these observations, real-time analysis techniques to make an effective use of these data are necessary. A representative study was made by Tsushima et al. (2009) that proposed a method to provide instant tsunami source prediction based on achieving tsunami waveform data. As time passes, the prediction is improved by using updated waveform data. After a tsunami source is predicted, tsunami waveforms are synthesized from pre-computed tsunami Green functions of linear long wave equations. Tsushima et al. (2014) updated the method by combining the tsunami waveform inversion with an instant inversion of coseismic crustal deformation and improved the prediction accuracy and speed in the early stages. For disaster mitigation purposes, real-time predictions of tsunami inundation are also important. In this study, we discuss the possibility of real-time tsunami inundation predictions, which require faster-than-real-time tsunami inundation simulation in addition to instant tsunami source analysis. Although the computational amount is large to solve non-linear shallow water equations for inundation predictions, it has become executable through the recent developments of high performance computing technologies. We conducted parallel computations of tsunami inundation and achieved 6.0 TFLOPS by using 19,000 CPU cores. We employed a leap-frog finite difference method with nested staggered grids of which resolution range from 405 m to 5 m. The resolution ratio of each nested domain was 1/3. Total number of grid points were 13 million, and the time step was 0.1 seconds. Tsunami sources of 2011 Tohoku-oki earthquake were tested. The inundation prediction up to 2 hours after the earthquake occurs took about 2 minutes, which would be sufficient for a practical tsunami inundation predictions. In the presentation, the computational performance of our faster-than-real-time tsunami inundation model will be shown, and preferable tsunami wave source analysis for an accurate inundation prediction will also be discussed.

Viscoelastic Finite Difference Modeling Using Graphics Processing Units

NASA Astrophysics Data System (ADS)

Fabien-Ouellet, G.; Gloaguen, E.; Giroux, B.

2014-12-01

Full waveform seismic modeling requires a huge amount of computing power that still challenges today's technology. This limits the applicability of powerful processing approaches in seismic exploration like full-waveform inversion. This paper explores the use of Graphics Processing Units (GPU) to compute a time based finite-difference solution to the viscoelastic wave equation. The aim is to investigate whether the adoption of the GPU technology is susceptible to reduce significantly the computing time of simulations. The code presented herein is based on the freely accessible software of Bohlen (2002) in 2D provided under a General Public License (GNU) licence. This implementation is based on a second order centred differences scheme to approximate time differences and staggered grid schemes with centred difference of order 2, 4, 6, 8, and 12 for spatial derivatives. The code is fully parallel and is written using the Message Passing Interface (MPI), and it thus supports simulations of vast seismic models on a cluster of CPUs. To port the code from Bohlen (2002) on GPUs, the OpenCl framework was chosen for its ability to work on both CPUs and GPUs and its adoption by most of GPU manufacturers. In our implementation, OpenCL works in conjunction with MPI, which allows computations on a cluster of GPU for large-scale model simulations. We tested our code for model sizes between 1002 and 60002 elements. Comparison shows a decrease in computation time of more than two orders of magnitude between the GPU implementation run on a AMD Radeon HD 7950 and the CPU implementation run on a 2.26 GHz Intel Xeon Quad-Core. The speed-up varies depending on the order of the finite difference approximation and generally increases for higher orders. Increasing speed-ups are also obtained for increasing model size, which can be explained by kernel overheads and delays introduced by memory transfers to and from the GPU through the PCI-E bus. Those tests indicate that the GPU memory size and the slow memory transfers are the limiting factors of our GPU implementation. Those results show the benefits of using GPUs instead of CPUs for time based finite-difference seismic simulations. The reductions in computation time and in hardware costs are significant and open the door for new approaches in seismic inversion.

Introduction to multigrid methods

NASA Technical Reports Server (NTRS)

Wesseling, P.

1995-01-01

These notes were written for an introductory course on the application of multigrid methods to elliptic and hyperbolic partial differential equations for engineers, physicists and applied mathematicians. The use of more advanced mathematical tools, such as functional analysis, is avoided. The course is intended to be accessible to a wide audience of users of computational methods. We restrict ourselves to finite volume and finite difference discretization. The basic principles are given. Smoothing methods and Fourier smoothing analysis are reviewed. The fundamental multigrid algorithm is studied. The smoothing and coarse grid approximation properties are discussed. Multigrid schedules and structured programming of multigrid algorithms are treated. Robustness and efficiency are considered.

Comparison of variational real-space representations of the kinetic energy operator

NASA Astrophysics Data System (ADS)

Skylaris, Chris-Kriton; Diéguez, Oswaldo; Haynes, Peter D.; Payne, Mike C.

2002-08-01

We present a comparison of real-space methods based on regular grids for electronic structure calculations that are designed to have basis set variational properties, using as a reference the conventional method of finite differences (a real-space method that is not variational) and the reciprocal-space plane-wave method which is fully variational. We find that a definition of the finite-difference method [P. Maragakis, J. Soler, and E. Kaxiras, Phys. Rev. B 64, 193101 (2001)] satisfies one of the two properties of variational behavior at the cost of larger errors than the conventional finite-difference method. On the other hand, a technique which represents functions in a number of plane waves which is independent of system size closely follows the plane-wave method and therefore also the criteria for variational behavior. Its application is only limited by the requirement of having functions strictly localized in regions of real space, but this is a characteristic of an increasing number of modern real-space methods, as they are designed to have a computational cost that scales linearly with system size.

Assessment of sub-grid scale dispersion closure with regularized deconvolution method in a particle-laden turbulent jet

NASA Astrophysics Data System (ADS)

Wang, Qing; Zhao, Xinyu; Ihme, Matthias

2017-11-01

Particle-laden turbulent flows are important in numerous industrial applications, such as spray combustion engines, solar energy collectors etc. It is of interests to study this type of flows numerically, especially using large-eddy simulations (LES). However, capturing the turbulence-particle interaction in LES remains challenging due to the insufficient representation of the effect of sub-grid scale (SGS) dispersion. In the present work, a closure technique for the SGS dispersion using regularized deconvolution method (RDM) is assessed. RDM was proposed as the closure for the SGS dispersion in a counterflow spray that is studied numerically using finite difference method on a structured mesh. A presumed form of LES filter is used in the simulations. In the present study, this technique has been extended to finite volume method with an unstructured mesh, where no presumption on the filter form is required. The method is applied to a series of particle-laden turbulent jets. Parametric analyses of the model performance are conducted for flows with different Stokes numbers and Reynolds numbers. The results from LES will be compared against experiments and direct numerical simulations (DNS).

Nonlinear flutter analysis of composite panels

NASA Astrophysics Data System (ADS)

An, Xiaomin; Wang, Yan

2018-05-01

Nonlinear panel flutter is an interesting subject of fluid-structure interaction. In this paper, nonlinear flutter characteristics of curved composite panels are studied in very low supersonic flow. The composite panel with geometric nonlinearity is modeled by a nonlinear finite element method; and the responses are computed by the nonlinear Newmark algorithm. An unsteady aerodynamic solver, which contains a flux splitting scheme and dual time marching technology, is employed in calculating the unsteady pressure of the motion of the panel. Based on a half-step staggered coupled solution, the aeroelastic responses of two composite panels with different radius of R = 5 and R = 2.5 are computed and compared with each other at different dynamic pressure for Ma = 1.05. The nonlinear flutter characteristics comprising limited cycle oscillations and chaos are analyzed and discussed.

Direct numerical simulations of fluid flow, heat transfer and phase changes

NASA Technical Reports Server (NTRS)

Juric, D.; Tryggvason, G.; Han, J.

1997-01-01

Direct numerical simulations of fluid flow, heat transfer, and phase changes are presented. The simulations are made possible by a recently developed finite difference/front tracking method based on the one-field formulation of the governing equations where a single set of conservation equations is written for all the phases involved. The conservation equations are solved on a fixed rectangular grid, but the phase boundaries are kept sharp by tracking them explicitly by a moving grid of lower dimension. The method is discussed and applications to boiling heat transfer and the solidification of drops colliding with a wall are shown.

A computer program to generate two-dimensional grids about airfoils and other shapes by the use of Poisson's equation

NASA Technical Reports Server (NTRS)

Sorenson, R. L.

1980-01-01

A method for generating two dimensional finite difference grids about airfoils and other shapes by the use of the Poisson differential equation is developed. The inhomogeneous terms are automatically chosen such that two important effects are imposed on the grid at both the inner and outer boundaries. The first effect is control of the spacing between mesh points along mesh lines intersecting the boundaries. The second effect is control of the angles with which mesh lines intersect the boundaries. A FORTRAN computer program has been written to use this method. A description of the program, a discussion of the control parameters, and a set of sample cases are included.

The fundamentals of adaptive grid movement

NASA Technical Reports Server (NTRS)

Eiseman, Peter R.

1990-01-01

Basic grid point movement schemes are studied. The schemes are referred to as adaptive grids. Weight functions and equidistribution in one dimension are treated. The specification of coefficients in the linear weight, attraction to a given grid or a curve, and evolutionary forces are considered. Curve by curve and finite volume methods are described. The temporal coupling of partial differential equations solvers and grid generators was discussed.

An Approach for Dynamic Grids

NASA Technical Reports Server (NTRS)

Slater, John W.; Liou, Meng-Sing; Hindman, Richard G.

1994-01-01

An approach is presented for the generation of two-dimensional, structured, dynamic grids. The grid motion may be due to the motion of the boundaries of the computational domain or to the adaptation of the grid to the transient, physical solution. A time-dependent grid is computed through the time integration of the grid speeds which are computed from a system of grid speed equations. The grid speed equations are derived from the time-differentiation of the grid equations so as to ensure that the dynamic grid maintains the desired qualities of the static grid. The grid equations are the Euler-Lagrange equations derived from a variational statement for the grid. The dynamic grid method is demonstrated for a model problem involving boundary motion, an inviscid flow in a converging-diverging nozzle during startup, and a viscous flow over a flat plate with an impinging shock wave. It is shown that the approach is more accurate for transient flows than an approach in which the grid speeds are computed using a finite difference with respect to time of the grid. However, the approach requires significantly more computational effort.

Metal nano-grids for transparent conduction in solar cells

DOE PAGES

Muzzillo, Christopher P.

2017-05-11

A general procedure for predicting metal grid performance in solar cells was developed. Unlike transparent conducting oxides (TCOs) or other homogeneous films, metal grids induce more resistance in the neighbor layer. The resulting balance of transmittance, neighbor and grid resistance was explored in light of cheap lithography advances that have enabled metal nano-grid (MNG) fabrication. The patterned MNGs have junction resistances and degradation rates that are more favorable than solution-synthesized metal nanowires. Neighbor series resistance was simulated by the finite element method, although a simpler analytical model was sufficient in most cases. Finite-difference frequency-domain transmittance simulations were performed for MNGsmore » with minimum wire width (w) of 50 nm, but deviations from aperture transmittance were small in magnitude. Depending on the process, MNGs can exhibit increased series resistance as w is decreased. However, numerous experimental reports have already achieved transmittance-MNG sheet resistance trade-offs comparable to TCOs. The transmittance, neighbor and MNG series resistances were used to parameterize a grid fill factor for a solar cell. In conclusion, this new figure of merit was used to demonstrate that although MNGs have only been employed in low efficiency solar cells, substantial gains in performance are predicted for decreased w in all high efficiency absorber technologies.« less

Metal nano-grids for transparent conduction in solar cells

DOE Office of Scientific and Technical Information (OSTI.GOV)

Muzzillo, Christopher P.

A general procedure for predicting metal grid performance in solar cells was developed. Unlike transparent conducting oxides (TCOs) or other homogeneous films, metal grids induce more resistance in the neighbor layer. The resulting balance of transmittance, neighbor and grid resistance was explored in light of cheap lithography advances that have enabled metal nano-grid (MNG) fabrication. The patterned MNGs have junction resistances and degradation rates that are more favorable than solution-synthesized metal nanowires. Neighbor series resistance was simulated by the finite element method, although a simpler analytical model was sufficient in most cases. Finite-difference frequency-domain transmittance simulations were performed for MNGsmore » with minimum wire width (w) of 50 nm, but deviations from aperture transmittance were small in magnitude. Depending on the process, MNGs can exhibit increased series resistance as w is decreased. However, numerous experimental reports have already achieved transmittance-MNG sheet resistance trade-offs comparable to TCOs. The transmittance, neighbor and MNG series resistances were used to parameterize a grid fill factor for a solar cell. In conclusion, this new figure of merit was used to demonstrate that although MNGs have only been employed in low efficiency solar cells, substantial gains in performance are predicted for decreased w in all high efficiency absorber technologies.« less

Generalized Fourier analyses of the advection-diffusion equation - Part I: one-dimensional domains

NASA Astrophysics Data System (ADS)

Christon, Mark A.; Martinez, Mario J.; Voth, Thomas E.

2004-07-01

This paper presents a detailed multi-methods comparison of the spatial errors associated with finite difference, finite element and finite volume semi-discretizations of the scalar advection-diffusion equation. The errors are reported in terms of non-dimensional phase and group speed, discrete diffusivity, artificial diffusivity, and grid-induced anisotropy. It is demonstrated that Fourier analysis provides an automatic process for separating the discrete advective operator into its symmetric and skew-symmetric components and characterizing the spectral behaviour of each operator. For each of the numerical methods considered, asymptotic truncation error and resolution estimates are presented for the limiting cases of pure advection and pure diffusion. It is demonstrated that streamline upwind Petrov-Galerkin and its control-volume finite element analogue, the streamline upwind control-volume method, produce both an artificial diffusivity and a concomitant phase speed adjustment in addition to the usual semi-discrete artifacts observed in the phase speed, group speed and diffusivity. The Galerkin finite element method and its streamline upwind derivatives are shown to exhibit super-convergent behaviour in terms of phase and group speed when a consistent mass matrix is used in the formulation. In contrast, the CVFEM method and its streamline upwind derivatives yield strictly second-order behaviour. In Part II of this paper, we consider two-dimensional semi-discretizations of the advection-diffusion equation and also assess the affects of grid-induced anisotropy observed in the non-dimensional phase speed, and the discrete and artificial diffusivities. Although this work can only be considered a first step in a comprehensive multi-methods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common analysis framework. Published in 2004 by John Wiley & Sons, Ltd.

Three-dimensional curved grid finite-difference modelling for non-planar rupture dynamics

NASA Astrophysics Data System (ADS)

Zhang, Zhenguo; Zhang, Wei; Chen, Xiaofei

2014-11-01

In this study, we present a new method for simulating the 3-D dynamic rupture process occurring on a non-planar fault. The method is based on the curved-grid finite-difference method (CG-FDM) proposed by Zhang & Chen and Zhang et al. to simulate the propagation of seismic waves in media with arbitrary irregular surface topography. While keeping the advantages of conventional FDM, that is computational efficiency and easy implementation, the CG-FDM also is flexible in modelling the complex fault model by using general curvilinear grids, and thus is able to model the rupture dynamics of a fault with complex geometry, such as oblique dipping fault, non-planar fault, fault with step-over, fault branching, even if irregular topography exists. The accuracy and robustness of this new method have been validated by comparing with the previous results of Day et al., and benchmarks for rupture dynamics simulations. Finally, two simulations of rupture dynamics with complex fault geometry, that is a non-planar fault and a fault rupturing a free surface with topography, are presented. A very interesting phenomenon was observed that topography can weaken the tendency for supershear transition to occur when rupture breaks out at a free surface. Undoubtedly, this new method provides an effective, at least an alternative, tool to simulate the rupture dynamics of a complex non-planar fault, and can be applied to model the rupture dynamics of a real earthquake with complex geometry.

Finite-density transition line for QCD with 695 MeV dynamical fermions

NASA Astrophysics Data System (ADS)

Greensite, Jeff; Höllwieser, Roman

2018-06-01

We apply the relative weights method to SU(3) gauge theory with staggered fermions of mass 695 MeV at a set of temperatures in the range 151 ≤T ≤267 MeV , to obtain an effective Polyakov line action at each temperature. We then apply a mean field method to search for phase transitions in the effective theory at finite densities. The result is a transition line in the plane of temperature and chemical potential, with an end point at high temperature, as expected, but also a second end point at a lower temperature. We cannot rule out the possibilities that a transition line reappears at temperatures lower than the range investigated, or that the second end point is absent for light quarks.

Efficient Computation of Atmospheric Flows with Tempest: Development of Next-Generation Climate and Weather Prediction Algorithms at Non-Hydrostatic Scales

NASA Astrophysics Data System (ADS)

Guerra, J. E.; Ullrich, P. A.

2015-12-01

Tempest is a next-generation global climate and weather simulation platform designed to allow experimentation with numerical methods at very high spatial resolutions. The atmospheric fluid equations are discretized by continuous / discontinuous finite elements in the horizontal and by a staggered nodal finite element method (SNFEM) in the vertical, coupled with implicit/explicit time integration. At global horizontal resolutions below 10km, many important questions remain on optimal techniques for solving the fluid equations. We present results from a suite of meso-scale test cases to validate the performance of the SNFEM applied in the vertical. Internal gravity wave, mountain wave, convective, and Cartesian baroclinic instability tests will be shown at various vertical orders of accuracy and compared with known results.

On-the-fly Numerical Surface Integration for Finite-Difference Poisson-Boltzmann Methods.

PubMed

Cai, Qin; Ye, Xiang; Wang, Jun; Luo, Ray

2011-11-01

Most implicit solvation models require the definition of a molecular surface as the interface that separates the solute in atomic detail from the solvent approximated as a continuous medium. Commonly used surface definitions include the solvent accessible surface (SAS), the solvent excluded surface (SES), and the van der Waals surface. In this study, we present an efficient numerical algorithm to compute the SES and SAS areas to facilitate the applications of finite-difference Poisson-Boltzmann methods in biomolecular simulations. Different from previous numerical approaches, our algorithm is physics-inspired and intimately coupled to the finite-difference Poisson-Boltzmann methods to fully take advantage of its existing data structures. Our analysis shows that the algorithm can achieve very good agreement with the analytical method in the calculation of the SES and SAS areas. Specifically, in our comprehensive test of 1,555 molecules, the average unsigned relative error is 0.27% in the SES area calculations and 1.05% in the SAS area calculations at the grid spacing of 1/2Å. In addition, a systematic correction analysis can be used to improve the accuracy for the coarse-grid SES area calculations, with the average unsigned relative error in the SES areas reduced to 0.13%. These validation studies indicate that the proposed algorithm can be applied to biomolecules over a broad range of sizes and structures. Finally, the numerical algorithm can also be adapted to evaluate the surface integral of either a vector field or a scalar field defined on the molecular surface for additional solvation energetics and force calculations.

Edge equilibrium code for tokamaks

DOE Office of Scientific and Technical Information (OSTI.GOV)

Li, Xujing; Zakharov, Leonid E.; Drozdov, Vladimir V.

2014-01-15

The edge equilibrium code (EEC) described in this paper is developed for simulations of the near edge plasma using the finite element method. It solves the Grad-Shafranov equation in toroidal coordinate and uses adaptive grids aligned with magnetic field lines. Hermite finite elements are chosen for the numerical scheme. A fast Newton scheme which is the same as implemented in the equilibrium and stability code (ESC) is applied here to adjust the grids.

Hyperbolic Prismatic Grid Generation and Solution of Euler Equations on Prismatic Grids

NASA Technical Reports Server (NTRS)

Pandya, S. A.; Chattot, JJ; Hafez, M. M.; Kutler, Paul (Technical Monitor)

1994-01-01

A hyperbolic grid generation method is used to generate prismatic grids and an approach using prismatic grids to solve the Euler equations is presented. The theory of the stability and feasibility of the hyperbolic grid generation method is presented. The hyperbolic grid generation method of Steger et al for structured grids is applied to a three dimensional triangularized surface definition to generate a grid that is unstructured on each successive layer. The grid, however, retains structure in the body-normal direction and has a computational cell shaped like a triangular prism. In order to take advantage of the structure in the normal direction, a finite-volume scheme that treats the unknowns along the normal direction implicitly is introduced and the flow over a sphere is simulated.

Time dependent wave envelope finite difference analysis of sound propagation

NASA Technical Reports Server (NTRS)

Baumeister, K. J.

1984-01-01

A transient finite difference wave envelope formulation is presented for sound propagation, without steady flow. Before the finite difference equations are formulated, the governing wave equation is first transformed to a form whose solution tends not to oscillate along the propagation direction. This transformation reduces the required number of grid points by an order of magnitude. Physically, the transformed pressure represents the amplitude of the conventional sound wave. The derivation for the wave envelope transient wave equation and appropriate boundary conditions are presented as well as the difference equations and stability requirements. To illustrate the method, example solutions are presented for sound propagation in a straight hard wall duct and in a two dimensional straight soft wall duct. The numerical results are in good agreement with exact analytical results.

A numerical simulation of finite-length Taylor-Couette flow

NASA Technical Reports Server (NTRS)

Streett, C. L.; Hussaini, M. Y.

1987-01-01

The processes leading to laminar-turbulent transition in finite-channel-length Taylor-Couette flow are investigated analytically, solving the unsteady incompressible Navier-Stokes equations by spectral-collocation methods. A time-split algorithm, implementable in both axisymmetric and fully three-dimensional time-accurate versions, and an algorithm based on the staggered-mesh discretization of Bernardi and Maday (1986) are described in detail, and results obtained by applying the axisymmetric version of the first algorithm and a steady-state version of the second are presented graphically and compared with published experimental data. The feasibility of full three-dimensional simulations of the progression through chaotic states to turbulence under the constraints of Taylor-Couette flow is demonstrated.

POSTPROCESSING MIXED FINITE ELEMENT METHODS FOR SOLVING CAHN-HILLIARD EQUATION: METHODS AND ERROR ANALYSIS

PubMed Central

Wang, Wansheng; Chen, Long; Zhou, Jie

2015-01-01

A postprocessing technique for mixed finite element methods for the Cahn-Hilliard equation is developed and analyzed. Once the mixed finite element approximations have been computed at a fixed time on the coarser mesh, the approximations are postprocessed by solving two decoupled Poisson equations in an enriched finite element space (either on a finer grid or a higher-order space) for which many fast Poisson solvers can be applied. The nonlinear iteration is only applied to a much smaller size problem and the computational cost using Newton and direct solvers is negligible compared with the cost of the linear problem. The analysis presented here shows that this technique remains the optimal rate of convergence for both the concentration and the chemical potential approximations. The corresponding error estimate obtained in our paper, especially the negative norm error estimates, are non-trivial and different with the existing results in the literatures. PMID:27110063

Computation of tightly-focused laser beams in the FDTD method

PubMed Central

Çapoğlu, İlker R.; Taflove, Allen; Backman, Vadim

2013-01-01

We demonstrate how a tightly-focused coherent TEMmn laser beam can be computed in the finite-difference time-domain (FDTD) method. The electromagnetic field around the focus is decomposed into a plane-wave spectrum, and approximated by a finite number of plane waves injected into the FDTD grid using the total-field/scattered-field (TF/SF) method. We provide an error analysis, and guidelines for the discrete approximation. We analyze the scattering of the beam from layered spaces and individual scatterers. The described method should be useful for the simulation of confocal microscopy and optical data storage. An implementation of the method can be found in our free and open source FDTD software (“Angora”). PMID:23388899

Computation of tightly-focused laser beams in the FDTD method.

PubMed

Capoğlu, Ilker R; Taflove, Allen; Backman, Vadim

2013-01-14

We demonstrate how a tightly-focused coherent TEMmn laser beam can be computed in the finite-difference time-domain (FDTD) method. The electromagnetic field around the focus is decomposed into a plane-wave spectrum, and approximated by a finite number of plane waves injected into the FDTD grid using the total-field/scattered-field (TF/SF) method. We provide an error analysis, and guidelines for the discrete approximation. We analyze the scattering of the beam from layered spaces and individual scatterers. The described method should be useful for the simulation of confocal microscopy and optical data storage. An implementation of the method can be found in our free and open source FDTD software ("Angora").

Accurate solutions for transonic viscous flow over finite wings

NASA Technical Reports Server (NTRS)

Vatsa, V. N.

1986-01-01

An explicit multistage Runge-Kutta type time-stepping scheme is used for solving the three-dimensional, compressible, thin-layer Navier-Stokes equations. A finite-volume formulation is employed to facilitate treatment of complex grid topologies encountered in three-dimensional calculations. Convergence to steady state is expedited through usage of acceleration techniques. Further numerical efficiency is achieved through vectorization of the computer code. The accuracy of the overall scheme is evaluated by comparing the computed solutions with the experimental data for a finite wing under different test conditions in the transonic regime. A grid refinement study ir conducted to estimate the grid requirements for adequate resolution of salient features of such flows.

