Gpu Implementation of a Viscous Flow Solver on Unstructured Grids

NASA Astrophysics Data System (ADS)

Xu, Tianhao; Chen, Long

2016-06-01

Graphics processing units have gained popularities in scientific computing over past several years due to their outstanding parallel computing capability. Computational fluid dynamics applications involve large amounts of calculations, therefore a latest GPU card is preferable of which the peak computing performance and memory bandwidth are much better than a contemporary high-end CPU. We herein focus on the detailed implementation of our GPU targeting Reynolds-averaged Navier-Stokes equations solver based on finite-volume method. The solver employs a vertex-centered scheme on unstructured grids for the sake of being capable of handling complex topologies. Multiple optimizations are carried out to improve the memory accessing performance and kernel utilization. Both steady and unsteady flow simulation cases are carried out using explicit Runge-Kutta scheme. The solver with GPU acceleration in this paper is demonstrated to have competitive advantages over the CPU targeting one.

Workload Characterization of CFD Applications Using Partial Differential Equation Solvers

NASA Technical Reports Server (NTRS)

Waheed, Abdul; Yan, Jerry; Saini, Subhash (Technical Monitor)

1998-01-01

Workload characterization is used for modeling and evaluating of computing systems at different levels of detail. We present workload characterization for a class of Computational Fluid Dynamics (CFD) applications that solve Partial Differential Equations (PDEs). This workload characterization focuses on three high performance computing platforms: SGI Origin2000, EBM SP-2, a cluster of Intel Pentium Pro bases PCs. We execute extensive measurement-based experiments on these platforms to gather statistics of system resource usage, which results in workload characterization. Our workload characterization approach yields a coarse-grain resource utilization behavior that is being applied for performance modeling and evaluation of distributed high performance metacomputing systems. In addition, this study enhances our understanding of interactions between PDE solver workloads and high performance computing platforms and is useful for tuning these applications.

Using Financial Calculators in a Business Mathematics Course.

ERIC Educational Resources Information Center

Heller, William H.; Taylor, Monty B.

2000-01-01

Discusses the authors' experiences with integrating financial calculators into a business mathematics course. Presents a brief overview of the operation of financial calculators, reviews some of the more common models, discusses how to use the equation solver utility on other calculators to emulate a financial calculator, and explores the…

Multiply scaled constrained nonlinear equation solvers. [for nonlinear heat conduction problems

NASA Technical Reports Server (NTRS)

Padovan, Joe; Krishna, Lala

1986-01-01

To improve the numerical stability of nonlinear equation solvers, a partitioned multiply scaled constraint scheme is developed. This scheme enables hierarchical levels of control for nonlinear equation solvers. To complement the procedure, partitioned convergence checks are established along with self-adaptive partitioning schemes. Overall, such procedures greatly enhance the numerical stability of the original solvers. To demonstrate and motivate the development of the scheme, the problem of nonlinear heat conduction is considered. In this context the main emphasis is given to successive substitution-type schemes. To verify the improved numerical characteristics associated with partitioned multiply scaled solvers, results are presented for several benchmark examples.

MIBPB: A software package for electrostatic analysis

PubMed Central

Chen, Duan; Chen, Zhan; Chen, Changjun; Geng, Weihua; Wei, Guo-Wei

2010-01-01

The Poisson-Boltzmann equation (PBE) is an established model for the electrostatic analysis of biomolecules. The development of advanced computational techniques for the solution of the PBE has been an important topic in the past two decades. This paper presents a matched interface and boundary (MIB) based PBE software package, the MIBPB solver, for electrostatic analysis. The MIBPB has a unique feature that it is the first interface technique based PBE solver that rigorously enforces the solution and flux continuity conditions at the dielectric interface between the biomolecule and the solvent. For protein molecular surfaces which may possess troublesome geometrical singularities, the MIB scheme makes the MIBPB by far the only existing PBE solver that is able to deliver the second order convergence, i.e., the accuracy increases four times when the mesh size is halved. The MIBPB method is also equipped with a Dirichlet-to-Neumann mapping (DNM) technique, that builds a Green's function approach to analytically resolve the singular charge distribution in biomolecules in order to obtain reliable solutions at meshes as coarse as 1Å — while it usually takes other traditional PB solvers 0.25Å to reach similar level of reliability. The present work further accelerates the rate of convergence of linear equation systems resulting from the MIBPB by utilizing the Krylov subspace (KS) techniques. Condition numbers of the MIBPB matrices are significantly reduced by using appropriate Krylov subspace solver and preconditioner combinations. Both linear and nonlinear PBE solvers in the MIBPB package are tested by protein-solvent solvation energy calculations and analysis of salt effects on protein-protein binding energies, respectively. PMID:20845420

An approximate Riemann solver for real gas parabolized Navier-Stokes equations

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

Urbano, Annafederica, E-mail: annafederica.urbano@uniroma1.it; Nasuti, Francesco, E-mail: francesco.nasuti@uniroma1.it

2013-01-15

Under specific assumptions, parabolized Navier-Stokes equations are a suitable mean to study channel flows. A special case is that of high pressure flow of real gases in cooling channels where large crosswise gradients of thermophysical properties occur. To solve the parabolized Navier-Stokes equations by a space marching approach, the hyperbolicity of the system of governing equations is obtained, even for very low Mach number flow, by recasting equations such that the streamwise pressure gradient is considered as a source term. For this system of equations an approximate Roe's Riemann solver is developed as the core of a Godunov type finitemore » volume algorithm. The properties of the approximated Riemann solver, which is a modification of Roe's Riemann solver for the parabolized Navier-Stokes equations, are presented and discussed with emphasis given to its original features introduced to handle fluids governed by a generic real gas EoS. Sample solutions are obtained for low Mach number high compressible flows of transcritical methane, heated in straight long channels, to prove the solver ability to describe flows dominated by complex thermodynamic phenomena.« less

Overcoming Challenges in Kinetic Modeling of Magnetized Plasmas and Vacuum Electronic Devices

NASA Astrophysics Data System (ADS)

Omelchenko, Yuri; Na, Dong-Yeop; Teixeira, Fernando

2017-10-01

We transform the state-of-the art of plasma modeling by taking advantage of novel computational techniques for fast and robust integration of multiscale hybrid (full particle ions, fluid electrons, no displacement current) and full-PIC models. These models are implemented in 3D HYPERS and axisymmetric full-PIC CONPIC codes. HYPERS is a massively parallel, asynchronous code. The HYPERS solver does not step fields and particles synchronously in time but instead executes local variable updates (events) at their self-adaptive rates while preserving fundamental conservation laws. The charge-conserving CONPIC code has a matrix-free explicit finite-element (FE) solver based on a sparse-approximate inverse (SPAI) algorithm. This explicit solver approximates the inverse FE system matrix (``mass'' matrix) using successive sparsity pattern orders of the original matrix. It does not reduce the set of Maxwell's equations to a vector-wave (curl-curl) equation of second order but instead utilizes the standard coupled first-order Maxwell's system. We discuss the ability of our codes to accurately and efficiently account for multiscale physical phenomena in 3D magnetized space and laboratory plasmas and axisymmetric vacuum electronic devices.

Evaluation of out-of-core computer programs for the solution of symmetric banded linear equations. [simultaneous equations

NASA Technical Reports Server (NTRS)

Dunham, R. S.

1976-01-01

FORTRAN coded out-of-core equation solvers that solve using direct methods symmetric banded systems of simultaneous algebraic equations. Banded, frontal and column (skyline) solvers were studied as well as solvers that can partition the working area and thus could fit into any available core. Comparison timings are presented for several typical two dimensional and three dimensional continuum type grids of elements with and without midside nodes. Extensive conclusions are also given.

Performance of Nonlinear Finite-Difference Poisson-Boltzmann Solvers

PubMed Central

Cai, Qin; Hsieh, Meng-Juei; Wang, Jun; Luo, Ray

2014-01-01

We implemented and optimized seven finite-difference solvers for the full nonlinear Poisson-Boltzmann equation in biomolecular applications, including four relaxation methods, one conjugate gradient method, and two inexact Newton methods. The performance of the seven solvers was extensively evaluated with a large number of nucleic acids and proteins. Worth noting is the inexact Newton method in our analysis. We investigated the role of linear solvers in its performance by incorporating the incomplete Cholesky conjugate gradient and the geometric multigrid into its inner linear loop. We tailored and optimized both linear solvers for faster convergence rate. In addition, we explored strategies to optimize the successive over-relaxation method to reduce its convergence failures without too much sacrifice in its convergence rate. Specifically we attempted to adaptively change the relaxation parameter and to utilize the damping strategy from the inexact Newton method to improve the successive over-relaxation method. Our analysis shows that the nonlinear methods accompanied with a functional-assisted strategy, such as the conjugate gradient method and the inexact Newton method, can guarantee convergence in the tested molecules. Especially the inexact Newton method exhibits impressive performance when it is combined with highly efficient linear solvers that are tailored for its special requirement. PMID:24723843

A High-Order Direct Solver for Helmholtz Equations with Neumann Boundary Conditions

NASA Technical Reports Server (NTRS)

Sun, Xian-He; Zhuang, Yu

1997-01-01

In this study, a compact finite-difference discretization is first developed for Helmholtz equations on rectangular domains. Special treatments are then introduced for Neumann and Neumann-Dirichlet boundary conditions to achieve accuracy and separability. Finally, a Fast Fourier Transform (FFT) based technique is used to yield a fast direct solver. Analytical and experimental results show this newly proposed solver is comparable to the conventional second-order elliptic solver when accuracy is not a primary concern, and is significantly faster than that of the conventional solver if a highly accurate solution is required. In addition, this newly proposed fourth order Helmholtz solver is parallel in nature. It is readily available for parallel and distributed computers. The compact scheme introduced in this study is likely extendible for sixth-order accurate algorithms and for more general elliptic equations.

How to Find a Bug in Ten Thousand Lines Transport Solver? Outline of Experiences from AN Advection-Diffusion Code Verification

NASA Astrophysics Data System (ADS)

Zamani, K.; Bombardelli, F.

2011-12-01

Almost all natural phenomena on Earth are highly nonlinear. Even simplifications to the equations describing nature usually end up being nonlinear partial differential equations. Transport (ADR) equation is a pivotal equation in atmospheric sciences and water quality. This nonlinear equation needs to be solved numerically for practical purposes so academicians and engineers thoroughly rely on the assistance of numerical codes. Thus, numerical codes require verification before they are utilized for multiple applications in science and engineering. Model verification is a mathematical procedure whereby a numerical code is checked to assure the governing equation is properly solved as it is described in the design document. CFD verification is not a straightforward and well-defined course. Only a complete test suite can uncover all the limitations and bugs. Results are needed to be assessed to make a distinction between bug-induced-defect and innate limitation of a numerical scheme. As Roache (2009) said, numerical verification is a state-of-the-art procedure. Sometimes novel tricks work out. This study conveys the synopsis of the experiences we gained during a comprehensive verification process which was done for a transport solver. A test suite was designed including unit tests and algorithmic tests. Tests were layered in complexity in several dimensions from simple to complex. Acceptance criteria defined for the desirable capabilities of the transport code such as order of accuracy, mass conservation, handling stiff source term, spurious oscillation, and initial shape preservation. At the begining, mesh convergence study which is the main craft of the verification is performed. To that end, analytical solution of ADR equation gathered. Also a new solution was derived. In the more general cases, lack of analytical solution could be overcome through Richardson Extrapolation and Manufactured Solution. Then, two bugs which were concealed during the mesh convergence study uncovered with the method of false injection and visualization of the results. Symmetry had dual functionality: there was a bug, which was hidden due to the symmetric nature of a test (it was detected afterward utilizing artificial false injection), on the other hand self-symmetry was used to design a new test, and in a case the analytical solution of the ADR equation was unknown. Assisting subroutines designed to check and post-process conservation of mass and oscillatory behavior. Finally, capability of the solver also checked for stiff reaction source term. The above test suite not only was a decent tool of error detection but also it provided a thorough feedback on the ADR solvers limitations. Such information is the crux of any rigorous numerical modeling for a modeler who deals with surface/subsurface pollution transport.

An iterative solver for the 3D Helmholtz equation

NASA Astrophysics Data System (ADS)

Belonosov, Mikhail; Dmitriev, Maxim; Kostin, Victor; Neklyudov, Dmitry; Tcheverda, Vladimir

2017-09-01

We develop a frequency-domain iterative solver for numerical simulation of acoustic waves in 3D heterogeneous media. It is based on the application of a unique preconditioner to the Helmholtz equation that ensures convergence for Krylov subspace iteration methods. Effective inversion of the preconditioner involves the Fast Fourier Transform (FFT) and numerical solution of a series of boundary value problems for ordinary differential equations. Matrix-by-vector multiplication for iterative inversion of the preconditioned matrix involves inversion of the preconditioner and pointwise multiplication of grid functions. Our solver has been verified by benchmarking against exact solutions and a time-domain solver.

Solving the Fluid Pressure Poisson Equation Using Multigrid-Evaluation and Improvements.

PubMed

Dick, Christian; Rogowsky, Marcus; Westermann, Rudiger

2016-11-01

In many numerical simulations of fluids governed by the incompressible Navier-Stokes equations, the pressure Poisson equation needs to be solved to enforce mass conservation. Multigrid solvers show excellent convergence in simple scenarios, yet they can converge slowly in domains where physically separated regions are combined at coarser scales. Moreover, existing multigrid solvers are tailored to specific discretizations of the pressure Poisson equation, and they cannot easily be adapted to other discretizations. In this paper we analyze the convergence properties of existing multigrid solvers for the pressure Poisson equation in different simulation domains, and we show how to further improve the multigrid convergence rate by using a graph-based extension to determine the coarse grid hierarchy. The proposed multigrid solver is generic in that it can be applied to different kinds of discretizations of the pressure Poisson equation, by using solely the specification of the simulation domain and pre-assembled computational stencils. We analyze the proposed solver in combination with finite difference and finite volume discretizations of the pressure Poisson equation. Our evaluations show that, despite the common assumption, multigrid schemes can exploit their potential even in the most complicated simulation scenarios, yet this behavior is obtained at the price of higher memory consumption.

Generalized conjugate-gradient methods for the Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Ajmani, Kumud; Ng, Wing-Fai; Liou, Meng-Sing

1991-01-01

A generalized conjugate-gradient method is used to solve the two-dimensional, compressible Navier-Stokes equations of fluid flow. The equations are discretized with an implicit, upwind finite-volume formulation. Preconditioning techniques are incorporated into the new solver to accelerate convergence of the overall iterative method. The superiority of the new solver is demonstrated by comparisons with a conventional line Gauss-Siedel Relaxation solver. Computational test results for transonic flow (trailing edge flow in a transonic turbine cascade) and hypersonic flow (M = 6.0 shock-on-shock phenoena on a cylindrical leading edge) are presented. When applied to the transonic cascade case, the new solver is 4.4 times faster in terms of number of iterations and 3.1 times faster in terms of CPU time than the Relaxation solver. For the hypersonic shock case, the new solver is 3.0 times faster in terms of number of iterations and 2.2 times faster in terms of CPU time than the Relaxation solver.

MODFLOW-2000, The U.S. Geological Survey Modular Ground-Water Model -- GMG Linear Equation Solver Package Documentation

USGS Publications Warehouse

Wilson, John D.; Naff, Richard L.

2004-01-01

A geometric multigrid solver (GMG), based in the preconditioned conjugate gradient algorithm, has been developed for solving systems of equations resulting from applying the cell-centered finite difference algorithm to flow in porous media. This solver has been adapted to the U.S. Geological Survey ground-water flow model MODFLOW-2000. The documentation herein is a description of the solver and the adaptation to MODFLOW-2000.

Navier-Stokes calculations on multi-element airfoils using a chimera-based solver

NASA Technical Reports Server (NTRS)

Jasper, Donald W.; Agrawal, Shreekant; Robinson, Brian A.

1993-01-01

A study of Navier-Stokes calculations of flows about multielement airfoils using a chimera grid approach is presented. The chimera approach utilizes structured, overlapped grids which allow great flexibility of grid arrangement and simplifies grid generation. Calculations are made for two-, three-, and four-element airfoils, and modeling of the effect of gap distance between elements is demonstrated for a two element case. Solutions are obtained using the thin-layer form of the Reynolds averaged Navier-Stokes equations with turbulence closure provided by the Baldwin-Lomax algebraic model or the Baldwin-Barth one equation model. The Baldwin-Barth turbulence model is shown to provide better agreement with experimental data and to dramatically improve convergence rates for some cases. Recently developed, improved farfield boundary conditions are incorporated into the solver for greater efficiency. Computed results show good comparison with experimental data which include aerodynamic forces, surface pressures, and boundary layer velocity profiles.

AQUASOL: An efficient solver for the dipolar Poisson–Boltzmann–Langevin equation

PubMed Central

Koehl, Patrice; Delarue, Marc

2010-01-01

The Poisson–Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson–Boltzmann–Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE solver. While both methods are not guaranteed to converge, numerical evidences suggest that they do and that their convergence is also superlinear. Both variants are significantly faster than the solver based on the exact Jacobian, with a much smaller memory footprint. All three methods have been implemented in a new code named AQUASOL, which is freely available. PMID:20151727

AQUASOL: An efficient solver for the dipolar Poisson-Boltzmann-Langevin equation.

PubMed

Koehl, Patrice; Delarue, Marc

2010-02-14

The Poisson-Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE solver. While both methods are not guaranteed to converge, numerical evidences suggest that they do and that their convergence is also superlinear. Both variants are significantly faster than the solver based on the exact Jacobian, with a much smaller memory footprint. All three methods have been implemented in a new code named AQUASOL, which is freely available.

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 application of the suite. The current models are accurate at low supersonic speed and reasonable for engineering approximation at hypersonic speeds. Improvements to extend the models fully into the hypersonic regime are given in the Recommendations section.

Application of fast Fourier transforms to the direct solution of a class of two-dimensional separable elliptic equations on the sphere

NASA Technical Reports Server (NTRS)

Moorthi, Shrinivas; Higgins, R. W.

1993-01-01

An efficient, direct, second-order solver for the discrete solution of a class of two-dimensional separable elliptic equations on the sphere (which generally arise in implicit and semi-implicit atmospheric models) is presented. The method involves a Fourier transformation in longitude and a direct solution of the resulting coupled second-order finite-difference equations in latitude. The solver is made efficient by vectorizing over longitudinal wave-number and by using a vectorized fast Fourier transform routine. It is evaluated using a prescribed solution method and compared with a multigrid solver and the standard direct solver from FISHPAK.

A high performance linear equation solver on the VPP500 parallel supercomputer

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

Nakanishi, Makoto; Ina, Hiroshi; Miura, Kenichi

1994-12-31

This paper describes the implementation of two high performance linear equation solvers developed for the Fujitsu VPP500, a distributed memory parallel supercomputer system. The solvers take advantage of the key architectural features of VPP500--(1) scalability for an arbitrary number of processors up to 222 processors, (2) flexible data transfer among processors provided by a crossbar interconnection network, (3) vector processing capability on each processor, and (4) overlapped computation and transfer. The general linear equation solver based on the blocked LU decomposition method achieves 120.0 GFLOPS performance with 100 processors in the LIN-PACK Highly Parallel Computing benchmark.

AFMPB: An adaptive fast multipole Poisson-Boltzmann solver for calculating electrostatics in biomolecular systems

NASA Astrophysics Data System (ADS)

Lu, Benzhuo; Cheng, Xiaolin; Huang, Jingfang; McCammon, J. Andrew

2010-06-01

A Fortran program package is introduced for rapid evaluation of the electrostatic potentials and forces in biomolecular systems modeled by the linearized Poisson-Boltzmann equation. The numerical solver utilizes a well-conditioned boundary integral equation (BIE) formulation, a node-patch discretization scheme, a Krylov subspace iterative solver package with reverse communication protocols, and an adaptive new version of fast multipole method in which the exponential expansions are used to diagonalize the multipole-to-local translations. The program and its full description, as well as several closely related libraries and utility tools are available at http://lsec.cc.ac.cn/~lubz/afmpb.html and a mirror site at http://mccammon.ucsd.edu/. This paper is a brief summary of the program: the algorithms, the implementation and the usage. Program summaryProgram title: AFMPB: Adaptive fast multipole Poisson-Boltzmann solver Catalogue identifier: AEGB_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGB_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GPL 2.0 No. of lines in distributed program, including test data, etc.: 453 649 No. of bytes in distributed program, including test data, etc.: 8 764 754 Distribution format: tar.gz Programming language: Fortran Computer: Any Operating system: Any RAM: Depends on the size of the discretized biomolecular system Classification: 3 External routines: Pre- and post-processing tools are required for generating the boundary elements and for visualization. Users can use MSMS ( http://www.scripps.edu/~sanner/html/msms_home.html) for pre-processing, and VMD ( http://www.ks.uiuc.edu/Research/vmd/) for visualization. Sub-programs included: An iterative Krylov subspace solvers package from SPARSKIT by Yousef Saad ( http://www-users.cs.umn.edu/~saad/software/SPARSKIT/sparskit.html), and the fast multipole methods subroutines from FMMSuite ( http://www.fastmultipole.org/). Nature of problem: Numerical solution of the linearized Poisson-Boltzmann equation that describes electrostatic interactions of molecular systems in ionic solutions. Solution method: A novel node-patch scheme is used to discretize the well-conditioned boundary integral equation formulation of the linearized Poisson-Boltzmann equation. Various Krylov subspace solvers can be subsequently applied to solve the resulting linear system, with a bounded number of iterations independent of the number of discretized unknowns. The matrix-vector multiplication at each iteration is accelerated by the adaptive new versions of fast multipole methods. The AFMPB solver requires other stand-alone pre-processing tools for boundary mesh generation, post-processing tools for data analysis and visualization, and can be conveniently coupled with different time stepping methods for dynamics simulation. Restrictions: Only three or six significant digits options are provided in this version. Unusual features: Most of the codes are in Fortran77 style. Memory allocation functions from Fortran90 and above are used in a few subroutines. Additional comments: The current version of the codes is designed and written for single core/processor desktop machines. Check http://lsec.cc.ac.cn/~lubz/afmpb.html and http://mccammon.ucsd.edu/ for updates and changes. Running time: The running time varies with the number of discretized elements ( N) in the system and their distributions. In most cases, it scales linearly as a function of N.

Analysis Tools for CFD Multigrid Solvers

NASA Technical Reports Server (NTRS)

Mineck, Raymond E.; Thomas, James L.; Diskin, Boris

2004-01-01

Analysis tools are needed to guide the development and evaluate the performance of multigrid solvers for the fluid flow equations. Classical analysis tools, such as local mode analysis, often fail to accurately predict performance. Two-grid analysis tools, herein referred to as Idealized Coarse Grid and Idealized Relaxation iterations, have been developed and evaluated within a pilot multigrid solver. These new tools are applicable to general systems of equations and/or discretizations and point to problem areas within an existing multigrid solver. Idealized Relaxation and Idealized Coarse Grid are applied in developing textbook-efficient multigrid solvers for incompressible stagnation flow problems.

An interactive adaptive remeshing algorithm for the two-dimensional Euler equations

NASA Technical Reports Server (NTRS)

Slack, David C.; Walters, Robert W.; Lohner, R.

1990-01-01

An interactive adaptive remeshing algorithm utilizing a frontal grid generator and a variety of time integration schemes for the two-dimensional Euler equations on unstructured meshes is presented. Several device dependent interactive graphics interfaces have been developed along with a device independent DI-3000 interface which can be employed on any computer that has the supporting software including the Cray-2 supercomputers Voyager and Navier. The time integration methods available include: an explicit four stage Runge-Kutta and a fully implicit LU decomposition. A cell-centered finite volume upwind scheme utilizing Roe's approximate Riemann solver is developed. To obtain higher order accurate results a monotone linear reconstruction procedure proposed by Barth is utilized. Results for flow over a transonic circular arc and flow through a supersonic nozzle are examined.

