Comparison of two Galerkin quadrature methods

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

Morel, Jim E.; Warsa, James; Franke, Brian C.

Here, we compare two methods for generating Galerkin quadratures. In method 1, the standard S N method is used to generate the moment-to-discrete matrix and the discrete-to-moment matrix is generated by inverting the moment-to-discrete matrix. This is a particular form of the original Galerkin quadrature method. In method 2, which we introduce here, the standard S N method is used to generate the discrete-to-moment matrix and the moment-to-discrete matrix is generated by inverting the discrete-to-moment matrix. With an N-point quadrature, method 1 has the advantage that it preserves N eigenvalues and N eigenvectors of the scattering operator in a pointwisemore » sense. With an N-point quadrature, method 2 has the advantage that it generates consistent angular moment equations from the corresponding S N equations while preserving N eigenvalues of the scattering operator. Our computational results indicate that these two methods are quite comparable for the test problem considered.« less

Adaptive Discontinuous Evolution Galerkin Method for Dry Atmospheric Flow

DTIC Science & Technology

2013-04-02

standard one-dimensional approximate Riemann solver used for the flux integration demonstrate better stability, accuracy as well as reliability of the...discontinuous evolution Galerkin method for dry atmospheric convection. Comparisons with the standard one-dimensional approximate Riemann solver used...instead of a standard one- dimensional approximate Riemann solver , the flux integration within the discontinuous Galerkin method is now realized by

Dual-scale Galerkin methods for Darcy flow

NASA Astrophysics Data System (ADS)

Wang, Guoyin; Scovazzi, Guglielmo; Nouveau, Léo; Kees, Christopher E.; Rossi, Simone; Colomés, Oriol; Main, Alex

2018-02-01

The discontinuous Galerkin (DG) method has found widespread application in elliptic problems with rough coefficients, of which the Darcy flow equations are a prototypical example. One of the long-standing issues of DG approximations is the overall computational cost, and many different strategies have been proposed, such as the variational multiscale DG method, the hybridizable DG method, the multiscale DG method, the embedded DG method, and the Enriched Galerkin method. In this work, we propose a mixed dual-scale Galerkin method, in which the degrees-of-freedom of a less computationally expensive coarse-scale approximation are linked to the degrees-of-freedom of a base DG approximation. We show that the proposed approach has always similar or improved accuracy with respect to the base DG method, with a considerable reduction in computational cost. For the specific definition of the coarse-scale space, we consider Raviart-Thomas finite elements for the mass flux and piecewise-linear continuous finite elements for the pressure. We provide a complete analysis of stability and convergence of the proposed method, in addition to a study on its conservation and consistency properties. We also present a battery of numerical tests to verify the results of the analysis, and evaluate a number of possible variations, such as using piecewise-linear continuous finite elements for the coarse-scale mass fluxes.

SUPG Finite Element Simulations of Compressible Flows

NASA Technical Reports Server (NTRS)

Kirk, Brnjamin, S.

2006-01-01

The Streamline-Upwind Petrov-Galerkin (SUPG) finite element simulations of compressible flows is presented. The topics include: 1) Introduction; 2) SUPG Galerkin Finite Element Methods; 3) Applications; and 4) Bibliography.

A weak Galerkin generalized multiscale finite element method

DOE PAGES

Mu, Lin; Wang, Junping; Ye, Xiu

2016-03-31

In this study, we propose a general framework for weak Galerkin generalized multiscale (WG-GMS) finite element method for the elliptic problems with rapidly oscillating or high contrast coefficients. This general WG-GMS method features in high order accuracy on general meshes and can work with multiscale basis derived by different numerical schemes. A special case is studied under this WG-GMS framework in which the multiscale basis functions are obtained by solving local problem with the weak Galerkin finite element method. Convergence analysis and numerical experiments are obtained for the special case.

A weak Galerkin generalized multiscale finite element method

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

Mu, Lin; Wang, Junping; Ye, Xiu

In this study, we propose a general framework for weak Galerkin generalized multiscale (WG-GMS) finite element method for the elliptic problems with rapidly oscillating or high contrast coefficients. This general WG-GMS method features in high order accuracy on general meshes and can work with multiscale basis derived by different numerical schemes. A special case is studied under this WG-GMS framework in which the multiscale basis functions are obtained by solving local problem with the weak Galerkin finite element method. Convergence analysis and numerical experiments are obtained for the special case.

Parallel Implementation of the Discontinuous Galerkin Method

NASA Technical Reports Server (NTRS)

Baggag, Abdalkader; Atkins, Harold; Keyes, David

1999-01-01

This paper describes a parallel implementation of the discontinuous Galerkin method. Discontinuous Galerkin is a spatially compact method that retains its accuracy and robustness on non-smooth unstructured grids and is well suited for time dependent simulations. Several parallelization approaches are studied and evaluated. The most natural and symmetric of the approaches has been implemented in all object-oriented code used to simulate aeroacoustic scattering. The parallel implementation is MPI-based and has been tested on various parallel platforms such as the SGI Origin, IBM SP2, and clusters of SGI and Sun workstations. The scalability results presented for the SGI Origin show slightly superlinear speedup on a fixed-size problem due to cache effects.

Local Discontinuous Galerkin Methods for the Cahn-Hilliard Type Equations

DTIC Science & Technology

2007-01-01

Kuramoto-Sivashinsky equations , the Ito-type coupled KdV equa- tions, the Kadomtsev - Petviashvili equation , and the Zakharov-Kuznetsov equation . A common...Local discontinuous Galerkin methods for the Cahn-Hilliard type equations Yinhua Xia∗, Yan Xu† and Chi-Wang Shu ‡ Abstract In this paper we develop...local discontinuous Galerkin (LDG) methods for the fourth-order nonlinear Cahn-Hilliard equation and system. The energy stability of the LDG methods is

Weak Galerkin method for the Biot’s consolidation model

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

Hu, Xiaozhe; Mu, Lin; Ye, Xiu

In this study, we develop a weak Galerkin (WG) finite element method for the Biot’s consolidation model in the classical displacement–pressure two-field formulation. Weak Galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations. Backward Euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme. We study the well-posedness of the linear system at each time step and also derive the overall optimal-order convergence of the WG formulation. Such WG scheme is designed on general shape regular polytopal meshes and provides stable and oscillation-free approximation for the pressure withoutmore » special treatment. Lastlyl, numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak Galerkin finite element method.« less

Weak Galerkin method for the Biot’s consolidation model

DOE PAGES

Hu, Xiaozhe; Mu, Lin; Ye, Xiu

2017-08-23

In this study, we develop a weak Galerkin (WG) finite element method for the Biot’s consolidation model in the classical displacement–pressure two-field formulation. Weak Galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations. Backward Euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme. We study the well-posedness of the linear system at each time step and also derive the overall optimal-order convergence of the WG formulation. Such WG scheme is designed on general shape regular polytopal meshes and provides stable and oscillation-free approximation for the pressure withoutmore » special treatment. Lastlyl, numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak Galerkin finite element method.« less

An Application of the Quadrature-Free Discontinuous Galerkin Method

NASA Technical Reports Server (NTRS)

Lockard, David P.; Atkins, Harold L.

2000-01-01

The process of generating a block-structured mesh with the smoothness required for high-accuracy schemes is still a time-consuming process often measured in weeks or months. Unstructured grids about complex geometries are more easily generated, and for this reason, methods using unstructured grids have gained favor for aerodynamic analyses. The discontinuous Galerkin (DG) method is a compact finite-element projection method that provides a practical framework for the development of a high-order method using unstructured grids. Higher-order accuracy is obtained by representing the solution as a high-degree polynomial whose time evolution is governed by a local Galerkin projection. The traditional implementation of the discontinuous Galerkin uses quadrature for the evaluation of the integral projections and is prohibitively expensive. Atkins and Shu introduced the quadrature-free formulation in which the integrals are evaluated a-priori and exactly for a similarity element. The approach has been demonstrated to possess the accuracy required for acoustics even in cases where the grid is not smooth. Other issues such as boundary conditions and the treatment of non-linear fluxes have also been studied in earlier work This paper describes the application of the quadrature-free discontinuous Galerkin method to a two-dimensional shear layer problem. First, a brief description of the method is given. Next, the problem is described and the solution is presented. Finally, the resources required to perform the calculations are given.

A multidimensional finite element method for CFD

NASA Technical Reports Server (NTRS)

Pepper, Darrell W.; Humphrey, Joseph W.

1991-01-01

A finite element method is used to solve the equations of motion for 2- and 3-D fluid flow. The time-dependent equations are solved explicitly using quadrilateral (2-D) and hexahedral (3-D) elements, mass lumping, and reduced integration. A Petrov-Galerkin technique is applied to the advection terms. The method requires a minimum of computational storage, executes quickly, and is scalable for execution on computer systems ranging from PCs to supercomputers.

A Discontinuous Galerkin Finite Element Method for Hamilton-Jacobi Equations

NASA Technical Reports Server (NTRS)

Hu, Changqing; Shu, Chi-Wang

1998-01-01

In this paper, we present a discontinuous Galerkin finite element method for solving the nonlinear Hamilton-Jacobi equations. This method is based on the Runge-Kutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples are given to illustrate the capability of the method.

The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems

NASA Technical Reports Server (NTRS)

Cockburn, Bernardo; Shu, Chi-Wang

1997-01-01

In this paper, we study the Local Discontinuous Galerkin methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge-Kutta Discontinuous Galerkin methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, their high-order formal accuracy, and their easy handling of complicated geometries, for convection dominated problems. It is proven that for scalar equations, the Local Discontinuous Galerkin methods are L(sup 2)-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are k-th order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.

A High Order Discontinuous Galerkin Method for 2D Incompressible Flows

NASA Technical Reports Server (NTRS)

Liu, Jia-Guo; Shu, Chi-Wang

1999-01-01

In this paper we introduce a high order discontinuous Galerkin method for two dimensional incompressible flow in vorticity streamfunction formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method The streamfunction is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability The method is suitable for inviscid or high Reynolds number flows. Optimal error estimates are proven and verified by numerical experiments.

A Moving Discontinuous Galerkin Finite Element Method for Flows with Interfaces

DTIC Science & Technology

2017-12-07

Naval Research Laboratory Washington, DC 20375-5320 NRL/MR/6040--17-9765 A Moving Discontinuous Galerkin Finite Element Method for Flows with...guidance to revise the method to ensure such properties. Acknowledgements This work was sponsored by the Office of Naval Research through the Naval...18. NUMBER OF PAGES 17. LIMITATION OF ABSTRACT A Moving Discontinuous Galerkin Finite Element Method for Flows with Interfaces Andrew Corrigan, Andrew

Discontinuous Galerkin Methods and High-Speed Turbulent Flows

NASA Astrophysics Data System (ADS)

Atak, Muhammed; Larsson, Johan; Munz, Claus-Dieter

2014-11-01

Discontinuous Galerkin methods gain increasing importance within the CFD community as they combine arbitrary high order of accuracy in complex geometries with parallel efficiency. Particularly the discontinuous Galerkin spectral element method (DGSEM) is a promising candidate for both the direct numerical simulation (DNS) and large eddy simulation (LES) of turbulent flows due to its excellent scaling attributes. In this talk, we present a DNS of a compressible turbulent boundary layer along a flat plate at a free-stream Mach number of M = 2.67 and assess the computational efficiency of the DGSEM at performing high-fidelity simulations of both transitional and turbulent boundary layers. We compare the accuracy of the results as well as the computational performance to results using a high order finite difference method.

Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

NASA Technical Reports Server (NTRS)

Yan, Jue; Shu, Chi-Wang; Bushnell, Dennis M. (Technical Monitor)

2002-01-01

In this paper we review the existing and develop new continuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develop new local discontinuous Galerkin methods for the time dependent bi-harmonic type equations involving fourth derivatives, and partial differential equations involving fifth derivatives. For these new methods we present correct interface numerical fluxes and prove L(exp 2) stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methods. Finally, we present new results on a post-processing technique, originally designed for methods with good negative-order error estimates, on the local discontinuous Galerkin methods applied to equations with higher derivatives. Numerical experiments show that this technique works as well for the new higher derivative cases, in effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh.

A variational multiscale method for particle-cloud tracking in turbomachinery flows

NASA Astrophysics Data System (ADS)

Corsini, A.; Rispoli, F.; Sheard, A. G.; Takizawa, K.; Tezduyar, T. E.; Venturini, P.

2014-11-01

We present a computational method for simulation of particle-laden flows in turbomachinery. The method is based on a stabilized finite element fluid mechanics formulation and a finite element particle-cloud tracking method. We focus on induced-draft fans used in process industries to extract exhaust gases in the form of a two-phase fluid with a dispersed solid phase. The particle-laden flow causes material wear on the fan blades, degrading their aerodynamic performance, and therefore accurate simulation of the flow would be essential in reliable computational turbomachinery analysis and design. The turbulent-flow nature of the problem is dealt with a Reynolds-Averaged Navier-Stokes model and Streamline-Upwind/Petrov-Galerkin/Pressure-Stabilizing/Petrov-Galerkin stabilization, the particle-cloud trajectories are calculated based on the flow field and closure models for the turbulence-particle interaction, and one-way dependence is assumed between the flow field and particle dynamics. We propose a closure model utilizing the scale separation feature of the variational multiscale method, and compare that to the closure utilizing the eddy viscosity model. We present computations for axial- and centrifugal-fan configurations, and compare the computed data to those obtained from experiments, analytical approaches, and other computational methods.

On cell entropy inequality for discontinuous Galerkin methods

NASA Technical Reports Server (NTRS)

Jiang, Guangshan; Shu, Chi-Wang

1993-01-01

We prove a cell entropy inequality for a class of high order discontinuous Galerkin finite element methods approximating conservation laws, which implies convergence for the one dimensional scalar convex case.

Galerkin Method for Nonlinear Dynamics

NASA Astrophysics Data System (ADS)

Noack, Bernd R.; Schlegel, Michael; Morzynski, Marek; Tadmor, Gilead

A Galerkin method is presented for control-oriented reduced-order models (ROM). This method generalizes linear approaches elaborated by M. Morzyński et al. for the nonlinear Navier-Stokes equation. These ROM are used as plants for control design in the chapters by G. Tadmor et al., S. Siegel, and R. King in this volume. Focus is placed on empirical ROM which compress flow data in the proper orthogonal decomposition (POD). The chapter shall provide a complete description for construction of straight-forward ROM as well as the physical understanding and teste

A Weak Galerkin Method for the Reissner–Mindlin Plate in Primary Form

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

Mu, Lin; Wang, Junping; Ye, Xiu

We developed a new finite element method for the Reissner–Mindlin equations in its primary form by using the weak Galerkin approach. Like other weak Galerkin finite element methods, this one is highly flexible and robust by allowing the use of discontinuous approximating functions on arbitrary shape of polygons and, at the same time, is parameter independent on its stability and convergence. Furthermore, error estimates of optimal order in mesh size h are established for the corresponding weak Galerkin approximations. Numerical experiments are conducted for verifying the convergence theory, as well as suggesting some superconvergence and a uniform convergence of themore » method with respect to the plate thickness.« less

A Weak Galerkin Method for the Reissner–Mindlin Plate in Primary Form

DOE PAGES

Mu, Lin; Wang, Junping; Ye, Xiu

2017-10-04

We developed a new finite element method for the Reissner–Mindlin equations in its primary form by using the weak Galerkin approach. Like other weak Galerkin finite element methods, this one is highly flexible and robust by allowing the use of discontinuous approximating functions on arbitrary shape of polygons and, at the same time, is parameter independent on its stability and convergence. Furthermore, error estimates of optimal order in mesh size h are established for the corresponding weak Galerkin approximations. Numerical experiments are conducted for verifying the convergence theory, as well as suggesting some superconvergence and a uniform convergence of themore » method with respect to the plate thickness.« less

Simplified Discontinuous Galerkin Methods for Systems of Conservation Laws with Convex Extension

NASA Technical Reports Server (NTRS)

Barth, Timothy J.

1999-01-01

Simplified forms of the space-time discontinuous Galerkin (DG) and discontinuous Galerkin least-squares (DGLS) finite element method are developed and analyzed. The new formulations exploit simplifying properties of entropy endowed conservation law systems while retaining the favorable energy properties associated with symmetric variable formulations.

Discontinuous Galerkin finite element methods for radiative transfer in spherical symmetry

NASA Astrophysics Data System (ADS)

Kitzmann, D.; Bolte, J.; Patzer, A. B. C.

2016-11-01

The discontinuous Galerkin finite element method (DG-FEM) is successfully applied to treat a broad variety of transport problems numerically. In this work, we use the full capacity of the DG-FEM to solve the radiative transfer equation in spherical symmetry. We present a discontinuous Galerkin method to directly solve the spherically symmetric radiative transfer equation as a two-dimensional problem. The transport equation in spherical atmospheres is more complicated than in the plane-parallel case owing to the appearance of an additional derivative with respect to the polar angle. The DG-FEM formalism allows for the exact integration of arbitrarily complex scattering phase functions, independent of the angular mesh resolution. We show that the discontinuous Galerkin method is able to describe accurately the radiative transfer in extended atmospheres and to capture discontinuities or complex scattering behaviour which might be present in the solution of certain radiative transfer tasks and can, therefore, cause severe numerical problems for other radiative transfer solution methods.

Adaptive Discontinuous Galerkin Methods in Multiwavelets Bases

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

Archibald, Richard K; Fann, George I; Shelton Jr, William Allison

2011-01-01

We use a multiwavelet basis with the Discontinuous Galerkin (DG) method to produce a multi-scale DG method. We apply this Multiwavelet DG method to convection and convection-diffusion problems in multiple dimensions. Merging the DG method with multiwavelets allows the adaptivity in the DG method to be resolved through manipulation of multiwavelet coefficients rather than grid manipulation. Additionally, the Multiwavelet DG method is tested on non-linear equations in one dimension and on the cubed sphere.

A Reconstructed Discontinuous Galerkin Method for the Euler Equations on Arbitrary Grids

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

Hong Luo; Luqing Luo; Robert Nourgaliev

2012-11-01

A reconstruction-based discontinuous Galerkin (RDG(P1P2)) method, a variant of P1P2 method, is presented for the solution of the compressible Euler equations on arbitrary grids. In this method, an in-cell reconstruction, designed to enhance the accuracy of the discontinuous Galerkin method, is used to obtain a quadratic polynomial solution (P2) from the underlying linear polynomial (P1) discontinuous Galerkin solution using a least-squares method. The stencils used in the reconstruction involve only the von Neumann neighborhood (face-neighboring cells) and are compact and consistent with the underlying DG method. The developed RDG method is used to compute a variety of flow problems onmore » arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG(P1P2) method is third-order accurate, and outperforms the third-order DG method (DG(P2)) in terms of both computing costs and storage requirements.« less

A hybrid perturbation-Galerkin method for differential equations containing a parameter

NASA Technical Reports Server (NTRS)

Geer, James F.; Andersen, Carl M.

1989-01-01

A two-step hybrid perturbation-Galerkin method to solve a variety of differential equations which involve a parameter is presented and discussed. The method consists of: (1) the use of a perturbation method to determine the asymptotic expansion of the solution about one or more values of the parameter; and (2) the use of some of the perturbation coefficient functions as trial functions in the classical Bubnov-Galerkin method. This hybrid method has the potential of overcoming some of the drawbacks of the perturbation method and the Bubnov-Galerkin method when they are applied by themselves, while combining some of the good features of both. The proposed method is illustrated first with a simple linear two-point boundary value problem and is then applied to a nonlinear two-point boundary value problem in lubrication theory. The results obtained from the hybrid method are compared with approximate solutions obtained by purely numerical methods. Some general features of the method, as well as some special tips for its implementation, are discussed. A survey of some current research application areas is presented and its degree of applicability to broader problem areas is discussed.

An enriched finite element method to fractional advection-diffusion equation

NASA Astrophysics Data System (ADS)

Luan, Shengzhi; Lian, Yanping; Ying, Yuping; Tang, Shaoqiang; Wagner, Gregory J.; Liu, Wing Kam

2017-08-01

In this paper, an enriched finite element method with fractional basis [ 1,x^{α }] for spatial fractional partial differential equations is proposed to obtain more stable and accurate numerical solutions. For pure fractional diffusion equation without advection, the enriched Galerkin finite element method formulation is demonstrated to simulate the exact solution successfully without any numerical oscillation, which is advantageous compared to the traditional Galerkin finite element method with integer basis [ 1,x] . For fractional advection-diffusion equation, the oscillatory behavior becomes complex due to the introduction of the advection term which can be characterized by a fractional element Peclet number. For the purpose of addressing the more complex numerical oscillation, an enriched Petrov-Galerkin finite element method is developed by using a dimensionless fractional stabilization parameter, which is formulated through a minimization of the residual of the nodal solution. The effectiveness and accuracy of the enriched finite element method are demonstrated by a series of numerical examples of fractional diffusion equation and fractional advection-diffusion equation, including both one-dimensional and two-dimensional, steady-state and time-dependent cases.

Super-convergence of Discontinuous Galerkin Method Applied to the Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Atkins, Harold L.

2009-01-01

The practical benefits of the hyper-accuracy properties of the discontinuous Galerkin method are examined. In particular, we demonstrate that some flow attributes exhibit super-convergence even in the absence of any post-processing technique. Theoretical analysis suggest that flow features that are dominated by global propagation speeds and decay or growth rates should be super-convergent. Several discrete forms of the discontinuous Galerkin method are applied to the simulation of unsteady viscous flow over a two-dimensional cylinder. Convergence of the period of the naturally occurring oscillation is examined and shown to converge at 2p+1, where p is the polynomial degree of the discontinuous Galerkin basis. Comparisons are made between the different discretizations and with theoretical analysis.

Note on coefficient matrices from stochastic Galerkin methods for random diffusion equations

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

Zhou Tao, E-mail: tzhou@lsec.cc.ac.c; Tang Tao, E-mail: ttang@hkbu.edu.h

2010-11-01

In a recent work by Xiu and Shen [D. Xiu, J. Shen, Efficient stochastic Galerkin methods for random diffusion equations, J. Comput. Phys. 228 (2009) 266-281], the Galerkin methods are used to solve stochastic diffusion equations in random media, where some properties for the coefficient matrix of the resulting system are provided. They also posed an open question on the properties of the coefficient matrix. In this work, we will provide some results related to the open question.

A coupled electro-thermal Discontinuous Galerkin method

NASA Astrophysics Data System (ADS)

Homsi, L.; Geuzaine, C.; Noels, L.

2017-11-01

This paper presents a Discontinuous Galerkin scheme in order to solve the nonlinear elliptic partial differential equations of coupled electro-thermal problems. In this paper we discuss the fundamental equations for the transport of electricity and heat, in terms of macroscopic variables such as temperature and electric potential. A fully coupled nonlinear weak formulation for electro-thermal problems is developed based on continuum mechanics equations expressed in terms of energetically conjugated pair of fluxes and fields gradients. The weak form can thus be formulated as a Discontinuous Galerkin method. The existence and uniqueness of the weak form solution are proved. The numerical properties of the nonlinear elliptic problems i.e., consistency and stability, are demonstrated under specific conditions, i.e. use of high enough stabilization parameter and at least quadratic polynomial approximations. Moreover the prior error estimates in the H1-norm and in the L2-norm are shown to be optimal in the mesh size with the polynomial approximation degree.

Discontinuous Galerkin Finite Element Method for Parabolic Problems

NASA Technical Reports Server (NTRS)

Kaneko, Hideaki; Bey, Kim S.; Hou, Gene J. W.

2004-01-01

In this paper, we develop a time and its corresponding spatial discretization scheme, based upon the assumption of a certain weak singularity of parallel ut(t) parallel Lz(omega) = parallel ut parallel2, for the discontinuous Galerkin finite element method for one-dimensional parabolic problems. Optimal convergence rates in both time and spatial variables are obtained. A discussion of automatic time-step control method is also included.

MIB Galerkin method for elliptic interface problems.

PubMed

Xia, Kelin; Zhan, Meng; Wei, Guo-Wei

2014-12-15

Material interfaces are omnipresent in the real-world structures and devices. Mathematical modeling of material interfaces often leads to elliptic partial differential equations (PDEs) with discontinuous coefficients and singular sources, which are commonly called elliptic interface problems. The development of high-order numerical schemes for elliptic interface problems has become a well defined field in applied and computational mathematics and attracted much attention in the past decades. Despite of significant advances, challenges remain in the construction of high-order schemes for nonsmooth interfaces, i.e., interfaces with geometric singularities, such as tips, cusps and sharp edges. The challenge of geometric singularities is amplified when they are associated with low solution regularities, e.g., tip-geometry effects in many fields. The present work introduces a matched interface and boundary (MIB) Galerkin method for solving two-dimensional (2D) elliptic PDEs with complex interfaces, geometric singularities and low solution regularities. The Cartesian grid based triangular elements are employed to avoid the time consuming mesh generation procedure. Consequently, the interface cuts through elements. To ensure the continuity of classic basis functions across the interface, two sets of overlapping elements, called MIB elements, are defined near the interface. As a result, differentiation can be computed near the interface as if there is no interface. Interpolation functions are constructed on MIB element spaces to smoothly extend function values across the interface. A set of lowest order interface jump conditions is enforced on the interface, which in turn, determines the interpolation functions. The performance of the proposed MIB Galerkin finite element method is validated by numerical experiments with a wide range of interface geometries, geometric singularities, low regularity solutions and grid resolutions. Extensive numerical studies confirm the

A Streaming Language Implementation of the Discontinuous Galerkin Method

NASA Technical Reports Server (NTRS)

Barth, Timothy; Knight, Timothy

2005-01-01

We present a Brook streaming language implementation of the 3-D discontinuous Galerkin method for compressible fluid flow on tetrahedral meshes. Efficient implementation of the discontinuous Galerkin method using the streaming model of computation introduces several algorithmic design challenges. Using a cycle-accurate simulator, performance characteristics have been obtained for the Stanford Merrimac stream processor. The current Merrimac design achieves 128 Gflops per chip and the desktop board is populated with 16 chips yielding a peak performance of 2 Teraflops. Total parts cost for the desktop board is less than $20K. Current cycle-accurate simulations for discretizations of the 3-D compressible flow equations yield approximately 40-50% of the peak performance of the Merrimac streaming processor chip. Ongoing work includes the assessment of the performance of the same algorithm on the 2 Teraflop desktop board with a target goal of achieving 1 Teraflop performance.

Tensor-product preconditioners for higher-order space-time discontinuous Galerkin methods

NASA Astrophysics Data System (ADS)

Diosady, Laslo T.; Murman, Scott M.

2017-02-01

A space-time discontinuous-Galerkin spectral-element discretization is presented for direct numerical simulation of the compressible Navier-Stokes equations. An efficient solution technique based on a matrix-free Newton-Krylov method is developed in order to overcome the stiffness associated with high solution order. The use of tensor-product basis functions is key to maintaining efficiency at high-order. Efficient preconditioning methods are presented which can take advantage of the tensor-product formulation. A diagonalized Alternating-Direction-Implicit (ADI) scheme is extended to the space-time discontinuous Galerkin discretization. A new preconditioner for the compressible Euler/Navier-Stokes equations based on the fast-diagonalization method is also presented. Numerical results demonstrate the effectiveness of these preconditioners for the direct numerical simulation of subsonic turbulent flows.

Tensor-Product Preconditioners for Higher-Order Space-Time Discontinuous Galerkin Methods

NASA Technical Reports Server (NTRS)

Diosady, Laslo T.; Murman, Scott M.

2016-01-01

space-time discontinuous-Galerkin spectral-element discretization is presented for direct numerical simulation of the compressible Navier-Stokes equat ions. An efficient solution technique based on a matrix-free Newton-Krylov method is developed in order to overcome the stiffness associated with high solution order. The use of tensor-product basis functions is key to maintaining efficiency at high order. Efficient preconditioning methods are presented which can take advantage of the tensor-product formulation. A diagonalized Alternating-Direction-Implicit (ADI) scheme is extended to the space-time discontinuous Galerkin discretization. A new preconditioner for the compressible Euler/Navier-Stokes equations based on the fast-diagonalization method is also presented. Numerical results demonstrate the effectiveness of these preconditioners for the direct numerical simulation of subsonic turbulent flows.

Stochastic Galerkin methods for the steady-state Navier–Stokes equations

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

Sousedík, Bedřich, E-mail: sousedik@umbc.edu; Elman, Howard C., E-mail: elman@cs.umd.edu

2016-07-01

We study the steady-state Navier–Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galerkin method, and we explore properties of the resulting stochastic solutions. We also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic (Galerkin) nonlinear iteration and demonstrate its effectiveness for solving a set of benchmarkmore » problems.« less

Stochastic Galerkin methods for the steady-state Navier–Stokes equations

DOE PAGES

Sousedík, Bedřich; Elman, Howard C.

2016-04-12

We study the steady-state Navier–Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galerkin method, and we explore properties of the resulting stochastic solutions. We also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic (Galerkin) nonlinear iteration and demonstrate its effectiveness for solving a set of benchmarkmore » problems.« less

Analysis and development of adjoint-based h-adaptive direct discontinuous Galerkin method for the compressible Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Cheng, Jian; Yue, Huiqiang; Yu, Shengjiao; Liu, Tiegang

2018-06-01

In this paper, an adjoint-based high-order h-adaptive direct discontinuous Galerkin method is developed and analyzed for the two dimensional steady state compressible Navier-Stokes equations. Particular emphasis is devoted to the analysis of the adjoint consistency for three different direct discontinuous Galerkin discretizations: including the original direct discontinuous Galerkin method (DDG), the direct discontinuous Galerkin method with interface correction (DDG(IC)) and the symmetric direct discontinuous Galerkin method (SDDG). Theoretical analysis shows the extra interface correction term adopted in the DDG(IC) method and the SDDG method plays a key role in preserving the adjoint consistency. To be specific, for the model problem considered in this work, we prove that the original DDG method is not adjoint consistent, while the DDG(IC) method and the SDDG method can be adjoint consistent with appropriate treatment of boundary conditions and correct modifications towards the underlying output functionals. The performance of those three DDG methods is carefully investigated and evaluated through typical test cases. Based on the theoretical analysis, an adjoint-based h-adaptive DDG(IC) method is further developed and evaluated, numerical experiment shows its potential in the applications of adjoint-based adaptation for simulating compressible flows.

High-Order Hyperbolic Residual-Distribution Schemes on Arbitrary Triangular Grids

DTIC Science & Technology

2015-06-22

Galerkin methodology formulated in the framework of the residual-distribution method. For both second- and third- 1. REPORT DATE (DD-MM-YYYY) 4. TITLE AND...construct these schemes based on the Low-Diffusion-A and the Streamwise-Upwind-Petrov-Galerkin methodology formulated in the framework of the residual...methodology formulated in the framework of the residual-distribution method. For both second- and third-order-schemes, we construct a fully implicit

Camellia v1.0 Manual: Part I

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

Roberts, Nathan V.

2016-09-28

Camellia began as an effort to simplify implementation of efficient solvers for the discontinuous Petrov-Galerkin (DPG) finite element methodology of Demkowicz and Gopalakrishnan. Since then, the feature set has expanded, to allow implementation of traditional continuous Galerkin methods, as well as discontinuous Galerkin (DG) methods, hybridizable DG (HDG) methods, first-order-system least squares (FOSLS), and the primal DPG method. This manual serves as an introduction to using Camellia. We begin, in Section 1.1, by describing some of the core features of Camellia. In Section 1.2 we provide an outline of the manual as a whole.

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

NASA Astrophysics Data System (ADS)

Li, Fengyan; Shu, Chi-Wang; Zhang, Yong-Tao; Zhao, Hongkai

2008-09-01

In this paper, we construct a second order fast sweeping method with a discontinuous Galerkin (DG) local solver for computing viscosity solutions of a class of static Hamilton-Jacobi equations, namely the Eikonal equations. Our piecewise linear DG local solver is built on a DG method developed recently [Y. Cheng, C.-W. Shu, A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations, Journal of Computational Physics 223 (2007) 398-415] for the time-dependent Hamilton-Jacobi equations. The causality property of Eikonal equations is incorporated into the design of this solver. The resulting local nonlinear system in the Gauss-Seidel iterations is a simple quadratic system and can be solved explicitly. The compactness of the DG method and the fast sweeping strategy lead to fast convergence of the new scheme for Eikonal equations. Extensive numerical examples verify efficiency, convergence and second order accuracy of the proposed method.

A Fourier spectral-discontinuous Galerkin method for time-dependent 3-D Schrödinger-Poisson equations with discontinuous potentials

NASA Astrophysics Data System (ADS)

Lu, Tiao; Cai, Wei

2008-10-01

In this paper, we propose a high order Fourier spectral-discontinuous Galerkin method for time-dependent Schrödinger-Poisson equations in 3-D spaces. The Fourier spectral Galerkin method is used for the two periodic transverse directions and a high order discontinuous Galerkin method for the longitudinal propagation direction. Such a combination results in a diagonal form for the differential operators along the transverse directions and a flexible method to handle the discontinuous potentials present in quantum heterojunction and supperlattice structures. As the derivative matrices are required for various time integration schemes such as the exponential time differencing and Crank Nicholson methods, explicit derivative matrices of the discontinuous Galerkin method of various orders are derived. Numerical results, using the proposed method with various time integration schemes, are provided to validate the method.

Supercomputer implementation of finite element algorithms for high speed compressible flows

NASA Technical Reports Server (NTRS)

Thornton, E. A.; Ramakrishnan, R.

1986-01-01

Prediction of compressible flow phenomena using the finite element method is of recent origin and considerable interest. Two shock capturing finite element formulations for high speed compressible flows are described. A Taylor-Galerkin formulation uses a Taylor series expansion in time coupled with a Galerkin weighted residual statement. The Taylor-Galerkin algorithms use explicit artificial dissipation, and the performance of three dissipation models are compared. A Petrov-Galerkin algorithm has as its basis the concepts of streamline upwinding. Vectorization strategies are developed to implement the finite element formulations on the NASA Langley VPS-32. The vectorization scheme results in finite element programs that use vectors of length of the order of the number of nodes or elements. The use of the vectorization procedure speeds up processing rates by over two orders of magnitude. The Taylor-Galerkin and Petrov-Galerkin algorithms are evaluated for 2D inviscid flows on criteria such as solution accuracy, shock resolution, computational speed and storage requirements. The convergence rates for both algorithms are enhanced by local time-stepping schemes. Extension of the vectorization procedure for predicting 2D viscous and 3D inviscid flows are demonstrated. Conclusions are drawn regarding the applicability of the finite element procedures for realistic problems that require hundreds of thousands of nodes.

Discontinuous Galerkin Methods for Turbulence Simulation

NASA Technical Reports Server (NTRS)

Collis, S. Scott

2002-01-01

A discontinuous Galerkin (DG) method is formulated, implemented, and tested for simulation of compressible turbulent flows. The method is applied to turbulent channel flow at low Reynolds number, where it is found to successfully predict low-order statistics with fewer degrees of freedom than traditional numerical methods. This reduction is achieved by utilizing local hp-refinement such that the computational grid is refined simultaneously in all three spatial coordinates with decreasing distance from the wall. Another advantage of DG is that Dirichlet boundary conditions can be enforced weakly through integrals of the numerical fluxes. Both for a model advection-diffusion problem and for turbulent channel flow, weak enforcement of wall boundaries is found to improve results at low resolution. Such weak boundary conditions may play a pivotal role in wall modeling for large-eddy simulation.

Non-oscillatory and non-diffusive solution of convection problems by the iteratively reweighted least-squares finite element method

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan

1993-01-01

A comparative description is presented for the least-squares FEM (LSFEM) for 2D steady-state pure convection problems. In addition to exhibiting better control of the streamline derivative than the streamline upwinding Petrov-Galerkin method, numerical convergence rates are obtained which show the LSFEM to be virtually optimal. The LSFEM is used as a framework for an iteratively reweighted LSFEM yielding nonoscillatory and nondiffusive solutions for problems with contact discontinuities; this method is shown to convect contact discontinuities without error when using triangular and bilinear elements.

Planet-disc interactions with Discontinuous Galerkin Methods using GPUs

NASA Astrophysics Data System (ADS)

Velasco Romero, David A.; Veiga, Maria Han; Teyssier, Romain; Masset, Frédéric S.

2018-05-01

We present a two-dimensional Cartesian code based on high order discontinuous Galerkin methods, implemented to run in parallel over multiple GPUs. A simple planet-disc setup is used to compare the behaviour of our code against the behaviour found using the FARGO3D code with a polar mesh. We make use of the time dependence of the torque exerted by the disc on the planet as a mean to quantify the numerical viscosity of the code. We find that the numerical viscosity of the Keplerian flow can be as low as a few 10-8r2Ω, r and Ω being respectively the local orbital radius and frequency, for fifth order schemes and resolution of ˜10-2r. Although for a single disc problem a solution of low numerical viscosity can be obtained at lower computational cost with FARGO3D (which is nearly an order of magnitude faster than a fifth order method), discontinuous Galerkin methods appear promising to obtain solutions of low numerical viscosity in more complex situations where the flow cannot be captured on a polar or spherical mesh concentric with the disc.

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

The dimension split element-free Galerkin method for three-dimensional potential problems

NASA Astrophysics Data System (ADS)

Meng, Z. J.; Cheng, H.; Ma, L. D.; Cheng, Y. M.

2018-06-01

This paper presents the dimension split element-free Galerkin (DSEFG) method for three-dimensional potential problems, and the corresponding formulae are obtained. The main idea of the DSEFG method is that a three-dimensional potential problem can be transformed into a series of two-dimensional problems. For these two-dimensional problems, the improved moving least-squares (IMLS) approximation is applied to construct the shape function, which uses an orthogonal function system with a weight function as the basis functions. The Galerkin weak form is applied to obtain a discretized system equation, and the penalty method is employed to impose the essential boundary condition. The finite difference method is selected in the splitting direction. For the purposes of demonstration, some selected numerical examples are solved using the DSEFG method. The convergence study and error analysis of the DSEFG method are presented. The numerical examples show that the DSEFG method has greater computational precision and computational efficiency than the IEFG method.

Numerical solution of the Navier-Stokes equations by discontinuous Galerkin method

NASA Astrophysics Data System (ADS)

Krasnov, M. M.; Kuchugov, P. A.; E Ladonkina, M.; E Lutsky, A.; Tishkin, V. F.

2017-02-01

Detailed unstructured grids and numerical methods of high accuracy are frequently used in the numerical simulation of gasdynamic flows in areas with complex geometry. Galerkin method with discontinuous basis functions or Discontinuous Galerkin Method (DGM) works well in dealing with such problems. This approach offers a number of advantages inherent to both finite-element and finite-difference approximations. Moreover, the present paper shows that DGM schemes can be viewed as Godunov method extension to piecewise-polynomial functions. As is known, DGM involves significant computational complexity, and this brings up the question of ensuring the most effective use of all the computational capacity available. In order to speed up the calculations, operator programming method has been applied while creating the computational module. This approach makes possible compact encoding of mathematical formulas and facilitates the porting of programs to parallel architectures, such as NVidia CUDA and Intel Xeon Phi. With the software package, based on DGM, numerical simulations of supersonic flow past solid bodies has been carried out. The numerical results are in good agreement with the experimental ones.

Numerical solution of the unsteady diffusion-convection-reaction equation based on improved spectral Galerkin method

NASA Astrophysics Data System (ADS)

Zhong, Jiaqi; Zeng, Cheng; Yuan, Yupeng; Zhang, Yuzhe; Zhang, Ye

2018-04-01

The aim of this paper is to present an explicit numerical algorithm based on improved spectral Galerkin method for solving the unsteady diffusion-convection-reaction equation. The principal characteristics of this approach give the explicit eigenvalues and eigenvectors based on the time-space separation method and boundary condition analysis. With the help of Fourier series and Galerkin truncation, we can obtain the finite-dimensional ordinary differential equations which facilitate the system analysis and controller design. By comparing with the finite element method, the numerical solutions are demonstrated via two examples. It is shown that the proposed method is effective.

Discontinuous Galerkin methods for Hamiltonian ODEs and PDEs

NASA Astrophysics Data System (ADS)

Tang, Wensheng; Sun, Yajuan; Cai, Wenjun

2017-02-01

In this article, we present a unified framework of discontinuous Galerkin (DG) discretizations for Hamiltonian ODEs and PDEs. We show that with appropriate numerical fluxes the numerical algorithms deduced from DG discretizations can be combined with the symplectic methods in time to derive the multi-symplectic PRK schemes. The resulting numerical discretizations are applied to the linear and nonlinear Schrödinger equations. Some conservative properties of the numerical schemes are investigated and confirmed in the numerical experiments.

POD/DEIM reduced-order strategies for efficient four dimensional variational data assimilation

NASA Astrophysics Data System (ADS)

Ştefănescu, R.; Sandu, A.; Navon, I. M.

2015-08-01

This work studies reduced order modeling (ROM) approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models. It is shown that a key requirement for a successful reduced order solution is that reduced order Karush-Kuhn-Tucker conditions accurately represent their full order counterparts. In particular, accurate reduced order approximations are needed for the forward and adjoint dynamical models, as well as for the reduced gradient. New strategies to construct reduced order based are developed for proper orthogonal decomposition (POD) ROM data assimilation using both Galerkin and Petrov-Galerkin projections. For the first time POD, tensorial POD, and discrete empirical interpolation method (DEIM) are employed to develop reduced data assimilation systems for a geophysical flow model, namely, the two dimensional shallow water equations. Numerical experiments confirm the theoretical framework for Galerkin projection. In the case of Petrov-Galerkin projection, stabilization strategies must be considered for the reduced order models. The new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time.

A full vectorial generalized discontinuous Galerkin beam propagation method (GDG-BPM) for nonsmooth electromagnetic fields in waveguides

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

Fan Kai; Cai Wei; Ji Xia

2008-07-20

In this paper, we propose a new full vectorial generalized discontinuous Galerkin beam propagation method (GDG-BPM) to accurately handle the discontinuities in electromagnetic fields associated with wave propagations in inhomogeneous optical waveguides. The numerical method is a combination of the traditional beam propagation method (BPM) with a newly developed generalized discontinuous Galerkin (GDG) method [K. Fan, W. Cai, X. Ji, A generalized discontinuous Galerkin method (GDG) for Schroedinger equations with nonsmooth solutions, J. Comput. Phys. 227 (2008) 2387-2410]. The GDG method is based on a reformulation, using distributional variables to account for solution jumps across material interfaces, of Schroedinger equationsmore » resulting from paraxial approximations of vector Helmholtz equations. Four versions of the GDG-BPM are obtained for either the electric or magnetic field components. Modeling of wave propagations in various optical fibers using the full vectorial GDG-BPM is included. Numerical results validate the high order accuracy and the flexibility of the method for various types of interface jump conditions.« less

A Lagrangian discontinuous Galerkin hydrodynamic method

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

Liu, Xiaodong; Morgan, Nathaniel Ray; Burton, Donald E.

Here, we present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for solving the two-dimensional gas dynamic equations on unstructured hybrid meshes. The physical conservation laws for the momentum and total energy are discretized using a DG method based on linear Taylor expansions. Three different approaches are investigated for calculating the density variation over the element. The first approach evolves a Taylor expansion of the specific volume field. The second approach follows certain finite element methods and uses the strong mass conservation to calculate the density field at a location inside the element or on the element surface. The thirdmore » approach evolves a Taylor expansion of the density field. The nodal velocity, and the corresponding forces, are explicitly calculated by solving a multidirectional approximate Riemann problem. An effective limiting strategy is presented that ensures monotonicity of the primitive variables. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. Results from a suite of test problems are presented to demonstrate the robustness and expected second-order accuracy of this new method.« less

A Lagrangian discontinuous Galerkin hydrodynamic method

DOE PAGES

Liu, Xiaodong; Morgan, Nathaniel Ray; Burton, Donald E.

2017-12-11

Here, we present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for solving the two-dimensional gas dynamic equations on unstructured hybrid meshes. The physical conservation laws for the momentum and total energy are discretized using a DG method based on linear Taylor expansions. Three different approaches are investigated for calculating the density variation over the element. The first approach evolves a Taylor expansion of the specific volume field. The second approach follows certain finite element methods and uses the strong mass conservation to calculate the density field at a location inside the element or on the element surface. The thirdmore » approach evolves a Taylor expansion of the density field. The nodal velocity, and the corresponding forces, are explicitly calculated by solving a multidirectional approximate Riemann problem. An effective limiting strategy is presented that ensures monotonicity of the primitive variables. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. Results from a suite of test problems are presented to demonstrate the robustness and expected second-order accuracy of this new method.« less

Discontinuous Galerkin finite element method for solving population density functions of cortical pyramidal and thalamic neuronal populations.

PubMed

Huang, Chih-Hsu; Lin, Chou-Ching K; Ju, Ming-Shaung

2015-02-01

Compared with the Monte Carlo method, the population density method is efficient for modeling collective dynamics of neuronal populations in human brain. In this method, a population density function describes the probabilistic distribution of states of all neurons in the population and it is governed by a hyperbolic partial differential equation. In the past, the problem was mainly solved by using the finite difference method. In a previous study, a continuous Galerkin finite element method was found better than the finite difference method for solving the hyperbolic partial differential equation; however, the population density function often has discontinuity and both methods suffer from a numerical stability problem. The goal of this study is to improve the numerical stability of the solution using discontinuous Galerkin finite element method. To test the performance of the new approach, interaction of a population of cortical pyramidal neurons and a population of thalamic neurons was simulated. The numerical results showed good agreement between results of discontinuous Galerkin finite element and Monte Carlo methods. The convergence and accuracy of the solutions are excellent. The numerical stability problem could be resolved using the discontinuous Galerkin finite element method which has total-variation-diminishing property. The efficient approach will be employed to simulate the electroencephalogram or dynamics of thalamocortical network which involves three populations, namely, thalamic reticular neurons, thalamocortical neurons and cortical pyramidal neurons. Copyright © 2014 Elsevier Ltd. All rights reserved.

Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrodinger equations

DOE PAGES

Liang, Xiao; Khaliq, Abdul Q. M.; Xing, Yulong

2015-01-23

In this paper, we study a local discontinuous Galerkin method combined with fourth order exponential time differencing Runge-Kutta time discretization and a fourth order conservative method for solving the nonlinear Schrödinger equations. Based on different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative local discontinuous Galerkin methods, and have proven the error estimates for the semi-discrete methods applied to linear Schrödinger equation. The numerical methods are proven to be highly efficient and stable for long-range soliton computations. Finally, extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.

Hybridizable discontinuous Galerkin method for the 2-D frequency-domain elastic wave equations

NASA Astrophysics Data System (ADS)

Bonnasse-Gahot, Marie; Calandra, Henri; Diaz, Julien; Lanteri, Stéphane

2018-04-01

Discontinuous Galerkin (DG) methods are nowadays actively studied and increasingly exploited for the simulation of large-scale time-domain (i.e. unsteady) seismic wave propagation problems. Although theoretically applicable to frequency-domain problems as well, their use in this context has been hampered by the potentially large number of coupled unknowns they incur, especially in the 3-D case, as compared to classical continuous finite element methods. In this paper, we address this issue in the framework of the so-called hybridizable discontinuous Galerkin (HDG) formulations. As a first step, we study an HDG method for the resolution of the frequency-domain elastic wave equations in the 2-D case. We describe the weak formulation of the method and provide some implementation details. The proposed HDG method is assessed numerically including a comparison with a classical upwind flux-based DG method, showing better overall computational efficiency as a result of the drastic reduction of the number of globally coupled unknowns in the resulting discrete HDG system.

Applications of Taylor-Galerkin finite element method to compressible internal flow problems

NASA Technical Reports Server (NTRS)

Sohn, Jeong L.; Kim, Yongmo; Chung, T. J.

1989-01-01

A two-step Taylor-Galerkin finite element method with Lapidus' artificial viscosity scheme is applied to several test cases for internal compressible inviscid flow problems. Investigations for the effect of supersonic/subsonic inlet and outlet boundary conditions on computational results are particularly emphasized.

A B-spline Galerkin method for the Dirac equation

NASA Astrophysics Data System (ADS)

Froese Fischer, Charlotte; Zatsarinny, Oleg

2009-06-01

The B-spline Galerkin method is first investigated for the simple eigenvalue problem, y=-λy, that can also be written as a pair of first-order equations y=λz, z=-λy. Expanding both y(r) and z(r) in the B basis results in many spurious solutions such as those observed for the Dirac equation. However, when y(r) is expanded in the B basis and z(r) in the dB/dr basis, solutions of the well-behaved second-order differential equation are obtained. From this analysis, we propose a stable method ( B,B) basis for the Dirac equation and evaluate its accuracy by comparing the computed and exact R-matrix for a wide range of nuclear charges Z and angular quantum numbers κ. When splines of the same order are used, many spurious solutions are found whereas none are found for splines of different order. Excellent agreement is obtained for the R-matrix and energies for bound states for low values of Z. For high Z, accuracy requires the use of a grid with many points near the nucleus. We demonstrate the accuracy of the bound-state wavefunctions by comparing integrals arising in hyperfine interaction matrix elements with exact analytic expressions. We also show that the Thomas-Reiche-Kuhn sum rule is not a good measure of the quality of the solutions obtained by the B-spline Galerkin method whereas the R-matrix is very sensitive to the appearance of pseudo-states.

WEAK GALERKIN METHODS FOR SECOND ORDER ELLIPTIC INTERFACE PROBLEMS

PubMed Central

MU, LIN; WANG, JUNPING; WEI, GUOWEI; YE, XIU; ZHAO, SHAN

2013-01-01

Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both L2 and L∞ norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order O(h) to O(h1.5) for the solution itself in L∞ norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order O(h1.75) to O(h2) in the L∞ norm for C1 or Lipschitz continuous interfaces associated with a C1 or H2 continuous solution. PMID:24072935

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.

An HP Adaptive Discontinuous Galerkin Method for Hyperbolic Conservation Laws. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Bey, Kim S.

1994-01-01

This dissertation addresses various issues for model classes of hyperbolic conservation laws. The basic approach developed in this work employs a new family of adaptive, hp-version, finite element methods based on a special discontinuous Galerkin formulation for hyperbolic problems. The discontinuous Galerkin formulation admits high-order local approximations on domains of quite general geometry, while providing a natural framework for finite element approximations and for theoretical developments. The use of hp-versions of the finite element method makes possible exponentially convergent schemes with very high accuracies in certain cases; the use of adaptive hp-schemes allows h-refinement in regions of low regularity and p-enrichment to deliver high accuracy, while keeping problem sizes manageable and dramatically smaller than many conventional approaches. The use of discontinuous Galerkin methods is uncommon in applications, but the methods rest on a reasonable mathematical basis for low-order cases and has local approximation features that can be exploited to produce very efficient schemes, especially in a parallel, multiprocessor environment. The place of this work is to first and primarily focus on a model class of linear hyperbolic conservation laws for which concrete mathematical results, methodologies, error estimates, convergence criteria, and parallel adaptive strategies can be developed, and to then briefly explore some extensions to more general cases. Next, we provide preliminaries to the study and a review of some aspects of the theory of hyperbolic conservation laws. We also provide a review of relevant literature on this subject and on the numerical analysis of these types of problems.

Tensor-product preconditioners for a space-time discontinuous Galerkin method

NASA Astrophysics Data System (ADS)

Diosady, Laslo T.; Murman, Scott M.

2014-10-01

A space-time discontinuous Galerkin spectral element discretization is presented for direct numerical simulation of the compressible Navier-Stokes equations. An efficient solution technique based on a matrix-free Newton-Krylov method is presented. A diagonalized alternating direction implicit preconditioner is extended to a space-time formulation using entropy variables. The effectiveness of this technique is demonstrated for the direct numerical simulation of turbulent flow in a channel.

What happens to Petrov classification, on horizons of axisymmetric dirty black holes

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

Tanatarov, I. V., E-mail: igor.tanatarov@gmail.com; Department of Physics and Technology, Kharkov V.N. Karazin National University, 4 Svoboda Square, Kharkov 61077; Zaslavskii, O. B., E-mail: zaslav@ukr.net

2014-02-15

We consider axisymmetric stationary dirty black holes with regular non-extremal or extremal horizons, and compute their on-horizon Petrov types. The Petrov type (PT) in the frame of the observer crossing the horizon can be different from that formally obtained in the usual (but singular in the horizon limit) frame of an observer on a circular orbit. We call this entity the boosted Petrov type (BPT), as the corresponding frame is obtained by a singular boost from the regular one. The PT off-horizon can be more general than PT on-horizon and that can be more general than the BPT on horizon.more » This is valid for all regular metrics, irrespective of the extremality of the horizon. We analyze and classify the possible relations between the three characteristics and discuss the nature and features of the underlying singular boost. The three Petrov types can be the same only for space-times of PT D and O off-horizon. The mutual alignment of principal null directions and the generator in the vicinity of the horizon is studied in detail. As an example, we also analyze a special class of metrics with utra-extremal horizons (for which the regularity conditions look different from the general case) and compare their off-horizon and on-horizon algebraic structure in both frames.« less

Effective implementation of wavelet Galerkin method

NASA Astrophysics Data System (ADS)

Finěk, Václav; Šimunková, Martina

2012-11-01

It was proved by W. Dahmen et al. that an adaptive wavelet scheme is asymptotically optimal for a wide class of elliptic equations. This scheme approximates the solution u by a linear combination of N wavelets and a benchmark for its performance is the best N-term approximation, which is obtained by retaining the N largest wavelet coefficients of the unknown solution. Moreover, the number of arithmetic operations needed to compute the approximate solution is proportional to N. The most time consuming part of this scheme is the approximate matrix-vector multiplication. In this contribution, we will introduce our implementation of wavelet Galerkin method for Poisson equation -Δu = f on hypercube with homogeneous Dirichlet boundary conditions. In our implementation, we identified nonzero elements of stiffness matrix corresponding to the above problem and we perform matrix-vector multiplication only with these nonzero elements.

A weak Galerkin least-squares finite element method for div-curl systems

NASA Astrophysics Data System (ADS)

Li, Jichun; Ye, Xiu; Zhang, Shangyou

2018-06-01

In this paper, we introduce a weak Galerkin least-squares method for solving div-curl problem. This finite element method leads to a symmetric positive definite system and has the flexibility to work with general meshes such as hybrid mesh, polytopal mesh and mesh with hanging nodes. Error estimates of the finite element solution are derived. The numerical examples demonstrate the robustness and flexibility of the proposed method.