High-order conservative finite difference GLM-MHD schemes for cell-centered MHD

NASA Astrophysics Data System (ADS)

Mignone, Andrea; Tzeferacos, Petros; Bodo, Gianluigi

2010-08-01

We present and compare third- as well as fifth-order accurate finite difference schemes for the numerical solution of the compressible ideal MHD equations in multiple spatial dimensions. The selected methods lean on four different reconstruction techniques based on recently improved versions of the weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving (MP) schemes as well as slope-limited polynomial reconstruction. The proposed numerical methods are highly accurate in smooth regions of the flow, avoid loss of accuracy in proximity of smooth extrema and provide sharp non-oscillatory transitions at discontinuities. We suggest a numerical formulation based on a cell-centered approach where all of the primary flow variables are discretized at the zone center. The divergence-free condition is enforced by augmenting the MHD equations with a generalized Lagrange multiplier yielding a mixed hyperbolic/parabolic correction, as in Dedner et al. [J. Comput. Phys. 175 (2002) 645-673]. The resulting family of schemes is robust, cost-effective and straightforward to implement. Compared to previous existing approaches, it completely avoids the CPU intensive workload associated with an elliptic divergence cleaning step and the additional complexities required by staggered mesh algorithms. Extensive numerical testing demonstrate the robustness and reliability of the proposed framework for computations involving both smooth and discontinuous features.

The Impact of Varying the Physics Grid Resolution Relative to the Dynamical Core Resolution in CAM-SE-CSLAM

NASA Astrophysics Data System (ADS)

Herrington, A. R.; Lauritzen, P. H.; Reed, K. A.

2017-12-01

The spectral element dynamical core of the Community Atmosphere Model (CAM) has recently been coupled to an approximately isotropic, finite-volume grid per implementation of the conservative semi-Lagrangian multi-tracer transport scheme (CAM-SE-CSLAM; Lauritzen et al. 2017). In this framework, the semi-Lagrangian transport of tracers are computed on the finite-volume grid, while the adiabatic dynamics are solved using the spectral element grid. The physical parameterizations are evaluated on the finite-volume grid, as opposed to the unevenly spaced Gauss-Lobatto-Legendre nodes of the spectral element grid. Computing the physics on the finite-volume grid reduces numerical artifacts such as grid imprinting, possibly because the forcing terms are no longer computed at element boundaries where the resolved dynamics are least smooth. The separation of the physics grid and the dynamics grid allows for a unique opportunity to understand the resolution sensitivity in CAM-SE-CSLAM. The observed large sensitivity of CAM to horizontal resolution is a poorly understood impediment to improved simulations of regional climate using global, variable resolution grids. Here, a series of idealized moist simulations are presented in which the finite-volume grid resolution is varied relative to the spectral element grid resolution in CAM-SE-CSLAM. The simulations are carried out at multiple spectral element grid resolutions, in part to provide a companion set of simulations, in which the spectral element grid resolution is varied relative to the finite-volume grid resolution, but more generally to understand if the sensitivity to the finite-volume grid resolution is consistent across a wider spectrum of resolved scales. Results are interpreted in the context of prior ideas regarding resolution sensitivity of global atmospheric models.

A rotationally biased upwind difference scheme for the Euler equations

NASA Technical Reports Server (NTRS)

Davis, S. F.

1983-01-01

The upwind difference schemes of Godunov, Osher, Roe and van Leer are able to resolve one dimensional steady shocks for the Euler equations within one or two mesh intervals. Unfortunately, this resolution is lost in two dimensions when the shock crosses the computing grid at an oblique angle. To correct this problem, a numerical scheme was developed which automatically locates the angle at which a shock might be expected to cross the computing grid and then constructs separate finite difference formulas for the flux components normal and tangential to this direction. Numerical results which illustrate the ability of this method to resolve steady oblique shocks are presented.

A coarse-grid projection method for accelerating incompressible flow computations

NASA Astrophysics Data System (ADS)

San, Omer; Staples, Anne

2011-11-01

We present a coarse-grid projection (CGP) algorithm for accelerating incompressible flow computations, which is applicable to methods involving Poisson equations as incompressibility constraints. CGP methodology is a modular approach that facilitates data transfer with simple interpolations and uses black-box solvers for the Poisson and advection-diffusion equations in the flow solver. Here, we investigate a particular CGP method for the vorticity-stream function formulation that uses the full weighting operation for mapping from fine to coarse grids, the third-order Runge-Kutta method for time stepping, and finite differences for the spatial discretization. After solving the Poisson equation on a coarsened grid, bilinear interpolation is used to obtain the fine data for consequent time stepping on the full grid. We compute several benchmark flows: the Taylor-Green vortex, a vortex pair merging, a double shear layer, decaying turbulence and the Taylor-Green vortex on a distorted grid. In all cases we use either FFT-based or V-cycle multigrid linear-cost Poisson solvers. Reducing the number of degrees of freedom of the Poisson solver by powers of two accelerates these computations while, for the first level of coarsening, retaining the same level of accuracy in the fine resolution vorticity field.

Mathematical modeling of polymer flooding using the unstructured Voronoi grid

NASA Astrophysics Data System (ADS)

Kireev, T. F.; Bulgakova, G. T.; Khatmullin, I. F.

2017-12-01

Effective recovery of unconventional oil reserves necessitates development of enhanced oil recovery techniques such as polymer flooding. The study investigated the model of polymer flooding with effects of adsorption and water salinity. The model takes into account six components that include elements of the classic black oil model. These components are polymer, salt, water, dead oil, dry gas and dissolved gas. Solution of the problem is obtained by finite volume method on unstructured Voronoi grid using fully implicit scheme and the Newton’s method. To compare several different grid configurations numerical simulation of polymer flooding is performed. The oil rates obtained by a hexagonal locally refined Voronoi grid are shown to be more accurate than the oil rates obtained by a rectangular grid with the same number of cells. The latter effect is caused by high solution accuracy near the wells due to the local grid refinement. Minimization of the grid orientation effect caused by the hexagonal pattern is also demonstrated. However, in the inter-well regions with large Voronoi cells flood front tends to flatten and the water breakthrough moment is smoothed.

A parallel finite-difference method for computational aerodynamics

NASA Technical Reports Server (NTRS)

Swisshelm, Julie M.

1989-01-01

A finite-difference scheme for solving complex three-dimensional aerodynamic flow on parallel-processing supercomputers is presented. The method consists of a basic flow solver with multigrid convergence acceleration, embedded grid refinements, and a zonal equation scheme. Multitasking and vectorization have been incorporated into the algorithm. Results obtained include multiprocessed flow simulations from the Cray X-MP and Cray-2. Speedups as high as 3.3 for the two-dimensional case and 3.5 for segments of the three-dimensional case have been achieved on the Cray-2. The entire solver attained a factor of 2.7 improvement over its unitasked version on the Cray-2. The performance of the parallel algorithm on each machine is analyzed.

Edge Equilibrium Code (EEC) For Tokamaks

DOE Office of Scientific and Technical Information (OSTI.GOV)

Li, Xujling

2014-02-24

The edge equilibrium code (EEC) described in this paper is developed for simulations of the near edge plasma using the finite element method. It solves the Grad-Shafranov equation in toroidal coordinate and uses adaptive grids aligned with magnetic field lines. Hermite finite elements are chosen for the numerical scheme. A fast Newton scheme which is the same as implemented in the equilibrium and stability code (ESC) is applied here to adjust the grids

Moment Inversion of the DPRK Nuclear Tests Using Finite-Difference Three-dimensional Strain Green's Tensors

NASA Astrophysics Data System (ADS)

Bao, X.; Shen, Y.; Wang, N.

2017-12-01

Accurate estimation of the source moment is important for discriminating underground explosions from earthquakes and other seismic sources. In this study, we invert for the full moment tensors of the recent seismic events (since 2016) at the Democratic People's Republic of Korea (PRRK) Punggye-ri test site. We use waveform data from broadband seismic stations located in China, Korea, and Japan in the inversion. Using a non-staggered-grid, finite-difference algorithm, we calculate the strain Green's tensors (SGT) based on one-dimensional (1D) and three-dimensional (3D) Earth models. Taking advantage of the source-receiver reciprocity, a SGT database pre-calculated and stored for the Punggye-ri test site is used in inversion for the source mechanism of each event. With the source locations estimated from cross-correlation using regional Pn and Pn-coda waveforms, we obtain the optimal source mechanism that best fits synthetics to the observed waveforms of both body and surface waves. The moment solutions of the first three events (2016-01-06, 2016-09-09, and 2017-09-03) show dominant isotropic components, as expected from explosions, though there are also notable non-isotropic components. The last event ( 8 minutes after the mb6.3 explosion in 2017) contained mainly implosive component, suggesting a collapse following the explosion. The solutions from the 3D model can better fit observed waveforms than the corresponding solutions from the 1D model. The uncertainty in the resulting moment solution is influenced by heterogeneities not resolved by the Earth model according to the waveform misfit. Using the moment solutions, we predict the peak ground acceleration at the Punggye-ri test site and compare the prediction with corresponding InSAR and other satellite images.

On the Interconnection of Incompatible Solid Finite Element Meshes Using Multipoint Constraints

NASA Technical Reports Server (NTRS)

Fox, G. L.

1985-01-01

Incompatible meshes, i.e., meshes that physically must have a common boundary, but do not necessarily have coincident grid points, can arise in the course of a finite element analysis. For example, two substructures may have been developed at different times for different purposes and it becomes necessary to interconnect the two models. A technique that uses only multipoint constraints, i.e., MPC cards (or MPCS cards in substructuring), is presented. Since the method uses only MPC's, the procedure may apply at any stage in an analysis; no prior planning or special data is necessary.

Modeling dam-break flows using finite volume method on unstructured grid

USDA-ARS?s Scientific Manuscript database

Two-dimensional shallow water models based on unstructured finite volume method and approximate Riemann solvers for computing the intercell fluxes have drawn growing attention because of their robustness, high adaptivity to complicated geometry and ability to simulate flows with mixed regimes and di...

Adaptive Grid Generation for Numerical Solution of Partial Differential Equations.

DTIC Science & Technology

1983-12-01

numerical solution of fluid dynamics problems is presented. However, the method is applicable to the numer- ical evaluation of any partial differential...emphasis is being placed on numerical solution of the governing differential equations by finite difference methods . In the past two decades, considerable...original equations presented in that paper. The solution of the second problem is more difficult. 2 The method of Thompson et al. provides control for

Solution of the neutronics code dynamic benchmark by finite element method

NASA Astrophysics Data System (ADS)

Avvakumov, A. V.; Vabishchevich, P. N.; Vasilev, A. O.; Strizhov, V. F.

2016-10-01

The objective is to analyze the dynamic benchmark developed by Atomic Energy Research for the verification of best-estimate neutronics codes. The benchmark scenario includes asymmetrical ejection of a control rod in a water-type hexagonal reactor at hot zero power. A simple Doppler feedback mechanism assuming adiabatic fuel temperature heating is proposed. The finite element method on triangular calculation grids is used to solve the three-dimensional neutron kinetics problem. The software has been developed using the engineering and scientific calculation library FEniCS. The matrix spectral problem is solved using the scalable and flexible toolkit SLEPc. The solution accuracy of the dynamic benchmark is analyzed by condensing calculation grid and varying degree of finite elements.

Discrete breathers dynamic in a model for DNA chain with a finite stacking enthalpy

NASA Astrophysics Data System (ADS)

Gninzanlong, Carlos Lawrence; Ndjomatchoua, Frank Thomas; Tchawoua, Clément

2018-04-01

The nonlinear dynamics of a homogeneous DNA chain based on site-dependent finite stacking and pairing enthalpies is studied. A new variant of extended discrete nonlinear Schrödinger equation describing the dynamics of modulated wave is derived. The regions of discrete modulational instability of plane carrier waves are studied, and it appears that these zones depend strongly on the phonon frequency of Fourier's mode. The staggered/unstaggered discrete breather (SDB/USDB) is obtained straightforwardly without the staggering transformation, and it is demonstrated that SDBs are less unstable than USDB. The instability of discrete multi-humped SDB/USDB solution does not depend on the number of peaks of the discrete breather (DB). By using the concept of Peierls-Nabarro energy barrier, it appears that the low-frequency DBs are more mobile.

Algebraic grid generation using tensor product B-splines. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Saunders, B. V.

1985-01-01

Finite difference methods are more successful if the accompanying grid has lines which are smooth and nearly orthogonal. The development of an algorithm which produces such a grid when given the boundary description. Topological considerations in structuring the grid generation mapping are discussed. The concept of the degree of a mapping and how it can be used to determine what requirements are necessary if a mapping is to produce a suitable grid is examined. The grid generation algorithm uses a mapping composed of bicubic B-splines. Boundary coefficients are chosen so that the splines produce Schoenberg's variation diminishing spline approximation to the boundary. Interior coefficients are initially chosen to give a variation diminishing approximation to the transfinite bilinear interpolant of the function mapping the boundary of the unit square onto the boundary grid. The practicality of optimizing the grid by minimizing a functional involving the Jacobian of the grid generation mapping at each interior grid point and the dot product of vectors tangent to the grid lines is investigated. Grids generated by using the algorithm are presented.

Scale effect and value criterion of the permeability of the interlayer staggered zones in the basalt of Jinsha River basin, China

NASA Astrophysics Data System (ADS)

Zhou, Zhifang; Lin, Mu; Guo, Qiaona; Chen, Meng

2018-05-01

The hydrogeological characteristics of structural planes are different to those of the associated bedrock. The permeability, and therefore hydraulic conductivity (K), of a structural plane can be significantly different at different scales. The interlayer staggered zones in the Emeishan Basalt of early Late Permian were studied; this formation is located in the Baihetan hydropower project area in Jinsha River Basin, China. The seepage flow distribution of a solid model and two generalized models (A and B) were computed using COMSOL. The K values of the interlayer staggered zones for all three models were calculated by both simulation and analytical methods. The results show that the calculated K results of the generalized models can reflect the variation trend of permeability in each section of the solid model, and the approximate analytical calculation of K can be taken into account in the calculation of K in the generalized models instead of that found by simulation. Further studies are needed to investigate permeability variation in the interlayer staggered zones under the condition of different scales, considering the scaling variation in each section of an interlayer staggered zone. The permeability of each section of an interlayer staggered zone presents a certain degree of dispersivity at small scales; however, the permeability values tends to converge to a similar value as the scale of each section increases. The regularity of each section of the interlayer staggered zones under the condition of different scales can provide a scientific basis for reasonable selection of different engineering options.

Use of upscaled elevation and surface roughness data in two-dimensional surface water models

USGS Publications Warehouse

Hughes, J.D.; Decker, J.D.; Langevin, C.D.

2011-01-01

In this paper, we present an approach that uses a combination of cell-block- and cell-face-averaging of high-resolution cell elevation and roughness data to upscale hydraulic parameters and accurately simulate surface water flow in relatively low-resolution numerical models. The method developed allows channelized features that preferentially connect large-scale grid cells at cell interfaces to be represented in models where these features are significantly smaller than the selected grid size. The developed upscaling approach has been implemented in a two-dimensional finite difference model that solves a diffusive wave approximation of the depth-integrated shallow surface water equations using preconditioned Newton–Krylov methods. Computational results are presented to show the effectiveness of the mixed cell-block and cell-face averaging upscaling approach in maintaining model accuracy, reducing model run-times, and how decreased grid resolution affects errors. Application examples demonstrate that sub-grid roughness coefficient variations have a larger effect on simulated error than sub-grid elevation variations.

An assessment of the adaptive unstructured tetrahedral grid, Euler Flow Solver Code FELISA

NASA Technical Reports Server (NTRS)

Djomehri, M. Jahed; Erickson, Larry L.

1994-01-01

A three-dimensional solution-adaptive Euler flow solver for unstructured tetrahedral meshes is assessed, and the accuracy and efficiency of the method for predicting sonic boom pressure signatures about simple generic models are demonstrated. Comparison of computational and wind tunnel data and enhancement of numerical solutions by means of grid adaptivity are discussed. The mesh generation is based on the advancing front technique. The FELISA code consists of two solvers, the Taylor-Galerkin and the Runge-Kutta-Galerkin schemes, both of which are spacially discretized by the usual Galerkin weighted residual finite-element methods but with different explicit time-marching schemes to steady state. The solution-adaptive grid procedure is based on either remeshing or mesh refinement techniques. An alternative geometry adaptive procedure is also incorporated.

Simulation of hot spots formation and evolution in HMX

NASA Astrophysics Data System (ADS)

Wang, Cheng; Yang, Tonghui

2017-06-01

In order to study the formation and evolution of hot spots under shock loading, HMX explosives were selected as the object of study for the two-dimensional finite difference numerical simulation. A fifth order finite difference weighted essentially non-oscillatory (WENO) scheme and a third order TVD Runge-Kutta method are utilized for the spatial discretization and the time advance, respectively. The governing equations are based on the fluid elasto-plastic control equations. The Mie-Gruneisen equation of state and the ideal gas equation of state are selected to use in the state equation of the solid explosives and gas material. In order to simplify the calculation of the model, the reaction can be considered to complete in one step. The calculated area is [ 3.0 ×10-5 m ] × [ 3.0 ×10-5 m ] . The radius is 0.6 ×10-5 m, and the internal gas is not involved in the reaction. The calculation area is divided into 300×300 grids and 10 grids are selected from the bottom of each column to give the particle velocity u as the initial condition. In the selected grid, different initial velocity 100m/s and 200m/s are loaded respectively to study the influence of hot spot formation and evolution in different impact intensity.

Stability analysis of cylinders with circular cutouts

NASA Technical Reports Server (NTRS)

Almroth, B. O.; Brogan, F. A.; Marlowe, M. B.

1973-01-01

The stability of axially compressed cylinders with circular cutouts is analyzed numerically. An extension of the finite-difference method is used which removes the requirement that displacement components be defined in the directions of the grid lines. The results of this nonlinear analysis are found to be in good agreement with earlier experimental results.

Development of a steady potential solver for use with linearized, unsteady aerodynamic analyses

NASA Technical Reports Server (NTRS)

Hoyniak, Daniel; Verdon, Joseph M.

1991-01-01

A full potential steady flow solver (SFLOW) developed explicitly for use with an inviscid unsteady aerodynamic analysis (LINFLO) is described. The steady solver uses the nonconservative form of the nonlinear potential flow equations together with an implicit, least squares, finite difference approximation to solve for the steady flow field. The difference equations were developed on a composite mesh which consists of a C grid embedded in a rectilinear (H grid) cascade mesh. The composite mesh is capable of resolving blade to blade and far field phenomena on the H grid, while accurately resolving local phenomena on the C grid. The resulting system of algebraic equations is arranged in matrix form using a sparse matrix package and solved by Newton's method. Steady and unsteady results are presented for two cascade configurations: a high speed compressor and a turbine with high exit Mach number.

Comparison of local grid refinement methods for MODFLOW

USGS Publications Warehouse

Mehl, S.; Hill, M.C.; Leake, S.A.

2006-01-01

Many ground water modeling efforts use a finite-difference method to solve the ground water flow equation, and many of these models require a relatively fine-grid discretization to accurately represent the selected process in limited areas of interest. Use of a fine grid over the entire domain can be computationally prohibitive; using a variably spaced grid can lead to cells with a large aspect ratio and refinement in areas where detail is not needed. One solution is to use local-grid refinement (LGR) whereby the grid is only refined in the area of interest. This work reviews some LGR methods and identifies advantages and drawbacks in test cases using MODFLOW-2000. The first test case is two dimensional and heterogeneous; the second is three dimensional and includes interaction with a meandering river. Results include simulations using a uniform fine grid, a variably spaced grid, a traditional method of LGR without feedback, and a new shared node method with feedback. Discrepancies from the solution obtained with the uniform fine grid are investigated. For the models tested, the traditional one-way coupled approaches produced discrepancies in head up to 6.8% and discrepancies in cell-to-cell fluxes up to 7.1%, while the new method has head and cell-to-cell flux discrepancies of 0.089% and 0.14%, respectively. Additional results highlight the accuracy, flexibility, and CPU time trade-off of these methods and demonstrate how the new method can be successfully implemented to model surface water-ground water interactions. Copyright ?? 2006 The Author(s).

Efficient Computation of Atmospheric Flows with Tempest: Validation of Next-Generation Climate and Weather Prediction Algorithms at Non-Hydrostatic Scales

NASA Astrophysics Data System (ADS)

Guerra, Jorge; Ullrich, Paul

2016-04-01

Tempest is a next-generation global climate and weather simulation platform designed to allow experimentation with numerical methods for a wide range of spatial resolutions. The atmospheric fluid equations are discretized by continuous / discontinuous finite elements in the horizontal and by a staggered nodal finite element method (SNFEM) in the vertical, coupled with implicit/explicit time integration. At horizontal resolutions below 10km, many important questions remain on optimal techniques for solving the fluid equations. We present results from a suite of idealized test cases to validate the performance of the SNFEM applied in the vertical with an emphasis on flow features and dynamic behavior. Internal gravity wave, mountain wave, convective bubble, and Cartesian baroclinic instability tests will be shown at various vertical orders of accuracy and compared with known results.

Automatic partitioning of unstructured meshes for the parallel solution of problems in computational mechanics

NASA Technical Reports Server (NTRS)

Farhat, Charbel; Lesoinne, Michel

1993-01-01

Most of the recently proposed computational methods for solving partial differential equations on multiprocessor architectures stem from the 'divide and conquer' paradigm and involve some form of domain decomposition. For those methods which also require grids of points or patches of elements, it is often necessary to explicitly partition the underlying mesh, especially when working with local memory parallel processors. In this paper, a family of cost-effective algorithms for the automatic partitioning of arbitrary two- and three-dimensional finite element and finite difference meshes is presented and discussed in view of a domain decomposed solution procedure and parallel processing. The influence of the algorithmic aspects of a solution method (implicit/explicit computations), and the architectural specifics of a multiprocessor (SIMD/MIMD, startup/transmission time), on the design of a mesh partitioning algorithm are discussed. The impact of the partitioning strategy on load balancing, operation count, operator conditioning, rate of convergence and processor mapping is also addressed. Finally, the proposed mesh decomposition algorithms are demonstrated with realistic examples of finite element, finite volume, and finite difference meshes associated with the parallel solution of solid and fluid mechanics problems on the iPSC/2 and iPSC/860 multiprocessors.

3D CSEM inversion based on goal-oriented adaptive finite element method

NASA Astrophysics Data System (ADS)

Zhang, Y.; Key, K.

2016-12-01

We present a parallel 3D frequency domain controlled-source electromagnetic inversion code name MARE3DEM. Non-linear inversion of observed data is performed with the Occam variant of regularized Gauss-Newton optimization. The forward operator is based on the goal-oriented finite element method that efficiently calculates the responses and sensitivity kernels in parallel using a data decomposition scheme where independent modeling tasks contain different frequencies and subsets of the transmitters and receivers. To accommodate complex 3D conductivity variation with high flexibility and precision, we adopt the dual-grid approach where the forward mesh conforms to the inversion parameter grid and is adaptively refined until the forward solution converges to the desired accuracy. This dual-grid approach is memory efficient, since the inverse parameter grid remains independent from fine meshing generated around the transmitter and receivers by the adaptive finite element method. Besides, the unstructured inverse mesh efficiently handles multiple scale structures and allows for fine-scale model parameters within the region of interest. Our mesh generation engine keeps track of the refinement hierarchy so that the map of conductivity and sensitivity kernel between the forward and inverse mesh is retained. We employ the adjoint-reciprocity method to calculate the sensitivity kernels which establish a linear relationship between changes in the conductivity model and changes in the modeled responses. Our code uses a direcy solver for the linear systems, so the adjoint problem is efficiently computed by re-using the factorization from the primary problem. Further computational efficiency and scalability is obtained in the regularized Gauss-Newton portion of the inversion using parallel dense matrix-matrix multiplication and matrix factorization routines implemented with the ScaLAPACK library. We show the scalability, reliability and the potential of the algorithm to deal with complex geological scenarios by applying it to the inversion of synthetic marine controlled source EM data generated for a complex 3D offshore model with significant seafloor topography.

Static aeroelastic analysis of wings using Euler/Navier-Stokes equations coupled with improved wing-box finite element structures

NASA Technical Reports Server (NTRS)

Guruswamy, Guru P.; MacMurdy, Dale E.; Kapania, Rakesh K.

1994-01-01

Strong interactions between flow about an aircraft wing and the wing structure can result in aeroelastic phenomena which significantly impact aircraft performance. Time-accurate methods for solving the unsteady Navier-Stokes equations have matured to the point where reliable results can be obtained with reasonable computational costs for complex non-linear flows with shock waves, vortices and separations. The ability to combine such a flow solver with a general finite element structural model is key to an aeroelastic analysis in these flows. Earlier work involved time-accurate integration of modal structural models based on plate elements. A finite element model was developed to handle three-dimensional wing boxes, and incorporated into the flow solver without the need for modal analysis. Static condensation is performed on the structural model to reduce the structural degrees of freedom for the aeroelastic analysis. Direct incorporation of the finite element wing-box structural model with the flow solver requires finding adequate methods for transferring aerodynamic pressures to the structural grid and returning deflections to the aerodynamic grid. Several schemes were explored for handling the grid-to-grid transfer of information. The complex, built-up nature of the wing-box complicated this transfer. Aeroelastic calculations for a sample wing in transonic flow comparing various simple transfer schemes are presented and discussed.