Scalable domain decomposition solvers for stochastic PDEs in high performance computing

DOE PAGES

Desai, Ajit; Khalil, Mohammad; Pettit, Chris; ...

2017-09-21

Stochastic spectral finite element models of practical engineering systems may involve solutions of linear systems or linearized systems for non-linear problems with billions of unknowns. For stochastic modeling, it is therefore essential to design robust, parallel and scalable algorithms that can efficiently utilize high-performance computing to tackle such large-scale systems. Domain decomposition based iterative solvers can handle such systems. And though these algorithms exhibit excellent scalabilities, significant algorithmic and implementational challenges exist to extend them to solve extreme-scale stochastic systems using emerging computing platforms. Intrusive polynomial chaos expansion based domain decomposition algorithms are extended here to concurrently handle high resolutionmore » in both spatial and stochastic domains using an in-house implementation. Sparse iterative solvers with efficient preconditioners are employed to solve the resulting global and subdomain level local systems through multi-level iterative solvers. We also use parallel sparse matrix–vector operations to reduce the floating-point operations and memory requirements. Numerical and parallel scalabilities of these algorithms are presented for the diffusion equation having spatially varying diffusion coefficient modeled by a non-Gaussian stochastic process. Scalability of the solvers with respect to the number of random variables is also investigated.« less

Scalable domain decomposition solvers for stochastic PDEs in high performance computing

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

Desai, Ajit; Khalil, Mohammad; Pettit, Chris

Stochastic spectral finite element models of practical engineering systems may involve solutions of linear systems or linearized systems for non-linear problems with billions of unknowns. For stochastic modeling, it is therefore essential to design robust, parallel and scalable algorithms that can efficiently utilize high-performance computing to tackle such large-scale systems. Domain decomposition based iterative solvers can handle such systems. And though these algorithms exhibit excellent scalabilities, significant algorithmic and implementational challenges exist to extend them to solve extreme-scale stochastic systems using emerging computing platforms. Intrusive polynomial chaos expansion based domain decomposition algorithms are extended here to concurrently handle high resolutionmore » in both spatial and stochastic domains using an in-house implementation. Sparse iterative solvers with efficient preconditioners are employed to solve the resulting global and subdomain level local systems through multi-level iterative solvers. We also use parallel sparse matrix–vector operations to reduce the floating-point operations and memory requirements. Numerical and parallel scalabilities of these algorithms are presented for the diffusion equation having spatially varying diffusion coefficient modeled by a non-Gaussian stochastic process. Scalability of the solvers with respect to the number of random variables is also investigated.« less

FoSSI: the family of simplified solver interfaces for the rapid development of parallel numerical atmosphere and ocean models

NASA Astrophysics Data System (ADS)

Frickenhaus, Stephan; Hiller, Wolfgang; Best, Meike

The portable software FoSSI is introduced that—in combination with additional free solver software packages—allows for an efficient and scalable parallel solution of large sparse linear equations systems arising in finite element model codes. FoSSI is intended to support rapid model code development, completely hiding the complexity of the underlying solver packages. In particular, the model developer need not be an expert in parallelization and is yet free to switch between different solver packages by simple modifications of the interface call. FoSSI offers an efficient and easy, yet flexible interface to several parallel solvers, most of them available on the web, such as PETSC, AZTEC, MUMPS, PILUT and HYPRE. FoSSI makes use of the concept of handles for vectors, matrices, preconditioners and solvers, that is frequently used in solver libraries. Hence, FoSSI allows for a flexible treatment of several linear equations systems and associated preconditioners at the same time, even in parallel on separate MPI-communicators. The second special feature in FoSSI is the task specifier, being a combination of keywords, each configuring a certain phase in the solver setup. This enables the user to control a solver over one unique subroutine. Furthermore, FoSSI has rather similar features for all solvers, making a fast solver intercomparison or exchange an easy task. FoSSI is a community software, proven in an adaptive 2D-atmosphere model and a 3D-primitive equation ocean model, both formulated in finite elements. The present paper discusses perspectives of an OpenMP-implementation of parallel iterative solvers based on domain decomposition methods. This approach to OpenMP solvers is rather attractive, as the code for domain-local operations of factorization, preconditioning and matrix-vector product can be readily taken from a sequential implementation that is also suitable to be used in an MPI-variant. Code development in this direction is in an advanced state under the name ScOPES: the Scalable Open Parallel sparse linear Equations Solver.

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

Bolding, Simon R.; Cleveland, Mathew Allen; Morel, Jim E.

In this paper, we have implemented a new high-order low-order (HOLO) algorithm for solving thermal radiative transfer problems. The low-order (LO) system is based on the spatial and angular moments of the transport equation and a linear-discontinuous finite-element spatial representation, producing equations similar to the standard S 2 equations. The LO solver is fully implicit in time and efficiently resolves the nonlinear temperature dependence at each time step. The high-order (HO) solver utilizes exponentially convergent Monte Carlo (ECMC) to give a globally accurate solution for the angular intensity to a fixed-source pure-absorber transport problem. This global solution is used tomore » compute consistency terms, which require the HO and LO solutions to converge toward the same solution. The use of ECMC allows for the efficient reduction of statistical noise in the Monte Carlo solution, reducing inaccuracies introduced through the LO consistency terms. Finally, we compare results with an implicit Monte Carlo code for one-dimensional gray test problems and demonstrate the efficiency of ECMC over standard Monte Carlo in this HOLO algorithm.« less

A High-Order Low-Order Algorithm with Exponentially Convergent Monte Carlo for Thermal Radiative Transfer

DOE PAGES

Bolding, Simon R.; Cleveland, Mathew Allen; Morel, Jim E.

2016-10-21

In this paper, we have implemented a new high-order low-order (HOLO) algorithm for solving thermal radiative transfer problems. The low-order (LO) system is based on the spatial and angular moments of the transport equation and a linear-discontinuous finite-element spatial representation, producing equations similar to the standard S 2 equations. The LO solver is fully implicit in time and efficiently resolves the nonlinear temperature dependence at each time step. The high-order (HO) solver utilizes exponentially convergent Monte Carlo (ECMC) to give a globally accurate solution for the angular intensity to a fixed-source pure-absorber transport problem. This global solution is used tomore » compute consistency terms, which require the HO and LO solutions to converge toward the same solution. The use of ECMC allows for the efficient reduction of statistical noise in the Monte Carlo solution, reducing inaccuracies introduced through the LO consistency terms. Finally, we compare results with an implicit Monte Carlo code for one-dimensional gray test problems and demonstrate the efficiency of ECMC over standard Monte Carlo in this HOLO algorithm.« less

The DANTE Boltzmann transport solver: An unstructured mesh, 3-D, spherical harmonics algorithm compatible with parallel computer architectures

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

McGhee, J.M.; Roberts, R.M.; Morel, J.E.

1997-06-01

A spherical harmonics research code (DANTE) has been developed which is compatible with parallel computer architectures. DANTE provides 3-D, multi-material, deterministic, transport capabilities using an arbitrary finite element mesh. The linearized Boltzmann transport equation is solved in a second order self-adjoint form utilizing a Galerkin finite element spatial differencing scheme. The core solver utilizes a preconditioned conjugate gradient algorithm. Other distinguishing features of the code include options for discrete-ordinates and simplified spherical harmonics angular differencing, an exact Marshak boundary treatment for arbitrarily oriented boundary faces, in-line matrix construction techniques to minimize memory consumption, and an effective diffusion based preconditioner formore » scattering dominated problems. Algorithm efficiency is demonstrated for a massively parallel SIMD architecture (CM-5), and compatibility with MPP multiprocessor platforms or workstation clusters is anticipated.« less

General purpose nonlinear system solver based on Newton-Krylov method.

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

2013-12-01

KINSOL is part of a software family called SUNDIALS: SUite of Nonlinear and Differential/Algebraic equation Solvers [1]. KINSOL is a general-purpose nonlinear system solver based on Newton-Krylov and fixed-point solver technologies [2].

A Lagrangian meshfree method applied to linear and nonlinear elasticity.

PubMed

Walker, Wade A

2017-01-01

The repeated replacement method (RRM) is a Lagrangian meshfree method which we have previously applied to the Euler equations for compressible fluid flow. In this paper we present new enhancements to RRM, and we apply the enhanced method to both linear and nonlinear elasticity. We compare the results of ten test problems to those of analytic solvers, to demonstrate that RRM can successfully simulate these elastic systems without many of the requirements of traditional numerical methods such as numerical derivatives, equation system solvers, or Riemann solvers. We also show the relationship between error and computational effort for RRM on these systems, and compare RRM to other methods to highlight its strengths and weaknesses. And to further explain the two elastic equations used in the paper, we demonstrate the mathematical procedure used to create Riemann and Sedov-Taylor solvers for them, and detail the numerical techniques needed to embody those solvers in code.

A Lagrangian meshfree method applied to linear and nonlinear elasticity

PubMed Central

2017-01-01

The repeated replacement method (RRM) is a Lagrangian meshfree method which we have previously applied to the Euler equations for compressible fluid flow. In this paper we present new enhancements to RRM, and we apply the enhanced method to both linear and nonlinear elasticity. We compare the results of ten test problems to those of analytic solvers, to demonstrate that RRM can successfully simulate these elastic systems without many of the requirements of traditional numerical methods such as numerical derivatives, equation system solvers, or Riemann solvers. We also show the relationship between error and computational effort for RRM on these systems, and compare RRM to other methods to highlight its strengths and weaknesses. And to further explain the two elastic equations used in the paper, we demonstrate the mathematical procedure used to create Riemann and Sedov-Taylor solvers for them, and detail the numerical techniques needed to embody those solvers in code. PMID:29045443

Final Report: Subcontract B623868 Algebraic Multigrid solvers for coupled PDE systems

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

Brannick, J.

The Pennsylvania State University (“Subcontractor”) continued to work on the design of algebraic multigrid solvers for coupled systems of partial differential equations (PDEs) arising in numerical modeling of various applications, with a main focus on solving the Dirac equation arising in Quantum Chromodynamics (QCD). The goal of the proposed work was to develop combined geometric and algebraic multilevel solvers that are robust and lend themselves to efficient implementation on massively parallel heterogeneous computers for these QCD systems. The research in these areas built on previous works, focusing on the following three topics: (1) the development of parallel full-multigrid (PFMG) andmore » non-Galerkin coarsening techniques in this frame work for solving the Wilson Dirac system; (2) the use of these same Wilson MG solvers for preconditioning the Overlap and Domain Wall formulations of the Dirac equation; and (3) the design and analysis of algebraic coarsening algorithms for coupled PDE systems including Stokes equation, Maxwell equation and linear elasticity.« less

Flutter and Forced Response Analyses of Cascades using a Two-Dimensional Linearized Euler Solver

NASA Technical Reports Server (NTRS)

Reddy, T. S. R.; Srivastava, R.; Mehmed, O.

1999-01-01

Flutter and forced response analyses for a cascade of blades in subsonic and transonic flow is presented. The structural model for each blade is a typical section with bending and torsion degrees of freedom. The unsteady aerodynamic forces due to bending and torsion motions. and due to a vortical gust disturbance are obtained by solving unsteady linearized Euler equations. The unsteady linearized equations are obtained by linearizing the unsteady nonlinear equations about the steady flow. The predicted unsteady aerodynamic forces include the effect of steady aerodynamic loading due to airfoil shape, thickness and angle of attack. The aeroelastic equations are solved in the frequency domain by coupling the un- steady aerodynamic forces to the aeroelastic solver MISER. The present unsteady aerodynamic solver showed good correlation with published results for both flutter and forced response predictions. Further improvements are required to use the unsteady aerodynamic solver in a design cycle.

A Matlab-based finite-difference solver for the Poisson problem with mixed Dirichlet-Neumann boundary conditions

NASA Astrophysics Data System (ADS)

Reimer, Ashton S.; Cheviakov, Alexei F.

2013-03-01

A Matlab-based finite-difference numerical solver for the Poisson equation for a rectangle and a disk in two dimensions, and a spherical domain in three dimensions, is presented. The solver is optimized for handling an arbitrary combination of Dirichlet and Neumann boundary conditions, and allows for full user control of mesh refinement. The solver routines utilize effective and parallelized sparse vector and matrix operations. Computations exhibit high speeds, numerical stability with respect to mesh size and mesh refinement, and acceptable error values even on desktop computers. Catalogue identifier: AENQ_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENQ_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License v3.0 No. of lines in distributed program, including test data, etc.: 102793 No. of bytes in distributed program, including test data, etc.: 369378 Distribution format: tar.gz Programming language: Matlab 2010a. Computer: PC, Macintosh. Operating system: Windows, OSX, Linux. RAM: 8 GB (8, 589, 934, 592 bytes) Classification: 4.3. Nature of problem: To solve the Poisson problem in a standard domain with “patchy surface”-type (strongly heterogeneous) Neumann/Dirichlet boundary conditions. Solution method: Finite difference with mesh refinement. Restrictions: Spherical domain in 3D; rectangular domain or a disk in 2D. Unusual features: Choice between mldivide/iterative solver for the solution of large system of linear algebraic equations that arise. Full user control of Neumann/Dirichlet boundary conditions and mesh refinement. Running time: Depending on the number of points taken and the geometry of the domain, the routine may take from less than a second to several hours to execute.

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.

Formal Solutions for Polarized Radiative Transfer. III. Stiffness and Instability

NASA Astrophysics Data System (ADS)

Janett, Gioele; Paganini, Alberto

2018-04-01

Efficient numerical approximation of the polarized radiative transfer equation is challenging because this system of ordinary differential equations exhibits stiff behavior, which potentially results in numerical instability. This negatively impacts the accuracy of formal solvers, and small step-sizes are often necessary to retrieve physical solutions. This work presents stability analyses of formal solvers for the radiative transfer equation of polarized light, identifies instability issues, and suggests practical remedies. In particular, the assumptions and the limitations of the stability analysis of Runge–Kutta methods play a crucial role. On this basis, a suitable and pragmatic formal solver is outlined and tested. An insightful comparison to the scalar radiative transfer equation is also presented.

Optical solver for a system of ordinary differential equations based on an external feedback assisted microring resonator.

PubMed

Hou, Jie; Dong, Jianji; Zhang, Xinliang

2017-06-15

Systems of ordinary differential equations (SODEs) are crucial for describing the dynamic behaviors in various systems such as modern control systems which require observability and controllability. In this Letter, we propose and experimentally demonstrate an all-optical SODE solver based on the silicon-on-insulator platform. We use an add/drop microring resonator to construct two different ordinary differential equations (ODEs) and then introduce two external feedback waveguides to realize the coupling between these ODEs, thus forming the SODE solver. A temporal coupled mode theory is used to deduce the expression of the SODE. A system experiment is carried out for further demonstration. For the input 10 GHz NRZ-like pulses, the measured output waveforms of the SODE solver agree well with the calculated results.

A Depth-Averaged 2-D Simulation for Coastal Barrier Breaching Processes

DTIC Science & Technology

2011-05-01

including bed change and variable flow density in the flow continuity and momentum equations. The model adopts the HLL approximate Riemann solver to handle...flow density in the flow continuity and momentum equations. The model adopts the HLL approximate Riemann solver to handle the mixed-regime flows near...18 547 Keulegan equation or the Bernoulli equation, and the breach morphological change is determined using simplified sediment transport models

Preconditioned conjugate-gradient methods for low-speed flow calculations

NASA Technical Reports Server (NTRS)

Ajmani, Kumud; Ng, Wing-Fai; Liou, Meng-Sing

1993-01-01

An investigation is conducted into the viability of using a generalized Conjugate Gradient-like method as an iterative solver to obtain steady-state solutions of very low-speed fluid flow problems. Low-speed flow at Mach 0.1 over a backward-facing step is chosen as a representative test problem. The unsteady form of the two dimensional, compressible Navier-Stokes equations is integrated in time using discrete time-steps. The Navier-Stokes equations are cast in an implicit, upwind finite-volume, flux split formulation. The new iterative solver is used to solve a linear system of equations at each step of the time-integration. Preconditioning techniques are used with the new solver to enhance the stability and convergence rate of the solver and are found to be critical to the overall success of the solver. A study of various preconditioners reveals that a preconditioner based on the Lower-Upper Successive Symmetric Over-Relaxation iterative scheme is more efficient than a preconditioner based on Incomplete L-U factorizations of the iteration matrix. The performance of the new preconditioned solver is compared with a conventional Line Gauss-Seidel Relaxation (LGSR) solver. Overall speed-up factors of 28 (in terms of global time-steps required to converge to a steady-state solution) and 20 (in terms of total CPU time on one processor of a CRAY-YMP) are found in favor of the new preconditioned solver, when compared with the LGSR solver.

Preconditioned Conjugate Gradient methods for low speed flow calculations

NASA Technical Reports Server (NTRS)

Ajmani, Kumud; Ng, Wing-Fai; Liou, Meng-Sing

1993-01-01

An investigation is conducted into the viability of using a generalized Conjugate Gradient-like method as an iterative solver to obtain steady-state solutions of very low-speed fluid flow problems. Low-speed flow at Mach 0.1 over a backward-facing step is chosen as a representative test problem. The unsteady form of the two dimensional, compressible Navier-Stokes equations are integrated in time using discrete time-steps. The Navier-Stokes equations are cast in an implicit, upwind finite-volume, flux split formulation. The new iterative solver is used to solve a linear system of equations at each step of the time-integration. Preconditioning techniques are used with the new solver to enhance the stability and the convergence rate of the solver and are found to be critical to the overall success of the solver. A study of various preconditioners reveals that a preconditioner based on the lower-upper (L-U)-successive symmetric over-relaxation iterative scheme is more efficient than a preconditioner based on incomplete L-U factorizations of the iteration matrix. The performance of the new preconditioned solver is compared with a conventional line Gauss-Seidel relaxation (LGSR) solver. Overall speed-up factors of 28 (in terms of global time-steps required to converge to a steady-state solution) and 20 (in terms of total CPU time on one processor of a CRAY-YMP) are found in favor of the new preconditioned solver, when compared with the LGSR solver.

Acceleration of Linear Finite-Difference Poisson-Boltzmann Methods on Graphics Processing Units.

PubMed

Qi, Ruxi; Botello-Smith, Wesley M; Luo, Ray

2017-07-11

Electrostatic interactions play crucial roles in biophysical processes such as protein folding and molecular recognition. Poisson-Boltzmann equation (PBE)-based models have emerged as widely used in modeling these important processes. Though great efforts have been put into developing efficient PBE numerical models, challenges still remain due to the high dimensionality of typical biomolecular systems. In this study, we implemented and analyzed commonly used linear PBE solvers for the ever-improving graphics processing units (GPU) for biomolecular simulations, including both standard and preconditioned conjugate gradient (CG) solvers with several alternative preconditioners. Our implementation utilizes the standard Nvidia CUDA libraries cuSPARSE, cuBLAS, and CUSP. Extensive tests show that good numerical accuracy can be achieved given that the single precision is often used for numerical applications on GPU platforms. The optimal GPU performance was observed with the Jacobi-preconditioned CG solver, with a significant speedup over standard CG solver on CPU in our diversified test cases. Our analysis further shows that different matrix storage formats also considerably affect the efficiency of different linear PBE solvers on GPU, with the diagonal format best suited for our standard finite-difference linear systems. Further efficiency may be possible with matrix-free operations and integrated grid stencil setup specifically tailored for the banded matrices in PBE-specific linear systems.

Preconditioned implicit solvers for the Navier-Stokes equations on distributed-memory machines

NASA Technical Reports Server (NTRS)

Ajmani, Kumud; Liou, Meng-Sing; Dyson, Rodger W.

1994-01-01

The GMRES method is parallelized, and combined with local preconditioning to construct an implicit parallel solver to obtain steady-state solutions for the Navier-Stokes equations of fluid flow on distributed-memory machines. The new implicit parallel solver is designed to preserve the convergence rate of the equivalent 'serial' solver. A static domain-decomposition is used to partition the computational domain amongst the available processing nodes of the parallel machine. The SPMD (Single-Program Multiple-Data) programming model is combined with message-passing tools to develop the parallel code on a 32-node Intel Hypercube and a 512-node Intel Delta machine. The implicit parallel solver is validated for internal and external flow problems, and is found to compare identically with flow solutions obtained on a Cray Y-MP/8. A peak computational speed of 2300 MFlops/sec has been achieved on 512 nodes of the Intel Delta machine,k for a problem size of 1024 K equations (256 K grid points).

Equation solvers for distributed-memory computers

NASA Technical Reports Server (NTRS)

Storaasli, Olaf O.

1994-01-01

A large number of scientific and engineering problems require the rapid solution of large systems of simultaneous equations. The performance of parallel computers in this area now dwarfs traditional vector computers by nearly an order of magnitude. This talk describes the major issues involved in parallel equation solvers with particular emphasis on the Intel Paragon, IBM SP-1 and SP-2 processors.

A semi-Lagrangian approach to the shallow water equation

NASA Technical Reports Server (NTRS)

Bates, J. R.; Mccormick, Stephen F.; Ruge, John; Sholl, David S.; Yavneh, Irad

1993-01-01

We present a formulation of the shallow water equations that emphasizes the conservation of potential vorticity. A locally conservative semi-Lagrangian time-stepping scheme is developed, which leads to a system of three coupled PDE's to be solved at each time level. We describe a smoothing analysis of these equations, on which an effective multigrid solver is constructed. Some results from applying this solver to the static version of these equations are presented.

Factorizable Schemes for the Equations of Fluid Flow

NASA Technical Reports Server (NTRS)

Sidilkover, David

1999-01-01

We present an upwind high-resolution factorizable (UHF) discrete scheme for the compressible Euler equations that allows to distinguish between full-potential and advection factors at the discrete level. The scheme approximates equations in their general conservative form and is related to the family of genuinely multidimensional upwind schemes developed previously and demonstrated to have good shock-capturing capabilities. A unique property of this scheme is that in addition to the aforementioned features it is also factorizable, i.e., it allows to distinguish between full-potential and advection factors at the discrete level. The latter property facilitates the construction of optimally efficient multigrid solvers. This is done through a relaxation procedure that utilizes the factorizability property.

A new mathematical adjoint for the modified SAAF -SN equations

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

Schunert, Sebastian; Wang, Yaqi; Martineau, Richard

2015-01-01

We present a new adjoint FEM weak form, which can be directly used for evaluating the mathematical adjoint, suitable for perturbation calculations, of the self-adjoint angular flux SN equations (SAAF -SN) without construction and transposition of the underlying coefficient matrix. Stabilization schemes incorporated in the described SAAF -SN method make the mathematical adjoint distinct from the physical adjoint, i.e. the solution of the continuous adjoint equation with SAAF -SN . This weak form is implemented into RattleSnake, the MOOSE (Multiphysics Object-Oriented Simulation Environment) based transport solver. Numerical results verify the correctness of the implementation and show its utility both formore » fixed source and eigenvalue problems.« less

An Equation of State for Polymethylpentene (TPX) including Multi-Shock Response

NASA Astrophysics Data System (ADS)

Aslam, Tariq; Gustavsen, Richard; Sanchez, Nathaniel; Bartram, Brian

2011-06-01

The equation of state (EOS) of polymethylpentene (TPX) is examined through both single shock Hugoniot data as well as more recent multi-shock compression and release experiments. Results from the recent multi-shock experiments on LANL's 2-stage gas gun will be presented. A simple conservative Lagrangian numerical scheme utilizing total-variation-diminishing interpolation and an approximate Riemann solver will be presented as well as the methodology of calibration. It is shown that a simple Mie-Gruneisen EOS based off a Keane fitting form for the isentrope can replicate both the single shock and multi-shock experiments.

An equation of state for polymethylpentene (TPX) including multi-shock response

NASA Astrophysics Data System (ADS)

Aslam, Tariq D.; Gustavsen, Rick; Sanchez, Nathaniel; Bartram, Brian D.