Study of flow over object problems by a nodal discontinuous Galerkin-lattice Boltzmann method

NASA Astrophysics Data System (ADS)

Wu, Jie; Shen, Meng; Liu, Chen

2018-04-01

The flow over object problems are studied by a nodal discontinuous Galerkin-lattice Boltzmann method (NDG-LBM) in this work. Different from the standard lattice Boltzmann method, the current method applies the nodal discontinuous Galerkin method into the streaming process in LBM to solve the resultant pure convection equation, in which the spatial discretization is completed on unstructured grids and the low-storage explicit Runge-Kutta scheme is used for time marching. The present method then overcomes the disadvantage of standard LBM for depending on the uniform meshes. Moreover, the collision process in the LBM is completed by using the multiple-relaxation-time scheme. After the validation of the NDG-LBM by simulating the lid-driven cavity flow, the simulations of flows over a fixed circular cylinder, a stationary airfoil and rotating-stationary cylinders are performed. Good agreement of present results with previous results is achieved, which indicates that the current NDG-LBM is accurate and effective for flow over object problems.

Numerical and experimental validation of a particle Galerkin method for metal grinding simulation

NASA Astrophysics Data System (ADS)

Wu, C. T.; Bui, Tinh Quoc; Wu, Youcai; Luo, Tzui-Liang; Wang, Morris; Liao, Chien-Chih; Chen, Pei-Yin; Lai, Yu-Sheng

2018-03-01

In this paper, a numerical approach with an experimental validation is introduced for modelling high-speed metal grinding processes in 6061-T6 aluminum alloys. The derivation of the present numerical method starts with an establishment of a stabilized particle Galerkin approximation. A non-residual penalty term from strain smoothing is introduced as a means of stabilizing the particle Galerkin method. Additionally, second-order strain gradients are introduced to the penalized functional for the regularization of damage-induced strain localization problem. To handle the severe deformation in metal grinding simulation, an adaptive anisotropic Lagrangian kernel is employed. Finally, the formulation incorporates a bond-based failure criterion to bypass the prospective spurious damage growth issues in material failure and cutting debris simulation. A three-dimensional metal grinding problem is analyzed and compared with the experimental results to demonstrate the effectiveness and accuracy of the proposed numerical approach.

A Reconstructed Discontinuous Galerkin Method for the Compressible Euler Equations on Arbitrary Grids

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

Hong Luo; Luquing Luo; Robert Nourgaliev

2009-06-01

A reconstruction-based discontinuous Galerkin (DG) method is presented for the solution of the compressible Euler equations on arbitrary grids. By taking advantage of handily available and yet invaluable information, namely the derivatives, in the context of the discontinuous Galerkin methods, a solution polynomial of one degree higher is reconstructed using a least-squares method. The stencils used in the reconstruction involve only the van Neumann neighborhood (face-neighboring cells) and are compact and consistent with the underlying DG method. The resulting DG method can be regarded as an improvement of a recovery-based DG method in the sense that it shares the samemore » nice features as the recovery-based DG method, such as high accuracy and efficiency, and yet overcomes some of its shortcomings such as a lack of flexibility, compactness, and robustness. The developed DG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate the accuracy and efficiency of the method. The numerical results indicate that this reconstructed DG method is able to obtain a third-order accurate solution at a slightly higher cost than its second-order DG method and provide an increase in performance over the third order DG method in terms of computing time and storage requirement.« less

A Meshless Method Using Radial Basis Functions for Beam Bending Problems

NASA Technical Reports Server (NTRS)

Raju, I. S.; Phillips, D. R.; Krishnamurthy, T.

2004-01-01

A meshless local Petrov-Galerkin (MLPG) method that uses radial basis functions (RBFs) as trial functions in the study of Euler-Bernoulli beam problems is presented. RBFs, rather than generalized moving least squares (GMLS) interpolations, are used to develop the trial functions. This choice yields a computationally simpler method as fewer matrix inversions and multiplications are required than when GMLS interpolations are used. Test functions are chosen as simple weight functions as they are in the conventional MLPG method. Compactly and noncompactly supported RBFs are considered. Noncompactly supported cubic RBFs are found to be preferable. Patch tests, mixed boundary value problems, and problems with complex loading conditions are considered. Results obtained from the radial basis MLPG method are either of comparable or better accuracy than those obtained when using the conventional MLPG method.

Galerkin-collocation domain decomposition method for arbitrary binary black holes

NASA Astrophysics Data System (ADS)

Barreto, W.; Clemente, P. C. M.; de Oliveira, H. P.; Rodriguez-Mueller, B.

2018-05-01

We present a new computational framework for the Galerkin-collocation method for double domain in the context of ADM 3 +1 approach in numerical relativity. This work enables us to perform high resolution calculations for initial sets of two arbitrary black holes. We use the Bowen-York method for binary systems and the puncture method to solve the Hamiltonian constraint. The nonlinear numerical code solves the set of equations for the spectral modes using the standard Newton-Raphson method, LU decomposition and Gaussian quadratures. We show convergence of our code for the conformal factor and the ADM mass. Thus, we display features of the conformal factor for different masses, spins and linear momenta.

A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations

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

Mu, Lin; Wang, Junping; Ye, Xiu

Here, in this article, we introduce a least-squares-based weak Galerkin finite element method for the second order elliptic equation. This new method is shown to provide very accurate numerical approximations for both the primal and the flux variables. In contrast to other existing least-squares finite element methods, this new method allows us to use discontinuous approximating functions on finite element partitions consisting of arbitrary polygon/polyhedron shapes. We also develop a Schur complement algorithm for the resulting discretization problem by eliminating all the unknowns that represent the solution information in the interior of each element. Optimal order error estimates for bothmore » the primal and the flux variables are established. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the least-squares-based weak Galerkin finite element method. Finally, the numerical examples cover a wide range of applied problems, including singularly perturbed reaction-diffusion equations and the flow of fluid in porous media with strong anisotropy and heterogeneity.« less

A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations

DOE PAGES

Mu, Lin; Wang, Junping; Ye, Xiu

2017-08-17

Here, in this article, we introduce a least-squares-based weak Galerkin finite element method for the second order elliptic equation. This new method is shown to provide very accurate numerical approximations for both the primal and the flux variables. In contrast to other existing least-squares finite element methods, this new method allows us to use discontinuous approximating functions on finite element partitions consisting of arbitrary polygon/polyhedron shapes. We also develop a Schur complement algorithm for the resulting discretization problem by eliminating all the unknowns that represent the solution information in the interior of each element. Optimal order error estimates for bothmore » the primal and the flux variables are established. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the least-squares-based weak Galerkin finite element method. Finally, the numerical examples cover a wide range of applied problems, including singularly perturbed reaction-diffusion equations and the flow of fluid in porous media with strong anisotropy and heterogeneity.« less

An improved wavelet-Galerkin method for dynamic response reconstruction and parameter identification of shear-type frames

NASA Astrophysics Data System (ADS)

Bu, Haifeng; Wang, Dansheng; Zhou, Pin; Zhu, Hongping

2018-04-01

An improved wavelet-Galerkin (IWG) method based on the Daubechies wavelet is proposed for reconstructing the dynamic responses of shear structures. The proposed method flexibly manages wavelet resolution level according to excitation, thereby avoiding the weakness of the wavelet-Galerkin multiresolution analysis (WGMA) method in terms of resolution and the requirement of external excitation. IWG is implemented by this work in certain case studies, involving single- and n-degree-of-freedom frame structures subjected to a determined discrete excitation. Results demonstrate that IWG performs better than WGMA in terms of accuracy and computation efficiency. Furthermore, a new method for parameter identification based on IWG and an optimization algorithm are also developed for shear frame structures, and a simultaneous identification of structural parameters and excitation is implemented. Numerical results demonstrate that the proposed identification method is effective for shear frame structures.

Uniqueness of Petrov Type D Spatially Inhomogeneous Irrotational Silent Models

NASA Astrophysics Data System (ADS)

Apostolopoulos, Pantelis S.; Carot, Jaume

The consistency of the constraint with the evolution equations for spatially inhomogeneous and irrotational silent (SIIS) models of Petrov type I, demands that the former are preserved along the timelike congruence represented by the velocity of the dust fluid, leading to new nontrivial constraints. This fact has been used to conjecture that the resulting models correspond to the spatially homogeneous (SH) models of Bianchi type I, at least for the case where the cosmological constant vanish. By exploiting the full set of the constraint equations as expressed in the 1+3 covariant formalism and using elements from the theory of the spacelike congruences, we provide a direct and simple proof of this conjecture for vacuum and dust fluid models, which shows that the Szekeres family of solutions represents the most general class of SIIS models. The suggested procedure also shows that, the uniqueness of the SIIS of the Petrov type D is not, in general, affected by the presence of a nonzero pressure fluid. Therefore, in order to allow a broader class of Petrov type I solutions apart from the SH models of Bianchi type I, one should consider more general "silent" configurations by relaxing the vanishing of the vorticity and the magnetic part of the Weyl tensor but maintaining their "silence" properties, i.e. the vanishing of the curls of Eab, Hab and the pressure p.

The Discontinuous Galerkin Method for the Multiscale Modeling of Dynamics of Crystalline Solids

DTIC Science & Technology

2007-08-26

number. 1. REPORT DATE 26 AUG 2007 2 . REPORT TYPE 3. DATES COVERED 00-00-2007 to 00-00-2007 4. TITLE AND SUBTITLE The Discontinuous Galerkin...Dynamics method (MAAD) [ 2 ], the bridging scale method [47], the bridging domain methods [48], the heterogeneous multiscale method (HMM) [23, 36, 24], and...method consists of three components, 1. a macro solver for the continuum model, 2 . a micro solver to equilibrate the atomistic system locally to the appro

Mass Conservation of the Unified Continuous and Discontinuous Element-Based Galerkin Methods on Dynamically Adaptive Grids with Application to Atmospheric Simulations

DTIC Science & Technology

2015-09-01

Discontinuous Element-Based Galerkin Methods on Dynamically Adaptive Grids with Application to Atmospheric Simulations 5a. CONTRACT NUMBER 5b. GRANT NUMBER...Discontinuous Element-Based Galerkin Methods on Dynamically Adaptive Grids with Application to Atmospheric Simulations. Michal A. Koperaa,∗, Francis X...mass conservation, as it is an important feature for many atmospheric applications . We believe this is a good metric because, for smooth solutions

Discontinuous Galerkin Method with Numerical Roe Flux for Spherical Shallow Water Equations

NASA Astrophysics Data System (ADS)

Yi, T.; Choi, S.; Kang, S.

2013-12-01

In developing the dynamic core of a numerical weather prediction model with discontinuous Galerkin method, a numerical flux at the boundaries of grid elements plays a vital role since it preserves the local conservation properties and has a significant impact on the accuracy and stability of numerical solutions. Due to these reasons, we developed the numerical Roe flux based on an approximate Riemann problem for spherical shallow water equations in Cartesian coordinates [1] to find out its stability and accuracy. In order to compare the performance with its counterpart flux, we used the Lax-Friedrichs flux, which has been used in many dynamic cores such as NUMA [1], CAM-DG [2] and MCore [3] because of its simplicity. The Lax-Friedrichs flux is implemented by a flux difference between left and right states plus the maximum characteristic wave speed across the boundaries of elements. It has been shown that the Lax-Friedrichs flux with the finite volume method is more dissipative and unstable than other numerical fluxes such as HLLC, AUSM+ and Roe. The Roe flux implemented in this study is based on the decomposition of flux difference over the element boundaries where the nonlinear equations are linearized. It is rarely used in dynamic cores due to its complexity and thus computational expensiveness. To compare the stability and accuracy of the Roe flux with the Lax-Friedrichs, two- and three-dimensional test cases are performed on a plane and cubed-sphere, respectively, with various numbers of element and polynomial order. For the two-dimensional case, the Gaussian bell is simulated on the plane with two different numbers of elements at the fixed polynomial orders. In three-dimensional cases on the cubed-sphere, we performed the test cases of a zonal flow over an isolated mountain and a Rossby-Haurwitz wave, of which initial conditions are the same as those of Williamson [4]. This study presented that the Roe flux with the discontinuous Galerkin method is less

An Advanced One-Dimensional Finite Element Model for Incompressible Thermally Expandable Flow

DOE PAGES

Hu, Rui

2017-03-27

Here, this paper provides an overview of a new one-dimensional finite element flow model for incompressible but thermally expandable flow. The flow model was developed for use in system analysis tools for whole-plant safety analysis of sodium fast reactors. Although the pressure-based formulation was implemented, the use of integral equations in the conservative form ensured the conservation laws of the fluid. A stabilization scheme based on streamline-upwind/Petrov-Galerkin and pressure-stabilizing/Petrov-Galerkin formulations is also introduced. The flow model and its implementation have been verified by many test problems, including density wave propagation, steep gradient problems, discharging between tanks, and the conjugate heatmore » transfer in a heat exchanger.« less

An Advanced One-Dimensional Finite Element Model for Incompressible Thermally Expandable Flow

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

Hu, Rui

Here, this paper provides an overview of a new one-dimensional finite element flow model for incompressible but thermally expandable flow. The flow model was developed for use in system analysis tools for whole-plant safety analysis of sodium fast reactors. Although the pressure-based formulation was implemented, the use of integral equations in the conservative form ensured the conservation laws of the fluid. A stabilization scheme based on streamline-upwind/Petrov-Galerkin and pressure-stabilizing/Petrov-Galerkin formulations is also introduced. The flow model and its implementation have been verified by many test problems, including density wave propagation, steep gradient problems, discharging between tanks, and the conjugate heatmore » transfer in a heat exchanger.« less

Individualized drug dosing using RBF-Galerkin method: Case of anemia management in chronic kidney disease.

PubMed

Mirinejad, Hossein; Gaweda, Adam E; Brier, Michael E; Zurada, Jacek M; Inanc, Tamer

2017-09-01

Anemia is a common comorbidity in patients with chronic kidney disease (CKD) and is frequently associated with decreased physical component of quality of life, as well as adverse cardiovascular events. Current treatment methods for renal anemia are mostly population-based approaches treating individual patients with a one-size-fits-all model. However, FDA recommendations stipulate individualized anemia treatment with precise control of the hemoglobin concentration and minimal drug utilization. In accordance with these recommendations, this work presents an individualized drug dosing approach to anemia management by leveraging the theory of optimal control. A Multiple Receding Horizon Control (MRHC) approach based on the RBF-Galerkin optimization method is proposed for individualized anemia management in CKD patients. Recently developed by the authors, the RBF-Galerkin method uses the radial basis function approximation along with the Galerkin error projection to solve constrained optimal control problems numerically. The proposed approach is applied to generate optimal dosing recommendations for individual patients. Performance of the proposed approach (MRHC) is compared in silico to that of a population-based anemia management protocol and an individualized multiple model predictive control method for two case scenarios: hemoglobin measurement with and without observational errors. In silico comparison indicates that hemoglobin concentration with MRHC method has less variation among the methods, especially in presence of measurement errors. In addition, the average achieved hemoglobin level from the MRHC is significantly closer to the target hemoglobin than that of the other two methods, according to the analysis of variance (ANOVA) statistical test. Furthermore, drug dosages recommended by the MRHC are more stable and accurate and reach the steady-state value notably faster than those generated by the other two methods. The proposed method is highly efficient for

A Galerkin formulation of the MIB method for three dimensional elliptic interface problems

PubMed Central

Xia, Kelin; Wei, Guo-Wei

2014-01-01

We develop a three dimensional (3D) Galerkin formulation of the matched interface and boundary (MIB) method for solving elliptic partial differential equations (PDEs) with discontinuous coefficients, i.e., the elliptic interface problem. The present approach builds up two sets of elements respectively on two extended subdomains which both include the interface. As a result, two sets of elements overlap each other near the interface. Fictitious solutions are defined on the overlapping part of the elements, so that the differentiation operations of the original PDEs can be discretized as if there was no interface. The extra coefficients of polynomial basis functions, which furnish the overlapping elements and solve the fictitious solutions, are determined by interface jump conditions. Consequently, the interface jump conditions are rigorously enforced on the interface. The present method utilizes Cartesian meshes to avoid the mesh generation in conventional finite element methods (FEMs). We implement the proposed MIB Galerkin method with three different elements, namely, rectangular prism element, five-tetrahedron element and six-tetrahedron element, which tile the Cartesian mesh without introducing any new node. The accuracy, stability and robustness of the proposed 3D MIB Galerkin are extensively validated over three types of elliptic interface problems. In the first type, interfaces are analytically defined by level set functions. These interfaces are relatively simple but admit geometric singularities. In the second type, interfaces are defined by protein surfaces, which are truly arbitrarily complex. The last type of interfaces originates from multiprotein complexes, such as molecular motors. Near second order accuracy has been confirmed for all of these problems. To our knowledge, it is the first time for an FEM to show a near second order convergence in solving the Poisson equation with realistic protein surfaces. Additionally, the present work offers the

Long-time stability effects of quadrature and artificial viscosity on nodal discontinuous Galerkin methods for gas dynamics

NASA Astrophysics Data System (ADS)

Durant, Bradford; Hackl, Jason; Balachandar, Sivaramakrishnan

2017-11-01

Nodal discontinuous Galerkin schemes present an attractive approach to robust high-order solution of the equations of fluid mechanics, but remain accompanied by subtle challenges in their consistent stabilization. The effect of quadrature choices (full mass matrix vs spectral elements), over-integration to manage aliasing errors, and explicit artificial viscosity on the numerical solution of a steady homentropic vortex are assessed over a wide range of resolutions and polynomial orders using quadrilateral elements. In both stagnant and advected vortices in periodic and non-periodic domains the need arises for explicit stabilization beyond the numerical surface fluxes of discontinuous Galerkin spectral elements. Artificial viscosity via the entropy viscosity method is assessed as a stabilizing mechanism. It is shown that the regularity of the artificial viscosity field is essential to its use for long-time stabilization of small-scale features in nodal discontinuous Galerkin solutions of the Euler equations of gas dynamics. Supported by the Department of Energy Predictive Science Academic Alliance Program Contract DE-NA0002378.

Discontinuous Galerkin Methods for NonLinear Differential Systems

NASA Technical Reports Server (NTRS)

Barth, Timothy; Mansour, Nagi (Technical Monitor)

2001-01-01

This talk considers simplified finite element discretization techniques for first-order systems of conservation laws equipped with a convex (entropy) extension. Using newly developed techniques in entropy symmetrization theory, simplified forms of the discontinuous Galerkin (DG) finite element method have been developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE (partial differential equation) system. Central to the development of the simplified DG methods is the Eigenvalue Scaling Theorem which characterizes right symmetrizers of an arbitrary first-order hyperbolic system in terms of scaled eigenvectors of the corresponding flux Jacobian matrices. A constructive proof is provided for the Eigenvalue Scaling Theorem with detailed consideration given to the Euler equations of gas dynamics and extended conservation law systems derivable as moments of the Boltzmann equation. Using results from kinetic Boltzmann moment closure theory, we then derive and prove energy stability for several approximate DG fluxes which have practical and theoretical merit.

Weak Galerkin finite element methods for Darcy flow: Anisotropy and heterogeneity

NASA Astrophysics Data System (ADS)

Lin, Guang; Liu, Jiangguo; Mu, Lin; Ye, Xiu

2014-11-01

This paper presents a family of weak Galerkin finite element methods (WGFEMs) for Darcy flow computation. The WGFEMs are new numerical methods that rely on the novel concept of discrete weak gradients. The WGFEMs solve for pressure unknowns both in element interiors and on the mesh skeleton. The numerical velocity is then obtained from the discrete weak gradient of the numerical pressure. The new methods are quite different than many existing numerical methods in that they are locally conservative by design, the resulting discrete linear systems are symmetric and positive-definite, and there is no need for tuning problem-dependent penalty factors. We test the WGFEMs on benchmark problems to demonstrate the strong potential of these new methods in handling strong anisotropy and heterogeneity in Darcy flow.

A discontinuous Galerkin method for two-dimensional PDE models of Asian options

NASA Astrophysics Data System (ADS)

Hozman, J.; Tichý, T.; Cvejnová, D.

2016-06-01

In our previous research we have focused on the problem of plain vanilla option valuation using discontinuous Galerkin method for numerical PDE solution. Here we extend a simple one-dimensional problem into two-dimensional one and design a scheme for valuation of Asian options, i.e. options with payoff depending on the average of prices collected over prespecified horizon. The algorithm is based on the approach combining the advantages of the finite element methods together with the piecewise polynomial generally discontinuous approximations. Finally, an illustrative example using DAX option market data is provided.

An h-p Taylor-Galerkin finite element method for compressible Euler equations

NASA Technical Reports Server (NTRS)

Demkowicz, L.; Oden, J. T.; Rachowicz, W.; Hardy, O.

1991-01-01

An extension of the familiar Taylor-Galerkin method to arbitrary h-p spatial approximations is proposed. Boundary conditions are analyzed, and a linear stability result for arbitrary meshes is given, showing the unconditional stability for the parameter of implicitness alpha not less than 0.5. The wedge and blunt body problems are solved with both linear, quadratic, and cubic elements and h-adaptivity, showing the feasibility of higher orders of approximation for problems with shocks.

Galerkin v. least-squares Petrov–Galerkin projection in nonlinear model reduction

DOE PAGES

Carlberg, Kevin Thomas; Barone, Matthew F.; Antil, Harbir

2016-10-20

Least-squares Petrov–Galerkin (LSPG) model-reduction techniques such as the Gauss–Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible flow problems where standard Galerkin techniques have failed. Furthermore, there has been limited comparative analysis of the two approaches. This is due in part to difficulties arising from the fact that Galerkin techniques perform optimal projection associated with residual minimization at the time-continuous level, while LSPG techniques do so at the time-discrete level. This work provides a detailed theoretical and computational comparison of the two techniques for two common classes of timemore » integrators: linear multistep schemes and Runge–Kutta schemes. We present a number of new findings, including conditions under which the LSPG ROM has a time-continuous representation, conditions under which the two techniques are equivalent, and time-discrete error bounds for the two approaches. Perhaps most surprisingly, we demonstrate both theoretically and computationally that decreasing the time step does not necessarily decrease the error for the LSPG ROM; instead, the time step should be ‘matched’ to the spectral content of the reduced basis. In numerical experiments carried out on a turbulent compressible-flow problem with over one million unknowns, we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the LSPG reduced-order model by an order of magnitude.« less

Galerkin v. least-squares Petrov–Galerkin projection in nonlinear model reduction

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

Carlberg, Kevin Thomas; Barone, Matthew F.; Antil, Harbir

Least-squares Petrov–Galerkin (LSPG) model-reduction techniques such as the Gauss–Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible flow problems where standard Galerkin techniques have failed. Furthermore, there has been limited comparative analysis of the two approaches. This is due in part to difficulties arising from the fact that Galerkin techniques perform optimal projection associated with residual minimization at the time-continuous level, while LSPG techniques do so at the time-discrete level. This work provides a detailed theoretical and computational comparison of the two techniques for two common classes of timemore » integrators: linear multistep schemes and Runge–Kutta schemes. We present a number of new findings, including conditions under which the LSPG ROM has a time-continuous representation, conditions under which the two techniques are equivalent, and time-discrete error bounds for the two approaches. Perhaps most surprisingly, we demonstrate both theoretically and computationally that decreasing the time step does not necessarily decrease the error for the LSPG ROM; instead, the time step should be ‘matched’ to the spectral content of the reduced basis. In numerical experiments carried out on a turbulent compressible-flow problem with over one million unknowns, we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the LSPG reduced-order model by an order of magnitude.« less

A third-order implicit discontinuous Galerkin method based on a Hermite WENO reconstruction for time-accurate solution of the compressible Navier-Stokes equations

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

Xia, Yidong; Liu, Xiaodong; Luo, Hong

2015-06-01

Here, a space and time third-order discontinuous Galerkin method based on a Hermite weighted essentially non-oscillatory reconstruction is presented for the unsteady compressible Euler and Navier–Stokes equations. At each time step, a lower-upper symmetric Gauss–Seidel preconditioned generalized minimal residual solver is used to solve the systems of linear equations arising from an explicit first stage, single diagonal coefficient, diagonally implicit Runge–Kutta time integration scheme. The performance of the developed method is assessed through a variety of unsteady flow problems. Numerical results indicate that this method is able to deliver the designed third-order accuracy of convergence in both space and time,more » while requiring remarkably less storage than the standard third-order discontinous Galerkin methods, and less computing time than the lower-order discontinous Galerkin methods to achieve the same level of temporal accuracy for computing unsteady flow problems.« less

Numerical Evaluation of P-Multigrid Method for the Solution of Discontinuous Galerkin Discretizations of Diffusive Equations

NASA Technical Reports Server (NTRS)

Atkins, H. L.; Helenbrook, B. T.

2005-01-01

This paper describes numerical experiments with P-multigrid to corroborate analysis, validate the present implementation, and to examine issues that arise in the implementations of the various combinations of relaxation schemes, discretizations and P-multigrid methods. The two approaches to implement P-multigrid presented here are equivalent for most high-order discretization methods such as spectral element, SUPG, and discontinuous Galerkin applied to advection; however it is discovered that the approach that mimics the common geometric multigrid implementation is less robust, and frequently unstable when applied to discontinuous Galerkin discretizations of di usion. Gauss-Seidel relaxation converges 40% faster than block Jacobi, as predicted by analysis; however, the implementation of Gauss-Seidel is considerably more expensive that one would expect because gradients in most neighboring elements must be updated. A compromise quasi Gauss-Seidel relaxation method that evaluates the gradient in each element twice per iteration converges at rates similar to those predicted for true Gauss-Seidel.

A fully Sinc-Galerkin method for Euler-Bernoulli beam models

NASA Technical Reports Server (NTRS)

Smith, R. C.; Bowers, K. L.; Lund, J.

1990-01-01

A fully Sinc-Galerkin method in both space and time is presented for fourth-order time-dependent partial differential equations with fixed and cantilever boundary conditions. The Sinc discretizations for the second-order temporal problem and the fourth-order spatial problems are presented. Alternate formulations for variable parameter fourth-order problems are given which prove to be especially useful when applying the forward techniques to parameter recovery problems. The discrete system which corresponds to the time-dependent partial differential equations of interest are then formulated. Computational issues are discussed and a robust and efficient algorithm for solving the resulting matrix system is outlined. Numerical results which highlight the method are given for problems with both analytic and singular solutions as well as fixed and cantilever boundary conditions.

Relaxation and Preconditioning for High Order Discontinuous Galerkin Methods with Applications to Aeroacoustics and High Speed Flows

NASA Technical Reports Server (NTRS)

Shu, Chi-Wang

2004-01-01

This project is about the investigation of the development of the discontinuous Galerkin finite element methods, for general geometry and triangulations, for solving convection dominated problems, with applications to aeroacoustics. Other related issues in high order WENO finite difference and finite volume methods have also been investigated. methods are two classes of high order, high resolution methods suitable for convection dominated simulations with possible discontinuous or sharp gradient solutions. In [18], we first review these two classes of methods, pointing out their similarities and differences in algorithm formulation, theoretical properties, implementation issues, applicability, and relative advantages. We then present some quantitative comparisons of the third order finite volume WENO methods and discontinuous Galerkin methods for a series of test problems to assess their relative merits in accuracy and CPU timing. In [3], we review the development of the Runge-Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge-Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier

Weak Galerkin finite element methods for Darcy flow: Anisotropy and heterogeneity

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

Lin, Guang; Liu, Jiangguo; Mu, Lin

2014-11-01

This paper presents a family of weak Galerkin finite element methods (WGFEMs) for Darcy flow computation. The WGFEMs are new numerical methods that rely on the novel concept of discrete weak gradients. The WGFEMs solve for pressure unknowns both in element interiors and on the mesh skeleton. The numerical velocity is then obtained from the discrete weak gradient of the numerical pressure. The new methods are quite different than many existing numerical methods in that they are locally conservative by design, the resulting discrete linear systems are symmetric and positive-definite, and there is no need for tuning problem-dependent penalty factors.more » We test the WGFEMs on benchmark problems to demonstrate the strong potential of these new methods in handling strong anisotropy and heterogeneity in Darcy flow.« less

Discontinuous Galerkin method with Gaussian artificial viscosity on graphical processing units for nonlinear acoustics

NASA Astrophysics Data System (ADS)

Tripathi, Bharat B.; Marchiano, Régis; Baskar, Sambandam; Coulouvrat, François

2015-10-01

Propagation of acoustical shock waves in complex geometry is a topic of interest in the field of nonlinear acoustics. For instance, simulation of Buzz Saw Noice requires the treatment of shock waves generated by the turbofan through the engines of aeroplanes with complex geometries and wall liners. Nevertheless, from a numerical point of view it remains a challenge. The two main hurdles are to take into account the complex geometry of the domain and to deal with the spurious oscillations (Gibbs phenomenon) near the discontinuities. In this work, first we derive the conservative hyperbolic system of nonlinear acoustics (up to quadratic nonlinear terms) using the fundamental equations of fluid dynamics. Then, we propose to adapt the classical nodal discontinuous Galerkin method to develop a high fidelity solver for nonlinear acoustics. The discontinuous Galerkin method is a hybrid of finite element and finite volume method and is very versatile to handle complex geometry. In order to obtain better performance, the method is parallelized on Graphical Processing Units. Like other numerical methods, discontinuous Galerkin method suffers with the problem of Gibbs phenomenon near the shock, which is a numerical artifact. Among the various ways to manage these spurious oscillations, we choose the method of parabolic regularization. Although, the introduction of artificial viscosity into the system is a popular way of managing shocks, we propose a new approach of introducing smooth artificial viscosity locally in each element, wherever needed. Firstly, a shock sensor using the linear coefficients of the spectral solution is used to locate the position of the discontinuities. Then, a viscosity coefficient depending on the shock sensor is introduced into the hyperbolic system of equations, only in the elements near the shock. The viscosity is applied as a two-dimensional Gaussian patch with its shape parameters depending on the element dimensions, referred here as Element

Galerkin Spectral Method for the 2D Solitary Waves of Boussinesq Paradigm Equation

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

Christou, M. A.; Christov, C. I.

2009-10-29

We consider the 2D stationary propagating solitary waves of the so-called Boussinesq Paradigm equation. The fourth- order elliptic boundary value problem on infinite interval is solved by a Galerkin spectral method. An iterative procedure based on artificial time ('false transients') and operator splitting is used. Results are obtained for the shapes of the solitary waves for different values of the dispersion parameters for both subcritical and supercritical phase speeds.

A Leap-Frog Discontinuous Galerkin Method for the Time-Domain Maxwell's Equations in Metamaterials

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

Li, J., Waters, J. W., Machorro, E. A.

2012-06-01

Numerical simulation of metamaterials play a very important role in the design of invisibility cloak, and sub-wavelength imaging. In this paper, we propose a leap-frog discontinuous Galerkin method to solve the time-dependent Maxwell’s equations in metamaterials. Conditional stability and error estimates are proved for the scheme. The proposed algorithm is implemented and numerical results supporting the analysis are provided.

Error Analysis of p-Version Discontinuous Galerkin Method for Heat Transfer in Built-up Structures

NASA Technical Reports Server (NTRS)

Kaneko, Hideaki; Bey, Kim S.

2004-01-01

The purpose of this paper is to provide an error analysis for the p-version of the discontinuous Galerkin finite element method for heat transfer in built-up structures. As a special case of the results in this paper, a theoretical error estimate for the numerical experiments recently conducted by James Tomey is obtained.

Lagrangian Particle Tracking in a Discontinuous Galerkin Method for Hypersonic Reentry Flows in Dusty Environments

NASA Astrophysics Data System (ADS)

Ching, Eric; Lv, Yu; Ihme, Matthias

2017-11-01

Recent interest in human-scale missions to Mars has sparked active research into high-fidelity simulations of reentry flows. A key feature of the Mars atmosphere is the high levels of suspended dust particles, which can not only enhance erosion of thermal protection systems but also transfer energy and momentum to the shock layer, increasing surface heat fluxes. Second-order finite-volume schemes are typically employed for hypersonic flow simulations, but such schemes suffer from a number of limitations. An attractive alternative is discontinuous Galerkin methods, which benefit from arbitrarily high spatial order of accuracy, geometric flexibility, and other advantages. As such, a Lagrangian particle method is developed in a discontinuous Galerkin framework to enable the computation of particle-laden hypersonic flows. Two-way coupling between the carrier and disperse phases is considered, and an efficient particle search algorithm compatible with unstructured curved meshes is proposed. In addition, variable thermodynamic properties are considered to accommodate high-temperature gases. The performance of the particle method is demonstrated in several test cases, with focus on the accurate prediction of particle trajectories and heating augmentation. Financial support from a Stanford Graduate Fellowship and the NASA Early Career Faculty program are gratefully acknowledged.

Mechanics of cantilever beam: Implementation and comparison of FEM and MLPG approach

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

Trobec, Roman

2016-06-08

Two weak form solution approaches for partial differential equations, the well known meshbased finite element method and the newer meshless local Petrov Galerkin method are described and compared on a standard test case - mechanics of cantilever beam. The implementation, solution accuracy and calculation complexity are addressed for both approaches. We found out that FEM is superior in most standard criteria, but MLPG has some advantages because of its flexibility that results from its general formulation.

A hybrid perturbation Galerkin technique with applications to slender body theory

NASA Technical Reports Server (NTRS)

Geer, James F.; Andersen, Carl M.

1989-01-01

A two-step hybrid perturbation-Galerkin method to solve a variety of applied mathematics problems which involve a small parameter is presented. The method consists of: (1) the use of a regular or singular perturbation method to determine the asymptotic expansion of the solution in terms of the small parameter; (2) construction of an approximate solution in the form of a sum of the perturbation coefficient functions multiplied by (unknown) amplitudes (gauge functions); and (3) the use of the classical Bubnov-Galerkin method to determine these amplitudes. This hybrid method has the potential of overcoming some of the drawbacks of the perturbation method and the Bubnov-Galerkin method when they are applied by themselves, while combining some of the good features of both. The proposed method is applied to some singular perturbation problems in slender body theory. The results obtained from the hybrid method are compared with approximate solutions obtained by other methods, and the degree of applicability of the hybrid method to broader problem areas is discussed.

A hybrid perturbation Galerkin technique with applications to slender body theory

NASA Technical Reports Server (NTRS)

Geer, James F.; Andersen, Carl M.

1987-01-01

A two step hybrid perturbation-Galerkin method to solve a variety of applied mathematics problems which involve a small parameter is presented. The method consists of: (1) the use of a regular or singular perturbation method to determine the asymptotic expansion of the solution in terms of the small parameter; (2) construction of an approximate solution in the form of a sum of the perturbation coefficient functions multiplied by (unknown) amplitudes (gauge functions); and (3) the use of the classical Bubnov-Galerkin method to determine these amplitudes. This hybrid method has the potential of overcoming some of the drawbacks of the perturbation method and the Bubnov-Galerkin method when they are applied by themselves, while combining some of the good features of both. The proposed method is applied to some singular perturbation problems in slender body theory. The results obtained from the hybrid method are compared with approximate solutions obtained by other methods, and the degree of applicability of the hybrid method to broader problem areas is discussed.

Analysis of Preconditioning and Relaxation Operators for the Discontinuous Galerkin Method Applied to Diffusion

NASA Technical Reports Server (NTRS)

Atkins, H. L.; Shu, Chi-Wang

2001-01-01

The explicit stability constraint of the discontinuous Galerkin method applied to the diffusion operator decreases dramatically as the order of the method is increased. Block Jacobi and block Gauss-Seidel preconditioner operators are examined for their effectiveness at accelerating convergence. A Fourier analysis for methods of order 2 through 6 reveals that both preconditioner operators bound the eigenvalues of the discrete spatial operator. Additionally, in one dimension, the eigenvalues are grouped into two or three regions that are invariant with order of the method. Local relaxation methods are constructed that rapidly damp high frequencies for arbitrarily large time step.

Solution of the 2-D steady-state radiative transfer equation in participating media with specular reflections using SUPG and DG finite elements

NASA Astrophysics Data System (ADS)

Le Hardy, D.; Favennec, Y.; Rousseau, B.

2016-08-01

The 2D radiative transfer equation coupled with specular reflection boundary conditions is solved using finite element schemes. Both Discontinuous Galerkin and Streamline-Upwind Petrov-Galerkin variational formulations are fully developed. These two schemes are validated step-by-step for all involved operators (transport, scattering, reflection) using analytical formulations. Numerical comparisons of the two schemes, in terms of convergence rate, reveal that the quadratic SUPG scheme proves efficient for solving such problems. This comparison constitutes the main issue of the paper. Moreover, the solution process is accelerated using block SOR-type iterative methods, for which the determination of the optimal parameter is found in a very cheap way.

An updated Lagrangian discontinuous Galerkin hydrodynamic method for gas dynamics

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

Wu, Tong; Shashkov, Mikhail Jurievich; Morgan, Nathaniel Ray

Here, we present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for gas dynamics. The new method evolves conserved unknowns in the current configuration, which obviates the Jacobi matrix that maps the element in a reference coordinate system or the initial coordinate system to the current configuration. The density, momentum, and total energy (ρ, ρu, E) are approximated with conservative higher-order Taylor expansions over the element and are limited toward a piecewise constant field near discontinuities using a limiter. Two new limiting methods are presented for enforcing the bounds on the primitive variables of density, velocity, and specific internal energymore » (ρ, u, e). The nodal velocity, and the corresponding forces, are calculated by solving an approximate Riemann problem at the element nodes. An explicit second-order method is used to temporally advance the solution. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. 1D Cartesian coordinates test problem results are presented to demonstrate the accuracy and convergence order of the new DG method with the new limiters.« less

An updated Lagrangian discontinuous Galerkin hydrodynamic method for gas dynamics

DOE PAGES

Wu, Tong; Shashkov, Mikhail Jurievich; Morgan, Nathaniel Ray; ...

2018-04-09

Here, we present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for gas dynamics. The new method evolves conserved unknowns in the current configuration, which obviates the Jacobi matrix that maps the element in a reference coordinate system or the initial coordinate system to the current configuration. The density, momentum, and total energy (ρ, ρu, E) are approximated with conservative higher-order Taylor expansions over the element and are limited toward a piecewise constant field near discontinuities using a limiter. Two new limiting methods are presented for enforcing the bounds on the primitive variables of density, velocity, and specific internal energymore » (ρ, u, e). The nodal velocity, and the corresponding forces, are calculated by solving an approximate Riemann problem at the element nodes. An explicit second-order method is used to temporally advance the solution. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. 1D Cartesian coordinates test problem results are presented to demonstrate the accuracy and convergence order of the new DG method with the new limiters.« less

Effective implementation of the weak Galerkin finite element methods for the biharmonic equation

DOE PAGES

Mu, Lin; Wang, Junping; Ye, Xiu

2017-07-06

The weak Galerkin (WG) methods have been introduced in [11, 12, 17] for solving the biharmonic equation. The purpose of this paper is to develop an algorithm to implement the WG methods effectively. This can be achieved by eliminating local unknowns to obtain a global system with significant reduction of size. In fact this reduced global system is equivalent to the Schur complements of the WG methods. The unknowns of the Schur complement of the WG method are those defined on the element boundaries. The equivalence of theWG method and its Schur complement is established. The numerical results demonstrate themore » effectiveness of this new implementation technique.« less

Effective implementation of the weak Galerkin finite element methods for the biharmonic equation

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

Mu, Lin; Wang, Junping; Ye, Xiu

The weak Galerkin (WG) methods have been introduced in [11, 12, 17] for solving the biharmonic equation. The purpose of this paper is to develop an algorithm to implement the WG methods effectively. This can be achieved by eliminating local unknowns to obtain a global system with significant reduction of size. In fact this reduced global system is equivalent to the Schur complements of the WG methods. The unknowns of the Schur complement of the WG method are those defined on the element boundaries. The equivalence of theWG method and its Schur complement is established. The numerical results demonstrate themore » effectiveness of this new implementation technique.« less

A hybrid perturbation-Galerkin technique for partial differential equations

NASA Technical Reports Server (NTRS)

Geer, James F.; Anderson, Carl M.

1990-01-01

A two-step hybrid perturbation-Galerkin technique for improving the usefulness of perturbation solutions to partial differential equations which contain a parameter is presented and discussed. In the first step of the method, the leading terms in the asymptotic expansion(s) of the solution about one or more values of the perturbation parameter are obtained using standard perturbation methods. In the second step, the perturbation functions obtained in the first step are used as trial functions in a Bubnov-Galerkin approximation. This semi-analytical, semi-numerical hybrid technique appears to overcome some of the drawbacks of the perturbation and Galerkin methods when they are applied by themselves, while combining some of the good features of each. The technique is illustrated first by a simple example. It is then applied to the problem of determining the flow of a slightly compressible fluid past a circular cylinder and to the problem of determining the shape of a free surface due to a sink above the surface. Solutions obtained by the hybrid method are compared with other approximate solutions, and its possible application to certain problems associated with domain decomposition is discussed.

Galerkin Models Enhancements for Flow Control

NASA Astrophysics Data System (ADS)

Tadmor, Gilead; Lehmann, Oliver; Noack, Bernd R.; Morzyński, Marek

Low order Galerkin models were originally introduced as an effective tool for stability analysis of fixed points and, later, of attractors, in nonlinear distributed systems. An evolving interest in their use as low complexity dynamical models, goes well beyond that original intent. It exposes often severe weaknesses of low order Galerkin models as dynamic predictors and has motivated efforts, spanning nearly three decades, to alleviate these shortcomings. Transients across natural and enforced variations in the operating point, unsteady inflow, boundary actuation and both aeroelastic and actuated boundary motion, are hallmarks of current and envisioned needs in feedback flow control applications, bringing these shortcomings to even higher prominence. Building on the discussion in our previous chapters, we shall now review changes in the Galerkin paradigm that aim to create a mathematically and physically consistent modeling framework, that remove what are otherwise intractable roadblocks. We shall then highlight some guiding design principles that are especially important in the context of these models. We shall continue to use the simple example of wake flow instabilities to illustrate the various issues, ideas and methods that will be discussed in this chapter.

Analysis of the discontinuous Galerkin method applied to the European option pricing problem

NASA Astrophysics Data System (ADS)

Hozman, J.

2013-12-01

In this paper we deal with a numerical solution of a one-dimensional Black-Scholes partial differential equation, an important scalar nonstationary linear convection-diffusion-reaction equation describing the pricing of European vanilla options. We present a derivation of the numerical scheme based on the space semidiscretization of the model problem by the discontinuous Galerkin method with nonsymmetric stabilization of diffusion terms and with the interior and boundary penalty. The main attention is paid to the investigation of a priori error estimates for the proposed scheme. The appended numerical experiments illustrate the theoretical results and the potency of the method, consequently.

An asymptotic-preserving stochastic Galerkin method for the radiative heat transfer equations with random inputs and diffusive scalings

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

Jin, Shi, E-mail: sjin@wisc.edu; Institute of Natural Sciences, Department of Mathematics, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240; Lu, Hanqing, E-mail: hanqing@math.wisc.edu

2017-04-01

In this paper, we develop an Asymptotic-Preserving (AP) stochastic Galerkin scheme for the radiative heat transfer equations with random inputs and diffusive scalings. In this problem the random inputs arise due to uncertainties in cross section, initial data or boundary data. We use the generalized polynomial chaos based stochastic Galerkin (gPC-SG) method, which is combined with the micro–macro decomposition based deterministic AP framework in order to handle efficiently the diffusive regime. For linearized problem we prove the regularity of the solution in the random space and consequently the spectral accuracy of the gPC-SG method. We also prove the uniform (inmore » the mean free path) linear stability for the space-time discretizations. Several numerical tests are presented to show the efficiency and accuracy of proposed scheme, especially in the diffusive regime.« less

Two-dimensional mesh embedding for Galerkin B-spline methods

NASA Technical Reports Server (NTRS)

Shariff, Karim; Moser, Robert D.

1995-01-01

A number of advantages result from using B-splines as basis functions in a Galerkin method for solving partial differential equations. Among them are arbitrary order of accuracy and high resolution similar to that of compact schemes but without the aliasing error. This work develops another property, namely, the ability to treat semi-structured embedded or zonal meshes for two-dimensional geometries. This can drastically reduce the number of grid points in many applications. Both integer and non-integer refinement ratios are allowed. The report begins by developing an algorithm for choosing basis functions that yield the desired mesh resolution. These functions are suitable products of one-dimensional B-splines. Finally, test cases for linear scalar equations such as the Poisson and advection equation are presented. The scheme is conservative and has uniformly high order of accuracy throughout the domain.

Analysis of the incomplete Galerkin method for modelling of smoothly-irregular transition between planar waveguides

NASA Astrophysics Data System (ADS)

Divakov, D.; Sevastianov, L.; Nikolaev, N.

2017-01-01

The paper deals with a numerical solution of the problem of waveguide propagation of polarized light in smoothly-irregular transition between closed regular waveguides using the incomplete Galerkin method. This method consists in replacement of variables in the problem of reduction of the Helmholtz equation to the system of differential equations by the Kantorovich method and in formulation of the boundary conditions for the resulting system. The formulation of the boundary problem for the ODE system is realized in computer algebra system Maple. The stated boundary problem is solved using Maples libraries of numerical methods.

Numerical simulation of electrophoresis separation processes

NASA Technical Reports Server (NTRS)

Ganjoo, D. K.; Tezduyar, T. E.

1986-01-01

A new Petrov-Galerkin finite element formulation has been proposed for transient convection-diffusion problems. Most Petrov-Galerkin formulations take into account the spatial discretization, and the weighting functions so developed give satisfactory solutions for steady state problems. Though these schemes can be used for transient problems, there is scope for improvement. The schemes proposed here, which consider temporal as well as spatial discretization, provide improved solutions. Electrophoresis, which involves the motion of charged entities under the influence of an applied electric field, is governed by equations similiar to those encountered in fluid flow problems, i.e., transient convection-diffusion equations. Test problems are solved in electrophoresis and fluid flow. The results obtained are satisfactory. It is also expected that these schemes, suitably adapted, will improve the numerical solutions of the compressible Euler and the Navier-Stokes equations.

Discontinuous Galerkin method for coupled problems of compressible flow and elastic structures

NASA Astrophysics Data System (ADS)

Kosík, A.; Feistauer, M.; Hadrava, M.; Horáček, J.

2013-10-01

This paper is concerned with the numerical simulation of the interaction of 2D compressible viscous flow and an elastic structure. We consider the model of dynamical linear elasticity. Each individual problem is discretized in space by the discontinuous Galerkin method (DGM). For the time discretization we can use either the BDF (backward difference formula) method or also the DGM. The time dependence of the domain occupied by the fluid is given by the deformation of the elastic structure adjacent to the flow domain. It is treated with the aid of the Arbitrary Lagrangian-Eulerian (ALE) method. The fluid-structure interaction, given by transient conditions, is realized by an iterative process. The developed method is applied to the simulation of the biomechanical problem containing the onset of the voice production.

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.

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.

A free energy satisfying discontinuous Galerkin method for one-dimensional Poisson-Nernst-Planck systems

NASA Astrophysics Data System (ADS)

Liu, Hailiang; Wang, Zhongming

2017-01-01

We design an arbitrary-order free energy satisfying discontinuous Galerkin (DG) method for solving time-dependent Poisson-Nernst-Planck systems. Both the semi-discrete and fully discrete DG methods are shown to satisfy the corresponding discrete free energy dissipation law for positive numerical solutions. Positivity of numerical solutions is enforced by an accuracy-preserving limiter in reference to positive cell averages. Numerical examples are presented to demonstrate the high resolution of the numerical algorithm and to illustrate the proven properties of mass conservation, free energy dissipation, as well as the preservation of steady states.

A hybrid-perturbation-Galerkin technique which combines multiple expansions

NASA Technical Reports Server (NTRS)

Geer, James F.; Andersen, Carl M.

1989-01-01

A two-step hybrid perturbation-Galerkin method for the solution of a variety of differential equations type problems is found to give better results when multiple perturbation expansions are employed. The method assumes that there is parameter in the problem formulation and that a perturbation method can be sued to construct one or more expansions in this perturbation coefficient functions multiplied by computed amplitudes. In step one, regular and/or singular perturbation methods are used to determine the perturbation coefficient functions. The results of step one are in the form of one or more expansions each expressed as a sum of perturbation coefficient functions multiplied by a priori known gauge functions. In step two the classical Bubnov-Galerkin method uses the perturbation coefficient functions computed in step one to determine a set of amplitudes which replace and improve upon the gauge functions. The hybrid method has the potential of overcoming some of the drawbacks of the perturbation and Galerkin methods as applied separately, while combining some of their better features. The proposed method is applied, with two perturbation expansions in each case, to a variety of model ordinary differential equations problems including: a family of linear two-boundary-value problems, a nonlinear two-point boundary-value problem, a quantum mechanical eigenvalue problem and a nonlinear free oscillation problem. The results obtained from the hybrid methods are compared with approximate solutions obtained by other methods, and the applicability of the hybrid method to broader problem areas is discussed.

An iterative phase-space explicit discontinuous Galerkin method for stellar radiative transfer in extended atmospheres

NASA Astrophysics Data System (ADS)

de Almeida, Valmor F.

2017-07-01

A phase-space discontinuous Galerkin (PSDG) method is presented for the solution of stellar radiative transfer problems. It allows for greater adaptivity than competing methods without sacrificing generality. The method is extensively tested on a spherically symmetric, static, inverse-power-law scattering atmosphere. Results for different sizes of atmospheres and intensities of scattering agreed with asymptotic values. The exponentially decaying behavior of the radiative field in the diffusive-transparent transition region, and the forward peaking behavior at the surface of extended atmospheres were accurately captured. The integrodifferential equation of radiation transfer is solved iteratively by alternating between the radiative pressure equation and the original equation with the integral term treated as an energy density source term. In each iteration, the equations are solved via an explicit, flux-conserving, discontinuous Galerkin method. Finite elements are ordered in wave fronts perpendicular to the characteristic curves so that elemental linear algebraic systems are solved quickly by sweeping the phase space element by element. Two implementations of a diffusive boundary condition at the origin are demonstrated wherein the finite discontinuity in the radiation intensity is accurately captured by the proposed method. This allows for a consistent mechanism to preserve photon luminosity. The method was proved to be robust and fast, and a case is made for the adequacy of parallel processing. In addition to classical two-dimensional plots, results of normalized radiation intensity were mapped onto a log-polar surface exhibiting all distinguishing features of the problem studied.

A Dynamic Eddy Viscosity Model for the Shallow Water Equations Solved by Spectral Element and Discontinuous Galerkin Methods

NASA Astrophysics Data System (ADS)

Marras, Simone; Suckale, Jenny; Giraldo, Francis X.; Constantinescu, Emil

2016-04-01

We present the solution of the viscous shallow water equations where viscosity is built as a residual-based subgrid scale model originally designed for large eddy simulation of compressible [1] and stratified flows [2]. The necessity of viscosity for a shallow water model not only finds motivation from mathematical analysis [3], but is supported by physical reasoning as can be seen by an analysis of the energetics of the solution. We simulated the flow of an idealized wave as it hits a set of obstacles. The kinetic energy spectrum of this flow shows that, although the inviscid Galerkin solutions -by spectral elements and discontinuous Galerkin [4]- preserve numerical stability in spite of the spurious oscillations in the proximity of the wave fronts, the slope of the energy cascade deviates from the theoretically expected values. We show that only a sufficiently small amount of dynamically adaptive viscosity removes the unwanted high-frequency modes while preserving the overall sharpness of the solution. In addition, it yields a physically plausible energy decay. This work is motivated by a larger interest in the application of a shallow water model to the solution of tsunami triggered coastal flows. In particular, coastal flows in regions around the world where coastal parks made of mitigation hills of different sizes and configurations are considered as a means to deviate the power of the incoming wave. References [1] M. Nazarov and J. Hoffman (2013) "Residual-based artificial viscosity for simulation of turbulent compressible flow using adaptive finite element methods" Int. J. Numer. Methods Fluids, 71:339-357 [2] S. Marras, M. Nazarov, F. X. Giraldo (2015) "Stabilized high-order Galerkin methods based on a parameter-free dynamic SGS model for LES" J. Comput. Phys. 301:77-101 [3] J. F. Gerbeau and B. Perthame (2001) "Derivation of the viscous Saint-Venant system for laminar shallow water; numerical validation" Discrete Contin. Dyn. Syst. Ser. B, 1:89?102 [4] F

Quadratic Finite Element Method for 1D Deterministic Transport

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

Tolar, Jr., D R; Ferguson, J M

2004-01-06

In the discrete ordinates, or SN, numerical solution of the transport equation, both the spatial ({und r}) and angular ({und {Omega}}) dependences on the angular flux {psi}{und r},{und {Omega}}are modeled discretely. While significant effort has been devoted toward improving the spatial discretization of the angular flux, we focus on improving the angular discretization of {psi}{und r},{und {Omega}}. Specifically, we employ a Petrov-Galerkin quadratic finite element approximation for the differencing of the angular variable ({mu}) in developing the one-dimensional (1D) spherical geometry S{sub N} equations. We develop an algorithm that shows faster convergence with angular resolution than conventional S{sub N} algorithms.

Meshless Local Petrov-Galerkin Method for Solving Contact, Impact and Penetration Problems

DTIC Science & Technology

2006-11-30

Crack Growth 3 point of view, this approach makes the full use of the ex- isting FE models to avoid any model regeneration , which is extremely high in...process, at point C, the pressure reduces to zero, but the volumet- ric strain does not go to zero due to the collapsed void volume. 2.2 Damage...lease rate to go beyond the critical strain energy release rate. Thus, the micro-cracks begin to growth inside these areas. At 10 micro-seconds, these

Robust and Accurate Shock Capturing Method for High-Order Discontinuous Galerkin Methods

NASA Technical Reports Server (NTRS)

Atkins, Harold L.; Pampell, Alyssa

2011-01-01

A simple yet robust and accurate approach for capturing shock waves using a high-order discontinuous Galerkin (DG) method is presented. The method uses the physical viscous terms of the Navier-Stokes equations as suggested by others; however, the proposed formulation of the numerical viscosity is continuous and compact by construction, and does not require the solution of an auxiliary diffusion equation. This work also presents two analyses that guided the formulation of the numerical viscosity and certain aspects of the DG implementation. A local eigenvalue analysis of the DG discretization applied to a shock containing element is used to evaluate the robustness of several Riemann flux functions, and to evaluate algorithm choices that exist within the underlying DG discretization. A second analysis examines exact solutions to the DG discretization in a shock containing element, and identifies a "model" instability that will inevitably arise when solving the Euler equations using the DG method. This analysis identifies the minimum viscosity required for stability. The shock capturing method is demonstrated for high-speed flow over an inviscid cylinder and for an unsteady disturbance in a hypersonic boundary layer. Numerical tests are presented that evaluate several aspects of the shock detection terms. The sensitivity of the results to model parameters is examined with grid and order refinement studies.

Studies of Coherent Synchrotron Radiation with the Discontinuous Galerkin Method

NASA Astrophysics Data System (ADS)

Bizzozero, David A.