An Implicit Characteristic Based Method for Electromagnetics

NASA Technical Reports Server (NTRS)

Beggs, John H.; Briley, W. Roger

2001-01-01

An implicit characteristic-based approach for numerical solution of Maxwell's time-dependent curl equations in flux conservative form is introduced. This method combines a characteristic based finite difference spatial approximation with an implicit lower-upper approximate factorization (LU/AF) time integration scheme. This approach is advantageous for three-dimensional applications because the characteristic differencing enables a two-factor approximate factorization that retains its unconditional stability in three space dimensions, and it does not require solution of tridiagonal systems. Results are given both for a Fourier analysis of stability, damping and dispersion properties, and for one-dimensional model problems involving propagation and scattering for free space and dielectric materials using both uniform and nonuniform grids. The explicit Finite Difference Time Domain Method (FDTD) algorithm is used as a convenient reference algorithm for comparison. The one-dimensional results indicate that for low frequency problems on a highly resolved uniform or nonuniform grid, this LU/AF algorithm can produce accurate solutions at Courant numbers significantly greater than one, with a corresponding improvement in efficiency for simulating a given period of time. This approach appears promising for development of dispersion optimized LU/AF schemes for three dimensional applications.

Effect of grid transparency and finite collector size on determining ion temperature and density by the retarding potential analyzer

NASA Technical Reports Server (NTRS)

Troy, B. E., Jr.; Maier, E. J.

1975-01-01

The effects of the grid transparency and finite collector size on the values of thermal ion density and temperature determined by the standard RPA (retarding potential analyzer) analysis method are investigated. The current-voltage curves calculated for varying RPA parameters and a given ion mass, temperature, and density are analyzed by the standard RPA method. It is found that only small errors in temperature and density are introduced for an RPA with typical dimensions, and that even when the density error is substantial for nontypical dimensions, the temperature error remains minimum.

Time-reversal imaging techniques applied to tremor waveforms near Cholame, California to locate tectonic tremor

NASA Astrophysics Data System (ADS)

Horstmann, T.; Harrington, R. M.; Cochran, E. S.

2012-12-01

Frequently, the lack of distinctive phase arrivals makes locating tectonic tremor more challenging than locating earthquakes. Classic location algorithms based on travel times cannot be directly applied because impulsive phase arrivals are often difficult to recognize. Traditional location algorithms are often modified to use phase arrivals identified from stacks of recurring low-frequency events (LFEs) observed within tremor episodes, rather than single events. Stacking the LFE waveforms improves the signal-to-noise ratio for the otherwise non-distinct phase arrivals. In this study, we apply a different method to locate tectonic tremor: a modified time-reversal imaging approach that potentially exploits the information from the entire tremor waveform instead of phase arrivals from individual LFEs. Time reversal imaging uses the waveforms of a given seismic source recorded by multiple seismometers at discrete points on the surface and a 3D velocity model to rebroadcast the waveforms back into the medium to identify the seismic source location. In practice, the method works by reversing the seismograms recorded at each of the stations in time, and back-propagating them from the receiver location individually into the sub-surface as a new source time function. We use a staggered-grid, finite-difference code with 2.5 ms time steps and a grid node spacing of 50 m to compute the rebroadcast wavefield. We calculate the time-dependent curl field at each grid point of the model volume for each back-propagated seismogram. To locate the tremor, we assume that the source time function back-propagated from each individual station produces a similar curl field at the source position. We then cross-correlate the time dependent curl field functions and calculate a median cross-correlation coefficient at each grid point. The highest median cross-correlation coefficient in the model volume is expected to represent the source location. For our analysis, we use the velocity model of Thurber et al. (2006) interpolated to a grid spacing of 50 m. Such grid spacing corresponds to frequencies of up to 8 Hz, which is suitable to calculate the wave propagation of tremor. Our dataset contains continuous broadband data from 13 STS-2 seismometers deployed from May 2010 to July 2011 along the Cholame segment of the San Andreas Fault as well as data from the HRSN and PBO networks. Initial synthetic results from tests on a 2D plane using a line of 15 receivers suggest that we are able to recover accurate event locations to within 100 m horizontally and 300 m depth. We conduct additional synthetic tests to determine the influence of signal-to-noise ratio, number of stations used, and the uncertainty in the velocity model on the location result by adding noise to the seismograms and perturbations to the velocity model. Preliminary results show accurate show location results to within 400 m with a median signal-to-noise ratio of 3.5 and 5% perturbations in the velocity model. The next steps will entail performing the synthetic tests on the 3D velocity model, and applying the method to tremor waveforms. Furthermore, we will determine the spatial and temporal distribution of the source locations and compare our results to those by Sumy and others.

Equivalent modulus method for finite element simulation of the sound absorption of anechoic coating backed with orthogonally rib-stiffened plate

NASA Astrophysics Data System (ADS)

Jin, Zhongkun; Yin, Yao; Liu, Bilong

2016-03-01

The finite element method is often used to investigate the sound absorption of anechoic coating backed with orthogonally rib-stiffened plate. Since the anechoic coating contains cavities, the number of grid nodes of a periodic unit cell is usually large. An equivalent modulus method is proposed to reduce the large amount of nodes by calculating an equivalent homogeneous layer. Applications of this method in several models show that the method can well predict the sound absorption coefficient of such structure in a wide frequency range. Based on the simulation results, the sound absorption performance of such structure and the influences of different backings on the first absorption peak are also discussed.

Interpolation methods and the accuracy of lattice-Boltzmann mesh refinement

DOE PAGES

Guzik, Stephen M.; Weisgraber, Todd H.; Colella, Phillip; ...

2013-12-10

A lattice-Boltzmann model to solve the equivalent of the Navier-Stokes equations on adap- tively refined grids is presented. A method for transferring information across interfaces between different grid resolutions was developed following established techniques for finite- volume representations. This new approach relies on a space-time interpolation and solving constrained least-squares problems to ensure conservation. The effectiveness of this method at maintaining the second order accuracy of lattice-Boltzmann is demonstrated through a series of benchmark simulations and detailed mesh refinement studies. These results exhibit smaller solution errors and improved convergence when compared with similar approaches relying only on spatial interpolation. Examplesmore » highlighting the mesh adaptivity of this method are also provided.« less

Finite-surface method for the Maxwell equations with corner singularities

NASA Technical Reports Server (NTRS)

Vinokur, Marcel; Yarrow, Maurice

1994-01-01

The finite-surface method for the two-dimensional Maxwell equations in generalized coordinates is extended to treat perfect conductor boundaries with sharp corners. Known singular forms of the grid and the electromagnetic fields in the neighborhood of each corner are used to obtain accurate approximations to the surface and line integrals appearing in the method. Numerical results are presented for a harmonic plane wave incident on a finite flat plate. Comparisons with exact solutions show good agreement.

A new approach to flow simulation in highly heterogeneous porous media

DOE Office of Scientific and Technical Information (OSTI.GOV)

Rame, M.; Killough, J.E.

In this paper, applications are presented for a new numerical method - operator splittings on multiple grids (OSMG) - devised for simulations in heterogeneous porous media. A coarse-grid, finite-element pressure solver is interfaced with a fine-grid timestepping scheme. The CPU time for the pressure solver is greatly reduced and concentration fronts have minimal numerical dispersion.

Comparing the GPR responses of real experiment and simulation of cavity

NASA Astrophysics Data System (ADS)

Yu, H.; Nam, M. J.; Kim, C.; Lee, D. K.

2017-12-01

Seoul, capital city of South Korea, has been suffering from ground subsidence mainly caused by cavities beneath the road. Urban subsidence usually brings serious social problems such as damages of human life, properties and so on. To prevent ground subsidence, Korea government embark much money in developing techniques to detect cavities in advance. Ground penetrating radar (GPR) is known as the most effective method among geophysical surveys in exploring underground cavitied but shallow ones only. For the study of GPR responses for underground cavities, real scale physical models have been made and GPR surveys are conducted. In simulating cavities with various sizes at various depths, spheres of polystyrene have been used since the electric permittivity of polystyrene has a similar value to that of the air. However, the real scale experiments only used simple shapes of cavities due to its expensive construction cost and further changing in shapes of cavities is limited once they are built. For not only comparison between field responses for the physical model and numerical responses but also for analyzing GPR responses for more various cavity shapes in numerous environments, we conducted numerical simulation of GPR responses using three-dimensional (3D) finite difference time domain (FDTD) GPR modeling algorithm employing staggered grid. We first construct numerical modeling for models similar to the physical models to confirm considering radiation pattern in numerical modeling of GPR responses which is critical to generate similar responses to field GPR data. Further, GPR responses computed for various shapes of cavities in several different environments determine not only additional construction of the physical cavities but also analyze the characteristics of GPR responses.

Updating finite element dynamic models using an element-by-element sensitivity methodology

NASA Technical Reports Server (NTRS)

Farhat, Charbel; Hemez, Francois M.

1993-01-01

A sensitivity-based methodology for improving the finite element model of a given structure using test modal data and a few sensors is presented. The proposed method searches for both the location and sources of the mass and stiffness errors and does not interfere with the theory behind the finite element model while correcting these errors. The updating algorithm is derived from the unconstrained minimization of the squared L sub 2 norms of the modal dynamic residuals via an iterative two-step staggered procedure. At each iteration, the measured mode shapes are first expanded assuming that the model is error free, then the model parameters are corrected assuming that the expanded mode shapes are exact. The numerical algorithm is implemented in an element-by-element fashion and is capable of 'zooming' on the detected error locations. Several simulation examples which demonstate the potential of the proposed methodology are discussed.

Advanced numerical methods for three dimensional two-phase flow calculations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Toumi, I.; Caruge, D.

1997-07-01

This paper is devoted to new numerical methods developed for both one and three dimensional two-phase flow calculations. These methods are finite volume numerical methods and are based on the use of Approximate Riemann Solvers concepts to define convective fluxes versus mean cell quantities. The first part of the paper presents the numerical method for a one dimensional hyperbolic two-fluid model including differential terms as added mass and interface pressure. This numerical solution scheme makes use of the Riemann problem solution to define backward and forward differencing to approximate spatial derivatives. The construction of this approximate Riemann solver uses anmore » extension of Roe`s method that has been successfully used to solve gas dynamic equations. As far as the two-fluid model is hyperbolic, this numerical method seems very efficient for the numerical solution of two-phase flow problems. The scheme was applied both to shock tube problems and to standard tests for two-fluid computer codes. The second part describes the numerical method in the three dimensional case. The authors discuss also some improvements performed to obtain a fully implicit solution method that provides fast running steady state calculations. Such a scheme is not implemented in a thermal-hydraulic computer code devoted to 3-D steady-state and transient computations. Some results obtained for Pressurised Water Reactors concerning upper plenum calculations and a steady state flow in the core with rod bow effect evaluation are presented. In practice these new numerical methods have proved to be stable on non staggered grids and capable of generating accurate non oscillating solutions for two-phase flow calculations.« less

Method to predict external store carriage characteristics at transonic speeds

NASA Technical Reports Server (NTRS)

Rosen, Bruce S.

1988-01-01

Development of a computational method for prediction of external store carriage characteristics at transonic speeds is described. The geometric flexibility required for treatment of pylon-mounted stores is achieved by computing finite difference solutions on a five-level embedded grid arrangement. A completely automated grid generation procedure facilitates applications. Store modeling capability consists of bodies of revolution with multiple fore and aft fins. A body-conforming grid improves the accuracy of the computed store body flow field. A nonlinear relaxation scheme developed specifically for modified transonic small disturbance flow equations enhances the method's numerical stability and accuracy. As a result, treatment of lower aspect ratio, more highly swept and tapered wings is possible. A limited supersonic freestream capability is also provided. Pressure, load distribution, and force/moment correlations show good agreement with experimental data for several test cases. A detailed computer program description for the Transonic Store Carriage Loads Prediction (TSCLP) Code is included.

A Mixed Finite Volume Element Method for Flow Calculations in Porous Media

NASA Technical Reports Server (NTRS)

Jones, Jim E.

1996-01-01

A key ingredient in the simulation of flow in porous media is the accurate determination of the velocities that drive the flow. The large scale irregularities of the geology, such as faults, fractures, and layers suggest the use of irregular grids in the simulation. Work has been done in applying the finite volume element (FVE) methodology as developed by McCormick in conjunction with mixed methods which were developed by Raviart and Thomas. The resulting mixed finite volume element discretization scheme has the potential to generate more accurate solutions than standard approaches. The focus of this paper is on a multilevel algorithm for solving the discrete mixed FVE equations. The algorithm uses a standard cell centered finite difference scheme as the 'coarse' level and the more accurate mixed FVE scheme as the 'fine' level. The algorithm appears to have potential as a fast solver for large size simulations of flow in porous media.

Solving the forward problem of magnetoacoustic tomography with magnetic induction by means of the finite element method

NASA Astrophysics Data System (ADS)

Li, Xun; Li, Xu; Zhu, Shanan; He, Bin

2009-05-01

Magnetoacoustic tomography with magnetic induction (MAT-MI) is a recently proposed imaging modality to image the electrical impedance of biological tissue. It combines the good contrast of electrical impedance tomography with the high spatial resolution of sonography. In this paper, a three-dimensional MAT-MI forward problem was investigated using the finite element method (FEM). The corresponding FEM formulae describing the forward problem are introduced. In the finite element analysis, magnetic induction in an object with conductivity values close to biological tissues was first carried out. The stimulating magnetic field was simulated as that generated from a three-dimensional coil. The corresponding acoustic source and field were then simulated. Computer simulation studies were conducted using both concentric and eccentric spherical conductivity models with different geometric specifications. In addition, the grid size for finite element analysis was evaluated for the model calibration and evaluation of the corresponding acoustic field.

Solving the Forward Problem of Magnetoacoustic Tomography with Magnetic Induction by Means of the Finite Element Method

PubMed Central

Li, Xun; Li, Xu; Zhu, Shanan; He, Bin

2010-01-01

Magnetoacoustic Tomography with Magnetic Induction (MAT-MI) is a recently proposed imaging modality to image the electrical impedance of biological tissue. It combines the good contrast of electrical impedance tomography with the high spatial resolution of sonography. In this paper, three-dimensional MAT-MI forward problem was investigated using the finite element method (FEM). The corresponding FEM formulas describing the forward problem are introduced. In the finite element analysis, magnetic induction in an object with conductivity values close to biological tissues was first carried out. The stimulating magnetic field was simulated as that generated from a three-dimensional coil. The corresponding acoustic source and field were then simulated. Computer simulation studies were conducted using both concentric and eccentric spherical conductivity models with different geometric specifications. In addition, the grid size for finite element analysis was evaluated for model calibration and evaluation of the corresponding acoustic field. PMID:19351978

Numerical Treatment of Degenerate Diffusion Equations via Feller's Boundary Classification, and Applications

NASA Technical Reports Server (NTRS)

Cacio, Emanuela; Cohn, Stephen E.; Spigler, Renato

2011-01-01

A numerical method is devised to solve a class of linear boundary-value problems for one-dimensional parabolic equations degenerate at the boundaries. Feller theory, which classifies the nature of the boundary points, is used to decide whether boundary conditions are needed to ensure uniqueness, and, if so, which ones they are. The algorithm is based on a suitable preconditioned implicit finite-difference scheme, grid, and treatment of the boundary data. Second-order accuracy, unconditional stability, and unconditional convergence of solutions of the finite-difference scheme to a constant as the time-step index tends to infinity are further properties of the method. Several examples, pertaining to financial mathematics, physics, and genetics, are presented for the purpose of illustration.

High-Order Accurate Solutions to the Helmholtz Equation in the Presence of Boundary Singularities

NASA Astrophysics Data System (ADS)

Britt, Darrell Steven, Jr.

Problems of time-harmonic wave propagation arise in important fields of study such as geological surveying, radar detection/evasion, and aircraft design. These often involve highfrequency waves, which demand high-order methods to mitigate the dispersion error. We propose a high-order method for computing solutions to the variable-coefficient inhomogeneous Helmholtz equation in two dimensions on domains bounded by piecewise smooth curves of arbitrary shape with a finite number of boundary singularities at known locations. We utilize compact finite difference (FD) schemes on regular structured grids to achieve highorder accuracy due to their efficiency and simplicity, as well as the capability to approximate variable-coefficient differential operators. In this work, a 4th-order compact FD scheme for the variable-coefficient Helmholtz equation on a Cartesian grid in 2D is derived and tested. The well known limitation of finite differences is that they lose accuracy when the boundary curve does not coincide with the discretization grid, which is a severe restriction on the geometry of the computational domain. Therefore, the algorithm presented in this work combines high-order FD schemes with the method of difference potentials (DP), which retains the efficiency of FD while allowing for boundary shapes that are not aligned with the grid without sacrificing the accuracy of the FD scheme. Additionally, the theory of DP allows for the universal treatment of the boundary conditions. One of the significant contributions of this work is the development of an implementation that accommodates general boundary conditions (BCs). In particular, Robin BCs with discontinuous coefficients are studied, for which we introduce a piecewise parameterization of the boundary curve. Problems with discontinuities in the boundary data itself are also studied. We observe that the design convergence rate suffers whenever the solution loses regularity due to the boundary conditions. This is because the FD scheme is only consistent for classical solutions of the PDE. For this reason, we implement the method of singularity subtraction as a means for restoring the design accuracy of the scheme in the presence of singularities at the boundary. While this method is well studied for low order methods and for problems in which singularities arise from the geometry (e.g., corners), we adapt it to our high-order scheme for curved boundaries via a conformal mapping and show that it can also be used to restore accuracy when the singularity arises from the BCs rather than the geometry. Altogether, the proposed methodology for 2D boundary value problems is computationally efficient, easily handles a wide class of boundary conditions and boundary shapes that are not aligned with the discretization grid, and requires little modification for solving new problems.

Quantitative Estimation of Seismic Velocity Changes Using Time-Lapse Seismic Data and Elastic-Wave Sensitivity Approach

NASA Astrophysics Data System (ADS)

Denli, H.; Huang, L.

2008-12-01

Quantitative monitoring of reservoir property changes is essential for safe geologic carbon sequestration. Time-lapse seismic surveys have the potential to effectively monitor fluid migration in the reservoir that causes geophysical property changes such as density, and P- and S-wave velocities. We introduce a novel method for quantitative estimation of seismic velocity changes using time-lapse seismic data. The method employs elastic sensitivity wavefields, which are the derivatives of elastic wavefield with respect to density, P- and S-wave velocities of a target region. We derive the elastic sensitivity equations from analytical differentiations of the elastic-wave equations with respect to seismic-wave velocities. The sensitivity equations are coupled with the wave equations in a way that elastic waves arriving in a target reservoir behave as a secondary source to sensitivity fields. We use a staggered-grid finite-difference scheme with perfectly-matched layers absorbing boundary conditions to simultaneously solve the elastic-wave equations and the elastic sensitivity equations. By elastic-wave sensitivities, a linear relationship between relative seismic velocity changes in the reservoir and time-lapse seismic data at receiver locations can be derived, which leads to an over-determined system of equations. We solve this system of equations using a least- square method for each receiver to obtain P- and S-wave velocity changes. We validate the method using both surface and VSP synthetic time-lapse seismic data for a multi-layered model and the elastic Marmousi model. Then we apply it to the time-lapse field VSP data acquired at the Aneth oil field in Utah. A total of 10.5K tons of CO2 was injected into the oil reservoir between the two VSP surveys for enhanced oil recovery. The synthetic and field data studies show that our new method can quantitatively estimate changes in seismic velocities within a reservoir due to CO2 injection/migration.

Interfacial characteristics of hybrid nanocomposite under thermomechanical loading

NASA Astrophysics Data System (ADS)

Choyal, Vijay; Kundalwal, Shailesh I.

2017-12-01

In this work, an improved shear lag model was developed to investigate the interfacial characteristics of three-phase hybrid nanocomposite which is reinforced with microscale fibers augmented with carbon nanotubes on their circumferential surfaces. The shear lag model accounts for (i) radial and axial deformations of different transversely isotropic constituents, (ii) thermomechanical loads on the representative volume element (RVE), and (iii) staggering effect of adjacent RVEs. The results from the current newly developed shear lag model are validated with the finite element simulations and found to be in good agreement. This study reveals that the reduction in the maximum value of the axial stress in the fiber and the interfacial shear stress along its length become more pronounced in the presence of applied thermomechanical loads on the staggered RVEs. The existence of shear tractions along the RVE length plays a significant role in the interfacial characteristics and cannot be ignored.

On Accuracy of Adaptive Grid Methods for Captured Shocks

NASA Technical Reports Server (NTRS)

Yamaleev, Nail K.; Carpenter, Mark H.

2002-01-01

The accuracy of two grid adaptation strategies, grid redistribution and local grid refinement, is examined by solving the 2-D Euler equations for the supersonic steady flow around a cylinder. Second- and fourth-order linear finite difference shock-capturing schemes, based on the Lax-Friedrichs flux splitting, are used to discretize the governing equations. The grid refinement study shows that for the second-order scheme, neither grid adaptation strategy improves the numerical solution accuracy compared to that calculated on a uniform grid with the same number of grid points. For the fourth-order scheme, the dominant first-order error component is reduced by the grid adaptation, while the design-order error component drastically increases because of the grid nonuniformity. As a result, both grid adaptation techniques improve the numerical solution accuracy only on the coarsest mesh or on very fine grids that are seldom found in practical applications because of the computational cost involved. Similar error behavior has been obtained for the pressure integral across the shock. A simple analysis shows that both grid adaptation strategies are not without penalties in the numerical solution accuracy. Based on these results, a new grid adaptation criterion for captured shocks is proposed.

YAP Version 4.0

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nelson, Eric M.

2004-05-20

The YAP software library computes (1) electromagnetic modes, (2) electrostatic fields, (3) magnetostatic fields and (4) particle trajectories in 2d and 3d models. The code employs finite element methods on unstructured grids of tetrahedral, hexahedral, prism and pyramid elements, with linear through cubic element shapes and basis functions to provide high accuracy. The novel particle tracker is robust, accurate and efficient, even on unstructured grids with discontinuous fields. This software library is a component of the MICHELLE 3d finite element gun code.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Herrnstein, Aaron R.

An ocean model with adaptive mesh refinement (AMR) capability is presented for simulating ocean circulation on decade time scales. The model closely resembles the LLNL ocean general circulation model with some components incorporated from other well known ocean models when appropriate. Spatial components are discretized using finite differences on a staggered grid where tracer and pressure variables are defined at cell centers and velocities at cell vertices (B-grid). Horizontal motion is modeled explicitly with leapfrog and Euler forward-backward time integration, and vertical motion is modeled semi-implicitly. New AMR strategies are presented for horizontal refinement on a B-grid, leapfrog time integration,more » and time integration of coupled systems with unequal time steps. These AMR capabilities are added to the LLNL software package SAMRAI (Structured Adaptive Mesh Refinement Application Infrastructure) and validated with standard benchmark tests. The ocean model is built on top of the amended SAMRAI library. The resulting model has the capability to dynamically increase resolution in localized areas of the domain. Limited basin tests are conducted using various refinement criteria and produce convergence trends in the model solution as refinement is increased. Carbon sequestration simulations are performed on decade time scales in domains the size of the North Atlantic and the global ocean. A suggestion is given for refinement criteria in such simulations. AMR predicts maximum pH changes and increases in CO 2 concentration near the injection sites that are virtually unattainable with a uniform high resolution due to extremely long run times. Fine scale details near the injection sites are achieved by AMR with shorter run times than the finest uniform resolution tested despite the need for enhanced parallel performance. The North Atlantic simulations show a reduction in passive tracer errors when AMR is applied instead of a uniform coarse resolution. No dramatic or persistent signs of error growth in the passive tracer outgassing or the ocean circulation are observed to result from AMR.« less

FDTD simulation of transmittance characteristics of one-dimensional conducting electrodes.

PubMed

Lee, Kilbock; Song, Seok Ho; Ahn, Jinho

2014-03-24

We investigated transparent conducting electrodes consisting of periodic one-dimensional Ag or Al grids with widths from 25 nm to 5 μm via the finite-difference time-domain method. To retain high transmittance, two grid configurations with opening ratios of 90% and 95% were simulated. Polarization-dependent characteristics of the transmission spectra revealed that the overall transmittance of micron-scale grid electrodes may be estimated by the sum of light power passing through the uncovered area and the light power penetrating the covered metal layer. However, several dominant physical phenomena significantly affect the transmission spectra of the nanoscale grids: Rayleigh anomaly, transmission decay in TE polarized mode, and localized surface plasmon resonance. We conclude that, for applications of transparent electrodes, the critical feature sizes of conducting 1D grids should not be less than the wavelength scale in order to maintain uniform and predictable transmission spectra and low electrical resistivity.

Numerical aerodynamic simulation facility. [for flows about three-dimensional configurations

NASA Technical Reports Server (NTRS)

Bailey, F. R.; Hathaway, A. W.

1978-01-01

Critical to the advancement of computational aerodynamics capability is the ability to simulate flows about three-dimensional configurations that contain both compressible and viscous effects, including turbulence and flow separation at high Reynolds numbers. Analyses were conducted of two solution techniques for solving the Reynolds averaged Navier-Stokes equations describing the mean motion of a turbulent flow with certain terms involving the transport of turbulent momentum and energy modeled by auxiliary equations. The first solution technique is an implicit approximate factorization finite-difference scheme applied to three-dimensional flows that avoids the restrictive stability conditions when small grid spacing is used. The approximate factorization reduces the solution process to a sequence of three one-dimensional problems with easily inverted matrices. The second technique is a hybrid explicit/implicit finite-difference scheme which is also factored and applied to three-dimensional flows. Both methods are applicable to problems with highly distorted grids and a variety of boundary conditions and turbulence models.