2012-03-01

The equation of state (EOS) of polymethylpentene (TPX) is examined through both single shock Hugoniot data as well as more recent multi-shock compression and release experiments. Results from the recent multi-shock experiments on LANL's two-stage gas gun will be presented. A simple conservative Lagrangian numerical scheme utilizing total variation diminishing interpolation and an approximate Riemann solver will be presented as well as the methodology of calibration. It is shown that a simple Mie-Grüneisen EOS based on a Keane fitting form for the isentrope can replicate both the single shock and multi-shock experiments.

A particle-particle hybrid method for kinetic and continuum equations

NASA Astrophysics Data System (ADS)

Tiwari, Sudarshan; Klar, Axel; Hardt, Steffen

2009-10-01

We present a coupling procedure for two different types of particle methods for the Boltzmann and the Navier-Stokes equations. A variant of the DSMC method is applied to simulate the Boltzmann equation, whereas a meshfree Lagrangian particle method, similar to the SPH method, is used for simulations of the Navier-Stokes equations. An automatic domain decomposition approach is used with the help of a continuum breakdown criterion. We apply adaptive spatial and time meshes. The classical Sod's 1D shock tube problem is solved for a large range of Knudsen numbers. Results from Boltzmann, Navier-Stokes and hybrid solvers are compared. The CPU time for the hybrid solver is 3-4 times faster than for the Boltzmann solver.

Numerical Simulations For the F-16XL Aircraft Configuration

NASA Technical Reports Server (NTRS)

Elmiligui, Alaa A.; Abdol-Hamid, Khaled; Cavallo, Peter A.; Parlette, Edward B.

2014-01-01

Numerical simulations of flow around the F-16XL are presented as a contribution to the Cranked Arrow Wing Aerodynamic Project International II (CAWAPI-II). The NASA Tetrahedral Unstructured Software System (TetrUSS) is used to perform numerical simulations. This CFD suite, developed and maintained by NASA Langley Research Center, includes an unstructured grid generation program called VGRID, a postprocessor named POSTGRID, and the flow solver USM3D. The CRISP CFD package is utilized to provide error estimates and grid adaption for verification of USM3D results. A subsonic high angle-of-attack case flight condition (FC) 25 is computed and analyzed. Three turbulence models are used in the calculations: the one-equation Spalart-Allmaras (SA), the two-equation shear stress transport (SST) and the ke turbulence models. Computational results, and surface static pressure profiles are presented and compared with flight data. Solution verification is performed using formal grid refinement studies, the solution of Error Transport Equations, and adaptive mesh refinement. The current study shows that the USM3D solver coupled with CRISP CFD can be used in an engineering environment in predicting vortex-flow physics on a complex configuration at flight Reynolds numbers.

Methods for Solving Gas Damping Problems in Perforated Microstructures Using a 2D Finite-Element Solver

PubMed Central

Veijola, Timo; Råback, Peter

2007-01-01

We present a straightforward method to solve gas damping problems for perforated structures in two dimensions (2D) utilising a Perforation Profile Reynolds (PPR) solver. The PPR equation is an extended Reynolds equation that includes additional terms modelling the leakage flow through the perforations, and variable diffusivity and compressibility profiles. The solution method consists of two phases: 1) determination of the specific admittance profile and relative diffusivity (and relative compressibility) profiles due to the perforation, and 2) solution of the PPR equation with a FEM solver in 2D. Rarefied gas corrections in the slip-flow region are also included. Analytic profiles for circular and square holes with slip conditions are presented in the paper. To verify the method, square perforated dampers with 16–64 holes were simulated with a three-dimensional (3D) Navier-Stokes solver, a homogenised extended Reynolds solver, and a 2D PPR solver. Cases for both translational (in normal to the surfaces) and torsional motion were simulated. The presented method extends the region of accurate simulation of perforated structures to cases where the homogenisation method is inaccurate and the full 3D Navier-Stokes simulation is too time-consuming.

Application of NASA General-Purpose Solver to Large-Scale Computations in Aeroacoustics

NASA Technical Reports Server (NTRS)

Watson, Willie R.; Storaasli, Olaf O.

2004-01-01

Of several iterative and direct equation solvers evaluated previously for computations in aeroacoustics, the most promising was the NASA-developed General-Purpose Solver (winner of NASA's 1999 software of the year award). This paper presents detailed, single-processor statistics of the performance of this solver, which has been tailored and optimized for large-scale aeroacoustic computations. The statistics, compiled using an SGI ORIGIN 2000 computer with 12 Gb available memory (RAM) and eight available processors, are the central processing unit time, RAM requirements, and solution error. The equation solver is capable of solving 10 thousand complex unknowns in as little as 0.01 sec using 0.02 Gb RAM, and 8.4 million complex unknowns in slightly less than 3 hours using all 12 Gb. This latter solution is the largest aeroacoustics problem solved to date with this technique. The study was unable to detect any noticeable error in the solution, since noise levels predicted from these solution vectors are in excellent agreement with the noise levels computed from the exact solution. The equation solver provides a means for obtaining numerical solutions to aeroacoustics problems in three dimensions.

Evolutionary algorithm based optimization of hydraulic machines utilizing a state-of-the-art block coupled CFD solver and parametric geometry and mesh generation tools

NASA Astrophysics Data System (ADS)

S, Kyriacou; E, Kontoleontos; S, Weissenberger; L, Mangani; E, Casartelli; I, Skouteropoulou; M, Gattringer; A, Gehrer; M, Buchmayr

2014-03-01

An efficient hydraulic optimization procedure, suitable for industrial use, requires an advanced optimization tool (EASY software), a fast solver (block coupled CFD) and a flexible geometry generation tool. EASY optimization software is a PCA-driven metamodel-assisted Evolutionary Algorithm (MAEA (PCA)) that can be used in both single- (SOO) and multiobjective optimization (MOO) problems. In MAEAs, low cost surrogate evaluation models are used to screen out non-promising individuals during the evolution and exclude them from the expensive, problem specific evaluation, here the solution of Navier-Stokes equations. For additional reduction of the optimization CPU cost, the PCA technique is used to identify dependences among the design variables and to exploit them in order to efficiently drive the application of the evolution operators. To further enhance the hydraulic optimization procedure, a very robust and fast Navier-Stokes solver has been developed. This incompressible CFD solver employs a pressure-based block-coupled approach, solving the governing equations simultaneously. This method, apart from being robust and fast, also provides a big gain in terms of computational cost. In order to optimize the geometry of hydraulic machines, an automatic geometry and mesh generation tool is necessary. The geometry generation tool used in this work is entirely based on b-spline curves and surfaces. In what follows, the components of the tool chain are outlined in some detail and the optimization results of hydraulic machine components are shown in order to demonstrate the performance of the presented optimization procedure.

Steady potential solver for unsteady aerodynamic analyses

NASA Technical Reports Server (NTRS)

Hoyniak, Dan

1994-01-01

Development of a steady flow solver for use with LINFLO was the objective of this report. The solver must be compatible with LINFLO, be composed of composite mesh, and have transonic capability. The approaches used were: (1) steady flow potential equations written in nonconservative form; (2) Newton's Method; (3) implicit, least-squares, interpolation method to obtain finite difference equations; and (4) matrix inversion routines from LINFLO. This report was given during the NASA LeRC Workshop on Forced Response in Turbomachinery in August of 1993.

Parallel-vector out-of-core equation solver for computational mechanics

NASA Technical Reports Server (NTRS)

Qin, J.; Agarwal, T. K.; Storaasli, O. O.; Nguyen, D. T.; Baddourah, M. A.

1993-01-01

A parallel/vector out-of-core equation solver is developed for shared-memory computers, such as the Cray Y-MP machine. The input/ output (I/O) time is reduced by using the a synchronous BUFFER IN and BUFFER OUT, which can be executed simultaneously with the CPU instructions. The parallel and vector capability provided by the supercomputers is also exploited to enhance the performance. Numerical applications in large-scale structural analysis are given to demonstrate the efficiency of the present out-of-core solver.

A fast Poisson solver for unsteady incompressible Navier-Stokes equations on the half-staggered grid

NASA Technical Reports Server (NTRS)

Golub, G. H.; Huang, L. C.; Simon, H.; Tang, W. -P.

1995-01-01

In this paper, a fast Poisson solver for unsteady, incompressible Navier-Stokes equations with finite difference methods on the non-uniform, half-staggered grid is presented. To achieve this, new algorithms for diagonalizing a semi-definite pair are developed. Our fast solver can also be extended to the three dimensional case. The motivation and related issues in using this second kind of staggered grid are also discussed. Numerical testing has indicated the effectiveness of this algorithm.

A general multiblock Euler code for propulsion integration. Volume 1: Theory document

NASA Technical Reports Server (NTRS)

Chen, H. C.; Su, T. Y.; Kao, T. J.

1991-01-01

A general multiblock Euler solver was developed for the analysis of flow fields over geometrically complex configurations either in free air or in a wind tunnel. In this approach, the external space around a complex configuration was divided into a number of topologically simple blocks, so that surface-fitted grids and an efficient flow solution algorithm could be easily applied in each block. The computational grid in each block is generated using a combination of algebraic and elliptic methods. A grid generation/flow solver interface program was developed to facilitate the establishment of block-to-block relations and the boundary conditions for each block. The flow solver utilizes a finite volume formulation and an explicit time stepping scheme to solve the Euler equations. A multiblock version of the multigrid method was developed to accelerate the convergence of the calculations. The generality of the method was demonstrated through the analysis of two complex configurations at various flow conditions. Results were compared to available test data. Two accompanying volumes, user manuals for the preparation of multi-block grids (vol. 2) and for the Euler flow solver (vol. 3), provide information on input data format and program execution.

An HLLC Riemann solver for resistive relativistic magnetohydrodynamics

NASA Astrophysics Data System (ADS)

Miranda-Aranguren, S.; Aloy, M. A.; Rembiasz, T.

2018-05-01

We present a new approximate Riemann solver for the augmented system of equations of resistive relativistic magnetohydrodynamics that belongs to the family of Harten-Lax-van Leer contact wave (HLLC) solvers. In HLLC solvers, the solution is approximated by two constant states flanked by two shocks separated by a contact wave. The accuracy of the new approximate solver is calibrated through 1D and 2D test problems.

A vectorized Poisson solver over a spherical shell and its application to the quasi-geostrophic omega-equation

NASA Technical Reports Server (NTRS)

Mullenmeister, Paul

1988-01-01

The quasi-geostrophic omega-equation in flux form is developed as an example of a Poisson problem over a spherical shell. Solutions of this equation are obtained by applying a two-parameter Chebyshev solver in vector layout for CDC 200 series computers. The performance of this vectorized algorithm greatly exceeds the performance of its scalar analog. The algorithm generates solutions of the omega-equation which are compared with the omega fields calculated with the aid of the mass continuity equation.

Solving regularly and singularly perturbed reaction-diffusion equations in three space dimensions

NASA Astrophysics Data System (ADS)

Moore, Peter K.

2007-06-01

In [P.K. Moore, Effects of basis selection and h-refinement on error estimator reliability and solution efficiency for higher-order methods in three space dimensions, Int. J. Numer. Anal. Mod. 3 (2006) 21-51] a fixed, high-order h-refinement finite element algorithm, Href, was introduced for solving reaction-diffusion equations in three space dimensions. In this paper Href is coupled with continuation creating an automatic method for solving regularly and singularly perturbed reaction-diffusion equations. The simple quasilinear Newton solver of Moore, (2006) is replaced by the nonlinear solver NITSOL [M. Pernice, H.F. Walker, NITSOL: a Newton iterative solver for nonlinear systems, SIAM J. Sci. Comput. 19 (1998) 302-318]. Good initial guesses for the nonlinear solver are obtained using continuation in the small parameter ɛ. Two strategies allow adaptive selection of ɛ. The first depends on the rate of convergence of the nonlinear solver and the second implements backtracking in ɛ. Finally a simple method is used to select the initial ɛ. Several examples illustrate the effectiveness of the algorithm.

New preconditioning strategy for Jacobian-free solvers for variably saturated flows with Richards’ equation

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

Lipnikov, Konstantin; Moulton, David; Svyatskiy, Daniil

2016-04-29

We develop a new approach for solving the nonlinear Richards’ equation arising in variably saturated flow modeling. The growing complexity of geometric models for simulation of subsurface flows leads to the necessity of using unstructured meshes and advanced discretization methods. Typically, a numerical solution is obtained by first discretizing PDEs and then solving the resulting system of nonlinear discrete equations with a Newton-Raphson-type method. Efficiency and robustness of the existing solvers rely on many factors, including an empiric quality control of intermediate iterates, complexity of the employed discretization method and a customized preconditioner. We propose and analyze a new preconditioningmore » strategy that is based on a stable discretization of the continuum Jacobian. We will show with numerical experiments for challenging problems in subsurface hydrology that this new preconditioner improves convergence of the existing Jacobian-free solvers 3-20 times. Furthermore, we show that the Picard method with this preconditioner becomes a more efficient nonlinear solver than a few widely used Jacobian-free solvers.« less

A 1D-2D Shallow Water Equations solver for discontinuous porosity field based on a Generalized Riemann Problem

NASA Astrophysics Data System (ADS)

Ferrari, Alessia; Vacondio, Renato; Dazzi, Susanna; Mignosa, Paolo

2017-09-01

A novel augmented Riemann Solver capable of handling porosity discontinuities in 1D and 2D Shallow Water Equation (SWE) models is presented. With the aim of accurately approximating the porosity source term, a Generalized Riemann Problem is derived by adding an additional fictitious equation to the SWEs system and imposing mass and momentum conservation across the porosity discontinuity. The modified Shallow Water Equations are theoretically investigated, and the implementation of an augmented Roe Solver in a 1D Godunov-type finite volume scheme is presented. Robust treatment of transonic flows is ensured by introducing an entropy fix based on the wave pattern of the Generalized Riemann Problem. An Exact Riemann Solver is also derived in order to validate the numerical model. As an extension of the 1D scheme, an analogous 2D numerical model is also derived and validated through test cases with radial symmetry. The capability of the 1D and 2D numerical models to capture different wave patterns is assessed against several Riemann Problems with different wave patterns.

A Tensor-Train accelerated solver for integral equations in complex geometries

NASA Astrophysics Data System (ADS)

Corona, Eduardo; Rahimian, Abtin; Zorin, Denis

2017-04-01

We present a framework using the Quantized Tensor Train (QTT) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the QTT decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations. For a broad range of problems, computational and storage costs of the inversion scheme are extremely modest O (log ⁡ N) and once the inverse is computed, it can be applied in O (Nlog ⁡ N) . We analyze the QTT ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as FMM and HSS. We prove that the QTT ranks are bounded for translation-invariant systems and argue that this behavior extends to non-translation invariant volume and boundary integrals. For volume integrals, the QTT decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers. We present results demonstrating the remarkable performance of the QTT-based solver when applied to both translation and non-translation invariant volume integrals in 3D. For boundary integral equations, we demonstrate that using a QTT decomposition to construct preconditioners for a Krylov subspace method leads to an efficient and robust solver with a small memory footprint. We test the QTT preconditioners in the iterative solution of an exterior elliptic boundary value problem (Laplace) formulated as a boundary integral equation in complex, multiply connected geometries.

An equation of state for polyurea aerogel based on multi-shock response

NASA Astrophysics Data System (ADS)

Aslam, T. D.; Gustavsen, R. L.; Bartram, B. D.

2014-05-01

The equation of state (EOS) of polyurea aerogel (PUA) is examined through both single shock Hugoniot data as well as more recent multi-shock compression experiments performed on the LANL 2-stage gas gun. A simple conservative Lagrangian numerical scheme, utilizing total variation diminishing (TVD) interpolation and an approximate Riemann solver, will be presented as well as the methodology of calibration. It will been demonstrated that a p-a model based on a Mie-Gruneisen fitting form for the solid material can reasonably replicate multi-shock compression response at a variety of initial densities; such a methodology will be presented for a commercially available polyurea aerogel.

Exact Riemann solutions of the Ripa model for flat and non-flat bottom topographies

NASA Astrophysics Data System (ADS)

Rehman, Asad; Ali, Ishtiaq; Qamar, Shamsul

2018-03-01

This article is concerned with the derivation of exact Riemann solutions for Ripa model considering flat and non-flat bottom topographies. The Ripa model is a system of shallow water equations accounting for horizontal temperature gradients. In the case of non-flat bottom topography, the mass, momentum and energy conservation principles are utilized to relate the left and right states across the step-type bottom topography. The resulting system of algebraic equations is solved iteratively. Different numerical case studies of physical interest are considered. The solutions obtained from developed exact Riemann solvers are compared with the approximate solutions of central upwind scheme.

General Equation Set Solver for Compressible and Incompressible Turbomachinery Flows

NASA Technical Reports Server (NTRS)

Sondak, Douglas L.; Dorney, Daniel J.

2002-01-01

Turbomachines for propulsion applications operate with many different working fluids and flow conditions. The flow may be incompressible, such as in the liquid hydrogen pump in a rocket engine, or supersonic, such as in the turbine which may drive the hydrogen pump. Separate codes have traditionally been used for incompressible and compressible flow solvers. The General Equation Set (GES) method can be used to solve both incompressible and compressible flows, and it is not restricted to perfect gases, as are many compressible-flow turbomachinery solvers. An unsteady GES turbomachinery flow solver has been developed and applied to both air and water flows through turbines. It has been shown to be an excellent alternative to maintaining two separate codes.

Albany/FELIX: A parallel, scalable and robust, finite element, first-order Stokes approximation ice sheet solver built for advanced analysis

DOE PAGES

Tezaur, I. K.; Perego, M.; Salinger, A. G.; ...

2015-04-27

This paper describes a new parallel, scalable and robust finite element based solver for the first-order Stokes momentum balance equations for ice flow. The solver, known as Albany/FELIX, is constructed using the component-based approach to building application codes, in which mature, modular libraries developed as a part of the Trilinos project are combined using abstract interfaces and template-based generic programming, resulting in a final code with access to dozens of algorithmic and advanced analysis capabilities. Following an overview of the relevant partial differential equations and boundary conditions, the numerical methods chosen to discretize the ice flow equations are described, alongmore » with their implementation. The results of several verification studies of the model accuracy are presented using (1) new test cases for simplified two-dimensional (2-D) versions of the governing equations derived using the method of manufactured solutions, and (2) canonical ice sheet modeling benchmarks. Model accuracy and convergence with respect to mesh resolution are then studied on problems involving a realistic Greenland ice sheet geometry discretized using hexahedral and tetrahedral meshes. Also explored as a part of this study is the effect of vertical mesh resolution on the solution accuracy and solver performance. The robustness and scalability of our solver on these problems is demonstrated. Lastly, we show that good scalability can be achieved by preconditioning the iterative linear solver using a new algebraic multilevel preconditioner, constructed based on the idea of semi-coarsening.« less

All-optical 1st- and 2nd-order differential equation solvers with large tuning ranges using Fabry-Pérot semiconductor optical amplifiers.

PubMed

Chen, Kaisheng; Hou, Jie; Huang, Zhuyang; Cao, Tong; Zhang, Jihua; Yu, Yuan; Zhang, Xinliang

2015-02-09

We experimentally demonstrate an all-optical temporal computation scheme for solving 1st- and 2nd-order linear ordinary differential equations (ODEs) with tunable constant coefficients by using Fabry-Pérot semiconductor optical amplifiers (FP-SOAs). By changing the injection currents of FP-SOAs, the constant coefficients of the differential equations are practically tuned. A quite large constant coefficient tunable range from 0.0026/ps to 0.085/ps is achieved for the 1st-order differential equation. Moreover, the constant coefficient p of the 2nd-order ODE solver can be continuously tuned from 0.0216/ps to 0.158/ps, correspondingly with the constant coefficient q varying from 0.0000494/ps(2) to 0.006205/ps(2). Additionally, a theoretical model that combining the carrier density rate equation of the semiconductor optical amplifier (SOA) with the transfer function of the Fabry-Pérot (FP) cavity is exploited to analyze the solving processes. For both 1st- and 2nd-order solvers, excellent agreements between the numerical simulations and the experimental results are obtained. The FP-SOAs based all-optical differential-equation solvers can be easily integrated with other optical components based on InP/InGaAsP materials, such as laser, modulator, photodetector and waveguide, which can motivate the realization of the complicated optical computing on a single integrated chip.

Effects of high-frequency damping on iterative convergence of implicit viscous solver

NASA Astrophysics Data System (ADS)

Nishikawa, Hiroaki; Nakashima, Yoshitaka; Watanabe, Norihiko

2017-11-01

This paper discusses effects of high-frequency damping on iterative convergence of an implicit defect-correction solver for viscous problems. The study targets a finite-volume discretization with a one parameter family of damped viscous schemes. The parameter α controls high-frequency damping: zero damping with α = 0, and larger damping for larger α (> 0). Convergence rates are predicted for a model diffusion equation by a Fourier analysis over a practical range of α. It is shown that the convergence rate attains its minimum at α = 1 on regular quadrilateral grids, and deteriorates for larger values of α. A similar behavior is observed for regular triangular grids. In both quadrilateral and triangular grids, the solver is predicted to diverge for α smaller than approximately 0.5. Numerical results are shown for the diffusion equation and the Navier-Stokes equations on regular and irregular grids. The study suggests that α = 1 and 4/3 are suitable values for robust and efficient computations, and α = 4 / 3 is recommended for the diffusion equation, which achieves higher-order accuracy on regular quadrilateral grids. Finally, a Jacobian-Free Newton-Krylov solver with the implicit solver (a low-order Jacobian approximately inverted by a multi-color Gauss-Seidel relaxation scheme) used as a variable preconditioner is recommended for practical computations, which provides robust and efficient convergence for a wide range of α.

Preconditioned conjugate gradient methods for the Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Ajmani, Kumud; Ng, Wing-Fai; Liou, Meng-Sing

1994-01-01

A preconditioned Krylov subspace method (GMRES) is used to solve the linear systems of equations formed at each time-integration step of the unsteady, two-dimensional, compressible Navier-Stokes equations of fluid flow. The Navier-Stokes equations are cast in an implicit, upwind finite-volume, flux-split formulation. Several preconditioning techniques are investigated to enhance the efficiency and convergence rate of the implicit solver based on the GMRES algorithm. The superiority of the new solver is established by comparisons with a conventional implicit solver, namely line Gauss-Seidel relaxation (LGSR). Computational test results for low-speed (incompressible flow over a backward-facing step at Mach 0.1), transonic flow (trailing edge flow in a transonic turbine cascade), and hypersonic flow (shock-on-shock interactions on a cylindrical leading edge at Mach 6.0) are presented. For the Mach 0.1 case, overall speedup factors of up to 17 (in terms of time-steps) and 15 (in terms of CPU time on a CRAY-YMP/8) are found in favor of the preconditioned GMRES solver, when compared with the LGSR solver. The corresponding speedup factors for the transonic flow case are 17 and 23, respectively. The hypersonic flow case shows slightly lower speedup factors of 9 and 13, respectively. The study of preconditioners conducted in this research reveals that a new LUSGS-type preconditioner is much more efficient than a conventional incomplete LU-type preconditioner.

Revisiting Parallel Cyclic Reduction and Parallel Prefix-Based Algorithms for Block Tridiagonal System of Equations

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

Seal, Sudip K; Perumalla, Kalyan S; Hirshman, Steven Paul

2013-01-01

Simulations that require solutions of block tridiagonal systems of equations rely on fast parallel solvers for runtime efficiency. Leading parallel solvers that are highly effective for general systems of equations, dense or sparse, are limited in scalability when applied to block tridiagonal systems. This paper presents scalability results as well as detailed analyses of two parallel solvers that exploit the special structure of block tridiagonal matrices to deliver superior performance, often by orders of magnitude. A rigorous analysis of their relative parallel runtimes is shown to reveal the existence of a critical block size that separates the parameter space spannedmore » by the number of block rows, the block size and the processor count, into distinct regions that favor one or the other of the two solvers. Dependence of this critical block size on the above parameters as well as on machine-specific constants is established. These formal insights are supported by empirical results on up to 2,048 cores of a Cray XT4 system. To the best of our knowledge, this is the highest reported scalability for parallel block tridiagonal solvers to date.« less

A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments.