In this thesis, we present methods for integrating Maxwell's equations in Frenet-Serret coordinates in several settings using discontinuous Galerkin (DG) finite element method codes in 1D, 2D, and 3D. We apply these routines to the study of coherent synchrotron radiation, an important topic in accelerator physics. We build upon the published computational work of T. Agoh and D. Zhou in solving Maxwell's equations in the frequency-domain using a paraxial approximation which reduces Maxwell's equations to a Schrodinger-like system. We also evolve Maxwell's equations in the time-domain using a Fourier series decomposition with 2D DG motivated by an experiment performed at the Canadian Light Source. A comparison between theory and experiment has been published (Phys. Rev. Lett. 114, 204801 (2015)). Lastly, we devise a novel approach to integrating Maxwell's equations with 3D DG using a Galilean transformation and demonstrate proof-of-concept. In the above studies, we examine the accuracy, efficiency, and convergence of DG.

A hybridized formulation for the weak Galerkin mixed finite element method

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

Mu, Lin; Wang, Junping; Ye, Xiu

This paper presents a hybridized formulation for the weak Galerkin mixed finite element method (WG-MFEM) which was introduced and analyzed in Wang and Ye (2014) for second order elliptic equations. The WG-MFEM method was designed by using discontinuous piecewise polynomials on finite element partitions consisting of polygonal or polyhedral elements of arbitrary shape. The key to WG-MFEM is the use of a discrete weak divergence operator which is defined and computed by solving inexpensive problems locally on each element. The hybridized formulation of this paper leads to a significantly reduced system of linear equations involving only the unknowns arising frommore » the Lagrange multiplier in hybridization. Optimal-order error estimates are derived for the hybridized WG-MFEM approximations. In conclusion, some numerical results are reported to confirm the theory and a superconvergence for the Lagrange multiplier.« less

A hybridized formulation for the weak Galerkin mixed finite element method

DOE PAGES

Mu, Lin; Wang, Junping; Ye, Xiu

2016-01-14

This paper presents a hybridized formulation for the weak Galerkin mixed finite element method (WG-MFEM) which was introduced and analyzed in Wang and Ye (2014) for second order elliptic equations. The WG-MFEM method was designed by using discontinuous piecewise polynomials on finite element partitions consisting of polygonal or polyhedral elements of arbitrary shape. The key to WG-MFEM is the use of a discrete weak divergence operator which is defined and computed by solving inexpensive problems locally on each element. The hybridized formulation of this paper leads to a significantly reduced system of linear equations involving only the unknowns arising frommore » the Lagrange multiplier in hybridization. Optimal-order error estimates are derived for the hybridized WG-MFEM approximations. In conclusion, some numerical results are reported to confirm the theory and a superconvergence for the Lagrange multiplier.« less

Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains

NASA Technical Reports Server (NTRS)

Barth, Timothy J.; Sethian, James A.

1997-01-01

Borrowing from techniques developed for conservation law equations, numerical schemes which discretize the Hamilton-Jacobi (H-J), level set, and Eikonal equations on triangulated domains are presented. The first scheme is a provably monotone discretization for certain forms of the H-J equations. Unfortunately, the basic scheme lacks proper Lipschitz continuity of the numerical Hamiltonian. By employing a virtual edge flipping technique, Lipschitz continuity of the numerical flux is restored on acute triangulations. Next, schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous Hamiltonians. These schemes possess a discrete maximum principle on arbitrary triangulations and naturally exhibit proper Lipschitz continuity of the numerical Hamiltonian. Finally, a class of Petrov-Galerkin approximations are considered. These schemes are stabilized via a least-squares bilinear form. The Petrov-Galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy.

An iterative phase-space explicit discontinuous Galerkin method for stellar radiative transfer in extended atmospheres

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

de Almeida, Valmor F.

In this work, a phase-space discontinuous Galerkin (PSDG) method is presented for the solution of stellar radiative transfer problems. It allows for greater adaptivity than competing methods without sacrificing generality. The method is extensively tested on a spherically symmetric, static, inverse-power-law scattering atmosphere. Results for different sizes of atmospheres and intensities of scattering agreed with asymptotic values. The exponentially decaying behavior of the radiative field in the diffusive-transparent transition region, and the forward peaking behavior at the surface of extended atmospheres were accurately captured. The integrodifferential equation of radiation transfer is solved iteratively by alternating between the radiative pressure equationmore » and the original equation with the integral term treated as an energy density source term. In each iteration, the equations are solved via an explicit, flux-conserving, discontinuous Galerkin method. Finite elements are ordered in wave fronts perpendicular to the characteristic curves so that elemental linear algebraic systems are solved quickly by sweeping the phase space element by element. Two implementations of a diffusive boundary condition at the origin are demonstrated wherein the finite discontinuity in the radiation intensity is accurately captured by the proposed method. This allows for a consistent mechanism to preserve photon luminosity. The method was proved to be robust and fast, and a case is made for the adequacy of parallel processing. In addition to classical two-dimensional plots, results of normalized radiation intensity were mapped onto a log-polar surface exhibiting all distinguishing features of the problem studied.« less

An iterative phase-space explicit discontinuous Galerkin method for stellar radiative transfer in extended atmospheres

DOE PAGES

de Almeida, Valmor F.

2017-04-19

In this work, a phase-space discontinuous Galerkin (PSDG) method is presented for the solution of stellar radiative transfer problems. It allows for greater adaptivity than competing methods without sacrificing generality. The method is extensively tested on a spherically symmetric, static, inverse-power-law scattering atmosphere. Results for different sizes of atmospheres and intensities of scattering agreed with asymptotic values. The exponentially decaying behavior of the radiative field in the diffusive-transparent transition region, and the forward peaking behavior at the surface of extended atmospheres were accurately captured. The integrodifferential equation of radiation transfer is solved iteratively by alternating between the radiative pressure equationmore » and the original equation with the integral term treated as an energy density source term. In each iteration, the equations are solved via an explicit, flux-conserving, discontinuous Galerkin method. Finite elements are ordered in wave fronts perpendicular to the characteristic curves so that elemental linear algebraic systems are solved quickly by sweeping the phase space element by element. Two implementations of a diffusive boundary condition at the origin are demonstrated wherein the finite discontinuity in the radiation intensity is accurately captured by the proposed method. This allows for a consistent mechanism to preserve photon luminosity. The method was proved to be robust and fast, and a case is made for the adequacy of parallel processing. In addition to classical two-dimensional plots, results of normalized radiation intensity were mapped onto a log-polar surface exhibiting all distinguishing features of the problem studied.« less

Very high order discontinuous Galerkin method in elliptic problems

NASA Astrophysics Data System (ADS)

Jaśkowiec, Jan

2017-09-01

The paper deals with high-order discontinuous Galerkin (DG) method with the approximation order that exceeds 20 and reaches 100 and even 1000 with respect to one-dimensional case. To achieve such a high order solution, the DG method with finite difference method has to be applied. The basis functions of this method are high-order orthogonal Legendre or Chebyshev polynomials. These polynomials are defined in one-dimensional space (1D), but they can be easily adapted to two-dimensional space (2D) by cross products. There are no nodes in the elements and the degrees of freedom are coefficients of linear combination of basis functions. In this sort of analysis the reference elements are needed, so the transformations of the reference element into the real one are needed as well as the transformations connected with the mesh skeleton. Due to orthogonality of the basis functions, the obtained matrices are sparse even for finite elements with more than thousands degrees of freedom. In consequence, the truncation errors are limited and very high-order analysis can be performed. The paper is illustrated with a set of benchmark examples of 1D and 2D for the elliptic problems. The example presents the great effectiveness of the method that can shorten the length of calculation over hundreds times.

Very high order discontinuous Galerkin method in elliptic problems

NASA Astrophysics Data System (ADS)

Jaśkowiec, Jan

2018-07-01

The paper deals with high-order discontinuous Galerkin (DG) method with the approximation order that exceeds 20 and reaches 100 and even 1000 with respect to one-dimensional case. To achieve such a high order solution, the DG method with finite difference method has to be applied. The basis functions of this method are high-order orthogonal Legendre or Chebyshev polynomials. These polynomials are defined in one-dimensional space (1D), but they can be easily adapted to two-dimensional space (2D) by cross products. There are no nodes in the elements and the degrees of freedom are coefficients of linear combination of basis functions. In this sort of analysis the reference elements are needed, so the transformations of the reference element into the real one are needed as well as the transformations connected with the mesh skeleton. Due to orthogonality of the basis functions, the obtained matrices are sparse even for finite elements with more than thousands degrees of freedom. In consequence, the truncation errors are limited and very high-order analysis can be performed. The paper is illustrated with a set of benchmark examples of 1D and 2D for the elliptic problems. The example presents the great effectiveness of the method that can shorten the length of calculation over hundreds times.

Efficient Implementations of the Quadrature-Free Discontinuous Galerkin Method

NASA Technical Reports Server (NTRS)

Lockard, David P.; Atkins, Harold L.

1999-01-01

The efficiency of the quadrature-free form of the dis- continuous Galerkin method in two dimensions, and briefly in three dimensions, is examined. Most of the work for constant-coefficient, linear problems involves the volume and edge integrations, and the transformation of information from the volume to the edges. These operations can be viewed as matrix-vector multiplications. Many of the matrices are sparse as a result of symmetry, and blocking and specialized multiplication routines are used to account for the sparsity. By optimizing these operations, a 35% reduction in total CPU time is achieved. For nonlinear problems, the calculation of the flux becomes dominant because of the cost associated with polynomial products and inversion. This component of the work can be reduced by up to 75% when the products are approximated by truncating terms. Because the cost is high for nonlinear problems on general elements, it is suggested that simplified physics and the most efficient element types be used over most of the domain.

High-order Discontinuous Element-based Schemes for the Inviscid Shallow Water Equations: Spectral Multidomain Penalty and Discontinuous Galerkin Methods

DTIC Science & Technology

2011-07-19

multidomain methods, Discontinuous Galerkin methods, interfacial treatment ∗ Jorge A. Escobar-Vargas, School of Civil and Environmental Engineering, Cornell...Click here to view linked References 1. Introduction Geophysical flows exhibit a complex structure and dynamics over a broad range of scales that...hyperbolic problems, where the interfacial patching was implemented with an upwind scheme based on a modified method of characteristics. This approach

A discontinuous Galerkin method for poroelastic wave propagation: The two-dimensional case

NASA Astrophysics Data System (ADS)

Dudley Ward, N. F.; Lähivaara, T.; Eveson, S.

2017-12-01

In this paper, we consider a high-order discontinuous Galerkin (DG) method for modelling wave propagation in coupled poroelastic-elastic media. The upwind numerical flux is derived as an exact solution for the Riemann problem including the poroelastic-elastic interface. Attenuation mechanisms in both Biot's low- and high-frequency regimes are considered. The current implementation supports non-uniform basis orders which can be used to control the numerical accuracy element by element. In the numerical examples, we study the convergence properties of the proposed DG scheme and provide experiments where the numerical accuracy of the scheme under consideration is compared to analytic and other numerical solutions.

Galerkin method for unsplit 3-D Dirac equation using atomically/kinetically balanced B-spline basis

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

Fillion-Gourdeau, F., E-mail: filliong@CRM.UMontreal.ca; Centre de Recherches Mathématiques, Université de Montréal, Montréal, H3T 1J4; Lorin, E., E-mail: elorin@math.carleton.ca

2016-02-15

A Galerkin method is developed to solve the time-dependent Dirac equation in prolate spheroidal coordinates for an electron–molecular two-center system. The initial state is evaluated from a variational principle using a kinetic/atomic balanced basis, which allows for an efficient and accurate determination of the Dirac spectrum and eigenfunctions. B-spline basis functions are used to obtain high accuracy. This numerical method is used to compute the energy spectrum of the two-center problem and then the evolution of eigenstate wavefunctions in an external electromagnetic field.

Republication of: A theorem on Petrov types

NASA Astrophysics Data System (ADS)

Goldberg, J. N.; Sachs, R. K.

2009-02-01

This is a republication of the paper “A Theorem on Petrov Types” by Goldberg and Sachs, Acta Phys. Pol. 22 (supplement), 13 (1962), in which they proved the Goldberg-Sachs theorem. The article has been selected for publication in the Golden Oldies series of General Relativity and Gravitation. Typographical errors of the original publication were corrected by the editor. The paper is accompanied by a Golden Oldie Editorial containing an editorial note written by Andrzej Krasiński and Maciej Przanowski and Goldberg’s brief autobiography. The editorial note explains some difficult parts of the proof of the theorem and discusses the influence of results of the paper on later research.

Chebyshev polynomial filtered subspace iteration in the discontinuous Galerkin method for large-scale electronic structure calculations

DOE PAGES

Banerjee, Amartya S.; Lin, Lin; Hu, Wei; ...

2016-10-21

The Discontinuous Galerkin (DG) electronic structure method employs an adaptive local basis (ALB) set to solve the Kohn-Sham equations of density functional theory in a discontinuous Galerkin framework. The adaptive local basis is generated on-the-fly to capture the local material physics and can systematically attain chemical accuracy with only a few tens of degrees of freedom per atom. A central issue for large-scale calculations, however, is the computation of the electron density (and subsequently, ground state properties) from the discretized Hamiltonian in an efficient and scalable manner. We show in this work how Chebyshev polynomial filtered subspace iteration (CheFSI) canmore » be used to address this issue and push the envelope in large-scale materials simulations in a discontinuous Galerkin framework. We describe how the subspace filtering steps can be performed in an efficient and scalable manner using a two-dimensional parallelization scheme, thanks to the orthogonality of the DG basis set and block-sparse structure of the DG Hamiltonian matrix. The on-the-fly nature of the ALB functions requires additional care in carrying out the subspace iterations. We demonstrate the parallel scalability of the DG-CheFSI approach in calculations of large-scale twodimensional graphene sheets and bulk three-dimensional lithium-ion electrolyte systems. In conclusion, employing 55 296 computational cores, the time per self-consistent field iteration for a sample of the bulk 3D electrolyte containing 8586 atoms is 90 s, and the time for a graphene sheet containing 11 520 atoms is 75 s.« less

DNS of Flows over Periodic Hills using a Discontinuous-Galerkin Spectral-Element Method

NASA Technical Reports Server (NTRS)

Diosady, Laslo T.; Murman, Scott M.

2014-01-01

Direct numerical simulation (DNS) of turbulent compressible flows is performed using a higher-order space-time discontinuous-Galerkin finite-element method. The numerical scheme is validated by performing DNS of the evolution of the Taylor-Green vortex and turbulent flow in a channel. The higher-order method is shown to provide increased accuracy relative to low-order methods at a given number of degrees of freedom. The turbulent flow over a periodic array of hills in a channel is simulated at Reynolds number 10,595 using an 8th-order scheme in space and a 4th-order scheme in time. These results are validated against previous large eddy simulation (LES) results. A preliminary analysis provides insight into how these detailed simulations can be used to improve Reynoldsaveraged Navier-Stokes (RANS) modeling

The use of Galerkin finite-element methods to solve mass-transport equations

USGS Publications Warehouse

Grove, David B.

1977-01-01

The partial differential equation that describes the transport and reaction of chemical solutes in porous media was solved using the Galerkin finite-element technique. These finite elements were superimposed over finite-difference cells used to solve the flow equation. Both convection and flow due to hydraulic dispersion were considered. Linear and Hermite cubic approximations (basis functions) provided satisfactory results: however, the linear functions were computationally more efficient for two-dimensional problems. Successive over relaxation (SOR) and iteration techniques using Tchebyschef polynomials were used to solve the sparce matrices generated using the linear and Hermite cubic functions, respectively. Comparisons of the finite-element methods to the finite-difference methods, and to analytical results, indicated that a high degree of accuracy may be obtained using the method outlined. The technique was applied to a field problem involving an aquifer contaminated with chloride, tritium, and strontium-90. (Woodard-USGS)

Dynamic Rupture Modeling in Three Dimensions on Unstructured Meshes Using a Discontinuous Galerkin Method

NASA Astrophysics Data System (ADS)

Pelties, C.; Käser, M.

2010-12-01

We will present recent developments concerning the extensions of the ADER-DG method to solve three dimensional dynamic rupture problems on unstructured tetrahedral meshes. The simulation of earthquake rupture dynamics and seismic wave propagation using a discontinuous Galerkin (DG) method in 2D was recently presented by J. de la Puente et al. (2009). A considerable feature of this study regarding spontaneous rupture problems was the combination of the DG scheme and a time integration method using Arbitrarily high-order DERivatives (ADER) to provide high accuracy in space and time with the discretization on unstructured meshes. In the resulting discrete velocity-stress formulation of the elastic wave equations variables are naturally discontinuous at the interfaces between elements. The so-called Riemann problem can then be solved to obtain well defined values of the variables at the discontinuity itself. This is in particular valid for the fault at which a certain friction law has to be evaluated. Hence, the fault’s geometry is honored by the computational mesh. This way, complex fault planes can be modeled adequately with small elements while fast mesh coarsening is possible with increasing distance from the fault. Due to the strict locality of the scheme using only direct neighbor communication, excellent parallel behavior can be observed. A further advantage of the scheme is that it avoids spurious high-frequency contributions in the slip rate spectra and therefore does not require artificial Kelvin-Voigt damping or filtering of synthetic seismograms. In order to test the accuracy of the ADER-DG method the Southern California Earthquake Center (SCEC) benchmark for spontaneous rupture simulations was employed. Reference: J. de la Puente, J.-P. Ampuero, and M. Käser (2009), Dynamic rupture modeling on unstructured meshes using a discontinuous Galerkin method, JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, B10302, doi:10.1029/2008JB006271

Assessment of Hybrid High-Order methods on curved meshes and comparison with discontinuous Galerkin methods

NASA Astrophysics Data System (ADS)

Botti, Lorenzo; Di Pietro, Daniele A.

2018-10-01

We propose and validate a novel extension of Hybrid High-Order (HHO) methods to meshes featuring curved elements. HHO methods are based on discrete unknowns that are broken polynomials on the mesh and its skeleton. We propose here the use of physical frame polynomials over mesh elements and reference frame polynomials over mesh faces. With this choice, the degree of face unknowns must be suitably selected in order to recover on curved meshes the same convergence rates as on straight meshes. We provide an estimate of the optimal face polynomial degree depending on the element polynomial degree and on the so-called effective mapping order. The estimate is numerically validated through specifically crafted numerical tests. All test cases are conducted considering two- and three-dimensional pure diffusion problems, and include comparisons with discontinuous Galerkin discretizations. The extension to agglomerated meshes with curved boundaries is also considered.

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

NASA Astrophysics Data System (ADS)

Läuter, Matthias; Giraldo, Francis X.; Handorf, Dörthe; Dethloff, Klaus

2008-12-01

A global model of the atmosphere is presented governed by the shallow water equations and discretized by a Runge-Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional surface in R3, are locally represented in terms of spherical triangular coordinates, the appropriate local coordinate mappings on triangles. On every triangular grid element, this leads to a two-dimensional representation of tangential momentum and therefore only two discrete momentum equations. The discontinuous Galerkin method consists of an integral formulation which requires both area (elements) and line (element faces) integrals. Here, we use a Rusanov numerical flux to resolve the discontinuous fluxes at the element faces. A strong stability-preserving third-order Runge-Kutta method is applied for the time discretization. The polynomial space of order k on each curved triangle of the grid is characterized by a Lagrange basis and requires high-order quadature rules for the integration over elements and element faces. For the presented method no mass matrix inversion is necessary, except in a preprocessing step. The validation of the atmospheric model has been done considering standard tests from Williamson et al. [D.L. Williamson, J.B. Drake, J.J. Hack, R. Jakob, P.N. Swarztrauber, A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys. 102 (1992) 211-224], unsteady analytical solutions of the nonlinear shallow water equations and a barotropic instability caused by an initial perturbation of a jet stream. A convergence rate of O(Δx) was observed in the model experiments. Furthermore, a numerical experiment is presented, for which the third-order time-integration method limits the model error. Thus, the time step Δt is restricted by both the CFL-condition and accuracy demands. Conservation of mass was shown up to machine precision and energy conservation

Comparison between results of solution of Burgers' equation and Laplace's equation by Galerkin and least-square finite element methods

NASA Astrophysics Data System (ADS)

Adib, Arash; Poorveis, Davood; Mehraban, Farid

2018-03-01

In this research, two equations are considered as examples of hyperbolic and elliptic equations. In addition, two finite element methods are applied for solving of these equations. The purpose of this research is the selection of suitable method for solving each of two equations. Burgers' equation is a hyperbolic equation. This equation is a pure advection (without diffusion) equation. This equation is one-dimensional and unsteady. A sudden shock wave is introduced to the model. This wave moves without deformation. In addition, Laplace's equation is an elliptical equation. This equation is steady and two-dimensional. The solution of Laplace's equation in an earth dam is considered. By solution of Laplace's equation, head pressure and the value of seepage in the directions X and Y are calculated in different points of earth dam. At the end, water table is shown in the earth dam. For Burgers' equation, least-square method can show movement of wave with oscillation but Galerkin method can not show it correctly (the best method for solving of the Burgers' equation is discrete space by least-square finite element method and discrete time by forward difference.). For Laplace's equation, Galerkin and least square methods can show water table correctly in earth dam.

Fully-Implicit Reconstructed Discontinuous Galerkin Method for Stiff Multiphysics Problems

NASA Astrophysics Data System (ADS)

Nourgaliev, Robert

2015-11-01

A new reconstructed Discontinuous Galerkin (rDG) method, based on orthogonal basis/test functions, is developed for fluid flows on unstructured meshes. Orthogonality of basis functions is essential for enabling robust and efficient fully-implicit Newton-Krylov based time integration. The method is designed for generic partial differential equations, including transient, hyperbolic, parabolic or elliptic operators, which are attributed to many multiphysics problems. We demonstrate the method's capabilities for solving compressible fluid-solid systems (in the low Mach number limit), with phase change (melting/solidification), as motivated by applications in Additive Manufacturing. We focus on the method's accuracy (in both space and time), as well as robustness and solvability of the system of linear equations involved in the linearization steps of Newton-based methods. The performance of the developed method is investigated for highly-stiff problems with melting/solidification, emphasizing the advantages from tight coupling of mass, momentum and energy conservation equations, as well as orthogonality of basis functions, which leads to better conditioning of the underlying (approximate) Jacobian matrices, and rapid convergence of the Krylov-based linear solver. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344, and funded by the LDRD at LLNL under project tracking code 13-SI-002.

Computer Program for Thin Wire Antenna over a Perfectly Conducting Ground Plane. [using Galerkins method and sinusoidal bases

NASA Technical Reports Server (NTRS)

Richmond, J. H.

1974-01-01

A computer program is presented for a thin-wire antenna over a perfect ground plane. The analysis is performed in the frequency domain, and the exterior medium is free space. The antenna may have finite conductivity and lumped loads. The output data includes the current distribution, impedance, radiation efficiency, and gain. The program uses sinusoidal bases and Galerkin's method.

A spectral hybridizable discontinuous Galerkin method for elastic-acoustic wave propagation

NASA Astrophysics Data System (ADS)

Terrana, S.; Vilotte, J. P.; Guillot, L.

2018-04-01

We introduce a time-domain, high-order in space, hybridizable discontinuous Galerkin (DG) spectral element method (HDG-SEM) for wave equations in coupled elastic-acoustic media. The method is based on a first-order hyperbolic velocity-strain formulation of the wave equations written in conservative form. This method follows the HDG approach by introducing a hybrid unknown, which is the approximation of the velocity on the elements boundaries, as the only globally (i.e. interelement) coupled degrees of freedom. In this paper, we first present a hybridized formulation of the exact Riemann solver at the element boundaries, taking into account elastic-elastic, acoustic-acoustic and elastic-acoustic interfaces. We then use this Riemann solver to derive an explicit construction of the HDG stabilization function τ for all the above-mentioned interfaces. We thus obtain an HDG scheme for coupled elastic-acoustic problems. This scheme is then discretized in space on quadrangular/hexahedral meshes using arbitrary high-order polynomial basis for both volumetric and hybrid fields, using an approach similar to the spectral element methods. This leads to a semi-discrete system of algebraic differential equations (ADEs), which thanks to the structure of the global conservativity condition can be reformulated easily as a classical system of first-order ordinary differential equations in time, allowing the use of classical explicit or implicit time integration schemes. When an explicit time scheme is used, the HDG method can be seen as a reformulation of a DG with upwind fluxes. The introduction of the velocity hybrid unknown leads to relatively simple computations at the element boundaries which, in turn, makes the HDG approach competitive with the DG-upwind methods. Extensive numerical results are provided to illustrate and assess the accuracy and convergence properties of this HDG-SEM. The approximate velocity is shown to converge with the optimal order of k + 1 in the L2-norm

An Investigation of Wave Propagations in Discontinuous Galerkin Method

NASA Technical Reports Server (NTRS)

Hu, Fang Q.

2004-01-01

Analysis of the discontinuous Galerkin method has been carried out for one- and two-dimensional system of hyperbolic equations. Analytical, as well as numerical, properties of wave propagation in a DGM scheme are derived and verified with direct numerical simulations. In addition to a systematic examination of the dissipation and dispersion errors, behaviours of a DG scheme at an interface of two different grid topologies are also studied. Under the same framework, a quantitative discrete analysis of various artificial boundary conditions is also conducted. Progress has been made in numerical boundary condition treatment that is closely related to the application of DGM in aeroacoustics problems. Finally, Fourier analysis of DGM for the Convective diffusion equation has also be studied in connection with the application of DG schemes for the Navier-Stokes equations. This research has resulted in five(5) publications, plus one additional manuscript in preparation, four(4) conference presentations, and three(3) departmental seminars, as summarized in part II. Abstracts of papers are given in part 111 of this report.

A discontinuous Galerkin method for gravity-driven viscous fingering instabilities in porous media

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

Scovazzi, G.; Gerstenberger, A.; Collis, S. S.

2013-01-01

We present a new approach to the simulation of gravity-driven viscous fingering instabilities in porous media flow. These instabilities play a very important role during carbon sequestration processes in brine aquifers. Our approach is based on a nonlinear implementation of the discontinuous Galerkin method, and possesses a number of key features. First, the method developed is inherently high order, and is therefore well suited to study unstable flow mechanisms. Secondly, it maintains high-order accuracy on completely unstructured meshes. The combination of these two features makes it a very appealing strategy in simulating the challenging flow patterns and very complex geometriesmore » of actual reservoirs and aquifers. This article includes an extensive set of verification studies on the stability and accuracy of the method, and also features a number of computations with unstructured grids and non-standard geometries.« less

A Discrete Analysis of Non-reflecting Boundary Conditions for Discontinuous Galerkin Method

NASA Technical Reports Server (NTRS)

Hu, Fang Q.; Atkins, Harold L.

2003-01-01

We present a discrete analysis of non-reflecting boundary conditions for the discontinuous Galerkin method. The boundary conditions considered in this paper include the recently proposed Perfectly Matched Layer absorbing boundary condition for the linearized Euler equation and two non-reflecting boundary conditions based on the characteristic decomposition of the flux on the boundary. The analyses for the three boundary conditions are carried out in a unifled way. In each case, eigensolutions of the discrete system are obtained and applied to compute the numerical reflection coefficients of a specified out-going wave. The dependencies of the reflections at the boundary on the out-going wave angle and frequency as well as the mesh sizes arc? studied. Comparisons with direct numerical simulation results are also presented.

A hybridizable discontinuous Galerkin method for modeling fluid-structure interaction

NASA Astrophysics Data System (ADS)

Sheldon, Jason P.; Miller, Scott T.; Pitt, Jonathan S.

2016-12-01

This work presents a novel application of the hybridizable discontinuous Galerkin (HDG) finite element method to the multi-physics simulation of coupled fluid-structure interaction (FSI) problems. Recent applications of the HDG method have primarily been for single-physics problems including both solids and fluids, which are necessary building blocks for FSI modeling. Utilizing these established models, HDG formulations for linear elastostatics, a nonlinear elastodynamic model, and arbitrary Lagrangian-Eulerian Navier-Stokes are derived. The elasticity formulations are written in a Lagrangian reference frame, with the nonlinear formulation restricted to hyperelastic materials. With these individual solid and fluid formulations, the remaining challenge in FSI modeling is coupling together their disparate mathematics on the fluid-solid interface. This coupling is presented, along with the resultant HDG FSI formulation. Verification of the component models, through the method of manufactured solutions, is performed and each model is shown to converge at the expected rate. The individual components, along with the complete FSI model, are then compared to the benchmark problems proposed by Turek and Hron [1]. The solutions from the HDG formulation presented in this work trend towards the benchmark as the spatial polynomial order and the temporal order of integration are increased.

A hybridizable discontinuous Galerkin method for modeling fluid–structure interaction

DOE PAGES

Sheldon, Jason P.; Miller, Scott T.; Pitt, Jonathan S.

2016-08-31

This study presents a novel application of the hybridizable discontinuous Galerkin (HDG) finite element method to the multi-physics simulation of coupled fluid–structure interaction (FSI) problems. Recent applications of the HDG method have primarily been for single-physics problems including both solids and fluids, which are necessary building blocks for FSI modeling. Utilizing these established models, HDG formulations for linear elastostatics, a nonlinear elastodynamic model, and arbitrary Lagrangian–Eulerian Navier–Stokes are derived. The elasticity formulations are written in a Lagrangian reference frame, with the nonlinear formulation restricted to hyperelastic materials. With these individual solid and fluid formulations, the remaining challenge in FSI modelingmore » is coupling together their disparate mathematics on the fluid–solid interface. This coupling is presented, along with the resultant HDG FSI formulation. Verification of the component models, through the method of manufactured solutions, is performed and each model is shown to converge at the expected rate. The individual components, along with the complete FSI model, are then compared to the benchmark problems proposed by Turek and Hron [1]. The solutions from the HDG formulation presented in this work trend towards the benchmark as the spatial polynomial order and the temporal order of integration are increased.« less

DEVELOPMENT OF SPLIT-OPERATOR, PETROV-GALERKIN METHODS TO SIMULATE TRANSPORT AND DIFFUSION PROBLEMS

EPA Science Inventory

The rate at which contaminants in groundwater undergo sorption and desorption is routinely described using diffusion models. Such approaches, when incorporated into transport models, lead to large systems of coupled equations, often nonlinear. This has restricted applications of ...

Solving Reynolds Equation in the Head-Disk Interface of Hard Disk Drives by Using a Meshless Method

NASA Astrophysics Data System (ADS)

Bao-Jun, Shi; Ting-Yi, Yang; Jian, Zhang; Yun-Dong, Du

2010-05-01

With the decrease of the flying height of the magnetic head/slider in hard disk drives (HDDs), Reynolds equation, which is used to describe the pressure distribution of the air bearing film in HDDs, must be modified to account for the rarefaction effect. Meshless local Petrov-Galerkin (MLPG) method has been successfully used in some fields of solid mechanics and fluid mechanics and was proven to be an efficacious method. No meshes are needed in MLPG method either for the interpolation of the trial and test functions, or for the integration of the weak form of the related differential equation. We solve Reynolds equation in the head-disk interface (HDI) of HDDs by using MLPG method. The pressure distribution of the air baring film by using MLPG method is obtained and compared with the exact solution and that obtained by using a least square finite difference (LSFD) method. We also investigate effects of the bearing number on the pressure value and the center of pressure based on this meshless method for different film-thickness ratios.

Galerkin's Method and the Double P$sub 1$ approximation for Thermal Flux Calculation; IL METODO DI GALERKIN E LA DOPPIA P$sub 1$ PER IL CALCOLO DEL FLUSSO TERMICO

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

Daneri, A.; Daneri, A.

1964-01-01

The program DESTHEC DP, in FORTRAN MONITOR for the IBM 7090, solves the transport equation for thermal neutrons in slab geometry. For the energy, Galerkin's method with the double P/sub 1/ approximation is used, Comparison shows good agreement between DESTHEC DP results and results obtained by the THERMOS program, which solves the transport equation in integral form. The theory is presented, and input and output are discussed. Numerical results are included, as well as the program listing. (D.C.W.)

Continuous and Discontinuous Galerkin Methods for a Scalable Three-Dimensional Nonhydrostatic Atmospheric Model: Limited-Area Mode

DTIC Science & Technology

2012-01-01

atmosphere model, Int. J . High Perform. Comput. Appl. 26 (1) (2012) 74–89. [8] J.M. Dennis, M. Levy, R.D. Nair, H.M. Tufo, T . Voran. Towards and efficient...26] A. Klockner, T . Warburton, J . Bridge, J.S, Hesthaven, Nodal discontinuous galerkin methods on graphics processors, J . Comput. Phys. 228 (21) (2009...mode James F. Kelly, Francis X. Giraldo ⇑ Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, United States a r t i c l e i n

A priori error estimates for an hp-version of the discontinuous Galerkin method for hyperbolic conservation laws

NASA Technical Reports Server (NTRS)

Bey, Kim S.; Oden, J. Tinsley

1993-01-01

A priori error estimates are derived for hp-versions of the finite element method for discontinuous Galerkin approximations of a model class of linear, scalar, first-order hyperbolic conservation laws. These estimates are derived in a mesh dependent norm in which the coefficients depend upon both the local mesh size h(sub K) and a number p(sub k) which can be identified with the spectral order of the local approximations over each element.

SUPG Finite Element Simulations of Compressible Flows for Aerothermodynamic Applications

NASA Technical Reports Server (NTRS)

Kirk, Benjamin S.

2007-01-01

This viewgraph presentation reviews the Streamline-Upwind Petrov-Galerkin (SUPG) Finite Element Simulation. It covers the background, governing equations, weak formulation, shock capturing, inviscid flux discretization, time discretization, linearization, and implicit solution strategies. It also reviews some applications such as Type IV Shock Interaction, Forward-Facing Cavity and AEDC Sharp Double Cone.

A new weak Galerkin finite element method for elliptic interface problems

DOE PAGES

Mu, Lin; Wang, Junping; Ye, Xiu; ...

2016-08-26

We introduce and analyze a new weak Galerkin (WG) finite element method in this paper for solving second order elliptic equations with discontinuous coefficients and interfaces. Comparing with the existing WG algorithm for solving the same type problems, the present WG method has a simpler variational formulation and fewer unknowns. Moreover, the new WG algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders. Optimal order error estimates in both H1 and L2 norms are established for the present WG finite element solutions. We conducted extensive numerical experiments inmore » order to examine the accuracy, flexibility, and robustness of the proposed WG interface approach. In solving regular elliptic interface problems, high order convergences are numerically confirmed by using piecewise polynomial basis functions of high degrees. Moreover, the WG method is shown to be able to accommodate very complicated interfaces, due to its flexibility in choosing finite element partitions. Finally, in dealing with challenging problems with low regularities, the piecewise linear WG method is capable of delivering a second order of accuracy in L∞ norm for both C1 and H2 continuous solutions.« less

A new weak Galerkin finite element method for elliptic interface problems

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

Mu, Lin; Wang, Junping; Ye, Xiu

We introduce and analyze a new weak Galerkin (WG) finite element method in this paper for solving second order elliptic equations with discontinuous coefficients and interfaces. Comparing with the existing WG algorithm for solving the same type problems, the present WG method has a simpler variational formulation and fewer unknowns. Moreover, the new WG algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders. Optimal order error estimates in both H1 and L2 norms are established for the present WG finite element solutions. We conducted extensive numerical experiments inmore » order to examine the accuracy, flexibility, and robustness of the proposed WG interface approach. In solving regular elliptic interface problems, high order convergences are numerically confirmed by using piecewise polynomial basis functions of high degrees. Moreover, the WG method is shown to be able to accommodate very complicated interfaces, due to its flexibility in choosing finite element partitions. Finally, in dealing with challenging problems with low regularities, the piecewise linear WG method is capable of delivering a second order of accuracy in L∞ norm for both C1 and H2 continuous solutions.« less

Positivity-preserving well-balanced discontinuous Galerkin methods for the shallow water flows in open channels

NASA Astrophysics Data System (ADS)

Qian, Shouguo; Li, Gang; Shao, Fengjing; Xing, Yulong

2018-05-01

We construct and study efficient high order discontinuous Galerkin methods for the shallow water flows in open channels with irregular geometry and a non-flat bottom topography in this paper. The proposed methods are well-balanced for the still water steady state solution, and can preserve the non-negativity of wet cross section numerically. The well-balanced property is obtained via a novel source term separation and discretization. A simple positivity-preserving limiter is employed to provide efficient and robust simulations near the wetting and drying fronts. Numerical examples are performed to verify the well-balanced property, the non-negativity of the wet cross section, and good performance for both continuous and discontinuous solutions.

Development of a Perfectly Matched Layer Technique for a Discontinuous-Galerkin Spectral-Element Method

NASA Technical Reports Server (NTRS)

Garai, Anirban; Diosady, Laslo T.; Murman, Scott M.; Madavan, Nateri K.

2016-01-01

The perfectly matched layer (PML) technique is developed in the context of a high- order spectral-element Discontinuous-Galerkin (DG) method. The technique is applied to a range of test cases and is shown to be superior compared to other approaches, such as those based on using characteristic boundary conditions and sponge layers, for treating the inflow and outflow boundaries of computational domains. In general, the PML technique improves the quality of the numerical results for simulations of practical flow configurations, but it also exhibits some instabilities for large perturbations. A preliminary analysis that attempts to understand the source of these instabilities is discussed.

Stabilized high-order Galerkin methods based on a parameter-free dynamic SGS model for LES

NASA Astrophysics Data System (ADS)

Marras, Simone; Nazarov, Murtazo; Giraldo, Francis X.

2015-11-01

The high order spectral element approximation of the Euler equations is stabilized via a dynamic sub-grid scale model (Dyn-SGS). This model was originally designed for linear finite elements to solve compressible flows at large Mach numbers. We extend its application to high-order spectral elements to solve the Euler equations of low Mach number stratified flows. The major justification of this work is twofold: stabilization and large eddy simulation are achieved via one scheme only. Because the diffusion coefficients of the regularization stresses obtained via Dyn-SGS are residual-based, the effect of the artificial diffusion is minimal in the regions where the solution is smooth. The direct consequence is that the nominal convergence rate of the high-order solution of smooth problems is not degraded. To our knowledge, this is the first application in atmospheric modeling of a spectral element model stabilized by an eddy viscosity scheme that, by construction, may fulfill stabilization requirements, can model turbulence via LES, and is completely free of a user-tunable parameter. From its derivation, it will be immediately clear that Dyn-SGS is independent of the numerical method; it could be implemented in a discontinuous Galerkin, finite volume, or other environments alike. Preliminary discontinuous Galerkin results are reported as well. The straightforward extension to non-linear scalar problems is also described. A suite of 1D, 2D, and 3D test cases is used to assess the method, with some comparison against the results obtained with the most known Lilly-Smagorinsky SGS model.

A Galerkin method for linear PDE systems in circular geometries with structural acoustic applications

NASA Technical Reports Server (NTRS)

Smith, Ralph C.

1994-01-01

A Galerkin method for systems of PDE's in circular geometries is presented with motivating problems being drawn from structural, acoustic, and structural acoustic applications. Depending upon the application under consideration, piecewise splines or Legendre polynomials are used when approximating the system dynamics with modifications included to incorporate the analytic solution decay near the coordinate singularity. This provides an efficient method which retains its accuracy throughout the circular domain without degradation at singularity. Because the problems under consideration are linear or weakly nonlinear with constant or piecewise constant coefficients, transform methods for the problems are not investigated. While the specific method is developed for the two dimensional wave equations on a circular domain and the equation of transverse motion for a thin circular plate, examples demonstrating the extension of the techniques to a fully coupled structural acoustic system are used to illustrate the flexibility of the method when approximating the dynamics of more complex systems.

Resonant frequency calculations using a hybrid perturbation-Galerkin technique

NASA Technical Reports Server (NTRS)

Geer, James F.; Andersen, Carl M.

1991-01-01

A two-step hybrid perturbation Galerkin technique is applied to the problem of determining the resonant frequencies of one or several degree of freedom nonlinear systems involving a parameter. In one step, the Lindstedt-Poincare method is used to determine perturbation solutions which are formally valid about one or more special values of the parameter (e.g., for large or small values of the parameter). In step two, a subset of the perturbation coordinate functions determined in step one is used in Galerkin type approximation. The technique is illustrated for several one degree of freedom systems, including the Duffing and van der Pol oscillators, as well as for the compound pendulum. For all of the examples considered, it is shown that the frequencies obtained by the hybrid technique using only a few terms from the perturbation solutions are significantly more accurate than the perturbation results on which they are based, and they compare very well with frequencies obtained by purely numerical methods.

On Formulations of Discontinuous Galerkin and Related Methods for Conservation Laws

NASA Technical Reports Server (NTRS)

Huynh, H. T.

2014-01-01

A formulation for the discontinuous Galerkin (DG) method that leads to solutions using the differential form of the equation (as opposed to the standard integral form) is presented. The formulation includes (a) a derivative calculation that involves only data within each cell with no data interaction among cells, and (b) for each cell, corrections to this derivative that deal with the jumps in fluxes at the cell boundaries and allow data across cells to interact. The derivative with no interaction is obtained by a projection, but for nodal-type methods, evaluating this derivative by interpolation at the nodal points is more economical. The corrections are derived using the approximate (Dirac) delta functions. The formulation results in a family of schemes: different approximate delta functions give rise to different methods. It is shown that the current formulation is essentially equivalent to the flux reconstruction (FR) formulation. Due to the use of approximate delta functions, an energy stability proof simpler than that of Vincent, Castonguay, and Jameson (2011) for a family of schemes is derived. Accuracy and stability of resulting schemes are discussed via Fourier analyses. Similar to FR, the current formulation provides a unifying framework for high-order methods by recovering the DG, spectral difference (SD), and spectral volume (SV) schemes. It also yields stable, accurate, and economical methods.

Developing Discontinuous Galerkin Methods for Solving Multiphysics Problems in General Relativity

NASA Astrophysics Data System (ADS)

Kidder, Lawrence; Field, Scott; Teukolsky, Saul; Foucart, Francois; SXS Collaboration

2016-03-01

Multi-messenger observations of the merger of black hole-neutron star and neutron star-neutron star binaries, and of supernova explosions will probe fundamental physics inaccessible to terrestrial experiments. Modeling these systems requires a relativistic treatment of hydrodynamics, including magnetic fields, as well as neutrino transport and nuclear reactions. The accuracy, efficiency, and robustness of current codes that treat all of these problems is not sufficient to keep up with the observational needs. We are building a new numerical code that uses the Discontinuous Galerkin method with a task-based parallelization strategy, a promising combination that will allow multiphysics applications to be treated both accurately and efficiently on petascale and exascale machines. The code will scale to more than 100,000 cores for efficient exploration of the parameter space of potential sources and allowed physics, and the high-fidelity predictions needed to realize the promise of multi-messenger astronomy. I will discuss the current status of the development of this new code.

Direct discontinuous Galerkin method and its variations for second order elliptic equations

DOE PAGES

Huang, Hongying; Chen, Zheng; Li, Jin; ...

2016-08-23

In this study, we study direct discontinuous Galerkin method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009) and its variations (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010; Vidden and Yan in J Comput Math 31(6):638–662, 2013; Yan in J Sci Comput 54(2–3):663–683, 2013) for 2nd order elliptic problems. A priori error estimate under energy norm is established for all four methods. Optimal error estimate under L 2 norm is obtained for DDG method with interface correction (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and symmetric DDG method (Vidden and Yan in J Comput Mathmore » 31(6):638–662, 2013). A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal (k+1)th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal (k+1)th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of Helmholz equation are well resolved.« less

Direct discontinuous Galerkin method and its variations for second order elliptic equations

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

Huang, Hongying; Chen, Zheng; Li, Jin

In this study, we study direct discontinuous Galerkin method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009) and its variations (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010; Vidden and Yan in J Comput Math 31(6):638–662, 2013; Yan in J Sci Comput 54(2–3):663–683, 2013) for 2nd order elliptic problems. A priori error estimate under energy norm is established for all four methods. Optimal error estimate under L 2 norm is obtained for DDG method with interface correction (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and symmetric DDG method (Vidden and Yan in J Comput Mathmore » 31(6):638–662, 2013). A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal (k+1)th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal (k+1)th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of Helmholz equation are well resolved.« less

De-Aliasing Through Over-Integration Applied to the Flux Reconstruction and Discontinuous Galerkin Methods

NASA Technical Reports Server (NTRS)

Spiegel, Seth C.; Huynh, H. T.; DeBonis, James R.

2015-01-01

High-order methods are quickly becoming popular for turbulent flows as the amount of computer processing power increases. The flux reconstruction (FR) method presents a unifying framework for a wide class of high-order methods including discontinuous Galerkin (DG), Spectral Difference (SD), and Spectral Volume (SV). It offers a simple, efficient, and easy way to implement nodal-based methods that are derived via the differential form of the governing equations. Whereas high-order methods have enjoyed recent success, they have been known to introduce numerical instabilities due to polynomial aliasing when applied to under-resolved nonlinear problems. Aliasing errors have been extensively studied in reference to DG methods; however, their study regarding FR methods has mostly been limited to the selection of the nodal points used within each cell. Here, we extend some of the de-aliasing techniques used for DG methods, primarily over-integration, to the FR framework. Our results show that over-integration does remove aliasing errors but may not remove all instabilities caused by insufficient resolution (for FR as well as DG).

Jacobi spectral Galerkin method for elliptic Neumann problems

NASA Astrophysics Data System (ADS)

Doha, E.; Bhrawy, A.; Abd-Elhameed, W.

2009-01-01

This paper is concerned with fast spectral-Galerkin Jacobi algorithms for solving one- and two-dimensional elliptic equations with homogeneous and nonhomogeneous Neumann boundary conditions. The paper extends the algorithms proposed by Shen (SIAM J Sci Comput 15:1489-1505, 1994) and Auteri et al. (J Comput Phys 185:427-444, 2003), based on Legendre polynomials, to Jacobi polynomials with arbitrary α and β. The key to the efficiency of our algorithms is to construct appropriate basis functions with zero slope at the endpoints, which lead to systems with sparse matrices for the discrete variational formulations. The direct solution algorithm developed for the homogeneous Neumann problem in two-dimensions relies upon a tensor product process. Nonhomogeneous Neumann data are accounted for by means of a lifting. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented.

Discontinuous Galerkin Approaches for Stokes Flow and Flow in Porous Media

NASA Astrophysics Data System (ADS)

Lehmann, Ragnar; Kaus, Boris; Lukacova, Maria

2014-05-01

Firstly, we present results of a study comparing two different numerical approaches for solving the Stokes equations with strongly varying viscosity: the continuous Galerkin (i.e., FEM) and the discontinuous Galerkin (DG) method. Secondly, we show how the latter method can be extended and applied to flow in porous media governed by Darcy's law. Nonlinearities in the viscosity or other material parameters can lead to discontinuities in the velocity-pressure solution that may not be approximated well with continuous elements. The DG method allows for discontinuities across interior edges of the underlying mesh. Furthermore, depending on the chosen basis functions, it naturally enforces local mass conservation, i.e., in every mesh cell. Computationally, it provides the capability to locally adapt the polynomial degree and needs communication only between directly adjacent mesh cells making it highly flexible and easy to parallelize. The methods are compared for several geophysically relevant benchmarking setups and discussed with respect to speed, accuracy, computational efficiency.

A discontinuous Galerkin method for numerical pricing of European options under Heston stochastic volatility

NASA Astrophysics Data System (ADS)

Hozman, J.; Tichý, T.

2016-12-01

The paper is based on the results from our recent research on multidimensional option pricing problems. We focus on European option valuation when the price movement of the underlying asset is driven by a stochastic volatility following a square root process proposed by Heston. The stochastic approach incorporates a new additional spatial variable into this model and makes it very robust, i.e. it provides a framework to price a variety of options that is closer to reality. The main topic is to present the numerical scheme arising from the concept of discontinuous Galerkin methods and applicable to the Heston option pricing model. The numerical results are presented on artificial benchmarks as well as on reference market data.

High-order local maximum principle preserving (MPP) discontinuous Galerkin finite element method for the transport equation

NASA Astrophysics Data System (ADS)

Anderson, R.; Dobrev, V.; Kolev, Tz.; Kuzmin, D.; Quezada de Luna, M.; Rieben, R.; Tomov, V.

2017-04-01

In this work we present a FCT-like Maximum-Principle Preserving (MPP) method to solve the transport equation. We use high-order polynomial spaces; in particular, we consider up to 5th order spaces in two and three dimensions and 23rd order spaces in one dimension. The method combines the concepts of positive basis functions for discontinuous Galerkin finite element spatial discretization, locally defined solution bounds, element-based flux correction, and non-linear local mass redistribution. We consider a simple 1D problem with non-smooth initial data to explain and understand the behavior of different parts of the method. Convergence tests in space indicate that high-order accuracy is achieved. Numerical results from several benchmarks in two and three dimensions are also reported.

Numerical simulation of fluid flow around a scramaccelerator projectile

NASA Technical Reports Server (NTRS)

Pepper, Darrell W.; Humphrey, Joseph W.; Sobota, Thomas H.

1991-01-01

Numerical simulations of the fluid motion and temperature distribution around a 'scramaccelerator' projectile are obtained for Mach numbers in the 5-10 range. A finite element method is used to solve the equations of motion for inviscid and viscous two-dimensional or axisymmetric compressible flow. The time-dependent equations are solved explicitly, using bilinear isoparametric quadrilateral elements, mass lumping, and a shock-capturing Petrov-Galerkin formulation. Computed results indicate that maintaining on-design performance for controlling and stabilizing oblique detonation waves is critically dependent on projectile shape and Mach number.

Discontinuous Galerkin methods for modeling Hurricane storm surge

NASA Astrophysics Data System (ADS)

Dawson, Clint; Kubatko, Ethan J.; Westerink, Joannes J.; Trahan, Corey; Mirabito, Christopher; Michoski, Craig; Panda, Nishant

2011-09-01

Storm surge due to hurricanes and tropical storms can result in significant loss of life, property damage, and long-term damage to coastal ecosystems and landscapes. Computer modeling of storm surge can be used for two primary purposes: forecasting of surge as storms approach land for emergency planning and evacuation of coastal populations, and hindcasting of storms for determining risk, development of mitigation strategies, coastal restoration and sustainability. Storm surge is modeled using the shallow water equations, coupled with wind forcing and in some events, models of wave energy. In this paper, we will describe a depth-averaged (2D) model of circulation in spherical coordinates. Tides, riverine forcing, atmospheric pressure, bottom friction, the Coriolis effect and wind stress are all important for characterizing the inundation due to surge. The problem is inherently multi-scale, both in space and time. To model these problems accurately requires significant investments in acquiring high-fidelity input (bathymetry, bottom friction characteristics, land cover data, river flow rates, levees, raised roads and railways, etc.), accurate discretization of the computational domain using unstructured finite element meshes, and numerical methods capable of capturing highly advective flows, wetting and drying, and multi-scale features of the solution. The discontinuous Galerkin (DG) method appears to allow for many of the features necessary to accurately capture storm surge physics. The DG method was developed for modeling shocks and advection-dominated flows on unstructured finite element meshes. It easily allows for adaptivity in both mesh ( h) and polynomial order ( p) for capturing multi-scale spatial events. Mass conservative wetting and drying algorithms can be formulated within the DG method. In this paper, we will describe the application of the DG method to hurricane storm surge. We discuss the general formulation, and new features which have been added to

A high order semi-Lagrangian discontinuous Galerkin method for Vlasov-Poisson simulations without operator splitting

NASA Astrophysics Data System (ADS)

Cai, Xiaofeng; Guo, Wei; Qiu, Jing-Mei

2018-02-01

In this paper, we develop a high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for nonlinear Vlasov-Poisson (VP) simulations without operator splitting. In particular, we combine two recently developed novel techniques: one is the high order non-splitting SLDG transport method (Cai et al. (2017) [4]), and the other is the high order characteristics tracing technique proposed in Qiu and Russo (2017) [29]. The proposed method with up to third order accuracy in both space and time is locally mass conservative, free of splitting error, positivity-preserving, stable and robust for large time stepping size. The SLDG VP solver is applied to classic benchmark test problems such as Landau damping and two-stream instabilities for VP simulations. Efficiency and effectiveness of the proposed scheme is extensively tested. Tremendous CPU savings are shown by comparisons between the proposed SL DG scheme and the classical Runge-Kutta DG method.

Numerical investigation of refractometric sensor elements based on side polished fibres using the Galerkin method

NASA Astrophysics Data System (ADS)

Karakoleva, E. I.; Andreev, A. Tz; Zafirova, B. S.

2006-12-01

The Galerkin method was applied to solve the vector wave equation in order to determine the propagation constants and the transverse electric fields of the modes propagating along side polished single-mode and two-mode optical fibres. The effective refractive indices of the modes were calculated depending on the values of the residual cladding (minimum distance between a fibre core and a polished surface) and the superstrate refractive index. The influence of the fibre parameters and working wavelength on the refractometric sensitivity was estimated in the case when a side polished fibre with inscribed in-fibre Bragg grating is used as a sensor element.

A fully Galerkin method for the recovery of stiffness and damping parameters in Euler-Bernoulli beam models

NASA Technical Reports Server (NTRS)

Smith, R. C.; Bowers, K. L.

1991-01-01

A fully Sinc-Galerkin method for recovering the spatially varying stiffness and damping parameters in Euler-Bernoulli beam models is presented. The forward problems are discretized with a sinc basis in both the spatial and temporal domains thus yielding an approximate solution which converges exponentially and is valid on the infinite time interval. Hence the method avoids the time-stepping which is characteristic of many of the forward schemes which are used in parameter recovery algorithms. Tikhonov regularization is used to stabilize the resulting inverse problem, and the L-curve method for determining an appropriate value of the regularization parameter is briefly discussed. Numerical examples are given which demonstrate the applicability of the method for both individual and simultaneous recovery of the material parameters.

On the stability of projection methods for the incompressible Navier-Stokes equations based on high-order discontinuous Galerkin discretizations

NASA Astrophysics Data System (ADS)

Fehn, Niklas; Wall, Wolfgang A.; Kronbichler, Martin

2017-12-01

The present paper deals with the numerical solution of the incompressible Navier-Stokes equations using high-order discontinuous Galerkin (DG) methods for discretization in space. For DG methods applied to the dual splitting projection method, instabilities have recently been reported that occur for small time step sizes. Since the critical time step size depends on the viscosity and the spatial resolution, these instabilities limit the robustness of the Navier-Stokes solver in case of complex engineering applications characterized by coarse spatial resolutions and small viscosities. By means of numerical investigation we give evidence that these instabilities are related to the discontinuous Galerkin formulation of the velocity divergence term and the pressure gradient term that couple velocity and pressure. Integration by parts of these terms with a suitable definition of boundary conditions is required in order to obtain a stable and robust method. Since the intermediate velocity field does not fulfill the boundary conditions prescribed for the velocity, a consistent boundary condition is derived from the convective step of the dual splitting scheme to ensure high-order accuracy with respect to the temporal discretization. This new formulation is stable in the limit of small time steps for both equal-order and mixed-order polynomial approximations. Although the dual splitting scheme itself includes inf-sup stabilizing contributions, we demonstrate that spurious pressure oscillations appear for equal-order polynomials and small time steps highlighting the necessity to consider inf-sup stability explicitly.