MODFLOW–USG version 1: An unstructured grid version of MODFLOW for simulating groundwater flow and tightly coupled processes using a control volume finite-difference formulation

USGS Publications Warehouse

Panday, Sorab; Langevin, Christian D.; Niswonger, Richard G.; Ibaraki, Motomu; Hughes, Joseph D.

2013-01-01

A new version of MODFLOW, called MODFLOW–USG (for UnStructured Grid), was developed to support a wide variety of structured and unstructured grid types, including nested grids and grids based on prismatic triangles, rectangles, hexagons, and other cell shapes. Flexibility in grid design can be used to focus resolution along rivers and around wells, for example, or to subdiscretize individual layers to better represent hydrostratigraphic units. MODFLOW–USG is based on an underlying control volume finite difference (CVFD) formulation in which a cell can be connected to an arbitrary number of adjacent cells. To improve accuracy of the CVFD formulation for irregular grid-cell geometries or nested grids, a generalized Ghost Node Correction (GNC) Package was developed, which uses interpolated heads in the flow calculation between adjacent connected cells. MODFLOW–USG includes a Groundwater Flow (GWF) Process, based on the GWF Process in MODFLOW–2005, as well as a new Connected Linear Network (CLN) Process to simulate the effects of multi-node wells, karst conduits, and tile drains, for example. The CLN Process is tightly coupled with the GWF Process in that the equations from both processes are formulated into one matrix equation and solved simultaneously. This robustness results from using an unstructured grid with unstructured matrix storage and solution schemes. MODFLOW–USG also contains an optional Newton-Raphson formulation, based on the formulation in MODFLOW–NWT, for improving solution convergence and avoiding problems with the drying and rewetting of cells. Because the existing MODFLOW solvers were developed for structured and symmetric matrices, they were replaced with a new Sparse Matrix Solver (SMS) Package developed specifically for MODFLOW–USG. The SMS Package provides several methods for resolving nonlinearities and multiple symmetric and asymmetric linear solution schemes to solve the matrix arising from the flow equations and the Newton-Raphson formulation, respectively.

Field Test of a Hybrid Finite-Difference and Analytic Element Regional Model.

PubMed

Abrams, D B; Haitjema, H M; Feinstein, D T; Hunt, R J

2016-01-01

Regional finite-difference models often have cell sizes that are too large to sufficiently model well-stream interactions. Here, a steady-state hybrid model is applied whereby the upper layer or layers of a coarse MODFLOW model are replaced by the analytic element model GFLOW, which represents surface waters and wells as line and point sinks. The two models are coupled by transferring cell-by-cell leakage obtained from the original MODFLOW model to the bottom of the GFLOW model. A real-world test of the hybrid model approach is applied on a subdomain of an existing model of the Lake Michigan Basin. The original (coarse) MODFLOW model consists of six layers, the top four of which are aggregated into GFLOW as a single layer, while the bottom two layers remain part of MODFLOW in the hybrid model. The hybrid model and a refined "benchmark" MODFLOW model simulate similar baseflows. The hybrid and benchmark models also simulate similar baseflow reductions due to nearby pumping when the well is located within the layers represented by GFLOW. However, the benchmark model requires refinement of the model grid in the local area of interest, while the hybrid approach uses a gridless top layer and is thus unaffected by grid discretization errors. The hybrid approach is well suited to facilitate cost-effective retrofitting of existing coarse grid MODFLOW models commonly used for regional studies because it leverages the strengths of both finite-difference and analytic element methods for predictions in mildly heterogeneous systems that can be simulated with steady-state conditions. © 2015, National Ground Water Association.

Thermo-mechanical analysis of an internal cooling system with various configurations of a combustion liner after shell

NASA Astrophysics Data System (ADS)

Moon, Hokyu; Kim, Kyung Min; Park, Jun Su; Kim, Beom Seok; Cho, Hyung Hee

2015-12-01

The after-shell section, which is part of the gas turbine combustion liner, is exposed to the hottest combustion gas. Various cooling schemes have been applied to protect against severe thermal load. However, there is a significant discrepancy in the thermal expansion with large temperature differences, resulting in thermo-mechanical crack formation. In this study, to reduce combustion liner damage, thermo-mechanical analysis was conducted on three after-shell section configurations: inline-discrete divider wall, staggered divider wall, and swirler wall arrays. These array components are well-known heat-transfer enhancement structures in the duct. In the numerical analyses, the heat transfer characteristics, temperature and thermo-mechanical stress distribution were evaluated using finite volume method and finite element method commercial codes. As a result, we demonstrated that the temperature and the thermo-mechanical stress distribution were readily dependent on the structural array for cooling effectiveness and structural support in each modified cooling system. Compared with the reference model, the swirler wall array was most effective in diminishing the thermo-mechanical stress concentration, especially on the inner ring that is vulnerable to crack formation.

Cyberinfrastructure for the Unified Study of Earth Structure and Earthquake Sources in Complex Geologic Environments

NASA Astrophysics Data System (ADS)

Zhao, L.; Chen, P.; Jordan, T. H.; Olsen, K. B.; Maechling, P.; Faerman, M.

2004-12-01

The Southern California Earthquake Center (SCEC) is developing a Community Modeling Environment (CME) to facilitate the computational pathways of physics-based seismic hazard analysis (Maechling et al., this meeting). Major goals are to facilitate the forward modeling of seismic wavefields in complex geologic environments, including the strong ground motions that cause earthquake damage, and the inversion of observed waveform data for improved models of Earth structure and fault rupture. Here we report on a unified approach to these coupled inverse problems that is based on the ability to generate and manipulate wavefields in densely gridded 3D Earth models. A main element of this approach is a database of receiver Green tensors (RGT) for the seismic stations, which comprises all of the spatial-temporal displacement fields produced by the three orthogonal unit impulsive point forces acting at each of the station locations. Once the RGT database is established, synthetic seismograms for any earthquake can be simply calculated by extracting a small, source-centered volume of the RGT from the database and applying the reciprocity principle. The partial derivatives needed for point- and finite-source inversions can be generated in the same way. Moreover, the RGT database can be employed in full-wave tomographic inversions launched from a 3D starting model, because the sensitivity (Fréchet) kernels for travel-time and amplitude anomalies observed at seismic stations in the database can be computed by convolving the earthquake-induced displacement field with the station RGTs. We illustrate all elements of this unified analysis with an RGT database for 33 stations of the California Integrated Seismic Network in and around the Los Angeles Basin, which we computed for the 3D SCEC Community Velocity Model (SCEC CVM3.0) using a fourth-order staggered-grid finite-difference code. For a spatial grid spacing of 200 m and a time resolution of 10 ms, the calculations took ~19,000 node-hours on the Linux cluster at USC's High-Performance Computing Center. The 33-station database with a volume of ~23.5 TB was archived in the SCEC digital library at the San Diego Supercomputer Center using the Storage Resource Broker (SRB). From a laptop, anyone with access to this SRB collection can compute synthetic seismograms for an arbitrary source in the CVM in a matter of minutes. Efficient approaches have been implemented to use this RGT database in the inversions of waveforms for centroid and finite moment tensors and tomographic inversions to improve the CVM. Our experience with these large problems suggests areas where the cyberinfrastructure currently available for geoscience computation needs to be improved.

Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations: Inviscid Fluxes

NASA Technical Reports Server (NTRS)

Diskin, Boris; Thomas, James L.

2010-01-01

Cell-centered and node-centered approaches have been compared for unstructured finite-volume discretization of inviscid fluxes. The grids range from regular grids to irregular grids, including mixed-element grids and grids with random perturbations of nodes. Accuracy, complexity, and convergence rates of defect-correction iterations are studied for eight nominally second-order accurate schemes: two node-centered schemes with weighted and unweighted least-squares (LSQ) methods for gradient reconstruction and six cell-centered schemes two node-averaging with and without clipping and four schemes that employ different stencils for LSQ gradient reconstruction. The cell-centered nearest-neighbor (CC-NN) scheme has the lowest complexity; a version of the scheme that involves smart augmentation of the LSQ stencil (CC-SA) has only marginal complexity increase. All other schemes have larger complexity; complexity of node-centered (NC) schemes are somewhat lower than complexity of cell-centered node-averaging (CC-NA) and full-augmentation (CC-FA) schemes. On highly anisotropic grids typical of those encountered in grid adaptation, discretization errors of five of the six cell-centered schemes converge with second order on all tested grids; the CC-NA scheme with clipping degrades solution accuracy to first order. The NC schemes converge with second order on regular and/or triangular grids and with first order on perturbed quadrilaterals and mixed-element grids. All schemes may produce large relative errors in gradient reconstruction on grids with perturbed nodes. Defect-correction iterations for schemes employing weighted least-square gradient reconstruction diverge on perturbed stretched grids. Overall, the CC-NN and CC-SA schemes offer the best options of the lowest complexity and secondorder discretization errors. On anisotropic grids over a curved body typical of turbulent flow simulations, the discretization errors converge with second order and are small for the CC-NN, CC-SA, and CC-FA schemes on all grids and for NC schemes on triangular grids; the discretization errors of the CC-NA scheme without clipping do not converge on irregular grids. Accurate gradient reconstruction can be achieved by introducing a local approximate mapping; without approximate mapping, only the NC scheme with weighted LSQ method provides accurate gradients. Defect correction iterations for the CC-NA scheme without clipping diverge; for the NC scheme with weighted LSQ method, the iterations either diverge or converge very slowly. The best option in curved geometries is the CC-SA scheme that offers low complexity, second-order discretization errors, and fast convergence.

Analysis of the cylinder’s movement characteristics after entering water based on CFD

NASA Astrophysics Data System (ADS)

Liu, Xianlong

2017-10-01

It’s a variable speed motion after the cylinder vertical entry the water. Using dynamic mesh is mostly unstructured grid, and the calculation results are not ideal and consume huge computing resources. CFD method is used to calculate the resistance of the cylinder at different velocities. Cubic spline interpolation method is used to obtain the resistance at fixed speeds. The finite difference method is used to solve the motion equation, and the acceleration, velocity, displacement and other physical quantities are obtained after the cylinder enters the water.

Scalability and performance of data-parallel pressure-based multigrid methods for viscous flows

DOE Office of Scientific and Technical Information (OSTI.GOV)

Blosch, E.L.; Shyy, W.

1996-05-01

A full-approximation storage multigrid method for solving the steady-state 2-d incompressible Navier-Stokes equations on staggered grids has been implemented in Fortran on the CM-5, using the array aliasing feature in CM-Fortran to avoid declaring fine-grid-sized arrays on all levels while still allowing a variable number of grid levels. Thus, the storage cost scales with the number of unknowns, allowing us to consider significantly larger problems than would otherwise be possible. Timings over a range of problem sizes and numbers of processors, up to 4096 x 4096 on 512 nodes, show that the smoothing procedure, a pressure-correction technique, is scalable andmore » that the restriction and prolongation steps are nearly so. The performance obtained for the multigrid method is 333 Mflops out of the theoretical peak 4 Gflops on a 32-node CM-5. In comparison, a single-grid computation obtained 420 Mflops. The decrease is due to the inefficiency of the smoothing iterations on the coarse grid levels. W cycles cost much more and are much less efficient than V cycles, due to the increased contribution from the coarse grids. The convergence rate characteristics of the pressure-correction multigrid method are investigated in a Re = 5000 lid-driven cavity flow and a Re = 300 symmetric backward-facing step flow, using either a defect-correction scheme or a second-order upwind scheme. A heuristic technique relating the convergence tolerances for the course grids to the truncation error of the discretization has been found effective and robust. With second-order upwinding on all grid levels, a 5-level 320 x 80 step flow solution was obtained in 20 V cycles, which corresponds to a smoothing rate of 0.7, and required 25 s on a 32-node CM-5. Overall, the convergence rates obtained in the present work are comparable to the most competitive findings reported in the literature. 62 refs., 13 figs.« less

Scalability and Performance of Data-Parallel Pressure-Based Multigrid Methods for Viscous Flows

NASA Astrophysics Data System (ADS)

Blosch, Edwin L.; Shyy, Wei

1996-05-01

A full-approximation storage multigrid method for solving the steady-state 2-dincompressible Navier-Stokes equations on staggered grids has been implemented in Fortran on the CM-5,using the array aliasing feature in CM-Fortran to avoid declaring fine-grid-sized arrays on all levels while still allowing a variable number of grid levels. Thus, the storage cost scales with the number of unknowns,allowing us to consider significantly larger problems than would otherwise be possible. Timings over a range of problem sizes and numbers of processors, up to 4096 × 4096 on 512 nodes, show that the smoothing procedure, a pressure-correction technique, is scalable and that the restriction and prolongation steps are nearly so. The performance obtained for the multigrid method is 333 Mflops out of the theoretical peak 4 Gflops on a 32-node CM-5. In comparison, a single-grid computation obtained 420 Mflops. The decrease is due to the inefficiency of the smoothing iterations on the coarse grid levels. W cycles cost much more and are much less efficient than V cycles, due to the increased contribution from the coarse grids. The convergence rate characteristics of the pressure-correction multigrid method are investigated in a Re = 5000 lid-driven cavity flow and a Re = 300 symmetric backward-facing step flow, using either a defect-correction scheme or a second-order upwind scheme. A heuristic technique relating the convergence tolerances for the coarse grids to the truncation error of the discretization has been found effective and robust. With second-order upwinding on all grid levels, a 5-level 320× 80 step flow solution was obtained in 20 V cycles, which corresponds to a smoothing rate of 0.7, and required 25 s on a 32-node CM-5. Overall, the convergence rates obtained in the present work are comparable to the most competitive findings reported in the literature.

On the effects of grid ill-conditioning in three dimensional finite element vector potential magnetostatic field computations

NASA Technical Reports Server (NTRS)

Wang, R.; Demerdash, N. A.

1990-01-01

The effects of finite element grid geometries and associated ill-conditioning were studied in single medium and multi-media (air-iron) three dimensional magnetostatic field computation problems. The sensitivities of these 3D field computations to finite element grid geometries were investigated. It was found that in single medium applications the unconstrained magnetic vector potential curl-curl formulation in conjunction with first order finite elements produce global results which are almost totally insensitive to grid geometries. However, it was found that in multi-media (air-iron) applications first order finite element results are sensitive to grid geometries and consequent elemental shape ill-conditioning. These sensitivities were almost totally eliminated by means of the use of second order finite elements in the field computation algorithms. Practical examples are given in this paper to demonstrate these aspects mentioned above.

Improved finite-difference computation of the van der Waals force: One-dimensional case

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pinto, Fabrizio

2009-10-15

We present an improved demonstration of the calculation of Casimir forces in one-dimensional systems based on the recently proposed numerical imaginary frequency Green's function computation approach. The dispersion force on two thick lossy dielectric slabs separated by an empty gap and placed within a perfectly conducting cavity is obtained from the Green's function of the modified Helmholtz equation by means of an ordinary finite-difference method. In order to demonstrate the possibility to develop algorithms to explore complex geometries in two and three dimensions to higher order in the mesh spacing, we generalize existing classical electromagnetism algebraic methods to generate themore » difference equations for dielectric boundaries not coinciding with any grid points. Diagnostic tests are presented to monitor the accuracy of our implementation of the method and follow-up applications in higher dimensions are introduced.« less

Refinement of Out of Circularity and Thickness Measurements of a Cylinder for Finite Element Analysis

DTIC Science & Technology

2016-09-01

UNCLASSIFIED UNCLASSIFIED Refinement of Out of Circularity and Thickness Measurements of a Cylinder for Finite Element Analysis...significant effect on the collapse strength and must be accurately represented in finite element analysis to obtain accurate results. Often it is necessary...to interpolate measurements from a relatively coarse grid to a refined finite element model and methods that have wide general acceptance are

Numerical Investigation of Pressure Profile in Hydrodynamic Lubrication Thrust Bearing.

PubMed

Najar, Farooq Ahmad; Harmain, G A

2014-01-01

Reynolds equation is solved using finite difference method (FDM) on the surface of the tilting pad to find the pressure distribution in the lubricant oil film. Different pressure profiles with grid independence are described. The present work evaluates pressure at various locations after performing a thorough grid refinement. In recent similar works, this aspect has not been addressed. However, present study shows that it can have significant effect on the pressure profile. Results of a sector shaped pad are presented and it is shown that the maximum average value of pressure is 12% (approximately) greater than the previous results. Grid independence occurs after 24 × 24 grids. A parameter "ψ" has been proposed to provide convenient indicator of obtaining grid independent results. ψ = |(P refinedgrid - P Refrence-grid)/P refinedgrid|, ψ ≤ ε, where "ε" can be fixed to a convenient value and a constant minimum film thickness value of 75 μm is used in present study. This important parameter is highlighted in the present work; the location of the peak pressure zone in terms of (r, θ) coordinates is getting shifted by changing the grid size which will help the designer and experimentalist to conveniently determine the position of pressure measurement probe.

Stability Test for Transient-Temperature Calculations

NASA Technical Reports Server (NTRS)

Campbell, W.

1984-01-01

Graphical test helps assure numerical stability of calculations of transient temperature or diffusion in composite medium. Rectangular grid forms basis of two-dimensional finite-difference model for heat conduction or other diffusion like phenomena. Model enables calculation of transient heat transfer among up to four different materials that meet at grid point.

A dynamically adaptive multigrid algorithm for the incompressible Navier-Stokes equations: Validation and model problems

NASA Technical Reports Server (NTRS)

Thompson, C. P.; Leaf, G. K.; Vanrosendale, J.

1991-01-01

An algorithm is described for the solution of the laminar, incompressible Navier-Stokes equations. The basic algorithm is a multigrid based on a robust, box-based smoothing step. Its most important feature is the incorporation of automatic, dynamic mesh refinement. This algorithm supports generalized simple domains. The program is based on a standard staggered-grid formulation of the Navier-Stokes equations for robustness and efficiency. Special grid transfer operators were introduced at grid interfaces in the multigrid algorithm to ensure discrete mass conservation. Results are presented for three models: the driven-cavity, a backward-facing step, and a sudden expansion/contraction.

The STAGGER-grid: A grid of 3D stellar atmosphere models. V. Synthetic stellar spectra and broad-band photometry

NASA Astrophysics Data System (ADS)

Chiavassa, A.; Casagrande, L.; Collet, R.; Magic, Z.; Bigot, L.; Thévenin, F.; Asplund, M.

2018-03-01

Context. The surface structures and dynamics of cool stars are characterised by the presence of convective motions and turbulent flows which shape the emergent spectrum. Aims: We used realistic three-dimensional (3D) radiative hydrodynamical simulations from the STAGGER-grid to calculate synthetic spectra with the radiative transfer code OPTIM3D for stars with different stellar parameters to predict photometric colours and convective velocity shifts. Methods: We calculated spectra from 1000 to 200 000 Å with a constant resolving power of λ/Δλ = 20 000 and from 8470 and 8710 Å (Gaia Radial Velocity Spectrometer - RVS - spectral range) with a constant resolving power of λ/Δλ = 300 000. Results: We used synthetic spectra to compute theoretical colours in the Johnson-Cousins UBV (RI)C, SDSS, 2MASS, Gaia, SkyMapper, Strömgren systems, and HST-WFC3. Our synthetic magnitudes are compared with those obtained using 1D hydrostatic models. We showed that 1D versus 3D differences are limited to a small percent except for the narrow filters that span the optical and UV region of the spectrum. In addition, we derived the effect of the convective velocity fields on selected Fe I lines. We found the overall convective shift for 3D simulations with respect to the reference 1D hydrostatic models, revealing line shifts of between -0.235 and +0.361 km s-1. We showed a net correlation of the convective shifts with the effective temperature: lower effective temperatures denote redshifts and higher effective temperatures denote blueshifts. We conclude that the extraction of accurate radial velocities from RVS spectra need an appropriate wavelength correction from convection shifts. Conclusions: The use of realistic 3D hydrodynamical stellar atmosphere simulations has a small but significant impact on the predicted photometry compared with classical 1D hydrostatic models for late-type stars. We make all the spectra publicly available for the community through the POLLUX database. Tables 5-8 are only available at the CDS and Table B.1 is also available at the CDS and via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/611/A11

Aerodynamic analysis for aircraft with nacelles, pylons, and winglets at transonic speeds

NASA Technical Reports Server (NTRS)

Boppe, Charles W.

1987-01-01

A computational method has been developed to provide an analysis for complex realistic aircraft configurations at transonic speeds. Wing-fuselage configurations with various combinations of pods, pylons, nacelles, and winglets can be analyzed along with simpler shapes such as airfoils, isolated wings, and isolated bodies. The flexibility required for the treatment of such diverse geometries is obtained by using a multiple nested grid approach in the finite-difference relaxation scheme. Aircraft components (and their grid systems) can be added or removed as required. As a result, the computational method can be used in the same manner as a wind tunnel to study high-speed aerodynamic interference effects. The multiple grid approach also provides high boundary point density/cost ratio. High resolution pressure distributions can be obtained. Computed results are correlated with wind tunnel and flight data using four different transport configurations. Experimental/computational component interference effects are included for cases where data are available. The computer code used for these comparisons is described in the appendices.

Incompressible flow simulations on regularized moving meshfree grids

NASA Astrophysics Data System (ADS)

Vasyliv, Yaroslav; Alexeev, Alexander

2017-11-01

A moving grid meshfree solver for incompressible flows is presented. To solve for the flow field, a semi-implicit approximate projection method is directly discretized on meshfree grids using General Finite Differences (GFD) with sharp interface stencil modifications. To maintain a regular grid, an explicit shift is used to relax compressed pseudosprings connecting a star node to its cloud of neighbors. The following test cases are used for validation: the Taylor-Green vortex decay, the analytic and modified lid-driven cavities, and an oscillating cylinder enclosed in a container for a range of Reynolds number values. We demonstrate that 1) the grid regularization does not impede the second order spatial convergence rate, 2) the Courant condition can be used for time marching but the projection splitting error reduces the convergence rate to first order, and 3) moving boundaries and arbitrary grid distortions can readily be handled. Financial support provided by the National Science Foundation (NSF) Graduate Research Fellowship, Grant No. DGE-1148903.

Combined Uncertainty and A-Posteriori Error Bound Estimates for CFD Calculations: Theory and Implementation

NASA Technical Reports Server (NTRS)

Barth, Timothy J.

2014-01-01

Simulation codes often utilize finite-dimensional approximation resulting in numerical error. Some examples include, numerical methods utilizing grids and finite-dimensional basis functions, particle methods using a finite number of particles. These same simulation codes also often contain sources of uncertainty, for example, uncertain parameters and fields associated with the imposition of initial and boundary data,uncertain physical model parameters such as chemical reaction rates, mixture model parameters, material property parameters, etc.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nutaro, James; Kuruganti, Teja

Numerical simulations of the wave equation that are intended to provide accurate time domain solutions require a computational mesh with grid points separated by a distance less than the wavelength of the source term and initial data. However, calculations of radio signal pathloss generally do not require accurate time domain solutions. This paper describes an approach for calculating pathloss by using the finite difference time domain and transmission line matrix models of wave propagation on a grid with points separated by distances much greater than the signal wavelength. The calculated pathloss can be kept close to the true value formore » freespace propagation with an appropriate selection of initial conditions. This method can also simulate diffraction with an error governed by the ratio of the signal wavelength to the grid spacing.« less

Compact cell-centered discretization stencils at fine-coarse block structured grid interfaces

NASA Astrophysics Data System (ADS)

Pletzer, Alexander; Jamroz, Ben; Crockett, Robert; Sides, Scott

2014-03-01

Different strategies for coupling fine-coarse grid patches are explored in the context of the adaptive mesh refinement (AMR) method. We show that applying linear interpolation to fill in the fine grid ghost values can produce a finite volume stencil of comparable accuracy to quadratic interpolation provided the cell volumes are adjusted. The volume of fine cells expands whereas the volume of neighboring coarse cells contracts. The amount by which the cells contract/expand depends on whether the interface is a face, an edge, or a corner. It is shown that quadratic or better interpolation is required when the conductivity is spatially varying, anisotropic, the refinement ratio is other than two, or when the fine-coarse interface is concave.

Experimental Study of Aligned and Staggered Wind Farms in a Convective Boundary Layer

NASA Astrophysics Data System (ADS)

Markfort, Corey; Zhang, Wei; Porte-Agel, Fernando

2011-11-01

Wind farm-atmosphere interaction is complicated by turbine configuration and thermal effects on momentum and kinetic energy fluxes. Wind farms of finite length have been modeled as increased surface roughness or as a sparse canopy; however it is not clear which approach is more appropriate. Experiments were conducted in a thermally controlled boundary layer wind tunnel, using a custom x-wire/cold wire and surface heat flux sensors, to understand the effect of aligned versus staggered turbine configurations on momentum absorption and flow adjustment in a convective boundary layer (CBL). Results for experiments of a large farm show the span-wise averaged flow statistics exhibit similar turbulent transport properties to that of canopy flows. The wake adjusts within and grows over the farm more quickly for a staggered compared to an aligned farm. Using canopy flow scaling, we show that the flow equilibrates faster and the overall momentum absorption is higher in a staggered compared to an aligned farm. Wake recovery behind a single turbine is facilitated by buoyancy in a CBL (Zhang et al. under review). We find a similar effect in wind farms resulting in reduced effective roughness and momentum absorption. We also find a reduction of surface heat flux for both wind farms, but greater for the staggered farm.