PubMed

Fisicaro, G; Genovese, L; Andreussi, O; Marzari, N; Goedecker, S

2016-01-07

The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into account the non-trivial electrostatic screening coming from the solvent and the electrolytes. As a consequence, the electrostatic potential has to be found by solving the generalized Poisson and the Poisson-Boltzmann equations for neutral and ionic solutions, respectively. In the present work, solvers for both problems have been developed. A preconditioned conjugate gradient method has been implemented for the solution of the generalized Poisson equation and the linear regime of the Poisson-Boltzmann, allowing to solve iteratively the minimization problem with some ten iterations of the ordinary Poisson equation solver. In addition, a self-consistent procedure enables us to solve the non-linear Poisson-Boltzmann problem. Both solvers exhibit very high accuracy and parallel efficiency and allow for the treatment of periodic, free, and slab boundary conditions. The solver has been integrated into the BigDFT and Quantum-ESPRESSO electronic-structure packages and will be released as an independent program, suitable for integration in other codes.

A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments

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

Fisicaro, G., E-mail: giuseppe.fisicaro@unibas.ch; Goedecker, S.; Genovese, L.

2016-01-07

The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into account the non-trivial electrostatic screening coming from the solvent and the electrolytes. As a consequence, the electrostatic potential has to be found by solving the generalized Poisson and the Poisson-Boltzmann equations for neutral and ionic solutions, respectively. In the present work, solvers for both problems have been developed. A preconditioned conjugate gradient method has been implemented for the solution of the generalized Poisson equation and themore » linear regime of the Poisson-Boltzmann, allowing to solve iteratively the minimization problem with some ten iterations of the ordinary Poisson equation solver. In addition, a self-consistent procedure enables us to solve the non-linear Poisson-Boltzmann problem. Both solvers exhibit very high accuracy and parallel efficiency and allow for the treatment of periodic, free, and slab boundary conditions. The solver has been integrated into the BigDFT and Quantum-ESPRESSO electronic-structure packages and will be released as an independent program, suitable for integration in other codes.« less

The solution of linear systems of equations with a structural analysis code on the NAS CRAY-2

NASA Technical Reports Server (NTRS)

Poole, Eugene L.; Overman, Andrea L.

1988-01-01

Two methods for solving linear systems of equations on the NAS Cray-2 are described. One is a direct method; the other is an iterative method. Both methods exploit the architecture of the Cray-2, particularly the vectorization, and are aimed at structural analysis applications. To demonstrate and evaluate the methods, they were installed in a finite element structural analysis code denoted the Computational Structural Mechanics (CSM) Testbed. A description of the techniques used to integrate the two solvers into the Testbed is given. Storage schemes, memory requirements, operation counts, and reformatting procedures are discussed. Finally, results from the new methods are compared with results from the initial Testbed sparse Choleski equation solver for three structural analysis problems. The new direct solvers described achieve the highest computational rates of the methods compared. The new iterative methods are not able to achieve as high computation rates as the vectorized direct solvers but are best for well conditioned problems which require fewer iterations to converge to the solution.

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

Thompson, S.

This report describes the use of several subroutines from the CORLIB core mathematical subroutine library for the solution of a model fluid flow problem. The model consists of the Euler partial differential equations. The equations are spatially discretized using the method of pseudo-characteristics. The resulting system of ordinary differential equations is then integrated using the method of lines. The stiff ordinary differential equation solver LSODE (2) from CORLIB is used to perform the time integration. The non-stiff solver ODE (4) is used to perform a related integration. The linear equation solver subroutines DECOMP and SOLVE are used to solve linearmore » systems whose solutions are required in the calculation of the time derivatives. The monotone cubic spline interpolation subroutines PCHIM and PCHFE are used to approximate water properties. The report describes the use of each of these subroutines in detail. It illustrates the manner in which modules from a standard mathematical software library such as CORLIB can be used as building blocks in the solution of complex problems of practical interest. 9 refs., 2 figs., 4 tabs.« less

A comparison of viscous-plastic sea ice solvers with and without replacement pressure

NASA Astrophysics Data System (ADS)

Kimmritz, Madlen; Losch, Martin; Danilov, Sergey

2017-07-01

Recent developments of the explicit elastic-viscous-plastic (EVP) solvers call for a new comparison with implicit solvers for the equations of viscous-plastic sea ice dynamics. In Arctic sea ice simulations, the modified and the adaptive EVP solvers, and the implicit Jacobian-free Newton-Krylov (JFNK) solver are compared against each other. The adaptive EVP method shows convergence rates that are generally similar or even better than those of the modified EVP method, but the convergence of the EVP methods is found to depend dramatically on the use of the replacement pressure (RP). Apparently, using the RP can affect the pseudo-elastic waves in the EVP methods by introducing extra non-physical oscillations so that, in the extreme case, convergence to the VP solution can be lost altogether. The JFNK solver also suffers from higher failure rates with RP implying that with RP the momentum equations are stiffer and more difficult to solve. For practical purposes, both EVP methods can be used efficiently with an unexpectedly low number of sub-cycling steps without compromising the solutions. The differences between the RP solutions and the NoRP solutions (when the RP is not being used) can be reduced with lower thresholds of viscous regularization at the cost of increasing stiffness of the equations, and hence the computational costs of solving them.

On unstructured grids and solvers

NASA Technical Reports Server (NTRS)

Barth, T. J.

1990-01-01

The fundamentals and the state-of-the-art technology for unstructured grids and solvers are highlighted. Algorithms and techniques pertinent to mesh generation are discussed. It is shown that grid generation and grid manipulation schemes rely on fast multidimensional searching. Flow solution techniques for the Euler equations, which can be derived from the integral form of the equations are discussed. Sample calculations are also provided.

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Barth, Timothy J.

1992-01-01

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology.

A fast direct solver for boundary value problems on locally perturbed geometries

NASA Astrophysics Data System (ADS)

Zhang, Yabin; Gillman, Adrianna

2018-03-01

Many applications including optimal design and adaptive discretization techniques involve solving several boundary value problems on geometries that are local perturbations of an original geometry. This manuscript presents a fast direct solver for boundary value problems that are recast as boundary integral equations. The idea is to write the discretized boundary integral equation on a new geometry as a low rank update to the discretized problem on the original geometry. Using the Sherman-Morrison formula, the inverse can be expressed in terms of the inverse of the original system applied to the low rank factors and the right hand side. Numerical results illustrate for problems where perturbation is localized the fast direct solver is three times faster than building a new solver from scratch.

Nonlinear Krylov and moving nodes in the method of lines

NASA Astrophysics Data System (ADS)

Miller, Keith

2005-11-01

We report on some successes and problem areas in the Method of Lines from our work with moving node finite element methods. First, we report on our "nonlinear Krylov accelerator" for the modified Newton's method on the nonlinear equations of our stiff ODE solver. Since 1990 it has been robust, simple, cheap, and automatic on all our moving node computations. We publicize further trials with it here because it should be of great general usefulness to all those solving evolutionary equations. Second, we discuss the need for reliable automatic choice of spatially variable time steps. Third, we discuss the need for robust and efficient iterative solvers for the difficult linearized equations (Jx=b) of our stiff ODE solver. Here, the 1997 thesis of Zulu Xaba has made significant progress.

Modeling of frequency-domain scalar wave equation with the average-derivative optimal scheme based on a multigrid-preconditioned iterative solver

NASA Astrophysics Data System (ADS)

Cao, Jian; Chen, Jing-Bo; Dai, Meng-Xue

2018-01-01

An efficient finite-difference frequency-domain modeling of seismic wave propagation relies on the discrete schemes and appropriate solving methods. The average-derivative optimal scheme for the scalar wave modeling is advantageous in terms of the storage saving for the system of linear equations and the flexibility for arbitrary directional sampling intervals. However, using a LU-decomposition-based direct solver to solve its resulting system of linear equations is very costly for both memory and computational requirements. To address this issue, we consider establishing a multigrid-preconditioned BI-CGSTAB iterative solver fit for the average-derivative optimal scheme. The choice of preconditioning matrix and its corresponding multigrid components is made with the help of Fourier spectral analysis and local mode analysis, respectively, which is important for the convergence. Furthermore, we find that for the computation with unequal directional sampling interval, the anisotropic smoothing in the multigrid precondition may affect the convergence rate of this iterative solver. Successful numerical applications of this iterative solver for the homogenous and heterogeneous models in 2D and 3D are presented where the significant reduction of computer memory and the improvement of computational efficiency are demonstrated by comparison with the direct solver. In the numerical experiments, we also show that the unequal directional sampling interval will weaken the advantage of this multigrid-preconditioned iterative solver in the computing speed or, even worse, could reduce its accuracy in some cases, which implies the need for a reasonable control of directional sampling interval in the discretization.

High fidelity CFD-CSD aeroelastic analysis of slender bladed horizontal-axis wind turbine

NASA Astrophysics Data System (ADS)

Sayed, M.; Lutz, Th.; Krämer, E.; Shayegan, Sh.; Ghantasala, A.; Wüchner, R.; Bletzinger, K.-U.

2016-09-01

The aeroelastic response of large multi-megawatt slender horizontal-axis wind turbine blades is investigated by means of a time-accurate CFD-CSD coupling approach. A loose coupling approach is implemented and used to perform the simulations. The block- structured CFD solver FLOWer is utilized to obtain the aerodynamic blade loads based on the time-accurate solution of the unsteady Reynolds-averaged Navier-Stokes equations. The CSD solver Carat++ is applied to acquire the blade elastic deformations based on non-linear beam elements. In this contribution, the presented coupling approach is utilized to study the aeroelastic response of the generic DTU 10MW wind turbine. Moreover, the effect of the coupled results on the wind turbine performance is discussed. The results are compared to the aeroelastic response predicted by FLOWer coupled to the MBS tool SIMPACK as well as the response predicted by SIMPACK coupled to a Blade Element Momentum code for aerodynamic predictions. A comparative study among the different modelling approaches for this coupled problem is discussed to quantify the coupling effects of the structural models on the aeroelastic response.

The Computer Bulletin Board.

ERIC Educational Resources Information Center

Batt, Russell H., Ed.

1990-01-01

Described is how spreadsheet and problem solver microcomputer programs may assist students in performing mathematical calculations. Discussed is the application of the equation solver "MathCAD" to various areas in the undergraduate curriculum. (KR)

A coarse-grid projection method for accelerating incompressible flow computations

NASA Astrophysics Data System (ADS)

San, Omer; Staples, Anne E.

2013-01-01

We present a coarse-grid projection (CGP) method for accelerating incompressible flow computations, which is applicable to methods involving Poisson equations as incompressibility constraints. The CGP methodology is a modular approach that facilitates data transfer with simple interpolations and uses black-box solvers for the Poisson and advection-diffusion equations in the flow solver. After solving the Poisson equation on a coarsened grid, an interpolation scheme is used to obtain the fine data for subsequent time stepping on the full grid. A particular version of the method is applied here to the vorticity-stream function, primitive variable, and vorticity-velocity formulations of incompressible Navier-Stokes equations. We compute several benchmark flow problems on two-dimensional Cartesian and non-Cartesian grids, as well as a three-dimensional flow problem. The method is found to accelerate these computations while retaining a level of accuracy close to that of the fine resolution field, which is significantly better than the accuracy obtained for a similar computation performed solely using a coarse grid. A linear acceleration rate is obtained for all the cases we consider due to the linear-cost elliptic Poisson solver used, with reduction factors in computational time between 2 and 42. The computational savings are larger when a suboptimal Poisson solver is used. We also find that the computational savings increase with increasing distortion ratio on non-Cartesian grids, making the CGP method a useful tool for accelerating generalized curvilinear incompressible flow solvers.

Numerical and experimental investigation of the 3D free surface flow in a model Pelton turbine

NASA Astrophysics Data System (ADS)

Fiereder, R.; Riemann, S.; Schilling, R.

2010-08-01

This investigation focuses on the numerical and experimental analysis of the 3D free surface flow in a Pelton turbine. In particular, two typical flow conditions occurring in a full scale Pelton turbine - a configuration with a straight inlet as well as a configuration with a 90 degree elbow upstream of the nozzle - are considered. Thereby, the effect of secondary flow due to the 90 degree bending of the upstream pipe on the characteristics of the jet is explored. The hybrid flow field consists of pure liquid flow within the conduit and free surface two component flow of the liquid jet emerging out of the nozzle into air. The numerical results are validated against experimental investigations performed in the laboratory of the Institute of Fluid Mechanics (FLM). For the numerical simulation of the flow the in-house unstructured fully parallelized finite volume solver solver3D is utilized. An advanced interface capturing model based on the classic Volume of Fluid method is applied. In order to ensure sharp interface resolution an additional convection term is added to the transport equation of the volume fraction. A collocated variable arrangement is used and the set of non-linear equations, containing fluid conservation equations and model equations for turbulence and volume fraction, are solved in a segregated manner. For pressure-velocity coupling the SIMPLE and PISO algorithms are implemented. Detailed analysis of the observed flow patterns in the jet and of the jet geometry are presented.

Assessment of Linear Finite-Difference Poisson-Boltzmann Solvers

PubMed Central

Wang, Jun; Luo, Ray

2009-01-01

CPU time and memory usage are two vital issues that any numerical solvers for the Poisson-Boltzmann equation have to face in biomolecular applications. In this study we systematically analyzed the CPU time and memory usage of five commonly used finite-difference solvers with a large and diversified set of biomolecular structures. Our comparative analysis shows that modified incomplete Cholesky conjugate gradient and geometric multigrid are the most efficient in the diversified test set. For the two efficient solvers, our test shows that their CPU times increase approximately linearly with the numbers of grids. Their CPU times also increase almost linearly with the negative logarithm of the convergence criterion at very similar rate. Our comparison further shows that geometric multigrid performs better in the large set of tested biomolecules. However, modified incomplete Cholesky conjugate gradient is superior to geometric multigrid in molecular dynamics simulations of tested molecules. We also investigated other significant components in numerical solutions of the Poisson-Boltzmann equation. It turns out that the time-limiting step is the free boundary condition setup for the linear systems for the selected proteins if the electrostatic focusing is not used. Thus, development of future numerical solvers for the Poisson-Boltzmann equation should balance all aspects of the numerical procedures in realistic biomolecular applications. PMID:20063271

Practical Aerodynamic Design Optimization Based on the Navier-Stokes Equations and a Discrete Adjoint Method

NASA Technical Reports Server (NTRS)

Grossman, Bernard

1999-01-01

The technical details are summarized below: Compressible and incompressible versions of a three-dimensional unstructured mesh Reynolds-averaged Navier-Stokes flow solver have been differentiated and resulting derivatives have been verified by comparisons with finite differences and a complex-variable approach. In this implementation, the turbulence model is fully coupled with the flow equations in order to achieve this consistency. The accuracy demonstrated in the current work represents the first time that such an approach has been successfully implemented. The accuracy of a number of simplifying approximations to the linearizations of the residual have been examined. A first-order approximation to the dependent variables in both the adjoint and design equations has been investigated. The effects of a "frozen" eddy viscosity and the ramifications of neglecting some mesh sensitivity terms were also examined. It has been found that none of the approximations yielded derivatives of acceptable accuracy and were often of incorrect sign. However, numerical experiments indicate that an incomplete convergence of the adjoint system often yield sufficiently accurate derivatives, thereby significantly lowering the time required for computing sensitivity information. The convergence rate of the adjoint solver relative to the flow solver has been examined. Inviscid adjoint solutions typically require one to four times the cost of a flow solution, while for turbulent adjoint computations, this ratio can reach as high as eight to ten. Numerical experiments have shown that the adjoint solver can stall before converging the solution to machine accuracy, particularly for viscous cases. A possible remedy for this phenomenon would be to include the complete higher-order linearization in the preconditioning step, or to employ a simple form of mesh sequencing to obtain better approximations to the solution through the use of coarser meshes. . An efficient surface parameterization based on a free-form deformation technique has been utilized and the resulting codes have been integrated with an optimization package. Lastly, sample optimizations have been shown for inviscid and turbulent flow over an ONERA M6 wing. Drag reductions have been demonstrated by reducing shock strengths across the span of the wing.

Efficient Implementation of Multigrid Solvers on Message-Passing Parrallel Systems

NASA Technical Reports Server (NTRS)

Lou, John

1994-01-01

We discuss our implementation strategies for finite difference multigrid partial differential equation (PDE) solvers on message-passing systems. Our target parallel architecture is Intel parallel computers: the Delta and Paragon system.

Integrated multidisciplinary CAD/CAE environment for micro-electro-mechanical systems (MEMS)

NASA Astrophysics Data System (ADS)

Przekwas, Andrzej J.

1999-03-01

Computational design of MEMS involves several strongly coupled physical disciplines, including fluid mechanics, heat transfer, stress/deformation dynamics, electronics, electro/magneto statics, calorics, biochemistry and others. CFDRC is developing a new generation multi-disciplinary CAD systems for MEMS using high-fidelity field solvers on unstructured, solution-adaptive grids for a full range of disciplines. The software system, ACE + MEMS, includes all essential CAD tools; geometry/grid generation for multi- discipline, multi-equation solvers, GUI, tightly coupled configurable 3D field solvers for FVM, FEM and BEM and a 3D visualization/animation tool. The flow/heat transfer/calorics/chemistry equations are solved with unstructured adaptive FVM solver, stress/deformation are computed with a FEM STRESS solver and a FAST BEM solver is used to solve linear heat transfer, electro/magnetostatics and elastostatics equations on adaptive polygonal surface grids. Tight multidisciplinary coupling and automatic interoperability between the tools was achieved by designing a comprehensive database structure and APIs for complete model definition. The virtual model definition is implemented in data transfer facility, a publicly available tool described in this paper. The paper presents overall description of the software architecture and MEMS design flow in ACE + MEMS. It describes current status, ongoing effort and future plans for the software. The paper also discusses new concepts of mixed-level and mixed- dimensionality capability in which 1D microfluidic networks are simulated concurrently with 3D high-fidelity models of discrete components.

Implementing a Loosely Coupled Fluid Structure Interaction Finite Element Model in PHASTA

NASA Astrophysics Data System (ADS)

Pope, David

Fluid Structure Interaction problems are an important multi-physics phenomenon in the design of aerospace vehicles and other engineering applications. A variety of computational fluid dynamics solvers capable of resolving the fluid dynamics exist. PHASTA is one such computational fluid dynamics solver. Enhancing the capability of PHASTA to resolve Fluid-Structure Interaction first requires implementing a structural dynamics solver. The implementation also requires a correction of the mesh used to solve the fluid equations to account for the deformation of the structure. This results in mesh motion and causes the need for an Arbitrary Lagrangian-Eulerian modification to the fluid dynamics equations currently implemented in PHASTA. With the implementation of both structural dynamics physics, mesh correction, and the Arbitrary Lagrangian-Eulerian modification of the fluid dynamics equations, PHASTA is made capable of solving Fluid-Structure Interaction problems.

Compact tunable silicon photonic differential-equation solver for general linear time-invariant systems.

PubMed

Wu, Jiayang; Cao, Pan; Hu, Xiaofeng; Jiang, Xinhong; Pan, Ting; Yang, Yuxing; Qiu, Ciyuan; Tremblay, Christine; Su, Yikai

2014-10-20

We propose and experimentally demonstrate an all-optical temporal differential-equation solver that can be used to solve ordinary differential equations (ODEs) characterizing general linear time-invariant (LTI) systems. The photonic device implemented by an add-drop microring resonator (MRR) with two tunable interferometric couplers is monolithically integrated on a silicon-on-insulator (SOI) wafer with a compact footprint of ~60 μm × 120 μm. By thermally tuning the phase shifts along the bus arms of the two interferometric couplers, the proposed device is capable of solving first-order ODEs with two variable coefficients. The operation principle is theoretically analyzed, and system testing of solving ODE with tunable coefficients is carried out for 10-Gb/s optical Gaussian-like pulses. The experimental results verify the effectiveness of the fabricated device as a tunable photonic ODE solver.

Textbook Multigrid Efficiency for Leading Edge Stagnation

NASA Technical Reports Server (NTRS)

Diskin, Boris; Thomas, James L.; Mineck, Raymond E.

2004-01-01

A multigrid solver is defined as having textbook multigrid efficiency (TME) if the solutions to the governing system of equations are attained in a computational work which is a small (less than 10) multiple of the operation count in evaluating the discrete residuals. TME in solving the incompressible inviscid fluid equations is demonstrated for leading-edge stagnation flows. The contributions of this paper include (1) a special formulation of the boundary conditions near stagnation allowing convergence of the Newton iterations on coarse grids, (2) the boundary relaxation technique to facilitate relaxation and residual restriction near the boundaries, (3) a modified relaxation scheme to prevent initial error amplification, and (4) new general analysis techniques for multigrid solvers. Convergence of algebraic errors below the level of discretization errors is attained by a full multigrid (FMG) solver with one full approximation scheme (FAS) cycle per grid. Asymptotic convergence rates of the FAS cycles for the full system of flow equations are very fast, approaching those for scalar elliptic equations.

Textbook Multigrid Efficiency for Leading Edge Stagnation

NASA Technical Reports Server (NTRS)

Diskin, Boris; Thomas, James L.; Mineck, Raymond E.

2004-01-01

A multigrid solver is defined as having textbook multigrid efficiency (TME) if the solutions to the governing system of equations are attained in a computational work which is a small (less than 10) multiple of the operation count in evaluating the discrete residuals. TME in solving the incompressible inviscid fluid equations is demonstrated for leading- edge stagnation flows. The contributions of this paper include (1) a special formulation of the boundary conditions near stagnation allowing convergence of the Newton iterations on coarse grids, (2) the boundary relaxation technique to facilitate relaxation and residual restriction near the boundaries, (3) a modified relaxation scheme to prevent initial error amplification, and (4) new general analysis techniques for multigrid solvers. Convergence of algebraic errors below the level of discretization errors is attained by a full multigrid (FMG) solver with one full approximation scheme (F.4S) cycle per grid. Asymptotic convergence rates of the F.4S cycles for the full system of flow equations are very fast, approaching those for scalar elliptic equations.

OVERSMART Reporting Tool for Flow Computations Over Large Grid Systems

NASA Technical Reports Server (NTRS)

Kao, David L.; Chan, William M.

2012-01-01

Structured grid solvers such as NASA's OVERFLOW compressible Navier-Stokes flow solver can generate large data files that contain convergence histories for flow equation residuals, turbulence model equation residuals, component forces and moments, and component relative motion dynamics variables. Most of today's large-scale problems can extend to hundreds of grids, and over 100 million grid points. However, due to the lack of efficient tools, only a small fraction of information contained in these files is analyzed. OVERSMART (OVERFLOW Solution Monitoring And Reporting Tool) provides a comprehensive report of solution convergence of flow computations over large, complex grid systems. It produces a one-page executive summary of the behavior of flow equation residuals, turbulence model equation residuals, and component forces and moments. Under the automatic option, a matrix of commonly viewed plots such as residual histograms, composite residuals, sub-iteration bar graphs, and component forces and moments is automatically generated. Specific plots required by the user can also be prescribed via a command file or a graphical user interface. Output is directed to the user s computer screen and/or to an html file for archival purposes. The current implementation has been targeted for the OVERFLOW flow solver, which is used to obtain a flow solution on structured overset grids. The OVERSMART framework allows easy extension to other flow solvers.

The Thin Oil Film Equation

NASA Technical Reports Server (NTRS)

Brown, James L.; Naughton, Jonathan W.