The ALE Discontinuous Galerkin Method for the Simulatio of Air Flow Through Pulsating Human Vocal Folds

NASA Astrophysics Data System (ADS)

Feistauer, Miloslav; Kučera, Václav; Prokopová, Jaroslav; Horáček, Jaromír

2010-09-01

The aim of this work is the simulation of viscous compressible flows in human vocal folds during phonation. The computational domain is a bounded subset of IR2, whose geometry mimics the shape of the human larynx. During phonation, parts of the solid impermeable walls are moving in a prescribed manner, thus simulating the opening and closing of the vocal chords. As the governing equations we take the compressible Navier-Stokes equations in ALE form. Space semidiscretization is carried out by the discontinuous Galerkin method combined with a linearized semi-implicit approach. Numerical experiments are performed with the resulting scheme.

Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods

NASA Astrophysics Data System (ADS)

Pazner, Will; Persson, Per-Olof

2018-02-01

In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require O (p2d) storage and O (p3d) computational work, where p is the degree of basis polynomials used, and d is the spatial dimension. Our SVD-based tensor-product preconditioner requires O (p d + 1) storage, O (p d + 1) work in two spatial dimensions, and O (p d + 2) work in three spatial dimensions. Combined with a matrix-free Newton-Krylov solver, these preconditioners allow for the solution of DG systems in linear time in p per degree of freedom in 2D, and reduce the computational complexity from O (p9) to O (p5) in 3D. Numerical results are shown in 2D and 3D for the advection, Euler, and Navier-Stokes equations, using polynomials of degree up to p = 30. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees p.

Comparison of reduced models for blood flow using Runge–Kutta discontinuous Galerkin methods

PubMed Central

Puelz, Charles; Čanić, Sunčica; Rivière, Béatrice; Rusin, Craig G.

2017-01-01

One–dimensional blood flow models take the general form of nonlinear hyperbolic systems but differ in their formulation. One class of models considers the physically conserved quantities of mass and momentum, while another class describes mass and velocity. Further, the averaging process employed in the model derivation requires the specification of the axial velocity profile; this choice differentiates models within each class. Discrepancies among differing models have yet to be investigated. In this paper, we comment on some theoretical differences among models and systematically compare them for physiologically relevant vessel parameters, network topology, and boundary data. In particular, the effect of the velocity profile is investigated in the cases of both smooth and discontinuous solutions, and a recommendation for a physiological model is provided. The models are discretized by a class of Runge–Kutta discontinuous Galerkin methods. PMID:29081563

Sequential limiting in continuous and discontinuous Galerkin methods for the Euler equations

NASA Astrophysics Data System (ADS)

Dobrev, V.; Kolev, Tz.; Kuzmin, D.; Rieben, R.; Tomov, V.

2018-03-01

We present a new predictor-corrector approach to enforcing local maximum principles in piecewise-linear finite element schemes for the compressible Euler equations. The new element-based limiting strategy is suitable for continuous and discontinuous Galerkin methods alike. In contrast to synchronized limiting techniques for systems of conservation laws, we constrain the density, momentum, and total energy in a sequential manner which guarantees positivity preservation for the pressure and internal energy. After the density limiting step, the total energy and momentum gradients are adjusted to incorporate the irreversible effect of density changes. Antidiffusive corrections to bounds-compatible low-order approximations are limited to satisfy inequality constraints for the specific total and kinetic energy. An accuracy-preserving smoothness indicator is introduced to gradually adjust lower bounds for the element-based correction factors. The employed smoothness criterion is based on a Hessian determinant test for the density. A numerical study is performed for test problems with smooth and discontinuous solutions.

Investigating a hybrid perturbation-Galerkin technique using computer algebra

NASA Technical Reports Server (NTRS)

Andersen, Carl M.; Geer, James F.

1988-01-01

A two-step hybrid perturbation-Galerkin method is presented for the solution of a variety of differential equations type problems which involve a scalar parameter. The resulting (approximate) solution has the form of a sum where each term consists of the product of two functions. The first function is a function of the independent field variable(s) x, and the second is a function of the parameter lambda. In step one the functions of x are determined by forming a perturbation expansion in lambda. In step two the functions of lambda are determined through the use of the classical Bubnov-Gelerkin method. The resulting hybrid method has the potential of overcoming some of the drawbacks of the perturbation and Bubnov-Galerkin methods applied separately, while combining some of the good features of each. In particular, the results can be useful well beyond the radius of convergence associated with the perturbation expansion. The hybrid method is applied with the aid of computer algebra to a simple two-point boundary value problem where the radius of convergence is finite and to a quantum eigenvalue problem where the radius of convergence is zero. For both problems the hybrid method apparently converges for an infinite range of the parameter lambda. The results obtained from the hybrid method are compared with approximate solutions obtained by other methods, and the applicability of the hybrid method to broader problem areas is discussed.

A set of parallel, implicit methods for a reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids

DOE PAGES

Xia, Yidong; Luo, Hong; Frisbey, Megan; ...

2014-07-01

A set of implicit methods are proposed for a third-order hierarchical WENO reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids. An attractive feature in these methods are the application of the Jacobian matrix based on the P1 element approximation, resulting in a huge reduction of memory requirement compared with DG (P2). Also, three approaches -- analytical derivation, divided differencing, and automatic differentiation (AD) are presented to construct the Jacobian matrix respectively, where the AD approach shows the best robustness. A variety of compressible flow problems are computed to demonstrate the fast convergence property of the implemented flowmore » solver. Furthermore, an SPMD (single program, multiple data) programming paradigm based on MPI is proposed to achieve parallelism. The numerical results on complex geometries indicate that this low-storage implicit method can provide a viable and attractive DG solution for complicated flows of practical importance.« less

ON THE ROLE OF INVOLUTIONS IN THE DISCONTINUOUS GALERKIN DISCRETIZATION OF MAXWELL AND MAGNETOHYDRODYNAMIC SYSTEMS

NASA Technical Reports Server (NTRS)

Barth, Timothy

2005-01-01

The role of involutions in energy stability of the discontinuous Galerkin (DG) discretization of Maxwell and magnetohydrodynamic (MHD) systems is examined. Important differences are identified in the symmetrization of the Maxwell and MHD systems that impact the construction of energy stable discretizations using the DG method. Specifically, general sufficient conditions to be imposed on the DG numerical flux and approximation space are given so that energy stability is retained These sufficient conditions reveal the favorable energy consequence of imposing continuity in the normal component of the magnetic induction field at interelement boundaries for MHD discretizations. Counterintuitively, this condition is not required for stability of Maxwell discretizations using the discontinuous Galerkin method.

Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials

NASA Astrophysics Data System (ADS)

Doha, E.; Bhrawy, A.

2006-06-01

It is well known that spectral methods (tau, Galerkin, collocation) have a condition number of ( is the number of retained modes of polynomial approximations). This paper presents some efficient spectral algorithms, which have a condition number of , based on the Jacobi?Galerkin methods of second-order elliptic equations in one and two space variables. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. The complexities of the algorithms are a small multiple of operations for a -dimensional domain with unknowns, while the convergence rates of the algorithms are exponentials with smooth solutions.

New algorithms for solving high even-order differential equations using third and fourth Chebyshev-Galerkin methods

NASA Astrophysics Data System (ADS)

Doha, E. H.; Abd-Elhameed, W. M.; Bassuony, M. A.

2013-03-01

This paper is concerned with spectral Galerkin algorithms for solving high even-order two point boundary value problems in one dimension subject to homogeneous and nonhomogeneous boundary conditions. The proposed algorithms are extended to solve two-dimensional high even-order differential equations. The key to the efficiency of these algorithms is to construct compact combinations of Chebyshev polynomials of the third and fourth kinds as basis functions. The algorithms lead to linear systems with specially structured matrices that can be efficiently inverted. Numerical examples are included to demonstrate the validity and applicability of the proposed algorithms, and some comparisons with some other methods are made.

Scalable direct Vlasov solver with discontinuous Galerkin method on unstructured mesh.

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

Xu, J.; Ostroumov, P. N.; Mustapha, B.

2010-12-01

This paper presents the development of parallel direct Vlasov solvers with discontinuous Galerkin (DG) method for beam and plasma simulations in four dimensions. Both physical and velocity spaces are in two dimesions (2P2V) with unstructured mesh. Contrary to the standard particle-in-cell (PIC) approach for kinetic space plasma simulations, i.e., solving Vlasov-Maxwell equations, direct method has been used in this paper. There are several benefits to solving a Vlasov equation directly, such as avoiding noise associated with a finite number of particles and the capability to capture fine structure in the plasma. The most challanging part of a direct Vlasov solvermore » comes from higher dimensions, as the computational cost increases as N{sup 2d}, where d is the dimension of the physical space. Recently, due to the fast development of supercomputers, the possibility has become more realistic. Many efforts have been made to solve Vlasov equations in low dimensions before; now more interest has focused on higher dimensions. Different numerical methods have been tried so far, such as the finite difference method, Fourier Spectral method, finite volume method, and spectral element method. This paper is based on our previous efforts to use the DG method. The DG method has been proven to be very successful in solving Maxwell equations, and this paper is our first effort in applying the DG method to Vlasov equations. DG has shown several advantages, such as local mass matrix, strong stability, and easy parallelization. These are particularly suitable for Vlasov equations. Domain decomposition in high dimensions has been used for parallelization; these include a highly scalable parallel two-dimensional Poisson solver. Benchmark results have been shown and simulation results will be reported.« less

Implicit and explicit subgrid-scale modeling in discontinuous Galerkin methods for large-eddy simulation

NASA Astrophysics Data System (ADS)

Fernandez, Pablo; Nguyen, Ngoc-Cuong; Peraire, Jaime

2017-11-01

Over the past few years, high-order discontinuous Galerkin (DG) methods for Large-Eddy Simulation (LES) have emerged as a promising approach to solve complex turbulent flows. Despite the significant research investment, the relation between the discretization scheme, the Riemann flux, the subgrid-scale (SGS) model and the accuracy of the resulting LES solver remains unclear. In this talk, we investigate the role of the Riemann solver and the SGS model in the ability to predict a variety of flow regimes, including transition to turbulence, wall-free turbulence, wall-bounded turbulence, and turbulence decay. The Taylor-Green vortex problem and the turbulent channel flow at various Reynolds numbers are considered. Numerical results show that DG methods implicitly introduce numerical dissipation in under-resolved turbulence simulations and, even in the high Reynolds number limit, this implicit dissipation provides a more accurate representation of the actual subgrid-scale dissipation than that by explicit models.

Implicit high-order discontinuous Galerkin method with HWENO type limiters for steady viscous flow simulations

NASA Astrophysics Data System (ADS)

Jiang, Zhen-Hua; Yan, Chao; Yu, Jian

2013-08-01

Two types of implicit algorithms have been improved for high order discontinuous Galerkin (DG) method to solve compressible Navier-Stokes (NS) equations on triangular grids. A block lower-upper symmetric Gauss-Seidel (BLU-SGS) approach is implemented as a nonlinear iterative scheme. And a modified LU-SGS (LLU-SGS) approach is suggested to reduce the memory requirements while retain the good convergence performance of the original LU-SGS approach. Both implicit schemes have the significant advantage that only the diagonal block matrix is stored. The resulting implicit high-order DG methods are applied, in combination with Hermite weighted essentially non-oscillatory (HWENO) limiters, to solve viscous flow problems. Numerical results demonstrate that the present implicit methods are able to achieve significant efficiency improvements over explicit counterparts and for viscous flows with shocks, and the HWENO limiters can be used to achieve the desired essentially non-oscillatory shock transition and the designed high-order accuracy simultaneously.

Finite element computation of compressible flows with the SUPG formulation

NASA Technical Reports Server (NTRS)

Le Beau, G. J.; Tezduyar, T. E.

1991-01-01

Finite element computation of compressible Euler equations is presented in the context of the streamline-upwind/Petrov-Galerkin (SUPG) formulation. The SUPG formulation, which is based on adding stabilizing terms to the Galerkin formulation, is further supplemented with a shock capturing operator which addresses the difficulty in maintaining a satisfactory solution near discontinuities in the solution field. The shock capturing operator, which has been derived from work done in entropy variables for a similar operator, is shown to lead to an appropriate level of additional stabilization near shocks, without resulting in excessive numerical diffusion. An implicit treatment of the impermeable wall boundary condition is also presented. This treatment of the no-penetration condition offers increased stability for large Courant numbers, and accelerated convergence of the computations for both implicit and explicit applications. Several examples are presented to demonstrate the ability of this method to solve the equations governing compressible fluid flow.

Numerical simulation of fluid flow through simplified blade cascade with prescribed harmonic motion using discontinuous Galerkin method

NASA Astrophysics Data System (ADS)

Vimmr, Jan; Bublík, Ondřej; Prausová, Helena; Hála, Jindřich; Pešek, Luděk

2018-06-01

This paper deals with a numerical simulation of compressible viscous fluid flow around three flat plates with prescribed harmonic motion. This arrangement presents a simplified blade cascade with forward wave motion. The aim of this simulation is to determine the aerodynamic forces acting on the flat plates. The mathematical model describing this problem is formed by Favre-averaged system of Navier-Stokes equations in arbitrary Lagrangian-Eulerian (ALE) formulation completed by one-equation Spalart-Allmaras turbulence model. The simulation was performed using the developed in-house CFD software based on discontinuous Galerkin method, which offers high order of accuracy.

Construction of reduced order models for the non-linear Navier-Stokes equations using the proper orthogonal fecomposition (POD)/Galerkin method.

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

Fike, Jeffrey A.

2013-08-01

The construction of stable reduced order models using Galerkin projection for the Euler or Navier-Stokes equations requires a suitable choice for the inner product. The standard L2 inner product is expected to produce unstable ROMs. For the non-linear Navier-Stokes equations this means the use of an energy inner product. In this report, Galerkin projection for the non-linear Navier-Stokes equations using the L2 inner product is implemented as a first step toward constructing stable ROMs for this set of physics.

A penalty-based nodal discontinuous Galerkin method for spontaneous rupture dynamics

NASA Astrophysics Data System (ADS)

Ye, R.; De Hoop, M. V.; Kumar, K.

2017-12-01

Numerical simulation of the dynamic rupture processes with slip is critical to understand the earthquake source process and the generation of ground motions. However, it can be challenging due to the nonlinear friction laws interacting with seismicity, coupled with the discontinuous boundary conditions across the rupture plane. In practice, the inhomogeneities in topography, fault geometry, elastic parameters and permiability add extra complexity. We develop a nodal discontinuous Galerkin method to simulate seismic wave phenomenon with slipping boundary conditions, including the fluid-solid boundaries and ruptures. By introducing a novel penalty flux, we avoid solving Riemann problems on interfaces, which makes our method capable for general anisotropic and poro-elastic materials. Based on unstructured tetrahedral meshes in 3D, the code can capture various geometries in geological model, and use polynomial expansion to achieve high-order accuracy. We consider the rate and state friction law, in the spontaneous rupture dynamics, as part of a nonlinear transmitting boundary condition, which is weakly enforced across the fault surface as numerical flux. An iterative coupling scheme is developed based on implicit time stepping, containing a constrained optimization process that accounts for the nonlinear part. To validate the method, we proof the convergence of the coupled system with error estimates. We test our algorithm on a well-established numerical example (TPV102) of the SCEC/USGS Spontaneous Rupture Code Verification Project, and benchmark with the simulation of PyLith and SPECFEM3D with agreeable results.

DNS of Flow in a Low-Pressure Turbine Cascade Using a Discontinuous-Galerkin Spectral-Element Method

NASA Technical Reports Server (NTRS)

Garai, Anirban; Diosady, Laslo Tibor; Murman, Scott; Madavan, Nateri

2015-01-01

A new computational capability under development for accurate and efficient high-fidelity direct numerical simulation (DNS) and large eddy simulation (LES) of turbomachinery is described. This capability is based on an entropy-stable Discontinuous-Galerkin spectral-element approach that extends to arbitrarily high orders of spatial and temporal accuracy and is implemented in a computationally efficient manner on a modern high performance computer architecture. A validation study using this method to perform DNS of flow in a low-pressure turbine airfoil cascade are presented. Preliminary results indicate that the method captures the main features of the flow. Discrepancies between the predicted results and the experiments are likely due to the effects of freestream turbulence not being included in the simulation and will be addressed in the final paper.

A locally conservative stabilized continuous Galerkin finite element method for two-phase flow in poroelastic subsurfaces

NASA Astrophysics Data System (ADS)

Deng, Q.; Ginting, V.; McCaskill, B.; Torsu, P.

2017-10-01

We study the application of a stabilized continuous Galerkin finite element method (CGFEM) in the simulation of multiphase flow in poroelastic subsurfaces. The system involves a nonlinear coupling between the fluid pressure, subsurface's deformation, and the fluid phase saturation, and as such, we represent this coupling through an iterative procedure. Spatial discretization of the poroelastic system employs the standard linear finite element in combination with a numerical diffusion term to maintain stability of the algebraic system. Furthermore, direct calculation of the normal velocities from pressure and deformation does not entail a locally conservative field. To alleviate this drawback, we propose an element based post-processing technique through which local conservation can be established. The performance of the method is validated through several examples illustrating the convergence of the method, the effectivity of the stabilization term, and the ability to achieve locally conservative normal velocities. Finally, the efficacy of the method is demonstrated through simulations of realistic multiphase flow in poroelastic subsurfaces.

Recovery Discontinuous Galerkin Jacobian-free Newton-Krylov Method for all-speed flows

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

HyeongKae Park; Robert Nourgaliev; Vincent Mousseau

2008-07-01

There is an increasing interest to develop the next generation simulation tools for the advanced nuclear energy systems. These tools will utilize the state-of-art numerical algorithms and computer science technology in order to maximize the predictive capability, support advanced reactor designs, reduce uncertainty and increase safety margins. In analyzing nuclear energy systems, we are interested in compressible low-Mach number, high heat flux flows with a wide range of Re, Ra, and Pr numbers. Under these conditions, the focus is placed on turbulent heat transfer, in contrast to other industries whose main interest is in capturing turbulent mixing. Our objective ismore » to develop singlepoint turbulence closure models for large-scale engineering CFD code, using Direct Numerical Simulation (DNS) or Large Eddy Simulation (LES) tools, requireing very accurate and efficient numerical algorithms. The focus of this work is placed on fully-implicit, high-order spatiotemporal discretization based on the discontinuous Galerkin method solving the conservative form of the compressible Navier-Stokes equations. The method utilizes a local reconstruction procedure derived from weak formulation of the problem, which is inspired by the recovery diffusion flux algorithm of van Leer and Nomura [?] and by the piecewise parabolic reconstruction [?] in the finite volume method. The developed methodology is integrated into the Jacobianfree Newton-Krylov framework [?] to allow a fully-implicit solution of the problem.« less

A Reconstructed Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations on Arbitrary Grids

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

Hong Luo; Luqing Luo; Robert Nourgaliev

2010-09-01

A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Navier–Stokes equations on arbitrary grids. The RDG method, originally developed for the compressible Euler equations, is extended to discretize viscous and heat fluxes in the Navier–Stokes equations using a so-called inter-cell reconstruction, where a smooth solution is locally reconstructed using a least-squares method from the underlying discontinuous DG solution. Similar to the recovery-based DG (rDG) methods, this reconstructed DG method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional DG methods. Unlike rDG methods, this RDG method does not need tomore » judiciously choose a proper form of a recovered polynomial, thus is simple, flexible, and robust, and can be used on arbitrary grids. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG method is able to deliver the same accuracy as the well-known Bassi–Rebay II scheme, at a half of its computing costs for the discretization of the viscous fluxes in the Navier–Stokes equations, clearly demonstrating its superior performance over the existing DG methods for solving the compressible Navier–Stokes equations.« less

A Reconstructed Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations on Arbitrary Grids

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

Hong Luo; Luqing Luo; Robert Nourgaliev

2010-01-01

A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Navier-Stokes equations on arbitrary grids. The RDG method, originally developed for the compressible Euler equations, is extended to discretize viscous and heat fluxes in the Navier-Stokes equations using a so-called inter-cell reconstruction, where a smooth solution is locally reconstructed using a least-squares method from the underlying discontinuous DG solution. Similar to the recovery-based DG (rDG) methods, this reconstructed DG method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional DG methods. Unlike rDG methods, this RDG method does not need tomore » judiciously choose a proper form of a recovered polynomial, thus is simple, flexible, and robust, and can be used on arbitrary grids. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG method is able to deliver the same accuracy as the well-known Bassi-Rebay II scheme, at a half of its computing costs for the discretization of the viscous fluxes in the Navier-Stokes equations, clearly demonstrating its superior performance over the existing DG methods for solving the compressible Navier-Stokes equations.« less

Local Analysis of Shock Capturing Using Discontinuous Galerkin Methodology

NASA Technical Reports Server (NTRS)

Atkins, H. L.

1997-01-01

The compact form of the discontinuous Galerkin method allows for a detailed local analysis of the method in the neighborhood of the shock for a non-linear model problem. Insight gained from the analysis leads to new flux formulas that are stable and that preserve the compactness of the method. Although developed for a model equation, the flux formulas are applicable to systems such as the Euler equations. This article presents the analysis for methods with a degree up to 5. The analysis is accompanied by supporting numerical experiments using Burgers' equation and the Euler equations.

DG-IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings

DOE PAGES

Chen, Zheng; Liu, Liu; Mu, Lin

2017-05-03

In this paper, we consider the linear transport equation under diffusive scaling and with random inputs. The method is based on the generalized polynomial chaos approach in the stochastic Galerkin framework. Several theoretical aspects will be addressed. Additionally, a uniform numerical stability with respect to the Knudsen number ϵ, and a uniform in ϵ error estimate is given. For temporal and spatial discretizations, we apply the implicit–explicit scheme under the micro–macro decomposition framework and the discontinuous Galerkin method, as proposed in Jang et al. (SIAM J Numer Anal 52:2048–2072, 2014) for deterministic problem. Lastly, we provide a rigorous proof ofmore » the stochastic asymptotic-preserving (sAP) property. Extensive numerical experiments that validate the accuracy and sAP of the method are conducted.« less

Asynchronous communication in spectral-element and discontinuous Galerkin methods for atmospheric dynamics - a case study using the High-Order Methods Modeling Environment (HOMME-homme_dg_branch)

NASA Astrophysics Data System (ADS)

Jamroz, Benjamin F.; Klöfkorn, Robert

2016-08-01

The scalability of computational applications on current and next-generation supercomputers is increasingly limited by the cost of inter-process communication. We implement non-blocking asynchronous communication in the High-Order Methods Modeling Environment for the time integration of the hydrostatic fluid equations using both the spectral-element and discontinuous Galerkin methods. This allows the overlap of computation with communication, effectively hiding some of the costs of communication. A novel detail about our approach is that it provides some data movement to be performed during the asynchronous communication even in the absence of other computations. This method produces significant performance and scalability gains in large-scale simulations.

The lowest-order weak Galerkin finite element method for the Darcy equation on quadrilateral and hybrid meshes

NASA Astrophysics Data System (ADS)

Liu, Jiangguo; Tavener, Simon; Wang, Zhuoran

2018-04-01

This paper investigates the lowest-order weak Galerkin finite element method for solving the Darcy equation on quadrilateral and hybrid meshes consisting of quadrilaterals and triangles. In this approach, the pressure is approximated by constants in element interiors and on edges. The discrete weak gradients of these constant basis functions are specified in local Raviart-Thomas spaces, specifically RT0 for triangles and unmapped RT[0] for quadrilaterals. These discrete weak gradients are used to approximate the classical gradient when solving the Darcy equation. The method produces continuous normal fluxes and is locally mass-conservative, regardless of mesh quality, and has optimal order convergence in pressure, velocity, and normal flux, when the quadrilaterals are asymptotically parallelograms. Implementation is straightforward and results in symmetric positive-definite discrete linear systems. We present numerical experiments and comparisons with other existing methods.

Semidiscrete Galerkin modelling of compressible viscous flow past a circular cone at incidence. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Meade, Andrew James, Jr.

1989-01-01

A numerical study of the laminar and compressible boundary layer, about a circular cone in a supersonic free stream, is presented. It is thought that if accurate and efficient numerical schemes can be produced to solve the boundary layer equations, they can be joined to numerical codes that solve the inviscid outer flow. The combination of these numerical codes is competitive with the accurate, but computationally expensive, Navier-Stokes schemes. The primary goal is to develop a finite element method for the calculation of 3-D compressible laminar boundary layer about a yawed cone. The proposed method can, in principle, be extended to apply to the 3-D boundary layer of pointed bodies of arbitrary cross section. The 3-D boundary layer equations governing supersonic free stream flow about a cone are examined. The 3-D partial differential equations are reduced to 2-D integral equations by applying the Howarth, Mangler, Crocco transformations, a linear relation between viscosity, and a Blasius-type of similarity variable. This is equivalent to a Dorodnitsyn-type formulation. The reduced equations are independent of density and curvature effects, and resemble the weak form of the 2-D incompressible boundary layer equations in Cartesian coordinates. In addition the coordinate normal to the wall has been stretched, which reduces the gradients across the layer and provides high resolution near the surface. Utilizing the parabolic nature of the boundary layer equations, a finite element method is applied to the Dorodnitsyn formulation. The formulation is presented in a Petrov-Galerkin finite element form and discretized across the layer using linear interpolation functions. The finite element discretization yields a system of ordinary differential equations in the circumferential direction. The circumferential derivatives are solved by an implicit and noniterative finite difference marching scheme. Solutions are presented for a 15 deg half angle cone at angles of attack of

Bound-preserving modified exponential Runge-Kutta discontinuous Galerkin methods for scalar hyperbolic equations with stiff source terms

NASA Astrophysics Data System (ADS)

Huang, Juntao; Shu, Chi-Wang

2018-05-01

In this paper, we develop bound-preserving modified exponential Runge-Kutta (RK) discontinuous Galerkin (DG) schemes to solve scalar hyperbolic equations with stiff source terms by extending the idea in Zhang and Shu [43]. Exponential strong stability preserving (SSP) high order time discretizations are constructed and then modified to overcome the stiffness and preserve the bound of the numerical solutions. It is also straightforward to extend the method to two dimensions on rectangular and triangular meshes. Even though we only discuss the bound-preserving limiter for DG schemes, it can also be applied to high order finite volume schemes, such as weighted essentially non-oscillatory (WENO) finite volume schemes as well.

Numerical algorithms based on Galerkin methods for the modeling of reactive interfaces in photoelectrochemical (PEC) solar cells

NASA Astrophysics Data System (ADS)

Harmon, Michael; Gamba, Irene M.; Ren, Kui

2016-12-01

This work concerns the numerical solution of a coupled system of self-consistent reaction-drift-diffusion-Poisson equations that describes the macroscopic dynamics of charge transport in photoelectrochemical (PEC) solar cells with reactive semiconductor and electrolyte interfaces. We present three numerical algorithms, mainly based on a mixed finite element and a local discontinuous Galerkin method for spatial discretization, with carefully chosen numerical fluxes, and implicit-explicit time stepping techniques, for solving the time-dependent nonlinear systems of partial differential equations. We perform computational simulations under various model parameters to demonstrate the performance of the proposed numerical algorithms as well as the impact of these parameters on the solution to the model.

Hybrid Fourier pseudospectral/discontinuous Galerkin time-domain method for wave propagation

NASA Astrophysics Data System (ADS)

Pagán Muñoz, Raúl; Hornikx, Maarten

2017-11-01

The Fourier Pseudospectral time-domain (Fourier PSTD) method was shown to be an efficient way of modelling acoustic propagation problems as described by the linearized Euler equations (LEE), but is limited to real-valued frequency independent boundary conditions and predominantly staircase-like boundary shapes. This paper presents a hybrid approach to solve the LEE, coupling Fourier PSTD with a nodal Discontinuous Galerkin (DG) method. DG exhibits almost no restrictions with respect to geometrical complexity or boundary conditions. The aim of this novel method is to allow the computation of complex geometries and to be a step towards the implementation of frequency dependent boundary conditions by using the benefits of DG at the boundaries, while keeping the efficient Fourier PSTD in the bulk of the domain. The hybridization approach is based on conformal meshes to avoid spatial interpolation of the DG solutions when transferring values from DG to Fourier PSTD, while the data transfer from Fourier PSTD to DG is done utilizing spectral interpolation of the Fourier PSTD solutions. The accuracy of the hybrid approach is presented for one- and two-dimensional acoustic problems and the main sources of error are investigated. It is concluded that the hybrid methodology does not introduce significant errors compared to the Fourier PSTD stand-alone solver. An example of a cylinder scattering problem is presented and accurate results have been obtained when using the proposed approach. Finally, no instabilities were found during long-time calculation using the current hybrid methodology on a two-dimensional domain.

A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials

NASA Astrophysics Data System (ADS)

Sun, Zheng; Carrillo, José A.; Shu, Chi-Wang

2018-01-01

We consider a class of time-dependent second order partial differential equations governed by a decaying entropy. The solution usually corresponds to a density distribution, hence positivity (non-negativity) is expected. This class of problems covers important cases such as Fokker-Planck type equations and aggregation models, which have been studied intensively in the past decades. In this paper, we design a high order discontinuous Galerkin method for such problems. If the interaction potential is not involved, or the interaction is defined by a smooth kernel, our semi-discrete scheme admits an entropy inequality on the discrete level. Furthermore, by applying the positivity-preserving limiter, our fully discretized scheme produces non-negative solutions for all cases under a time step constraint. Our method also applies to two dimensional problems on Cartesian meshes. Numerical examples are given to confirm the high order accuracy for smooth test cases and to demonstrate the effectiveness for preserving long time asymptotics.

Studies of Plasma Instabilities using Unstructured Discontinuous Galerkin Method with the Two-Fluid Plasma Model

NASA Astrophysics Data System (ADS)

Song, Yang; Srinivasan, Bhuvana

2017-10-01

The discontinuous Galerkin (DG) method has the advantage of resolving shocks and sharp gradients that occur in neutral fluids and plasmas. An unstructured DG code has been developed in this work to study plasma instabilities using the two-fluid plasma model. Unstructured meshes are known to produce small and randomized grid errors compared to traditional structured meshes. Computational tests for Rayleigh-Taylor instabilities in radially-converging flows are performed using the MHD model. Choice of grid geometry is not obvious for simulations of instabilities in these circular configurations. Comparisons of the effects for different grids are made. A 2D magnetic nozzle simulation using the two-fluid plasma model is also performed. A vacuum boundary condition technique is applied to accurately solve the Riemann problem on the edge of the plume.

A velocity-pressure integrated, mixed interpolation, Galerkin finite element method for high Reynolds number laminar flows

NASA Technical Reports Server (NTRS)

Kim, Sang-Wook

1988-01-01

A velocity-pressure integrated, mixed interpolation, Galerkin finite element method for the Navier-Stokes equations is presented. In the method, the velocity variables were interpolated using complete quadratic shape functions and the pressure was interpolated using linear shape functions. For the two dimensional case, the pressure is defined on a triangular element which is contained inside the complete biquadratic element for velocity variables; and for the three dimensional case, the pressure is defined on a tetrahedral element which is again contained inside the complete tri-quadratic element. Thus the pressure is discontinuous across the element boundaries. Example problems considered include: a cavity flow for Reynolds number of 400 through 10,000; a laminar backward facing step flow; and a laminar flow in a square duct of strong curvature. The computational results compared favorable with those of the finite difference methods as well as experimental data available. A finite elememt computer program for incompressible, laminar flows is presented.

Magnetic resonance electrical impedance tomography (MREIT) based on the solution of the convection equation using FEM with stabilization.

PubMed

Oran, Omer Faruk; Ider, Yusuf Ziya

2012-08-21

Most algorithms for magnetic resonance electrical impedance tomography (MREIT) concentrate on reconstructing the internal conductivity distribution of a conductive object from the Laplacian of only one component of the magnetic flux density (∇²B(z)) generated by the internal current distribution. In this study, a new algorithm is proposed to solve this ∇²B(z)-based MREIT problem which is mathematically formulated as the steady-state scalar pure convection equation. Numerical methods developed for the solution of the more general convection-diffusion equation are utilized. It is known that the solution of the pure convection equation is numerically unstable if sharp variations of the field variable (in this case conductivity) exist or if there are inconsistent boundary conditions. Various stabilization techniques, based on introducing artificial diffusion, are developed to handle such cases and in this study the streamline upwind Petrov-Galerkin (SUPG) stabilization method is incorporated into the Galerkin weighted residual finite element method (FEM) to numerically solve the MREIT problem. The proposed algorithm is tested with simulated and also experimental data from phantoms. Successful conductivity reconstructions are obtained by solving the related convection equation using the Galerkin weighted residual FEM when there are no sharp variations in the actual conductivity distribution. However, when there is noise in the magnetic flux density data or when there are sharp variations in conductivity, it is found that SUPG stabilization is beneficial.

A new family of stable elements for the Stokes problem based on a mixed Galerkin/least-squares finite element formulation

NASA Technical Reports Server (NTRS)

Franca, Leopoldo P.; Loula, Abimael F. D.; Hughes, Thomas J. R.; Miranda, Isidoro

1989-01-01

Adding to the classical Hellinger-Reissner formulation, a residual form of the equilibrium equation, a new Galerkin/least-squares finite element method is derived. It fits within the framework of a mixed finite element method and is stable for rather general combinations of stress and velocity interpolations, including equal-order discontinuous stress and continuous velocity interpolations which are unstable within the Galerkin approach. Error estimates are presented based on a generalization of the Babuska-Brezzi theory. Numerical results (not presented herein) have confirmed these estimates as well as the good accuracy and stability of the method.

Recovery Discontinuous Galerkin Jacobian-Free Newton-Krylov Method for All-Speed Flows

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

HyeongKae Park; Robert Nourgaliev; Vincent Mousseau

2008-07-01

A novel numerical algorithm (rDG-JFNK) for all-speed fluid flows with heat conduction and viscosity is introduced. The rDG-JFNK combines the Discontinuous Galerkin spatial discretization with the implicit Runge-Kutta time integration under the Jacobian-free Newton-Krylov framework. We solve fully-compressible Navier-Stokes equations without operator-splitting of hyperbolic, diffusion and reaction terms, which enables fully-coupled high-order temporal discretization. The stability constraint is removed due to the L-stable Explicit, Singly Diagonal Implicit Runge-Kutta (ESDIRK) scheme. The governing equations are solved in the conservative form, which allows one to accurately compute shock dynamics, as well as low-speed flows. For spatial discretization, we develop a “recovery” familymore » of DG, exhibiting nearly-spectral accuracy. To precondition the Krylov-based linear solver (GMRES), we developed an “Operator-Split”-(OS) Physics Based Preconditioner (PBP), in which we transform/simplify the fully-coupled system to a sequence of segregated scalar problems, each can be solved efficiently with Multigrid method. Each scalar problem is designed to target/cluster eigenvalues of the Jacobian matrix associated with a specific physics.« less

Parachordatortilis n. g. (Nematoda: Spirurida: Acuariidae), with a redescription of P. mathevossianae (Petrov & Chertkova, 1950) n. comb., a parasite of Falco tinnunculus L. (Aves: Falconiformes).

PubMed

Mutafchiev, Yasen; Santoro, Mario; Georgiev, Boyko B

2010-07-01

Parachordatortilis n. g. is proposed as a monotypic genus for Parachordatortilis mathevossianae (Petrov & Chertkova, 1950) n. comb. (Spirurida, Acuariidae), previously Dispharynx mathevossianae Petrov & Chertkova, 1950. The new genus can be distinguished from all other acuariid genera by having cordons with a sinistral torsion. Parachordatortilis possesses cordons consisting of two rows of delicate cuticular plates delimiting a longitudinal canal. This pattern of cordon structure suggests that Parachordatortilis is closely related to Dispharynx Railliet, Henry & Sisoff, 1912, Synhimantus Railliet, Henry & Sisoff, 1912 and Chordatortilis Machado de Mendonça & Olivera de Rodrigues, 1985. P. mathevossianae is redescribed on the basis of light and scanning electron microscopical observations of specimens collected from a common kestrel Falco tinnunculus L. in southern Italy.

The nonlinear Galerkin method: A multi-scale method applied to the simulation of homogeneous turbulent flows

NASA Technical Reports Server (NTRS)

Debussche, A.; Dubois, T.; Temam, R.

1993-01-01

Using results of Direct Numerical Simulation (DNS) in the case of two-dimensional homogeneous isotropic flows, the behavior of the small and large scales of Kolmogorov like flows at moderate Reynolds numbers are first analyzed in detail. Several estimates on the time variations of the small eddies and the nonlinear interaction terms were derived; those terms play the role of the Reynolds stress tensor in the case of LES. Since the time step of a numerical scheme is determined as a function of the energy-containing eddies of the flow, the variations of the small scales and of the nonlinear interaction terms over one iteration can become negligible by comparison with the accuracy of the computation. Based on this remark, a multilevel scheme which treats differently the small and the large eddies was proposed. Using mathematical developments, estimates of all the parameters involved in the algorithm, which then becomes a completely self-adaptive procedure were derived. Finally, realistic simulations of (Kolmorov like) flows over several eddy-turnover times were performed. The results are analyzed in detail and a parametric study of the nonlinear Galerkin method is performed.

Bound-Preserving Discontinuous Galerkin Methods for Conservative Phase Space Advection in Curvilinear Coordinates

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

Mezzacappa, Anthony; Endeve, Eirik; Hauck, Cory D.

We extend the positivity-preserving method of Zhang & Shu [49] to simulate the advection of neutral particles in phase space using curvilinear coordinates. The ability to utilize these coordinates is important for non-equilibrium transport problems in general relativity and also in science and engineering applications with specific geometries. The method achieves high-order accuracy using Discontinuous Galerkin (DG) discretization of phase space and strong stabilitypreserving, Runge-Kutta (SSP-RK) time integration. Special care in taken to ensure that the method preserves strict bounds for the phase space distribution function f; i.e., f ϵ [0, 1]. The combination of suitable CFL conditions and themore » use of the high-order limiter proposed in [49] is su cient to ensure positivity of the distribution function. However, to ensure that the distribution function satisfies the upper bound, the discretization must, in addition, preserve the divergencefree property of the phase space ow. Proofs that highlight the necessary conditions are presented for general curvilinear coordinates, and the details of these conditions are worked out for some commonly used coordinate systems (i.e., spherical polar spatial coordinates in spherical symmetry and cylindrical spatial coordinates in axial symmetry, both with spherical momentum coordinates). Results from numerical experiments - including one example in spherical symmetry adopting the Schwarzschild metric - demonstrate that the method achieves high-order accuracy and that the distribution function satisfies the maximum principle.« less

Asynchronous communication in spectral-element and discontinuous Galerkin methods for atmospheric dynamics – a case study using the High-Order Methods Modeling Environment (HOMME-homme_dg_branch)

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

Jamroz, Benjamin F.; Klofkorn, Robert

The scalability of computational applications on current and next-generation supercomputers is increasingly limited by the cost of inter-process communication. We implement non-blocking asynchronous communication in the High-Order Methods Modeling Environment for the time integration of the hydrostatic fluid equations using both the spectral-element and discontinuous Galerkin methods. This allows the overlap of computation with communication, effectively hiding some of the costs of communication. A novel detail about our approach is that it provides some data movement to be performed during the asynchronous communication even in the absence of other computations. This method produces significant performance and scalability gains in large-scalemore » simulations.« less

Asynchronous communication in spectral-element and discontinuous Galerkin methods for atmospheric dynamics – a case study using the High-Order Methods Modeling Environment (HOMME-homme_dg_branch)

DOE PAGES

Jamroz, Benjamin F.; Klofkorn, Robert

2016-08-26

The scalability of computational applications on current and next-generation supercomputers is increasingly limited by the cost of inter-process communication. We implement non-blocking asynchronous communication in the High-Order Methods Modeling Environment for the time integration of the hydrostatic fluid equations using both the spectral-element and discontinuous Galerkin methods. This allows the overlap of computation with communication, effectively hiding some of the costs of communication. A novel detail about our approach is that it provides some data movement to be performed during the asynchronous communication even in the absence of other computations. This method produces significant performance and scalability gains in large-scalemore » simulations.« less

Numerical Analysis of an H 1-Galerkin Mixed Finite Element Method for Time Fractional Telegraph Equation

PubMed Central

Wang, Jinfeng; Zhao, Meng; Zhang, Min; Liu, Yang; Li, Hong

2014-01-01

We discuss and analyze an H 1-Galerkin mixed finite element (H 1-GMFE) method to look for the numerical solution of time fractional telegraph equation. We introduce an auxiliary variable to reduce the original equation into lower-order coupled equations and then formulate an H 1-GMFE scheme with two important variables. We discretize the Caputo time fractional derivatives using the finite difference methods and approximate the spatial direction by applying the H 1-GMFE method. Based on the discussion on the theoretical error analysis in L 2-norm for the scalar unknown and its gradient in one dimensional case, we obtain the optimal order of convergence in space-time direction. Further, we also derive the optimal error results for the scalar unknown in H 1-norm. Moreover, we derive and analyze the stability of H 1-GMFE scheme and give the results of a priori error estimates in two- or three-dimensional cases. In order to verify our theoretical analysis, we give some results of numerical calculation by using the Matlab procedure. PMID:25184148

New algorithms for solving third- and fifth-order two point boundary value problems based on nonsymmetric generalized Jacobi Petrov–Galerkin method

PubMed Central

Doha, E.H.; Abd-Elhameed, W.M.; Youssri, Y.H.

2014-01-01

Two families of certain nonsymmetric generalized Jacobi polynomials with negative integer indexes are employed for solving third- and fifth-order two point boundary value problems governed by homogeneous and nonhomogeneous boundary conditions using a dual Petrov–Galerkin method. The idea behind our method is to use trial functions satisfying the underlying boundary conditions of the differential equations and the test functions satisfying the dual boundary conditions. The resulting linear systems from the application of our method are specially structured and they can be efficiently inverted. The use of generalized Jacobi polynomials simplify the theoretical and numerical analysis of the method and also leads to accurate and efficient numerical algorithms. The presented numerical results indicate that the proposed numerical algorithms are reliable and very efficient. PMID:26425358

Study on the wind field and pollutant dispersion in street canyons using a stable numerical method.

PubMed

Xia, Ji-Yang; Leung, Dennis Y C

2005-01-01

A stable finite element method for the time dependent Navier-Stokes equations was used for studying the wind flow and pollutant dispersion within street canyons. A three-step fractional method was used to solve the velocity field and the pressure field separately from the governing equations. The Streamline Upwind Petrov-Galerkin (SUPG) method was used to get stable numerical results. Numerical oscillation was minimized and satisfactory results can be obtained for flows at high Reynolds numbers. Simulating the flow over a square cylinder within a wide range of Reynolds numbers validates the wind field model. The Strouhal numbers obtained from the numerical simulation had a good agreement with those obtained from experiment. The wind field model developed in the present study is applied to simulate more complex flow phenomena in street canyons with two different building configurations. The results indicated that the flow at rooftop of buildings might not be assumed parallel to the ground as some numerical modelers did. A counter-clockwise rotating vortex may be found in street canyons with an inflow from the left to right. In addition, increasing building height can increase velocity fluctuations in the street canyon under certain circumstances, which facilitate pollutant dispersion. At high Reynolds numbers, the flow regimes in street canyons do not change with inflow velocity.

Model Adaptation in Parametric Space for POD-Galerkin Models

NASA Astrophysics Data System (ADS)

Gao, Haotian; Wei, Mingjun

2017-11-01

The development of low-order POD-Galerkin models is largely motivated by the expectation to use the model developed with a set of parameters at their native values to predict the dynamic behaviors of the same system under different parametric values, in other words, a successful model adaptation in parametric space. However, most of time, even small deviation of parameters from their original value may lead to large deviation or unstable results. It has been shown that adding more information (e.g. a steady state, mean value of a different unsteady state, or an entire different set of POD modes) may improve the prediction of flow with other parametric states. For a simple case of the flow passing a fixed cylinder, an orthogonal mean mode at a different Reynolds number may stabilize the POD-Galerkin model when Reynolds number is changed. For a more complicated case of the flow passing an oscillatory cylinder, a global POD-Galerkin model is first applied to handle the moving boundaries, then more information (e.g. more POD modes) is required to predicate the flow under different oscillatory frequencies. Supported by ARL.

A Runge-Kutta discontinuous Galerkin approach to solve reactive flows: The hyperbolic operator

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

Billet, G., E-mail: billet@onera.f; Ryan, J., E-mail: ryan@onera.f

2011-02-20

A Runge-Kutta discontinuous Galerkin method to solve the hyperbolic part of reactive Navier-Stokes equations written in conservation form is presented. Complex thermodynamics laws are taken into account. Particular care has been taken to solve the stiff gaseous interfaces correctly with no restrictive hypothesis. 1D and 2D test cases are presented.

Runge-Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes

NASA Astrophysics Data System (ADS)

Zhu, Jun; Zhong, Xinghui; Shu, Chi-Wang; Qiu, Jianxian

2013-09-01

In this paper we generalize a new type of limiters based on the weighted essentially non-oscillatory (WENO) finite volume methodology for the Runge-Kutta discontinuous Galerkin (RKDG) methods solving nonlinear hyperbolic conservation laws, which were recently developed in [32] for structured meshes, to two-dimensional unstructured triangular meshes. The key idea of such limiters is to use the entire polynomials of the DG solutions from the troubled cell and its immediate neighboring cells, and then apply the classical WENO procedure to form a convex combination of these polynomials based on smoothness indicators and nonlinear weights, with suitable adjustments to guarantee conservation. The main advantage of this new limiter is its simplicity in implementation, especially for the unstructured meshes considered in this paper, as only information from immediate neighbors is needed and the usage of complicated geometric information of the meshes is largely avoided. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good performance of this procedure.

Seismic waves in heterogeneous material: subcell resolution of the discontinuous Galerkin method

NASA Astrophysics Data System (ADS)

Castro, Cristóbal E.; Käser, Martin; Brietzke, Gilbert B.

2010-07-01

We present an important extension of the arbitrary high-order discontinuous Galerkin (DG) finite-element method to model 2-D elastic wave propagation in highly heterogeneous material. In this new approach we include space-variable coefficients to describe smooth or discontinuous material variations inside each element using the same numerical approximation strategy as for the velocity-stress variables in the formulation of the elastic wave equation. The combination of the DG method with a time integration scheme based on the solution of arbitrary accuracy derivatives Riemann problems still provides an explicit, one-step scheme which achieves arbitrary high-order accuracy in space and time. Compared to previous formulations the new scheme contains two additional terms in the form of volume integrals. We show that the increasing computational cost per element can be overcompensated due to the improved material representation inside each element as coarser meshes can be used which reduces the total number of elements and therefore computational time to reach a desired error level. We confirm the accuracy of the proposed scheme performing convergence tests and several numerical experiments considering smooth and highly heterogeneous material. As the approximation of the velocity and stress variables in the wave equation and of the material properties in the model can be chosen independently, we investigate the influence of the polynomial material representation on the accuracy of the synthetic seismograms with respect to computational cost. Moreover, we study the behaviour of the new method on strong material discontinuities, in the case where the mesh is not aligned with such a material interface. In this case second-order linear material approximation seems to be the best choice, with higher-order intra-cell approximation leading to potential instable behaviour. For all test cases we validate our solution against the well-established standard fourth-order finite

Semi-analytical discontinuous Galerkin finite element method for the calculation of dispersion properties of guided waves in plates.

PubMed

Hebaz, Salah-Eddine; Benmeddour, Farouk; Moulin, Emmanuel; Assaad, Jamal

2018-01-01

The development of reliable guided waves inspection systems is conditioned by an accurate knowledge of their dispersive properties. The semi-analytical finite element method has been proven to be very practical for modeling wave propagation in arbitrary cross-section waveguides. However, when it comes to computations on complex geometries to a given accuracy, it still has a major drawback: the high consumption of resources. Recently, discontinuous Galerkin finite element method (DG-FEM) has been found advantageous over the standard finite element method when applied as well in the frequency domain. In this work, a high-order method for the computation of Lamb mode characteristics in plates is proposed. The problem is discretised using a class of DG-FEM, namely, the interior penalty methods family. The analytical validation is performed through the homogeneous isotropic case with traction-free boundary conditions. Afterwards, functionally graded material plates are analysed and a numerical example is presented. It was found that the obtained results are in good agreement with those found in the literature.

Overset meshing coupled with hybridizable discontinuous Galerkin finite elements

DOE PAGES

Kauffman, Justin A.; Sheldon, Jason P.; Miller, Scott T.

2017-03-01

We introduce the use of hybridizable discontinuous Galerkin (HDG) finite element methods on overlapping (overset) meshes. Overset mesh methods are advantageous for solving problems on complex geometrical domains. We also combine geometric flexibility of overset methods with the advantages of HDG methods: arbitrarily high-order accuracy, reduced size of the global discrete problem, and the ability to solve elliptic, parabolic, and/or hyperbolic problems with a unified form of discretization. This approach to developing the ‘overset HDG’ method is to couple the global solution from one mesh to the local solution on the overset mesh. We present numerical examples for steady convection–diffusionmore » and static elasticity problems. The examples demonstrate optimal order convergence in all primal fields for an arbitrary amount of overlap of the underlying meshes.« less

A stable and high-order accurate discontinuous Galerkin based splitting method for the incompressible Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Piatkowski, Marian; Müthing, Steffen; Bastian, Peter

2018-03-01

In this paper we consider discontinuous Galerkin (DG) methods for the incompressible Navier-Stokes equations in the framework of projection methods. In particular we employ symmetric interior penalty DG methods within the second-order rotational incremental pressure correction scheme. The major focus of the paper is threefold: i) We propose a modified upwind scheme based on the Vijayasundaram numerical flux that has favourable properties in the context of DG. ii) We present a novel postprocessing technique in the Helmholtz projection step based on H (div) reconstruction of the pressure correction that is computed locally, is a projection in the discrete setting and ensures that the projected velocity satisfies the discrete continuity equation exactly. As a consequence it also provides local mass conservation of the projected velocity. iii) Numerical results demonstrate the properties of the scheme for different polynomial degrees applied to two-dimensional problems with known solution as well as large-scale three-dimensional problems. In particular we address second-order convergence in time of the splitting scheme as well as its long-time stability.

A Parallel Implicit Reconstructed Discontinuous Galerkin Method for Compressible Flows on Hybrid Grids

NASA Astrophysics Data System (ADS)

Xia, Yidong

The objective this work is to develop a parallel, implicit reconstructed discontinuous Galerkin (RDG) method using Taylor basis for the solution of the compressible Navier-Stokes equations on 3D hybrid grids. This third-order accurate RDG method is based on a hierarchical weighed essentially non- oscillatory reconstruction scheme, termed as HWENO(P1P 2) to indicate that a quadratic polynomial solution is obtained from the underlying linear polynomial DG solution via a hierarchical WENO reconstruction. The HWENO(P1P2) is designed not only to enhance the accuracy of the underlying DG(P1) method but also to ensure non-linear stability of the RDG method. In this reconstruction scheme, a quadratic polynomial (P2) solution is first reconstructed using a least-squares approach from the underlying linear (P1) discontinuous Galerkin solution. The final quadratic solution is then obtained using a Hermite WENO reconstruction, which is necessary to ensure the linear stability of the RDG method on 3D unstructured grids. The first derivatives of the quadratic polynomial solution are then reconstructed using a WENO reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities, thus ensuring the non-linear stability of the RDG method. The parallelization in the RDG method is based on a message passing interface (MPI) programming paradigm, where the METIS library is used for the partitioning of a mesh into subdomain meshes of approximately the same size. Both multi-stage explicit Runge-Kutta and simple implicit backward Euler methods are implemented for time advancement in the RDG method. In the implicit method, three approaches: analytical differentiation, divided differencing (DD), and automatic differentiation (AD) are developed and implemented to obtain the resulting flux Jacobian matrices. The automatic differentiation is a set of techniques based on the mechanical application of the chain rule to obtain derivatives of a function given as

Introduction to COFFE: The Next-Generation HPCMP CREATE-AV CFD Solver

NASA Technical Reports Server (NTRS)

Glasby, Ryan S.; Erwin, J. Taylor; Stefanski, Douglas L.; Allmaras, Steven R.; Galbraith, Marshall C.; Anderson, W. Kyle; Nichols, Robert H.

2016-01-01

HPCMP CREATE-AV Conservative Field Finite Element (COFFE) is a modular, extensible, robust numerical solver for the Navier-Stokes equations that invokes modularity and extensibility from its first principles. COFFE implores a flexible, class-based hierarchy that provides a modular approach consisting of discretization, physics, parallelization, and linear algebra components. These components are developed with modern software engineering principles to ensure ease of uptake from a user's or developer's perspective. The Streamwise Upwind/Petrov-Galerkin (SU/PG) method is utilized to discretize the compressible Reynolds-Averaged Navier-Stokes (RANS) equations tightly coupled with a variety of turbulence models. The mathematics and the philosophy of the methodology that makes up COFFE are presented.

A fully-coupled discontinuous Galerkin spectral element method for two-phase flow in petroleum reservoirs

NASA Astrophysics Data System (ADS)

Taneja, Ankur; Higdon, Jonathan

2018-01-01

A high-order spectral element discontinuous Galerkin method is presented for simulating immiscible two-phase flow in petroleum reservoirs. The governing equations involve a coupled system of strongly nonlinear partial differential equations for the pressure and fluid saturation in the reservoir. A fully implicit method is used with a high-order accurate time integration using an implicit Rosenbrock method. Numerical tests give the first demonstration of high order hp spatial convergence results for multiphase flow in petroleum reservoirs with industry standard relative permeability models. High order convergence is shown formally for spectral elements with up to 8th order polynomials for both homogeneous and heterogeneous permeability fields. Numerical results are presented for multiphase fluid flow in heterogeneous reservoirs with complex geometric or geologic features using up to 11th order polynomials. Robust, stable simulations are presented for heterogeneous geologic features, including globally heterogeneous permeability fields, anisotropic permeability tensors, broad regions of low-permeability, high-permeability channels, thin shale barriers and thin high-permeability fractures. A major result of this paper is the demonstration that the resolution of the high order spectral element method may be exploited to achieve accurate results utilizing a simple cartesian mesh for non-conforming geological features. Eliminating the need to mesh to the boundaries of geological features greatly simplifies the workflow for petroleum engineers testing multiple scenarios in the face of uncertainty in the subsurface geology.