Flow analysis for efficient design of wavy structured microchannel mixing devices

NASA Astrophysics Data System (ADS)

Kanchan, Mithun; Maniyeri, Ranjith

2018-04-01

Microfluidics is a rapidly growing field of applied research which is strongly driven by demands of bio-technology and medical innovation. Lab-on-chip (LOC) is one such application which deals with integrating bio-laboratory on micro-channel based single fluidic chip. Since fluid flow in such devices is restricted to laminar regime, designing an efficient passive modulator to induce chaotic mixing for such diffusion based flow is a major challenge. In the present work two-dimensional numerical simulation of viscous incompressible flow is carried out using immersed boundary method (IBM) to obtain an efficient design for wavy structured micro-channel mixing devices. The continuity and Navier-Stokes equations governing the flow are solved by fractional step based finite volume method on a staggered Cartesian grid system. IBM uses Eulerian co-ordinates to describe fluid flow and Lagrangian co-ordinates to describe solid boundary. Dirac delta function is used to couple both these co-ordinate variables. A tether forcing term is used to impose the no-slip boundary condition on the wavy structure and fluid interface. Fluid flow analysis by varying Reynolds number is carried out for four wavy structure models and one straight line model. By analyzing fluid accumulation zones and flow velocities, it can be concluded that straight line structure performs better mixing for low Reynolds number and Model 2 for higher Reynolds number. Thus wavy structures can be incorporated in micro-channels to improve mixing efficiency.

Fast Running Urban Dispersion Model for Radiological Dispersal Device (RDD) Releases: Model Description and Validation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gowardhan, Akshay; Neuscamman, Stephanie; Donetti, John

Aeolus is an efficient three-dimensional computational fluid dynamics code based on finite volume method developed for predicting transport and dispersion of contaminants in a complex urban area. It solves the time dependent incompressible Navier-Stokes equation on a regular Cartesian staggered grid using a fractional step method. It also solves a scalar transport equation for temperature and using the Boussinesq approximation. The model also includes a Lagrangian dispersion model for predicting the transport and dispersion of atmospheric contaminants. The model can be run in an efficient Reynolds Average Navier-Stokes (RANS) mode with a run time of several minutes, or a moremore » detailed Large Eddy Simulation (LES) mode with run time of hours for a typical simulation. This report describes the model components, including details on the physics models used in the code, as well as several model validation efforts. Aeolus wind and dispersion predictions are compared to field data from the Joint Urban Field Trials 2003 conducted in Oklahoma City (Allwine et al 2004) including both continuous and instantaneous releases. Newly implemented Aeolus capabilities include a decay chain model and an explosive Radiological Dispersal Device (RDD) source term; these capabilities are described. Aeolus predictions using the buoyant explosive RDD source are validated against two experimental data sets: the Green Field explosive cloud rise experiments conducted in Israel (Sharon et al 2012) and the Full-Scale RDD Field Trials conducted in Canada (Green et al 2016).« less

Modal element method for potential flow in non-uniform ducts: Combining closed form analysis with CFD

NASA Technical Reports Server (NTRS)

Baumeister, Kenneth J.; Baumeister, Joseph F.

1994-01-01

An analytical procedure is presented, called the modal element method, that combines numerical grid based algorithms with eigenfunction expansions developed by separation of variables. A modal element method is presented for solving potential flow in a channel with two-dimensional cylindrical like obstacles. The infinite computational region is divided into three subdomains; the bounded finite element domain, which is characterized by the cylindrical obstacle and the surrounding unbounded uniform channel entrance and exit domains. The velocity potential is represented approximately in the grid based domain by a finite element solution and is represented analytically by an eigenfunction expansion in the uniform semi-infinite entrance and exit domains. The calculated flow fields are in excellent agreement with exact analytical solutions. By eliminating the grid surrounding the obstacle, the modal element method reduces the numerical grid size, employs a more precise far field boundary condition, as well as giving theoretical insight to the interaction of the obstacle with the mean flow. Although the analysis focuses on a specific geometry, the formulation is general and can be applied to a variety of problems as seen by a comparison to companion theories in aeroacoustics and electromagnetics.

Multi-GPU three dimensional Stokes solver for simulating glacier flow

NASA Astrophysics Data System (ADS)

Licul, Aleksandar; Herman, Frédéric; Podladchikov, Yuri; Räss, Ludovic; Omlin, Samuel

2016-04-01

Here we present how we have recently developed a three-dimensional Stokes solver on the GPUs and apply it to a glacier flow. We numerically solve the Stokes momentum balance equations together with the incompressibility equation, while also taking into account strong nonlinearities for ice rheology. We have developed a fully three-dimensional numerical MATLAB application based on an iterative finite difference scheme with preconditioning of residuals. Differential equations are discretized on a regular staggered grid. We have ported it to C-CUDA to run it on GPU's in parallel, using MPI. We demonstrate the accuracy and efficiency of our developed model by manufactured analytical solution test for three-dimensional Stokes ice sheet models (Leng et al.,2013) and by comparison with other well-established ice sheet models on diagnostic ISMIP-HOM benchmark experiments (Pattyn et al., 2008). The results show that our developed model is capable to accurately and efficiently solve Stokes system of equations in a variety of different test scenarios, while preserving good parallel efficiency on up to 80 GPU's. For example, in 3D test scenarios with 250000 grid points our solver converges in around 3 minutes for single precision computations and around 10 minutes for double precision computations. We have also optimized the developed code to efficiently run on our newly acquired state-of-the-art GPU cluster octopus. This allows us to solve our problem on more than 20 million grid points, by just increasing the number of GPU used, while keeping the computation time the same. In future work we will apply our solver to real world applications and implement the free surface evolution capabilities. REFERENCES Leng,W.,Ju,L.,Gunzburger,M. & Price,S., 2013. Manufactured solutions and the verification of three-dimensional stokes ice-sheet models. Cryosphere 7,19-29. Pattyn, F., Perichon, L., Aschwanden, A., Breuer, B., de Smedt, B., Gagliardini, O., Gudmundsson,G.H., Hindmarsh, R.C.A., Hubbard, A., Johnson, J.V., Kleiner, T., Konovalov,Y., Martin, C., Payne, A.J., Pollard, D., Price, S., Rckamp, M., Saito, F., Souk, O.,Sugiyama, S. & Zwinger, T., 2008. Benchmark experiments for higher-order and full-stokes ice sheet models (ismiphom). The Cryosphere 2, 95-108.

Bounded Error Schemes for the Wave Equation on Complex Domains

NASA Technical Reports Server (NTRS)

Abarbanel, Saul; Ditkowski, Adi; Yefet, Amir

1998-01-01

This paper considers the application of the method of boundary penalty terms ("SAT") to the numerical solution of the wave equation on complex shapes with Dirichlet boundary conditions. A theory is developed, in a semi-discrete setting, that allows the use of a Cartesian grid on complex geometries, yet maintains the order of accuracy with only a linear temporal error-bound. A numerical example, involving the solution of Maxwell's equations inside a 2-D circular wave-guide demonstrates the efficacy of this method in comparison to others (e.g. the staggered Yee scheme) - we achieve a decrease of two orders of magnitude in the level of the L2-error.

A novel consistent and well-balanced algorithm for simulations of multiphase flows on unstructured grids

NASA Astrophysics Data System (ADS)

Patel, Jitendra Kumar; Natarajan, Ganesh

2017-12-01

We discuss the development and assessment of a robust numerical algorithm for simulating multiphase flows with complex interfaces and high density ratios on arbitrary polygonal meshes. The algorithm combines the volume-of-fluid method with an incremental projection approach for incompressible multiphase flows in a novel hybrid staggered/non-staggered framework. The key principles that characterise the algorithm are the consistent treatment of discrete mass and momentum transport and the similar discretisation of force terms appearing in the momentum equation. The former is achieved by invoking identical schemes for convective transport of volume fraction and momentum in the respective discrete equations while the latter is realised by representing the gravity and surface tension terms as gradients of suitable scalars which are then discretised in identical fashion resulting in a balanced formulation. The hybrid staggered/non-staggered framework employed herein solves for the scalar normal momentum at the cell faces, while the volume fraction is computed at the cell centroids. This is shown to naturally lead to similar terms for pressure and its correction in the momentum and pressure correction equations respectively, which are again treated discretely in a similar manner. We show that spurious currents that corrupt the solution may arise both from an unbalanced formulation where forces (gravity and surface tension) are discretised in dissimilar manner and from an inconsistent approach where different schemes are used to convect the mass and momentum, with the latter prominent in flows which are convection-dominant with high density ratios. Interestingly, the inconsistent approach is shown to perform as well as the consistent approach even for high density ratio flows in some cases while it exhibits anomalous behaviour for other scenarios, even at low density ratios. Using a plethora of test problems of increasing complexity, we conclusively demonstrate that the consistent transport and balanced force treatment results in a numerically stable solution procedure and physically consistent results. The algorithm proposed in this study qualifies as a robust approach to simulate multiphase flows with high density ratios on unstructured meshes and may be realised in existing flow solvers with relative ease.

Multigrid Method for Modeling Multi-Dimensional Combustion with Detailed Chemistry

NASA Technical Reports Server (NTRS)

Zheng, Xiaoqing; Liu, Chaoqun; Liao, Changming; Liu, Zhining; McCormick, Steve

1996-01-01

A highly accurate and efficient numerical method is developed for modeling 3-D reacting flows with detailed chemistry. A contravariant velocity-based governing system is developed for general curvilinear coordinates to maintain simplicity of the continuity equation and compactness of the discretization stencil. A fully-implicit backward Euler technique and a third-order monotone upwind-biased scheme on a staggered grid are used for the respective temporal and spatial terms. An efficient semi-coarsening multigrid method based on line-distributive relaxation is used as the flow solver. The species equations are solved in a fully coupled way and the chemical reaction source terms are treated implicitly. Example results are shown for a 3-D gas turbine combustor with strong swirling inflows.

Finite difference elastic wave modeling with an irregular free surface using ADER scheme

NASA Astrophysics Data System (ADS)

Almuhaidib, Abdulaziz M.; Nafi Toksöz, M.

2015-06-01

In numerical modeling of seismic wave propagation in the earth, we encounter two important issues: the free surface and the topography of the surface (i.e. irregularities). In this study, we develop a 2D finite difference solver for the elastic wave equation that combines a 4th- order ADER scheme (Arbitrary high-order accuracy using DERivatives), which is widely used in aeroacoustics, with the characteristic variable method at the free surface boundary. The idea is to treat the free surface boundary explicitly by using ghost values of the solution for points beyond the free surface to impose the physical boundary condition. The method is based on the velocity-stress formulation. The ultimate goal is to develop a numerical solver for the elastic wave equation that is stable, accurate and computationally efficient. The solver treats smooth arbitrary-shaped boundaries as simple plane boundaries. The computational cost added by treating the topography is negligible compared to flat free surface because only a small number of grid points near the boundary need to be computed. In the presence of topography, using 10 grid points per shortest shear-wavelength, the solver yields accurate results. Benchmark numerical tests using several complex models that are solved by our method and other independent accurate methods show an excellent agreement, confirming the validity of the method for modeling elastic waves with an irregular free surface.

An assessment of unstructured grid technology for timely CFD analysis

NASA Technical Reports Server (NTRS)

Kinard, Tom A.; Schabowski, Deanne M.

1995-01-01

An assessment of two unstructured methods is presented in this paper. A tetrahedral unstructured method USM3D, developed at NASA Langley Research Center is compared to a Cartesian unstructured method, SPLITFLOW, developed at Lockheed Fort Worth Company. USM3D is an upwind finite volume solver that accepts grids generated primarily from the Vgrid grid generator. SPLITFLOW combines an unstructured grid generator with an implicit flow solver in one package. Both methods are exercised on three test cases, a wing, and a wing body, and a fully expanded nozzle. The results for the first two runs are included here and compared to the structured grid method TEAM and to available test data. On each test case, the set up procedure are described, including any difficulties that were encountered. Detailed descriptions of the solvers are not included in this paper.

Two-Dimensional Grids About Airfoils and Other Shapes

NASA Technical Reports Server (NTRS)

Sorenson, R.

1982-01-01

GRAPE computer program generates two-dimensional finite-difference grids about airfoils and other shapes by use of Poisson differential equation. GRAPE can be used with any boundary shape, even one specified by tabulated points and including limited number of sharp corners. Numerically stable and computationally fast, GRAPE provides aerodynamic analyst with efficient and consistant means of grid generation.

Adaptive hierarchical grid model of water-borne pollutant dispersion

NASA Astrophysics Data System (ADS)

Borthwick, A. G. L.; Marchant, R. D.; Copeland, G. J. M.

Water pollution by industrial and agricultural waste is an increasingly major public health issue. It is therefore important for water engineers and managers to be able to predict accurately the local behaviour of water-borne pollutants. This paper describes the novel and efficient coupling of dynamically adaptive hierarchical grids with standard solvers of the advection-diffusion equation. Adaptive quadtree grids are able to focus on regions of interest such as pollutant fronts, while retaining economy in the total number of grid elements through selective grid refinement. Advection is treated using Lagrangian particle tracking. Diffusion is solved separately using two grid-based methods; one is by explicit finite differences, the other a diffusion-velocity approach. Results are given in two dimensions for pure diffusion of an initially Gaussian plume, advection-diffusion of the Gaussian plume in the rotating flow field of a forced vortex, and the transport of species in a rectangular channel with side wall boundary layers. Close agreement is achieved with analytical solutions of the advection-diffusion equation and simulations from a Lagrangian random walk model. An application to Sepetiba Bay, Brazil is included to demonstrate the method with complex flows and topography.

Boundary conditions for the solution of compressible Navier-Stokes equations by an implicit factored method

NASA Technical Reports Server (NTRS)

Shih, T. I.-P.; Smith, G. E.; Springer, G. S.; Rimon, Y.

1983-01-01

A method is presented for formulating the boundary conditions in implicit finite-difference form needed for obtaining solutions to the compressible Navier-Stokes equations by the Beam and Warming implicit factored method. The usefulness of the method was demonstrated (a) by establishing the boundary conditions applicable to the analysis of the flow inside an axisymmetric piston-cylinder configuration and (b) by calculating velocities and mass fractions inside the cylinder for different geometries and different operating conditions. Stability, selection of time step and grid sizes, and computer time requirements are discussed in reference to the piston-cylinder problem analyzed.

On some theoretical and practical aspects of multigrid methods. [to solve finite element systems from elliptic equations

NASA Technical Reports Server (NTRS)

Nicolaides, R. A.

1979-01-01

A description and explanation of a simple multigrid algorithm for solving finite element systems is given. Numerical results for an implementation are reported for a number of elliptic equations, including cases with singular coefficients and indefinite equations. The method shows the high efficiency, essentially independent of the grid spacing, predicted by the theory.

Wavelet-based adaptation methodology combined with finite difference WENO to solve ideal magnetohydrodynamics

NASA Astrophysics Data System (ADS)

Do, Seongju; Li, Haojun; Kang, Myungjoo

2017-06-01

In this paper, we present an accurate and efficient wavelet-based adaptive weighted essentially non-oscillatory (WENO) scheme for hydrodynamics and ideal magnetohydrodynamics (MHD) equations arising from the hyperbolic conservation systems. The proposed method works with the finite difference weighted essentially non-oscillatory (FD-WENO) method in space and the third order total variation diminishing (TVD) Runge-Kutta (RK) method in time. The philosophy of this work is to use the lifted interpolating wavelets as not only detector for singularities but also interpolator. Especially, flexible interpolations can be performed by an inverse wavelet transformation. When the divergence cleaning method introducing auxiliary scalar field ψ is applied to the base numerical schemes for imposing divergence-free condition to the magnetic field in a MHD equation, the approximations to derivatives of ψ require the neighboring points. Moreover, the fifth order WENO interpolation requires large stencil to reconstruct high order polynomial. In such cases, an efficient interpolation method is necessary. The adaptive spatial differentiation method is considered as well as the adaptation of grid resolutions. In order to avoid the heavy computation of FD-WENO, in the smooth regions fixed stencil approximation without computing the non-linear WENO weights is used, and the characteristic decomposition method is replaced by a component-wise approach. Numerical results demonstrate that with the adaptive method we are able to resolve the solutions that agree well with the solution of the corresponding fine grid.

The terminal area simulation system. Volume 1: Theoretical formulation

NASA Technical Reports Server (NTRS)

Proctor, F. H.

1987-01-01

A three-dimensional numerical cloud model was developed for the general purpose of studying convective phenomena. The model utilizes a time splitting integration procedure in the numerical solution of the compressible nonhydrostatic primitive equations. Turbulence closure is achieved by a conventional first-order diagnostic approximation. Open lateral boundaries are incorporated which minimize wave reflection and which do not induce domain-wide mass trends. Microphysical processes are governed by prognostic equations for potential temperature water vapor, cloud droplets, ice crystals, rain, snow, and hail. Microphysical interactions are computed by numerous Orville-type parameterizations. A diagnostic surface boundary layer is parameterized assuming Monin-Obukhov similarity theory. The governing equation set is approximated on a staggered three-dimensional grid with quadratic-conservative central space differencing. Time differencing is approximated by the second-order Adams-Bashforth method. The vertical grid spacing may be either linear or stretched. The model domain may translate along with a convective cell, even at variable speeds.

Theory and simulation of time-fractional fluid diffusion in porous media

NASA Astrophysics Data System (ADS)

Carcione, José M.; Sanchez-Sesma, Francisco J.; Luzón, Francisco; Perez Gavilán, Juan J.

2013-08-01

We simulate a fluid flow in inhomogeneous anisotropic porous media using a time-fractional diffusion equation and the staggered Fourier pseudospectral method to compute the spatial derivatives. A fractional derivative of the order of 0 < ν < 2 replaces the first-order time derivative in the classical diffusion equation. It implies a time-dependent permeability tensor having a power-law time dependence, which describes memory effects and accounts for anomalous diffusion. We provide a complete analysis of the physics based on plane waves. The concepts of phase, group and energy velocities are analyzed to describe the location of the diffusion front, and the attenuation and quality factors are obtained to quantify the amplitude decay. We also obtain the frequency-domain Green function. The time derivative is computed with the Grünwald-Letnikov summation, which is a finite-difference generalization of the standard finite-difference operator to derivatives of fractional order. The results match the analytical solution obtained from the Green function. An example of the pressure field generated by a fluid injection in a heterogeneous sandstone illustrates the performance of the algorithm for different values of ν. The calculation requires storing the whole pressure field in the computer memory since anomalous diffusion ‘recalls the past’.

Application of discontinuous Galerkin method for solving a compressible five-equation two-phase flow model

NASA Astrophysics Data System (ADS)

Saleem, M. Rehan; Ali, Ishtiaq; Qamar, Shamsul

2018-03-01

In this article, a reduced five-equation two-phase flow model is numerically investigated. The formulation of the model is based on the conservation and energy exchange laws. The model is non-conservative and the governing equations contain two equations for the mass conservation, one for the over all momentum and one for the total energy. The fifth equation is the energy equation for one of the two phases that includes a source term on the right hand side for incorporating energy exchange between the two fluids in the form of mechanical and thermodynamical works. A Runge-Kutta discontinuous Galerkin finite element method is applied to solve the model equations. The main attractive features of the proposed method include its formal higher order accuracy, its nonlinear stability, its ability to handle complicated geometries, and its ability to capture sharp discontinuities or strong gradients in the solutions without producing spurious oscillations. The proposed method is robust and well suited for large-scale time-dependent computational problems. Several case studies of two-phase flows are presented. For validation and comparison of the results, the same model equations are also solved by using a staggered central scheme. It was found that discontinuous Galerkin scheme produces better results as compared to the staggered central scheme.

Incompressible Navier-Stokes and parabolized Navier-Stokes solution procedures and computational techniques

NASA Technical Reports Server (NTRS)

Rubin, S. G.

1982-01-01

Recent developments with finite-difference techniques are emphasized. The quotation marks reflect the fact that any finite discretization procedure can be included in this category. Many so-called finite element collocation and galerkin methods can be reproduced by appropriate forms of the differential equations and discretization formulas. Many of the difficulties encountered in early Navier-Stokes calculations were inherent not only in the choice of the different equations (accuracy), but also in the method of solution or choice of algorithm (convergence and stability, in the manner in which the dependent variables or discretized equations are related (coupling), in the manner that boundary conditions are applied, in the manner that the coordinate mesh is specified (grid generation), and finally, in recognizing that for many high Reynolds number flows not all contributions to the Navier-Stokes equations are necessarily of equal importance (parabolization, preferred direction, pressure interaction, asymptotic and mathematical character). It is these elements that are reviewed. Several Navier-Stokes and parabolized Navier-Stokes formulations are also presented.

A Simple Algebraic Grid Adaptation Scheme with Applications to Two- and Three-dimensional Flow Problems

NASA Technical Reports Server (NTRS)

Hsu, Andrew T.; Lytle, John K.

1989-01-01

An algebraic adaptive grid scheme based on the concept of arc equidistribution is presented. The scheme locally adjusts the grid density based on gradients of selected flow variables from either finite difference or finite volume calculations. A user-prescribed grid stretching can be specified such that control of the grid spacing can be maintained in areas of known flowfield behavior. For example, the grid can be clustered near a wall for boundary layer resolution and made coarse near the outer boundary of an external flow. A grid smoothing technique is incorporated into the adaptive grid routine, which is found to be more robust and efficient than the weight function filtering technique employed by other researchers. Since the present algebraic scheme requires no iteration or solution of differential equations, the computer time needed for grid adaptation is trivial, making the scheme useful for three-dimensional flow problems. Applications to two- and three-dimensional flow problems show that a considerable improvement in flowfield resolution can be achieved by using the proposed adaptive grid scheme. Although the scheme was developed with steady flow in mind, it is a good candidate for unsteady flow computations because of its efficiency.

The construction of high-accuracy schemes for acoustic equations

NASA Technical Reports Server (NTRS)

Tang, Lei; Baeder, James D.

1995-01-01

An accuracy analysis of various high order schemes is performed from an interpolation point of view. The analysis indicates that classical high order finite difference schemes, which use polynomial interpolation, hold high accuracy only at nodes and are therefore not suitable for time-dependent problems. Thus, some schemes improve their numerical accuracy within grid cells by the near-minimax approximation method, but their practical significance is degraded by maintaining the same stencil as classical schemes. One-step methods in space discretization, which use piecewise polynomial interpolation and involve data at only two points, can generate a uniform accuracy over the whole grid cell and avoid spurious roots. As a result, they are more accurate and efficient than multistep methods. In particular, the Cubic-Interpolated Psuedoparticle (CIP) scheme is recommended for computational acoustics.

Fourier Collocation Approach With Mesh Refinement Method for Simulating Transit-Time Ultrasonic Flowmeters Under Multiphase Flow Conditions.

PubMed

Simurda, Matej; Duggen, Lars; Basse, Nils T; Lassen, Benny

2018-02-01

A numerical model for transit-time ultrasonic flowmeters operating under multiphase flow conditions previously presented by us is extended by mesh refinement and grid point redistribution. The method solves modified first-order stress-velocity equations of elastodynamics with additional terms to account for the effect of the background flow. Spatial derivatives are calculated by a Fourier collocation scheme allowing the use of the fast Fourier transform, while the time integration is realized by the explicit third-order Runge-Kutta finite-difference scheme. The method is compared against analytical solutions and experimental measurements to verify the benefit of using mapped grids. Additionally, a study of clamp-on and in-line ultrasonic flowmeters operating under multiphase flow conditions is carried out.

An unstaggered central scheme on nonuniform grids for the simulation of a compressible two-phase flow model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Touma, Rony; Zeidan, Dia

In this paper we extend a central finite volume method on nonuniform grids to the case of drift-flux two-phase flow problems. The numerical base scheme is an unstaggered, non oscillatory, second-order accurate finite volume scheme that evolves a piecewise linear numerical solution on a single grid and uses dual cells intermediately while updating the numerical solution to avoid the resolution of the Riemann problems arising at the cell interfaces. We then apply the numerical scheme and solve a classical drift-flux problem. The obtained results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potentialmore » of the proposed scheme.« less

Modeling Poroelastic Wave Propagation in a Real 2-D Complex Geological Structure Obtained via Self-Organizing Maps

NASA Astrophysics Data System (ADS)

Itzá Balam, Reymundo; Iturrarán-Viveros, Ursula; Parra, Jorge O.