1999-01-01

A thin film of oil on a surface responds primarily to the wall shear stress generated on that surface by a three-dimensional flow. The oil film is also subject to wall pressure gradients, surface tension effects and gravity. The partial differential equation governing the oil film flow is shown to be related to Burgers' equation. Analytical and numerical methods for solving the thin oil film equation are presented. A direct numerical solver is developed where the wall shear stress variation on the surface is known and which solves for the oil film thickness spatial and time variation on the surface. An inverse numerical solver is also developed where the oil film thickness spatial variation over the surface at two discrete times is known and which solves for the wall shear stress variation over the test surface. A One-Time-Level inverse solver is also demonstrated. The inverse numerical solver provides a mathematically rigorous basis for an improved form of a wall shear stress instrument suitable for application to complex three-dimensional flows. To demonstrate the complexity of flows for which these oil film methods are now suitable, extensive examination is accomplished for these analytical and numerical methods as applied to a thin oil film in the vicinity of a three-dimensional saddle of separation.

An implicit numerical scheme for the simulation of internal viscous flows on unstructured grids

NASA Technical Reports Server (NTRS)

Jorgenson, Philip C. E.; Pletcher, Richard H.

1994-01-01

The Navier-Stokes equations are solved numerically for two-dimensional steady viscous laminar flows. The grids are generated based on the method of Delaunay triangulation. A finite-volume approach is used to discretize the conservation law form of the compressible flow equations written in terms of primitive variables. A preconditioning matrix is added to the equations so that low Mach number flows can be solved economically. The equations are time marched using either an implicit Gauss-Seidel iterative procedure or a solver based on a conjugate gradient like method. A four color scheme is employed to vectorize the block Gauss-Seidel relaxation procedure. This increases the memory requirements minimally and decreases the computer time spent solving the resulting system of equations substantially. A factor of 7.6 speed up in the matrix solver is typical for the viscous equations. Numerical results are obtained for inviscid flow over a bump in a channel at subsonic and transonic conditions for validation with structured solvers. Viscous results are computed for developing flow in a channel, a symmetric sudden expansion, periodic tandem cylinders in a cross-flow, and a four-port valve. Comparisons are made with available results obtained by other investigators.

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.

Nonlinear Solver Approaches for the Diffusive Wave Approximation to the Shallow Water Equations

NASA Astrophysics Data System (ADS)

Collier, N.; Knepley, M.

2015-12-01

The diffusive wave approximation to the shallow water equations (DSW) is a doubly-degenerate, nonlinear, parabolic partial differential equation used to model overland flows. Despite its challenges, the DSW equation has been extensively used to model the overland flow component of various integrated surface/subsurface models. The equation's complications become increasingly problematic when ponding occurs, a feature which becomes pervasive when solving on large domains with realistic terrain. In this talk I discuss the various forms and regularizations of the DSW equation and highlight their effect on the solvability of the nonlinear system. In addition to this analysis, I present results of a numerical study which tests the applicability of a class of composable nonlinear algebraic solvers recently added to the Portable, Extensible, Toolkit for Scientific Computation (PETSc).

An Unsplit Monte-Carlo solver for the resolution of the linear Boltzmann equation coupled to (stiff) Bateman equations

NASA Astrophysics Data System (ADS)

Bernede, Adrien; Poëtte, Gaël

2018-02-01

In this paper, we are interested in the resolution of the time-dependent problem of particle transport in a medium whose composition evolves with time due to interactions. As a constraint, we want to use of Monte-Carlo (MC) scheme for the transport phase. A common resolution strategy consists in a splitting between the MC/transport phase and the time discretization scheme/medium evolution phase. After going over and illustrating the main drawbacks of split solvers in a simplified configuration (monokinetic, scalar Bateman problem), we build a new Unsplit MC (UMC) solver improving the accuracy of the solutions, avoiding numerical instabilities, and less sensitive to time discretization. The new solver is essentially based on a Monte Carlo scheme with time dependent cross sections implying the on-the-fly resolution of a reduced model for each MC particle describing the time evolution of the matter along their flight path.

Textbook Multigrid Efficiency for the Steady Euler Equations

NASA Technical Reports Server (NTRS)

Roberts, Thomas W.; Sidilkover, David; Swanson, R. C.

2004-01-01

A fast multigrid solver for the steady incompressible Euler equations is presented. Unlike time-marching schemes, this approach uses relaxation of the steady equations. Application of this method results in a discretization that correctly distinguishes between the advection and elliptic parts of the operator, allowing efficient smoothers to be constructed. Solvers for both unstructured triangular grids and structured quadrilateral grids have been written. Computations for channel flow and flow over a nonlifting airfoil have computed. Using Gauss-Seidel relaxation ordered in the flow direction, textbook multigrid convergence rates of nearly one order-of-magnitude residual reduction per multigrid cycle are achieved, independent of the grid spacing. This approach also may be applied to the compressible Euler equations and the incompressible Navier-Stokes equations.

Bypass Transitional Flow Calculations Using a Navier-Stokes Solver and Two-Equation Models

NASA Technical Reports Server (NTRS)

Liuo, William W.; Shih, Tsan-Hsing; Povinelli, L. A. (Technical Monitor)

2000-01-01

Bypass transitional flows over a flat plate were simulated using a Navier-Stokes solver and two equation models. A new model for the bypass transition, which occurs in cases with high free stream turbulence intensity (TI), is described. The new transition model is developed by including an intermittency correction function to an existing two-equation turbulence model. The advantages of using Navier-Stokes equations, as opposed to boundary-layer equations, in bypass transition simulations are also illustrated. The results for two test flows over a flat plate with different levels of free stream turbulence intensity are reported. Comparisons with the experimental measurements show that the new model can capture very well both the onset and the length of bypass transition.

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

NASA Technical Reports Server (NTRS)

Hoyniak, Daniel; Verdon, Joseph M.

1991-01-01

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

Parallel Element Agglomeration Algebraic Multigrid and Upscaling Library

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

Barker, Andrew T.; Benson, Thomas R.; Lee, Chak Shing

ParELAG is a parallel C++ library for numerical upscaling of finite element discretizations and element-based algebraic multigrid solvers. It provides optimal complexity algorithms to build multilevel hierarchies and solvers that can be used for solving a wide class of partial differential equations (elliptic, hyperbolic, saddle point problems) on general unstructured meshes. Additionally, a novel multilevel solver for saddle point problems with divergence constraint is implemented.

On multigrid solution of the implicit equations of hydrodynamics. Experiments for the compressible Euler equations in general coordinates

NASA Astrophysics Data System (ADS)

Kifonidis, K.; Müller, E.

2012-08-01

Aims: We describe and study a family of new multigrid iterative solvers for the multidimensional, implicitly discretized equations of hydrodynamics. Schemes of this class are free of the Courant-Friedrichs-Lewy condition. They are intended for simulations in which widely differing wave propagation timescales are present. A preferred solver in this class is identified. Applications to some simple stiff test problems that are governed by the compressible Euler equations, are presented to evaluate the convergence behavior, and the stability properties of this solver. Algorithmic areas are determined where further work is required to make the method sufficiently efficient and robust for future application to difficult astrophysical flow problems. Methods: The basic equations are formulated and discretized on non-orthogonal, structured curvilinear meshes. Roe's approximate Riemann solver and a second-order accurate reconstruction scheme are used for spatial discretization. Implicit Runge-Kutta (ESDIRK) schemes are employed for temporal discretization. The resulting discrete equations are solved with a full-coarsening, non-linear multigrid method. Smoothing is performed with multistage-implicit smoothers. These are applied here to the time-dependent equations by means of dual time stepping. Results: For steady-state problems, our results show that the efficiency of the present approach is comparable to the best implicit solvers for conservative discretizations of the compressible Euler equations that can be found in the literature. The use of red-black as opposed to symmetric Gauss-Seidel iteration in the multistage-smoother is found to have only a minor impact on multigrid convergence. This should enable scalable parallelization without having to seriously compromise the method's algorithmic efficiency. For time-dependent test problems, our results reveal that the multigrid convergence rate degrades with increasing Courant numbers (i.e. time step sizes). Beyond a Courant number of nine thousand, even complete multigrid breakdown is observed. Local Fourier analysis indicates that the degradation of the convergence rate is associated with the coarse-grid correction algorithm. An implicit scheme for the Euler equations that makes use of the present method was, nevertheless, able to outperform a standard explicit scheme on a time-dependent problem with a Courant number of order 1000. Conclusions: For steady-state problems, the described approach enables the construction of parallelizable, efficient, and robust implicit hydrodynamics solvers. The applicability of the method to time-dependent problems is presently restricted to cases with moderately high Courant numbers. This is due to an insufficient coarse-grid correction of the employed multigrid algorithm for large time steps. Further research will be required to help us to understand and overcome the observed multigrid convergence difficulties for time-dependent problems.

Practical Aerodynamic Design Optimization Based on the Navier-Stokes Equations and a Discrete Adjoint Method

NASA Technical Reports Server (NTRS)

Grossman, Bernard

1999-01-01

Compressible and incompressible versions of a three-dimensional unstructured mesh Reynolds-averaged Navier-Stokes flow solver have been differentiated and resulting derivatives have been verified by comparisons with finite differences and a complex-variable approach. In this implementation, the turbulence model is fully coupled with the flow equations in order to achieve this consistency. The accuracy demonstrated in the current work represents the first time that such an approach has been successfully implemented. The accuracy of a number of simplifying approximations to the linearizations of the residual have been examined. A first-order approximation to the dependent variables in both the adjoint and design equations has been investigated. The effects of a "frozen" eddy viscosity and the ramifications of neglecting some mesh sensitivity terms were also examined. It has been found that none of the approximations yielded derivatives of acceptable accuracy and were often of incorrect sign. However, numerical experiments indicate that an incomplete convergence of the adjoint system often yield sufficiently accurate derivatives, thereby significantly lowering the time required for computing sensitivity information. The convergence rate of the adjoint solver relative to the flow solver has been examined. Inviscid adjoint solutions typically require one to four times the cost of a flow solution, while for turbulent adjoint computations, this ratio can reach as high as eight to ten. Numerical experiments have shown that the adjoint solver can stall before converging the solution to machine accuracy, particularly for viscous cases. A possible remedy for this phenomenon would be to include the complete higher-order linearization in the preconditioning step, or to employ a simple form of mesh sequencing to obtain better approximations to the solution through the use of coarser meshes. An efficient surface parameterization based on a free-form deformation technique has been utilized and the resulting codes have been integrated with an optimization package. Lastly, sample optimizations have been shown for inviscid and turbulent flow over an ONERA M6 wing. Drag reductions have been demonstrated by reducing shock strengths across the span of the wing. In order for large scale optimization to become routine, the benefits of parallel architectures should be exploited. Although the flow solver has been parallelized using compiler directives. The parallel efficiency is under 50 percent. Clearly, parallel versions of the codes will have an immediate impact on the ability to design realistic configurations on fine meshes, and this effort is currently underway.

CUDA GPU based full-Stokes finite difference modelling of glaciers

NASA Astrophysics Data System (ADS)

Brædstrup, C. F.; Egholm, D. L.

2012-04-01

Many have stressed the limitations of using the shallow shelf and shallow ice approximations when modelling ice streams or surging glaciers. Using a full-stokes approach requires either large amounts of computer power or time and is therefore seldom an option for most glaciologists. Recent advances in graphics card (GPU) technology for high performance computing have proven extremely efficient in accelerating many large scale scientific computations. The general purpose GPU (GPGPU) technology is cheap, has a low power consumption and fits into a normal desktop computer. It could therefore provide a powerful tool for many glaciologists. Our full-stokes ice sheet model implements a Red-Black Gauss-Seidel iterative linear solver to solve the full stokes equations. This technique has proven very effective when applied to the stokes equation in geodynamics problems, and should therefore also preform well in glaciological flow probems. The Gauss-Seidel iterator is known to be robust but several other linear solvers have a much faster convergence. To aid convergence, the solver uses a multigrid approach where values are interpolated and extrapolated between different grid resolutions to minimize the short wavelength errors efficiently. This reduces the iteration count by several orders of magnitude. The run-time is further reduced by using the GPGPU technology where each card has up to 448 cores. Researchers utilizing the GPGPU technique in other areas have reported between 2 - 11 times speedup compared to multicore CPU implementations on similar problems. The goal of these initial investigations into the possible usage of GPGPU technology in glacial modelling is to apply the enhanced resolution of a full-stokes solver to ice streams and surging glaciers. This is a area of growing interest because ice streams are the main drainage conjugates for large ice sheets. It is therefore crucial to understand this streaming behavior and it's impact up-ice.

A Fast Solver for Implicit Integration of the Vlasov--Poisson System in the Eulerian Framework

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

Garrett, C. Kristopher; Hauck, Cory D.

In this paper, we present a domain decomposition algorithm to accelerate the solution of Eulerian-type discretizations of the linear, steady-state Vlasov equation. The steady-state solver then forms a key component in the implementation of fully implicit or nearly fully implicit temporal integrators for the nonlinear Vlasov--Poisson system. The solver relies on a particular decomposition of phase space that enables the use of sweeping techniques commonly used in radiation transport applications. The original linear system for the phase space unknowns is then replaced by a smaller linear system involving only unknowns on the boundary between subdomains, which can then be solvedmore » efficiently with Krylov methods such as GMRES. Steady-state solves are combined to form an implicit Runge--Kutta time integrator, and the Vlasov equation is coupled self-consistently to the Poisson equation via a linearized procedure or a nonlinear fixed-point method for the electric field. Finally, numerical results for standard test problems demonstrate the efficiency of the domain decomposition approach when compared to the direct application of an iterative solver to the original linear system.« less

SMPBS: Web server for computing biomolecular electrostatics using finite element solvers of size modified Poisson-Boltzmann equation.

PubMed

Xie, Yang; Ying, Jinyong; Xie, Dexuan

2017-03-30

SMPBS (Size Modified Poisson-Boltzmann Solvers) is a web server for computing biomolecular electrostatics using finite element solvers of the size modified Poisson-Boltzmann equation (SMPBE). SMPBE not only reflects ionic size effects but also includes the classic Poisson-Boltzmann equation (PBE) as a special case. Thus, its web server is expected to have a broader range of applications than a PBE web server. SMPBS is designed with a dynamic, mobile-friendly user interface, and features easily accessible help text, asynchronous data submission, and an interactive, hardware-accelerated molecular visualization viewer based on the 3Dmol.js library. In particular, the viewer allows computed electrostatics to be directly mapped onto an irregular triangular mesh of a molecular surface. Due to this functionality and the fast SMPBE finite element solvers, the web server is very efficient in the calculation and visualization of electrostatics. In addition, SMPBE is reconstructed using a new objective electrostatic free energy, clearly showing that the electrostatics and ionic concentrations predicted by SMPBE are optimal in the sense of minimizing the objective electrostatic free energy. SMPBS is available at the URL: smpbs.math.uwm.edu © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.

A dynamic-solver-consistent minimum action method: With an application to 2D Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Wan, Xiaoliang; Yu, Haijun

2017-02-01

This paper discusses the necessity and strategy to unify the development of a dynamic solver and a minimum action method (MAM) for a spatially extended system when employing the large deviation principle (LDP) to study the effects of small random perturbations. A dynamic solver is used to approximate the unperturbed system, and a minimum action method is used to approximate the LDP, which corresponds to solving an Euler-Lagrange equation related to but more complicated than the unperturbed system. We will clarify possible inconsistencies induced by independent numerical approximations of the unperturbed system and the LDP, based on which we propose to define both the dynamic solver and the MAM on the same approximation space for spatial discretization. The semi-discrete LDP can then be regarded as the exact LDP of the semi-discrete unperturbed system, which is a finite-dimensional ODE system. We achieve this methodology for the two-dimensional Navier-Stokes equations using a divergence-free approximation space. The method developed can be used to study the nonlinear instability of wall-bounded parallel shear flows, and be generalized straightforwardly to three-dimensional cases. Numerical experiments are presented.

A Fast Solver for Implicit Integration of the Vlasov--Poisson System in the Eulerian Framework

DOE PAGES

Garrett, C. Kristopher; Hauck, Cory D.

2018-04-05

In this paper, we present a domain decomposition algorithm to accelerate the solution of Eulerian-type discretizations of the linear, steady-state Vlasov equation. The steady-state solver then forms a key component in the implementation of fully implicit or nearly fully implicit temporal integrators for the nonlinear Vlasov--Poisson system. The solver relies on a particular decomposition of phase space that enables the use of sweeping techniques commonly used in radiation transport applications. The original linear system for the phase space unknowns is then replaced by a smaller linear system involving only unknowns on the boundary between subdomains, which can then be solvedmore » efficiently with Krylov methods such as GMRES. Steady-state solves are combined to form an implicit Runge--Kutta time integrator, and the Vlasov equation is coupled self-consistently to the Poisson equation via a linearized procedure or a nonlinear fixed-point method for the electric field. Finally, numerical results for standard test problems demonstrate the efficiency of the domain decomposition approach when compared to the direct application of an iterative solver to the original linear system.« less

A coarse-grid projection method for accelerating incompressible flow computations

NASA Astrophysics Data System (ADS)

San, Omer; Staples, Anne

2011-11-01

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

Development of axisymmetric lattice Boltzmann flux solver for complex multiphase flows

NASA Astrophysics Data System (ADS)

Wang, Yan; Shu, Chang; Yang, Li-Ming; Yuan, Hai-Zhuan

2018-05-01

This paper presents an axisymmetric lattice Boltzmann flux solver (LBFS) for simulating axisymmetric multiphase flows. In the solver, the two-dimensional (2D) multiphase LBFS is applied to reconstruct macroscopic fluxes excluding axisymmetric effects. Source terms accounting for axisymmetric effects are introduced directly into the governing equations. As compared to conventional axisymmetric multiphase lattice Boltzmann (LB) method, the present solver has the kinetic feature for flux evaluation and avoids complex derivations of external forcing terms. In addition, the present solver also saves considerable computational efforts in comparison with three-dimensional (3D) computations. The capability of the proposed solver in simulating complex multiphase flows is demonstrated by studying single bubble rising in a circular tube. The obtained results compare well with the published data.

A higher-order conservation element solution element method for solving hyperbolic differential equations on unstructured meshes

NASA Astrophysics Data System (ADS)

Bilyeu, David

This dissertation presents an extension of the Conservation Element Solution Element (CESE) method from second- to higher-order accuracy. The new method retains the favorable characteristics of the original second-order CESE scheme, including (i) the use of the space-time integral equation for conservation laws, (ii) a compact mesh stencil, (iii) the scheme will remain stable up to a CFL number of unity, (iv) a fully explicit, time-marching integration scheme, (v) true multidimensionality without using directional splitting, and (vi) the ability to handle two- and three-dimensional geometries by using unstructured meshes. This algorithm has been thoroughly tested in one, two and three spatial dimensions and has been shown to obtain the desired order of accuracy for solving both linear and non-linear hyperbolic partial differential equations. The scheme has also shown its ability to accurately resolve discontinuities in the solutions. Higher order unstructured methods such as the Discontinuous Galerkin (DG) method and the Spectral Volume (SV) methods have been developed for one-, two- and three-dimensional application. Although these schemes have seen extensive development and use, certain drawbacks of these methods have been well documented. For example, the explicit versions of these two methods have very stringent stability criteria. This stability criteria requires that the time step be reduced as the order of the solver increases, for a given simulation on a given mesh. The research presented in this dissertation builds upon the work of Chang, who developed a fourth-order CESE scheme to solve a scalar one-dimensional hyperbolic partial differential equation. The completed research has resulted in two key deliverables. The first is a detailed derivation of a high-order CESE methods on unstructured meshes for solving the conservation laws in two- and three-dimensional spaces. The second is the code implementation of these numerical methods in a computer code. For code development, a one-dimensional solver for the Euler equations was developed. This work is an extension of Chang's work on the fourth-order CESE method for solving a one-dimensional scalar convection equation. A generic formulation for the nth-order CESE method, where n ≥ 4, was derived. Indeed, numerical implementation of the scheme confirmed that the order of convergence was consistent with the order of the scheme. For the two- and three-dimensional solvers, SOLVCON was used as the basic framework for code implementation. A new solver kernel for the fourth-order CESE method has been developed and integrated into the framework provided by SOLVCON. The main part of SOLVCON, which deals with unstructured meshes and parallel computing, remains intact. The SOLVCON code for data transmission between computer nodes for High Performance Computing (HPC). To validate and verify the newly developed high-order CESE algorithms, several one-, two- and three-dimensional simulations where conducted. For the arbitrary order, one-dimensional, CESE solver, three sets of governing equations were selected for simulation: (i) the linear convection equation, (ii) the linear acoustic equations, (iii) the nonlinear Euler equations. All three systems of equations were used to verify the order of convergence through mesh refinement. In addition the Euler equations were used to solve the Shu-Osher and Blastwave problems. These two simulations demonstrated that the new high-order CESE methods can accurately resolve discontinuities in the flow field.For the two-dimensional, fourth-order CESE solver, the Euler equation was employed in four different test cases. The first case was used to verify the order of convergence through mesh refinement. The next three cases demonstrated the ability of the new solver to accurately resolve discontinuities in the flows. This was demonstrated through: (i) the interaction between acoustic waves and an entropy pulse, (ii) supersonic flow over a circular blunt body, (iii) supersonic flow over a guttered wedge. To validate and verify the three-dimensional, fourth-order CESE solver, two different simulations where selected. The first used the linear convection equations to demonstrate fourth-order convergence. The second used the Euler equations to simulate supersonic flow over a spherical body to demonstrate the scheme's ability to accurately resolve shocks. All test cases used are well known benchmark problems and as such, there are multiple sources available to validate the numerical results. Furthermore, the simulations showed that the high-order CESE solver was stable at a CFL number near unity.

Solving Coupled Gross--Pitaevskii Equations on a Cluster of PlayStation 3 Computers

NASA Astrophysics Data System (ADS)

Edwards, Mark; Heward, Jeffrey; Clark, C. W.

2009-05-01

At Georgia Southern University we have constructed an 8+1--node cluster of Sony PlayStation 3 (PS3) computers with the intention of using this computing resource to solve problems related to the behavior of ultra--cold atoms in general with a particular emphasis on studying bose--bose and bose--fermi mixtures confined in optical lattices. As a first project that uses this computing resource, we have implemented a parallel solver of the coupled time--dependent, one--dimensional Gross--Pitaevskii (TDGP) equations. These equations govern the behavior of dual-- species bosonic mixtures. We chose the split--operator/FFT to solve the coupled 1D TDGP equations. The fast Fourier transform component of this solver can be readily parallelized on the PS3 cpu known as the Cell Broadband Engine (CellBE). Each CellBE chip contains a single 64--bit PowerPC Processor Element known as the PPE and eight ``Synergistic Processor Element'' identified as the SPE's. We report on this algorithm and compare its performance to a non--parallel solver as applied to modeling evaporative cooling in dual--species bosonic mixtures.

A Conformal, Fully-Conservative Approach for Predicting Blast Effects on Ground Vehicles

DTIC Science & Technology

2014-04-01

time integration  Approximate Riemann Fluxes (HLLE, HLLC) ◦ Robust mixture model for multi-material flows  Multiple Equations of State ◦ Perfect Gas...Loci/CHEM: Chemically reacting compressible flow solver . ◦ Currently in production use by NASA for the simulation of rocket motors, plumes, and...vehicles  Loci/DROPLET: Eulerian and Lagrangian multiphase solvers  Loci/STREAM: pressure-based solver ◦ Developed by Streamline Numerics and

PBEQ-Solver for online visualization of electrostatic potential of biomolecules.