A New Runge-Kutta Discontinuous Galerkin Method with Conservation Constraint to Improve CFL Condition for Solving Conservation Laws

PubMed Central

Xu, Zhiliang; Chen, Xu-Yan; Liu, Yingjie

2014-01-01

We present a new formulation of the Runge-Kutta discontinuous Galerkin (RKDG) method [9, 8, 7, 6] for solving conservation Laws with increased CFL numbers. The new formulation requires the computed RKDG solution in a cell to satisfy additional conservation constraint in adjacent cells and does not increase the complexity or change the compactness of the RKDG method. Numerical computations for solving one-dimensional and two-dimensional scalar and systems of nonlinear hyperbolic conservation laws are performed with approximate solutions represented by piecewise quadratic and cubic polynomials, respectively. The hierarchical reconstruction [17, 33] is applied as a limiter to eliminate spurious oscillations in discontinuous solutions. From both numerical experiments and the analytic estimate of the CFL number of the newly formulated method, we find that: 1) this new formulation improves the CFL number over the original RKDG formulation by at least three times or more and thus reduces the overall computational cost; and 2) the new formulation essentially does not compromise the resolution of the numerical solutions of shock wave problems compared with ones computed by the RKDG method. PMID:25414520

Discontinuous Galerkin method for multicomponent chemically reacting flows and combustion

NASA Astrophysics Data System (ADS)

Lv, Yu; Ihme, Matthias

2014-08-01

This paper presents the development of a discontinuous Galerkin (DG) method for application to chemically reacting flows in subsonic and supersonic regimes under the consideration of variable thermo-viscous-diffusive transport properties, detailed and stiff reaction chemistry, and shock capturing. A hybrid-flux formulation is developed for treatment of the convective fluxes, combining a conservative Riemann-solver and an extended double-flux scheme. A computationally efficient splitting scheme is proposed, in which advection and diffusion operators are solved in the weak form, and the chemically stiff substep is advanced in the strong form using a time-implicit scheme. The discretization of the viscous-diffusive transport terms follows the second form of Bassi and Rebay, and the WENO-based limiter due to Zhong and Shu is extended to multicomponent systems. Boundary conditions are developed for subsonic and supersonic flow conditions, and the algorithm is coupled to thermochemical libraries to account for detailed reaction chemistry and complex transport. The resulting DG method is applied to a series of test cases of increasing physico-chemical complexity. Beginning with one- and two-dimensional multispecies advection and shock-fluid interaction problems, computational efficiency, convergence, and conservation properties are demonstrated. This study is followed by considering a series of detonation and supersonic combustion problems to investigate the convergence-rate and the shock-capturing capability in the presence of one- and multistep reaction chemistry. The DG algorithm is then applied to diffusion-controlled deflagration problems. By examining convergence properties for polynomial order and spatial resolution, and comparing these with second-order finite-volume solutions, it is shown that optimal convergence is achieved and that polynomial refinement provides advantages in better resolving the localized flame structure and complex flow-field features

Discontinuous Galerkin algorithms for fully kinetic plasmas

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

Juno, J.; Hakim, A.; TenBarge, J.

Here, we present a new algorithm for the discretization of the non-relativistic Vlasov–Maxwell system of equations for the study of plasmas in the kinetic regime. Using the discontinuous Galerkin finite element method for the spatial discretization, we obtain a high order accurate solution for the plasma's distribution function. Time stepping for the distribution function is done explicitly with a third order strong-stability preserving Runge–Kutta method. Since the Vlasov equation in the Vlasov–Maxwell system is a high dimensional transport equation, up to six dimensions plus time, we take special care to note various features we have implemented to reduce the costmore » while maintaining the integrity of the solution, including the use of a reduced high-order basis set. A series of benchmarks, from simple wave and shock calculations, to a five dimensional turbulence simulation, are presented to verify the efficacy of our set of numerical methods, as well as demonstrate the power of the implemented features.« less

Discontinuous Galerkin algorithms for fully kinetic plasmas

DOE PAGES

Juno, J.; Hakim, A.; TenBarge, J.; ...

2017-10-10

Here, we present a new algorithm for the discretization of the non-relativistic Vlasov–Maxwell system of equations for the study of plasmas in the kinetic regime. Using the discontinuous Galerkin finite element method for the spatial discretization, we obtain a high order accurate solution for the plasma's distribution function. Time stepping for the distribution function is done explicitly with a third order strong-stability preserving Runge–Kutta method. Since the Vlasov equation in the Vlasov–Maxwell system is a high dimensional transport equation, up to six dimensions plus time, we take special care to note various features we have implemented to reduce the costmore » while maintaining the integrity of the solution, including the use of a reduced high-order basis set. A series of benchmarks, from simple wave and shock calculations, to a five dimensional turbulence simulation, are presented to verify the efficacy of our set of numerical methods, as well as demonstrate the power of the implemented features.« less

An arbitrary-order Runge–Kutta discontinuous Galerkin approach to reinitialization for banded conservative level sets

DOE PAGES

Jibben, Zechariah Joel; Herrmann, Marcus

2017-08-24

Here, we present a Runge-Kutta discontinuous Galerkin method for solving conservative reinitialization in the context of the conservative level set method. This represents an extension of the method recently proposed by Owkes and Desjardins [21], by solving the level set equations on the refined level set grid and projecting all spatially-dependent variables into the full basis used by the discontinuous Galerkin discretization. By doing so, we achieve the full k+1 order convergence rate in the L1 norm of the level set field predicted for RKDG methods given kth degree basis functions when the level set profile thickness is held constantmore » with grid refinement. Shape and volume errors for the 0.5-contour of the level set, on the other hand, are found to converge between first and second order. We show a variety of test results, including the method of manufactured solutions, reinitialization of a circle and sphere, Zalesak's disk, and deforming columns and spheres, all showing substantial improvements over the high-order finite difference traditional level set method studied for example by Herrmann. We also demonstrate the need for kth order accurate normal vectors, as lower order normals are found to degrade the convergence rate of the method.« less

Explicit filtering in large eddy simulation using a discontinuous Galerkin method

NASA Astrophysics Data System (ADS)

Brazell, Matthew J.

The discontinuous Galerkin (DG) method is a formulation of the finite element method (FEM). DG provides the ability for a high order of accuracy in complex geometries, and allows for highly efficient parallelization algorithms. These attributes make the DG method attractive for solving the Navier-Stokes equations for large eddy simulation (LES). The main goal of this work is to investigate the feasibility of adopting an explicit filter in the numerical solution of the Navier-Stokes equations with DG. Explicit filtering has been shown to increase the numerical stability of under-resolved simulations and is needed for LES with dynamic sub-grid scale (SGS) models. The explicit filter takes advantage of DG's framework where the solution is approximated using a polyno- mial basis where the higher modes of the solution correspond to a higher order polynomial basis. By removing high order modes, the filtered solution contains low order frequency content much like an explicit low pass filter. The explicit filter implementation is tested on a simple 1-D solver with an initial condi- tion that has some similarity to turbulent flows. The explicit filter does restrict the resolution as well as remove accumulated energy in the higher modes from aliasing. However, the ex- plicit filter is unable to remove numerical errors causing numerical dissipation. A second test case solves the 3-D Navier-Stokes equations of the Taylor-Green vortex flow (TGV). The TGV is useful for SGS model testing because it is initially laminar and transitions into a fully turbulent flow. The SGS models investigated include the constant coefficient Smagorinsky model, dynamic Smagorinsky model, and dynamic Heinz model. The constant coefficient Smagorinsky model is over dissipative, this is generally not desirable however it does add stability. The dynamic Smagorinsky model generally performs better, especially during the laminar-turbulent transition region as expected. The dynamic Heinz model which is

A Posteriori Error Estimation for Discontinuous Galerkin Approximations of Hyperbolic Systems

NASA Technical Reports Server (NTRS)

Larson, Mats G.; Barth, Timothy J.

1999-01-01

This article considers a posteriori error estimation of specified functionals for first-order systems of conservation laws discretized using the discontinuous Galerkin (DG) finite element method. Using duality techniques, we derive exact error representation formulas for both linear and nonlinear functionals given an associated bilinear or nonlinear variational form. Weighted residual approximations of the exact error representation formula are then proposed and numerically evaluated for Ringleb flow, an exact solution of the 2-D Euler equations.

Sub-grid scale models for discontinuous Galerkin methods based on the Mori-Zwanzig formalism

NASA Astrophysics Data System (ADS)

Parish, Eric; Duraisamy, Karthk

2017-11-01

The optimal prediction framework of Chorin et al., which is a reformulation of the Mori-Zwanzig (M-Z) formalism of non-equilibrium statistical mechanics, provides a framework for the development of mathematically-derived closure models. The M-Z formalism provides a methodology to reformulate a high-dimensional Markovian dynamical system as a lower-dimensional, non-Markovian (non-local) system. In this lower-dimensional system, the effects of the unresolved scales on the resolved scales are non-local and appear as a convolution integral. The non-Markovian system is an exact statement of the original dynamics and is used as a starting point for model development. In this work, we investigate the development of M-Z-based closures model within the context of the Variational Multiscale Method (VMS). The method relies on a decomposition of the solution space into two orthogonal subspaces. The impact of the unresolved subspace on the resolved subspace is shown to be non-local in time and is modeled through the M-Z-formalism. The models are applied to hierarchical discontinuous Galerkin discretizations. Commonalities between the M-Z closures and conventional flux schemes are explored. This work was supported in part by AFOSR under the project ''LES Modeling of Non-local effects using Statistical Coarse-graining'' with Dr. Jean-Luc Cambier as the technical monitor.

Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes

DOE PAGES

Chen, Zheng; Huang, Hongying; Yan, Jue

2015-12-21

We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8], [9], [19] and [21] for convection diffusion equations on unstructured triangular mesh. We carefully calculate the normal derivative numerical flux across element edges and prove that, with proper choice of parameter pair (β 0,β 1) in the numerical flux formula, the quadratic polynomial solution satisfies strict maximum principle. The polynomial solution is bounded within the given range and third order accuracy is maintained. There is no geometric restriction on the meshes and obtuse triangles are allowed in the partition. As a result, a sequence of numerical examples are carried outmore » to demonstrate the accuracy and capability of the maximum-principle-satisfying limiter.« less

Comparisons of Particle Tracking Techniques and Galerkin Finite Element Methods in Flow Simulations on Watershed Scales

NASA Astrophysics Data System (ADS)

Shih, D.; Yeh, G.

2009-12-01

This paper applies two numerical approximations, the particle tracking technique and Galerkin finite element method, to solve the diffusive wave equation in both one-dimensional and two-dimensional flow simulations. The finite element method is one of most commonly approaches in numerical problems. It can obtain accurate solutions, but calculation times may be rather extensive. The particle tracking technique, using either single-velocity or average-velocity tracks to efficiently perform advective transport, could use larger time-step sizes than the finite element method to significantly save computational time. Comparisons of the alternative approximations are examined in this poster. We adapt the model WASH123D to examine the work. WASH123D is an integrated multimedia, multi-processes, physics-based computational model suitable for various spatial-temporal scales, was first developed by Yeh et al., at 1998. The model has evolved in design capability and flexibility, and has been used for model calibrations and validations over the course of many years. In order to deliver a locally hydrological model in Taiwan, the Taiwan Typhoon and Flood Research Institute (TTFRI) is working with Prof. Yeh to develop next version of WASH123D. So, the work of our preliminary cooperationx is also sketched in this poster.

PHYSICAL-CONSTRAINT-PRESERVING CENTRAL DISCONTINUOUS GALERKIN METHODS FOR SPECIAL RELATIVISTIC HYDRODYNAMICS WITH A GENERAL EQUATION OF STATE

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

Wu, Kailiang; Tang, Huazhong, E-mail: wukl@pku.edu.cn, E-mail: hztang@math.pku.edu.cn

The ideal gas equation of state (EOS) with a constant adiabatic index is a poor approximation for most relativistic astrophysical flows, although it is commonly used in relativistic hydrodynamics (RHD). This paper develops high-order accurate, physical-constraints-preserving (PCP), central, discontinuous Galerkin (DG) methods for the one- and two-dimensional special RHD equations with a general EOS. It is built on our theoretical analysis of the admissible states for RHD and the PCP limiting procedure that enforce the admissibility of central DG solutions. The convexity, scaling invariance, orthogonal invariance, and Lax–Friedrichs splitting property of the admissible state set are first proved with themore » aid of its equivalent form. Then, the high-order central DG methods with the PCP limiting procedure and strong stability-preserving time discretization are proved, to preserve the positivity of the density, pressure, specific internal energy, and the bound of the fluid velocity, maintain high-order accuracy, and be L {sup 1}-stable. The accuracy, robustness, and effectiveness of the proposed methods are demonstrated by several 1D and 2D numerical examples involving large Lorentz factor, strong discontinuities, or low density/pressure, etc.« less

High-Order Discontinuous Galerkin Level Set Method for Interface Tracking and Re-Distancing on Unstructured Meshes

NASA Astrophysics Data System (ADS)

Greene, Patrick; Nourgaliev, Robert; Schofield, Sam

2015-11-01

A new sharp high-order interface tracking method for multi-material flow problems on unstructured meshes is presented. The method combines the marker-tracking algorithm with a discontinuous Galerkin (DG) level set method to implicitly track interfaces. DG projection is used to provide a mapping from the Lagrangian marker field to the Eulerian level set field. For the level set re-distancing, we developed a novel marching method that takes advantage of the unique features of the DG representation of the level set. The method efficiently marches outward from the zero level set with values in the new cells being computed solely from cell neighbors. Results are presented for a number of different interface geometries including ones with sharp corners and multiple hierarchical level sets. The method can robustly handle the level set discontinuities without explicit utilization of solution limiters. Results show that the expected high order (3rd and higher) of convergence for the DG representation of the level set is obtained for smooth solutions on unstructured meshes. High-order re-distancing on irregular meshes is a must for applications were the interfacial curvature is important for underlying physics, such as surface tension, wetting and detonation shock dynamics. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Information management release number LLNL-ABS-675636.

Development and application of discontinuous Galerkin method for the solution of two-dimensional Maxwell equations

NASA Astrophysics Data System (ADS)

Wong, See-Cheuk

We inhabit an environment of electromagnetic (EM) waves. The waves within the EM spectrum---whether light, radio, or microwaves---all obey the same physical laws. A band in the spectrum is designated to the microwave frequencies (30MHz--300GHz), at which radar systems operate. The precise modeling of the scattered EM-ields about a target, as well as the numerical prediction of the radar return is the crux of the computational electromagnetics (CEM) problems. The signature or return from a target observed by radar is commonly provided in the form of radar cross section (RCS). Incidentally, the efforts in the reduction of such return forms the basis of stealth aircraft design. The object of this dissertation is to extend Discontinuous Galerkin (DG) method to solve numerically the Maxwell equations for scatterings from perfect electric conductor (PEC) objects. The governing equations are derived by writing the Maxwell equations in conservation-law form for scattered field quantities. The transverse magnetic (TM) and the transverse electric (TE) waveforms of the Maxwell equations are considered. A finite-element scheme is developed with proper representations for the electric and magnetic fluxes at a cell interface to account for variations in properties, in both space and time. A characteristic sub-path integration process, known as the "Riemann solver" is involved. An explicit Runge-Kutta Discontinuous Galerkin (RKDG) upwind scheme, which is fourth-order accurate in time and second-order in space, is employed to solve the TM and TE equations. Arbitrary cross-sectioned bodies are modeled, around which computational grids using random triangulation are generated. The RKDG method, in its development stage, was constructed and studied for solving hyperbolic conservation equations numerically. It was later extended to multidimensional nonlinear systems of conservation laws. The algorithms are described, including the formulations and treatments to the numerical fluxes

Sinc-Galerkin estimation of diffusivity in parabolic problems

NASA Technical Reports Server (NTRS)

Smith, Ralph C.; Bowers, Kenneth L.

1991-01-01

A fully Sinc-Galerkin method for the numerical recovery of spatially varying diffusion coefficients in linear partial differential equations is presented. Because the parameter recovery problems are inherently ill-posed, an output error criterion in conjunction with Tikhonov regularization is used to formulate them as infinite-dimensional minimization problems. The forward problems are discretized with a sinc basis in both the spatial and temporal domains thus yielding an approximate solution which displays an exponential convergence rate and is valid on the infinite time interval. The minimization problems are then solved via a quasi-Newton/trust region algorithm. The L-curve technique for determining an approximate value of the regularization parameter is briefly discussed, and numerical examples are given which show the applicability of the method both for problems with noise-free data as well as for those whose data contains white noise.

An interior penalty stabilised incompressible discontinuous Galerkin-Fourier solver for implicit large eddy simulations

NASA Astrophysics Data System (ADS)

Ferrer, Esteban

2017-11-01

We present an implicit Large Eddy Simulation (iLES) h / p high order (≥2) unstructured Discontinuous Galerkin-Fourier solver with sliding meshes. The solver extends the laminar version of Ferrer and Willden, 2012 [34], to enable the simulation of turbulent flows at moderately high Reynolds numbers in the incompressible regime. This solver allows accurate flow solutions of the laminar and turbulent 3D incompressible Navier-Stokes equations on moving and static regions coupled through a high order sliding interface. The spatial discretisation is provided by the Symmetric Interior Penalty Discontinuous Galerkin (IP-DG) method in the x-y plane coupled with a purely spectral method that uses Fourier series and allows efficient computation of spanwise periodic three-dimensional flows. Since high order methods (e.g. discontinuous Galerkin and Fourier) are unable to provide enough numerical dissipation to enable under-resolved high Reynolds computations (i.e. as necessary in the iLES approach), we adapt the laminar version of the solver to increase (controllably) the dissipation and enhance the stability in under-resolved simulations. The novel stabilisation relies on increasing the penalty parameter included in the DG interior penalty (IP) formulation. The latter penalty term is included when discretising the linear viscous terms in the incompressible Navier-Stokes equations. These viscous penalty fluxes substitute the stabilising effect of non-linear fluxes, which has been the main trend in implicit LES discontinuous Galerkin approaches. The IP-DG penalty term provides energy dissipation, which is controlled by the numerical jumps at element interfaces (e.g. large in under-resolved regions) such as to stabilise under-resolved high Reynolds number flows. This dissipative term has minimal impact in well resolved regions and its implicit treatment does not restrict the use of large time steps, thus providing an efficient stabilization mechanism for iLES. The IP

GNuMe: A Galerkin-based Numerical Modeliing Environment for modeling geophysical fluid dynamics applications ranging from the Atmosphere to the Ocean

NASA Astrophysics Data System (ADS)

Giraldo, Francis; Abdi, Daniel; Kopera, Michal

2017-04-01

We have built a Galerkin-based Numerical Modeling Environment (GNuMe) for non hydrostatic atmospheric and ocean processes. GNuMe uses continuous Galerkin and Discontinuous Galerkin (CG/DG) discetizations as well as non-conforming adaptive mesh refinement (AMR), along with advanced time-integration methods that exploits both CG/DG and AMR capabilities. GNuMe currently solves the compressible and incompressible Navier-Stokes equations, the shallow water equations (with wetting and drying), and work is underway for inclusion of other types of equations. Moreover, GNuMe can run in both 2D and 3D modes on any type of accelerator hardware such as Nvidia GPUs and Intel KNL, and on standard X86 cores. In this talk, we shall present representative solutions obtained with GNuMe and will discuss where we think such a modeling framework could fit within standard Earth Systems Models. For further information on GNuMe please visit: http://frankgiraldo.wixsite.com/mysite/gnume.

Integral equation and discontinuous Galerkin methods for the analysis of light-matter interaction

NASA Astrophysics Data System (ADS)

Baczewski, Andrew David

Light-matter interaction is among the most enduring interests of the physical sciences. The understanding and control of this physics is of paramount importance to the design of myriad technologies ranging from stained glass, to molecular sensing and characterization techniques, to quantum computers. The development of complex engineered systems that exploit this physics is predicated at least partially upon in silico design and optimization that properly capture the light-matter coupling. In this thesis, the details of computational frameworks that enable this type of analysis, based upon both Integral Equation and Discontinuous Galerkin formulations will be explored. There will be a primary focus on the development of efficient and accurate software, with results corroborating both. The secondary focus will be on the use of these tools in the analysis of a number of exemplary systems.

Application of the Finite Element Method to Rotary Wing Aeroelasticity

NASA Technical Reports Server (NTRS)

Straub, F. K.; Friedmann, P. P.

1982-01-01

A finite element method for the spatial discretization of the dynamic equations of equilibrium governing rotary-wing aeroelastic problems is presented. Formulation of the finite element equations is based on weighted Galerkin residuals. This Galerkin finite element method reduces algebraic manipulative labor significantly, when compared to the application of the global Galerkin method in similar problems. The coupled flap-lag aeroelastic stability boundaries of hingeless helicopter rotor blades in hover are calculated. The linearized dynamic equations are reduced to the standard eigenvalue problem from which the aeroelastic stability boundaries are obtained. The convergence properties of the Galerkin finite element method are studied numerically by refining the discretization process. Results indicate that four or five elements suffice to capture the dynamics of the blade with the same accuracy as the global Galerkin method.

Numerical study of the stress-strain state of reinforced plate on an elastic foundation by the Bubnov-Galerkin method

NASA Astrophysics Data System (ADS)

Beskopylny, Alexey; Kadomtseva, Elena; Strelnikov, Grigory

2017-10-01

The stress-strain state of a rectangular slab resting on an elastic foundation is considered. The slab material is isotropic. The slab has stiffening ribs that directed parallel to both sides of the plate. Solving equations are obtained for determining the deflection for various mechanical and geometric characteristics of the stiffening ribs which are parallel to different sides of the plate, having different rigidity for bending and torsion. The calculation scheme assumes an orthotropic slab having different cylindrical stiffness in two mutually perpendicular directions parallel to the reinforcing ribs. An elastic foundation is adopted by Winkler model. To determine the deflection the Bubnov-Galerkin method is used. The deflection is taken in the form of an expansion in a series with unknown coefficients by special polynomials, which are a combination of Legendre polynomials.

Galerkin v. discrete-optimal projection in nonlinear model reduction

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

Carlberg, Kevin Thomas; Barone, Matthew Franklin; Antil, Harbir

Discrete-optimal model-reduction techniques such as the Gauss{Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible ow problems where standard Galerkin techniques have failed. However, there has been limited comparative analysis of the two approaches. This is due in part to difficulties arising from the fact that Galerkin techniques perform projection at the time-continuous level, while discrete-optimal techniques do so at the time-discrete level. This work provides a detailed theoretical and experimental comparison of the two techniques for two common classes of time integrators: linear multistep schemes and Runge{Kutta schemes.more » We present a number of new ndings, including conditions under which the discrete-optimal ROM has a time-continuous representation, conditions under which the two techniques are equivalent, and time-discrete error bounds for the two approaches. Perhaps most surprisingly, we demonstrate both theoretically and experimentally that decreasing the time step does not necessarily decrease the error for the discrete-optimal ROM; instead, the time step should be `matched' to the spectral content of the reduced basis. In numerical experiments carried out on a turbulent compressible- ow problem with over one million unknowns, we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the discrete-optimal reduced-order model by an order of magnitude.« less

Accurate traveltime computation in complex anisotropic media with discontinuous Galerkin method

NASA Astrophysics Data System (ADS)

Le Bouteiller, P.; Benjemaa, M.; Métivier, L.; Virieux, J.

2017-12-01

Travel time computation is of major interest for a large range of geophysical applications, among which source localization and characterization, phase identification, data windowing and tomography, from decametric scale up to global Earth scale.Ray-tracing tools, being essentially 1D Lagrangian integration along a path, have been used for their efficiency but present some drawbacks, such as a rather difficult control of the medium sampling. Moreover, they do not provide answers in shadow zones. Eikonal solvers, based on an Eulerian approach, have attracted attention in seismology with the pioneering work of Vidale (1988), while such approach has been proposed earlier by Riznichenko (1946). They have been used now for first-arrival travel-time tomography at various scales (Podvin & Lecomte (1991). The framework for solving this non-linear partial differential equation is now well understood and various finite-difference approaches have been proposed, essentially for smooth media. We propose a novel finite element approach which builds a precise solution for strongly heterogeneous anisotropic medium (still in the limit of Eikonal validity). The discontinuous Galerkin method we have developed allows local refinement of the mesh and local high orders of interpolation inside elements. High precision of the travel times and its spatial derivatives is obtained through this formulation. This finite element method also honors boundary conditions, such as complex topographies and absorbing boundaries for mimicking an infinite medium. Applications from travel-time tomography, slope tomography are expected, but also for migration and take-off angles estimation, thanks to the accuracy obtained when computing first-arrival times.References:Podvin, P. and Lecomte, I., 1991. Finite difference computation of traveltimes in very contrasted velocity model: a massively parallel approach and its associated tools, Geophys. J. Int., 105, 271-284.Riznichenko, Y., 1946. Geometrical

Spectral methods on arbitrary grids

NASA Technical Reports Server (NTRS)

Carpenter, Mark H.; Gottlieb, David

1995-01-01

Stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations. These new methods are equivalent to conventional spectral methods but do not rely on specific grid distributions. Specifically, we show how to implement Legendre Galerkin, Legendre collocation, and Laguerre Galerkin methodology on arbitrary grids.

Numerical analysis on interactions between fluid flow and structure deformation in plate-fin heat exchanger by Galerkin method

NASA Astrophysics Data System (ADS)

Liu, Jing-cheng; Wei, Xiu-ting; Zhou, Zhi-yong; Wei, Zhen-wen

2018-03-01

The fluid-structure interaction performance of plate-fin heat exchanger (PFHE) with serrated fins in large scale air-separation equipment was investigated in this paper. The stress and deformation of fins were analyzed, besides, the interaction equations were deduced by Galerkin method. The governing equations of fluid flow and heat transfer in PFHE were deduced by finite volume method (FVM). The distribution of strain and stress were calculated in large scale air separation equipment and the coupling situation of serrated fins under laminar situation was analyzed. The results indicated that the interactions between fins and fluid flow in the exchanger have significant impacts on heat transfer enhancement, meanwhile, the strain and stress of fins includes dynamic pressure of the sealing head and flow impact with the increase of flow velocity. The impacts are especially significant at the conjunction of two fins because of the non-alignment fins. It can be concluded that the soldering process and channel width led to structure deformation of fins in the exchanger, and degraded heat transfer efficiency.

Galerkin methods for Boltzmann-Poisson transport with reflection conditions on rough boundaries

NASA Astrophysics Data System (ADS)

Morales Escalante, José A.; Gamba, Irene M.

2018-06-01

We consider in this paper the mathematical and numerical modeling of reflective boundary conditions (BC) associated to Boltzmann-Poisson systems, including diffusive reflection in addition to specularity, in the context of electron transport in semiconductor device modeling at nano scales, and their implementation in Discontinuous Galerkin (DG) schemes. We study these BC on the physical boundaries of the device and develop a numerical approximation to model an insulating boundary condition, or equivalently, a pointwise zero flux mathematical condition for the electron transport equation. Such condition balances the incident and reflective momentum flux at the microscopic level, pointwise at the boundary, in the case of a more general mixed reflection with momentum dependant specularity probability p (k →). We compare the computational prediction of physical observables given by the numerical implementation of these different reflection conditions in our DG scheme for BP models, and observe that the diffusive condition influences the kinetic moments over the whole domain in position space.

A hybrid incremental projection method for thermal-hydraulics applications

NASA Astrophysics Data System (ADS)

Christon, Mark A.; Bakosi, Jozsef; Nadiga, Balasubramanya T.; Berndt, Markus; Francois, Marianne M.; Stagg, Alan K.; Xia, Yidong; Luo, Hong

2016-07-01

A new second-order accurate, hybrid, incremental projection method for time-dependent incompressible viscous flow is introduced in this paper. The hybrid finite-element/finite-volume discretization circumvents the well-known Ladyzhenskaya-Babuška-Brezzi conditions for stability, and does not require special treatment to filter pressure modes by either Rhie-Chow interpolation or by using a Petrov-Galerkin finite element formulation. The use of a co-velocity with a high-resolution advection method and a linearly consistent edge-based treatment of viscous/diffusive terms yields a robust algorithm for a broad spectrum of incompressible flows. The high-resolution advection method is shown to deliver second-order spatial convergence on mixed element topology meshes, and the implicit advective treatment significantly increases the stable time-step size. The algorithm is robust and extensible, permitting the incorporation of features such as porous media flow, RANS and LES turbulence models, and semi-/fully-implicit time stepping. A series of verification and validation problems are used to illustrate the convergence properties of the algorithm. The temporal stability properties are demonstrated on a range of problems with 2 ≤ CFL ≤ 100. The new flow solver is built using the Hydra multiphysics toolkit. The Hydra toolkit is written in C++ and provides a rich suite of extensible and fully-parallel components that permit rapid application development, supports multiple discretization techniques, provides I/O interfaces, dynamic run-time load balancing and data migration, and interfaces to scalable popular linear solvers, e.g., in open-source packages such as HYPRE, PETSc, and Trilinos.

Finite element procedures for time-dependent convection-diffusion-reaction systems

NASA Technical Reports Server (NTRS)

Tezduyar, T. E.; Park, Y. J.; Deans, H. A.

1988-01-01

New finite element procedures based on the streamline-upwind/Petrov-Galerkin formulations are developed for time-dependent convection-diffusion-reaction equations. These procedures minimize spurious oscillations for convection-dominated and reaction-dominated problems. The results obtained for representative numerical examples are accurate with minimal oscillations. As a special application problem, the single-well chemical tracer test (a procedure for measuring oil remaining in a depleted field) is simulated numerically. The results show the importance of temperature effects on the interpreted value of residual oil saturation from such tests.

Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

NASA Astrophysics Data System (ADS)

Abdi, Daniel S.; Giraldo, Francis X.

2016-09-01

A unified approach for the numerical solution of the 3D hyperbolic Euler equations using high order methods, namely continuous Galerkin (CG) and discontinuous Galerkin (DG) methods, is presented. First, we examine how classical CG that uses a global storage scheme can be constructed within the DG framework using constraint imposition techniques commonly used in the finite element literature. Then, we implement and test a simplified version in the Non-hydrostatic Unified Model of the Atmosphere (NUMA) for the case of explicit time integration and a diagonal mass matrix. Constructing CG within the DG framework allows CG to benefit from the desirable properties of DG such as, easier hp-refinement, better stability etc. Moreover, this representation allows for regional mixing of CG and DG depending on the flow regime in an area. The different flavors of CG and DG in the unified implementation are then tested for accuracy and performance using a suite of benchmark problems representative of cloud-resolving scale, meso-scale and global-scale atmospheric dynamics. The value of our unified approach is that we are able to show how to carry both CG and DG methods within the same code and also offer a simple recipe for modifying an existing CG code to DG and vice versa.

Discrete maximum principle for the P1 - P0 weak Galerkin finite element approximations

NASA Astrophysics Data System (ADS)

Wang, Junping; Ye, Xiu; Zhai, Qilong; Zhang, Ran

2018-06-01

This paper presents two discrete maximum principles (DMP) for the numerical solution of second order elliptic equations arising from the weak Galerkin finite element method. The results are established by assuming an h-acute angle condition for the underlying finite element triangulations. The mathematical theory is based on the well-known De Giorgi technique adapted in the finite element context. Some numerical results are reported to validate the theory of DMP.

Regionally Implicit Discontinuous Galerkin Methods for Solving the Relativistic Vlasov-Maxwell System Submitted to Iowa State University

NASA Astrophysics Data System (ADS)

Guthrey, Pierson Tyler

The relativistic Vlasov-Maxwell system (RVM) models the behavior of collisionless plasma, where electrons and ions interact via the electromagnetic fields they generate. In the RVM system, electrons could accelerate to significant fractions of the speed of light. An idea that is actively being pursued by several research groups around the globe is to accelerate electrons to relativistic speeds by hitting a plasma with an intense laser beam. As the laser beam passes through the plasma it creates plasma wakes, much like a ship passing through water, which can trap electrons and push them to relativistic speeds. Such setups are known as laser wakefield accelerators, and have the potential to yield particle accelerators that are significantly smaller than those currently in use. Ultimately, the goal of such research is to harness the resulting electron beams to generate electromagnetic waves that can be used in medical imaging applications. High-order accurate numerical discretizations of kinetic Vlasov plasma models are very effective at yielding low-noise plasma simulations, but are computationally expensive to solve because of the high dimensionality. In addition to the general difficulties inherent to numerically simulating Vlasov models, the relativistic Vlasov-Maxwell system has unique challenges not present in the non-relativistic case. One such issue is that operator splitting of the phase gradient leads to potential instabilities, thus we require an alternative to operator splitting of the phase. The goal of the current work is to develop a new class of high-order accurate numerical methods for solving kinetic Vlasov models of plasma. The main discretization in configuration space is handled via a high-order finite element method called the discontinuous Galerkin method (DG). One difficulty is that standard explicit time-stepping methods for DG suffer from time-step restrictions that are significantly worse than what a simple Courant-Friedrichs-Lewy (CFL

A Galerkin least squares approach to viscoelastic flow.

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

Rao, Rekha R.; Schunk, Peter Randall

2015-10-01

A Galerkin/least-squares stabilization technique is applied to a discrete Elastic Viscous Stress Splitting formulation of for viscoelastic flow. From this, a possible viscoelastic stabilization method is proposed. This method is tested with the flow of an Oldroyd-B fluid past a rigid cylinder, where it is found to produce inaccurate drag coefficients. Furthermore, it fails for relatively low Weissenberg number indicating it is not suited for use as a general algorithm. In addition, a decoupled approach is used as a way separating the constitutive equation from the rest of the system. A Pressure Poisson equation is used when the velocity andmore » pressure are sought to be decoupled, but this fails to produce a solution when inflow/outflow boundaries are considered. However, a coupled pressure-velocity equation with a decoupled constitutive equation is successful for the flow past a rigid cylinder and seems to be suitable as a general-use algorithm.« less

A well-posed and stable stochastic Galerkin formulation of the incompressible Navier–Stokes equations with random data

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

Pettersson, Per, E-mail: per.pettersson@uib.no; Nordström, Jan, E-mail: jan.nordstrom@liu.se; Doostan, Alireza, E-mail: alireza.doostan@colorado.edu

2016-02-01

We present a well-posed stochastic Galerkin formulation of the incompressible Navier–Stokes equations with uncertainty in model parameters or the initial and boundary conditions. The stochastic Galerkin method involves representation of the solution through generalized polynomial chaos expansion and projection of the governing equations onto stochastic basis functions, resulting in an extended system of equations. A relatively low-order generalized polynomial chaos expansion is sufficient to capture the stochastic solution for the problem considered. We derive boundary conditions for the continuous form of the stochastic Galerkin formulation of the velocity and pressure equations. The resulting problem formulation leads to an energy estimatemore » for the divergence. With suitable boundary data on the pressure and velocity, the energy estimate implies zero divergence of the velocity field. Based on the analysis of the continuous equations, we present a semi-discretized system where the spatial derivatives are approximated using finite difference operators with a summation-by-parts property. With a suitable choice of dissipative boundary conditions imposed weakly through penalty terms, the semi-discrete scheme is shown to be stable. Numerical experiments in the laminar flow regime corroborate the theoretical results and we obtain high-order accurate results for the solution variables and the velocity divergence converges to zero as the mesh is refined.« less

Robustness of controllers designed using Galerkin type approximations

NASA Technical Reports Server (NTRS)

Morris, K. A.

1990-01-01

One of the difficulties in designing controllers for infinite-dimensional systems arises from attempting to calculate a state for the system. It is shown that Galerkin type approximations can be used to design controllers which will perform as designed when implemented on the original infinite-dimensional system. No assumptions, other than those typically employed in numerical analysis, are made on the approximating scheme.

Second derivative time integration methods for discontinuous Galerkin solutions of unsteady compressible flows

NASA Astrophysics Data System (ADS)

Nigro, A.; De Bartolo, C.; Crivellini, A.; Bassi, F.

2017-12-01

In this paper we investigate the possibility of using the high-order accurate A (α) -stable Second Derivative (SD) schemes proposed by Enright for the implicit time integration of the Discontinuous Galerkin (DG) space-discretized Navier-Stokes equations. These multistep schemes are A-stable up to fourth-order, but their use results in a system matrix difficult to compute. Furthermore, the evaluation of the nonlinear function is computationally very demanding. We propose here a Matrix-Free (MF) implementation of Enright schemes that allows to obtain a method without the costs of forming, storing and factorizing the system matrix, which is much less computationally expensive than its matrix-explicit counterpart, and which performs competitively with other implicit schemes, such as the Modified Extended Backward Differentiation Formulae (MEBDF). The algorithm makes use of the preconditioned GMRES algorithm for solving the linear system of equations. The preconditioner is based on the ILU(0) factorization of an approximated but computationally cheaper form of the system matrix, and it has been reused for several time steps to improve the efficiency of the MF Newton-Krylov solver. We additionally employ a polynomial extrapolation technique to compute an accurate initial guess to the implicit nonlinear system. The stability properties of SD schemes have been analyzed by solving a linear model problem. For the analysis on the Navier-Stokes equations, two-dimensional inviscid and viscous test cases, both with a known analytical solution, are solved to assess the accuracy properties of the proposed time integration method for nonlinear autonomous and non-autonomous systems, respectively. The performance of the SD algorithm is compared with the ones obtained by using an MF-MEBDF solver, in order to evaluate its effectiveness, identifying its limitations and suggesting possible further improvements.

The Discontinuous Galerkin Finite Element Method for Solving the MEG and the Combined MEG/EEG Forward Problem

PubMed Central

Piastra, Maria Carla; Nüßing, Andreas; Vorwerk, Johannes; Bornfleth, Harald; Oostenveld, Robert; Engwer, Christian; Wolters, Carsten H.

2018-01-01

In Electro- (EEG) and Magnetoencephalography (MEG), one important requirement of source reconstruction is the forward model. The continuous Galerkin finite element method (CG-FEM) has become one of the dominant approaches for solving the forward problem over the last decades. Recently, a discontinuous Galerkin FEM (DG-FEM) EEG forward approach has been proposed as an alternative to CG-FEM (Engwer et al., 2017). It was shown that DG-FEM preserves the property of conservation of charge and that it can, in certain situations such as the so-called skull leakages, be superior to the standard CG-FEM approach. In this paper, we developed, implemented, and evaluated two DG-FEM approaches for the MEG forward problem, namely a conservative and a non-conservative one. The subtraction approach was used as source model. The validation and evaluation work was done in statistical investigations in multi-layer homogeneous sphere models, where an analytic solution exists, and in a six-compartment realistically shaped head volume conductor model. In agreement with the theory, the conservative DG-FEM approach was found to be superior to the non-conservative DG-FEM implementation. This approach also showed convergence with increasing resolution of the hexahedral meshes. While in the EEG case, in presence of skull leakages, DG-FEM outperformed CG-FEM, in MEG, DG-FEM achieved similar numerical errors as the CG-FEM approach, i.e., skull leakages do not play a role for the MEG modality. In particular, for the finest mesh resolution of 1 mm sources with a distance of 1.59 mm from the brain-CSF surface, DG-FEM yielded mean topographical errors (relative difference measure, RDM%) of 1.5% and mean magnitude errors (MAG%) of 0.1% for the magnetic field. However, if the goal is a combined source analysis of EEG and MEG data, then it is highly desirable to employ the same forward model for both EEG and MEG data. Based on these results, we conclude that the newly presented conservative DG-FEM can

A Galerkin discretisation-based identification for parameters in nonlinear mechanical systems

NASA Astrophysics Data System (ADS)

Liu, Zuolin; Xu, Jian

2018-04-01

In the paper, a new parameter identification method is proposed for mechanical systems. Based on the idea of Galerkin finite-element method, the displacement over time history is approximated by piecewise linear functions, and the second-order terms in model equation are eliminated by integrating by parts. In this way, the lost function of integration form is derived. Being different with the existing methods, the lost function actually is a quadratic sum of integration over the whole time history. Then for linear or nonlinear systems, the optimisation of the lost function can be applied with traditional least-squares algorithm or the iterative one, respectively. Such method could be used to effectively identify parameters in linear and arbitrary nonlinear mechanical systems. Simulation results show that even under the condition of sparse data or low sampling frequency, this method could still guarantee high accuracy in identifying linear and nonlinear parameters.

Element free Galerkin formulation of composite beam with longitudinal slip

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

Ahmad, Dzulkarnain; Mokhtaram, Mokhtazul Haizad; Badli, Mohd Iqbal

2015-05-15

Behaviour between two materials in composite beam is assumed partially interact when longitudinal slip at its interfacial surfaces is considered. Commonly analysed by the mesh-based formulation, this study used meshless formulation known as Element Free Galerkin (EFG) method in the beam partial interaction analysis, numerically. As meshless formulation implies that the problem domain is discretised only by nodes, the EFG method is based on Moving Least Square (MLS) approach for shape functions formulation with its weak form is developed using variational method. The essential boundary conditions are enforced by Langrange multipliers. The proposed EFG formulation gives comparable results, after beenmore » verified by analytical solution, thus signify its application in partial interaction problems. Based on numerical test results, the Cubic Spline and Quartic Spline weight functions yield better accuracy for the EFG formulation, compares to other proposed weight functions.« less

A Galerkin approximation for linear elastic shallow shells

NASA Astrophysics Data System (ADS)

Figueiredo, I. N.; Trabucho, L.

1992-03-01

This work is a generalization to shallow shell models of previous results for plates by B. Miara (1989). Using the same basis functions as in the plate case, we construct a Galerkin approximation of the three-dimensional linearized elasticity problem, and establish some error estimates as a function of the thickness, the curvature, the geometry of the shell, the forces and the Lamé costants.

A hybrid Pade-Galerkin technique for differential equations

NASA Technical Reports Server (NTRS)

Geer, James F.; Andersen, Carl M.

1993-01-01

A three-step hybrid analysis technique, which successively uses the regular perturbation expansion method, the Pade expansion method, and then a Galerkin approximation, is presented and applied to some model boundary value problems. In the first step of the method, the regular perturbation method is used to construct an approximation to the solution in the form of a finite power series in a small parameter epsilon associated with the problem. In the second step of the method, the series approximation obtained in step one is used to construct a Pade approximation in the form of a rational function in the parameter epsilon. In the third step, the various powers of epsilon which appear in the Pade approximation are replaced by new (unknown) parameters (delta(sub j)). These new parameters are determined by requiring that the residual formed by substituting the new approximation into the governing differential equation is orthogonal to each of the perturbation coordinate functions used in step one. The technique is applied to model problems involving ordinary or partial differential equations. In general, the technique appears to provide good approximations to the solution even when the perturbation and Pade approximations fail to do so. The method is discussed and topics for future investigations are indicated.

An adaptive simplex cut-cell method for high-order discontinuous Galerkin discretizations of elliptic interface problems and conjugate heat transfer problems

NASA Astrophysics Data System (ADS)

Sun, Huafei; Darmofal, David L.

2014-12-01

In this paper we propose a new high-order solution framework for interface problems on non-interface-conforming meshes. The framework consists of a discontinuous Galerkin (DG) discretization, a simplex cut-cell technique, and an output-based adaptive scheme. We first present a DG discretization with a dual-consistent output evaluation for elliptic interface problems on interface-conforming meshes, and then extend the method to handle multi-physics interface problems, in particular conjugate heat transfer (CHT) problems. The method is then applied to non-interface-conforming meshes using a cut-cell technique, where the interface definition is completely separate from the mesh generation process. No assumption is made on the interface shape (other than Lipschitz continuity). We then equip our strategy with an output-based adaptive scheme for an accurate output prediction. Through numerical examples, we demonstrate high-order convergence for elliptic interface problems and CHT problems with both smooth and non-smooth interface shapes.

High performance computing aspects of a dimension independent semi-Lagrangian discontinuous Galerkin code

NASA Astrophysics Data System (ADS)

Einkemmer, Lukas

2016-05-01

The recently developed semi-Lagrangian discontinuous Galerkin approach is used to discretize hyperbolic partial differential equations (usually first order equations). Since these methods are conservative, local in space, and able to limit numerical diffusion, they are considered a promising alternative to more traditional semi-Lagrangian schemes (which are usually based on polynomial or spline interpolation). In this paper, we consider a parallel implementation of a semi-Lagrangian discontinuous Galerkin method for distributed memory systems (so-called clusters). Both strong and weak scaling studies are performed on the Vienna Scientific Cluster 2 (VSC-2). In the case of weak scaling we observe a parallel efficiency above 0.8 for both two and four dimensional problems and up to 8192 cores. Strong scaling results show good scalability to at least 512 cores (we consider problems that can be run on a single processor in reasonable time). In addition, we study the scaling of a two dimensional Vlasov-Poisson solver that is implemented using the framework provided. All of the simulations are conducted in the context of worst case communication overhead; i.e., in a setting where the CFL (Courant-Friedrichs-Lewy) number increases linearly with the problem size. The framework introduced in this paper facilitates a dimension independent implementation of scientific codes (based on C++ templates) using both an MPI and a hybrid approach to parallelization. We describe the essential ingredients of our implementation.

The Reverse Time Migration technique coupled with Interior Penalty Discontinuous Galerkin method.

NASA Astrophysics Data System (ADS)

Baldassari, C.; Barucq, H.; Calandra, H.; Denel, B.; Diaz, J.

2009-04-01

Seismic imaging is based on the seismic reflection method which produces an image of the subsurface from reflected waves recordings by using a tomography process and seismic migration is the industrial standard to improve the quality of the images. The migration process consists in replacing the recorded wavefields at their actual place by using various mathematical and numerical methods but each of them follows the same schedule, according to the pioneering idea of Claerbout: numerical propagation of the source function (propagation) and of the recorded wavefields (retropropagation) and next, construction of the image by applying an imaging condition. The retropropagation step can be realized accouting for the time reversibility of the wave equation and the resulting algorithm is currently called Reverse Time Migration (RTM). To be efficient, especially in three dimensional domain, the RTM requires the solution of the full wave equation by fast numerical methods. Finite element methods are considered as the best discretization method for solving the wave equation, even if they lead to the solution of huge systems with several millions of degrees of freedom, since they use meshes adapted to the domain topography and the boundary conditions are naturally taken into account in the variational formulation. Among the different finite element families, the spectral element one (SEM) is very interesting because it leads to a diagonal mass matrix which dramatically reduces the cost of the numerical computation. Moreover this method is very accurate since it allows the use of high order finite elements. However, SEM uses meshes of the domain made of quadrangles in 2D or hexaedra in 3D which are difficult to compute and not always suitable for complex topographies. Recently, Grote et al. applied the IPDG (Interior Penalty Discontinuous Galerkin) method to the wave equation. This approach is very interesting since it relies on meshes with triangles in 2D or tetrahedra in 3D

Convergence of Galerkin approximations for operator Riccati equations: A nonlinear evolution equation approach

NASA Technical Reports Server (NTRS)

Rosen, I. G.

1988-01-01

An approximation and convergence theory was developed for Galerkin approximations to infinite dimensional operator Riccati differential equations formulated in the space of Hilbert-Schmidt operators on a separable Hilbert space. The Riccati equation was treated as a nonlinear evolution equation with dynamics described by a nonlinear monotone perturbation of a strongly coercive linear operator. A generic approximation result was proven for quasi-autonomous nonlinear evolution system involving accretive operators which was then used to demonstrate the Hilbert-Schmidt norm convergence of Galerkin approximations to the solution of the Riccati equation. The application of the results was illustrated in the context of a linear quadratic optimal control problem for a one dimensional heat equation.

U.S.-Soviet Collaborative Geological and Geophysical Survey of the Mid-Atlantic Ridge near 31 degrees N, the Petrov Fracture Zone

USGS Publications Warehouse

Klitgord, Kim D.; Dmitriev, Leonard V.; Casey, John F.; Silantiev, Sergei; Johnson, Kevin

1993-01-01

IntroductionIn February 1989, the first formal U.S.-Soviet joint marine geologic-geophysical study in 10 years was undertaken along the Mid-Atlantic Ridge near 31°N on the 12th Cruise of the RN Akademik Boris Petrov of the Vernadsky Institute of Geochemistry (USSR Academy of Sciences, Moscow). This survey was initiated as part of the U.S.S.R.-U.S. cooperative research project "Mid-Atlantic Ridge Crest Processes" within the framework of the Soviet-U.S. bilateral Ocean Studies Agreement (Ostenso, 1989). U.S. scientists from the U.S. Geological Survey, University of Houston, and Woods Hole Oceanographic Institution participated in this program with Soviet scientists from the Vernadsky Institute of Geochemistry, Institute of Geology, and Schmidt Institute of Physics of the Earth, all institutes of the USSR Academy of Sciences in Moscow (Appendix 1 ). The ship departed from Rotterdam, Nederlands on February 2, 1989 and docked in Bridgetown, Barbados on February 28, 1989. A log of the ship's schedule during this cruise is given in Appendix 2. This study involved a limited multibeam-bathymetric, gravity, magnetic, and seismic- reflection survey. and dredging program of a short-offset transform fault named the Petrov Fracture Zone near 31 °N, located just north of the Atlantis Fracture Zone on the Mid Atlantic Ridge. A site survey at King's Trough in the northeast Atlantic for a MIR submersible program in June 1989 was originally planned as part of this program, but bad weather and the resultant poor quality geophysical data forced this work to be terminated after only one day. Nearly 6000 km of geophysical profile data and 13 dredge stations were completed during this cruise. A description of the geophysical systems aboard the RN Petrov is given in Appendices 3 and 4. All geophysical data were recorded on magnetic tape in data formats described in Appendix 5. Dredge locales and description summaries only are presented in Appendix 6. Detailed descriptions of dredge

Application of the Galerkin/least-squares formulation to the analysis of hypersonic flows. II - Flow past a double ellipse

NASA Technical Reports Server (NTRS)

Chalot, F.; Hughes, T. J. R.; Johan, Z.; Shakib, F.

1991-01-01

A finite element method for the compressible Navier-Stokes equations is introduced. The discretization is based on entropy variables. The methodology is developed within the framework of a Galerkin/least-squares formulation to which a discontinuity-capturing operator is added. Results for four test cases selected among those of the Workshop on Hypersonic Flows for Reentry Problems are presented.

Mass-conservative reconstruction of Galerkin velocity fields for transport simulations

NASA Astrophysics Data System (ADS)

Scudeler, C.; Putti, M.; Paniconi, C.

2016-08-01

Accurate calculation of mass-conservative velocity fields from numerical solutions of Richards' equation is central to reliable surface-subsurface flow and transport modeling, for example in long-term tracer simulations to determine catchment residence time distributions. In this study we assess the performance of a local Larson-Niklasson (LN) post-processing procedure for reconstructing mass-conservative velocities from a linear (P1) Galerkin finite element solution of Richards' equation. This approach, originally proposed for a-posteriori error estimation, modifies the standard finite element velocities by imposing local conservation on element patches. The resulting reconstructed flow field is characterized by continuous fluxes on element edges that can be efficiently used to drive a second order finite volume advective transport model. Through a series of tests of increasing complexity that compare results from the LN scheme to those using velocity fields derived directly from the P1 Galerkin solution, we show that a locally mass-conservative velocity field is necessary to obtain accurate transport results. We also show that the accuracy of the LN reconstruction procedure is comparable to that of the inherently conservative mixed finite element approach, taken as a reference solution, but that the LN scheme has much lower computational costs. The numerical tests examine steady and unsteady, saturated and variably saturated, and homogeneous and heterogeneous cases along with initial and boundary conditions that include dry soil infiltration, alternating solute and water injection, and seepage face outflow. Typical problems that arise with velocities derived from P1 Galerkin solutions include outgoing solute flux from no-flow boundaries, solute entrapment in zones of low hydraulic conductivity, and occurrences of anomalous sources and sinks. In addition to inducing significant mass balance errors, such manifestations often lead to oscillations in concentration

Discontinuous Galerkin finite element method for the nonlinear hyperbolic problems with entropy-based artificial viscosity stabilization

NASA Astrophysics Data System (ADS)

Zingan, Valentin Nikolaevich

This work develops a discontinuous Galerkin finite element discretization of non- linear hyperbolic conservation equations with efficient and robust high order stabilization built on an entropy-based artificial viscosity approximation. The solutions of equations are represented by elementwise polynomials of an arbitrary degree p > 0 which are continuous within each element but discontinuous on the boundaries. The discretization of equations in time is done by means of high order explicit Runge-Kutta methods identified with respective Butcher tableaux. To stabilize a numerical solution in the vicinity of shock waves and simultaneously preserve the smooth parts from smearing, we add some reasonable amount of artificial viscosity in accordance with the physical principle of entropy production in the interior of shock waves. The viscosity coefficient is proportional to the local size of the residual of an entropy equation and is bounded from above by the first-order artificial viscosity defined by a local wave speed. Since the residual of an entropy equation is supposed to be vanishingly small in smooth regions (of the order of the Local Truncation Error) and arbitrarily large in shocks, the entropy viscosity is almost zero everywhere except the shocks, where it reaches the first-order upper bound. One- and two-dimensional benchmark test cases are presented for nonlinear hyperbolic scalar conservation laws and the system of compressible Euler equations. These tests demonstrate the satisfactory stability properties of the method and optimal convergence rates as well. All numerical solutions to the test problems agree well with the reference solutions found in the literature. We conclude that the new method developed in the present work is a valuable alternative to currently existing techniques of viscous stabilization.

Parallel discontinuous Galerkin FEM for computing hyperbolic conservation law on unstructured grids

NASA Astrophysics Data System (ADS)

Ma, Xinrong; Duan, Zhijian

2018-04-01

High-order resolution Discontinuous Galerkin finite element methods (DGFEM) has been known as a good method for solving Euler equations and Navier-Stokes equations on unstructured grid, but it costs too much computational resources. An efficient parallel algorithm was presented for solving the compressible Euler equations. Moreover, the multigrid strategy based on three-stage three-order TVD Runge-Kutta scheme was used in order to improve the computational efficiency of DGFEM and accelerate the convergence of the solution of unsteady compressible Euler equations. In order to make each processor maintain load balancing, the domain decomposition method was employed. Numerical experiment performed for the inviscid transonic flow fluid problems around NACA0012 airfoil and M6 wing. The results indicated that our parallel algorithm can improve acceleration and efficiency significantly, which is suitable for calculating the complex flow fluid.

A Galerkin method for the estimation of parameters in hybrid systems governing the vibration of flexible beams with tip bodies

NASA Technical Reports Server (NTRS)

Banks, H. T.; Rosen, I. G.

1985-01-01

An approximation scheme is developed for the identification of hybrid systems describing the transverse vibrations of flexible beams with attached tip bodies. In particular, problems involving the estimation of functional parameters are considered. The identification problem is formulated as a least squares fit to data subject to the coupled system of partial and ordinary differential equations describing the transverse displacement of the beam and the motion of the tip bodies respectively. A cubic spline-based Galerkin method applied to the state equations in weak form and the discretization of the admissible parameter space yield a sequence of approximating finite dimensional identification problems. It is shown that each of the approximating problems admits a solution and that from the resulting sequence of optimal solutions a convergent subsequence can be extracted, the limit of which is a solution to the original identification problem. The approximating identification problems can be solved using standard techniques and readily available software.