2018-03-01

Two main stages of seismic modeling are geological model building and numerical computation of seismic response for the model. The quality of the computed seismic response is partly related to the type of model that is built. Therefore, the model building approaches become as important as seismic forward numerical methods. For this purpose, three petrophysical facies (sands, shales and limestones) are extracted from reflection seismic data and some seismic attributes via the clustering method called Self-Organizing Maps (SOM), which, in this context, serves as a geological model building tool. This model with all its properties is the input to the Optimal Implicit Staggered Finite Difference (OISFD) algorithm to create synthetic seismograms for poroelastic, poroacoustic and elastic media. The results show a good agreement between observed and 2-D synthetic seismograms. This demonstrates that the SOM classification method enables us to extract facies from seismic data and allows us to integrate the lithology at the borehole scale with the 2-D seismic data.

Uncertainty based pressure reconstruction from velocity measurement with generalized least squares

NASA Astrophysics Data System (ADS)

Zhang, Jiacheng; Scalo, Carlo; Vlachos, Pavlos

2017-11-01

A method using generalized least squares reconstruction of instantaneous pressure field from velocity measurement and velocity uncertainty is introduced and applied to both planar and volumetric flow data. Pressure gradients are computed on a staggered grid from flow acceleration. The variance-covariance matrix of the pressure gradients is evaluated from the velocity uncertainty by approximating the pressure gradient error to a linear combination of velocity errors. An overdetermined system of linear equations which relates the pressure and the computed pressure gradients is formulated and then solved using generalized least squares with the variance-covariance matrix of the pressure gradients. By comparing the reconstructed pressure field against other methods such as solving the pressure Poisson equation, the omni-directional integration, and the ordinary least squares reconstruction, generalized least squares method is found to be more robust to the noise in velocity measurement. The improvement on pressure result becomes more remarkable when the velocity measurement becomes less accurate and more heteroscedastic. The uncertainty of the reconstructed pressure field is also quantified and compared across the different methods.

Design and Fabrication of nanowire-grid polarizer in near-infrared broadband

NASA Astrophysics Data System (ADS)

Jin, Qiufeng; Liu, Quan; Wu, Jianhong; Cheng, Yu

2012-11-01

The infrared polarizers are widely used in the infrared imaging systems as the core components, such as infrared stealth, target acquisition and mine detection, automobile night-vision instrument and other systems. For the requirements of near-infrared imaging systems, a nanowire-grid is designed by Finite Difference Time Domain (FDTD) method. Herein, considering the high reflection of metal aluminum in the manufacturing process, we propose a structure with aluminum-copper nanowire-grid. FDTD method is adapted to analyze the effects of the thickness of aluminumcopper in different combinations on the TM and TE polarization transmission efficiency as well as the extinction ratio when the grating's period is 300nm. Numerical results and theoretical analysis show that: the reflection on the substrate is suppressed with the optimal thickness of the Cu layer. Considering the resist-substrate reflectivity and the final performance of the polarizer, the structure with an 120nm Al layer, and a 50nm anti-reflection Cu layer is chosen; and the TM transmission efficiency is more than 71%, and the extinction ratio is more than 25dB. At last we used Holographic lithography and IBE to fabricate a prototype of the nanowire-grid.

AN ACCURATE AND EFFICIENT ALGORITHM FOR NUMERICAL SIMULATION OF CONDUCTION-TYPE PROBLEMS. (R824801)

EPA Science Inventory

Abstract

A modification of the finite analytic numerical method for conduction-type (diffusion) problems is presented. The finite analytic discretization scheme is derived by means of the Fourier series expansion for the most general case of nonuniform grid and variabl...

Modeling of the 2011 Tohoku-oki Tsunami and its Impacts on Hawaii

NASA Astrophysics Data System (ADS)

Cheung, K.; Yamazaki, Y.; Roeber, V.; Lay, T.

2011-12-01

The 2011 Tohoku-oki great earthquake (Mw 9.0) generated a destructive tsunami along the entire Pacific coast of northeastern Japan. The tsunami, which registered 6.7 m amplitude at a coastal GPS gauge and 1.75 m at an open-ocean DART buoy, triggered warnings across the Pacific. The waves reached Hawaii 7 hours after the earthquake and caused localized damage and persistent coastal oscillations along the island chain. Several tide gauges and a DART buoy west of Hawaii Island recorded clear signals of the tsunami. The Tsunami Observer Program of Hawaii State Civil Defense immediately conducted field surveys to gather runup and inundation data on Kauai, Oahu, Maui, and Hawaii Island. The extensive global seismic networks and geodetic instruments allows evaluation and validation of finite fault solutions for the tsunami modeling. We reconstruct the 2011 Tohoku-oki tsunami using the long-wave model NEOWAVE (Non-hydrostatic Evolution of Ocean WAVEs) and a finite fault solution based on inversion of teleseismic P waves. The depth-integrated model describes dispersive waves through the non-hydrostatic pressure and vertical velocity, which also account for tsunami generation from time histories of seafloor deformation. The semi-implicit, staggered finite difference model captures flow discontinuities associated with bores or hydraulic jumps through the momentum-conserved advection scheme. Four levels of two-way nested grids in spherical coordinates allow description of tsunami evolution processes of different time and spatial scales for investigation of the impacts around the Hawaiian Islands. The model results are validated with DART data across the Pacific as well as tide gauge and runup measurements in Hawaii. Spectral analysis of the computed surface elevation reveals a series of resonance modes over the insular shelf and slope complex along the archipelago. Resonance oscillations provide an explanation for the localized impacts and the persistent wave activities in the aftermath. The model results provide insights into effects of fringing reefs, which are present along 70% of Hawaii's coastlines, on tsunami transformation and runup processes. This case study improves our understanding of tsunamis in tropical island environment and validates the modeling capability to predict their impacts for hazard mitigation and emergency management.

Multigrid Techniques for Highly Indefinite Equations

NASA Technical Reports Server (NTRS)

Shapira, Yair

1996-01-01

A multigrid method for the solution of finite difference approximations of elliptic PDE's is introduced. A parallelizable version of it, suitable for two and multi level analysis, is also defined, and serves as a theoretical tool for deriving a suitable implementation for the main version. For indefinite Helmholtz equations, this analysis provides a suitable mesh size for the coarsest grid used. Numerical experiments show that the method is applicable to diffusion equations with discontinuous coefficients and highly indefinite Helmholtz equations.

Theoretical and numerical studies of chaotic mixing

NASA Astrophysics Data System (ADS)

Kim, Ho Jun

Theoretical and numerical studies of chaotic mixing are performed to circumvent the difficulties of efficient mixing, which come from the lack of turbulence in microfluidic devices. In order to carry out efficient and accurate parametric studies and to identify a fully chaotic state, a spectral element algorithm for solution of the incompressible Navier-Stokes and species transport equations is developed. Using Taylor series expansions in time marching, the new algorithm employs an algebraic factorization scheme on multi-dimensional staggered spectral element grids, and extends classical conforming Galerkin formulations to nonconforming spectral elements. Lagrangian particle tracking methods are utilized to study particle dispersion in the mixing device using spectral element and fourth order Runge-Kutta discretizations in space and time, respectively. Comparative studies of five different techniques commonly employed to identify the chaotic strength and mixing efficiency in microfluidic systems are presented to demonstrate the competitive advantages and shortcomings of each method. These are the stirring index based on the box counting method, Poincare sections, finite time Lyapunov exponents, the probability density function of the stretching field, and mixing index inverse, based on the standard deviation of scalar species distribution. Series of numerical simulations are performed by varying the Peclet number (Pe) at fixed kinematic conditions. The mixing length (lm) is characterized as function of the Pe number, and lm ∝ ln(Pe) scaling is demonstrated for fully chaotic cases. Employing the aforementioned techniques, optimum kinematic conditions and the actuation frequency of the stirrer that result in the highest mixing/stirring efficiency are identified in a zeta potential patterned straight micro channel, where a continuous flow is generated by superposition of a steady pressure driven flow and time periodic electroosmotic flow induced by a stream-wise AC electric field. Finally, it is shown that the invariant manifold of hyperbolic periodic point determines the geometry of fast mixing zones in oscillatory flows in two-dimensional cavity.

Multi-hole seismic modeling in 3-D space and cross-hole seismic tomography analysis for boulder detection

NASA Astrophysics Data System (ADS)

Cheng, Fei; Liu, Jiangping; Wang, Jing; Zong, Yuquan; Yu, Mingyu

2016-11-01

A boulder stone, a common geological feature in south China, is referred to the remnant of a granite body which has been unevenly weathered. Undetected boulders could adversely impact the schedule and safety of subway construction when using tunnel boring machine (TBM) method. Therefore, boulder detection has always been a key issue demanded to be solved before the construction. Nowadays, cross-hole seismic tomography is a high resolution technique capable of boulder detection, however, the method can only solve for velocity in a 2-D slice between two wells, and the size and central position of the boulder are generally difficult to be accurately obtained. In this paper, the authors conduct a multi-hole wave field simulation and characteristic analysis of a boulder model based on the 3-D elastic wave staggered-grid finite difference theory, and also a 2-D imaging analysis based on first arrival travel time. The results indicate that (1) full wave field records could be obtained from multi-hole seismic wave simulations. Simulation results describe that the seismic wave propagation pattern in cross-hole high-velocity spherical geological bodies is more detailed and can serve as a basis for the wave field analysis. (2) When a cross-hole seismic section cuts through the boulder, the proposed method provides satisfactory cross-hole tomography results; however, when the section is closely positioned to the boulder, such high-velocity object in the 3-D space would impact on the surrounding wave field. The received diffracted wave interferes with the primary wave and in consequence the picked first arrival travel time is not derived from the profile, which results in a false appearance of high-velocity geology features. Finally, the results of 2-D analysis in 3-D modeling space are comparatively analyzed with the physical model test vis-a-vis the effect of high velocity body on the seismic tomographic measurements.

Nonuniform depth grids in parabolic equation solutions.

PubMed

Sanders, William M; Collins, Michael D

2013-04-01

The parabolic wave equation is solved using a finite-difference solution in depth that involves a nonuniform grid. The depth operator is discretized using Galerkin's method with asymmetric hat functions. Examples are presented to illustrate that this approach can be used to improve efficiency for problems in ocean acoustics and seismo-acoustics. For shallow water problems, accuracy is sensitive to the precise placement of the ocean bottom interface. This issue is often addressed with the inefficient approach of using a fine grid spacing over all depth. Efficiency may be improved by using a relatively coarse grid with nonuniform sampling to precisely position the interface. Efficiency may also be improved by reducing the sampling in the sediment and in an absorbing layer that is used to truncate the computational domain. Nonuniform sampling may also be used to improve the implementation of a single-scattering approximation for sloping fluid-solid interfaces.

Time domain simulation of harmonic ultrasound images and beam patterns in 3D using the k-space pseudospectral method.

PubMed

Treeby, Bradley E; Tumen, Mustafa; Cox, B T

2011-01-01

A k-space pseudospectral model is developed for the fast full-wave simulation of nonlinear ultrasound propagation through heterogeneous media. The model uses a novel equation of state to account for nonlinearity in addition to power law absorption. The spectral calculation of the spatial gradients enables a significant reduction in the number of required grid nodes compared to finite difference methods. The model is parallelized using a graphical processing unit (GPU) which allows the simulation of individual ultrasound scan lines using a 256 x 256 x 128 voxel grid in less than five minutes. Several numerical examples are given, including the simulation of harmonic ultrasound images and beam patterns using a linear phased array transducer.

A High-Order Finite Spectral Volume Method for Conservation Laws on Unstructured Grids

NASA Technical Reports Server (NTRS)

Wang, Z. J.; Liu, Yen; Kwak, Dochan (Technical Monitor)

2001-01-01

A time accurate, high-order, conservative, yet efficient method named Finite Spectral Volume (FSV) is developed for conservation laws on unstructured grids. The concept of a 'spectral volume' is introduced to achieve high-order accuracy in an efficient manner similar to spectral element and multi-domain spectral methods. In addition, each spectral volume is further sub-divided into control volumes (CVs), and cell-averaged data from these control volumes is used to reconstruct a high-order approximation in the spectral volume. Riemann solvers are used to compute the fluxes at spectral volume boundaries. Then cell-averaged state variables in the control volumes are updated independently. Furthermore, TVD (Total Variation Diminishing) and TVB (Total Variation Bounded) limiters are introduced in the FSV method to remove/reduce spurious oscillations near discontinuities. A very desirable feature of the FSV method is that the reconstruction is carried out only once, and analytically, and is the same for all cells of the same type, and that the reconstruction stencil is always non-singular, in contrast to the memory and CPU-intensive reconstruction in a high-order finite volume (FV) method. Discussions are made concerning why the FSV method is significantly more efficient than high-order finite volume and the Discontinuous Galerkin (DG) methods. Fundamental properties of the FSV method are studied and high-order accuracy is demonstrated for several model problems with and without discontinuities.

The rotational barrier in ethane: a molecular orbital study.

PubMed

Quijano-Quiñones, Ramiro F; Quesadas-Rojas, Mariana; Cuevas, Gabriel; Mena-Rejón, Gonzalo J

2012-04-20

The energy change on each Occupied Molecular Orbital as a function of rotation about the C-C bond in ethane was studied using the B3LYP, mPWB95 functional and MP2 methods with different basis sets. Also, the effect of the ZPE on rotational barrier was analyzed. We have found that σ and π energies contribution stabilize a staggered conformation. The σ(s) molecular orbital stabilizes the staggered conformation while the stabilizes the eclipsed conformation and destabilize the staggered conformation. The π(z) and molecular orbitals stabilize both the eclipsed and staggered conformations, which are destabilized by the π(v) and molecular orbitals. The results show that the method of calculation has the effect of changing the behavior of the energy change in each Occupied Molecular Orbital energy as a function of the angle of rotation about the C-C bond in ethane. Finally, we found that if the molecular orbital energy contribution is deleted from the rotational energy, an inversion in conformational preference occurs.

Application of finite difference techniques to noise propagation in jet engine ducts

NASA Technical Reports Server (NTRS)

Baumeister, K. J.

1973-01-01

A finite difference formulation is presented for wave propagation in a rectangular two-dimensional duct without steady flow. The difference technique, which should be used in the study of acoustically treated inlet and exhausts ducts used in turbofan engines, can readily handle acoustical flow field complications such as axial variations in wall impedance and cross-section area. In the numerical analysis, the continuous acoustic field is lumped into a series of grid points in which the pressure and velocity at each grid point are separated into real and imaginary terms. An example calculation is also presented for the sound attenuation in a two-dimensional straight soft-walled suppressor.

Application of finite difference techniques to noise propagation in jet engine ducts

NASA Technical Reports Server (NTRS)

Baumeister, K. J.

1973-01-01

A finite difference formulation is presented for wave propagation in a rectangular two-dimensional duct without steady flow. The difference technique, which should be useful in the study of acoustically treated inlet and exhausts ducts used in turbofan engines, can readily handle acoustical flow field complications such as axial variations in wall impedance and cross section area. In the numerical analysis, the continuous acoustic field is lumped into a series of grid points in which the pressure and velocity at each grid point are separated into real and imaginary terms. An example calculation is also presented for the sound attenuation in a two-dimensional straight soft-walled suppressor.

Comments regarding two upwind methods for solving two-dimensional external flows using unstructured grids

NASA Technical Reports Server (NTRS)

Kleb, W. L.

1994-01-01

Steady flow over the leading portion of a multicomponent airfoil section is studied using computational fluid dynamics (CFD) employing an unstructured grid. To simplify the problem, only the inviscid terms are retained from the Reynolds-averaged Navier-Stokes equations - leaving the Euler equations. The algorithm is derived using the finite-volume approach, incorporating explicit time-marching of the unsteady Euler equations to a time-asymptotic, steady-state solution. The inviscid fluxes are obtained through either of two approximate Riemann solvers: Roe's flux difference splitting or van Leer's flux vector splitting. Results are presented which contrast the solutions given by the two flux functions as a function of Mach number and grid resolution. Additional information is presented concerning code verification techniques, flow recirculation regions, convergence histories, and computational resources.

Modifications made to ModelMuse to add support for the Saturated-Unsaturated Transport model (SUTRA)

USGS Publications Warehouse

Winston, Richard B.

2014-01-01

This report (1) describes modifications to ModelMuse,as described in U.S. Geological Survey (USGS) Techniques and Methods (TM) 6–A29 (Winston, 2009), to add support for the Saturated-Unsaturated Transport model (SUTRA) (Voss and Provost, 2002; version of September 22, 2010) and (2) supplements USGS TM 6–A29. Modifications include changes to the main ModelMuse window where the model is designed, addition of methods for generating a finite-element mesh suitable for SUTRA, defining how some functions shouldapply when using a finite-element mesh rather than a finite-difference grid (as originally programmed in ModelMuse), and applying spatial interpolation to angles. In addition, the report describes ways of handling objects on the front view of the model and displaying data. A tabulation contains a summary of the new or modified dialog boxes.

A partially penalty immersed Crouzeix-Raviart finite element method for interface problems.

PubMed

An, Na; Yu, Xijun; Chen, Huanzhen; Huang, Chaobao; Liu, Zhongyan

2017-01-01

The elliptic equations with discontinuous coefficients are often used to describe the problems of the multiple materials or fluids with different densities or conductivities or diffusivities. In this paper we develop a partially penalty immersed finite element (PIFE) method on triangular grids for anisotropic flow models, in which the diffusion coefficient is a piecewise definite-positive matrix. The standard linear Crouzeix-Raviart type finite element space is used on non-interface elements and the piecewise linear Crouzeix-Raviart type immersed finite element (IFE) space is constructed on interface elements. The piecewise linear functions satisfying the interface jump conditions are uniquely determined by the integral averages on the edges as degrees of freedom. The PIFE scheme is given based on the symmetric, nonsymmetric or incomplete interior penalty discontinuous Galerkin formulation. The solvability of the method is proved and the optimal error estimates in the energy norm are obtained. Numerical experiments are presented to confirm our theoretical analysis and show that the newly developed PIFE method has optimal-order convergence in the [Formula: see text] norm as well. In addition, numerical examples also indicate that this method is valid for both the isotropic and the anisotropic elliptic interface problems.

Numerical Investigation of Pressure Profile in Hydrodynamic Lubrication Thrust Bearing

PubMed Central

Najar, Farooq Ahmad; Harmain, G. A.

2014-01-01

Reynolds equation is solved using finite difference method (FDM) on the surface of the tilting pad to find the pressure distribution in the lubricant oil film. Different pressure profiles with grid independence are described. The present work evaluates pressure at various locations after performing a thorough grid refinement. In recent similar works, this aspect has not been addressed. However, present study shows that it can have significant effect on the pressure profile. Results of a sector shaped pad are presented and it is shown that the maximum average value of pressure is 12% (approximately) greater than the previous results. Grid independence occurs after 24 × 24 grids. A parameter “ψ” has been proposed to provide convenient indicator of obtaining grid independent results. ψ = |(P refinedgrid − P Refrence-grid)/P refinedgrid|, ψ ≤ ε, where “ε” can be fixed to a convenient value and a constant minimum film thickness value of 75 μm is used in present study. This important parameter is highlighted in the present work; the location of the peak pressure zone in terms of (r, θ) coordinates is getting shifted by changing the grid size which will help the designer and experimentalist to conveniently determine the position of pressure measurement probe. PMID:27350977

A freestream-preserving fourth-order finite-volume method in mapped coordinates with adaptive-mesh refinement

DOE PAGES

Guzik, Stephen M.; Gao, Xinfeng; Owen, Landon D.; ...

2015-12-20

We present a fourth-order accurate finite-volume method for solving time-dependent hyperbolic systems of conservation laws on mapped grids that are adaptively refined in space and time. Some novel considerations for formulating the semi-discrete system of equations in computational space are combined with detailed mechanisms for accommodating the adapting grids. Furthermore, these considerations ensure that conservation is maintained and that the divergence of a constant vector field is always zero (freestream-preservation property). The solution in time is advanced with a fourth-order Runge-Kutta method. A series of tests verifies that the expected accuracy is achieved in smooth flows and the solution ofmore » a Mach reflection problem demonstrates the effectiveness of the algorithm in resolving strong discontinuities.« less

Implementation of Implicit Adaptive Mesh Refinement in an Unstructured Finite-Volume Flow Solver

NASA Technical Reports Server (NTRS)

Schwing, Alan M.; Nompelis, Ioannis; Candler, Graham V.

2013-01-01

This paper explores the implementation of adaptive mesh refinement in an unstructured, finite-volume solver. Unsteady and steady problems are considered. The effect on the recovery of high-order numerics is explored and the results are favorable. Important to this work is the ability to provide a path for efficient, implicit time advancement. A method using a simple refinement sensor based on undivided differences is discussed and applied to a practical problem: a shock-shock interaction on a hypersonic, inviscid double-wedge. Cases are compared to uniform grids without the use of adapted meshes in order to assess error and computational expense. Discussion of difficulties, advances, and future work prepare this method for additional research. The potential for this method in more complicated flows is described.

Technical report series on global modeling and data assimilation. Volume 2: Direct solution of the implicit formulation of fourth order horizontal diffusion for gridpoint models on the sphere

NASA Technical Reports Server (NTRS)

Li, Yong; Moorthi, S.; Bates, J. Ray; Suarez, Max J.

1994-01-01

High order horizontal diffusion of the form K Delta(exp 2m) is widely used in spectral models as a means of preventing energy accumulation at the shortest resolved scales. In the spectral context, an implicit formation of such diffusion is trivial to implement. The present note describes an efficient method of implementing implicit high order diffusion in global finite difference models. The method expresses the high order diffusion equation as a sequence of equations involving Delta(exp 2). The solution is obtained by combining fast Fourier transforms in longitude with a finite difference solver for the second order ordinary differential equation in latitude. The implicit diffusion routine is suitable for use in any finite difference global model that uses a regular latitude/longitude grid. The absence of a restriction on the timestep makes it particularly suitable for use in semi-Lagrangian models. The scale selectivity of the high order diffusion gives it an advantage over the uncentering method that has been used to control computational noise in two-time-level semi-Lagrangian models.

A locally refined rectangular grid finite element method - Application to computational fluid dynamics and computational physics

NASA Technical Reports Server (NTRS)

Young, David P.; Melvin, Robin G.; Bieterman, Michael B.; Johnson, Forrester T.; Samant, Satish S.

1991-01-01

The present FEM technique addresses both linear and nonlinear boundary value problems encountered in computational physics by handling general three-dimensional regions, boundary conditions, and material properties. The box finite elements used are defined by a Cartesian grid independent of the boundary definition, and local refinements proceed by dividing a given box element into eight subelements. Discretization employs trilinear approximations on the box elements; special element stiffness matrices are included for boxes cut by any boundary surface. Illustrative results are presented for representative aerodynamics problems involving up to 400,000 elements.

3D Orthorhombic Elastic Wave Propagation Pre-Test Simulation of SPE DAG-1 Test

NASA Astrophysics Data System (ADS)

Jensen, R. P.; Preston, L. A.

2017-12-01

A more realistic representation of many geologic media can be characterized as a dense system of vertically-aligned microfractures superimposed on a finely-layered horizontal geology found in shallow crustal rocks. This seismic anisotropy representation lends itself to being modeled as an orthorhombic elastic medium comprising three mutually orthogonal symmetry planes containing nine independent moduli. These moduli can be determined by observing (or prescribing) nine independent P-wave and S-wave phase speeds along different propagation directions. We have developed an explicit time-domain finite-difference (FD) algorithm for simulating 3D elastic wave propagation in a heterogeneous orthorhombic medium. The components of the particle velocity vector and the stress tensor are governed by a set of nine, coupled, first-order, linear, partial differential equations (PDEs) called the velocity-stress system. All time and space derivatives are discretized with centered and staggered FD operators possessing second- and fourth-order numerical accuracy, respectively. Additionally, we have implemented novel perfectly matched layer (PML) absorbing boundary conditions, specifically designed for orthorhombic media, to effectively suppress grid boundary reflections. In support of the Source Physics Experiment (SPE) Phase II, a series of underground chemical explosions at the Nevada National Security Site, the code has been used to perform pre-test estimates of the Dry Alluvium Geology - Experiment 1 (DAG-1). Based on literature searches, realistic geologic structure and values for orthorhombic P-wave and S-wave speeds have been estimated. Results and predictions from the simulations are presented.

Combustion chamber analysis code

NASA Technical Reports Server (NTRS)

Przekwas, A. J.; Lai, Y. G.; Krishnan, A.; Avva, R. K.; Giridharan, M. G.

1993-01-01

A three-dimensional, time dependent, Favre averaged, finite volume Navier-Stokes code has been developed to model compressible and incompressible flows (with and without chemical reactions) in liquid rocket engines. The code has a non-staggered formulation with generalized body-fitted-coordinates (BFC) capability. Higher order differencing methodologies such as MUSCL and Osher-Chakravarthy schemes are available. Turbulent flows can be modeled using any of the five turbulent models present in the code. A two-phase, two-liquid, Lagrangian spray model has been incorporated into the code. Chemical equilibrium and finite rate reaction models are available to model chemically reacting flows. The discrete ordinate method is used to model effects of thermal radiation. The code has been validated extensively against benchmark experimental data and has been applied to model flows in several propulsion system components of the SSME and the STME.

Preparing CAM-SE for Multi-Tracer Applications: CAM-SE-Cslam

NASA Astrophysics Data System (ADS)

Lauritzen, P. H.; Taylor, M.; Goldhaber, S.