PubMed

Jo, Sunhwan; Vargyas, Miklos; Vasko-Szedlar, Judit; Roux, Benoît; Im, Wonpil

2008-07-01

PBEQ-Solver provides a web-based graphical user interface to read biomolecular structures, solve the Poisson-Boltzmann (PB) equations and interactively visualize the electrostatic potential. PBEQ-Solver calculates (i) electrostatic potential and solvation free energy, (ii) protein-protein (DNA or RNA) electrostatic interaction energy and (iii) pKa of a selected titratable residue. All the calculations can be performed in both aqueous solvent and membrane environments (with a cylindrical pore in the case of membrane). PBEQ-Solver uses the PBEQ module in the biomolecular simulation program CHARMM to solve the finite-difference PB equation of molecules specified by users. Users can interactively inspect the calculated electrostatic potential on the solvent-accessible surface as well as iso-electrostatic potential contours using a novel online visualization tool based on MarvinSpace molecular visualization software, a Java applet integrated within CHARMM-GUI (http://www.charmm-gui.org). To reduce the computational time on the server, and to increase the efficiency in visualization, all the PB calculations are performed with coarse grid spacing (1.5 A before and 1 A after focusing). PBEQ-Solver suggests various physical parameters for PB calculations and users can modify them if necessary. PBEQ-Solver is available at http://www.charmm-gui.org/input/pbeqsolver.

Algebraic Construction of Exact Difference Equations from Symmetry of Equations

NASA Astrophysics Data System (ADS)

Itoh, Toshiaki

2009-09-01

Difference equations or exact numerical integrations, which have general solutions, are treated algebraically. Eliminating the symmetries of the equation, we can construct difference equations (DCE) or numerical integrations equivalent to some ODEs or PDEs that means both have the same solution functions. When arbitrary functions are given, whether we can construct numerical integrations that have solution functions equal to given function or not are treated in this work. Nowadays, Lie's symmetries solver for ODE and PDE has been implemented in many symbolic software. Using this solver we can construct algebraic DCEs or numerical integrations which are correspond to some ODEs or PDEs. In this work, we treated exact correspondence between ODE or PDE and DCE or numerical integration with Gröbner base and Janet base from the view of Lie's symmetries.

A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

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

Pelanti, Marica, E-mail: Marica.Pelanti@ens.f; Bouchut, Francois, E-mail: francois.bouchut@univ-mlv.f; Mangeney, Anne, E-mail: mangeney@ipgp.jussieu.f

2011-02-01

We present a Riemann solver derived by a relaxation technique for classical single-phase shallow flow equations and for a two-phase shallow flow model describing a mixture of solid granular material and fluid. Our primary interest is the numerical approximation of this two-phase solid/fluid model, whose complexity poses numerical difficulties that cannot be efficiently addressed by existing solvers. In particular, we are concerned with ensuring a robust treatment of dry bed states. The relaxation system used by the proposed solver is formulated by introducing auxiliary variables that replace the momenta in the spatial gradients of the original model systems. The resultingmore » relaxation solver is related to Roe solver in that its Riemann solution for the flow height and relaxation variables is formally computed as Roe's Riemann solution. The relaxation solver has the advantage of a certain degree of freedom in the specification of the wave structure through the choice of the relaxation parameters. This flexibility can be exploited to handle robustly vacuum states, which is a well known difficulty of standard Roe's method, while maintaining Roe's low diffusivity. For the single-phase model positivity of flow height is rigorously preserved. For the two-phase model positivity of volume fractions in general is not ensured, and a suitable restriction on the CFL number might be needed. Nonetheless, numerical experiments suggest that the proposed two-phase flow solver efficiently models wet/dry fronts and vacuum formation for a large range of flow conditions. As a corollary of our study, we show that for single-phase shallow flow equations the relaxation solver is formally equivalent to the VFRoe solver with conservative variables of Gallouet and Masella [T. Gallouet, J.-M. Masella, Un schema de Godunov approche C.R. Acad. Sci. Paris, Serie I, 323 (1996) 77-84]. The relaxation interpretation allows establishing positivity conditions for this VFRoe method.« less

An Upwind Solver for the National Combustion Code

NASA Technical Reports Server (NTRS)

Sockol, Peter M.

2011-01-01

An upwind solver is presented for the unstructured grid National Combustion Code (NCC). The compressible Navier-Stokes equations with time-derivative preconditioning and preconditioned flux-difference splitting of the inviscid terms are used. First order derivatives are computed on cell faces and used to evaluate the shear stresses and heat fluxes. A new flux limiter uses these same first order derivatives in the evaluation of left and right states used in the flux-difference splitting. The k-epsilon turbulence equations are solved with the same second-order method. The new solver has been installed in a recent version of NCC and the resulting code has been tested successfully in 2D on two laminar cases with known solutions and one turbulent case with experimental data.

Code Samples Used for Complexity and Control

NASA Astrophysics Data System (ADS)

Ivancevic, Vladimir G.; Reid, Darryn J.

2015-11-01

The following sections are included: * MathematicaⓇ Code * Generic Chaotic Simulator * Vector Differential Operators * NLS Explorer * 2C++ Code * C++ Lambda Functions for Real Calculus * Accelerometer Data Processor * Simple Predictor-Corrector Integrator * Solving the BVP with the Shooting Method * Linear Hyperbolic PDE Solver * Linear Elliptic PDE Solver * Method of Lines for a Set of the NLS Equations * C# Code * Iterative Equation Solver * Simulated Annealing: A Function Minimum * Simple Nonlinear Dynamics * Nonlinear Pendulum Simulator * Lagrangian Dynamics Simulator * Complex-Valued Crowd Attractor Dynamics * Freeform Fortran Code * Lorenz Attractor Simulator * Complex Lorenz Attractor * Simple SGE Soliton * Complex Signal Presentation * Gaussian Wave Packet * Hermitian Matrices * Euclidean L2-Norm * Vector/Matrix Operations * Plain C-Code: Levenberg-Marquardt Optimizer * Free Basic Code: 2D Crowd Dynamics with 3000 Agents

Axisymmetric Powell-Eyring fluid flow over a stretching sheet with a convective boundary condition and suction effects

NASA Astrophysics Data System (ADS)

Nasir, Nor Ain Azeany Mohd; Ishak, Anuar; Pop, Ioan

2018-04-01

In this paper, the heat and mass transfer of an axisymmetric Powell-Eyring fluid flow over a stretching sheet with a convective boundary condition and suction effects are investigated. An appropriate similarity transformation is used to reduce the highly non-linear partial differential equation into second and third order non-linear ordinary differential equations. Numerical solutions of the reduced governing equations are computed numerically by utilizing the MATLAB's built-in boundary value problem solver, bvp4c. The physical significance of various parameters such as Biot number, fluid parameters and Prandtl number on the velocity and temperature evolution profiles are illustrated graphically. The effects of these governing parameters on the skin friction coefficient and the local Nusselt number are also displayed graphically. It is noticed that the Powell-Eyring fluid parameter gives significant influence on the rates of heat and mass transfer of the fluid.

Scaling Relations and Self-Similarity of 3-Dimensional Reynolds-Averaged Navier-Stokes Equations.

PubMed

Ercan, Ali; Kavvas, M Levent

2017-07-25

Scaling conditions to achieve self-similar solutions of 3-Dimensional (3D) Reynolds-Averaged Navier-Stokes Equations, as an initial and boundary value problem, are obtained by utilizing Lie Group of Point Scaling Transformations. By means of an open-source Navier-Stokes solver and the derived self-similarity conditions, we demonstrated self-similarity within the time variation of flow dynamics for a rigid-lid cavity problem under both up-scaled and down-scaled domains. The strength of the proposed approach lies in its ability to consider the underlying flow dynamics through not only from the governing equations under consideration but also from the initial and boundary conditions, hence allowing to obtain perfect self-similarity in different time and space scales. The proposed methodology can be a valuable tool in obtaining self-similar flow dynamics under preferred level of detail, which can be represented by initial and boundary value problems under specific assumptions.

STELLTRANS: A Transport Analysis Suite for Stellarators

NASA Astrophysics Data System (ADS)

Mittelstaedt, Joseph; Lazerson, Samuel; Pablant, Novimir; Weir, Gavin; W7-X Team

2016-10-01

The stellarator transport code STELLTRANS allows us to better analyze the power balance in W7-X. Although profiles of temperature and density are measured experimentally, geometrical factors are needed in conjunction with these measurements to properly analyze heat flux densities in stellarators. The STELLTRANS code interfaces with VMEC to find an equilibrium flux surface configuration and with TRAVIS to determine the RF heating and current drive in the plasma. Stationary transport equations are then considered which are solved using a boundary value differential equation solver. The equations and quantities considered are averaged over flux surfaces to reduce the system to an essentially one dimensional problem. We have applied this code to data from W-7X and were able to calculate the heat flux coefficients. We will also present extensions of the code to a predictive capacity which would utilize DKES to find neoclassical transport coefficients to update the temperature and density profiles.

Progress in high-lift aerodynamic calculations

NASA Technical Reports Server (NTRS)

Rogers, Stuart E.

1993-01-01

The current work presents progress in the effort to numerically simulate the flow over high-lift aerodynamic components, namely, multi-element airfoils and wings in either a take-off or a landing configuration. The computational approach utilizes an incompressible flow solver and an overlaid chimera grid approach. A detailed grid resolution study is presented for flow over a three-element airfoil. Two turbulence models, a one-equation Baldwin-Barth model and a two equation k-omega model are compared. Excellent agreement with experiment is obtained for the lift coefficient at all angles of attack, including the prediction of maximum lift when using the two-equation model. Results for two other flap riggings are shown. Three-dimensional results are presented for a wing with a square wing-tip as a validation case. Grid generation and topology is discussed for computing the flow over a T-39 Sabreliner wing with flap deployed and the initial calculations for this geometry are presented.

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

Weston, Brian T.

This dissertation focuses on the development of a fully-implicit, high-order compressible ow solver with phase change. The work is motivated by laser-induced phase change applications, particularly by the need to develop large-scale multi-physics simulations of the selective laser melting (SLM) process in metal additive manufacturing (3D printing). Simulations of the SLM process require precise tracking of multi-material solid-liquid-gas interfaces, due to laser-induced melting/ solidi cation and evaporation/condensation of metal powder in an ambient gas. These rapid density variations and phase change processes tightly couple the governing equations, requiring a fully compressible framework to robustly capture the rapid density variations ofmore » the ambient gas and the melting/evaporation of the metal powder. For non-isothermal phase change, the velocity is gradually suppressed through the mushy region by a variable viscosity and Darcy source term model. The governing equations are discretized up to 4th-order accuracy with our reconstructed Discontinuous Galerkin spatial discretization scheme and up to 5th-order accuracy with L-stable fully implicit time discretization schemes (BDF2 and ESDIRK3-5). The resulting set of non-linear equations is solved using a robust Newton-Krylov method, with the Jacobian-free version of the GMRES solver for linear iterations. Due to the sti nes associated with the acoustic waves and thermal and viscous/material strength e ects, preconditioning the GMRES solver is essential. A robust and scalable approximate block factorization preconditioner was developed, which utilizes the velocity-pressure (vP) and velocity-temperature (vT) Schur complement systems. This multigrid block reduction preconditioning technique converges for high CFL/Fourier numbers and exhibits excellent parallel and algorithmic scalability on classic benchmark problems in uid dynamics (lid-driven cavity ow and natural convection heat transfer) as well as for laser-induced phase change problems in 2D and 3D.« less

Suite of Benchmark Tests to Conduct Mesh-Convergence Analysis of Nonlinear and Non-constant Coefficient Transport Codes

NASA Astrophysics Data System (ADS)

Zamani, K.; Bombardelli, F. A.

2014-12-01

Verification of geophysics codes is imperative to avoid serious academic as well as practical consequences. In case that access to any given source code is not possible, the Method of Manufactured Solution (MMS) cannot be employed in code verification. In contrast, employing the Method of Exact Solution (MES) has several practical advantages. In this research, we first provide four new one-dimensional analytical solutions designed for code verification; these solutions are able to uncover the particular imperfections of the Advection-diffusion-reaction equation, such as nonlinear advection, diffusion or source terms, as well as non-constant coefficient equations. After that, we provide a solution of Burgers' equation in a novel setup. Proposed solutions satisfy the continuity of mass for the ambient flow, which is a crucial factor for coupled hydrodynamics-transport solvers. Then, we use the derived analytical solutions for code verification. To clarify gray-literature issues in the verification of transport codes, we designed a comprehensive test suite to uncover any imperfection in transport solvers via a hierarchical increase in the level of tests' complexity. The test suite includes hundreds of unit tests and system tests to check vis-a-vis the portions of the code. Examples for checking the suite start by testing a simple case of unidirectional advection; then, bidirectional advection and tidal flow and build up to nonlinear cases. We design tests to check nonlinearity in velocity, dispersivity and reactions. The concealing effect of scales (Peclet and Damkohler numbers) on the mesh-convergence study and appropriate remedies are also discussed. For the cases in which the appropriate benchmarks for mesh convergence study are not available, we utilize symmetry. Auxiliary subroutines for automation of the test suite and report generation are designed. All in all, the test package is not only a robust tool for code verification but it also provides comprehensive insight on the ADR solvers capabilities. Such information is essential for any rigorous computational modeling of ADR equation for surface/subsurface pollution transport. We also convey our experiences in finding several errors which were not detectable with routine verification techniques.

An explicit predictor-corrector solver with applications to Burgers' equation

NASA Technical Reports Server (NTRS)

Dey, S. K.; Dey, C.

1983-01-01

Forward Euler's explicit, finite-difference formula of extrapolation, is used as a predictor and a convex formula as a corrector to integrate differential equations numerically. An application has been made to Burger's equation.

Energy Optimization Modeling of Geothermal Power Plant (Case Study: Darajat Geothermal Field Unit III)

NASA Astrophysics Data System (ADS)

Sinaga, R. H. M.; Darmanto, P. S.

2016-09-01

Darajat unit III geothermal power plant is developed by PT. Chevron Geothermal Indonesia (CGI). The plant capacity is 121 MW and load 110%. The greatest utilization power is consumed by Hot Well Pump (HWP) and Cooling Tower Fan (CTF). Reducing the utility power can be attempted by utilizing the wet bulb temperature fluctuation. In this study, a modelling process is developed by using Engineering Equation Solver (EES) software version 9.430.The possibility of energy saving is indicated by Specific Steam Consumption (SSC) net in relation to wet bulb temperature fluctuation from 9°C up to 20.5°C. Result shows that the existing daily operation reaches its optimum condition. The installation of Variable Frequency Drive (VFD) could be applied to optimize both utility power of HWP and CTF. The highest gain is obtained by VFD HWP installation as much as 0.80% when wet bulb temperature 18.5 °C.

A fast Cauchy-Riemann solver. [differential equation solution for boundary conditions by finite difference approximation

NASA Technical Reports Server (NTRS)

Ghil, M.; Balgovind, R.

1979-01-01

The inhomogeneous Cauchy-Riemann equations in a rectangle are discretized by a finite difference approximation. Several different boundary conditions are treated explicitly, leading to algorithms which have overall second-order accuracy. All boundary conditions with either u or v prescribed along a side of the rectangle can be treated by similar methods. The algorithms presented here have nearly minimal time and storage requirements and seem suitable for development into a general-purpose direct Cauchy-Riemann solver for arbitrary boundary conditions.

QED multi-dimensional vacuum polarization finite-difference solver

NASA Astrophysics Data System (ADS)

Carneiro, Pedro; Grismayer, Thomas; Silva, Luís; Fonseca, Ricardo

2015-11-01

The Extreme Light Infrastructure (ELI) is expected to deliver peak intensities of 1023 - 1024 W/cm2 allowing to probe nonlinear Quantum Electrodynamics (QED) phenomena in an unprecedented regime. Within the framework of QED, the second order process of photon-photon scattering leads to a set of extended Maxwell's equations [W. Heisenberg and H. Euler, Z. Physik 98, 714] effectively creating nonlinear polarization and magnetization terms that account for the nonlinear response of the vacuum. To model this in a self-consistent way, we present a multi dimensional generalized Maxwell equation finite difference solver with significantly enhanced dispersive properties, which was implemented in the OSIRIS particle-in-cell code [R.A. Fonseca et al. LNCS 2331, pp. 342-351, 2002]. We present a detailed numerical analysis of this electromagnetic solver. As an illustration of the properties of the solver, we explore several examples in extreme conditions. We confirm the theoretical prediction of vacuum birefringence of a pulse propagating in the presence of an intense static background field [arXiv:1301.4918 [quant-ph

A multigrid solver for the semiconductor equations

NASA Technical Reports Server (NTRS)

Bachmann, Bernhard

1993-01-01

We present a multigrid solver for the exponential fitting method. The solver is applied to the current continuity equations of semiconductor device simulation in two dimensions. The exponential fitting method is based on a mixed finite element discretization using the lowest-order Raviart-Thomas triangular element. This discretization method yields a good approximation of front layers and guarantees current conservation. The corresponding stiffness matrix is an M-matrix. 'Standard' multigrid solvers, however, cannot be applied to the resulting system, as this is dominated by an unsymmetric part, which is due to the presence of strong convection in part of the domain. To overcome this difficulty, we explore the connection between Raviart-Thomas mixed methods and the nonconforming Crouzeix-Raviart finite element discretization. In this way we can construct nonstandard prolongation and restriction operators using easily computable weighted L(exp 2)-projections based on suitable quadrature rules and the upwind effects of the discretization. The resulting multigrid algorithm shows very good results, even for real-world problems and for locally refined grids.

RELATIVISTIC MAGNETOHYDRODYNAMICS: RENORMALIZED EIGENVECTORS AND FULL WAVE DECOMPOSITION RIEMANN SOLVER

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

Anton, Luis; MartI, Jose M; Ibanez, Jose M

2010-05-01

We obtain renormalized sets of right and left eigenvectors of the flux vector Jacobians of the relativistic MHD equations, which are regular and span a complete basis in any physical state including degenerate ones. The renormalization procedure relies on the characterization of the degeneracy types in terms of the normal and tangential components of the magnetic field to the wave front in the fluid rest frame. Proper expressions of the renormalized eigenvectors in conserved variables are obtained through the corresponding matrix transformations. Our work completes previous analysis that present different sets of right eigenvectors for non-degenerate and degenerate states, andmore » can be seen as a relativistic generalization of earlier work performed in classical MHD. Based on the full wave decomposition (FWD) provided by the renormalized set of eigenvectors in conserved variables, we have also developed a linearized (Roe-type) Riemann solver. Extensive testing against one- and two-dimensional standard numerical problems allows us to conclude that our solver is very robust. When compared with a family of simpler solvers that avoid the knowledge of the full characteristic structure of the equations in the computation of the numerical fluxes, our solver turns out to be less diffusive than HLL and HLLC, and comparable in accuracy to the HLLD solver. The amount of operations needed by the FWD solver makes it less efficient computationally than those of the HLL family in one-dimensional problems. However, its relative efficiency increases in multidimensional simulations.« less

An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension)

NASA Technical Reports Server (NTRS)

Powell, Kenneth G.

1994-01-01

An approximate Riemann solver is developed for the governing equations of ideal magnetohydrodynamics (MHD). The Riemann solver has an eight-wave structure, where seven of the waves are those used in previous work on upwind schemes for MHD, and the eighth wave is related to the divergence of the magnetic field. The structure of the eighth wave is not immediately obvious from the governing equations as they are usually written, but arises from a modification of the equations that is presented in this paper. The addition of the eighth wave allows multidimensional MHD problems to be solved without the use of staggered grids or a projection scheme, one or the other of which was necessary in previous work on upwind schemes for MHD. A test problem made up of a shock tube with rotated initial conditions is solved to show that the two-dimensional code yields answers consistent with the one-dimensional methods developed previously.

SIERRA - A 3-D device simulator for reliability modeling

NASA Astrophysics Data System (ADS)

Chern, Jue-Hsien; Arledge, Lawrence A., Jr.; Yang, Ping; Maeda, John T.

1989-05-01

SIERRA is a three-dimensional general-purpose semiconductor-device simulation program which serves as a foundation for investigating integrated-circuit (IC) device and reliability issues. This program solves the Poisson and continuity equations in silicon under dc, transient, and small-signal conditions. Executing on a vector/parallel minisupercomputer, SIERRA utilizes a matrix solver which uses an incomplete LU (ILU) preconditioned conjugate gradient square (CGS, BCG) method. The ILU-CGS method provides a good compromise between memory size and convergence rate. The authors have observed a 5x to 7x speedup over standard direct methods in simulations of transient problems containing highly coupled Poisson and continuity equations such as those found in reliability-oriented simulations. The application of SIERRA to parasitic CMOS latchup and dynamic random-access memory single-event-upset studies is described.

A GPU-based incompressible Navier-Stokes solver on moving overset grids

NASA Astrophysics Data System (ADS)

Chandar, Dominic D. J.; Sitaraman, Jayanarayanan; Mavriplis, Dimitri J.

2013-07-01

In pursuit of obtaining high fidelity solutions to the fluid flow equations in a short span of time, graphics processing units (GPUs) which were originally intended for gaming applications are currently being used to accelerate computational fluid dynamics (CFD) codes. With a high peak throughput of about 1 TFLOPS on a PC, GPUs seem to be favourable for many high-resolution computations. One such computation that involves a lot of number crunching is computing time accurate flow solutions past moving bodies. The aim of the present paper is thus to discuss the development of a flow solver on unstructured and overset grids and its implementation on GPUs. In its present form, the flow solver solves the incompressible fluid flow equations on unstructured/hybrid/overset grids using a fully implicit projection method. The resulting discretised equations are solved using a matrix-free Krylov solver using several GPU kernels such as gradient, Laplacian and reduction. Some of the simple arithmetic vector calculations are implemented using the CU++: An Object Oriented Framework for Computational Fluid Dynamics Applications using Graphics Processing Units, Journal of Supercomputing, 2013, doi:10.1007/s11227-013-0985-9 approach where GPU kernels are automatically generated at compile time. Results are presented for two- and three-dimensional computations on static and moving grids.

A Robust and Efficient Method for Steady State Patterns in Reaction-Diffusion Systems

PubMed Central

Lo, Wing-Cheong; Chen, Long; Wang, Ming; Nie, Qing

2012-01-01

An inhomogeneous steady state pattern of nonlinear reaction-diffusion equations with no-flux boundary conditions is usually computed by solving the corresponding time-dependent reaction-diffusion equations using temporal schemes. Nonlinear solvers (e.g., Newton’s method) take less CPU time in direct computation for the steady state; however, their convergence is sensitive to the initial guess, often leading to divergence or convergence to spatially homogeneous solution. Systematically numerical exploration of spatial patterns of reaction-diffusion equations under different parameter regimes requires that the numerical method be efficient and robust to initial condition or initial guess, with better likelihood of convergence to an inhomogeneous pattern. Here, a new approach that combines the advantages of temporal schemes in robustness and Newton’s method in fast convergence in solving steady states of reaction-diffusion equations is proposed. In particular, an adaptive implicit Euler with inexact solver (AIIE) method is found to be much more efficient than temporal schemes and more robust in convergence than typical nonlinear solvers (e.g., Newton’s method) in finding the inhomogeneous pattern. Application of this new approach to two reaction-diffusion equations in one, two, and three spatial dimensions, along with direct comparisons to several other existing methods, demonstrates that AIIE is a more desirable method for searching inhomogeneous spatial patterns of reaction-diffusion equations in a large parameter space. PMID:22773849

THOR: an open-source exo-GCM

NASA Astrophysics Data System (ADS)

Grosheintz, Luc; Mendonça, João; Käppeli, Roger; Lukas Grimm, Simon; Mishra, Siddhartha; Heng, Kevin

2015-12-01

In this talk, I will present THOR, the first fully conservative, GPU-accelerated exo-GCM (general circulation model) on a nearly uniform, global grid that treats shocks and is non-hydrostatic. THOR will be freely available to the community as a standard tool.Unlike most GCMs THOR solves the full, non-hydrostatic Euler equations instead of the primitive equations. The equations are solved on a global three-dimensional icosahedral grid by a second order Finite Volume Method (FVM). Icosahedral grids are nearly uniform refinements of an icosahedron. We've implemented three different versions of this grid. FVM conserves the prognostic variables (density, momentum and energy) exactly and doesn't require a diffusion term (artificial viscosity) in the Euler equations to stabilize our solver. Historically FVM was designed to treat discontinuities correctly. Hence it excels at resolving shocks, including those present in hot exoplanetary atmospheres.Atmospheres are generally in near hydrostatic equilibrium. We therefore implement a well-balancing technique recently developed at the ETH Zurich. This well-balancing ensures that our FVM maintains hydrostatic equilibrium to machine precision. Better yet, it is able to resolve pressure perturbations from this equilibrium as small as one part in 100'000. It is important to realize that these perturbations are significantly smaller than the truncation error of the same scheme without well-balancing. If during the course of the simulation (due to forcing) the atmosphere becomes non-hydrostatic, our solver continues to function correctly.THOR just passed an important mile stone. We've implemented the explicit part of the solver. The explicit solver is useful to study instabilities or local problems on relatively short time scales. I'll show some nice properties of the explicit THOR. An explicit solver is not appropriate for climate study because the time step is limited by the sound speed. Therefore, we are working on the first fully implicit GCM. By ESS3, I hope to present results for the advection equation.THOR is part of the Exoclimes Simulation Platform (ESP), a set of open-source community codes for simulating and understanding the atmospheres of exoplanets. The ESP also includes tools for radiative transfer and retrieval (HELIOS), an opacity calculator (HELIOS-K), and a chemical kinetics solver (VULCAN). We expect to publicly release an initial version of THOR in 2016 on www.exoclime.org.