A mass-conserving mixed Fourier-Galerkin B-Spline-collocation method for Direct Numerical Simulation of the variable-density Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Reuter, Bryan; Oliver, Todd; Lee, M. K.; Moser, Robert

2017-11-01

We present an algorithm for a Direct Numerical Simulation of the variable-density Navier-Stokes equations based on the velocity-vorticity approach introduced by Kim, Moin, and Moser (1987). In the current work, a Helmholtz decomposition of the momentum is performed. Evolution equations for the curl and the Laplacian of the divergence-free portion are formulated by manipulation of the momentum equations and the curl-free portion is reconstructed by enforcing continuity. The solution is expanded in Fourier bases in the homogeneous directions and B-Spline bases in the inhomogeneous directions. Discrete equations are obtained through a mixed Fourier-Galerkin and collocation weighted residual method. The scheme is designed such that the numerical solution conserves mass locally and globally by ensuring the discrete divergence projection is exact through the use of higher order splines in the inhomogeneous directions. The formulation is tested on multiple variable-density flow problems.

Energy Stable Flux Formulas For The Discontinuous Galerkin Discretization Of First Order Nonlinear Conservation Laws

NASA Technical Reports Server (NTRS)

Barth, Timothy; Charrier, Pierre; Mansour, Nagi N. (Technical Monitor)

2001-01-01

We consider the discontinuous Galerkin (DG) finite element discretization of first order systems of conservation laws derivable as moments of the kinetic Boltzmann equation. This includes well known conservation law systems such as the Euler For the class of first order nonlinear conservation laws equipped with an entropy extension, an energy analysis of the DG method for the Cauchy initial value problem is developed. Using this DG energy analysis, several new variants of existing numerical flux functions are derived and shown to be energy stable.

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.

Numerical stability analysis of two-dimensional solute transport along a discrete fracture in a porous rock matrix

NASA Astrophysics Data System (ADS)

Watanabe, Norihiro; Kolditz, Olaf

2015-07-01

This work reports numerical stability conditions in two-dimensional solute transport simulations including discrete fractures surrounded by an impermeable rock matrix. We use an advective-dispersive problem described in Tang et al. (1981) and examine the stability of the Crank-Nicolson Galerkin finite element method (CN-GFEM). The stability conditions are analyzed in terms of the spatial discretization length perpendicular to the fracture, the flow velocity, the diffusion coefficient, the matrix porosity, the fracture aperture, and the fracture longitudinal dispersivity. In addition, we verify applicability of the recently developed finite element method-flux corrected transport (FEM-FCT) method by Kuzmin () to suppress oscillations in the hybrid system, with a comparison to the commonly utilized Streamline Upwinding/Petrov-Galerkin (SUPG) method. Major findings of this study are (1) the mesh von Neumann number (Fo) ≥ 0.373 must be satisfied to avoid undershooting in the matrix, (2) in addition to an upper bound, the Courant number also has a lower bound in the fracture in cases of low dispersivity, and (3) the FEM-FCT method can effectively suppress the oscillations in both the fracture and the matrix. The results imply that, in cases of low dispersivity, prerefinement of a numerical mesh is not sufficient to avoid the instability in the hybrid system if a problem involves evolutionary flow fields and dynamic material parameters. Applying the FEM-FCT method to such problems is recommended if negative concentrations cannot be tolerated and computing time is not a strong issue.

Numerical model for dendritic solidification of binary alloys

NASA Technical Reports Server (NTRS)

Felicelli, S. D.; Heinrich, J. C.; Poirier, D. R.

1993-01-01

A finite element model capable of simulating solidification of binary alloys and the formation of freckles is presented. It uses a single system of equations to deal with the all-liquid region, the dendritic region, and the all-solid region. The dendritic region is treated as an anisotropic porous medium. The algorithm uses the bilinear isoparametric element, with a penalty function approximation and a Petrov-Galerkin formulation. Numerical simulations are shown in which an NH4Cl-H2O mixture and a Pb-Sn alloy melt are cooled. The solidification process is followed in time. Instabilities in the process can be clearly observed and the final compositions obtained.

Topics in spectral methods

NASA Technical Reports Server (NTRS)

Gottlieb, D.; Turkel, E.

1985-01-01

After detailing the construction of spectral approximations to time-dependent mixed initial boundary value problems, a study is conducted of differential equations of the form 'partial derivative of u/partial derivative of t = Lu + f', where for each t, u(t) belongs to a Hilbert space such that u satisfies homogeneous boundary conditions. For the sake of simplicity, it is assumed that L is an unbounded, time-independent linear operator. Attention is given to Fourier methods of both Galerkin and pseudospectral method types, the Galerkin method, the pseudospectral Chebyshev and Legendre methods, the error equation, hyperbolic partial differentiation equations, and time discretization and iterative methods.

A thermodynamically consistent discontinuous Galerkin formulation for interface separation

DOE PAGES

Versino, Daniele; Mourad, Hashem M.; Dávila, Carlos G.; ...

2015-07-31

Our paper describes the formulation of an interface damage model, based on the discontinuous Galerkin (DG) method, for the simulation of failure and crack propagation in laminated structures. The DG formulation avoids common difficulties associated with cohesive elements. Specifically, it does not introduce any artificial interfacial compliance and, in explicit dynamic analysis, it leads to a stable time increment size which is unaffected by the presence of stiff massless interfaces. This proposed method is implemented in a finite element setting. Convergence and accuracy are demonstrated in Mode I and mixed-mode delamination in both static and dynamic analyses. Significantly, numerical resultsmore » obtained using the proposed interface model are found to be independent of the value of the penalty factor that characterizes the DG formulation. By contrast, numerical results obtained using a classical cohesive method are found to be dependent on the cohesive penalty stiffnesses. The proposed approach is shown to yield more accurate predictions pertaining to crack propagation under mixed-mode fracture because of the advantage. Furthermore, in explicit dynamic analysis, the stable time increment size calculated with the proposed method is found to be an order of magnitude larger than the maximum allowable value for classical cohesive elements.« less

Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes

NASA Astrophysics Data System (ADS)

Liu, Yong; Shu, Chi-Wang; Zhang, Mengping

2018-02-01

We present a discontinuous Galerkin (DG) scheme with suitable quadrature rules [15] for ideal compressible magnetohydrodynamic (MHD) equations on structural meshes. The semi-discrete scheme is analyzed to be entropy stable by using the symmetrizable version of the equations as introduced by Godunov [32], the entropy stable DG framework with suitable quadrature rules [15], the entropy conservative flux in [14] inside each cell and the entropy dissipative approximate Godunov type numerical flux at cell interfaces to make the scheme entropy stable. The main difficulty in the generalization of the results in [15] is the appearance of the non-conservative "source terms" added in the modified MHD model introduced by Godunov [32], which do not exist in the general hyperbolic system studied in [15]. Special care must be taken to discretize these "source terms" adequately so that the resulting DG scheme satisfies entropy stability. Total variation diminishing / bounded (TVD/TVB) limiters and bound-preserving limiters are applied to control spurious oscillations. We demonstrate the accuracy and robustness of this new scheme on standard MHD examples.

New Galerkin operational matrices for solving Lane-Emden type equations

NASA Astrophysics Data System (ADS)

Abd-Elhameed, W. M.; Doha, E. H.; Saad, A. S.; Bassuony, M. A.

2016-04-01

Lane-Emden type equations model many phenomena in mathematical physics and astrophysics, such as thermal explosions. This paper is concerned with introducing third and fourth kind Chebyshev-Galerkin operational matrices in order to solve such problems. The principal idea behind the suggested algorithms is based on converting the linear or nonlinear Lane-Emden problem, through the application of suitable spectral methods, into a system of linear or nonlinear equations in the expansion coefficients, which can be efficiently solved. The main advantage of the proposed algorithm in the linear case is that the resulting linear systems are specially structured, and this of course reduces the computational effort required to solve such systems. As an application, we consider the solar model polytrope with n=3 to show that the suggested solutions in this paper are in good agreement with the numerical results.

Finite element techniques for the Navier-Stokes equations in the primitive variable formulation and the vorticity stream-function formulation

NASA Technical Reports Server (NTRS)

Glaisner, F.; Tezduyar, T. E.

1987-01-01

Finite element procedures for the Navier-Stokes equations in the primitive variable formulation and the vorticity stream-function formulation have been implemented. For both formulations, streamline-upwind/Petrov-Galerkin techniques are used for the discretization of the transport equations. The main problem associated with the vorticity stream-function formulation is the lack of boundary conditions for vorticity at solid surfaces. Here an implicit treatment of the vorticity at no-slip boundaries is incorporated in a predictor-multicorrector time integration scheme. For the primitive variable formulation, mixed finite-element approximations are used. A nine-node element and a four-node + bubble element have been implemented. The latter is shown to exhibit a checkerboard pressure mode and a numerical treatment for this spurious pressure mode is proposed. The two methods are compared from the points of view of simulating internal and external flows and the possibilities of extensions to three dimensions.

A new finite element formulation for computational fluid dynamics. IX - Fourier analysis of space-time Galerkin/least-squares algorithms

NASA Technical Reports Server (NTRS)

Shakib, Farzin; Hughes, Thomas J. R.

1991-01-01

A Fourier stability and accuracy analysis of the space-time Galerkin/least-squares method as applied to a time-dependent advective-diffusive model problem is presented. Two time discretizations are studied: a constant-in-time approximation and a linear-in-time approximation. Corresponding space-time predictor multi-corrector algorithms are also derived and studied. The behavior of the space-time algorithms is compared to algorithms based on semidiscrete formulations.

ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics

NASA Astrophysics Data System (ADS)

Fambri, F.; Dumbser, M.; Köppel, S.; Rezzolla, L.; Zanotti, O.

2018-07-01

We present a new class of high-order accurate numerical algorithms for solving the equations of general-relativistic ideal magnetohydrodynamics in curved space-times. In this paper, we assume the background space-time to be given and static, i.e. we make use of the Cowling approximation. The governing partial differential equations are solved via a new family of fully discrete and arbitrary high-order accurate path-conservative discontinuous Galerkin (DG) finite-element methods combined with adaptive mesh refinement and time accurate local time-stepping. In order to deal with shock waves and other discontinuities, the high-order DG schemes are supplemented with a novel a posteriori subcell finite-volume limiter, which makes the new algorithms as robust as classical second-order total-variation diminishing finite-volume methods at shocks and discontinuities, but also as accurate as unlimited high-order DG schemes in smooth regions of the flow. We show the advantages of this new approach by means of various classical two- and three-dimensional benchmark problems on fixed space-times. Finally, we present a performance and accuracy comparisons between Runge-Kutta DG schemes and ADER high-order finite-volume schemes, showing the higher efficiency of DG schemes.

ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics

NASA Astrophysics Data System (ADS)

Fambri, F.; Dumbser, M.; Köppel, S.; Rezzolla, L.; Zanotti, O.

2018-03-01

We present a new class of high-order accurate numerical algorithms for solving the equations of general-relativistic ideal magnetohydrodynamics in curved spacetimes. In this paper we assume the background spacetime to be given and static, i.e. we make use of the Cowling approximation. The governing partial differential equations are solved via a new family of fully-discrete and arbitrary high-order accurate path-conservative discontinuous Galerkin (DG) finite-element methods combined with adaptive mesh refinement and time accurate local timestepping. In order to deal with shock waves and other discontinuities, the high-order DG schemes are supplemented with a novel a-posteriori subcell finite-volume limiter, which makes the new algorithms as robust as classical second-order total-variation diminishing finite-volume methods at shocks and discontinuities, but also as accurate as unlimited high-order DG schemes in smooth regions of the flow. We show the advantages of this new approach by means of various classical two- and three-dimensional benchmark problems on fixed spacetimes. Finally, we present a performance and accuracy comparisons between Runge-Kutta DG schemes and ADER high-order finite-volume schemes, showing the higher efficiency of DG schemes.

DNS of Low-Pressure Turbine Cascade Flows with Elevated Inflow Turbulence Using a Discontinuous-Galerkin Spectral-Element Method

NASA Technical Reports Server (NTRS)

Garai, Anirban; Diosady, Laslo T.; Murman, Scott M.; Madavan, Nateri K.

2016-01-01

Recent progress towards developing a new computational capability for accurate and efficient high-fidelity direct numerical simulation (DNS) and large-eddy simulation (LES) of turbomachinery is described. This capability is based on an entropy- stable Discontinuous-Galerkin spectral-element approach that extends to arbitrarily high orders of spatial and temporal accuracy, and is implemented in a computationally efficient manner on a modern high performance computer architecture. An inflow turbulence generation procedure based on a linear forcing approach has been incorporated in this framework and DNS conducted to study the effect of inflow turbulence on the suction- side separation bubble in low-pressure turbine (LPT) cascades. The T106 series of airfoil cascades in both lightly (T106A) and highly loaded (T106C) configurations at exit isentropic Reynolds numbers of 60,000 and 80,000, respectively, are considered. The numerical simulations are performed using 8th-order accurate spatial and 4th-order accurate temporal discretization. The changes in separation bubble topology due to elevated inflow turbulence is captured by the present method and the physical mechanisms leading to the changes are explained. The present results are in good agreement with prior numerical simulations but some expected discrepancies with the experimental data for the T106C case are noted and discussed.

Various methods of determining the natural frequencies and damping of composite cantilever plates. 2. Approximate solution by Galerkin's method for the trinomial model of damping

NASA Astrophysics Data System (ADS)

Ekel'chik, V. S.; Ryabov, V. M.

1997-01-01

The application of Kantorovich's method to a trinomial model of deformation taking into account transverse bending of a plate leads to a connected system of three ordinary differential equations of fourth order with respect to three unknown functions of the longitudinal coordinate and to the coresponding boundary conditions for them at the fixed end and on the free edge. For the approximate calculation of the frequencies and forms of natural vibrations Galerkin's method is used, and as coordinate functions we chose orthogonal Jacobi polynomials with weight function. The dimensionless frequencies depend on the magnitude of the four dimensionless complexes, three of which characterize the anisotropy of the elastic properties of the composite. For the fibrous composites used at present we determined the possible range of change of the dimensionless complexes d16 and d26 attained by oblique placement. The article examines the influence of the angle of reinforcement on some first dimensionless frequencies of a plate made of unidirectional carbon reinforced plastic. It also analyzes the asymptotics of the frequencies when the length of the plate is increased, and it shows that for strongly anisotropic material with the structure [ϕ]T the frequencies of the flexural as well as of the torsional vibrations may be substantially lower when flexural-torsional interaction is taken into account.

The Blended Finite Element Method for Multi-fluid Plasma Modeling

DTIC Science & Technology

2016-07-01

Briefing Charts 3. DATES COVERED (From - To) 07 June 2016 - 01 July 2016 4. TITLE AND SUBTITLE The Blended Finite Element Method for Multi-fluid Plasma...BLENDED FINITE ELEMENT METHOD FOR MULTI-FLUID PLASMA MODELING Éder M. Sousa1, Uri Shumlak2 1ERC INC., IN-SPACE PROPULSION BRANCH (RQRS) AIR FORCE RESEARCH...MULTI-FLUID PLASMA MODEL 2 BLENDED FINITE ELEMENT METHOD Blended Finite Element Method Nodal Continuous Galerkin Modal Discontinuous Galerkin Model

Multiphysics modeling of non-linear laser-matter interactions for optically active semiconductors

NASA Astrophysics Data System (ADS)

Kraczek, Brent; Kanp, Jaroslaw

Development of photonic devices for sensors and communications devices has been significantly enhanced by computational modeling. We present a new computational method for modelling laser propagation in optically-active semiconductors within the paraxial wave approximation (PWA). Light propagation is modeled using the Streamline-upwind/Petrov-Galerkin finite element method (FEM). Material response enters through the non-linear polarization, which serves as the right-hand side of the FEM calculation. Maxwell's equations for classical light propagation within the PWA can be written solely in terms of the electric field, producing a wave equation that is a form of the advection-diffusion-reaction equations (ADREs). This allows adaptation of the computational machinery developed for solving ADREs in fluid dynamics to light-propagation modeling. The non-linear polarization is incorporated using a flexible framework to enable the use of multiple methods for carrier-carrier interactions (e.g. relaxation-time-based or Monte Carlo) to enter through the non-linear polarization, as appropriate to the material type. We demonstrate using a simple carrier-carrier model approximating the response of GaN. Supported by ARL Materials Enterprise.

VizieR Online Data Catalog: Optically Bright extragalactic Radio Sources II (Petrov, 2013)

NASA Astrophysics Data System (ADS)

Petrov, L.

2014-06-01

The first VLBI (Very Long Baseline Interferometry) observing campaign in 2007 resulted in the detection of 398 targets with the European VLBI Network (EVN; Bourda et al., 2010, cat. J/A+A/520/A113). During the second observing campaign, a subset of 105 sources detected in the previous campaign was observed (Bourda et al., 2011, cat. J/A+A/526/A102). Their positions were derived by Petrov (2011, cat. J/AJ/142/105) and formed the OBRS-1 (Optically Bright extragalactic Radio Sources) catalog. The remaining sources were observed in the third campaign, called OBRS-2. During the OBRS-2 campaign, there were three observing sessions with 10 VLBA (Very Long Baseline Array) stations and 5-6 EVN stations from this list: EFLSBERG, MEDICINA, ONSALA60, YEBES40M, DSS63, HARTRAO, and NOTO. Observations were made on 2010 Mar 23 (session ID gc034a), on 2011 Nov 8 (gc034bcd), and on 2011 Mar 15 (gc034ef). The OBRS-2 catalog presents precise positions of the 295 extragalactic radio sources as well as median correlated flux densities at 8.4 and 2.2GHz at baseline lengths shorter than 900km and at baseline lengths longer than 5000km. (1 data file).

A New Ice-sheet / Ocean Interaction Model for Greenland Fjords using High-Order Discontinuous Galerkin Methods

NASA Astrophysics Data System (ADS)

Kopera, M. A.; Maslowski, W.; Giraldo, F.

2015-12-01

One of the key outstanding challenges in modeling of climate change and sea-level rise is the ice-sheet/ocean interaction in narrow, elongated and geometrically complicated fjords around Greenland. To address this challenge we propose a new approach, a separate fjord model using discontinuous Galerkin (DG) methods, or FDG. The goal of this project is to build a separate, high-resolution module for use in Earth System Models (ESMs) to realistically represent the fjord bathymetry, coastlines, exchanges with the outside ocean, circulation and fine-scale processes occurring within the fjord and interactions at the ice shelf interface. FDG is currently at the first stage of development. The DG method provides FDG with high-order accuracy as well as geometrical flexibility, including the capacity to handle non-conforming adaptive mesh refinement to resolve the processes occurring near the ice-sheet/ocean interface without introducing prohibitive computational costs. Another benefit of this method is its excellent performance on multi- and many-core architectures, which allows for utilizing modern high performance computing systems for high-resolution simulations. The non-hydrostatic model of the incompressible Navier-Stokes equation will account for the stationary ice-shelf with sub-shelf ocean interaction, basal melting and subglacial meltwater influx and with boundary conditions at the surface to account for floating sea ice. The boundary conditions will be provided to FDG via a flux coupler to emulate the integration with an ESM. Initially, FDG will be tested for the Sermilik Fjord settings, using real bathymetry, boundary and initial conditions, and evaluated against available observations and other model results for this fjord. The overarching goal of the project is to be able to resolve the ice-sheet/ocean interactions around the entire coast of Greenland and two-way coupling with regional and global climate models such as the Regional Arctic System Model (RASM

Large deformation of uniaxially loaded slender microbeams on the basis of modified couple stress theory: Analytical solution and Galerkin-based method

NASA Astrophysics Data System (ADS)

Kiani, Keivan

2017-09-01

Large deformation regime of micro-scale slender beam-like structures subjected to axially pointed loads is of high interest to nanotechnologists and applied mechanics community. Herein, size-dependent nonlinear governing equations are derived by employing modified couple stress theory. Under various boundary conditions, analytical relations between axially applied loads and deformations are presented. Additionally, a novel Galerkin-based assumed mode method (AMM) is established to solve the highly nonlinear equations. In some particular cases, the predicted results by the analytical approach are also checked with those of AMM and a reasonably good agreement is reported. Subsequently, the key role of the material length scale on the load-deformation of microbeams is discussed and the deficiencies of the classical elasticity theory in predicting such a crucial mechanical behavior are explained in some detail. The influences of slenderness ratio and thickness of the microbeam on the obtained results are also examined. The present work could be considered as a pivotal step in better realizing the postbuckling behavior of nano-/micro- electro-mechanical systems consist of microbeams.

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

NASA Astrophysics Data System (ADS)

Wintermeyer, Niklas; Winters, Andrew R.; Gassner, Gregor J.; Kopriva, David A.

2017-07-01

We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretization exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretization of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem.

An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes - III. Viscoelastic attenuation

NASA Astrophysics Data System (ADS)

Käser, Martin; Dumbser, Michael; de la Puente, Josep; Igel, Heiner

2007-01-01

We present a new numerical method to solve the heterogeneous anelastic, seismic wave equations with arbitrary high order accuracy in space and time on 3-D unstructured tetrahedral meshes. Using the velocity-stress formulation provides a linear hyperbolic system of equations with source terms that is completed by additional equations for the anelastic functions including the strain history of the material. These additional equations result from the rheological model of the generalized Maxwell body and permit the incorporation of realistic attenuation properties of viscoelastic material accounting for the behaviour of elastic solids and viscous fluids. The proposed method combines the Discontinuous Galerkin (DG) finite element (FE) method with the ADER approach using Arbitrary high order DERivatives for flux calculations. The DG approach, in contrast to classical FE methods, uses a piecewise polynomial approximation of the numerical solution which allows for discontinuities at element interfaces. Therefore, the well-established theory of numerical fluxes across element interfaces obtained by the solution of Riemann problems can be applied as in the finite volume framework. The main idea of the ADER time integration approach is a Taylor expansion in time in which all time derivatives are replaced by space derivatives using the so-called Cauchy-Kovalewski procedure which makes extensive use of the governing PDE. Due to the ADER time integration technique the same approximation order in space and time is achieved automatically and the method is a one-step scheme advancing the solution for one time step without intermediate stages. To this end, we introduce a new unrolled recursive algorithm for efficiently computing the Cauchy-Kovalewski procedure by making use of the sparsity of the system matrices. The numerical convergence analysis demonstrates that the new schemes provide very high order accuracy even on unstructured tetrahedral meshes while computational cost and

A Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin for Diffusion

NASA Technical Reports Server (NTRS)

Huynh, H. T.

2009-01-01

We introduce a new approach to high-order accuracy for the numerical solution of diffusion problems by solving the equations in differential form using a reconstruction technique. The approach has the advantages of simplicity and economy. It results in several new high-order methods including a simplified version of discontinuous Galerkin (DG). It also leads to new definitions of common value and common gradient quantities at each interface shared by the two adjacent cells. In addition, the new approach clarifies the relations among the various choices of new and existing common quantities. Fourier stability and accuracy analyses are carried out for the resulting schemes. Extensions to the case of quadrilateral meshes are obtained via tensor products. For the two-point boundary value problem (steady state), it is shown that these schemes, which include most popular DG methods, yield exact common interface quantities as well as exact cell average solutions for nearly all cases.

Numerical analysis of heat treatment of TiCN coated AA7075 aluminium alloy

NASA Astrophysics Data System (ADS)

Srinath, M. K.; Prasad, M. S. Ganesha

2018-04-01

The Numerical analysis of heat treatments of TiCN coated AA7075 aluminium alloys is presented in this paper. The Convection-Diffusion-Reaction (CDR) equation with solutions in the Streamlined-Upward Petrov-Galerkin (SUPG) method for different parameters is provided for the understanding of the process. An experimental process to improve the surface properties of AA-7075 aluminium alloy was attempted through the coatings of TiCN and subsequent heat treatments. From the experimental process, optimized temperature and time was obtained which gave the maximum surface hardness and corrosion resistance. The paper gives an understanding and use of the CDR equation for application of the process. Expression to determine convection, diffusion and reaction parameters are provided which is used to obtain the overall expression of the heat treatment process. With the substitution of the optimized temperature and time, the governing equation may be obtained. Additionally, the total energy consumed during the heat treatment process is also developed to give a mathematical formulation of the energy consumed.

Extending the Riemann-Solver-Free High-Order Space-Time Discontinuous Galerkin Cell Vertex Scheme (DG-CVS) to Solve Compressible Magnetohydrodynamics Equations

DTIC Science & Technology

2016-06-08

forces. Plasmas in hypersonic and astrophysical flows are one of the most typical examples of such conductive fluids. Though MHD models are a low...remain powerful tools in helping researchers to understand the complex physical processes in the geospace environment. For example, the ideal MHD...vertex level within each physical time step. For this reason and the method’s DG ingredient, the method was named as the space-time discontinuous Galerkin

An accurate discontinuous Galerkin method for solving point-source Eikonal equation in 2-D heterogeneous anisotropic media

NASA Astrophysics Data System (ADS)

Le Bouteiller, P.; Benjemaa, M.; Métivier, L.; Virieux, J.

2018-03-01

Accurate numerical computation of wave traveltimes in heterogeneous media is of major interest for a large range of applications in seismics, such as phase identification, data windowing, traveltime tomography and seismic imaging. A high level of precision is needed for traveltimes and their derivatives in applications which require quantities such as amplitude or take-off angle. Even more challenging is the anisotropic case, where the general Eikonal equation is a quartic in the derivatives of traveltimes. Despite their efficiency on Cartesian meshes, finite-difference solvers are inappropriate when dealing with unstructured meshes and irregular topographies. Moreover, reaching high orders of accuracy generally requires wide stencils and high additional computational load. To go beyond these limitations, we propose a discontinuous-finite-element-based strategy which has the following advantages: (1) the Hamiltonian formalism is general enough for handling the full anisotropic Eikonal equations; (2) the scheme is suitable for any desired high-order formulation or mixing of orders (p-adaptivity); (3) the solver is explicit whatever Hamiltonian is used (no need to find the roots of the quartic); (4) the use of unstructured meshes provides the flexibility for handling complex boundary geometries such as topographies (h-adaptivity) and radiation boundary conditions for mimicking an infinite medium. The point-source factorization principles are extended to this discontinuous Galerkin formulation. Extensive tests in smooth analytical media demonstrate the high accuracy of the method. Simulations in strongly heterogeneous media illustrate the solver robustness to realistic Earth-sciences-oriented applications.

Simulation d'ecoulements internes compressibles laminaires et turbulents par une methode d'elements finis

NASA Astrophysics Data System (ADS)

Rebaine, Ali

1997-08-01

Ce travail consiste en la simulation numerique des ecoulements internes compressibles bidimensionnels laminaires et turbulents. On s'interesse, particulierement, aux ecoulements dans les ejecteurs supersoniques. Les equations de Navier-Stokes sont formulees sous forme conservative et utilisent, comme variables independantes, les variables dites enthalpiques a savoir: la pression statique, la quantite de mouvement et l'enthalpie totale specifique. Une formulation variationnelle stable des equations de Navier-Stokes est utilisee. Elle est base sur la methode SUPG (Streamline Upwinding Petrov Galerkin) et utilise un operateur de capture des forts gradients. Un modele de turbulence, pour la simulation des ecoulements dans les ejecteurs, est mis au point. Il consiste a separer deux regions distinctes: une region proche de la paroi solide, ou le modele de Baldwin et Lomax est utilise et l'autre, loin de la paroi, ou une formulation nouvelle, basee sur le modele de Schlichting pour les jets, est proposee. Une technique de calcul de la viscosite turbulente, sur un maillage non structure, est implementee. La discretisation dans l'espace de la forme variationnelle est faite a l'aide de la methode des elements finis en utilisant une approximation mixte: quadratique pour les composantes de la quantite de mouvement et de la vitesse et lineaire pour le reste des variables. La discretisation temporelle est effectuee par une methode de differences finies en utilisant le schema d'Euler implicite. Le systeme matriciel, resultant de la discretisation spatio-temporelle, est resolu a l'aide de l'algorithme GMRES en utilisant un preconditionneur diagonal. Les validations numeriques ont ete menees sur plusieurs types de tuyeres et ejecteurs. La principale validation consiste en la simulation de l'ecoulement dans l'ejecteur teste au centre de recherche NASA Lewis. Les resultats obtenus sont tres comparables avec ceux des travaux anterieurs et sont nettement superieurs concernant les

OpenACC acceleration of an unstructured CFD solver based on a reconstructed discontinuous Galerkin method for compressible flows

DOE PAGES

Xia, Yidong; Lou, Jialin; Luo, Hong; ...

2015-02-09

Here, an OpenACC directive-based graphics processing unit (GPU) parallel scheme is presented for solving the compressible Navier–Stokes equations on 3D hybrid unstructured grids with a third-order reconstructed discontinuous Galerkin method. The developed scheme requires the minimum code intrusion and algorithm alteration for upgrading a legacy solver with the GPU computing capability at very little extra effort in programming, which leads to a unified and portable code development strategy. A face coloring algorithm is adopted to eliminate the memory contention because of the threading of internal and boundary face integrals. A number of flow problems are presented to verify the implementationmore » of the developed scheme. Timing measurements were obtained by running the resulting GPU code on one Nvidia Tesla K20c GPU card (Nvidia Corporation, Santa Clara, CA, USA) and compared with those obtained by running the equivalent Message Passing Interface (MPI) parallel CPU code on a compute node (consisting of two AMD Opteron 6128 eight-core CPUs (Advanced Micro Devices, Inc., Sunnyvale, CA, USA)). Speedup factors of up to 24× and 1.6× for the GPU code were achieved with respect to one and 16 CPU cores, respectively. The numerical results indicate that this OpenACC-based parallel scheme is an effective and extensible approach to port unstructured high-order CFD solvers to GPU computing.« less

Fully-Implicit Orthogonal Reconstructed Discontinuous Galerkin for Fluid Dynamics with Phase Change

DOE PAGES

Nourgaliev, R.; Luo, H.; Weston, B.; ...

2015-11-11

A new reconstructed Discontinuous Galerkin (rDG) method, based on orthogonal basis/test functions, is developed for fluid flows on unstructured meshes. Orthogonality of basis functions is essential for enabling robust and efficient fully-implicit Newton-Krylov based time integration. The method is designed for generic partial differential equations, including transient, hyperbolic, parabolic or elliptic operators, which are attributed to many multiphysics problems. We demonstrate the method’s capabilities for solving compressible fluid-solid systems (in the low Mach number limit), with phase change (melting/solidification), as motivated by applications in Additive Manufacturing (AM). We focus on the method’s accuracy (in both space and time), as wellmore » as robustness and solvability of the system of linear equations involved in the linearization steps of Newton-based methods. The performance of the developed method is investigated for highly-stiff problems with melting/solidification, emphasizing the advantages from tight coupling of mass, momentum and energy conservation equations, as well as orthogonality of basis functions, which leads to better conditioning of the underlying (approximate) Jacobian matrices, and rapid convergence of the Krylov-based linear solver.« less

A mass-energy preserving Galerkin FEM for the coupled nonlinear fractional Schrödinger equations

NASA Astrophysics Data System (ADS)

Zhang, Guoyu; Huang, Chengming; Li, Meng

2018-04-01

We consider the numerical simulation of the coupled nonlinear space fractional Schrödinger equations. Based on the Galerkin finite element method in space and the Crank-Nicolson (CN) difference method in time, a fully discrete scheme is constructed. Firstly, we focus on a rigorous analysis of conservation laws for the discrete system. The definitions of discrete mass and energy here correspond with the original ones in physics. Then, we prove that the fully discrete system is uniquely solvable. Moreover, we consider the unconditionally convergent properties (that is to say, we complete the error estimates without any mesh ratio restriction). We derive L2-norm error estimates for the nonlinear equations and L^{∞}-norm error estimates for the linear equations. Finally, some numerical experiments are included showing results in agreement with the theoretical predictions.

A new limiting procedure for discontinuous Galerkin methods applied to compressible multiphase flows with shocks and interfaces

NASA Astrophysics Data System (ADS)

Henry de Frahan, Marc T.; Varadan, Sreenivas; Johnsen, Eric

2015-01-01

Although the Discontinuous Galerkin (DG) method has seen widespread use for compressible flow problems in a single fluid with constant material properties, it has yet to be implemented in a consistent fashion for compressible multiphase flows with shocks and interfaces. Specifically, it is challenging to design a scheme that meets the following requirements: conservation, high-order accuracy in smooth regions and non-oscillatory behavior at discontinuities (in particular, material interfaces). Following the interface-capturing approach of Abgrall [1], we model flows of multiple fluid components or phases using a single equation of state with variable material properties; discontinuities in these properties correspond to interfaces. To represent compressible phenomena in solids, liquids, and gases, we present our analysis for equations of state belonging to the Mie-Grüneisen family. Within the DG framework, we propose a conservative, high-order accurate, and non-oscillatory limiting procedure, verified with simple multifluid and multiphase problems. We show analytically that two key elements are required to prevent spurious pressure oscillations at interfaces and maintain conservation: (i) the transport equation(s) describing the material properties must be solved in a non-conservative weak form, and (ii) the suitable variables must be limited (density, momentum, pressure, and appropriate properties entering the equation of state), coupled with a consistent reconstruction of the energy. Further, we introduce a physics-based discontinuity sensor to apply limiting in a solution-adaptive fashion. We verify this approach with one- and two-dimensional problems with shocks and interfaces, including high pressure and density ratios, for fluids obeying different equations of state to illustrate the robustness and versatility of the method. The algorithm is implemented on parallel graphics processing units (GPU) to achieve high speedup.

MPSalsa Version 1.5: A Finite Element Computer Program for Reacting Flow Problems: Part 1 - Theoretical Development

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

Devine, K.D.; Hennigan, G.L.; Hutchinson, S.A.

1999-01-01

The theoretical background for the finite element computer program, MPSalsa Version 1.5, is presented in detail. MPSalsa is designed to solve laminar or turbulent low Mach number, two- or three-dimensional incompressible and variable density reacting fluid flows on massively parallel computers, using a Petrov-Galerkin finite element formulation. The code has the capability to solve coupled fluid flow (with auxiliary turbulence equations), heat transport, multicomponent species transport, and finite-rate chemical reactions, and to solve coupled multiple Poisson or advection-diffusion-reaction equations. The program employs the CHEMKIN library to provide a rigorous treatment of multicomponent ideal gas kinetics and transport. Chemical reactions occurringmore » in the gas phase and on surfaces are treated by calls to CHEMKIN and SURFACE CHEMK3N, respectively. The code employs unstructured meshes, using the EXODUS II finite element database suite of programs for its input and output files. MPSalsa solves both transient and steady flows by using fully implicit time integration, an inexact Newton method and iterative solvers based on preconditioned Krylov methods as implemented in the Aztec. solver library.« less

Simulations on Monitoring and Evaluation of Plasticity-Driven Material Damage Based on Second Harmonic of S0 Mode Lamb Waves in Metallic Plates

PubMed Central

Sun, Xiaoqiang; Liu, Xuyang; Liu, Yaolu; Hu, Ning; Zhao, Youxuan; Ding, Xiangyan; Qin, Shiwei; Zhang, Jianyu; Zhang, Jun; Liu, Feng; Fu, Shaoyun

2017-01-01

In this study, a numerical approach—the discontinuous Meshless Local Petrov-Galerkin-Eshelby Method (MLPGEM)—was adopted to simulate and measure material plasticity in an Al 7075-T651 plate. The plate was modeled in two dimensions by assemblies of small particles that interact with each other through bonding stiffness. The material plasticity of the model loaded to produce different levels of strain is evaluated with the Lamb waves of S0 mode. A tone burst at the center frequency of 200 kHz was used as excitation. Second-order nonlinear wave was extracted from the spectrogram of a signal receiving point. Tensile-driven plastic deformation and cumulative second harmonic generation of S0 mode were observed in the simulation. Simulated measurement of the acoustic nonlinearity increased monotonically with the level of tensile-driven plastic strain captured by MLPGEM, whereas achieving this state by other numerical methods is comparatively more difficult. This result indicates that the second harmonics of S0 mode can be employed to monitor and evaluate the material or structural early-stage damage induced by plasticity. PMID:28773188

On the efficacy of stochastic collocation, stochastic Galerkin, and stochastic reduced order models for solving stochastic problems

DOE PAGES

Richard V. Field, Jr.; Emery, John M.; Grigoriu, Mircea Dan

2015-05-19

The stochastic collocation (SC) and stochastic Galerkin (SG) methods are two well-established and successful approaches for solving general stochastic problems. A recently developed method based on stochastic reduced order models (SROMs) can also be used. Herein we provide a comparison of the three methods for some numerical examples; our evaluation only holds for the examples considered in the paper. The purpose of the comparisons is not to criticize the SC or SG methods, which have proven very useful for a broad range of applications, nor is it to provide overall ratings of these methods as compared to the SROM method.more » Furthermore, our objectives are to present the SROM method as an alternative approach to solving stochastic problems and provide information on the computational effort required by the implementation of each method, while simultaneously assessing their performance for a collection of specific problems.« less

Local Discontinuous Galerkin (LDG) Method for Advection of Active Compositional Fields with Discontinuous Boundaries: Demonstration and Comparison with Other Methods in the Mantle Convection Code ASPECT

NASA Astrophysics Data System (ADS)

He, Y.; Billen, M. I.; Puckett, E. G.

2015-12-01

Flow in the Earth's mantle is driven by thermo-chemical convection in which the properties and geochemical signatures of rocks vary depending on their origin and composition. For example, tectonic plates are composed of compositionally-distinct layers of crust, residual lithosphere and fertile mantle, while in the lower-most mantle there are large compositionally distinct "piles" with thinner lenses of different material. Therefore, tracking of active or passive fields with distinct compositional, geochemical or rheologic properties is important for incorporating physical realism into mantle convection simulations, and for investigating the long term mixing properties of the mantle. The difficulty in numerically advecting fields arises because they are non-diffusive and have sharp boundaries, and therefore require different methods than usually used for temperature. Previous methods for tracking fields include the marker-chain, tracer particle, and field-correction (e.g., the Lenardic Filter) methods: each of these has different advantages or disadvantages, trading off computational speed with accuracy in tracking feature boundaries. Here we present a method for modeling active fields in mantle dynamics simulations using a new solver implemented in the deal.II package that underlies the ASPECT software. The new solver for the advection-diffusion equation uses a Local Discontinuous Galerkin (LDG) algorithm, which combines features of both finite element and finite volume methods, and is particularly suitable for problems with a dominant first-order term and discontinuities. Furthermore, we have applied a post-processing technique to insure that the solution satisfies a global maximum/minimum. One potential drawback for the LDG method is that the total number of degrees of freedom is larger than the finite element method. To demonstrate the capabilities of this new method we present results for two benchmarks used previously: a falling cube with distinct buoyancy and

Application of wall-models to discontinuous Galerkin LES

NASA Astrophysics Data System (ADS)

Frère, Ariane; Carton de Wiart, Corentin; Hillewaert, Koen; Chatelain, Philippe; Winckelmans, Grégoire

2017-08-01

Wall-resolved Large-Eddy Simulations (LES) are still limited to moderate Reynolds number flows due to the high computational cost required to capture the inner part of the boundary layer. Wall-modeled LES (WMLES) provide more affordable LES by modeling the near-wall layer. Wall function-based WMLES solve LES equations up to the wall, where the coarse mesh resolution essentially renders the calculation under-resolved. This makes the accuracy of WMLES very sensitive to the behavior of the numerical method. Therefore, best practice rules regarding the use and implementation of WMLES cannot be directly transferred from one methodology to another regardless of the type of discretization approach. Whilst numerous studies present guidelines on the use of WMLES, there is a lack of knowledge for discontinuous finite-element-like high-order methods. Incidentally, these methods are increasingly used on the account of their high accuracy on unstructured meshes and their strong computational efficiency. The present paper proposes best practice guidelines for the use of WMLES in these methods. The study is based on sensitivity analyses of turbulent channel flow simulations by means of a Discontinuous Galerkin approach. It appears that good results can be obtained without the use of a spatial or temporal averaging. The study confirms the importance of the wall function input data location and suggests to take it at the bottom of the second off-wall element. These data being available through the ghost element, the suggested method prevents the loss of computational scalability experienced in unstructured WMLES. The study also highlights the influence of the polynomial degree used in the wall-adjacent element. It should preferably be of even degree as using polynomials of degree two in the first off-wall element provides, surprisingly, better results than using polynomials of degree three.

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.

Parallel adaptive discontinuous Galerkin approximation for thin layer avalanche modeling

NASA Astrophysics Data System (ADS)

Patra, A. K.; Nichita, C. C.; Bauer, A. C.; Pitman, E. B.; Bursik, M.; Sheridan, M. F.

2006-08-01

This paper describes the development of highly accurate adaptive discontinuous Galerkin schemes for the solution of the equations arising from a thin layer type model of debris flows. Such flows have wide applicability in the analysis of avalanches induced by many natural calamities, e.g. volcanoes, earthquakes, etc. These schemes are coupled with special parallel solution methodologies to produce a simulation tool capable of very high-order numerical accuracy. The methodology successfully replicates cold rock avalanches at Mount Rainier, Washington and hot volcanic particulate flows at Colima Volcano, Mexico.

Propel: A Discontinuous-Galerkin Finite Element Code for Solving the Reacting Navier-Stokes Equations

NASA Astrophysics Data System (ADS)

Johnson, Ryan; Kercher, Andrew; Schwer, Douglas; Corrigan, Andrew; Kailasanath, Kazhikathra

2017-11-01

This presentation focuses on the development of a Discontinuous Galerkin (DG) method for application to chemically reacting flows. The in-house code, called Propel, was developed by the Laboratory of Computational Physics and Fluid Dynamics at the Naval Research Laboratory. It was designed specifically for developing advanced multi-dimensional algorithms to run efficiently on new and innovative architectures such as GPUs. For these results, Propel solves for convection and diffusion simultaneously with detailed transport and thermodynamics. Chemistry is currently solved in a time-split approach using Strang-splitting with finite element DG time integration of chemical source terms. Results presented here show canonical unsteady reacting flow cases, such as co-flow and splitter plate, and we report performance for higher order DG on CPU and GPUs.

Discontinuous Galerkin (DG) Method for solving time dependent convection-diffusion type temperature equation : Demonstration and Comparison with Other Methods in the Mantle Convection Code ASPECT

NASA Astrophysics Data System (ADS)

He, Y.; Puckett, E. G.; Billen, M. I.; Kellogg, L. H.

2016-12-01

For a convection-dominated system, like convection in the Earth's mantle, accurate modeling of the temperature field in terms of the interaction between convective and diffusive processes is one of the most common numerical challenges. In the geodynamics community using Finite Element Method (FEM) with artificial entropy viscosity is a popular approach to resolve this difficulty, but introduce numerical diffusion. The extra artificial viscosity added into the temperature system will not only oversmooth the temperature field where the convective process dominates, but also change the physical properties by increasing the local material conductivity, which will eventually change the local conservation of energy. Accurate modeling of temperature is especially important in the mantle, where material properties are strongly dependent on temperature. In subduction zones, for example, the rheology of the cold sinking slab depends nonlinearly on the temperature, and physical processes such as slab detachment, rollback, and melting all are sensitively dependent on temperature and rheology. Therefore methods that overly smooth the temperature may inaccurately represent the physical processes governing subduction, lithospheric instabilities, plume generation and other aspects of mantle convection. Here we present a method for modeling the temperature field in mantle dynamics simulations using a new solver implemented in the ASPECT software. The new solver for the temperature equation uses a Discontinuous Galerkin (DG) approach, which combines features of both finite element and finite volume methods, and is particularly suitable for problems satisfying the conservation law, and the solution has a large variation locally. Furthermore, we have applied a post-processing technique to insure that the solution satisfies a local discrete maximum principle in order to eliminate the overshoots and undershoots in the temperature locally. To demonstrate the capabilities of this new

Modelling uncertainty in incompressible flow simulation using Galerkin based generalized ANOVA

NASA Astrophysics Data System (ADS)

Chakraborty, Souvik; Chowdhury, Rajib

2016-11-01

This paper presents a new algorithm, referred to here as Galerkin based generalized analysis of variance decomposition (GG-ANOVA) for modelling input uncertainties and its propagation in incompressible fluid flow. The proposed approach utilizes ANOVA to represent the unknown stochastic response. Further, the unknown component functions of ANOVA are represented using the generalized polynomial chaos expansion (PCE). The resulting functional form obtained by coupling the ANOVA and PCE is substituted into the stochastic Navier-Stokes equation (NSE) and Galerkin projection is employed to decompose it into a set of coupled deterministic 'Navier-Stokes alike' equations. Temporal discretization of the set of coupled deterministic equations is performed by employing Adams-Bashforth scheme for convective term and Crank-Nicolson scheme for diffusion term. Spatial discretization is performed by employing finite difference scheme. Implementation of the proposed approach has been illustrated by two examples. In the first example, a stochastic ordinary differential equation has been considered. This example illustrates the performance of proposed approach with change in nature of random variable. Furthermore, convergence characteristics of GG-ANOVA has also been demonstrated. The second example investigates flow through a micro channel. Two case studies, namely the stochastic Kelvin-Helmholtz instability and stochastic vortex dipole, have been investigated. For all the problems results obtained using GG-ANOVA are in excellent agreement with benchmark solutions.

A collocation--Galerkin finite element model of cardiac action potential propagation.

PubMed

Rogers, J M; McCulloch, A D

1994-08-01

A new computational method was developed for modeling the effects of the geometric complexity, nonuniform muscle fiber orientation, and material inhomogeneity of the ventricular wall on cardiac impulse propagation. The method was used to solve a modification to the FitzHugh-Nagumo system of equations. The geometry, local muscle fiber orientation, and material parameters of the domain were defined using linear Lagrange or cubic Hermite finite element interpolation. Spatial variations of time-dependent excitation and recovery variables were approximated using cubic Hermite finite element interpolation, and the governing finite element equations were assembled using the collocation method. To overcome the deficiencies of conventional collocation methods on irregular domains, Galerkin equations for the no-flux boundary conditions were used instead of collocation equations for the boundary degrees-of-freedom. The resulting system was evolved using an adaptive Runge-Kutta method. Converged two-dimensional simulations of normal propagation showed that this method requires less CPU time than a traditional finite difference discretization. The model also reproduced several other physiologic phenomena known to be important in arrhythmogenesis including: Wenckebach periodicity, slowed propagation and unidirectional block due to wavefront curvature, reentry around a fixed obstacle, and spiral wave reentry. In a new result, we observed wavespeed variations and block due to nonuniform muscle fiber orientation. The findings suggest that the finite element method is suitable for studying normal and pathological cardiac activation and has significant advantages over existing techniques.

Dynamic mesh adaptation for front evolution using discontinuous Galerkin based weighted condition number relaxation

DOE PAGES

Greene, Patrick T.; Schofield, Samuel P.; Nourgaliev, Robert

2017-01-27

A new mesh smoothing method designed to cluster cells near a dynamically evolving interface is presented. The method is based on weighted condition number mesh relaxation with the weight function computed from a level set representation of the interface. The weight function is expressed as a Taylor series based discontinuous Galerkin projection, which makes the computation of the derivatives of the weight function needed during the condition number optimization process a trivial matter. For cases when a level set is not available, a fast method for generating a low-order level set from discrete cell-centered fields, such as a volume fractionmore » or index function, is provided. Results show that the low-order level set works equally well as the actual level set for mesh smoothing. Meshes generated for a number of interface geometries are presented, including cases with multiple level sets. Lastly, dynamic cases with moving interfaces show the new method is capable of maintaining a desired resolution near the interface with an acceptable number of relaxation iterations per time step, which demonstrates the method's potential to be used as a mesh relaxer for arbitrary Lagrangian Eulerian (ALE) methods.« less

Dynamic Mesh Adaptation for Front Evolution Using Discontinuous Galerkin Based Weighted Condition Number Mesh Relaxation

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

Greene, Patrick T.; Schofield, Samuel P.; Nourgaliev, Robert

2016-06-21

A new mesh smoothing method designed to cluster mesh cells near a dynamically evolving interface is presented. The method is based on weighted condition number mesh relaxation with the weight function being computed from a level set representation of the interface. The weight function is expressed as a Taylor series based discontinuous Galerkin projection, which makes the computation of the derivatives of the weight function needed during the condition number optimization process a trivial matter. For cases when a level set is not available, a fast method for generating a low-order level set from discrete cell-centered elds, such as amore » volume fraction or index function, is provided. Results show that the low-order level set works equally well for the weight function as the actual level set. Meshes generated for a number of interface geometries are presented, including cases with multiple level sets. Dynamic cases for moving interfaces are presented to demonstrate the method's potential usefulness to arbitrary Lagrangian Eulerian (ALE) methods.« less

Galerkin analysis of kinematic dynamos in the von Kármán geometry

NASA Astrophysics Data System (ADS)

Marié, L.; Normand, C.; Daviaud, F.

2006-01-01

We investigate dynamo action by solving the kinematic dynamo problem for velocity fields of the von Kármán type between two coaxial counter-rotating propellers in a cylinder. A Galerkin method is implemented that takes advantage of the symmetries of the flow and their subsequent influence on the nature of the magnetic field at the dynamo threshold. Distinct modes of instability have been identified that differ by their spatial and temporal behaviors. Our calculations give the result that a stationary and antisymmetric mode prevails at the dynamo threshold. We then present a quantitative analysis of the results based on the parametric study of four interaction coefficients obtained by reduction of our initially large eigenvalue problem. We propose these coefficients to measure the relative importance of the different mechanisms at play in the von Kármán kinematic dynamo.

An adaptive discontinuous Galerkin solver for aerodynamic flows

NASA Astrophysics Data System (ADS)

Burgess, Nicholas K.

This work considers the accuracy, efficiency, and robustness of an unstructured high-order accurate discontinuous Galerkin (DG) solver for computational fluid dynamics (CFD). Recently, there has been a drive to reduce the discretization error of CFD simulations using high-order methods on unstructured grids. However, high-order methods are often criticized for lacking robustness and having high computational cost. The goal of this work is to investigate methods that enhance the robustness of high-order discontinuous Galerkin (DG) methods on unstructured meshes, while maintaining low computational cost and high accuracy of the numerical solutions. This work investigates robustness enhancement of high-order methods by examining effective non-linear solvers, shock capturing methods, turbulence model discretizations and adaptive refinement techniques. The goal is to develop an all encompassing solver that can simulate a large range of physical phenomena, where all aspects of the solver work together to achieve a robust, efficient and accurate solution strategy. The components and framework for a robust high-order accurate solver that is capable of solving viscous, Reynolds Averaged Navier-Stokes (RANS) and shocked flows is presented. In particular, this work discusses robust discretizations of the turbulence model equation used to close the RANS equations, as well as stable shock capturing strategies that are applicable across a wide range of discretization orders and applicable to very strong shock waves. Furthermore, refinement techniques are considered as both efficiency and robustness enhancement strategies. Additionally, efficient non-linear solvers based on multigrid and Krylov subspace methods are presented. The accuracy, efficiency, and robustness of the solver is demonstrated using a variety of challenging aerodynamic test problems, which include turbulent high-lift and viscous hypersonic flows. Adaptive mesh refinement was found to play a critical role in

Finite-dimensional compensators for infinite-dimensional systems via Galerkin-type approximation

NASA Technical Reports Server (NTRS)

Ito, Kazufumi

1990-01-01

In this paper existence and construction of stabilizing compensators for linear time-invariant systems defined on Hilbert spaces are discussed. An existence result is established using Galkerin-type approximations in which independent basis elements are used instead of the complete set of eigenvectors. A design procedure based on approximate solutions of the optimal regulator and optimal observer via Galerkin-type approximation is given and the Schumacher approach is used to reduce the dimension of compensators. A detailed discussion for parabolic and hereditary differential systems is included.

A discontinuous Galerkin method with a bound preserving limiter for the advection of non-diffusive fields in solid Earth geodynamics

NASA Astrophysics Data System (ADS)

He, Ying; Puckett, Elbridge Gerry; Billen, Magali I.

2017-02-01

Mineral composition has a strong effect on the properties of rocks and is an essentially non-diffusive property in the context of large-scale mantle convection. Due to the non-diffusive nature and the origin of compositionally distinct regions in the Earth the boundaries between distinct regions can be nearly discontinuous. While there are different methods for tracking rock composition in numerical simulations of mantle convection, one must consider trade-offs between computational cost, accuracy or ease of implementation when choosing an appropriate method. Existing methods can be computationally expensive, cause over-/undershoots, smear sharp boundaries, or are not easily adapted to tracking multiple compositional fields. Here we present a Discontinuous Galerkin method with a bound preserving limiter (abbreviated as DG-BP) using a second order Runge-Kutta, strong stability-preserving time discretization method for the advection of non-diffusive fields. First, we show that the method is bound-preserving for a point-wise divergence free flow (e.g., a prescribed circular flow in a box). However, using standard adaptive mesh refinement (AMR) there is an over-shoot error (2%) because the cell average is not preserved during mesh coarsening. The effectiveness of the algorithm for convection-dominated flows is demonstrated using the falling box problem. We find that the DG-BP method maintains sharper compositional boundaries (3-5 elements) as compared to an artificial entropy-viscosity method (6-15 elements), although the over-/undershoot errors are similar. When used with AMR the DG-BP method results in fewer degrees of freedom due to smaller regions of mesh refinement in the neighborhood of the discontinuity. However, using Taylor-Hood elements and a uniform mesh there is an over-/undershoot error on the order of 0.0001%, but this error increases to 0.01-0.10% when using AMR. Therefore, for research problems in which a continuous field method is desired the DG

Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation

NASA Technical Reports Server (NTRS)

Helenbrook, B. T.; Atkins, H. L.

2006-01-01

We investigate p-multigrid as a solution method for several different discontinuous Galerkin (DG) formulations of the Poisson equation. Different combinations of relaxation schemes and basis sets have been combined with the DG formulations to find the best performing combination. The damping factors of the schemes have been determined using Fourier analysis for both one and two-dimensional problems. One important finding is that when using DG formulations, the standard approach of forming the coarse p matrices separately for each level of multigrid is often unstable. To ensure stability the coarse p matrices must be constructed from the fine grid matrices using algebraic multigrid techniques. Of the relaxation schemes, we find that the combination of Jacobi relaxation with the spectral element basis is fairly effective. The results using this combination are p sensitive in both one and two dimensions, but reasonable convergence rates can still be achieved for moderate values of p and isotropic meshes. A competitive alternative is a block Gauss-Seidel relaxation. This actually out performs a more expensive line relaxation when the mesh is isotropic. When the mesh becomes highly anisotropic, the implicit line method and the Gauss-Seidel implicit line method are the only effective schemes. Adding the Gauss-Seidel terms to the implicit line method gives a significant improvement over the line relaxation method.