2014-12-01

The NCAR-DOE spectral element (SE) dynamical core comes from the HOMME (High-Order Modeling Environment; Dennis et al., 2012) and it is available in CAM. The CAM-SE dynamical core is designed with intrinsic mimetic properties guaranteeing total energy conservation (to time-truncation errors) and mass-conservation, and has demonstrated excellent scalability on massively parallel compute platforms (Taylor, 2011). For applications involving many tracers such as chemistry and biochemistry modeling, CAM-SE has been found to be significantly more computationally costly than the current "workhorse" model CAM-FV (Finite-Volume; Lin 2004). Hence a multi-tracer efficient scheme, called the CSLAM (Conservative Semi-Lagrangian Multi-tracer; Lauritzen et al., 2011) scheme, has been implemented in the HOMME (Erath et al., 2012). The CSLAM scheme has recently been cast in flux-form in HOMME so that it can be coupled to the SE dynamical core through conventional flux-coupling methods where the SE dynamical core provides background air mass fluxes to CSLAM. Since the CSLAM scheme makes use of a finite-volume gnomonic cubed-sphere grid and hence does not operate on the SE quadrature grid, the capability of running tracer advection, the physical parameterization suite and dynamics on separate grids has been implemented in CAM-SE. The default CAM-SE-CSLAM setup is to run physics on the quasi-equal area CSLAM grid. The capability of running physics on a different grid than the SE dynamical core may provide a more consistent coupling since the physics grid option operates with quasi-equal-area cell average values rather than non-equi-distant grid-point (SE quadrature point) values. Preliminary results on the performance of CAM-SE-CSLAM will be presented.

Distributed Relaxation Multigrid and Defect Correction Applied to the Compressible Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Thomas, J. L.; Diskin, B.; Brandt, A.

1999-01-01

The distributed-relaxation multigrid and defect- correction methods are applied to the two- dimensional compressible Navier-Stokes equations. The formulation is intended for high Reynolds number applications and several applications are made at a laminar Reynolds number of 10,000. A staggered- grid arrangement of variables is used; the coupled pressure and internal energy equations are solved together with multigrid, requiring a block 2x2 matrix solution. Textbook multigrid efficiencies are attained for incompressible and slightly compressible simulations of the boundary layer on a flat plate. Textbook efficiencies are obtained for compressible simulations up to Mach numbers of 0.7 for a viscous wake simulation.

A WENO-solver combined with adaptive momentum discretization for the Wigner transport equation and its application to resonant tunneling diodes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dorda, Antonius, E-mail: dorda@tugraz.at; Schürrer, Ferdinand, E-mail: ferdinand.schuerrer@tugraz.at

2015-03-01

We present a novel numerical scheme for the deterministic solution of the Wigner transport equation, especially suited to deal with situations in which strong quantum effects are present. The unique feature of the algorithm is the expansion of the Wigner function in local basis functions, similar to finite element or finite volume methods. This procedure yields a discretization of the pseudo-differential operator that conserves the particle density on arbitrarily chosen grids. The high flexibility in refining the grid spacing together with the weighted essentially non-oscillatory (WENO) scheme for the advection term allows for an accurate and well-resolved simulation of themore » phase space dynamics. A resonant tunneling diode is considered as test case and a detailed convergence study is given by comparing the results to a non-equilibrium Green's functions calculation. The impact of the considered domain size and of the grid spacing is analyzed. The obtained convergence of the results towards a quasi-exact agreement of the steady state Wigner and Green's functions computations demonstrates the accuracy of the scheme, as well as the high flexibility to adjust to different physical situations.« less

A WENO-solver combined with adaptive momentum discretization for the Wigner transport equation and its application to resonant tunneling diodes

PubMed Central

Dorda, Antonius; Schürrer, Ferdinand

2015-01-01

We present a novel numerical scheme for the deterministic solution of the Wigner transport equation, especially suited to deal with situations in which strong quantum effects are present. The unique feature of the algorithm is the expansion of the Wigner function in local basis functions, similar to finite element or finite volume methods. This procedure yields a discretization of the pseudo-differential operator that conserves the particle density on arbitrarily chosen grids. The high flexibility in refining the grid spacing together with the weighted essentially non-oscillatory (WENO) scheme for the advection term allows for an accurate and well-resolved simulation of the phase space dynamics. A resonant tunneling diode is considered as test case and a detailed convergence study is given by comparing the results to a non-equilibrium Green's functions calculation. The impact of the considered domain size and of the grid spacing is analyzed. The obtained convergence of the results towards a quasi-exact agreement of the steady state Wigner and Green's functions computations demonstrates the accuracy of the scheme, as well as the high flexibility to adjust to different physical situations. PMID:25892748

A WENO-solver combined with adaptive momentum discretization for the Wigner transport equation and its application to resonant tunneling diodes.

PubMed

Dorda, Antonius; Schürrer, Ferdinand

2015-03-01

We present a novel numerical scheme for the deterministic solution of the Wigner transport equation, especially suited to deal with situations in which strong quantum effects are present. The unique feature of the algorithm is the expansion of the Wigner function in local basis functions, similar to finite element or finite volume methods. This procedure yields a discretization of the pseudo-differential operator that conserves the particle density on arbitrarily chosen grids. The high flexibility in refining the grid spacing together with the weighted essentially non-oscillatory (WENO) scheme for the advection term allows for an accurate and well-resolved simulation of the phase space dynamics. A resonant tunneling diode is considered as test case and a detailed convergence study is given by comparing the results to a non-equilibrium Green's functions calculation. The impact of the considered domain size and of the grid spacing is analyzed. The obtained convergence of the results towards a quasi-exact agreement of the steady state Wigner and Green's functions computations demonstrates the accuracy of the scheme, as well as the high flexibility to adjust to different physical situations.

An algebraic homotopy method for generating quasi-three-dimensional grids for high-speed configurations

NASA Technical Reports Server (NTRS)

Moitra, Anutosh

1989-01-01

A fast and versatile procedure for algebraically generating boundary conforming computational grids for use with finite-volume Euler flow solvers is presented. A semi-analytic homotopic procedure is used to generate the grids. Grids generated in two-dimensional planes are stacked to produce quasi-three-dimensional grid systems. The body surface and outer boundary are described in terms of surface parameters. An interpolation scheme is used to blend between the body surface and the outer boundary in order to determine the field points. The method, albeit developed for analytically generated body geometries is equally applicable to other classes of geometries. The method can be used for both internal and external flow configurations, the only constraint being that the body geometries be specified in two-dimensional cross-sections stationed along the longitudinal axis of the configuration. Techniques for controlling various grid parameters, e.g., clustering and orthogonality are described. Techniques for treating problems arising in algebraic grid generation for geometries with sharp corners are addressed. A set of representative grid systems generated by this method is included. Results of flow computations using these grids are presented for validation of the effectiveness of the method.

Study of the Nankai seismogenic fault using dynamic wave propagation modelling of digital rock from the Nobeoka Fault

NASA Astrophysics Data System (ADS)

Eng, Chandoeun; Ikeda, Tatsunori; Tsuji, Takeshi

2018-10-01

To understand the characteristics of the Nankai seismogenic fault in the plate convergent margin, we calculated the P- and S-wave velocities (VP and VS) of digital rock models constructed from core samples of an ancient plate boundary fault at Nobeoka, Kyushu Island, Japan. We first constructed 3D digital rock models from microcomputed tomography images and identified their heterogeneous textures such as cracks and veins. We replaced the cracks and veins with air, water, quartz, calcite and other materials with different bulk and shear moduli. Using the Rotated Staggered Grid Finite-Difference Method, we performed dynamic wave propagation simulations and quantified the effective VP, VS and the ratio of VP to VS (VP/VS) of the 3D digital rock models with different crack-filling minerals. Our results demonstrate that the water-saturated cracks considerably decreased the seismic velocity and increased VP/VS. The VP/VS of the quartz-filled rock model was lower than that in the water-saturated case and in the calcite-filled rock model. By comparing the elastic properties derived from the digital rock models with the seismic velocities (e.g. VP and VP/VS) around the seismogenic fault estimated from field seismic data, we characterised the evolution process of the deep seismogenic fault. The high VP/VS and low VP observed at the transition from aseismic to coseismic regimes in the Nankai Trough can be explained by open cracks (or fractures), while the low VP/VS and high VP observed at the deeper coseismic fault zone suggests quartz-filled cracks. The quartz-rich fault zone characterised as low VP/VS and high VP in this study could partially relate to the coseismic behaviour as suggested by previous studies, because quartz exhibits slip-weakening behaviour (i.e. unstable coseismic slip).

Development of a Regional Structured and Unstructured Grid Methodology for Chemically Reactive Turbulent Flows

NASA Astrophysics Data System (ADS)

Stefanski, Douglas Lawrence

A finite volume method for solving the Reynolds Averaged Navier-Stokes (RANS) equations on unstructured hybrid grids is presented. Capabilities for handling arbitrary mixtures of reactive gas species within the unstructured framework are developed. The modeling of turbulent effects is carried out via the 1998 Wilcox k -- o model. This unstructured solver is incorporated within VULCAN -- a multi-block structured grid code -- as part of a novel patching procedure in which non-matching interfaces between structured blocks are replaced by transitional unstructured grids. This approach provides a fully-conservative alternative to VULCAN's non-conservative patching methods for handling such interfaces. In addition, the further development of the standalone unstructured solver toward large-eddy simulation (LES) applications is also carried out. Dual time-stepping using a Crank-Nicholson formulation is added to recover time-accuracy, and modeling of sub-grid scale effects is incorporated to provide higher fidelity LES solutions for turbulent flows. A switch based on the work of Ducros, et al., is implemented to transition from a monotonicity-preserving flux scheme near shocks to a central-difference method in vorticity-dominated regions in order to better resolve small-scale turbulent structures. The updated unstructured solver is used to carry out large-eddy simulations of a supersonic constrained mixing layer.

On Calculation Methods and Results for Straight Cylindrical Roller Bearing Deflection, Stiffness, and Stress

NASA Technical Reports Server (NTRS)

Krantz, Timothy L.

2011-01-01

The purpose of this study was to assess some calculation methods for quantifying the relationships of bearing geometry, material properties, load, deflection, stiffness, and stress. The scope of the work was limited to two-dimensional modeling of straight cylindrical roller bearings. Preparations for studies of dynamic response of bearings with damaged surfaces motivated this work. Studies were selected to exercise and build confidence in the numerical tools. Three calculation methods were used in this work. Two of the methods were numerical solutions of the Hertz contact approach. The third method used was a combined finite element surface integral method. Example calculations were done for a single roller loaded between an inner and outer raceway for code verification. Next, a bearing with 13 rollers and all-steel construction was used as an example to do additional code verification, including an assessment of the leading order of accuracy of the finite element and surface integral method. Results from that study show that the method is at least first-order accurate. Those results also show that the contact grid refinement has a more significant influence on precision as compared to the finite element grid refinement. To explore the influence of material properties, the 13-roller bearing was modeled as made from Nitinol 60, a material with very different properties from steel and showing some potential for bearing applications. The codes were exercised to compare contact areas and stress levels for steel and Nitinol 60 bearings operating at equivalent power density. As a step toward modeling the dynamic response of bearings having surface damage, static analyses were completed to simulate a bearing with a spall or similar damage.

Three-dimensional forward modeling of DC resistivity using the aggregation-based algebraic multigrid method

NASA Astrophysics Data System (ADS)

Chen, Hui; Deng, Ju-Zhi; Yin, Min; Yin, Chang-Chun; Tang, Wen-Wu

2017-03-01

To speed up three-dimensional (3D) DC resistivity modeling, we present a new multigrid method, the aggregation-based algebraic multigrid method (AGMG). We first discretize the differential equation of the secondary potential field with mixed boundary conditions by using a seven-point finite-difference method to obtain a large sparse system of linear equations. Then, we introduce the theory behind the pairwise aggregation algorithms for AGMG and use the conjugate-gradient method with the V-cycle AGMG preconditioner (AGMG-CG) to solve the linear equations. We use typical geoelectrical models to test the proposed AGMG-CG method and compare the results with analytical solutions and the 3DDCXH algorithm for 3D DC modeling (3DDCXH). In addition, we apply the AGMG-CG method to different grid sizes and geoelectrical models and compare it to different iterative methods, such as ILU-BICGSTAB, ILU-GCR, and SSOR-CG. The AGMG-CG method yields nearly linearly decreasing errors, whereas the number of iterations increases slowly with increasing grid size. The AGMG-CG method is precise and converges fast, and thus can improve the computational efficiency in forward modeling of three-dimensional DC resistivity.

Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations. Part 1; Viscous Fluxes

NASA Technical Reports Server (NTRS)

Diskin, Boris; Thomas, James L.; Nielsen, Eric J.; Nishikawa, Hiroaki; White, Jeffery A.

2009-01-01

Discretization of the viscous terms in current finite-volume unstructured-grid schemes are compared using node-centered and cell-centered approaches in two dimensions. Accuracy and efficiency are studied for six nominally second-order accurate schemes: a node-centered scheme, cell-centered node-averaging schemes with and without clipping, and cell-centered schemes with unweighted, weighted, and approximately mapped least-square face gradient reconstruction. The grids considered range from structured (regular) grids to irregular grids composed of arbitrary mixtures of triangles and quadrilaterals, including random perturbations of the grid points to bring out the worst possible behavior of the solution. Two classes of tests are considered. The first class of tests involves smooth manufactured solutions on both isotropic and highly anisotropic grids with discontinuous metrics, typical of those encountered in grid adaptation. The second class concerns solutions and grids varying strongly anisotropically over a curved body, typical of those encountered in high-Reynolds number turbulent flow simulations. Results from the first class indicate the face least-square methods, the node-averaging method without clipping, and the node-centered method demonstrate second-order convergence of discretization errors with very similar accuracies per degree of freedom. The second class of tests are more discriminating. The node-centered scheme is always second order with an accuracy and complexity in linearization comparable to the best of the cell-centered schemes. In comparison, the cell-centered node-averaging schemes are less accurate, have a higher complexity in linearization, and can fail to converge to the exact solution when clipping of the node-averaged values is used. The cell-centered schemes using least-square face gradient reconstruction have more compact stencils with a complexity similar to the complexity of the node-centered scheme. For simulations on highly anisotropic curved grids, the least-square methods have to be amended either by introducing a local mapping of the surface anisotropy or modifying the scheme stencil to reflect the direction of strong coupling.

Numerical simulation of steady three-dimensional flows in axial turbomachinery bladerows

NASA Astrophysics Data System (ADS)

Basson, Anton Herman

The formulation for and application of a numerical model for low Mach number steady three-dimensional flows in axial turbomachinery blade rows is presented. The formulation considered here includes an efficient grid generation scheme (particularly suited to computational grids for the analysis of turbulent turbomachinery flows) and a semi-implicit, pressure-based computational fluid dynamics scheme that directly includes artificial dissipation, applicable to viscous and inviscid flows. The grid generation technique uses a combination of algebraic and elliptic methods, in conjunction with the Minimal Residual Method, to economically generate smooth structured grids. For typical H-grids in turbomachinery bladerows, when compared to a purely elliptic grid generation scheme, the presented grid generation scheme produces grids with much improved smoothness near the leading and trailing edges, allows the use of small near wall grid spacing required by low Reynolds number turbulence models, and maintains orthogonality of the grid near the solid boundaries even for high flow angle cascades. A specialized embedded H-grid for application particularly to tip clearance flows is presented. This topology smoothly discretizes the domain without modifying the tip shape, while requiring only minor modifications to H-grid flow solvers. Better quantitative modeling of the tip clearance vortex structure than that obtained with a pinched tip approximation is demonstrated. The formulation of artificial dissipation terms for a semi-implicit, pressure-based (SIMPLE type) flow solver, is presented. It is applied to both the Euler and the Navier-Stokes equations, expressed in generalized coordinates using a non-staggered grid. This formulation is compared to some SIMPLE and time marching formulations, revealing the artificial dissipation inherent in some commonly used semi-implicit formulations. The effect of the amount of dissipation on the accuracy of the solution and the convergence rate is quantitatively demonstrated for a number of flow cases. The ability of the formulation to model complex steady turbomachinery flows is demonstrated, e.g. for pressure driven secondary flows, turbine nozzle wakes, turbulent boundary layers. The formulation's modeling of blade surface heat transfer is assessed. The numerical model is used to investigate the structure of phenomena associated with tip clearance flows in a turbine nozzle.

A cut-cell finite volume – finite element coupling approach for fluid–structure interaction in compressible flow

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pasquariello, Vito, E-mail: vito.pasquariello@tum.de; Hammerl, Georg; Örley, Felix

2016-02-15

We present a loosely coupled approach for the solution of fluid–structure interaction problems between a compressible flow and a deformable structure. The method is based on staggered Dirichlet–Neumann partitioning. The interface motion in the Eulerian frame is accounted for by a conservative cut-cell Immersed Boundary method. The present approach enables sub-cell resolution by considering individual cut-elements within a single fluid cell, which guarantees an accurate representation of the time-varying solid interface. The cut-cell procedure inevitably leads to non-matching interfaces, demanding for a special treatment. A Mortar method is chosen in order to obtain a conservative and consistent load transfer. Wemore » validate our method by investigating two-dimensional test cases comprising a shock-loaded rigid cylinder and a deformable panel. Moreover, the aeroelastic instability of a thin plate structure is studied with a focus on the prediction of flutter onset. Finally, we propose a three-dimensional fluid–structure interaction test case of a flexible inflated thin shell interacting with a shock wave involving large and complex structural deformations.« less

On the superconvergence of Galerkin methods for hyperbolic IBVP

NASA Technical Reports Server (NTRS)

Gottlieb, David; Gustafsson, Bertil; Olsson, Pelle; Strand, BO

1993-01-01

Finite element Galerkin methods for periodic first order hyperbolic equations exhibit superconvergence on uniform grids at the nodes, i.e., there is an error estimate 0(h(sup 2r)) instead of the expected approximation order 0(h(sup r)). It will be shown that no matter how the approximating subspace S(sup h) is chosen, the superconvergence property is lost if there are characteristics leaving the domain. The implications of this result when constructing compact implicit difference schemes is also discussed.

Adaptive finite element method for turbulent flow near a propeller

NASA Astrophysics Data System (ADS)

Pelletier, Dominique; Ilinca, Florin; Hetu, Jean-Francois

1994-11-01

This paper presents an adaptive finite element method based on remeshing to solve incompressible turbulent free shear flow near a propeller. Solutions are obtained in primitive variables using a highly accurate finite element approximation on unstructured grids. Turbulence is modeled by a mixing length formulation. Two general purpose error estimators, which take into account swirl and the variation of the eddy viscosity, are presented and applied to the turbulent wake of a propeller. Predictions compare well with experimental measurements. The proposed adaptive scheme is robust, reliable and cost effective.

Variational formulation of high performance finite elements: Parametrized variational principles

NASA Technical Reports Server (NTRS)

Felippa, Carlos A.; Militello, Carmello

1991-01-01

High performance elements are simple finite elements constructed to deliver engineering accuracy with coarse arbitrary grids. This is part of a series on the variational basis of high-performance elements, with emphasis on those constructed with the free formulation (FF) and assumed natural strain (ANS) methods. Parametrized variational principles that provide a foundation for the FF and ANS methods, as well as for a combination of both are presented.

Schnek: A C++ library for the development of parallel simulation codes on regular grids

NASA Astrophysics Data System (ADS)

Schmitz, Holger

2018-05-01

A large number of algorithms across the field of computational physics are formulated on grids with a regular topology. We present Schnek, a library that enables fast development of parallel simulations on regular grids. Schnek contains a number of easy-to-use modules that greatly reduce the amount of administrative code for large-scale simulation codes. The library provides an interface for reading simulation setup files with a hierarchical structure. The structure of the setup file is translated into a hierarchy of simulation modules that the developer can specify. The reader parses and evaluates mathematical expressions and initialises variables or grid data. This enables developers to write modular and flexible simulation codes with minimal effort. Regular grids of arbitrary dimension are defined as well as mechanisms for defining physical domain sizes, grid staggering, and ghost cells on these grids. Ghost cells can be exchanged between neighbouring processes using MPI with a simple interface. The grid data can easily be written into HDF5 files using serial or parallel I/O.

A MULTIPLE GRID ALGORITHM FOR ONE-DIMENSIONAL TRANSIENT OPEN CHANNEL FLOWS. (R825200)

EPA Science Inventory

Numerical modeling of open channel flows with shocks using explicit finite difference schemes is constrained by the choice of time step, which is limited by the CFL stability criteria. To overcome this limitation, in this work we introduce the application of a multiple grid al...

Numerical Modeling of Poroelastic-Fluid Systems Using High-Resolution Finite Volume Methods

NASA Astrophysics Data System (ADS)

Lemoine, Grady

Poroelasticity theory models the mechanics of porous, fluid-saturated, deformable solids. It was originally developed by Maurice Biot to model geophysical problems, such as seismic waves in oil reservoirs, but has also been applied to modeling living bone and other porous media. Poroelastic media often interact with fluids, such as in ocean bottom acoustics or propagation of waves from soft tissue into bone. This thesis describes the development and testing of high-resolution finite volume numerical methods, and simulation codes implementing these methods, for modeling systems of poroelastic media and fluids in two and three dimensions. These methods operate on both rectilinear grids and logically rectangular mapped grids. To allow the use of these methods, Biot's equations of poroelasticity are formulated as a first-order hyperbolic system with a source term; this source term is incorporated using operator splitting. Some modifications are required to the classical high-resolution finite volume method. Obtaining correct solutions at interfaces between poroelastic media and fluids requires a novel transverse propagation scheme and the removal of the classical second-order correction term at the interface, and in three dimensions a new wave limiting algorithm is also needed to correctly limit shear waves. The accuracy and convergence rates of the methods of this thesis are examined for a variety of analytical solutions, including simple plane waves, reflection and transmission of waves at an interface between different media, and scattering of acoustic waves by a poroelastic cylinder. Solutions are also computed for a variety of test problems from the computational poroelasticity literature, as well as some original test problems designed to mimic possible applications for the simulation code.

Modal density of rectangular structures in a wide frequency range

NASA Astrophysics Data System (ADS)

Parrinello, A.; Ghiringhelli, G. L.

2018-04-01

A novel approach to investigate the modal density of a rectangular structure in a wide frequency range is presented. First, the modal density is derived, in the whole frequency range of interest, on the basis of sound transmission through the infinite counterpart of the structure; then, it is corrected by means of the low-frequency modal behavior of the structure, taking into account actual size and boundary conditions. A statistical analysis reveals the connection between the modal density of the structure and the transmission of sound through its thickness. A transfer matrix approach is used to compute the required acoustic parameters, making it possible to deal with structures having arbitrary stratifications of different layers. A finite element method is applied on coarse grids to derive the first few eigenfrequencies required to correct the modal density. Both the transfer matrix approach and the coarse grids involved in the finite element analysis grant high efficiency. Comparison with alternative formulations demonstrates the effectiveness of the proposed methodology.

Simulating ground water-lake interactions: Approaches and insights

USGS Publications Warehouse

Hunt, R.J.; Haitjema, H.M.; Krohelski, J.T.; Feinstein, D.T.

2003-01-01

Approaches for modeling lake-ground water interactions have evolved significantly from early simulations that used fixed lake stages specified as constant head to sophisticated LAK packages for MODFLOW. Although model input can be complex, the LAK package capabilities and output are superior to methods that rely on a fixed lake stage and compare well to other simple methods where lake stage can be calculated. Regardless of the approach, guidelines presented here for model grid size, location of three-dimensional flow, and extent of vertical capture can facilitate the construction of appropriately detailed models that simulate important lake-ground water interactions without adding unnecessary complexity. In addition to MODFLOW approaches, lake simulation has been formulated in terms of analytic elements. The analytic element lake package had acceptable agreement with a published LAK1 problem, even though there were differences in the total lake conductance and number of layers used in the two models. The grid size used in the original LAK1 problem, however, violated a grid size guideline presented in this paper. Grid sensitivity analyses demonstrated that an appreciable discrepancy in the distribution of stream and lake flux was related to the large grid size used in the original LAK1 problem. This artifact is expected regardless of MODFLOW LAK package used. When the grid size was reduced, a finite-difference formulation approached the analytic element results. These insights and guidelines can help ensure that the proper lake simulation tool is being selected and applied.

Effect of Finite Particle Size on Convergence of Point Particle Models in Euler-Lagrange Multiphase Dispersed Flow

NASA Astrophysics Data System (ADS)

Nili, Samaun; Park, Chanyoung; Haftka, Raphael T.; Kim, Nam H.; Balachandar, S.