A finite-volume HLLC-based scheme for compressible interfacial flows with surface tension

NASA Astrophysics Data System (ADS)

Garrick, Daniel P.; Owkes, Mark; Regele, Jonathan D.

2017-06-01

Shock waves are often used in experiments to create a shear flow across liquid droplets to study secondary atomization. Similar behavior occurs inside of supersonic combustors (scramjets) under startup conditions, but it is challenging to study these conditions experimentally. In order to investigate this phenomenon further, a numerical approach is developed to simulate compressible multiphase flows under the effects of surface tension forces. The flow field is solved via the compressible multicomponent Euler equations (i.e., the five equation model) discretized with the finite volume method on a uniform Cartesian grid. The solver utilizes a total variation diminishing (TVD) third-order Runge-Kutta method for time-marching and second order TVD spatial reconstruction. Surface tension is incorporated using the Continuum Surface Force (CSF) model. Fluxes are upwinded with a modified Harten-Lax-van Leer Contact (HLLC) approximate Riemann solver. An interface compression scheme is employed to counter numerical diffusion of the interface. The present work includes modifications to both the HLLC solver and the interface compression scheme to account for capillary force terms and the associated pressure jump across the gas-liquid interface. A simple method for numerically computing the interface curvature is developed and an acoustic scaling of the surface tension coefficient is proposed for the non-dimensionalization of the model. The model captures the surface tension induced pressure jump exactly if the exact curvature is known and is further verified with an oscillating elliptical droplet and Mach 1.47 and 3 shock-droplet interaction problems. The general characteristics of secondary atomization at a range of Weber numbers are also captured in a series of simulations.

A finite-volume HLLC-based scheme for compressible interfacial flows with surface tension

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

Garrick, Daniel P.; Owkes, Mark; Regele, Jonathan D., E-mail: jregele@iastate.edu

Shock waves are often used in experiments to create a shear flow across liquid droplets to study secondary atomization. Similar behavior occurs inside of supersonic combustors (scramjets) under startup conditions, but it is challenging to study these conditions experimentally. In order to investigate this phenomenon further, a numerical approach is developed to simulate compressible multiphase flows under the effects of surface tension forces. The flow field is solved via the compressible multicomponent Euler equations (i.e., the five equation model) discretized with the finite volume method on a uniform Cartesian grid. The solver utilizes a total variation diminishing (TVD) third-order Runge–Kuttamore » method for time-marching and second order TVD spatial reconstruction. Surface tension is incorporated using the Continuum Surface Force (CSF) model. Fluxes are upwinded with a modified Harten–Lax–van Leer Contact (HLLC) approximate Riemann solver. An interface compression scheme is employed to counter numerical diffusion of the interface. The present work includes modifications to both the HLLC solver and the interface compression scheme to account for capillary force terms and the associated pressure jump across the gas–liquid interface. A simple method for numerically computing the interface curvature is developed and an acoustic scaling of the surface tension coefficient is proposed for the non-dimensionalization of the model. The model captures the surface tension induced pressure jump exactly if the exact curvature is known and is further verified with an oscillating elliptical droplet and Mach 1.47 and 3 shock-droplet interaction problems. The general characteristics of secondary atomization at a range of Weber numbers are also captured in a series of simulations.« less

Scalable High Performance Computing: Direct and Large-Eddy Turbulent Flow Simulations Using Massively Parallel Computers

NASA Technical Reports Server (NTRS)

Morgan, Philip E.

2004-01-01

This final report contains reports of research related to the tasks "Scalable High Performance Computing: Direct and Lark-Eddy Turbulent FLow Simulations Using Massively Parallel Computers" and "Devleop High-Performance Time-Domain Computational Electromagnetics Capability for RCS Prediction, Wave Propagation in Dispersive Media, and Dual-Use Applications. The discussion of Scalable High Performance Computing reports on three objectives: validate, access scalability, and apply two parallel flow solvers for three-dimensional Navier-Stokes flows; develop and validate a high-order parallel solver for Direct Numerical Simulations (DNS) and Large Eddy Simulation (LES) problems; and Investigate and develop a high-order Reynolds averaged Navier-Stokes turbulence model. The discussion of High-Performance Time-Domain Computational Electromagnetics reports on five objectives: enhancement of an electromagnetics code (CHARGE) to be able to effectively model antenna problems; utilize lessons learned in high-order/spectral solution of swirling 3D jets to apply to solving electromagnetics project; transition a high-order fluids code, FDL3DI, to be able to solve Maxwell's Equations using compact-differencing; develop and demonstrate improved radiation absorbing boundary conditions for high-order CEM; and extend high-order CEM solver to address variable material properties. The report also contains a review of work done by the systems engineer.

DL_MG: A Parallel Multigrid Poisson and Poisson-Boltzmann Solver for Electronic Structure Calculations in Vacuum and Solution.

PubMed

Womack, James C; Anton, Lucian; Dziedzic, Jacek; Hasnip, Phil J; Probert, Matt I J; Skylaris, Chris-Kriton

2018-03-13

The solution of the Poisson equation is a crucial step in electronic structure calculations, yielding the electrostatic potential-a key component of the quantum mechanical Hamiltonian. In recent decades, theoretical advances and increases in computer performance have made it possible to simulate the electronic structure of extended systems in complex environments. This requires the solution of more complicated variants of the Poisson equation, featuring nonhomogeneous dielectric permittivities, ionic concentrations with nonlinear dependencies, and diverse boundary conditions. The analytic solutions generally used to solve the Poisson equation in vacuum (or with homogeneous permittivity) are not applicable in these circumstances, and numerical methods must be used. In this work, we present DL_MG, a flexible, scalable, and accurate solver library, developed specifically to tackle the challenges of solving the Poisson equation in modern large-scale electronic structure calculations on parallel computers. Our solver is based on the multigrid approach and uses an iterative high-order defect correction method to improve the accuracy of solutions. Using two chemically relevant model systems, we tested the accuracy and computational performance of DL_MG when solving the generalized Poisson and Poisson-Boltzmann equations, demonstrating excellent agreement with analytic solutions and efficient scaling to ∼10 9 unknowns and 100s of CPU cores. We also applied DL_MG in actual large-scale electronic structure calculations, using the ONETEP linear-scaling electronic structure package to study a 2615 atom protein-ligand complex with routinely available computational resources. In these calculations, the overall execution time with DL_MG was not significantly greater than the time required for calculations using a conventional FFT-based solver.

Algorithms and Application of Sparse Matrix Assembly and Equation Solvers for Aeroacoustics

NASA Technical Reports Server (NTRS)

Watson, W. R.; Nguyen, D. T.; Reddy, C. J.; Vatsa, V. N.; Tang, W. H.

2001-01-01

An algorithm for symmetric sparse equation solutions on an unstructured grid is described. Efficient, sequential sparse algorithms for degree-of-freedom reordering, supernodes, symbolic/numerical factorization, and forward backward solution phases are reviewed. Three sparse algorithms for the generation and assembly of symmetric systems of matrix equations are presented. The accuracy and numerical performance of the sequential version of the sparse algorithms are evaluated over the frequency range of interest in a three-dimensional aeroacoustics application. Results show that the solver solutions are accurate using a discretization of 12 points per wavelength. Results also show that the first assembly algorithm is impractical for high-frequency noise calculations. The second and third assembly algorithms have nearly equal performance at low values of source frequencies, but at higher values of source frequencies the third algorithm saves CPU time and RAM. The CPU time and the RAM required by the second and third assembly algorithms are two orders of magnitude smaller than that required by the sparse equation solver. A sequential version of these sparse algorithms can, therefore, be conveniently incorporated into a substructuring for domain decomposition formulation to achieve parallel computation, where different substructures are handles by different parallel processors.

SSE-based Thomas algorithm for quasi-block-tridiagonal linear equation systems, optimized for small dense blocks

NASA Astrophysics Data System (ADS)

Barnaś, Dawid; Bieniasz, Lesław K.

2017-07-01

We have recently developed a vectorized Thomas solver for quasi-block tridiagonal linear algebraic equation systems using Streaming SIMD Extensions (SSE) and Advanced Vector Extensions (AVX) in operations on dense blocks [D. Barnaś and L. K. Bieniasz, Int. J. Comput. Meth., accepted]. The acceleration caused by vectorization was observed for large block sizes, but was less satisfactory for small blocks. In this communication we report on another version of the solver, optimized for small blocks of size up to four rows and/or columns.

Multidisciplinary Simulation Acceleration using Multiple Shared-Memory Graphical Processing Units

NASA Astrophysics Data System (ADS)

Kemal, Jonathan Yashar

For purposes of optimizing and analyzing turbomachinery and other designs, the unsteady Favre-averaged flow-field differential equations for an ideal compressible gas can be solved in conjunction with the heat conduction equation. We solve all equations using the finite-volume multiple-grid numerical technique, with the dual time-step scheme used for unsteady simulations. Our numerical solver code targets CUDA-capable Graphical Processing Units (GPUs) produced by NVIDIA. Making use of MPI, our solver can run across networked compute notes, where each MPI process can use either a GPU or a Central Processing Unit (CPU) core for primary solver calculations. We use NVIDIA Tesla C2050/C2070 GPUs based on the Fermi architecture, and compare our resulting performance against Intel Zeon X5690 CPUs. Solver routines converted to CUDA typically run about 10 times faster on a GPU for sufficiently dense computational grids. We used a conjugate cylinder computational grid and ran a turbulent steady flow simulation using 4 increasingly dense computational grids. Our densest computational grid is divided into 13 blocks each containing 1033x1033 grid points, for a total of 13.87 million grid points or 1.07 million grid points per domain block. To obtain overall speedups, we compare the execution time of the solver's iteration loop, including all resource intensive GPU-related memory copies. Comparing the performance of 8 GPUs to that of 8 CPUs, we obtain an overall speedup of about 6.0 when using our densest computational grid. This amounts to an 8-GPU simulation running about 39.5 times faster than running than a single-CPU simulation.

Extension of lattice Boltzmann flux solver for simulation of compressible multi-component flows

NASA Astrophysics Data System (ADS)

Yang, Li-Ming; Shu, Chang; Yang, Wen-Ming; Wang, Yan

2018-05-01

The lattice Boltzmann flux solver (LBFS), which was presented by Shu and his coworkers for solving compressible fluid flow problems, is extended to simulate compressible multi-component flows in this work. To solve the two-phase gas-liquid problems, the model equations with stiffened gas equation of state are adopted. In this model, two additional non-conservative equations are introduced to represent the material interfaces, apart from the classical Euler equations. We first convert the interface equations into the full conservative form by applying the mass equation. After that, we calculate the numerical fluxes of the classical Euler equations by the existing LBFS and the numerical fluxes of the interface equations by the passive scalar approach. Once all the numerical fluxes at the cell interface are obtained, the conservative variables at cell centers can be updated by marching the equations in time and the material interfaces can be identified via the distributions of the additional variables. The numerical accuracy and stability of present scheme are validated by its application to several compressible multi-component fluid flow problems.

Multiscale solvers and systematic upscaling in computational physics

NASA Astrophysics Data System (ADS)

Brandt, A.

2005-07-01

Multiscale algorithms can overcome the scale-born bottlenecks that plague most computations in physics. These algorithms employ separate processing at each scale of the physical space, combined with interscale iterative interactions, in ways which use finer scales very sparingly. Having been developed first and well known as multigrid solvers for partial differential equations, highly efficient multiscale techniques have more recently been developed for many other types of computational tasks, including: inverse PDE problems; highly indefinite (e.g., standing wave) equations; Dirac equations in disordered gauge fields; fast computation and updating of large determinants (as needed in QCD); fast integral transforms; integral equations; astrophysics; molecular dynamics of macromolecules and fluids; many-atom electronic structures; global and discrete-state optimization; practical graph problems; image segmentation and recognition; tomography (medical imaging); fast Monte-Carlo sampling in statistical physics; and general, systematic methods of upscaling (accurate numerical derivation of large-scale equations from microscopic laws).

An AMR capable finite element diffusion solver for ALE hydrocodes [An AMR capable diffusion solver for ALE-AMR

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

Fisher, A. C.; Bailey, D. S.; Kaiser, T. B.

2015-02-01

Here, we present a novel method for the solution of the diffusion equation on a composite AMR mesh. This approach is suitable for including diffusion based physics modules to hydrocodes that support ALE and AMR capabilities. To illustrate, we proffer our implementations of diffusion based radiation transport and heat conduction in a hydrocode called ALE-AMR. Numerical experiments conducted with the diffusion solver and associated physics packages yield 2nd order convergence in the L 2 norm.

Convergence Acceleration of a Navier-Stokes Solver for Efficient Static Aeroelastic Computations

NASA Technical Reports Server (NTRS)

Obayashi, Shigeru; Guruswamy, Guru P.

1995-01-01

New capabilities have been developed for a Navier-Stokes solver to perform steady-state simulations more efficiently. The flow solver for solving the Navier-Stokes equations is based on a combination of the lower-upper factored symmetric Gauss-Seidel implicit method and the modified Harten-Lax-van Leer-Einfeldt upwind scheme. A numerically stable and efficient pseudo-time-marching method is also developed for computing steady flows over flexible wings. Results are demonstrated for transonic flows over rigid and flexible wings.

A positivity preserving and conservative variational scheme for phase-field modeling of two-phase flows

NASA Astrophysics Data System (ADS)

Joshi, Vaibhav; Jaiman, Rajeev K.

2018-05-01

We present a positivity preserving variational scheme for the phase-field modeling of incompressible two-phase flows with high density ratio. The variational finite element technique relies on the Allen-Cahn phase-field equation for capturing the phase interface on a fixed Eulerian mesh with mass conservative and energy-stable discretization. The mass conservation is achieved by enforcing a Lagrange multiplier which has both temporal and spatial dependence on the underlying solution of the phase-field equation. To make the scheme energy-stable in a variational sense, we discretize the spatial part of the Lagrange multiplier in the phase-field equation by the mid-point approximation. The proposed variational technique is designed to reduce the spurious and unphysical oscillations in the solution while maintaining the second-order accuracy of both spatial and temporal discretizations. We integrate the Allen-Cahn phase-field equation with the incompressible Navier-Stokes equations for modeling a broad range of two-phase flow and fluid-fluid interface problems. The coupling of the implicit discretizations corresponding to the phase-field and the incompressible flow equations is achieved via nonlinear partitioned iterative procedure. Comparison of results between the standard linear stabilized finite element method and the present variational formulation shows a remarkable reduction of oscillations in the solution while retaining the boundedness of the phase-indicator field. We perform a standalone test to verify the accuracy and stability of the Allen-Cahn two-phase solver. We examine the convergence and accuracy properties of the coupled phase-field solver through the standard benchmarks of the Laplace-Young law and a sloshing tank problem. Two- and three-dimensional dam break problems are simulated to assess the capability of the phase-field solver for complex air-water interfaces involving topological changes on unstructured meshes. Finally, we demonstrate the phase-field solver for a practical offshore engineering application of wave-structure interaction.

Implementation of the Jacobian-free Newton-Krylov method for solving the for solving the first-order ice sheet momentum balance

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

Salinger, Andy; Evans, Katherine J; Lemieux, Jean-Francois

2011-01-01

We have implemented the Jacobian-free Newton-Krylov (JFNK) method for solving the rst-order ice sheet momentum equation in order to improve the numerical performance of the Community Ice Sheet Model (CISM), the land ice component of the Community Earth System Model (CESM). Our JFNK implementation is based on signicant re-use of existing code. For example, our physics-based preconditioner uses the original Picard linear solver in CISM. For several test cases spanning a range of geometries and boundary conditions, our JFNK implementation is 1.84-3.62 times more efficient than the standard Picard solver in CISM. Importantly, this computational gain of JFNK over themore » Picard solver increases when rening the grid. Global convergence of the JFNK solver has been signicantly improved by rescaling the equation for the basal boundary condition and through the use of an inexact Newton method. While a diverse set of test cases show that our JFNK implementation is usually robust, for some problems it may fail to converge with increasing resolution (as does the Picard solver). Globalization through parameter continuation did not remedy this problem and future work to improve robustness will explore a combination of Picard and JFNK and the use of homotopy methods.« less

Large-Eddy/Reynolds-Averaged Navier-Stokes Simulation of Shock-Train Development in a Coil-Laser Diffuser

DTIC Science & Technology

2014-09-06

as the Riemann solver . The primitive-variable vector Ts kTwvupW ],,,,,,[ ω= is used in the reconstruction. The initial step in the PPM...University’s (NCSU) REACTMB flow solver is used in the present effort. REACTMB solves the Navier-Stokes equations governing a multi-component

The Mixed Finite Element Multigrid Method for Stokes Equations

PubMed Central

Muzhinji, K.; Shateyi, S.; Motsa, S. S.

2015-01-01

The stable finite element discretization of the Stokes problem produces a symmetric indefinite system of linear algebraic equations. A variety of iterative solvers have been proposed for such systems in an attempt to construct efficient, fast, and robust solution techniques. This paper investigates one of such iterative solvers, the geometric multigrid solver, to find the approximate solution of the indefinite systems. The main ingredient of the multigrid method is the choice of an appropriate smoothing strategy. This study considers the application of different smoothers and compares their effects in the overall performance of the multigrid solver. We study the multigrid method with the following smoothers: distributed Gauss Seidel, inexact Uzawa, preconditioned MINRES, and Braess-Sarazin type smoothers. A comparative study of the smoothers shows that the Braess-Sarazin smoothers enhance good performance of the multigrid method. We study the problem in a two-dimensional domain using stable Hood-Taylor Q 2-Q 1 pair of finite rectangular elements. We also give the main theoretical convergence results. We present the numerical results to demonstrate the efficiency and robustness of the multigrid method and confirm the theoretical results. PMID:25945361

Performance of a parallel thermal-hydraulics code TEMPEST

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

Fann, G.I.; Trent, D.S.

The authors describe the parallelization of the Tempest thermal-hydraulics code. The serial version of this code is used for production quality 3-D thermal-hydraulics simulations. Good speedup was obtained with a parallel diagonally preconditioned BiCGStab non-symmetric linear solver, using a spatial domain decomposition approach for the semi-iterative pressure-based and mass-conserved algorithm. The test case used here to illustrate the performance of the BiCGStab solver is a 3-D natural convection problem modeled using finite volume discretization in cylindrical coordinates. The BiCGStab solver replaced the LSOR-ADI method for solving the pressure equation in TEMPEST. BiCGStab also solves the coupled thermal energy equation. Scalingmore » performance of 3 problem sizes (221220 nodes, 358120 nodes, and 701220 nodes) are presented. These problems were run on 2 different parallel machines: IBM-SP and SGI PowerChallenge. The largest problem attains a speedup of 68 on an 128 processor IBM-SP. In real terms, this is over 34 times faster than the fastest serial production time using the LSOR-ADI solver.« less

A universal concept based on cellular neural networks for ultrafast and flexible solving of differential equations.

PubMed

Chedjou, Jean Chamberlain; Kyamakya, Kyandoghere

2015-04-01

This paper develops and validates a comprehensive and universally applicable computational concept for solving nonlinear differential equations (NDEs) through a neurocomputing concept based on cellular neural networks (CNNs). High-precision, stability, convergence, and lowest-possible memory requirements are ensured by the CNN processor architecture. A significant challenge solved in this paper is that all these cited computing features are ensured in all system-states (regular or chaotic ones) and in all bifurcation conditions that may be experienced by NDEs.One particular quintessence of this paper is to develop and demonstrate a solver concept that shows and ensures that CNN processors (realized either in hardware or in software) are universal solvers of NDE models. The solving logic or algorithm of given NDEs (possible examples are: Duffing, Mathieu, Van der Pol, Jerk, Chua, Rössler, Lorenz, Burgers, and the transport equations) through a CNN processor system is provided by a set of templates that are computed by our comprehensive templates calculation technique that we call nonlinear adaptive optimization. This paper is therefore a significant contribution and represents a cutting-edge real-time computational engineering approach, especially while considering the various scientific and engineering applications of this ultrafast, energy-and-memory-efficient, and high-precise NDE solver concept. For illustration purposes, three NDE models are demonstratively solved, and related CNN templates are derived and used: the periodically excited Duffing equation, the Mathieu equation, and the transport equation.

A systematic approach to numerical dispersion in Maxwell solvers

NASA Astrophysics Data System (ADS)

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

2018-03-01

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

Scalable smoothing strategies for a geometric multigrid method for the immersed boundary equations

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

Bhalla, Amneet Pal Singh; Knepley, Matthew G.; Adams, Mark F.

2016-12-20

The immersed boundary (IB) method is a widely used approach to simulating fluid-structure interaction (FSI). Although explicit versions of the IB method can suffer from severe time step size restrictions, these methods remain popular because of their simplicity and generality. In prior work (Guy et al., Adv Comput Math, 2015), some of us developed a geometric multigrid preconditioner for a stable semi-implicit IB method under Stokes flow conditions; however, this solver methodology used a Vanka-type smoother that presented limited opportunities for parallelization. This work extends this Stokes-IB solver methodology by developing smoothing techniques that are suitable for parallel implementation. Specifically,more » we demonstrate that an additive version of the Vanka smoother can yield an effective multigrid preconditioner for the Stokes-IB equations, and we introduce an efficient Schur complement-based smoother that is also shown to be effective for the Stokes-IB equations. We investigate the performance of these solvers for a broad range of material stiffnesses, both for Stokes flows and flows at nonzero Reynolds numbers, and for thick and thin structural models. We show here that linear solver performance degrades with increasing Reynolds number and material stiffness, especially for thin interface cases. Nonetheless, the proposed approaches promise to yield effective solution algorithms, especially at lower Reynolds numbers and at modest-to-high elastic stiffnesses.« less

A New LES/PDF Method for Computational Modeling of Turbulent Reacting Flows

NASA Astrophysics Data System (ADS)

Turkeri, Hasret; Muradoglu, Metin; Pope, Stephen B.