Weak solution concept and Galerkin's matrix for the exterior of an oblate ellipsoid of revolution in the representation of the Earth's gravity potential by buried masses

NASA Astrophysics Data System (ADS)

Holota, Petr; Nesvadba, Otakar

2017-04-01

The paper is motivated by the role of boundary value problems in Earth's gravity field studies. The discussion focuses on Neumann's problem formulated for the exterior of an oblate ellipsoid of revolution as this is considered a basis for an iteration solution of the linear gravimetric boundary value problem in the determination of the disturbing potential. The approach follows the concept of the weak solution and Galerkin's approximations are applied. This means that the solution of the problem is approximated by linear combinations of basis functions with scalar coefficients. The construction of Galerkin's matrix for basis functions generated by elementary potentials (point masses) is discussed. Ellipsoidal harmonics are used as a natural tool and the elementary potentials are expressed by means of series of ellipsoidal harmonics. The problem, however, is the summation of the series that represent the entries of Galerkin's matrix. It is difficult to reduce the number of summation indices since in the ellipsoidal case there is no analogue to the addition theorem known for spherical harmonics. Therefore, the straightforward application of series of ellipsoidal harmonics is complemented by deeper relations contained in the theory of ordinary differential equations of second order and in the theory of Legendre's functions. Subsequently, also hypergeometric functions and series are used. Moreover, within some approximations the entries are split into parts. Some of the resulting series may be summed relatively easily, apart from technical tricks. For the remaining series the summation was converted to elliptic integrals. The approach made it possible to deduce a closed (though approximate) form representation of the entries in Galerkin's matrix. The result rests on concepts and methods of mathematical analysis. In the paper it is confronted with a direct numerical approach applied for the implementation of Legendre's functions. The computation of the entries is more

A comparative study of Rosenbrock-type and implicit Runge-Kutta time integration for discontinuous Galerkin method for unsteady 3D compressible Navier-Stokes equations

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

Liu, Xiaodong; Xia, Yidong; Luo, Hong

A comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flowsmore » to DNS of turbulent flows, are presented to assess the performance of these schemes. Here, numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.« less

A comparative study of Rosenbrock-type and implicit Runge-Kutta time integration for discontinuous Galerkin method for unsteady 3D compressible Navier-Stokes equations

DOE PAGES

Liu, Xiaodong; Xia, Yidong; Luo, Hong; ...

2016-10-05

A comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flowsmore » to DNS of turbulent flows, are presented to assess the performance of these schemes. Here, numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.« less

Application of variational and Galerkin equations to linear and nonlinear finite element analysis

NASA Technical Reports Server (NTRS)

Yu, Y.-Y.

1974-01-01

The paper discusses the application of the variational equation to nonlinear finite element analysis. The problem of beam vibration with large deflection is considered. The variational equation is shown to be flexible in both the solution of a general problem and in the finite element formulation. Difficulties are shown to arise when Galerkin's equations are used in the consideration of the finite element formulation of two-dimensional linear elasticity and of the linear classical beam.

Galerkin finite difference Laplacian operators on isolated unstructured triangular meshes by linear combinations

NASA Technical Reports Server (NTRS)

Baumeister, Kenneth J.

1990-01-01

The Galerkin weighted residual technique using linear triangular weight functions is employed to develop finite difference formulae in Cartesian coordinates for the Laplacian operator on isolated unstructured triangular grids. The weighted residual coefficients associated with the weak formulation of the Laplacian operator along with linear combinations of the residual equations are used to develop the algorithm. The algorithm was tested for a wide variety of unstructured meshes and found to give satisfactory results.

FFR analysis of blood flow through a stenosed Left Anterior Descending Artery

NASA Astrophysics Data System (ADS)

Pasupathi, Jawahar; Arul Prakash, K.

2017-11-01

The numerical analyisis of blood flow through a stenosed tapering Left Anterior Descending (LAD) artery was done using Streamwise Upwind Petrov Galerkin (SUPG) method to obtain the clinical parameters such as Fractional Flow reserve (FFR) and Wall Shear Stress (WSS). The geometry was considered to be a straight tapering cylindrical duct with the severity of stenosis modeled using a curve equation based on the reduction in diameter at the stenosed region. Poiseuille velocity profile was given at the inlet such that at each time step the product of mean velocity and the inlet area gives the realistic flow rate through the LAD. The simulation was done for 30,50 and 70 percent reduction in cross-section of LAD. The average pressure values across the stenosis was used to quantify FFR. The FFR increased with higher pressure ratio across the stenosis, which is a result of increased severity of stenosis. The velocity gradients that are responsible for the shear stress at the walls were found to be dependent on the shape of the stenosis, i.e., the diameter and its length.

Modeling of high pressure arc-discharge with a fully-implicit Navier-Stokes stabilized finite element flow solver

NASA Astrophysics Data System (ADS)

Sahai, A.; Mansour, N. N.; Lopez, B.; Panesi, M.

2017-05-01

This work addresses the modeling of high pressure electric discharge in an arc-heated wind tunnel. The combined numerical solution of Poisson’s equation, radiative transfer equations, and the set of Favre-averaged thermochemical nonequilibrium Navier-Stokes equations allows for the determination of the electric, radiation, and flow fields, accounting for their mutual interaction. Semi-classical statistical thermodynamics is used to determine the plasma thermodynamic properties, while transport properties are obtained from kinetic principles with the Chapman-Enskog method. A multi-temperature formulation is used to account for thermal non-equilibrium. Finally, the turbulence closure of the flow equations is obtained by means of the Spalart-Allmaras model, which requires the solution of an additional scalar transport equation. A Streamline upwind Petrov-Galerkin stabilized finite element formulation is employed to solve the Navier-Stokes equation. The electric field equation is solved using the standard Galerkin formulation. A stable formulation for the radiative transfer equations is obtained using the least-squares finite element method. The developed simulation framework has been applied to investigate turbulent plasma flows in the 20 MW Aerodynamic Heating Facility at NASA Ames Research Center. The current model is able to predict the process of energy addition and re-distribution due to Joule heating and thermal radiation, resulting in a hot central core surrounded by colder flow. The use of an unsteady three-dimensional treatment also allows the asymmetry due to a dynamic electric arc attachment point in the cathode chamber to be captured accurately. The current work paves the way for detailed estimation of operating characteristics for arc-heated wind tunnels which are critical in testing thermal protection systems.

CosmosDG: An hp-adaptive Discontinuous Galerkin Code for Hyper-resolved Relativistic MHD

NASA Astrophysics Data System (ADS)

Anninos, Peter; Bryant, Colton; Fragile, P. Chris; Holgado, A. Miguel; Lau, Cheuk; Nemergut, Daniel

2017-08-01

We have extended Cosmos++, a multidimensional unstructured adaptive mesh code for solving the covariant Newtonian and general relativistic radiation magnetohydrodynamic (MHD) equations, to accommodate both discrete finite volume and arbitrarily high-order finite element structures. The new finite element implementation, called CosmosDG, is based on a discontinuous Galerkin (DG) formulation, using both entropy-based artificial viscosity and slope limiting procedures for the regularization of shocks. High-order multistage forward Euler and strong-stability preserving Runge-Kutta time integration options complement high-order spatial discretization. We have also added flexibility in the code infrastructure allowing for both adaptive mesh and adaptive basis order refinement to be performed separately or simultaneously in a local (cell-by-cell) manner. We discuss in this report the DG formulation and present tests demonstrating the robustness, accuracy, and convergence of our numerical methods applied to special and general relativistic MHD, although we note that an equivalent capability currently also exists in CosmosDG for Newtonian systems.

CosmosDG: An hp -adaptive Discontinuous Galerkin Code for Hyper-resolved Relativistic MHD

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

Anninos, Peter; Lau, Cheuk; Bryant, Colton

We have extended Cosmos++, a multidimensional unstructured adaptive mesh code for solving the covariant Newtonian and general relativistic radiation magnetohydrodynamic (MHD) equations, to accommodate both discrete finite volume and arbitrarily high-order finite element structures. The new finite element implementation, called CosmosDG, is based on a discontinuous Galerkin (DG) formulation, using both entropy-based artificial viscosity and slope limiting procedures for the regularization of shocks. High-order multistage forward Euler and strong-stability preserving Runge–Kutta time integration options complement high-order spatial discretization. We have also added flexibility in the code infrastructure allowing for both adaptive mesh and adaptive basis order refinement to be performedmore » separately or simultaneously in a local (cell-by-cell) manner. We discuss in this report the DG formulation and present tests demonstrating the robustness, accuracy, and convergence of our numerical methods applied to special and general relativistic MHD, although we note that an equivalent capability currently also exists in CosmosDG for Newtonian systems.« less

Assessment of a Hybrid Continuous/Discontinuous Galerkin Finite Element Code for Geothermal Reservoir Simulations

DOE PAGES

Xia, Yidong; Podgorney, Robert; Huang, Hai

2016-03-17

FALCON (“Fracturing And Liquid CONvection”) is a hybrid continuous / discontinuous Galerkin finite element geothermal reservoir simulation code based on the MOOSE (“Multiphysics Object-Oriented Simulation Environment”) framework being developed and used for multiphysics applications. In the present work, a suite of verification and validation (“V&V”) test problems for FALCON was defined to meet the design requirements, and solved to the interests of enhanced geothermal system (“EGS”) design. Furthermore, the intent for this test problem suite is to provide baseline comparison data that demonstrates the performance of the FALCON solution methods. The simulation problems vary in complexity from singly mechanical ormore » thermo process, to coupled thermo-hydro-mechanical processes in geological porous media. Numerical results obtained by FALCON agreed well with either the available analytical solution or experimental data, indicating the verified and validated implementation of these capabilities in FALCON. Some form of solution verification has been attempted to identify sensitivities in the solution methods, where possible, and suggest best practices when using the FALCON code.« less

Semi-discrete Galerkin solution of the compressible boundary-layer equations with viscous-inviscid interaction

NASA Technical Reports Server (NTRS)

Day, Brad A.; Meade, Andrew J., Jr.

1993-01-01

A semi-discrete Galerkin (SDG) method is under development to model attached, turbulent, and compressible boundary layers for transonic airfoil analysis problems. For the boundary-layer formulation the method models the spatial variable normal to the surface with linear finite elements and the time-like variable with finite differences. A Dorodnitsyn transformed system of equations is used to bound the infinite spatial domain thereby providing high resolution near the wall and permitting the use of a uniform finite element grid which automatically follows boundary-layer growth. The second-order accurate Crank-Nicholson scheme is applied along with a linearization method to take advantage of the parabolic nature of the boundary-layer equations and generate a non-iterative marching routine. The SDG code can be applied to any smoothly-connected airfoil shape without modification and can be coupled to any inviscid flow solver. In this analysis, a direct viscous-inviscid interaction is accomplished between the Euler and boundary-layer codes through the application of a transpiration velocity boundary condition. Results are presented for compressible turbulent flow past RAE 2822 and NACA 0012 airfoils at various freestream Mach numbers, Reynolds numbers, and angles of attack.

Galerkin approximation for inverse problems for nonautonomous nonlinear distributed systems

NASA Technical Reports Server (NTRS)

Banks, H. T.; Reich, Simeon; Rosen, I. G.

1988-01-01

An abstract framework and convergence theory is developed for Galerkin approximation for inverse problems involving the identification of nonautonomous nonlinear distributed parameter systems. A set of relatively easily verified conditions is provided which are sufficient to guarantee the existence of optimal solutions and their approximation by a sequence of solutions to a sequence of approximating finite dimensional identification problems. The approach is based on the theory of monotone operators in Banach spaces and is applicable to a reasonably broad class of nonlinear distributed systems. Operator theoretic and variational techniques are used to establish a fundamental convergence result. An example involving evolution systems with dynamics described by nonstationary quasilinear elliptic operators along with some applications are presented and discussed.

A nodal discontinuous Galerkin approach to 3-D viscoelastic wave propagation in complex geological media

NASA Astrophysics Data System (ADS)

Lambrecht, L.; Lamert, A.; Friederich, W.; Möller, T.; Boxberg, M. S.

2018-03-01

A nodal discontinuous Galerkin (NDG) approach is developed and implemented for the computation of viscoelastic wavefields in complex geological media. The NDG approach combines unstructured tetrahedral meshes with an element-wise, high-order spatial interpolation of the wavefield based on Lagrange polynomials. Numerical fluxes are computed from an exact solution of the heterogeneous Riemann problem. Our implementation offers capabilities for modelling viscoelastic wave propagation in 1-D, 2-D and 3-D settings of very different spatial scale with little logistical overhead. It allows the import of external tetrahedral meshes provided by independent meshing software and can be run in a parallel computing environment. Computation of adjoint wavefields and an interface for the computation of waveform sensitivity kernels are offered. The method is validated in 2-D and 3-D by comparison to analytical solutions and results from a spectral element method. The capabilities of the NDG method are demonstrated through a 3-D example case taken from tunnel seismics which considers high-frequency elastic wave propagation around a curved underground tunnel cutting through inclined and faulted sedimentary strata. The NDG method was coded into the open-source software package NEXD and is available from GitHub.

A discontinuous Galerkin conservative level set scheme for interface capturing in multiphase flows

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

Owkes, Mark, E-mail: mfc86@cornell.edu; Desjardins, Olivier

2013-09-15

The accurate conservative level set (ACLS) method of Desjardins et al. [O. Desjardins, V. Moureau, H. Pitsch, An accurate conservative level set/ghost fluid method for simulating turbulent atomization, J. Comput. Phys. 227 (18) (2008) 8395–8416] is extended by using a discontinuous Galerkin (DG) discretization. DG allows for the scheme to have an arbitrarily high order of accuracy with the smallest possible computational stencil resulting in an accurate method with good parallel scaling. This work includes a DG implementation of the level set transport equation, which moves the level set with the flow field velocity, and a DG implementation of themore » reinitialization equation, which is used to maintain the shape of the level set profile to promote good mass conservation. A near second order converging interface curvature is obtained by following a height function methodology (common amongst volume of fluid schemes) in the context of the conservative level set. Various numerical experiments are conducted to test the properties of the method and show excellent results, even on coarse meshes. The tests include Zalesak’s disk, two-dimensional deformation of a circle, time evolution of a standing wave, and a study of the Kelvin–Helmholtz instability. Finally, this novel methodology is employed to simulate the break-up of a turbulent liquid jet.« less

On the need of mode interpolation for data-driven Galerkin models of a transient flow around a sphere

NASA Astrophysics Data System (ADS)

Stankiewicz, Witold; Morzyński, Marek; Kotecki, Krzysztof; Noack, Bernd R.

2017-04-01

We present a low-dimensional Galerkin model with state-dependent modes capturing linear and nonlinear dynamics. Departure point is a direct numerical simulation of the three-dimensional incompressible flow around a sphere at Reynolds numbers 400. This solution starts near the unstable steady Navier-Stokes solution and converges to a periodic limit cycle. The investigated Galerkin models are based on the dynamic mode decomposition (DMD) and derive the dynamical system from first principles, the Navier-Stokes equations. A DMD model with training data from the initial linear transient fails to predict the limit cycle. Conversely, a model from limit-cycle data underpredicts the initial growth rate roughly by a factor 5. Key enablers for uniform accuracy throughout the transient are a continuous mode interpolation between both oscillatory fluctuations and the addition of a shift mode. This interpolated model is shown to capture both the transient growth of the oscillation and the limit cycle.

Eigensolution analysis of spectral/hp continuous Galerkin approximations to advection-diffusion problems: Insights into spectral vanishing viscosity

NASA Astrophysics Data System (ADS)

Moura, R. C.; Sherwin, S. J.; Peiró, J.

2016-02-01

This study addresses linear dispersion-diffusion analysis for the spectral/hp continuous Galerkin (CG) formulation in one dimension. First, numerical dispersion and diffusion curves are obtained for the advection-diffusion problem and the role of multiple eigencurves peculiar to spectral/hp methods is discussed. From the eigencurves' behaviour, we observe that CG might feature potentially undesirable non-smooth dispersion/diffusion characteristics for under-resolved simulations of problems strongly dominated by either convection or diffusion. Subsequently, the linear advection equation augmented with spectral vanishing viscosity (SVV) is analysed. Dispersion and diffusion characteristics of CG with SVV-based stabilization are verified to display similar non-smooth features in flow regions where convection is much stronger than dissipation or vice-versa, owing to a dependency of the standard SVV operator on a local Péclet number. First a modification is proposed to the traditional SVV scaling that enforces a globally constant Péclet number so as to avoid the previous issues. In addition, a new SVV kernel function is suggested and shown to provide a more regular behaviour for the eigencurves along with a consistent increase in resolution power for higher-order discretizations, as measured by the extent of the wavenumber range where numerical errors are negligible. The dissipation characteristics of CG with the SVV modifications suggested are then verified to be broadly equivalent to those obtained through upwinding in the discontinuous Galerkin (DG) scheme. Nevertheless, for the kernel function proposed, the full upwind DG scheme is found to have a slightly higher resolution power for the same dissipation levels. These results show that improved CG-SVV characteristics can be pursued via different kernel functions with the aid of optimization algorithms.

HPC Institutional Computing Project: W15_lesreactiveflow KIVA-hpFE Development: A Robust and Accurate Engine Modeling Software

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

Carrington, David Bradley; Waters, Jiajia

KIVA-hpFE is a high performance computer software for solving the physics of multi-species and multiphase turbulent reactive flow in complex geometries having immersed moving parts. The code is written in Fortran 90/95 and can be used on any computer platform with any popular complier. The code is in two versions, a serial version and a parallel version utilizing MPICH2 type Message Passing Interface (MPI or Intel MPI) for solving distributed domains. The parallel version is at least 30x faster than the serial version and much faster than our previous generation of parallel engine modeling software, by many factors. The 5thmore » generation algorithm construction is a Galerkin type Finite Element Method (FEM) solving conservative momentum, species, and energy transport equations along with two-equation turbulent model k-ω Reynolds Averaged Navier-Stokes (RANS) model and a Vreman type dynamic Large Eddy Simulation (LES) method. The LES method is capable modeling transitional flow from laminar to fully turbulent; therefore, this LES method does not require special hybrid or blending to walls. The FEM projection method also uses a Petrov-Galerkin (P-G) stabilization along with pressure stabilization. We employ hierarchical basis sets, constructed on the fly with enrichment in areas associated with relatively larger error as determined by error estimation methods. In addition, when not using the hp-adaptive module, the code employs Lagrangian basis or shape functions. The shape functions are constructed for hexahedral, prismatic and tetrahedral elements. The software is designed to solve many types of reactive flow problems, from burners to internal combustion engines and turbines. In addition, the formulation allows for direct integration of solid bodies (conjugate heat transfer), as in heat transfer through housings, parts, cylinders. It can also easily be extended to stress modeling of solids, used in fluid structure interactions problems, solidification, porous

Upwind methods for the Baer–Nunziato equations and higher-order reconstruction using artificial viscosity

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

Fraysse, F., E-mail: francois.fraysse@rs2n.eu; E. T. S. de Ingeniería Aeronáutica y del Espacio, Universidad Politécnica de Madrid, Madrid; Redondo, C.

This article is devoted to the numerical discretisation of the hyperbolic two-phase flow model of Baer and Nunziato. A special attention is paid on the discretisation of intercell flux functions in the framework of Finite Volume and Discontinuous Galerkin approaches, where care has to be taken to efficiently approximate the non-conservative products inherent to the model equations. Various upwind approximate Riemann solvers have been tested on a bench of discontinuous test cases. New discretisation schemes are proposed in a Discontinuous Galerkin framework following the criterion of Abgrall and the path-conservative formalism. A stabilisation technique based on artificial viscosity is appliedmore » to the high-order Discontinuous Galerkin method and compared against classical TVD-MUSCL Finite Volume flux reconstruction.« less

Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems

NASA Astrophysics Data System (ADS)

Mabuza, Sibusiso; Shadid, John N.; Kuzmin, Dmitri

2018-05-01

The objective of this paper is to present a local bounds preserving stabilized finite element scheme for hyperbolic systems on unstructured meshes based on continuous Galerkin (CG) discretization in space. A CG semi-discrete scheme with low order artificial dissipation that satisfies the local extremum diminishing (LED) condition for systems is used to discretize a system of conservation equations in space. The low order artificial diffusion is based on approximate Riemann solvers for hyperbolic conservation laws. In this case we consider both Rusanov and Roe artificial diffusion operators. In the Rusanov case, two designs are considered, a nodal based diffusion operator and a local projection stabilization operator. The result is a discretization that is LED and has first order convergence behavior. To achieve high resolution, limited antidiffusion is added back to the semi-discrete form where the limiter is constructed from a linearity preserving local projection stabilization operator. The procedure follows the algebraic flux correction procedure usually used in flux corrected transport algorithms. To further deal with phase errors (or terracing) common in FCT type methods, high order background dissipation is added to the antidiffusive correction. The resulting stabilized semi-discrete scheme can be discretized in time using a wide variety of time integrators. Numerical examples involving nonlinear scalar Burgers equation, and several shock hydrodynamics simulations for the Euler system are considered to demonstrate the performance of the method. For time discretization, Crank-Nicolson scheme and backward Euler scheme are utilized.

Galerkin CFD solvers for use in a multi-disciplinary suite for modeling advanced flight vehicles

NASA Astrophysics Data System (ADS)

Moffitt, Nicholas J.

This work extends existing Galerkin CFD solvers for use in a multi-disciplinary suite. The suite is proposed as a means of modeling advanced flight vehicles, which exhibit strong coupling between aerodynamics, structural dynamics, controls, rigid body motion, propulsion, and heat transfer. Such applications include aeroelastics, aeroacoustics, stability and control, and other highly coupled applications. The suite uses NASA STARS for modeling structural dynamics and heat transfer. Aerodynamics, propulsion, and rigid body dynamics are modeled in one of the five CFD solvers below. Euler2D and Euler3D are Galerkin CFD solvers created at OSU by Cowan (2003). These solvers are capable of modeling compressible inviscid aerodynamics with modal elastics and rigid body motion. This work reorganized these solvers to improve efficiency during editing and at run time. Simple and efficient propulsion models were added, including rocket, turbojet, and scramjet engines. Viscous terms were added to the previous solvers to create NS2D and NS3D. The viscous contributions were demonstrated in the inertial and non-inertial frames. Variable viscosity (Sutherland's equation) and heat transfer boundary conditions were added to both solvers but not verified in this work. Two turbulence models were implemented in NS2D and NS3D: Spalart-Allmarus (SA) model of Deck, et al. (2002) and Menter's SST model (1994). A rotation correction term (Shur, et al., 2000) was added to the production of turbulence. Local time stepping and artificial dissipation were adapted to each model. CFDsol is a Taylor-Galerkin solver with an SA turbulence model. This work improved the time accuracy, far field stability, viscous terms, Sutherland?s equation, and SA model with NS3D as a guideline and added the propulsion models from Euler3D to CFDsol. Simple geometries were demonstrated to utilize current meshing and processing capabilities. Air-breathing hypersonic flight vehicles (AHFVs) represent the ultimate

On Hilbert-Schmidt norm convergence of Galerkin approximation for operator Riccati equations

NASA Technical Reports Server (NTRS)

Rosen, I. G.

1988-01-01

An abstract approximation framework for the solution of operator algebraic Riccati equations is developed. The approach taken is based on a formulation of the Riccati equation as an abstract nonlinear operator equation on the space of Hilbert-Schmidt operators. Hilbert-Schmidt norm convergence of solutions to generic finite dimensional Galerkin approximations to the Riccati equation to the solution of the original infinite dimensional problem is argued. The application of the general theory is illustrated via an operator Riccati equation arising in the linear-quadratic design of an optimal feedback control law for a 1-D heat/diffusion equation. Numerical results demonstrating the convergence of the associated Hilbert-Schmidt kernels are included.

A Streamline-Upwind Petrov-Galerkin Finite Element Scheme for Non-Ionized Hypersonic Flows in Thermochemical Nonequilibrium

NASA Technical Reports Server (NTRS)

Kirk, Benjamin S.; Bova, Stephen W.; Bond, Ryan B.

2011-01-01

Presentation topics include background and motivation; physical modeling including governing equations and thermochemistry; finite element formulation; results of inviscid thermal nonequilibrium chemically reacting flow and viscous thermal equilibrium chemical reacting flow; and near-term effort.

A blended continuous–discontinuous finite element method for solving the multi-fluid plasma model

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

Sousa, E.M., E-mail: sousae@uw.edu; Shumlak, U., E-mail: shumlak@uw.edu

The multi-fluid plasma model represents electrons, multiple ion species, and multiple neutral species as separate fluids that interact through short-range collisions and long-range electromagnetic fields. The model spans a large range of temporal and spatial scales, which renders the model stiff and presents numerical challenges. To address the large range of timescales, a blended continuous and discontinuous Galerkin method is proposed, where the massive ion and neutral species are modeled using an explicit discontinuous Galerkin method while the electrons and electromagnetic fields are modeled using an implicit continuous Galerkin method. This approach is able to capture large-gradient ion and neutralmore » physics like shock formation, while resolving high-frequency electron dynamics in a computationally efficient manner. The details of the Blended Finite Element Method (BFEM) are presented. The numerical method is benchmarked for accuracy and tested using two-fluid one-dimensional soliton problem and electromagnetic shock problem. The results are compared to conventional finite volume and finite element methods, and demonstrate that the BFEM is particularly effective in resolving physics in stiff problems involving realistic physical parameters, including realistic electron mass and speed of light. The benefit is illustrated by computing a three-fluid plasma application that demonstrates species separation in multi-component plasmas.« less

EXPLICIT LEAST-DEGREE BOUNDARY FILTERS FOR DISCONTINUOUS GALERKIN.

PubMed

Nguyen, Dang-Manh; Peters, Jörg

2017-01-01

Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improve both smoothness and accuracy of the output. At domain boundaries, these filters have to be one-sided for non-periodic boundary conditions. Recently, position-dependent smoothness-increasing accuracy-preserving (PSIAC) filters were shown to be a superset of the well-known one-sided RLKV and SRV filters. Since PSIAC filters can be formulated symbolically, PSIAC filtering amounts to forming linear products with local DG output and so offers a more stable and efficient implementation. The paper introduces a new class of PSIAC filters NP 0 that have small support and are piecewise constant. Extensive numerical experiments for the canonical hyperbolic test equation show NP 0 filters outperform the more complex known boundary filters. NP 0 filters typically reduce the L ∞ error in the boundary region below that of the interior where optimally superconvergent symmetric filters of the same support are applied. NP 0 filtering can be implemented as forming linear combinations of the data with short rational weights. Exact derivatives of the convolved output are easy to compute.

EXPLICIT LEAST-DEGREE BOUNDARY FILTERS FOR DISCONTINUOUS GALERKIN*

PubMed Central

Nguyen, Dang-Manh; Peters, Jörg

2017-01-01

Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improve both smoothness and accuracy of the output. At domain boundaries, these filters have to be one-sided for non-periodic boundary conditions. Recently, position-dependent smoothness-increasing accuracy-preserving (PSIAC) filters were shown to be a superset of the well-known one-sided RLKV and SRV filters. Since PSIAC filters can be formulated symbolically, PSIAC filtering amounts to forming linear products with local DG output and so offers a more stable and efficient implementation. The paper introduces a new class of PSIAC filters NP0 that have small support and are piecewise constant. Extensive numerical experiments for the canonical hyperbolic test equation show NP0 filters outperform the more complex known boundary filters. NP0 filters typically reduce the L∞ error in the boundary region below that of the interior where optimally superconvergent symmetric filters of the same support are applied. NP0 filtering can be implemented as forming linear combinations of the data with short rational weights. Exact derivatives of the convolved output are easy to compute. PMID:29081643

The semi-discrete Galerkin finite element modelling of compressible viscous flow past an airfoil

NASA Technical Reports Server (NTRS)

Meade, Andrew J., Jr.

1992-01-01

A method is developed to solve the two-dimensional, steady, compressible, turbulent boundary-layer equations and is coupled to an existing Euler solver for attached transonic airfoil analysis problems. The boundary-layer formulation utilizes the semi-discrete Galerkin (SDG) method to model the spatial variable normal to the surface with linear finite elements and the time-like variable with finite differences. A Dorodnitsyn transformed system of equations is used to bound the infinite spatial domain thereby permitting the use of a uniform finite element grid which provides high resolution near the wall and automatically follows boundary-layer growth. The second-order accurate Crank-Nicholson scheme is applied along with a linearization method to take advantage of the parabolic nature of the boundary-layer equations and generate a non-iterative marching routine. The SDG code can be applied to any smoothly-connected airfoil shape without modification and can be coupled to any inviscid flow solver. In this analysis, a direct viscous-inviscid interaction is accomplished between the Euler and boundary-layer codes, through the application of a transpiration velocity boundary condition. Results are presented for compressible turbulent flow past NACA 0012 and RAE 2822 airfoils at various freestream Mach numbers, Reynolds numbers, and angles of attack. All results show good agreement with experiment, and the coupled code proved to be a computationally-efficient and accurate airfoil analysis tool.

The Bassi Rebay 1 scheme is a special case of the Symmetric Interior Penalty formulation for discontinuous Galerkin discretisations with Gauss-Lobatto points

NASA Astrophysics Data System (ADS)

Manzanero, Juan; Rueda-Ramírez, Andrés M.; Rubio, Gonzalo; Ferrer, Esteban

2018-06-01

In the discontinuous Galerkin (DG) community, several formulations have been proposed to solve PDEs involving second-order spatial derivatives (e.g. elliptic problems). In this paper, we show that, when the discretisation is restricted to the usage of Gauss-Lobatto points, there are important similarities between two common choices: the Bassi-Rebay 1 (BR1) method, and the Symmetric Interior Penalty (SIP) formulation. This equivalence enables the extrapolation of properties from one scheme to the other: a sharper estimation of the minimum penalty parameter for the SIP stability (compared to the more general estimate proposed by Shahbazi [1]), more efficient implementations of the BR1 scheme, and the compactness of the BR1 method for straight quadrilateral and hexahedral meshes.

Off-fault plasticity in three-dimensional dynamic rupture simulations using a modal Discontinuous Galerkin method on unstructured meshes: Implementation, verification, and application

NASA Astrophysics Data System (ADS)

Wollherr, Stephanie; Gabriel, Alice-Agnes; Uphoff, Carsten

2018-05-01

The dynamics and potential size of earthquakes depend crucially on rupture transfers between adjacent fault segments. To accurately describe earthquake source dynamics, numerical models can account for realistic fault geometries and rheologies such as nonlinear inelastic processes off the slip interface. We present implementation, verification, and application of off-fault Drucker-Prager plasticity in the open source software SeisSol (www.seissol.org). SeisSol is based on an arbitrary high-order derivative modal Discontinuous Galerkin (ADER-DG) method using unstructured, tetrahedral meshes specifically suited for complex geometries. Two implementation approaches are detailed, modelling plastic failure either employing sub-elemental quadrature points or switching to nodal basis coefficients. At fine fault discretizations the nodal basis approach is up to 6 times more efficient in terms of computational costs while yielding comparable accuracy. Both methods are verified in community benchmark problems and by three dimensional numerical h- and p-refinement studies with heterogeneous initial stresses. We observe no spectral convergence for on-fault quantities with respect to a given reference solution, but rather discuss a limitation to low-order convergence for heterogeneous 3D dynamic rupture problems. For simulations including plasticity, a high fault resolution may be less crucial than commonly assumed, due to the regularization of peak slip rate and an increase of the minimum cohesive zone width. In large-scale dynamic rupture simulations based on the 1992 Landers earthquake, we observe high rupture complexity including reverse slip, direct branching, and dynamic triggering. The spatio-temporal distribution of rupture transfers are altered distinctively by plastic energy absorption, correlated with locations of geometrical fault complexity. Computational cost increases by 7% when accounting for off-fault plasticity in the demonstrating application. Our results

A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes

NASA Astrophysics Data System (ADS)

Dumbser, Michael; Loubère, Raphaël

2016-08-01

In this paper we propose a simple, robust and accurate nonlinear a posteriori stabilization of the Discontinuous Galerkin (DG) finite element method for the solution of nonlinear hyperbolic PDE systems on unstructured triangular and tetrahedral meshes in two and three space dimensions. This novel a posteriori limiter, which has been recently proposed for the simple Cartesian grid case in [62], is able to resolve discontinuities at a sub-grid scale and is substantially extended here to general unstructured simplex meshes in 2D and 3D. It can be summarized as follows: At the beginning of each time step, an approximation of the local minimum and maximum of the discrete solution is computed for each cell, taking into account also the vertex neighbors of an element. Then, an unlimited discontinuous Galerkin scheme of approximation degree N is run for one time step to produce a so-called candidate solution. Subsequently, an a posteriori detection step checks the unlimited candidate solution at time t n + 1 for positivity, absence of floating point errors and whether the discrete solution has remained within or at least very close to the bounds given by the local minimum and maximum computed in the first step. Elements that do not satisfy all the previously mentioned detection criteria are flagged as troubled cells. For these troubled cells, the candidate solution is discarded as inappropriate and consequently needs to be recomputed. Within these troubled cells the old discrete solution at the previous time tn is scattered onto small sub-cells (Ns = 2 N + 1 sub-cells per element edge), in order to obtain a set of sub-cell averages at time tn. Then, a more robust second order TVD finite volume scheme is applied to update the sub-cell averages within the troubled DG cells from time tn to time t n + 1. The new sub-grid data at time t n + 1 are finally gathered back into a valid cell-centered DG polynomial of degree N by using a classical conservative and higher order

Continuum approach for aerothermal flow through ablative porous material using discontinuous Galerkin discretization.

NASA Astrophysics Data System (ADS)

Schrooyen, Pierre; Chatelain, Philippe; Hillewaert, Koen; Magin, Thierry E.

2014-11-01

The atmospheric entry of spacecraft presents several challenges in simulating the aerothermal flow around the heat shield. Predicting an accurate heat-flux is a complex task, especially regarding the interaction between the flow in the free stream and the erosion of the thermal protection material. To capture this interaction, a continuum approach is developed to go progressively from the region fully occupied by fluid to a receding porous medium. The volume averaged Navier-Stokes equations are used to model both phases in the same computational domain considering a single set of conservation laws. The porosity is itself a variable of the computation, allowing to take volumetric ablation into account through adequate source terms. This approach is implemented within a computational tool based on a high-order discontinuous Galerkin discretization. The multi-dimensional tool has already been validated and has proven its efficient parallel implementation. Within this platform, a fully implicit method was developed to simulate multi-phase reacting flows. Numerical results to verify and validate the methodology are considered within this work. Interactions between the flow and the ablated geometry are also presented. Supported by Fund for Research Training in Industry and Agriculture.

An efficient Adaptive Mesh Refinement (AMR) algorithm for the Discontinuous Galerkin method: Applications for the computation of compressible two-phase flows

NASA Astrophysics Data System (ADS)

Papoutsakis, Andreas; Sazhin, Sergei S.; Begg, Steven; Danaila, Ionut; Luddens, Francky

2018-06-01

We present an Adaptive Mesh Refinement (AMR) method suitable for hybrid unstructured meshes that allows for local refinement and de-refinement of the computational grid during the evolution of the flow. The adaptive implementation of the Discontinuous Galerkin (DG) method introduced in this work (ForestDG) is based on a topological representation of the computational mesh by a hierarchical structure consisting of oct- quad- and binary trees. Adaptive mesh refinement (h-refinement) enables us to increase the spatial resolution of the computational mesh in the vicinity of the points of interest such as interfaces, geometrical features, or flow discontinuities. The local increase in the expansion order (p-refinement) at areas of high strain rates or vorticity magnitude results in an increase of the order of accuracy in the region of shear layers and vortices. A graph of unitarian-trees, representing hexahedral, prismatic and tetrahedral elements is used for the representation of the initial domain. The ancestral elements of the mesh can be split into self-similar elements allowing each tree to grow branches to an arbitrary level of refinement. The connectivity of the elements, their genealogy and their partitioning are described by linked lists of pointers. An explicit calculation of these relations, presented in this paper, facilitates the on-the-fly splitting, merging and repartitioning of the computational mesh by rearranging the links of each node of the tree with a minimal computational overhead. The modal basis used in the DG implementation facilitates the mapping of the fluxes across the non conformal faces. The AMR methodology is presented and assessed using a series of inviscid and viscous test cases. Also, the AMR methodology is used for the modelling of the interaction between droplets and the carrier phase in a two-phase flow. This approach is applied to the analysis of a spray injected into a chamber of quiescent air, using the Eulerian

Application of the Galerkin/least-squares formulation to the analysis of hypersonic flows. I - Flow over a two-dimensional ramp

NASA Technical Reports Server (NTRS)

Chalot, F.; Hughes, T. J. R.; Johan, Z.; Shakib, F.

1991-01-01

An FEM for the compressible Navier-Stokes equations is introduced. The discretization is based on entropy variables. The methodology is developed within the framework of a Galerkin/least-squares formulation to which a discontinuity-capturing operator is added. Results for three test cases selected among those of the Workshop on Hypersonic Flows for Reentry Problems are presented.

A discontinuous Galerkin approach for conservative modeling of fully nonlinear and weakly dispersive wave transformations

NASA Astrophysics Data System (ADS)

Sharifian, Mohammad Kazem; Kesserwani, Georges; Hassanzadeh, Yousef

2018-05-01

This work extends a robust second-order Runge-Kutta Discontinuous Galerkin (RKDG2) method to solve the fully nonlinear and weakly dispersive flows, within a scope to simultaneously address accuracy, conservativeness, cost-efficiency and practical needs. The mathematical model governing such flows is based on a variant form of the Green-Naghdi (GN) equations decomposed as a hyperbolic shallow water system with an elliptic source term. Practical features of relevance (i.e. conservative modeling over irregular terrain with wetting and drying and local slope limiting) have been restored from an RKDG2 solver to the Nonlinear Shallow Water (NSW) equations, alongside new considerations to integrate elliptic source terms (i.e. via a fourth-order local discretization of the topography) and to enable local capturing of breaking waves (i.e. via adding a detector for switching off the dispersive terms). Numerical results are presented, demonstrating the overall capability of the proposed approach in achieving realistic prediction of nearshore wave processes involving both nonlinearity and dispersion effects within a single model.

An accurate boundary element method for the exterior elastic scattering problem in two dimensions

NASA Astrophysics Data System (ADS)

Bao, Gang; Xu, Liwei; Yin, Tao

2017-11-01

This paper is concerned with a Galerkin boundary element method solving the two dimensional exterior elastic wave scattering problem. The original problem is first reduced to the so-called Burton-Miller [1] boundary integral formulation, and essential mathematical features of its variational form are discussed. In numerical implementations, a newly-derived and analytically accurate regularization formula [2] is employed for the numerical evaluation of hyper-singular boundary integral operator. A new computational approach is employed based on the series expansions of Hankel functions for the computation of weakly-singular boundary integral operators during the reduction of corresponding Galerkin equations into a discrete linear system. The effectiveness of proposed numerical methods is demonstrated using several numerical examples.

The moving-least-squares-particle hydrodynamics method (MLSPH)

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

Dilts, G.

1997-12-31

An enhancement of the smooth-particle hydrodynamics (SPH) method has been developed using the moving-least-squares (MLS) interpolants of Lancaster and Salkauskas which simultaneously relieves the method of several well-known undesirable behaviors, including spurious boundary effects, inaccurate strain and rotation rates, pressure spikes at impact boundaries, and the infamous tension instability. The classical SPH method is derived in a novel manner by means of a Galerkin approximation applied to the Lagrangian equations of motion for continua using as basis functions the SPH kernel function multiplied by the particle volume. This derivation is then modified by simply substituting the MLS interpolants for themore » SPH Galerkin basis, taking care to redefine the particle volume and mass appropriately. The familiar SPH kernel approximation is now equivalent to a colocation-Galerkin method. Both classical conservative and recent non-conservative formulations of SPH can be derived and emulated. The non-conservative forms can be made conservative by adding terms that are zero within the approximation at the expense of boundary-value considerations. The familiar Monaghan viscosity is used. Test calculations of uniformly expanding fluids, the Swegle example, spinning solid disks, impacting bars, and spherically symmetric flow illustrate the superiority of the technique over SPH. In all cases it is seen that the marvelous ability of the MLS interpolants to add up correctly everywhere civilizes the noisy, unpredictable nature of SPH. Being a relatively minor perturbation of the SPH method, it is easily retrofitted into existing SPH codes. On the down side, computational expense at this point is significant, the Monaghan viscosity undoes the contribution of the MLS interpolants, and one-point quadrature (colocation) is not accurate enough. Solutions to these difficulties are being pursued vigorously.« less

Dynamic Rupture Benchmarking of the ADER-DG Method

NASA Astrophysics Data System (ADS)

Gabriel, Alice; Pelties, Christian

2013-04-01

We will verify the arbitrary high-order derivative Discontinuous Galerkin (ADER-DG) method in various test cases of the 'SCEC/USGS Dynamic Earthquake Rupture Code Verification Exercise' benchmark suite (Harris et al. 2009). The ADER-DG scheme is able to solve the spontaneous rupture problem with high-order accuracy in space and time on three-dimensional unstructured tetrahedral meshes. Strong mesh coarsening or refinement at areas of interest can be applied to keep the computational costs feasible. Moreover, the method does not generate spurious high-frequency contributions in the slip rate spectra and therefore does not require any artificial damping as demonstrated in previous presentations and publications (Pelties et al. 2010 and 2012). We will show that the mentioned features hold also for more advanced setups as e.g. a branching fault system, heterogeneous background stresses and bimaterial faults. The advanced geometrical flexibility combined with an enhanced accuracy will make the ADER-DG method a useful tool to study earthquake dynamics on complex fault systems in realistic rheologies. References: Harris, R.A., M. Barall, R. Archuleta, B. Aagaard, J.-P. Ampuero, H. Bhat, V. Cruz-Atienza, L. Dalguer, P. Dawson, S. Day, B. Duan, E. Dunham, G. Ely, Y. Kaneko, Y. Kase, N. Lapusta, Y. Liu, S. Ma, D. Oglesby, K. Olsen, A. Pitarka, S. Song, and E. Templeton, The SCEC/USGS Dynamic Earthquake Rupture Code Verification Exercise, Seismological Research Letters, vol. 80, no. 1, pages 119-126, 2009 Pelties, C., J. de la Puente, and M. Kaeser, Dynamic Rupture Modeling in Three Dimensions on Unstructured Meshes Using a Discontinuous Galerkin Method, AGU 2010 Fall Meeting, abstract #S21C-2068 Pelties, C., J. de la Puente, J.-P. Ampuero, G. Brietzke, and M. Kaeser, Three-Dimensional Dynamic Rupture Simulation with a High-order Discontinuous Galerkin Method on Unstructured Tetrahedral Meshes, JGR. - Solid Earth, VOL. 117, B02309, 2012

Dynamic Rupture Benchmarking of the ADER-DG Method

NASA Astrophysics Data System (ADS)

Pelties, C.; Gabriel, A.

2012-12-01

We will verify the arbitrary high-order derivative Discontinuous Galerkin (ADER-DG) method in various test cases of the 'SCEC/USGS Dynamic Earthquake Rupture Code Verification Exercise' benchmark suite (Harris et al. 2009). The ADER-DG scheme is able to solve the spontaneous rupture problem with high-order accuracy in space and time on three-dimensional unstructured tetrahedral meshes. Strong mesh coarsening or refinement at areas of interest can be applied to keep the computational costs feasible. Moreover, the method does not generate spurious high-frequency contributions in the slip rate spectra and therefore does not require any artificial damping as demonstrated in previous presentations and publications (Pelties et al. 2010 and 2012). We will show that the mentioned features hold also for more advanced setups as e.g. a branching fault system, heterogeneous background stresses and bimaterial faults. The advanced geometrical flexibility combined with an enhanced accuracy will make the ADER-DG method a useful tool to study earthquake dynamics on complex fault systems in realistic rheologies. References: Harris, R.A., M. Barall, R. Archuleta, B. Aagaard, J.-P. Ampuero, H. Bhat, V. Cruz-Atienza, L. Dalguer, P. Dawson, S. Day, B. Duan, E. Dunham, G. Ely, Y. Kaneko, Y. Kase, N. Lapusta, Y. Liu, S. Ma, D. Oglesby, K. Olsen, A. Pitarka, S. Song, and E. Templeton, The SCEC/USGS Dynamic Earthquake Rupture Code Verification Exercise, Seismological Research Letters, vol. 80, no. 1, pages 119-126, 2009 Pelties, C., J. de la Puente, and M. Kaeser, Dynamic Rupture Modeling in Three Dimensions on Unstructured Meshes Using a Discontinuous Galerkin Method, AGU 2010 Fall Meeting, abstract #S21C-2068 Pelties, C., J. de la Puente, J.-P. Ampuero, G. Brietzke, and M. Kaeser, Three-Dimensional Dynamic Rupture Simulation with a High-order Discontinuous Galerkin Method on Unstructured Tetrahedral Meshes, JGR. - Solid Earth, VOL. 117, B02309, 2012

EIT image reconstruction based on a hybrid FE-EFG forward method and the complete-electrode model.

PubMed

Hadinia, M; Jafari, R; Soleimani, M

2016-06-01

This paper presents the application of the hybrid finite element-element free Galerkin (FE-EFG) method for the forward and inverse problems of electrical impedance tomography (EIT). The proposed method is based on the complete electrode model. Finite element (FE) and element-free Galerkin (EFG) methods are accurate numerical techniques. However, the FE technique has meshing task problems and the EFG method is computationally expensive. In this paper, the hybrid FE-EFG method is applied to take both advantages of FE and EFG methods, the complete electrode model of the forward problem is solved, and an iterative regularized Gauss-Newton method is adopted to solve the inverse problem. The proposed method is applied to compute Jacobian in the inverse problem. Utilizing 2D circular homogenous models, the numerical results are validated with analytical and experimental results and the performance of the hybrid FE-EFG method compared with the FE method is illustrated. Results of image reconstruction are presented for a human chest experimental phantom.

A hybridized discontinuous Galerkin framework for high-order particle-mesh operator splitting of the incompressible Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Maljaars, Jakob M.; Labeur, Robert Jan; Möller, Matthias

2018-04-01

A generic particle-mesh method using a hybridized discontinuous Galerkin (HDG) framework is presented and validated for the solution of the incompressible Navier-Stokes equations. Building upon particle-in-cell concepts, the method is formulated in terms of an operator splitting technique in which Lagrangian particles are used to discretize an advection operator, and an Eulerian mesh-based HDG method is employed for the constitutive modeling to account for the inter-particle interactions. Key to the method is the variational framework provided by the HDG method. This allows to formulate the projections between the Lagrangian particle space and the Eulerian finite element space in terms of local (i.e. cellwise) ℓ2-projections efficiently. Furthermore, exploiting the HDG framework for solving the constitutive equations results in velocity fields which excellently approach the incompressibility constraint in a local sense. By advecting the particles through these velocity fields, the particle distribution remains uniform over time, obviating the need for additional quality control. The presented methodology allows for a straightforward extension to arbitrary-order spatial accuracy on general meshes. A range of numerical examples shows that optimal convergence rates are obtained in space and, given the particular time stepping strategy, second-order accuracy is obtained in time. The model capabilities are further demonstrated by presenting results for the flow over a backward facing step and for the flow around a cylinder.

Development of a Perfectly Matched Layer Technique for a Discontinuous-Galerkin Spectral-Element Method

NASA Technical Reports Server (NTRS)

Garai, Anirban; Murman, Scott M.; Madavan, Nateri K.

2016-01-01

used involves modeling the pressure fluctuations as acoustic waves propagating in the far-field relative to a single noise-source inside the buffer region. This approach treats vorticity-induced pressure fluctuations the same as acoustic waves. Another popular approach, often referred to as the "sponge layer," attempts to dampen the flow perturbations by introducing artificial dissipation in the buffer region. Although the artificial dissipation removes all perturbations inside the sponge layer, incoming waves are still reflected from the interface boundary between the computational domain and the sponge layer. The effect of these refkections can be somewhat mitigated by appropriately selecting the artificial dissipation strength and the extent of the sponge layer. One of the most promising variants on the buffer region approach is the Perfectly Matched Layer (PML) technique. The PML technique mitigates spurious reflections from boundaries and interfaces by dampening the perturbation modes inside the buffer region such that their eigenfunctions remain unchanged. The technique was first developed by Berenger for application to problems involving electromagnetic wave propagation. It was later extended to the linearized Euler, Euler and Navier-Stokes equations by Hu and his coauthors. The PML technique ensures the no-reflection property for all waves, irrespective of incidence angle, wavelength, and propagation direction. Although the technique requires the solution of a set of auxiliary equations, the computational overhead is easily justified since it allows smaller domain sizes and can provide better accuracy, stability, and convergence of the numerical solution. In this paper, the PML technique is developed in the context of a high-order spectral-element Discontinuous Galerkin (DG) method. The technique is compared to other approaches to treating the in flow and out flow boundary, such as those based on using characteristic boundary conditions and sponge layers. The

Hyperbolic heat conduction problems involving non-Fourier effects - Numerical simulations via explicit Lax-Wendroff/Taylor-Galerkin finite element formulations

NASA Technical Reports Server (NTRS)

Tamma, Kumar K.; Namburu, Raju R.

1989-01-01

Numerical simulations are presented for hyperbolic heat-conduction problems that involve non-Fourier effects, using explicit, Lax-Wendroff/Taylor-Galerkin FEM formulations as the principal computational tool. Also employed are smoothing techniques which stabilize the numerical noise and accurately predict the propagating thermal disturbances. The accurate capture of propagating thermal disturbances at characteristic time-step values is achieved; numerical test cases are presented which validate the proposed hyperbolic heat-conduction problem concepts.

Calculation of compressible boundary layer flow about airfoils by a finite element/finite difference method

NASA Technical Reports Server (NTRS)

Strong, Stuart L.; Meade, Andrew J., Jr.

1992-01-01

Preliminary results are presented of a finite element/finite difference method (semidiscrete Galerkin method) used to calculate compressible boundary layer flow about airfoils, in which the group finite element scheme is applied to the Dorodnitsyn formulation of the boundary layer equations. The semidiscrete Galerkin (SDG) method promises to be fast, accurate and computationally efficient. The SDG method can also be applied to any smoothly connected airfoil shape without modification and possesses the potential capability of calculating boundary layer solutions beyond flow separation. Results are presented for low speed laminar flow past a circular cylinder and past a NACA 0012 airfoil at zero angle of attack at a Mach number of 0.5. Also shown are results for compressible flow past a flat plate for a Mach number range of 0 to 10 and results for incompressible turbulent flow past a flat plate. All numerical solutions assume an attached boundary layer.

Spectral methods for time dependent partial differential equations

NASA Technical Reports Server (NTRS)

Gottlieb, D.; Turkel, E.

1983-01-01

The theory of spectral methods for time dependent partial differential equations is reviewed. When the domain is periodic Fourier methods are presented while for nonperiodic problems both Chebyshev and Legendre methods are discussed. The theory is presented for both hyperbolic and parabolic systems using both Galerkin and collocation procedures. While most of the review considers problems with constant coefficients the extension to nonlinear problems is also discussed. Some results for problems with shocks are presented.

Methods for High-Order Multi-Scale and Stochastic Problems Analysis, Algorithms, and Applications

DTIC Science & Technology

2016-10-17

finite volume schemes, discontinuous Galerkin finite element method, and related methods, for solving computational fluid dynamics (CFD) problems and...approximation for finite element methods. (3) The development of methods of simulation and analysis for the study of large scale stochastic systems of...laws, finite element method, Bernstein-Bezier finite elements , weakly interacting particle systems, accelerated Monte Carlo, stochastic networks 16

Finite element analysis of flowfield in the single hole film cooling technique.

PubMed

Bazdidi-Tehrani, F; Mahmoodi, A A

2001-05-01

Film cooling is currently used in gas turbine hot sections, such as the combustor wall and the turbine blades, to prevent those sections from failing at elevated temperatures. In the single hole film cooling method, coolant air is injected from a hole into the mainstream and thus the flow is naturally three dimensional. In this paper, the Navier-Stokes and the energy equations are solved on a flat plate by the Finite Element Method (FEM) using brick elements. Algebraic equations are obtained by use of the Petrov-Galerkin method. The pressure term is removed from the momentum equations, by employing the Penalty method. The governing equations are transient and the flow is incompressible and turbulent. The model of turbulence in the near wall region is the wall function method, and in the fully turbulent region is the k-epsilon model. The system of the algebraic equations are solved by the Frontal method. The coolant injection angle and the blowing rate are among the parameters which are studied. In order to examine the present computer code, the results are compared with the Blasius (exact) solution and also with the empirical 1/7th power-law and good agreement is shown. Also, the optimum cooling performance is shown to be at 35 degree angle of coolant injection and the optimum blowing rate is 0.5. The film cooling effectiveness data, at the optimum conditions, is directly compared with the experimental results of Goldstein et al. and good agreement is demonstrated.

Nonlocal and Mixed-Locality Multiscale Finite Element Methods

DOE PAGES

Costa, Timothy B.; Bond, Stephen D.; Littlewood, David J.

2018-03-27

In many applications the resolution of small-scale heterogeneities remains a significant hurdle to robust and reliable predictive simulations. In particular, while material variability at the mesoscale plays a fundamental role in processes such as material failure, the resolution required to capture mechanisms at this scale is often computationally intractable. Multiscale methods aim to overcome this difficulty through judicious choice of a subscale problem and a robust manner of passing information between scales. One promising approach is the multiscale finite element method, which increases the fidelity of macroscale simulations by solving lower-scale problems that produce enriched multiscale basis functions. Here, inmore » this study, we present the first work toward application of the multiscale finite element method to the nonlocal peridynamic theory of solid mechanics. This is achieved within the context of a discontinuous Galerkin framework that facilitates the description of material discontinuities and does not assume the existence of spatial derivatives. Analysis of the resulting nonlocal multiscale finite element method is achieved using the ambulant Galerkin method, developed here with sufficient generality to allow for application to multiscale finite element methods for both local and nonlocal models that satisfy minimal assumptions. Finally, we conclude with preliminary results on a mixed-locality multiscale finite element method in which a nonlocal model is applied at the fine scale and a local model at the coarse scale.« less

Nonlocal and Mixed-Locality Multiscale Finite Element Methods

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

Costa, Timothy B.; Bond, Stephen D.; Littlewood, David J.