2017-11-01

Point particle methods are extensively used in simulating Euler-Lagrange multiphase dispersed flow. When particles are much smaller than the Eulerian grid the point particle model is on firm theoretical ground. However, this standard approach of evaluating the gas-particle coupling at the particle center fails to converge as the Eulerian grid is reduced below particle size. We present an approach to model the interaction between particles and fluid for finite size particles that permits convergence. We use the generalized Faxen form to compute the force on a particle and compare the results against traditional point particle method. We apportion the different force components on the particle to fluid cells based on the fraction of particle volume or surface in the cell. The application is to a one-dimensional model of shock propagation through a particle-laden field at moderate volume fraction, where the convergence is achieved for a well-formulated force model and back coupling for finite size particles. Comparison with 3D direct fully resolved numerical simulations will be used to check if the approach also improves accuracy compared to the point particle model. Work supported by the U.S. Department of Energy, National Nuclear Security Administration, Advanced Simulation and Computing Program, as a Cooperative Agreement under the Predictive Science Academic Alliance Program, under Contract No. DE-NA0002378.

Decomposition of the Seismic Source Using Numerical Simulations and Observations of Nuclear Explosions

DTIC Science & Technology

2017-05-31

SUBJECT TERMS nonlinear finite element calculations, nuclear explosion monitoring, topography 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT 18...3D North Korea calculations........ Figure 6. The CRAM 3D finite element outer grid (left) is rectangular......................... Figure 7. Stress...Figure 6. The CRAM 3D finite element outer grid (left) is rectangular. The inner grid (center) is shaped to match the shape of the explosion shock wave

Can we estimate total magnetization directions from aeromagnetic data using Helbig's integrals?

USGS Publications Warehouse

Phillips, J.D.

2005-01-01

An algorithm that implements Helbig's (1963) integrals for estimating the vector components (mx, my, mz) of tile magnetic dipole moment from the first order moments of the vector magnetic field components (??X, ??Y, ??Z) is tested on real and synthetic data. After a grid of total field aeromagnetic data is converted to vector component grids using Fourier filtering, Helbig's infinite integrals are evaluated as finite integrals in small moving windows using a quadrature algorithm based on the 2-D trapezoidal rule. Prior to integration, best-fit planar surfaces must be removed from the component data within the data windows in order to make the results independent of the coordinate system origin. Two different approaches are described for interpreting the results of the integration. In the "direct" method, results from pairs of different window sizes are compared to identify grid nodes where the angular difference between solutions is small. These solutions provide valid estimates of total magnetization directions for compact sources such as spheres or dipoles, but not for horizontally elongated or 2-D sources. In the "indirect" method, which is more forgiving of source geometry, results of the quadrature analysis are scanned for solutions that are parallel to a specified total magnetization direction.

Improvements of the two-dimensional FDTD method for the simulation of normal- and superconducting planar waveguides using time series analysis

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hofschen, S.; Wolff, I.

1996-08-01

Time-domain simulation results of two-dimensional (2-D) planar waveguide finite-difference time-domain (FDTD) analysis are normally analyzed using Fourier transform. The introduced method of time series analysis to extract propagation and attenuation constants reduces the desired computation time drastically. Additionally, a nonequidistant discretization together with an adequate excitation technique is used to reduce the number of spatial grid points. Therefore, it is possible to reduce the number of spatial grid points. Therefore, it is possible to simulate normal- and superconducting planar waveguide structures with very thin conductors and small dimensions, as they are used in MMIC technology. The simulation results are comparedmore » with measurements and show good agreement.« less

Arbitrary-Lagrangian-Eulerian Discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes

NASA Astrophysics Data System (ADS)

Boscheri, Walter; Dumbser, Michael

2017-10-01

We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may also include parabolic terms in order to model dissipative transport processes, like molecular viscosity or heat conduction. High order piecewise polynomials of degree N are adopted to represent the discrete solution at each time level and within each spatial control volume of the computational grid, while high order of accuracy in time is achieved by the ADER approach, making use of an element-local space-time Galerkin finite element predictor. A novel nodal solver algorithm based on the HLL flux is derived to compute the velocity for each nodal degree of freedom that describes the current mesh geometry. In our algorithm the spatial mesh configuration can be defined in two different ways: either by an isoparametric approach that generates curved control volumes, or by a piecewise linear decomposition of each spatial control volume into simplex sub-elements. Each technique generates a corresponding number of geometrical degrees of freedom needed to describe the current mesh configuration and which must be considered by the nodal solver for determining the grid velocity. The connection of the old mesh configuration at time tn with the new one at time t n + 1 provides the space-time control volumes on which the governing equations have to be integrated in order to obtain the time evolution of the discrete solution. Our numerical method belongs to the category of so-called direct Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation formulation of the governing PDE system is considered and which already takes into account the new grid geometry (including a possible rezoning step) directly during the computation of the numerical fluxes. We emphasize that our method is a moving mesh method, as opposed to total Lagrangian formulations that are based on a fixed computational grid and which instead evolve the mapping of the reference configuration to the current one. Our new Lagrangian-type DG scheme adopts the novel a posteriori sub-cell finite volume limiter method recently developed in [62] for fixed unstructured grids. In this approach, the validity of the candidate solution produced in each cell by an unlimited ADER-DG scheme is verified against a set of physical and numerical detection criteria, such as the positivity of pressure and density, the absence of floating point errors (NaN) and the satisfaction of a relaxed discrete maximum principle (DMP) in the sense of polynomials. Those cells which do not satisfy all of the above criteria are flagged as troubled cells and are recomputed at the aid of a more robust second order TVD finite volume scheme. To preserve the subcell resolution capability of the original DG scheme, the FV limiter is run on a sub-grid that is 2 N + 1 times finer compared to the mesh of the original unlimited DG scheme. The new subcell averages are then gathered back into a high order DG polynomial by a usual conservative finite volume reconstruction operator. The numerical convergence rates of the new ALE ADER-DG schemes are studied up to fourth order in space and time and several test problems are simulated in order to check the accuracy and the robustness of the proposed numerical method in the context of the Euler and Navier-Stokes equations for compressible gas dynamics, considering both inviscid and viscous fluids. Finally, an application inspired by Inertial Confinement Fusion (ICF) type flows is considered by solving the Euler equations and the PDE of viscous and resistive magnetohydrodynamics (VRMHD).

Acoustic wave simulation using an overset grid for the global monitoring system

NASA Astrophysics Data System (ADS)

Kushida, N.; Le Bras, R.

2017-12-01

The International Monitoring System of the Comprehensive Nuclear-Test-Ban Treaty Organization (CTBTO) has been monitoring hydro-acoustic and infrasound waves over the globe. Because of the complex natures of the oceans and the atmosphere, computer simulation can play an important role in understanding the observed signals. In this regard, methods which depend on partial differential equations and require minimum modelling, are preferable. So far, to our best knowledge, acoustic wave propagation simulations based on partial differential equations on such a large scale have not been performed (pp 147 - 161 of ref [1], [2]). The main difficulties in building such simulation codes are: (1) considering the inhomogeneity of medium including background flows, (2) high aspect ratio of computational domain, (3) stability during long time integration. To overcome these difficulties, we employ a two-dimensional finite different (FDM) scheme on spherical coordinates with the Yin-Yang overset grid[3] solving the governing equation of acoustic waves introduces by Ostashev et. al.[4]. The comparison with real recording examples in hydro-acoustic will be presented at the conference. [1] Paul C. Etter: Underwater Acoustic Modeling and Simulation, Fourth Edition, CRC Press, 2013. [2] LIAN WANG et. al.: REVIEW OF UNDERWATER ACOUSTIC PROPAGATION MODELS, NPL Report AC 12, 2014. [3] A. Kageyama and T. Sato: "Yin-Yang grid": An overset grid in spherical geometry, Geochem. Geophys. Geosyst., 5, Q09005, 2004. [4] Vladimir E. Ostashev et. al: Equations for finite-difference, time-domain simulation of sound propagation in moving inhomogeneous media and numerical implementation, Acoustical Society of America. DOI: 10.1121/1.1841531, 2005.

Three-dimensional unsteady Euler equations solutions on dynamic grids

NASA Technical Reports Server (NTRS)

Belk, D. M.; Janus, J. M.; Whitfield, D. L.

1985-01-01

A method is presented for solving the three-dimensional unsteady Euler equations on dynamic grids based on flux vector splitting. The equations are cast in curvilinear coordinates and a finite volume discretization is used for handling arbitrary geometries. The discretized equations are solved using an explicit upwind second-order predictor corrector scheme that is stable for a CFL of 2. Characteristic variable boundary conditions are developed and used for unsteady impermeable surfaces and for the far-field boundary. Dynamic-grid results are presented for an oscillating air-foil and for a store separating from a reflection plate. For the cases considered of stores separating from a reflection plate, the unsteady aerodynamic forces on the store are significantly different from forces obtained by steady-state aerodynamics with the body inclination angle changed to account for plunge velocity.

Integration of the shallow water equations on the sphere using a vector semi-Lagrangian scheme with a multigrid solver

NASA Technical Reports Server (NTRS)

Bates, J. R.; Semazzi, F. H. M.; Higgins, R. W.; Barros, Saulo R. M.

1990-01-01

A vector semi-Lagrangian semi-implicit two-time-level finite-difference integration scheme for the shallow water equations on the sphere is presented. A C-grid is used for the spatial differencing. The trajectory-centered discretization of the momentum equation in vector form eliminates pole problems and, at comparable cost, gives greater accuracy than a previous semi-Lagrangian finite-difference scheme which used a rotated spherical coordinate system. In terms of the insensitivity of the results to increasing timestep, the new scheme is as successful as recent spectral semi-Lagrangian schemes. In addition, the use of a multigrid method for solving the elliptic equation for the geopotential allows efficient integration with an operation count which, at high resolution, is of lower order than in the case of the spectral models. The properties of the new scheme should allow finite-difference models to compete with spectral models more effectively than has previously been possible.

Investigating the influences of two position (non-staggered and staggered) of wind turbine arrays to produce power in a wind farm

NASA Astrophysics Data System (ADS)

Ismail, Kamal, Samsul; Purnomo, Sarjiya

2016-06-01

This investigation was conducted to identify the influences of the two positions (non-staggered and staggered) of wind turbine arrays. Identification on down-scaled size wind turbine arrays was carried out in an open circuit, suction-type wind tunnel. Based on the results of the experiment, empirical relations for the centreline velocity deficit, tipline velocity deficit and wake radius are proposed. The non-staggered position results are larger power generated than that of the staggered position, this influenced by the trend deficit in velocity that makes wind turbine generated power difference between staggered position and non-stagger position. The area used non-staggered position larger than staggered position. Result staggered position has become one of the solutions to harness wind farms confined areas.

Study on the Geomagnetic Short Period Variations of the Northwestern Yunnan

NASA Astrophysics Data System (ADS)

Yuan, Y.; Li, Q.; Cai, J.

2015-12-01

The Northwestern Yunnan is located in the interaction area between the Eurasian plate and the India plate. This area has been the ideal place for the research of continental dynamics and the prediction for risk region of strong earthquake for its complex tectonic environment and frequent seismic activity. Therefore the study on the geomagnetic short period variations is of great significance in the exploration of deep electrical structure, analysis of the seismic origin and deep geodynamics in the Northwestern Yunnan of China . This paper is based on the geomagnetic data from the magnetometer array with 8 sites built in the northwestern Yunnan to explore the deep electrical structure by the method of geomagnetic depth sounding. Firstly, we selected a total of 183 geomagnetic short period events at the range of 6min to 120min period. And we found a north northwest dividing line, of which two sides has the opposite value in the vertical component variation amplitude, which indicates the obvious conductivity anomaly underground. Secondly, the contour maps of the ratio of vertical component and horizontal component variation amplitude ΔZ/ΔH in different periods reflects the changes of a high conductivity belt's direction and position. In addition, the induction arrows maps within the period of 2 - 256min also shows that on the two sides of the dividing line the induction vectors deviate from each other, and the amplitude and direction of vectors varies with periods regularly. In the light of this, we infer that a high conductivity belt probably exists, which stretches from the deep crust to uppermost mantle and changes with depth constantly with the reference of magnetotelluric sounding. In the end of this paper, the staggered grid finite difference method is used to model the simplified three-dimensional high conductivity anomaly, and the result shows magnetic field distributions are consistent with the observed geomagnetic short period variations characteristics in different periods, which confirms the existence of the high conductivity belt. According to the characteristics of the short period geomagnetic variation above, in combination with the results of previous studies, the synthetic action of partial melting and fluid might be the origin of the belt.

Solution of the advection-dispersion equation by a finite-volume eulerian-lagrangian local adjoint method

USGS Publications Warehouse

Healy, R.W.; Russell, T.F.

1992-01-01

A finite-volume Eulerian-Lagrangian local adjoint method for solution of the advection-dispersion equation is developed and discussed. The method is mass conservative and can solve advection-dominated ground-water solute-transport problems accurately and efficiently. An integrated finite-difference approach is used in the method. A key component of the method is that the integral representing the mass-storage term is evaluated numerically at the current time level. Integration points, and the mass associated with these points, are then forward tracked up to the next time level. The number of integration points required to reach a specified level of accuracy is problem dependent and increases as the sharpness of the simulated solute front increases. Integration points are generally equally spaced within each grid cell. For problems involving variable coefficients it has been found to be advantageous to include additional integration points at strategic locations in each well. These locations are determined by backtracking. Forward tracking of boundary fluxes by the method alleviates problems that are encountered in the backtracking approaches of most characteristic methods. A test problem is used to illustrate that the new method offers substantial advantages over other numerical methods for a wide range of problems.

Parallel solution of high-order numerical schemes for solving incompressible flows

NASA Technical Reports Server (NTRS)

Milner, Edward J.; Lin, Avi; Liou, May-Fun; Blech, Richard A.

1993-01-01

A new parallel numerical scheme for solving incompressible steady-state flows is presented. The algorithm uses a finite-difference approach to solving the Navier-Stokes equations. The algorithms are scalable and expandable. They may be used with only two processors or with as many processors as are available. The code is general and expandable. Any size grid may be used. Four processors of the NASA LeRC Hypercluster were used to solve for steady-state flow in a driven square cavity. The Hypercluster was configured in a distributed-memory, hypercube-like architecture. By using a 50-by-50 finite-difference solution grid, an efficiency of 74 percent (a speedup of 2.96) was obtained.

Parallel Newton-Krylov-Schwarz algorithms for the transonic full potential equation

NASA Technical Reports Server (NTRS)

Cai, Xiao-Chuan; Gropp, William D.; Keyes, David E.; Melvin, Robin G.; Young, David P.

1996-01-01

We study parallel two-level overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, Newton-Krylov-Schwarz (NKS), employs an inexact finite-difference Newton method and a Krylov space iterative method, with a two-level overlapping Schwarz method as a preconditioner. We demonstrate that NKS, combined with a density upwinding continuation strategy for problems with weak shocks, is robust and, economical for this class of mixed elliptic-hyperbolic nonlinear partial differential equations, with proper specification of several parameters. We study upwinding parameters, inner convergence tolerance, coarse grid density, subdomain overlap, and the level of fill-in in the incomplete factorization, and report their effect on numerical convergence rate, overall execution time, and parallel efficiency on a distributed-memory parallel computer.

A High Order Finite Difference Scheme with Sharp Shock Resolution for the Euler Equations

NASA Technical Reports Server (NTRS)

Gerritsen, Margot; Olsson, Pelle

1996-01-01

We derive a high-order finite difference scheme for the Euler equations that satisfies a semi-discrete energy estimate, and present an efficient strategy for the treatment of discontinuities that leads to sharp shock resolution. The formulation of the semi-discrete energy estimate is based on a symmetrization of the Euler equations that preserves the homogeneity of the flux vector, a canonical splitting of the flux derivative vector, and the use of difference operators that satisfy a discrete analogue to the integration by parts procedure used in the continuous energy estimate. Around discontinuities or sharp gradients, refined grids are created on which the discrete equations are solved after adding a newly constructed artificial viscosity. The positioning of the sub-grids and computation of the viscosity are aided by a detection algorithm which is based on a multi-scale wavelet analysis of the pressure grid function. The wavelet theory provides easy to implement mathematical criteria to detect discontinuities, sharp gradients and spurious oscillations quickly and efficiently.

Composite scheme using localized relaxation with non-standard finite difference method for hyperbolic conservation laws

NASA Astrophysics Data System (ADS)

Kumar, Vivek; Raghurama Rao, S. V.

2008-04-01

Non-standard finite difference methods (NSFDM) introduced by Mickens [ Non-standard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994] are interesting alternatives to the traditional finite difference and finite volume methods. When applied to linear hyperbolic conservation laws, these methods reproduce exact solutions. In this paper, the NSFDM is first extended to hyperbolic systems of conservation laws, by a novel utilization of the decoupled equations using characteristic variables. In the second part of this paper, the NSFDM is studied for its efficacy in application to nonlinear scalar hyperbolic conservation laws. The original NSFDMs introduced by Mickens (1994) were not in conservation form, which is an important feature in capturing discontinuities at the right locations. Mickens [Construction and analysis of a non-standard finite difference scheme for the Burgers-Fisher equations, Journal of Sound and Vibration 257 (4) (2002) 791-797] recently introduced a NSFDM in conservative form. This method captures the shock waves exactly, without any numerical dissipation. In this paper, this algorithm is tested for the case of expansion waves with sonic points and is found to generate unphysical expansion shocks. As a remedy to this defect, we use the strategy of composite schemes [R. Liska, B. Wendroff, Composite schemes for conservation laws, SIAM Journal of Numerical Analysis 35 (6) (1998) 2250-2271] in which the accurate NSFDM is used as the basic scheme and localized relaxation NSFDM is used as the supporting scheme which acts like a filter. Relaxation schemes introduced by Jin and Xin [The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications in Pure and Applied Mathematics 48 (1995) 235-276] are based on relaxation systems which replace the nonlinear hyperbolic conservation laws by a semi-linear system with a stiff relaxation term. The relaxation parameter ( λ) is chosen locally on the three point stencil of grid which makes the proposed method more efficient. This composite scheme overcomes the problem of unphysical expansion shocks and captures the shock waves with an accuracy better than the upwind relaxation scheme, as demonstrated by the test cases, together with comparisons with popular numerical methods like Roe scheme and ENO schemes.

Finite element method for calculating spectral and optical characteristics of axially symmetric quantum dots

NASA Astrophysics Data System (ADS)

Gusev, A. A.; Chuluunbaatar, O.; Vinitsky, S. I.; Derbov, V. L.; Hai, L. L.; Kazaryan, E. M.; Sarkisyan, H. A.

2018-04-01

We present new calculation schemes using high-order finite element method implemented on unstructured grids with triangle elements for solving boundary-value problems that describe axially symmetric quantum dots. The efficiency of the algorithms and software is demonstrated by benchmark calculations of the energy spectrum, the envelope eigenfunctions of electron, hole and exciton states, and the direct interband light absorption in conical and spheroidal impenetrable quantum dots.

Ideal evolution of magnetohydrodynamic turbulence when imposing Taylor-Green symmetries.

PubMed

Brachet, M E; Bustamante, M D; Krstulovic, G; Mininni, P D; Pouquet, A; Rosenberg, D

2013-01-01

We investigate the ideal and incompressible magnetohydrodynamic (MHD) equations in three space dimensions for the development of potentially singular structures. The methodology consists in implementing the fourfold symmetries of the Taylor-Green vortex generalized to MHD, leading to substantial computer time and memory savings at a given resolution; we also use a regridding method that allows for lower-resolution runs at early times, with no loss of spectral accuracy. One magnetic configuration is examined at an equivalent resolution of 6144(3) points and three different configurations on grids of 4096(3) points. At the highest resolution, two different current and vorticity sheet systems are found to collide, producing two successive accelerations in the development of small scales. At the latest time, a convergence of magnetic field lines to the location of maximum current is probably leading locally to a strong bending and directional variability of such lines. A novel analytical method, based on sharp analysis inequalities, is used to assess the validity of the finite-time singularity scenario. This method allows one to rule out spurious singularities by evaluating the rate at which the logarithmic decrement of the analyticity-strip method goes to zero. The result is that the finite-time singularity scenario cannot be ruled out, and the singularity time could be somewhere between t=2.33 and t=2.70. More robust conclusions will require higher resolution runs and grid-point interpolation measurements of maximum current and vorticity.

Calculating depths to shallow magnetic sources using aeromagnetic data from the Tucson Basin

USGS Publications Warehouse

Casto, Daniel W.

2001-01-01

Using gridded high-resolution aeromagnetic data, the performance of several automated 3-D depth-to-source methods was evaluated over shallow control sources based on how close their depth estimates came to the actual depths to the tops of the sources. For all three control sources, only the simple analytic signal method, the local wavenumber method applied to the vertical integral of the magnetic field, and the horizontal gradient method applied to the pseudo-gravity field provided median depth estimates that were close (-11% to +14% error) to the actual depths. Careful attention to data processing was required in order to calculate a sufficient number of depth estimates and to reduce the occurrence of false depth estimates. For example, to eliminate sampling bias, high-frequency noise and interference from deeper sources, it was necessary to filter the data before calculating derivative grids and subsequent depth estimates. To obtain smooth spatial derivative grids using finite differences, the data had to be gridded at intervals less than one percent of the anomaly wavelength. Before finding peak values in the derived signal grids, it was necessary to remove calculation noise by applying a low-pass filter in the grid-line directions and to re-grid at an interval that enabled the search window to encompass only the peaks of interest. Using the methods that worked best over the control sources, depth estimates over geologic sites of interest suggested the possible occurrence of volcanics nearly 170 meters beneath a city landfill. Also, a throw of around 2 kilometers was determined for a detachment fault that has a displacement of roughly 6 kilometers.

Antiferromagnetic order in the Hubbard model on the Penrose lattice

NASA Astrophysics Data System (ADS)

Koga, Akihisa; Tsunetsugu, Hirokazu

2017-12-01

We study an antiferromagnetic order in the ground state of the half-filled Hubbard model on the Penrose lattice and investigate the effects of quasiperiodic lattice structure. In the limit of infinitesimal Coulomb repulsion U →+0 , the staggered magnetizations persist to be finite, and their values are determined by confined states, which are strictly localized with thermodynamics degeneracy. The magnetizations exhibit an exotic spatial pattern, and have the same sign in each of cluster regions, the size of which ranges from 31 sites to infinity. With increasing U , they continuously evolve to those of the corresponding spin model in the U =∞ limit. In both limits of U , local magnetizations exhibit a fairly intricate spatial pattern that reflects the quasiperiodic structure, but the pattern differs between the two limits. We have analyzed this pattern change by a mode analysis by the singular value decomposition method for the fractal-like magnetization pattern projected into the perpendicular space.

P1 Nonconforming Finite Element Method for the Solution of Radiation Transport Problems

NASA Technical Reports Server (NTRS)

Kang, Kab S.

2002-01-01

The simulation of radiation transport in the optically thick flux-limited diffusion regime has been identified as one of the most time-consuming tasks within large simulation codes. Due to multimaterial complex geometry, the radiation transport system must often be solved on unstructured grids. In this paper, we investigate the behavior and the benefits of the unstructured P(sub 1) nonconforming finite element method, which has proven to be flexible and effective on related transport problems, in solving unsteady implicit nonlinear radiation diffusion problems using Newton and Picard linearization methods. Key words. nonconforrning finite elements, radiation transport, inexact Newton linearization, multigrid preconditioning

Development and application of a volume penalization immersed boundary method for the computation of blood flow and shear stresses in cerebral vessels and aneurysms.

PubMed

Mikhal, Julia; Geurts, Bernard J

2013-12-01

A volume-penalizing immersed boundary method is presented for the simulation of laminar incompressible flow inside geometrically complex blood vessels in the human brain. We concentrate on cerebral aneurysms and compute flow in curved brain vessels with and without spherical aneurysm cavities attached. We approximate blood as an incompressible Newtonian fluid and simulate the flow with the use of a skew-symmetric finite-volume discretization and explicit time-stepping. A key element of the immersed boundary method is the so-called masking function. This is a binary function with which we identify at any location in the domain whether it is 'solid' or 'fluid', allowing to represent objects immersed in a Cartesian grid. We compare three definitions of the masking function for geometries that are non-aligned with the grid. In each case a 'staircase' representation is used in which a grid cell is either 'solid' or 'fluid'. Reliable findings are obtained with our immersed boundary method, even at fairly coarse meshes with about 16 grid cells across a velocity profile. The validation of the immersed boundary method is provided on the basis of classical Poiseuille flow in a cylindrical pipe. We obtain first order convergence for the velocity and the shear stress, reflecting the fact that in our approach the solid-fluid interface is localized with an accuracy on the order of a grid cell. Simulations for curved vessels and aneurysms are done for different flow regimes, characterized by different values of the Reynolds number (Re). The validation is performed for laminar flow at Re = 250, while the flow in more complex geometries is studied at Re = 100 and Re = 250, as suggested by physiological conditions pertaining to flow of blood in the circle of Willis.

A Fourier collocation time domain method for numerically solving Maxwell's equations

NASA Technical Reports Server (NTRS)

Shebalin, John V.

1991-01-01

A new method for solving Maxwell's equations in the time domain for arbitrary values of permittivity, conductivity, and permeability is presented. Spatial derivatives are found by a Fourier transform method and time integration is performed using a second order, semi-implicit procedure. Electric and magnetic fields are collocated on the same grid points, rather than on interleaved points, as in the Finite Difference Time Domain (FDTD) method. Numerical results are presented for the propagation of a 2-D Transverse Electromagnetic (TEM) mode out of a parallel plate waveguide and into a dielectric and conducting medium.