In many applications the resolution of small-scale heterogeneities remains a significant hurdle to robust and reliable predictive simulations. In particular, while material variability at the mesoscale plays a fundamental role in processes such as material failure, the resolution required to capture mechanisms at this scale is often computationally intractable. Multiscale methods aim to overcome this difficulty through judicious choice of a subscale problem and a robust manner of passing information between scales. One promising approach is the multiscale finite element method, which increases the fidelity of macroscale simulations by solving lower-scale problems that produce enriched multiscale basis functions. Here, inmore » this study, we present the first work toward application of the multiscale finite element method to the nonlocal peridynamic theory of solid mechanics. This is achieved within the context of a discontinuous Galerkin framework that facilitates the description of material discontinuities and does not assume the existence of spatial derivatives. Analysis of the resulting nonlocal multiscale finite element method is achieved using the ambulant Galerkin method, developed here with sufficient generality to allow for application to multiscale finite element methods for both local and nonlocal models that satisfy minimal assumptions. Finally, we conclude with preliminary results on a mixed-locality multiscale finite element method in which a nonlocal model is applied at the fine scale and a local model at the coarse scale.« less

Effect of boundary representation on viscous, separated flows in a discontinuous-Galerkin Navier-Stokes solver

NASA Astrophysics Data System (ADS)

Nelson, Daniel A.; Jacobs, Gustaaf B.; Kopriva, David A.

2016-08-01

The effect of curved-boundary representation on the physics of the separated flow over a NACA 65(1)-412 airfoil is thoroughly investigated. A method is presented to approximate curved boundaries with a high-order discontinuous-Galerkin spectral element method for the solution of the Navier-Stokes equations. Multiblock quadrilateral element meshes are constructed with the grid generation software GridPro. The boundary of a NACA 65(1)-412 airfoil, defined by a cubic natural spline, is piecewise-approximated by isoparametric polynomial interpolants that represent the edges of boundary-fitted elements. Direct numerical simulation of the airfoil is performed on a coarse mesh and fine mesh with polynomial orders ranging from four to twelve. The accuracy of the curve fitting is investigated by comparing the flows computed on curved-sided meshes with those given by straight-sided meshes. Straight-sided meshes yield irregular wakes, whereas curved-sided meshes produce a regular Karman street wake. Straight-sided meshes also produce lower lift and higher viscous drag as compared with curved-sided meshes. When the mesh is refined by reducing the sizes of the elements, the lift decrease and viscous drag increase are less pronounced. The differences in the aerodynamic performance between the straight-sided meshes and the curved-sided meshes are concluded to be the result of artificial surface roughness introduced by the piecewise-linear boundary approximation provided by the straight-sided meshes.

Thermosolutal convection and macrosegregation in dendritic alloys

NASA Technical Reports Server (NTRS)

Poirier, David R.; Heinrich, J. C.

1993-01-01

A mathematical model of solidification, that simulates the formation of channel segregates or freckles, is presented. The model simulates the entire solidification process, starting with the initial melt to the solidified cast, and the resulting segregation is predicted. Emphasis is given to the initial transient, when the dendritic zone begins to develop and the conditions for the possible nucleation of channels are established. The mechanisms that lead to the creation and eventual growth or termination of channels are explained in detail and illustrated by several numerical examples. A finite element model is used for the simulations. It uses a single system of equations to deal with the all-liquid region, the dendritic region, and the all-solid region. The dendritic region is treated as an anisotropic porous medium. The algorithm uses the bilinear isoparametric element, with a penalty function approximation and a Petrov-Galerkin formulation. The major task was to develop the solidification model. In addition, other tasks that were performed in conjunction with the modeling of dendritic solidification are briefly described.

Singularity Preserving Numerical Methods for Boundary Integral Equations

NASA Technical Reports Server (NTRS)

Kaneko, Hideaki (Principal Investigator)

1996-01-01

In the past twelve months (May 8, 1995 - May 8, 1996), under the cooperative agreement with Division of Multidisciplinary Optimization at NASA Langley, we have accomplished the following five projects: a note on the finite element method with singular basis functions; numerical quadrature for weakly singular integrals; superconvergence of degenerate kernel method; superconvergence of the iterated collocation method for Hammersteion equations; and singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel. This final report consists of five papers describing these projects. Each project is preceeded by a brief abstract.

High order spectral difference lattice Boltzmann method for incompressible hydrodynamics

NASA Astrophysics Data System (ADS)

Li, Weidong

2017-09-01

This work presents a lattice Boltzmann equation (LBE) based high order spectral difference method for incompressible flows. In the present method, the spectral difference (SD) method is adopted to discretize the convection and collision term of the LBE to obtain high order (≥3) accuracy. Because the SD scheme represents the solution as cell local polynomials and the solution polynomials have good tensor-product property, the present spectral difference lattice Boltzmann method (SD-LBM) can be implemented on arbitrary unstructured quadrilateral meshes for effective and efficient treatment of complex geometries. Thanks to only first oder PDEs involved in the LBE, no special techniques, such as hybridizable discontinuous Galerkin method (HDG), local discontinuous Galerkin method (LDG) and so on, are needed to discrete diffusion term, and thus, it simplifies the algorithm and implementation of the high order spectral difference method for simulating viscous flows. The proposed SD-LBM is validated with four incompressible flow benchmarks in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the lid-driven cavity flow without singularity at the two top corners-Burggraf flow; and (c) the unsteady Taylor-Green vortex flow; (d) the Blasius boundary-layer flow past a flat plate. Computational results are compared with analytical solutions of these cases and convergence studies of these cases are also given. The designed accuracy of the proposed SD-LBM is clearly verified.

A radial basis function Galerkin method for inhomogeneous nonlocal diffusion

DOE PAGES

Lehoucq, Richard B.; Rowe, Stephen T.

2016-02-01

We introduce a discretization for a nonlocal diffusion problem using a localized basis of radial basis functions. The stiffness matrix entries are assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, sparse, symmetric positive definite stiffness matrix. We demonstrate that both the continuum and discrete problems are well-posed and present numerical results for the convergence behavior of the radial basis function method. As a result, we explore approximating the solution to anisotropic differential equations by solving anisotropic nonlocal integral equations using the radial basis function method.

High Order Discontinuous Gelerkin Methods for Convection Dominated Problems with Application to Aeroacoustics

NASA Technical Reports Server (NTRS)

Shu, Chi-Wang

2000-01-01

This project is about the investigation of the development of the discontinuous Galerkin finite element methods, for general geometry and triangulations, for solving convection dominated problems, with applications to aeroacoustics. On the analysis side, we have studied the efficient and stable discontinuous Galerkin framework for small second derivative terms, for example in Navier-Stokes equations, and also for related equations such as the Hamilton-Jacobi equations. This is a truly local discontinuous formulation where derivatives are considered as new variables. On the applied side, we have implemented and tested the efficiency of different approaches numerically. Related issues in high order ENO and WENO finite difference methods and spectral methods have also been investigated. Jointly with Hu, we have presented a discontinuous Galerkin finite element method for solving the nonlinear Hamilton-Jacobi equations. This method is based on the RungeKutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples are given to illustrate the capability of the method. Jointly with Hu, we have constructed third and fourth order WENO schemes on two dimensional unstructured meshes (triangles) in the finite volume formulation. The third order schemes are based on a combination of linear polynomials with nonlinear weights, and the fourth order schemes are based on combination of quadratic polynomials with nonlinear weights. We have addressed several difficult issues associated with high order WENO schemes on unstructured mesh, including the choice of linear and nonlinear weights, what to do with negative weights, etc. Numerical examples are shown to demonstrate the accuracies and robustness of the

Multi-scale and Multi-physics Numerical Methods for Modeling Transport in Mesoscopic Systems

DTIC Science & Technology

2014-10-13

function and wide band Fast multipole methods for Hankel waves. (2) a new linear scaling discontinuous Galerkin density functional theory, which provide a...inflow boundary condition for Wigner quantum transport equations. Also, a book titled "Computational Methods for Electromagnetic Phenomena...equationsin layered media with FMM for Bessel functions , Science China Mathematics, (12 2013): 2561. doi: TOTAL: 6 Number of Papers published in peer

Generalized multiscale finite-element method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media

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

Gao, Kai; Fu, Shubin; Gibson, Richard L.

It is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs. However, the computational cost of conventional numerical modeling methods, such as finite-difference method and finite-element method, becomes prohibitively expensive when applied to very large models. We propose a Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media, where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element. The application of multiscale basis functions can capture the fine scale mediummore » property variations, and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite-element method for wave equation, while restricting the error to low values. We formulate the continuous Galerkin and discontinuous Galerkin formulation of the multiscale method, both of which have pros and cons. Applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system.« less

Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media

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

Gao, Kai, E-mail: kaigao87@gmail.com; Fu, Shubin, E-mail: shubinfu89@gmail.com; Gibson, Richard L., E-mail: gibson@tamu.edu

It is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs. However, the computational cost of conventional numerical modeling methods, such as finite-difference method and finite-element method, becomes prohibitively expensive when applied to very large models. We propose a Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media, where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element. The application of multiscale basis functions can capture the fine scale mediummore » property variations, and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite-element method for wave equation, while restricting the error to low values. We formulate the continuous Galerkin and discontinuous Galerkin formulation of the multiscale method, both of which have pros and cons. Applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system.« less

Generalized multiscale finite-element method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media

DOE PAGES

Gao, Kai; Fu, Shubin; Gibson, Richard L.; ...

2015-04-14

It is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs. However, the computational cost of conventional numerical modeling methods, such as finite-difference method and finite-element method, becomes prohibitively expensive when applied to very large models. We propose a Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media, where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element. The application of multiscale basis functions can capture the fine scale mediummore » property variations, and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite-element method for wave equation, while restricting the error to low values. We formulate the continuous Galerkin and discontinuous Galerkin formulation of the multiscale method, both of which have pros and cons. Applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system.« less

On the eddy-resolving capability of high-order discontinuous Galerkin approaches to implicit LES / under-resolved DNS of Euler turbulence

NASA Astrophysics Data System (ADS)

Moura, R. C.; Mengaldo, G.; Peiró, J.; Sherwin, S. J.

2017-02-01

We present estimates of spectral resolution power for under-resolved turbulent Euler flows obtained with high-order discontinuous Galerkin (DG) methods. The '1% rule' based on linear dispersion-diffusion analysis introduced by Moura et al. (2015) [10] is here adapted for 3D energy spectra and validated through the inviscid Taylor-Green vortex problem. The 1% rule estimates the wavenumber beyond which numerical diffusion induces an artificial dissipation range on measured energy spectra. As the original rule relies on standard upwinding, different Riemann solvers are tested. Very good agreement is found for solvers which treat the different physical waves in a consistent manner. Relatively good agreement is still found for simpler solvers. The latter however displayed spurious features attributed to the inconsistent treatment of different physical waves. It is argued that, in the limit of vanishing viscosity, such features might have a significant impact on robustness and solution quality. The estimates proposed are regarded as useful guidelines for no-model DG-based simulations of free turbulence at very high Reynolds numbers.

Numerical study of a multigrid method with four smoothing methods for the incompressible Navier-Stokes equations in general coordinates

NASA Technical Reports Server (NTRS)

Zeng, S.; Wesseling, P.

1993-01-01

The performance of a linear multigrid method using four smoothing methods, called SCGS (Symmetrical Coupled GauBeta-Seidel), CLGS (Collective Line GauBeta-Seidel), SILU (Scalar ILU), and CILU (Collective ILU), is investigated for the incompressible Navier-Stokes equations in general coordinates, in association with Galerkin coarse grid approximation. Robustness and efficiency are measured and compared by application to test problems. The numerical results show that CILU is the most robust, SILU the least, with CLGS and SCGS in between. CLGS is the best in efficiency, SCGS and CILU follow, and SILU is the worst.

Efficient searching in meshfree methods

NASA Astrophysics Data System (ADS)

Olliff, James; Alford, Brad; Simkins, Daniel C.

2018-04-01

Meshfree methods such as the Reproducing Kernel Particle Method and the Element Free Galerkin method have proven to be excellent choices for problems involving complex geometry, evolving topology, and large deformation, owing to their ability to model the problem domain without the constraints imposed on the Finite Element Method (FEM) meshes. However, meshfree methods have an added computational cost over FEM that come from at least two sources: increased cost of shape function evaluation and the determination of adjacency or connectivity. The focus of this paper is to formally address the types of adjacency information that arises in various uses of meshfree methods; a discussion of available techniques for computing the various adjacency graphs; propose a new search algorithm and data structure; and finally compare the memory and run time performance of the methods.

High-Order Space-Time Methods for Conservation Laws

NASA Technical Reports Server (NTRS)

Huynh, H. T.

2013-01-01

Current high-order methods such as discontinuous Galerkin and/or flux reconstruction can provide effective discretization for the spatial derivatives. Together with a time discretization, such methods result in either too small a time step size in the case of an explicit scheme or a very large system in the case of an implicit one. To tackle these problems, two new high-order space-time schemes for conservation laws are introduced: the first is explicit and the second, implicit. The explicit method here, also called the moment scheme, achieves a Courant-Friedrichs-Lewy (CFL) condition of 1 for the case of one-spatial dimension regardless of the degree of the polynomial approximation. (For standard explicit methods, if the spatial approximation is of degree p, then the time step sizes are typically proportional to 1/p(exp 2)). Fourier analyses for the one and two-dimensional cases are carried out. The property of super accuracy (or super convergence) is discussed. The implicit method is a simplified but optimal version of the discontinuous Galerkin scheme applied to time. It reduces to a collocation implicit Runge-Kutta (RK) method for ordinary differential equations (ODE) called Radau IIA. The explicit and implicit schemes are closely related since they employ the same intermediate time levels, and the former can serve as a key building block in an iterative procedure for the latter. A limiting technique for the piecewise linear scheme is also discussed. The technique can suppress oscillations near a discontinuity while preserving accuracy near extrema. Preliminary numerical results are shown

An All-Fragments Grammar for Simple and Accurate Parsing

DTIC Science & Technology

2012-03-21

Tsujii. Probabilistic CFG with latent annotations. In Proceedings of ACL, 2005. Slav Petrov and Dan Klein. Improved Inference for Unlexicalized Parsing. In...Proceedings of NAACL-HLT, 2007. Slav Petrov and Dan Klein. Sparse Multi-Scale Grammars for Discriminative Latent Variable Parsing. In Proceedings of...EMNLP, 2008. Slav Petrov, Leon Barrett, Romain Thibaux, and Dan Klein. Learning Accurate, Compact, and Interpretable Tree Annotation. In Proceedings

The crowding factor method applied to parafoveal vision

PubMed Central

Ghahghaei, Saeideh; Walker, Laura

2016-01-01

Crowding increases with eccentricity and is most readily observed in the periphery. During natural, active vision, however, central vision plays an important role. Measures of critical distance to estimate crowding are difficult in central vision, as these distances are small. Any overlap of flankers with the target may create an overlay masking confound. The crowding factor method avoids this issue by simultaneously modulating target size and flanker distance and using a ratio to compare crowded to uncrowded conditions. This method was developed and applied in the periphery (Petrov & Meleshkevich, 2011b). In this work, we apply the method to characterize crowding in parafoveal vision (<3.5 visual degrees) with spatial uncertainty. We find that eccentricity and hemifield have less impact on crowding than in the periphery, yet radial/tangential asymmetries are clearly preserved. There are considerable idiosyncratic differences observed between participants. The crowding factor method provides a powerful tool for examining crowding in central and peripheral vision, which will be useful in future studies that seek to understand visual processing under natural, active viewing conditions. PMID:27690170

High speed inviscid compressible flow by the finite element method

NASA Technical Reports Server (NTRS)

Zienkiewicz, O. C.; Loehner, R.; Morgan, K.

1984-01-01

The finite element method and an explicit time stepping algorithm which is based on Taylor-Galerkin schemes with an appropriate artificial viscosity is combined with an automatic mesh refinement process which is designed to produce accurate steady state solutions to problems of inviscid compressible flow in two dimensions. The results of two test problems are included which demonstrate the excellent performance characteristics of the proposed procedures.

Discontinuous Spectral Difference Method for Conservation Laws on Unstructured Grids

NASA Technical Reports Server (NTRS)

Liu, Yen; Vinokur, Marcel; Wang, Z. J.

2004-01-01

A new, high-order, conservative, and efficient 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. A discussion on the Discontinuous Spectral Difference (SD) Method, locations of the unknowns and flux points and numerical results are also presented.

Numerical treatment for solving two-dimensional space-fractional advection-dispersion equation using meshless method

NASA Astrophysics Data System (ADS)

Cheng, Rongjun; Sun, Fengxin; Wei, Qi; Wang, Jufeng

2018-02-01

Space-fractional advection-dispersion equation (SFADE) can describe particle transport in a variety of fields more accurately than the classical models of integer-order derivative. Because of nonlocal property of integro-differential operator of space-fractional derivative, it is very challenging to deal with fractional model, and few have been reported in the literature. In this paper, a numerical analysis of the two-dimensional SFADE is carried out by the element-free Galerkin (EFG) method. The trial functions for the SFADE are constructed by the moving least-square (MLS) approximation. By the Galerkin weak form, the energy functional is formulated. Employing the energy functional minimization procedure, the final algebraic equations system is obtained. The Riemann-Liouville operator is discretized by the Grünwald formula. With center difference method, EFG method and Grünwald formula, the fully discrete approximation schemes for SFADE are established. Comparing with exact results and available results by other well-known methods, the computed approximate solutions are presented in the format of tables and graphs. The presented results demonstrate the validity, efficiency and accuracy of the proposed techniques. Furthermore, the error is computed and the proposed method has reasonable convergence rates in spatial and temporal discretizations.

Gravitation, Symmetry and Undergraduates

NASA Astrophysics Data System (ADS)

Jorgensen, Jamie

2001-04-01

This talk will discuss "Project Petrov" Which is designed to investigate gravitational fields with symmetry. Project Petrov represents a collaboration involving physicists, mathematicians as well as graduate and undergraduate math and physics students. An overview of Project Petrov will be given, with an emphasis on students' contributions, including software to classify and generate Lie algebras, to classify isometry groups, and to compute the isometry group of a given metric.

Rotor dynamic simulation and system identification methods for application to vacuum whirl data

NASA Technical Reports Server (NTRS)

Berman, A.; Giansante, N.; Flannelly, W. G.

1980-01-01

Methods of using rotor vacuum whirl data to improve the ability to model helicopter rotors were developed. The work consisted of the formulation of the equations of motion of elastic blades on a hub using a Galerkin method; the development of a general computer program for simulation of these equations; the study and implementation of a procedure for determining physical parameters based on measured data; and the application of a method for computing the normal modes and natural frequencies based on test data.

Multiple methods integration for structural mechanics analysis and design

NASA Technical Reports Server (NTRS)

Housner, J. M.; Aminpour, M. A.

1991-01-01

A new research area of multiple methods integration is proposed for joining diverse methods of structural mechanics analysis which interact with one another. Three categories of multiple methods are defined: those in which a physical interface are well defined; those in which a physical interface is not well-defined, but selected; and those in which the interface is a mathematical transformation. Two fundamental integration procedures are presented that can be extended to integrate various methods (e.g., finite elements, Rayleigh Ritz, Galerkin, and integral methods) with one another. Since the finite element method will likely be the major method to be integrated, its enhanced robustness under element distortion is also examined and a new robust shell element is demonstrated.

Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid-structure interaction, and free surface flow: Part I

NASA Astrophysics Data System (ADS)

Saye, Robert

2017-09-01

In this two-part paper, a high-order accurate implicit mesh discontinuous Galerkin (dG) framework is developed for fluid interface dynamics, facilitating precise computation of interfacial fluid flow in evolving geometries. The framework uses implicitly defined meshes-wherein a reference quadtree or octree grid is combined with an implicit representation of evolving interfaces and moving domain boundaries-and allows physically prescribed interfacial jump conditions to be imposed or captured with high-order accuracy. Part one discusses the design of the framework, including: (i) high-order quadrature for implicitly defined elements and faces; (ii) high-order accurate discretisation of scalar and vector-valued elliptic partial differential equations with interfacial jumps in ellipticity coefficient, leading to optimal-order accuracy in the maximum norm and discrete linear systems that are symmetric positive (semi)definite; (iii) the design of incompressible fluid flow projection operators, which except for the influence of small penalty parameters, are discretely idempotent; and (iv) the design of geometric multigrid methods for elliptic interface problems on implicitly defined meshes and their use as preconditioners for the conjugate gradient method. Also discussed is a variety of aspects relating to moving interfaces, including: (v) dG discretisations of the level set method on implicitly defined meshes; (vi) transferring state between evolving implicit meshes; (vii) preserving mesh topology to accurately compute temporal derivatives; (viii) high-order accurate reinitialisation of level set functions; and (ix) the integration of adaptive mesh refinement. In part two, several applications of the implicit mesh dG framework in two and three dimensions are presented, including examples of single phase flow in nontrivial geometry, surface tension-driven two phase flow with phase-dependent fluid density and viscosity, rigid body fluid-structure interaction, and free

Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid-structure interaction, and free surface flow: Part II

NASA Astrophysics Data System (ADS)

Saye, Robert

2017-09-01

In this two-part paper, a high-order accurate implicit mesh discontinuous Galerkin (dG) framework is developed for fluid interface dynamics, facilitating precise computation of interfacial fluid flow in evolving geometries. The framework uses implicitly defined meshes-wherein a reference quadtree or octree grid is combined with an implicit representation of evolving interfaces and moving domain boundaries-and allows physically prescribed interfacial jump conditions to be imposed or captured with high-order accuracy. Part one discusses the design of the framework, including: (i) high-order quadrature for implicitly defined elements and faces; (ii) high-order accurate discretisation of scalar and vector-valued elliptic partial differential equations with interfacial jumps in ellipticity coefficient, leading to optimal-order accuracy in the maximum norm and discrete linear systems that are symmetric positive (semi)definite; (iii) the design of incompressible fluid flow projection operators, which except for the influence of small penalty parameters, are discretely idempotent; and (iv) the design of geometric multigrid methods for elliptic interface problems on implicitly defined meshes and their use as preconditioners for the conjugate gradient method. Also discussed is a variety of aspects relating to moving interfaces, including: (v) dG discretisations of the level set method on implicitly defined meshes; (vi) transferring state between evolving implicit meshes; (vii) preserving mesh topology to accurately compute temporal derivatives; (viii) high-order accurate reinitialisation of level set functions; and (ix) the integration of adaptive mesh refinement. In part two, several applications of the implicit mesh dG framework in two and three dimensions are presented, including examples of single phase flow in nontrivial geometry, surface tension-driven two phase flow with phase-dependent fluid density and viscosity, rigid body fluid-structure interaction, and free

Reliable and efficient a posteriori error estimation for adaptive IGA boundary element methods for weakly-singular integral equations

PubMed Central

Feischl, Michael; Gantner, Gregor; Praetorius, Dirk

2015-01-01

We consider the Galerkin boundary element method (BEM) for weakly-singular integral equations of the first-kind in 2D. We analyze some residual-type a posteriori error estimator which provides a lower as well as an upper bound for the unknown Galerkin BEM error. The required assumptions are weak and allow for piecewise smooth parametrizations of the boundary, local mesh-refinement, and related standard piecewise polynomials as well as NURBS. In particular, our analysis gives a first contribution to adaptive BEM in the frame of isogeometric analysis (IGABEM), for which we formulate an adaptive algorithm which steers the local mesh-refinement and the multiplicity of the knots. Numerical experiments underline the theoretical findings and show that the proposed adaptive strategy leads to optimal convergence. PMID:26085698

Proteus-MOC: A 3D deterministic solver incorporating 2D method of characteristics

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

Marin-Lafleche, A.; Smith, M. A.; Lee, C.

2013-07-01

A new transport solution methodology was developed by combining the two-dimensional method of characteristics with the discontinuous Galerkin method for the treatment of the axial variable. The method, which can be applied to arbitrary extruded geometries, was implemented in PROTEUS-MOC and includes parallelization in group, angle, plane, and space using a top level GMRES linear algebra solver. Verification tests were performed to show accuracy and stability of the method with the increased number of angular directions and mesh elements. Good scalability with parallelism in angle and axial planes is displayed. (authors)

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

Domain Decomposition Algorithms for First-Order System Least Squares Methods

NASA Technical Reports Server (NTRS)

Pavarino, Luca F.

1996-01-01

Least squares methods based on first-order systems have been recently proposed and analyzed for second-order elliptic equations and systems. They produce symmetric and positive definite discrete systems by using standard finite element spaces, which are not required to satisfy the inf-sup condition. In this paper, several domain decomposition algorithms for these first-order least squares methods are studied. Some representative overlapping and substructuring algorithms are considered in their additive and multiplicative variants. The theoretical and numerical results obtained show that the classical convergence bounds (on the iteration operator) for standard Galerkin discretizations are also valid for least squares methods.

Ablative Thermal Response Analysis Using the Finite Element Method

NASA Technical Reports Server (NTRS)

Dec John A.; Braun, Robert D.

2009-01-01

A review of the classic techniques used to solve ablative thermal response problems is presented. The advantages and disadvantages of both the finite element and finite difference methods are described. As a first step in developing a three dimensional finite element based ablative thermal response capability, a one dimensional computer tool has been developed. The finite element method is used to discretize the governing differential equations and Galerkin's method of weighted residuals is used to derive the element equations. A code to code comparison between the current 1-D tool and the 1-D Fully Implicit Ablation and Thermal Response Program (FIAT) has been performed.

A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow

NASA Astrophysics Data System (ADS)

Krank, Benjamin; Fehn, Niklas; Wall, Wolfgang A.; Kronbichler, Martin

2017-11-01

We present an efficient discontinuous Galerkin scheme for simulation of the incompressible Navier-Stokes equations including laminar and turbulent flow. We consider a semi-explicit high-order velocity-correction method for time integration as well as nodal equal-order discretizations for velocity and pressure. The non-linear convective term is treated explicitly while a linear system is solved for the pressure Poisson equation and the viscous term. The key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under-resolved high-Reynolds-number flows as well as small time steps. This penalty method is similar to the grad-div stabilization widely used in continuous finite elements. We further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations. The solver is specifically designed for large-scale computations through matrix-free linear solvers including efficient preconditioning strategies and tensor-product elements, which have allowed us to scale this code up to 34.4 billion degrees of freedom and 147,456 CPU cores. We validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large-eddy simulation of turbulent channel flow at Reτ = 180 as well as 590.

Parallel, adaptive finite element methods for conservation laws

NASA Technical Reports Server (NTRS)

Biswas, Rupak; Devine, Karen D.; Flaherty, Joseph E.

1994-01-01

We construct parallel finite element methods for the solution of hyperbolic conservation laws in one and two dimensions. Spatial discretization is performed by a discontinuous Galerkin finite element method using a basis of piecewise Legendre polynomials. Temporal discretization utilizes a Runge-Kutta method. Dissipative fluxes and projection limiting prevent oscillations near solution discontinuities. A posteriori estimates of spatial errors are obtained by a p-refinement technique using superconvergence at Radau points. The resulting method is of high order and may be parallelized efficiently on MIMD computers. We compare results using different limiting schemes and demonstrate parallel efficiency through computations on an NCUBE/2 hypercube. We also present results using adaptive h- and p-refinement to reduce the computational cost of the method.

Parallelization of an Object-Oriented Unstructured Aeroacoustics Solver

NASA Technical Reports Server (NTRS)

Baggag, Abdelkader; Atkins, Harold; Oezturan, Can; Keyes, David

1999-01-01

A computational aeroacoustics code based on the discontinuous Galerkin method is ported to several parallel platforms using MPI. The discontinuous Galerkin method is a compact high-order method that retains its accuracy and robustness on non-smooth unstructured meshes. In its semi-discrete form, the discontinuous Galerkin method can be combined with explicit time marching methods making it well suited to time accurate computations. The compact nature of the discontinuous Galerkin method also makes it well suited for distributed memory parallel platforms. The original serial code was written using an object-oriented approach and was previously optimized for cache-based machines. The port to parallel platforms was achieved simply by treating partition boundaries as a type of boundary condition. Code modifications were minimal because boundary conditions were abstractions in the original program. Scalability results are presented for the SCI Origin, IBM SP2, and clusters of SGI and Sun workstations. Slightly superlinear speedup is achieved on a fixed-size problem on the Origin, due to cache effects.

Optimal least-squares finite element method for elliptic problems

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Povinelli, Louis A.

1991-01-01

An optimal least squares finite element method is proposed for two dimensional and three dimensional elliptic problems and its advantages are discussed over the mixed Galerkin method and the usual least squares finite element method. In the usual least squares finite element method, the second order equation (-Delta x (Delta u) + u = f) is recast as a first order system (-Delta x p + u = f, Delta u - p = 0). The error analysis and numerical experiment show that, in this usual least squares finite element method, the rate of convergence for flux p is one order lower than optimal. In order to get an optimal least squares method, the irrotationality Delta x p = 0 should be included in the first order system.

Domain decomposition methods for nonconforming finite element spaces of Lagrange-type

NASA Technical Reports Server (NTRS)

Cowsar, Lawrence C.

1993-01-01

In this article, we consider the application of three popular domain decomposition methods to Lagrange-type nonconforming finite element discretizations of scalar, self-adjoint, second order elliptic equations. The additive Schwarz method of Dryja and Widlund, the vertex space method of Smith, and the balancing method of Mandel applied to nonconforming elements are shown to converge at a rate no worse than their applications to the standard conforming piecewise linear Galerkin discretization. Essentially, the theory for the nonconforming elements is inherited from the existing theory for the conforming elements with only modest modification by constructing an isomorphism between the nonconforming finite element space and a space of continuous piecewise linear functions.

Galerkin finite element scheme for magnetostrictive structures and composites

NASA Astrophysics Data System (ADS)

Kannan, Kidambi Srinivasan

The ever increasing-role of magnetostrictives in actuation and sensing applications is an indication of their importance in the emerging field of smart structures technology. As newer, and more complex, applications are developed, there is a growing need for a reliable computational tool that can effectively address the magneto-mechanical interactions and other nonlinearities in these materials and in structures incorporating them. This thesis presents a continuum level quasi-static, three-dimensional finite element computational scheme for modeling the nonlinear behavior of bulk magnetostrictive materials and particulate magnetostrictive composites. Models for magnetostriction must deal with two sources of nonlinearities-nonlinear body forces/moments in equilibrium equations governing magneto-mechanical interactions in deformable and magnetized bodies; and nonlinear coupled magneto-mechanical constitutive models for the material of interest. In the present work, classical differential formulations for nonlinear magneto-mechanical interactions are recast in integral form using the weighted-residual method. A discretized finite element form is obtained by applying the Galerkin technique. The finite element formulation is based upon three dimensional eight-noded (isoparametric) brick element interpolation functions and magnetostatic infinite elements at the boundary. Two alternative possibilities are explored for establishing the nonlinear incremental constitutive model-characterization in terms of magnetic field or in terms of magnetization. The former methodology is the one most commonly used in the literature. In this work, a detailed comparative study of both methodologies is carried out. The computational scheme is validated, qualitatively and quantitatively, against experimental measurements published in the literature on structures incorporating the magnetostrictive material Terfenol-D. The influence of nonlinear body forces and body moments of magnetic origin

kappa-Version of Finite Element Method: A New Mathematical and Computational Framework for BVP and IVP

DTIC Science & Technology

2007-01-01

differentiability, fluid-solid interaction, error estimation, re-discretization, moving meshes 16. SECURITY CLASSIFICATION OF: 17 . LIMITATION OF 18. NUMBER...method the weight function is an indepen- dent function v = 0 6 4Ph , with v = 0 on F, if W = W0 on F1. 2. Galerkin method (GM): If Wh is an approximation...This can be demonstrated by considering a simple I-D case (like described above) in which the discretization 17 is uniform with characteristic length

A symmetric Trefftz-DG formulation based on a local boundary element method for the solution of the Helmholtz equation

NASA Astrophysics Data System (ADS)

Barucq, H.; Bendali, A.; Fares, M.; Mattesi, V.; Tordeux, S.

2017-02-01

A general symmetric Trefftz Discontinuous Galerkin method is built for solving the Helmholtz equation with piecewise constant coefficients. The construction of the corresponding local solutions to the Helmholtz equation is based on a boundary element method. A series of numerical experiments displays an excellent stability of the method relatively to the penalty parameters, and more importantly its outstanding ability to reduce the instabilities known as the "pollution effect" in the literature on numerical simulations of long-range wave propagation.

On the error propagation of semi-Lagrange and Fourier methods for advection problems☆

PubMed Central

Einkemmer, Lukas; Ostermann, Alexander

2015-01-01

In this paper we study the error propagation of numerical schemes for the advection equation in the case where high precision is desired. The numerical methods considered are based on the fast Fourier transform, polynomial interpolation (semi-Lagrangian methods using a Lagrange or spline interpolation), and a discontinuous Galerkin semi-Lagrangian approach (which is conservative and has to store more than a single value per cell). We demonstrate, by carrying out numerical experiments, that the worst case error estimates given in the literature provide a good explanation for the error propagation of the interpolation-based semi-Lagrangian methods. For the discontinuous Galerkin semi-Lagrangian method, however, we find that the characteristic property of semi-Lagrangian error estimates (namely the fact that the error increases proportionally to the number of time steps) is not observed. We provide an explanation for this behavior and conduct numerical simulations that corroborate the different qualitative features of the error in the two respective types of semi-Lagrangian methods. The method based on the fast Fourier transform is exact but, due to round-off errors, susceptible to a linear increase of the error in the number of time steps. We show how to modify the Cooley–Tukey algorithm in order to obtain an error growth that is proportional to the square root of the number of time steps. Finally, we show, for a simple model, that our conclusions hold true if the advection solver is used as part of a splitting scheme. PMID:25844018

Lessons Learned in Part-of-Speech Tagging of Conversational Speech

DTIC Science & Technology

2010-10-01

for conversational speech recognition. In Plenary Meeting and Symposium on Prosody and Speech Processing. Slav Petrov and Dan Klein. 2007. Improved...inference for unlexicalized parsing. In HLT-NAACL. Slav Petrov. 2010. Products of random latent variable grammars. In HLT-NAACL. Brian Roark, Yang Liu

Discontinuous Galerkin modeling of the Columbia River's coupled estuary-plume dynamics

NASA Astrophysics Data System (ADS)

Vallaeys, Valentin; Kärnä, Tuomas; Delandmeter, Philippe; Lambrechts, Jonathan; Baptista, António M.; Deleersnijder, Eric; Hanert, Emmanuel

2018-04-01

The Columbia River (CR) estuary is characterized by high river discharge and strong tides that generate high velocity flows and sharp density gradients. Its dynamics strongly affects the coastal ocean circulation. Tidal straining in turn modulates the stratification in the estuary. Simulating the hydrodynamics of the CR estuary and plume therefore requires a multi-scale model as both shelf and estuarine circulations are coupled. Such a model has to keep numerical dissipation as low as possible in order to correctly represent the plume propagation and the salinity intrusion in the estuary. Here, we show that the 3D baroclinic discontinuous Galerkin finite element model SLIM 3D is able to reproduce the main features of the CR estuary-to-ocean continuum. We introduce new vertical discretization and mode splitting that allow us to model a region characterized by complex bathymetry and sharp density and velocity gradients. Our model takes into account the major forcings, i.e. tides, surface wind stress and river discharge, on a single multi-scale grid. The simulation period covers the end of spring-early summer of 2006, a period of high river flow and strong changes in the wind regime. SLIM 3D is validated with in-situ data on the shelf and at multiple locations in the estuary and compared with an operational implementation of SELFE. The model skill in the estuary and on the shelf indicate that SLIM 3D is able to reproduce the key processes driving the river plume dynamics, such as the occurrence of bidirectional plumes or reversals of the inner shelf coastal currents.

Efficiency trade-offs of steady-state methods using FEM and FDM. [iterative solutions for nonlinear flow equations

NASA Technical Reports Server (NTRS)

Gartling, D. K.; Roache, P. J.

1978-01-01

The efficiency characteristics of finite element and finite difference approximations for the steady-state solution of the Navier-Stokes equations are examined. The finite element method discussed is a standard Galerkin formulation of the incompressible, steady-state Navier-Stokes equations. The finite difference formulation uses simple centered differences that are O(delta x-squared). Operation counts indicate that a rapidly converging Newton-Raphson-Kantorovitch iteration scheme is generally preferable over a Picard method. A split NOS Picard iterative algorithm for the finite difference method was most efficient.

Advanced Fluid Reduced Order Models for Compressible Flow.

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

Tezaur, Irina Kalashnikova; Fike, Jeffrey A.; Carlberg, Kevin Thomas

This report summarizes fiscal year (FY) 2017 progress towards developing and implementing within the SPARC in-house finite volume flow solver advanced fluid reduced order models (ROMs) for compressible captive-carriage flow problems of interest to Sandia National Laboratories for the design and qualification of nuclear weapons components. The proposed projection-based model order reduction (MOR) approach, known as the Proper Orthogonal Decomposition (POD)/Least- Squares Petrov-Galerkin (LSPG) method, can substantially reduce the CPU-time requirement for these simulations, thereby enabling advanced analyses such as uncertainty quantification and de- sign optimization. Following a description of the project objectives and FY17 targets, we overview briefly themore » POD/LSPG approach to model reduction implemented within SPARC . We then study the viability of these ROMs for long-time predictive simulations in the context of a two-dimensional viscous laminar cavity problem, and describe some FY17 enhancements to the proposed model reduction methodology that led to ROMs with improved predictive capabilities. Also described in this report are some FY17 efforts pursued in parallel to the primary objective of determining whether the ROMs in SPARC are viable for the targeted application. These include the implemen- tation and verification of some higher-order finite volume discretization methods within SPARC (towards using the code to study the viability of ROMs on three-dimensional cavity problems) and a novel structure-preserving constrained POD/LSPG formulation that can improve the accuracy of projection-based reduced order models. We conclude the report by summarizing the key takeaways from our FY17 findings, and providing some perspectives for future work.« less

Construction of energy-stable Galerkin reduced order models.

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

Kalashnikova, Irina; Barone, Matthew Franklin; Arunajatesan, Srinivasan

2013-05-01

This report aims to unify several approaches for building stable projection-based reduced order models (ROMs). Attention is focused on linear time-invariant (LTI) systems. The model reduction procedure consists of two steps: the computation of a reduced basis, and the projection of the governing partial differential equations (PDEs) onto this reduced basis. Two kinds of reduced bases are considered: the proper orthogonal decomposition (POD) basis and the balanced truncation basis. The projection step of the model reduction can be done in two ways: via continuous projection or via discrete projection. First, an approach for building energy-stable Galerkin ROMs for linear hyperbolicmore » or incompletely parabolic systems of PDEs using continuous projection is proposed. The idea is to apply to the set of PDEs a transformation induced by the Lyapunov function for the system, and to build the ROM in the transformed variables. The resulting ROM will be energy-stable for any choice of reduced basis. It is shown that, for many PDE systems, the desired transformation is induced by a special weighted L2 inner product, termed the %E2%80%9Csymmetry inner product%E2%80%9D. Attention is then turned to building energy-stable ROMs via discrete projection. A discrete counterpart of the continuous symmetry inner product, a weighted L2 inner product termed the %E2%80%9CLyapunov inner product%E2%80%9D, is derived. The weighting matrix that defines the Lyapunov inner product can be computed in a black-box fashion for a stable LTI system arising from the discretization of a system of PDEs in space. It is shown that a ROM constructed via discrete projection using the Lyapunov inner product will be energy-stable for any choice of reduced basis. Connections between the Lyapunov inner product and the inner product induced by the balanced truncation algorithm are made. Comparisons are also made between the symmetry inner product and the Lyapunov inner product. The performance of ROMs

Uncertainty Propagation for Turbulent, Compressible Flow in a Quasi-1D Nozzle Using Stochastic Methods

NASA Technical Reports Server (NTRS)

Zang, Thomas A.; Mathelin, Lionel; Hussaini, M. Yousuff; Bataille, Francoise

2003-01-01

This paper describes a fully spectral, Polynomial Chaos method for the propagation of uncertainty in numerical simulations of compressible, turbulent flow, as well as a novel stochastic collocation algorithm for the same application. The stochastic collocation method is key to the efficient use of stochastic methods on problems with complex nonlinearities, such as those associated with the turbulence model equations in compressible flow and for CFD schemes requiring solution of a Riemann problem. Both methods are applied to compressible flow in a quasi-one-dimensional nozzle. The stochastic collocation method is roughly an order of magnitude faster than the fully Galerkin Polynomial Chaos method on the inviscid problem.

Exploiting Superconvergence in Discontinuous Galerkin Methods for Improved Time-Stepping and Visualization

DTIC Science & Technology

2016-09-08

Accuracy Conserving (SIAC) filter when applied to nonuniform meshes; 2) Theoretically and numerical demonstration of the 2k+1 order accuracy of the SIAC...Establishing a more theoretical and numerical understanding of a computationally efficient scaling for the SIAC filter for nonuniform meshes [7]; 2...Li, “SIAC Filtering of DG Methods – Boundary and Nonuniform Mesh”, International Conference on Spectral and Higher Order Methods (ICOSAHOM

Modeling radium and radon transport through soil and vegetation

USGS Publications Warehouse

Kozak, J.A.; Reeves, H.W.; Lewis, B.A.

2003-01-01

A one-dimensional flow and transport model was developed to describe the movement of two fluid phases, gas and water, within a porous medium and the transport of 226Ra and 222Rn within and between these two phases. Included in this model is the vegetative uptake of water and aqueous 226Ra and 222Rn that can be extracted from the soil via the transpiration stream. The mathematical model is formulated through a set of phase balance equations and a set of species balance equations. Mass exchange, sink terms and the dependence of physical properties upon phase composition couple the two sets of equations. Numerical solution of each set, with iteration between the sets, is carried out leading to a set-iterative compositional model. The Petrov-Galerkin finite element approach is used to allow for upstream weighting if required for a given simulation. Mass lumping improves solution convergence and stability behavior. The resulting numerical model was applied to four problems and was found to produce accurate, mass conservative solutions when compared to published experimental and numerical results and theoretical column experiments. Preliminary results suggest that the model can be used as an investigative tool to determine the feasibility of phytoremediating radium and radon-contaminated soil. ?? 2003 Elsevier Science B.V. All rights reserved.

A nodally condensed SUPG formulation for free-surface computation of steady-state flows constrained by unilateral contact - Application to rolling

NASA Astrophysics Data System (ADS)

Arora, Shitij; Fourment, Lionel

2018-05-01

In the context of the simulation of industrial hot forming processes, the resultant time-dependent thermo-mechanical multi-field problem (v →,p ,σ ,ɛ ) can be sped up by 10-50 times using the steady-state methods while compared to the conventional incremental methods. Though the steady-state techniques have been used in the past, but only on simple configurations and with structured meshes, and the modern-days problems are in the framework of complex configurations, unstructured meshes and parallel computing. These methods remove time dependency from the equations, but introduce an additional unknown into the problem: the steady-state shape. This steady-state shape x → can be computed as a geometric correction t → on the domain X → by solving the weak form of the steady-state equation v →.n →(t →)=0 using a Streamline Upwind Petrov Galerkin (SUPG) formulation. There exists a strong coupling between the domain shape and the material flow, hence, a two-step fixed point iterative resolution algorithm was proposed that involves (1) the computation of flow field from the resolution of thermo-mechanical equations on a prescribed domain shape and (2) the computation of steady-state shape for an assumed velocity field. The contact equations are introduced in the penalty form both during the flow computation as well as during the free-surface correction. The fact that the contact description is inhomogeneous, i.e., it is defined in the nodal form in the former, and in the weighted residual form in the latter, is assumed to be critical to the convergence of certain problems. Thus, the notion of nodal collocation is invoked in the weak form of the surface correction equation to homogenize the contact coupling. The surface correction algorithm is tested on certain analytical test cases and the contact coupling is tested with some hot rolling problems.

Edge reconstruction in armchair phosphorene nanoribbons revealed by discontinuous Galerkin density functional theory.

PubMed

Hu, Wei; Lin, Lin; Yang, Chao

2015-12-21

With the help of our recently developed massively parallel DGDFT (Discontinuous Galerkin Density Functional Theory) methodology, we perform large-scale Kohn-Sham density functional theory calculations on phosphorene nanoribbons with armchair edges (ACPNRs) containing a few thousands to ten thousand atoms. The use of DGDFT allows us to systematically achieve a conventional plane wave basis set type of accuracy, but with a much smaller number (about 15) of adaptive local basis (ALB) functions per atom for this system. The relatively small number of degrees of freedom required to represent the Kohn-Sham Hamiltonian, together with the use of the pole expansion the selected inversion (PEXSI) technique that circumvents the need to diagonalize the Hamiltonian, results in a highly efficient and scalable computational scheme for analyzing the electronic structures of ACPNRs as well as their dynamics. The total wall clock time for calculating the electronic structures of large-scale ACPNRs containing 1080-10,800 atoms is only 10-25 s per self-consistent field (SCF) iteration, with accuracy fully comparable to that obtained from conventional planewave DFT calculations. For the ACPNR system, we observe that the DGDFT methodology can scale to 5000-50,000 processors. We use DGDFT based ab initio molecular dynamics (AIMD) calculations to study the thermodynamic stability of ACPNRs. Our calculations reveal that a 2 × 1 edge reconstruction appears in ACPNRs at room temperature.

A mixed finite difference/Galerkin method for three-dimensional Rayleigh-Benard convection

NASA Technical Reports Server (NTRS)

Buell, Jeffrey C.

1988-01-01

A fast and accurate numerical method, for nonlinear conservation equation systems whose solutions are periodic in two of the three spatial dimensions, is presently implemented for the case of Rayleigh-Benard convection between two rigid parallel plates in the parameter region where steady, three-dimensional convection is known to be stable. High-order streamfunctions secure the reduction of the system of five partial differential equations to a system of only three. Numerical experiments are presented which verify both the expected convergence rates and the absolute accuracy of the method.

[Neo-urethroclitoroplasty according to Petrovic].

PubMed

Trombetta, Carlo; Liguori, Giovanni; Benvenuto, Sara; Petrovic, Milos; Napoli, Renata; Umari, Paolo; Rizzo, Michele; Zordani, Alessio

2011-01-01

We present a refinement to our original technique in MtF gender reassignment surgery. Our goal was to construct a neoclitoris, which is wet and covered with urethral neoprepuce. Since 1995 more than 300 transgender MtF patients have been operated at our institution. Our refinement has been applied to 12 cases and showed both excellent functional and cosmetic results during midterm follow-up. During 2010 several sex reassignment surgeries have been performed using our new technique that includes: bilateral orchiectomy, removal of corpora cavernosa of the penis, formation of the neourethra with neomeatus, neovaginoplasty by inversion of penoscrotal skin flaps, construction of the neoclitoris with preservation of the neurovascular bundle and exterior vulva formation. The refinement consists in creating a neoclitoris embedded in urethral mucosa using urethral flaps. These flaps are in continuity with the previously spatulated urethra. The urethral plate is further incised distally in a Y fashion. The urethral flaps are sutured around the neoclitoris to form a neo-urethroclitoris covered by urethral neoprepuce, which resembles a real female clitoris. The neoclitoris is positioned in the anatomical position of the male suspensory ligament of the penis that is also the natural anatomical position of the female clitoris. With this method we are able to construct a clitoris with a normal sensitivity embedded in urethral mucosa that remains wet and hairless. It can be easily stimulated during sexual intercourse, as most of the patients reported great satisfaction and ability to reach orgasm. We want to emphasize how both the cosmetic results and functionality of the neovagina and neoclitoris are important in this type of surgery for the quality of life of our patients. We are still far from a perfect surgical solution, but we are further improving our technique and follow our aims step by step.

Solution of second order quasi-linear boundary value problems by a wavelet method

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

Zhang, Lei; Zhou, Youhe; Wang, Jizeng, E-mail: jzwang@lzu.edu.cn

2015-03-10

A wavelet Galerkin method based on expansions of Coiflet-like scaling function bases is applied to solve second order quasi-linear boundary value problems which represent a class of typical nonlinear differential equations. Two types of typical engineering problems are selected as test examples: one is about nonlinear heat conduction and the other is on bending of elastic beams. Numerical results are obtained by the proposed wavelet method. Through comparing to relevant analytical solutions as well as solutions obtained by other methods, we find that the method shows better efficiency and accuracy than several others, and the rate of convergence can evenmore » reach orders of 5.8.« less

Equivalence between the Energy Stable Flux Reconstruction and Filtered Discontinuous Galerkin Schemes

NASA Astrophysics Data System (ADS)

Zwanenburg, Philip; Nadarajah, Siva

2016-02-01

The aim of this paper is to demonstrate the equivalence between filtered Discontinuous Galerkin (DG) schemes and the Energy Stable Flux Reconstruction (ESFR) schemes, expanding on previous demonstrations in 1D [1] and for straight-sided elements in 3D [2]. We first derive the DG and ESFR schemes in strong form and compare the respective flux penalization terms while highlighting the implications of the fundamental assumptions for stability in the ESFR formulations, notably that all ESFR scheme correction fields can be interpreted as modally filtered DG correction fields. We present the result in the general context of all higher dimensional curvilinear element formulations. Through a demonstration that there exists a weak form of the ESFR schemes which is both discretely and analytically equivalent to the strong form, we then extend the results obtained for the strong formulations to demonstrate that ESFR schemes can be interpreted as a DG scheme in weak form where discontinuous edge flux is substituted for numerical edge flux correction. Theoretical derivations are then verified with numerical results obtained from a 2D Euler testcase with curved boundaries. Given the current choice of high-order DG-type schemes and the question as to which might be best to use for a specific application, the main significance of this work is the bridge that it provides between them. Clearly outlining the similarities between the schemes results in the important conclusion that it is always less efficient to use ESFR schemes, as opposed to the weak DG scheme, when solving problems implicitly.

Quaternary Narrow-Band Semiconductors (HgTe)x(InSb)1-X for Far-Infrared Detectors.

DTIC Science & Technology

1986-08-31

A. L. Petrov, and G. A. Kuzmina . Sov. Phys. Semicond., 13, 480 (1979). 34. A. V. Voitsekhovskii and V. P. Drobyazko. Inorg. Materials, 3, 1976 (1967...801, 807 (1973); Izv. VUZ Fiz. USSR, 16, 76, 82 (1973). 37. A. L. Petrov and G. A. Kuzmina . Sov. Phys. Semicond., 14, . 604 (1980). 38. G. I

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.

High-Order Methods for Computational Fluid Dynamics: A Brief Review of Compact Differential Formulations on Unstructured Grids

NASA Technical Reports Server (NTRS)

Huynh, H. T.; Wang, Z. J.; Vincent, P. E.

2013-01-01

Popular high-order schemes with compact stencils for Computational Fluid Dynamics (CFD) include Discontinuous Galerkin (DG), Spectral Difference (SD), and Spectral Volume (SV) methods. The recently proposed Flux Reconstruction (FR) approach or Correction Procedure using Reconstruction (CPR) is based on a differential formulation and provides a unifying framework for these high-order schemes. Here we present a brief review of recent developments for the FR/CPR schemes as well as some pacing items.

Numerical evaluation of discontinuous and nonconforming finite element methods in nonlinear solid mechanics

NASA Astrophysics Data System (ADS)

Bayat, Hamid Reza; Krämer, Julian; Wunderlich, Linus; Wulfinghoff, Stephan; Reese, Stefanie; Wohlmuth, Barbara; Wieners, Christian

2018-03-01

This work presents a systematic study of discontinuous and nonconforming finite element methods for linear elasticity, finite elasticity, and small strain plasticity. In particular, we consider new hybrid methods with additional degrees of freedom on the skeleton of the mesh and allowing for a local elimination of the element-wise degrees of freedom. We show that this process leads to a well-posed approximation scheme. The quality of the new methods with respect to locking and anisotropy is compared with standard and in addition locking-free conforming methods as well as established (non-) symmetric discontinuous Galerkin methods with interior penalty. For several benchmark configurations, we show that all methods converge asymptotically for fine meshes and that in many cases the hybrid methods are more accurate for a fixed size of the discrete system.

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

Galerkin projection for geometrically-exact multilayer beams allowing for ply drop-off

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

Vu-Quoc, L.; Deng, H.

1995-12-31

Focusing on the static case in the present work, we develop a Galerkin projection of the resulting nonlinear governing equations of equilibrium for geometrically exact sandwich beams and 1-D plates developed. In the proposed theory, each layer in the beam can have different thickness and length. As such one can use the present formulation to model an important class of multilayer structures having ply drop-off. No restriction is imposed on the magnitude of the displacement field, whose continuity across the layer interfaces is exactly enforced. The layer cross section in the deformed beam is assumed to remain straight, but notmore » orthogonal to the layer centroidal line, thus shear deformation in each layer is accounted for. Also no restriction is imposed on the rotation of a layer cross section. It follows that the overall cross section in the deformed beam is continuous piecewise linear, and can be best thought of as a chain of rigid links, connected by hinges. The overall deformation of a multilayer beam can be described by the deformation of a reference layer. The unknown kinematic quantities are therefore the two displacement components of the deformed centroidal line of a reference layer, and the finite rotations of the layers. The present theory can be used to analyze large deformation in sandwich beams. Numerical examples, such as roll-up maneuver and sandwich beam with ply drop-off, which underline the salient features of the formulation are presented. Saint-Venant principle is demonstrated for very short sandwich beams. The readers are referred to the paper for detail.« less

Modeling Storm Surges Using Discontinuous Galerkin Methods

DTIC Science & Technology

2016-06-01

devastating impact on coastlines throughout the United States. In order to accurately understand the impacts of storm surges there needs to be an effective ...model. One of the governing systems of equations used to model storm surges’ effects is the Shallow Water Equations (SWE). In this thesis, we solve the...closer to the shoreline. In our simulation, we also learned of the effects spurious waves can have on the results. Due to boundary conditions, a

Ultrasound Tomography by Galerkin or Moment Methods,

DTIC Science & Technology

1983-05-05

in terms of i(x) . Let (31,32) gj~x) - J gji ~ix and G() W i(x) where i(x) is given by (24). Thus, by (25) the coefficients gji an Gqji are givenby...4 yK (mh,nh) y, (mh,nh) gji qji and i(mn)(X) Thus on factoring, we obtain f(S) (f - fu)). 2 (3) i i 4. i 0 ~ ko y,,Qxj)f 0 (,x,)g.. + A qjG bsil. 2

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

Methods for the calculation of axial wave numbers in lined ducts with mean flow

NASA Technical Reports Server (NTRS)

Eversman, W.

1981-01-01

A survey is made of the methods available for the calculation of axial wave numbers in lined ducts. Rectangular and circular ducts with both uniform and non-uniform flow are considered as are ducts with peripherally varying liners. A historical perspective is provided by a discussion of the classical methods for computing attenuation when no mean flow is present. When flow is present these techniques become either impractical or impossible. A number of direct eigenvalue determination schemes which have been used when flow is present are discussed. Methods described are extensions of the classical no-flow technique, perturbation methods based on the no-flow technique, direct integration methods for solution of the eigenvalue equation, an integration-iteration method based on the governing differential equation for acoustic transmission, Galerkin methods, finite difference methods, and finite element methods.

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

NASA Technical Reports Server (NTRS)

Djomehri, M. Jahed; Erickson, Larry L.