2013-11-01

A new LES/PDF method is developed for computational modeling of turbulent reacting flows. The open source package, OpenFOAM, is adopted as the LES solver and combined with the particle-based Monte Carlo method to solve the LES/PDF model equations. The dynamic Smagorinsky model is employed to account for the subgrid-scale motions. The LES solver is first validated for the Sandia Flame D using a steady flamelet method in which the chemical compositions, density and temperature fields are parameterized by the mean mixture fraction and its variance. In this approach, the modeled transport equations for the mean mixture fraction and the square of the mixture fraction are solved and the variance is then computed from its definition. The results are found to be in a good agreement with the experimental data. Then the LES solver is combined with the particle-based Monte Carlo algorithm to form a complete solver for the LES/PDF model equations. The in situ adaptive tabulation (ISAT) algorithm is incorporated into the LES/PDF method for efficient implementation of detailed chemical kinetics. The LES/PDF method is also applied to the Sandia Flame D using the GRI-Mech 3.0 chemical mechanism and the results are compared with the experimental data and the earlier PDF simulations. The Scientific and Technical Research Council of Turkey (TUBITAK), Grant No. 111M067.

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

NASA Technical Reports Server (NTRS)

Sockol, Peter M.

1993-01-01

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

Shock-capturing parabolized Navier-Stokes model /SCIPVIS/ for the analysis of turbulent underexpanded jets

NASA Technical Reports Server (NTRS)

Dash, S. M.; Wolf, D. E.

1983-01-01

A new computational model, SCIPVIS, has been developed to predict the multiple-cell wave/shock structure in under or over-expanded turbulent jets. SCIPVIS solves the parabolized Navier-Stokes jet mixing equations utilizing a shock-capturing approach in supersonic regions of the jet and a pressure-split approach in subsonic regions. Turbulence processes are represented by the solution of compressibility corrected two-equation turbulence models. The formation of Mach discs in the jet and the interactive turbulent mixing process occurring behind the disc are handled in a detailed fashion. SCIPVIS presently analyzes jets exhausting into a quiescent or supersonic external stream for which a single-pass spatial marching solution can be obtained. The iterative coupling of SCIPVIS with a potential flow solver for the analysis of subsonic/transonic external streams is under development.

Implementation of Advanced Two Equation Turbulence Models in the USM3D Unstructured Flow Solver

NASA Technical Reports Server (NTRS)

Wang, Qun-Zhen; Massey, Steven J.; Abdol-Hamid, Khaled S.

2000-01-01

USM3D is a widely-used unstructured flow solver for simulating inviscid and viscous flows over complex geometries. The current version (version 5.0) of USM3D, however, does not have advanced turbulence models to accurately simulate complicated flow. We have implemented two modified versions of the original Jones and Launder k-epsilon "two-equation" turbulence model and the Girimaji algebraic Reynolds stress model in USM3D. Tests have been conducted for three flat plate boundary layer cases, a RAE2822 airfoil and an ONERA M6 wing. The results are compared with those from direct numerical simulation, empirical formulae, theoretical results, and the existing Spalart-Allmaras one-equation model.

Hierarchically partitioned nonlinear equation solvers

NASA Technical Reports Server (NTRS)

Padovan, Joseph

1987-01-01

By partitioning solution space into a number of subspaces, a new multiply constrained partitioned Newton-Raphson nonlinear equation solver is developed. Specifically, for a given iteration, each of the various separate partitions are individually and simultaneously controlled. Due to the generality of the scheme, a hierarchy of partition levels can be employed. For finite-element-type applications, this includes the possibility of degree-of-freedom, nodal, elemental, geometric substructural, material and kinematically nonlinear group controls. It is noted that such partitioning can be continuously updated, depending on solution conditioning. In this context, convergence is ascertained at the individual partition level.

Development of high-accuracy convection schemes for sequential solvers

NASA Technical Reports Server (NTRS)

Thakur, Siddharth; Shyy, Wei

1993-01-01

An exploration is conducted of the applicability of such high resolution schemes as TVD to the resolving of sharp flow gradients using a sequential solution approach borrowed from pressure-based algorithms. It is shown that by extending these high-resolution shock-capturing schemes to a sequential solver that treats the equations as a collection of scalar conservation equations, the speed of signal propagation in the solution has to be coordinated by assigning the local convection speed as the characteristic speed for the entire system. A higher amount of dissipation is therefore needed to eliminate oscillations near discontinuities.

A comparison of turbulence models in computing multi-element airfoil flows

NASA Technical Reports Server (NTRS)

Rogers, Stuart E.; Menter, Florian; Durbin, Paul A.; Mansour, Nagi N.

1994-01-01

Four different turbulence models are used to compute the flow over a three-element airfoil configuration. These models are the one-equation Baldwin-Barth model, the one-equation Spalart-Allmaras model, a two-equation k-omega model, and a new one-equation Durbin-Mansour model. The flow is computed using the INS2D two-dimensional incompressible Navier-Stokes solver. An overset Chimera grid approach is utilized. Grid resolution tests are presented, and manual solution-adaptation of the grid was performed. The performance of each of the models is evaluated for test cases involving different angles-of-attack, Reynolds numbers, and flap riggings. The resulting surface pressure coefficients, skin friction, velocity profiles, and lift, drag, and moment coefficients are compared with experimental data. The models produce very similar results in most cases. Excellent agreement between computational and experimental surface pressures was observed, but only moderately good agreement was seen in the velocity profile data. In general, the difference between the predictions of the different models was less than the difference between the computational and experimental data.

Mathematical and Numerical Aspects of the Adaptive Fast Multipole Poisson-Boltzmann Solver

DOE PAGES

Zhang, Bo; Lu, Benzhuo; Cheng, Xiaolin; ...

2013-01-01

This paper summarizes the mathematical and numerical theories and computational elements of the adaptive fast multipole Poisson-Boltzmann (AFMPB) solver. We introduce and discuss the following components in order: the Poisson-Boltzmann model, boundary integral equation reformulation, surface mesh generation, the nodepatch discretization approach, Krylov iterative methods, the new version of fast multipole methods (FMMs), and a dynamic prioritization technique for scheduling parallel operations. For each component, we also remark on feasible approaches for further improvements in efficiency, accuracy and applicability of the AFMPB solver to large-scale long-time molecular dynamics simulations. Lastly, the potential of the solver is demonstrated with preliminary numericalmore » results.« less

Computationally efficient simulation of unsteady aerodynamics using POD on the fly

NASA Astrophysics Data System (ADS)

Moreno-Ramos, Ruben; Vega, José M.; Varas, Fernando

2016-12-01

Modern industrial aircraft design requires a large amount of sufficiently accurate aerodynamic and aeroelastic simulations. Current computational fluid dynamics (CFD) solvers with aeroelastic capabilities, such as the NASA URANS unstructured solver FUN3D, require very large computational resources. Since a very large amount of simulation is necessary, the CFD cost is just unaffordable in an industrial production environment and must be significantly reduced. Thus, a more inexpensive, yet sufficiently precise solver is strongly needed. An opportunity to approach this goal could follow some recent results (Terragni and Vega 2014 SIAM J. Appl. Dyn. Syst. 13 330-65 Rapun et al 2015 Int. J. Numer. Meth. Eng. 104 844-68) on an adaptive reduced order model that combines ‘on the fly’ a standard numerical solver (to compute some representative snapshots), proper orthogonal decomposition (POD) (to extract modes from the snapshots), Galerkin projection (onto the set of POD modes), and several additional ingredients such as projecting the equations using a limited amount of points and fairly generic mode libraries. When applied to the complex Ginzburg-Landau equation, the method produces acceleration factors (comparing with standard numerical solvers) of the order of 20 and 300 in one and two space dimensions, respectively. Unfortunately, the extension of the method to unsteady, compressible flows around deformable geometries requires new approaches to deal with deformable meshes, high-Reynolds numbers, and compressibility. A first step in this direction is presented considering the unsteady compressible, two-dimensional flow around an oscillating airfoil using a CFD solver in a rigidly moving mesh. POD on the Fly gives results whose accuracy is comparable to that of the CFD solver used to compute the snapshots.

Parallel Cartesian grid refinement for 3D complex flow simulations

NASA Astrophysics Data System (ADS)

Angelidis, Dionysios; Sotiropoulos, Fotis

2013-11-01

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

A SEMI-LAGRANGIAN TWO-LEVEL PRECONDITIONED NEWTON-KRYLOV SOLVER FOR CONSTRAINED DIFFEOMORPHIC IMAGE REGISTRATION.

PubMed

Mang, Andreas; Biros, George

2017-01-01

We propose an efficient numerical algorithm for the solution of diffeomorphic image registration problems. We use a variational formulation constrained by a partial differential equation (PDE), where the constraints are a scalar transport equation. We use a pseudospectral discretization in space and second-order accurate semi-Lagrangian time stepping scheme for the transport equations. We solve for a stationary velocity field using a preconditioned, globalized, matrix-free Newton-Krylov scheme. We propose and test a two-level Hessian preconditioner. We consider two strategies for inverting the preconditioner on the coarse grid: a nested preconditioned conjugate gradient method (exact solve) and a nested Chebyshev iterative method (inexact solve) with a fixed number of iterations. We test the performance of our solver in different synthetic and real-world two-dimensional application scenarios. We study grid convergence and computational efficiency of our new scheme. We compare the performance of our solver against our initial implementation that uses the same spatial discretization but a standard, explicit, second-order Runge-Kutta scheme for the numerical time integration of the transport equations and a single-level preconditioner. Our improved scheme delivers significant speedups over our original implementation. As a highlight, we observe a 20 × speedup for a two dimensional, real world multi-subject medical image registration problem.

A Nonlinear Modal Aeroelastic Solver for FUN3D

NASA Technical Reports Server (NTRS)

Goldman, Benjamin D.; Bartels, Robert E.; Biedron, Robert T.; Scott, Robert C.

2016-01-01

A nonlinear structural solver has been implemented internally within the NASA FUN3D computational fluid dynamics code, allowing for some new aeroelastic capabilities. Using a modal representation of the structure, a set of differential or differential-algebraic equations are derived for general thin structures with geometric nonlinearities. ODEPACK and LAPACK routines are linked with FUN3D, and the nonlinear equations are solved at each CFD time step. The existing predictor-corrector method is retained, whereby the structural solution is updated after mesh deformation. The nonlinear solver is validated using a test case for a flexible aeroshell at transonic, supersonic, and hypersonic flow conditions. Agreement with linear theory is seen for the static aeroelastic solutions at relatively low dynamic pressures, but structural nonlinearities limit deformation amplitudes at high dynamic pressures. No flutter was found at any of the tested trajectory points, though LCO may be possible in the transonic regime.

Advanced computational simulations of water waves interacting with wave energy converters

NASA Astrophysics Data System (ADS)

Pathak, Ashish; Freniere, Cole; Raessi, Mehdi

2017-03-01

Wave energy converter (WEC) devices harness the renewable ocean wave energy and convert it into useful forms of energy, e.g. mechanical or electrical. This paper presents an advanced 3D computational framework to study the interaction between water waves and WEC devices. The computational tool solves the full Navier-Stokes equations and considers all important effects impacting the device performance. To enable large-scale simulations in fast turnaround times, the computational solver was developed in an MPI parallel framework. A fast multigrid preconditioned solver is introduced to solve the computationally expensive pressure Poisson equation. The computational solver was applied to two surface-piercing WEC geometries: bottom-hinged cylinder and flap. Their numerically simulated response was validated against experimental data. Additional simulations were conducted to investigate the applicability of Froude scaling in predicting full-scale WEC response from the model experiments.

Parallel Computation of the Jacobian Matrix for Nonlinear Equation Solvers Using MATLAB

NASA Technical Reports Server (NTRS)

Rose, Geoffrey K.; Nguyen, Duc T.; Newman, Brett A.

2017-01-01

Demonstrating speedup for parallel code on a multicore shared memory PC can be challenging in MATLAB due to underlying parallel operations that are often opaque to the user. This can limit potential for improvement of serial code even for the so-called embarrassingly parallel applications. One such application is the computation of the Jacobian matrix inherent to most nonlinear equation solvers. Computation of this matrix represents the primary bottleneck in nonlinear solver speed such that commercial finite element (FE) and multi-body-dynamic (MBD) codes attempt to minimize computations. A timing study using MATLAB's Parallel Computing Toolbox was performed for numerical computation of the Jacobian. Several approaches for implementing parallel code were investigated while only the single program multiple data (spmd) method using composite objects provided positive results. Parallel code speedup is demonstrated but the goal of linear speedup through the addition of processors was not achieved due to PC architecture.

Verification and Validation Studies for the LAVA CFD Solver

NASA Technical Reports Server (NTRS)

Moini-Yekta, Shayan; Barad, Michael F; Sozer, Emre; Brehm, Christoph; Housman, Jeffrey A.; Kiris, Cetin C.

2013-01-01

The verification and validation of the Launch Ascent and Vehicle Aerodynamics (LAVA) computational fluid dynamics (CFD) solver is presented. A modern strategy for verification and validation is described incorporating verification tests, validation benchmarks, continuous integration and version control methods for automated testing in a collaborative development environment. The purpose of the approach is to integrate the verification and validation process into the development of the solver and improve productivity. This paper uses the Method of Manufactured Solutions (MMS) for the verification of 2D Euler equations, 3D Navier-Stokes equations as well as turbulence models. A method for systematic refinement of unstructured grids is also presented. Verification using inviscid vortex propagation and flow over a flat plate is highlighted. Simulation results using laminar and turbulent flow past a NACA 0012 airfoil and ONERA M6 wing are validated against experimental and numerical data.

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

Dumbser, Michael, E-mail: michael.dumbser@unitn.it; Balsara, Dinshaw S., E-mail: dbalsara@nd.edu

In this paper a new, simple and universal formulation of the HLLEM Riemann solver (RS) is proposed that works for general conservative and non-conservative systems of hyperbolic equations. For non-conservative PDE, a path-conservative formulation of the HLLEM RS is presented for the first time in this paper. The HLLEM Riemann solver is built on top of a novel and very robust path-conservative HLL method. It thus naturally inherits the positivity properties and the entropy enforcement of the underlying HLL scheme. However, with just the slight additional cost of evaluating eigenvectors and eigenvalues of intermediate characteristic fields, we can represent linearlymore » degenerate intermediate waves with a minimum of smearing. For conservative systems, our paper provides the easiest and most seamless path for taking a pre-existing HLL RS and quickly and effortlessly converting it to a RS that provides improved results, comparable with those of an HLLC, HLLD, Osher or Roe-type RS. This is done with minimal additional computational complexity, making our variant of the HLLEM RS also a very fast RS that can accurately represent linearly degenerate discontinuities. Our present HLLEM RS also transparently extends these advantages to non-conservative systems. For shallow water-type systems, the resulting method is proven to be well-balanced. Several test problems are presented for shallow water-type equations and two-phase flow models, as well as for gas dynamics with real equation of state, magnetohydrodynamics (MHD & RMHD), and nonlinear elasticity. Since our new formulation accommodates multiple intermediate waves and has a broader applicability than the original HLLEM method, it could alternatively be called the HLLI Riemann solver, where the “I” stands for the intermediate characteristic fields that can be accounted for. -- Highlights: •New simple and general path-conservative formulation of the HLLEM Riemann solver. •Application to general conservative and non-conservative hyperbolic systems. •Inclusion of sub-structure and resolution of intermediate characteristic fields. •Well-balanced for single- and two-layer shallow water equations and multi-phase flows. •Euler equations with real equation of state, MHD equations, nonlinear elasticity.« less

Turbulence modeling of free shear layers for high-performance aircraft

NASA Technical Reports Server (NTRS)

Sondak, Douglas L.

1993-01-01

The High Performance Aircraft (HPA) Grand Challenge of the High Performance Computing and Communications (HPCC) program involves the computation of the flow over a high performance aircraft. A variety of free shear layers, including mixing layers over cavities, impinging jets, blown flaps, and exhaust plumes, may be encountered in such flowfields. Since these free shear layers are usually turbulent, appropriate turbulence models must be utilized in computations in order to accurately simulate these flow features. The HPCC program is relying heavily on parallel computers. A Navier-Stokes solver (POVERFLOW) utilizing the Baldwin-Lomax algebraic turbulence model was developed and tested on a 128-node Intel iPSC/860. Algebraic turbulence models run very fast, and give good results for many flowfields. For complex flowfields such as those mentioned above, however, they are often inadequate. It was therefore deemed that a two-equation turbulence model will be required for the HPA computations. The k-epsilon two-equation turbulence model was implemented on the Intel iPSC/860. Both the Chien low-Reynolds-number model and a generalized wall-function formulation were included.

Nearly Interactive Parabolized Navier-Stokes Solver for High Speed Forebody and Inlet Flows

NASA Technical Reports Server (NTRS)

Benson, Thomas J.; Liou, May-Fun; Jones, William H.; Trefny, Charles J.

2009-01-01

A system of computer programs is being developed for the preliminary design of high speed inlets and forebodies. The system comprises four functions: geometry definition, flow grid generation, flow solver, and graphics post-processor. The system runs on a dedicated personal computer using the Windows operating system and is controlled by graphical user interfaces written in MATLAB (The Mathworks, Inc.). The flow solver uses the Parabolized Navier-Stokes equations to compute millions of mesh points in several minutes. Sample two-dimensional and three-dimensional calculations are demonstrated in the paper.

Improved numerical methods for turbulent viscous recirculating flows

NASA Technical Reports Server (NTRS)

Turan, A.; Vandoormaal, J. P.

1988-01-01

The performance of discrete methods for the prediction of fluid flows can be enhanced by improving the convergence rate of solvers and by increasing the accuracy of the discrete representation of the equations of motion. This report evaluates the gains in solver performance that are available when various acceleration methods are applied. Various discretizations are also examined and two are recommended because of their accuracy and robustness. Insertion of the improved discretization and solver accelerator into a TEACH mode, that has been widely applied to combustor flows, illustrates the substantial gains to be achieved.

GSRP/David Marshall: Fully Automated Cartesian Grid CFD Application for MDO in High Speed Flows

NASA Technical Reports Server (NTRS)

2003-01-01

With the renewed interest in Cartesian gridding methodologies for the ease and speed of gridding complex geometries in addition to the simplicity of the control volumes used in the computations, it has become important to investigate ways of extending the existing Cartesian grid solver functionalities. This includes developing methods of modeling the viscous effects in order to utilize Cartesian grids solvers for accurate drag predictions and addressing the issues related to the distributed memory parallelization of Cartesian solvers. This research presents advances in two areas of interest in Cartesian grid solvers, viscous effects modeling and MPI parallelization. The development of viscous effects modeling using solely Cartesian grids has been hampered by the widely varying control volume sizes associated with the mesh refinement and the cut cells associated with the solid surface. This problem is being addressed by using physically based modeling techniques to update the state vectors of the cut cells and removing them from the finite volume integration scheme. This work is performed on a new Cartesian grid solver, NASCART-GT, with modifications to its cut cell functionality. The development of MPI parallelization addresses issues associated with utilizing Cartesian solvers on distributed memory parallel environments. This work is performed on an existing Cartesian grid solver, CART3D, with modifications to its parallelization methodology.

An Equation of State for Foamed Divinylbenzene (DVB) Based on Multi-Shock Response

NASA Astrophysics Data System (ADS)

Aslam, Tariq; Schroen, Diana; Gustavsen, Richard; Bartram, Brian

2013-06-01

The methodology for making foamed Divinylbenzene (DVB) is described. For a variety of initial densities, foamed DVB is examined through multi-shock compression and release experiments. Results from multi-shock experiments on LANL's 2-stage gas gun will be presented. A simple conservative Lagrangian numerical scheme, utilizing total-variation-diminishing interpolation and an approximate Riemann solver, will be presented as well as the methodology of calibration. It has been previously demonstrated that a single Mie-Gruneisen fitting form can replicate foam multi-shock compression response at a variety of initial densities; such a methodology will be presented for foamed DVB.

Discrete sensitivity derivatives of the Navier-Stokes equations with a parallel Krylov solver

NASA Technical Reports Server (NTRS)

Ajmani, Kumud; Taylor, Arthur C., III

1994-01-01

This paper solves an 'incremental' form of the sensitivity equations derived by differentiating the discretized thin-layer Navier Stokes equations with respect to certain design variables of interest. The equations are solved with a parallel, preconditioned Generalized Minimal RESidual (GMRES) solver on a distributed-memory architecture. The 'serial' sensitivity analysis code is parallelized by using the Single Program Multiple Data (SPMD) programming model, domain decomposition techniques, and message-passing tools. Sensitivity derivatives are computed for low and high Reynolds number flows over a NACA 1406 airfoil on a 32-processor Intel Hypercube, and found to be identical to those computed on a single-processor Cray Y-MP. It is estimated that the parallel sensitivity analysis code has to be run on 40-50 processors of the Intel Hypercube in order to match the single-processor processing time of a Cray Y-MP.

NAS Experiences of Porting CM Fortran Codes to HPF on IBM SP2 and SGI Power Challenge

NASA Technical Reports Server (NTRS)

Saini, Subhash

1995-01-01

Current Connection Machine (CM) Fortran codes developed for the CM-2 and the CM-5 represent an important class of parallel applications. Several users have employed CM Fortran codes in production mode on the CM-2 and the CM-5 for the last five to six years, constituting a heavy investment in terms of cost and time. With Thinking Machines Corporation's decision to withdraw from the hardware business and with the decommissioning of many CM-2 and CM-5 machines, the best way to protect the substantial investment in CM Fortran codes is to port the codes to High Performance Fortran (HPF) on highly parallel systems. HPF is very similar to CM Fortran and thus represents a natural transition. Conversion issues involved in porting CM Fortran codes on the CM-5 to HPF are presented. In particular, the differences between data distribution directives and the CM Fortran Utility Routines Library, as well as the equivalent functionality in the HPF Library are discussed. Several CM Fortran codes (Cannon algorithm for matrix-matrix multiplication, Linear solver Ax=b, 1-D convolution for 2-D datasets, Laplace's Equation solver, and Direct Simulation Monte Carlo (DSMC) codes have been ported to Subset HPF on the IBM SP2 and the SGI Power Challenge. Speedup ratios versus number of processors for the Linear solver and DSMC code are presented.

Shallow-water sloshing in a moving vessel with variable cross-section and wetting-drying using an extension of George's well-balanced finite volume solver

NASA Astrophysics Data System (ADS)

Alemi Ardakani, Hamid; Bridges, Thomas J.; Turner, Matthew R.

2016-06-01

A class of augmented approximate Riemann solvers due to George (2008) [12] is extended to solve the shallow-water equations in a moving vessel with variable bottom topography and variable cross-section with wetting and drying. A class of Roe-type upwind solvers for the system of balance laws is derived which respects the steady-state solutions. The numerical solutions of the new adapted augmented f-wave solvers are validated against the Roe-type solvers. The theory is extended to solve the shallow-water flows in moving vessels with arbitrary cross-section with influx-efflux boundary conditions motivated by the shallow-water sloshing in the ocean wave energy converter (WEC) proposed by Offshore Wave Energy Ltd. (OWEL) [1]. A fractional step approach is used to handle the time-dependent forcing functions. The numerical solutions are compared to an extended new Roe-type solver for the system of balance laws with a time-dependent source function. The shallow-water sloshing finite volume solver can be coupled to a Runge-Kutta integrator for the vessel motion.

A Flexible CUDA LU-based Solver for Small, Batched Linear Systems

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

Tumeo, Antonino; Gawande, Nitin A.; Villa, Oreste

This chapter presents the implementation of a batched CUDA solver based on LU factorization for small linear systems. This solver may be used in applications such as reactive flow transport models, which apply the Newton-Raphson technique to linearize and iteratively solve the sets of non linear equations that represent the reactions for ten of thousands to millions of physical locations. The implementation exploits somewhat counterintuitive GPGPU programming techniques: it assigns the solution of a matrix (representing a system) to a single CUDA thread, does not exploit shared memory and employs dynamic memory allocation on the GPUs. These techniques enable ourmore » implementation to simultaneously solve sets of systems with over 100 equations and to employ LU decomposition with complete pivoting, providing the higher numerical accuracy required by certain applications. Other currently available solutions for batched linear solvers are limited by size and only support partial pivoting, although they may result faster in certain conditions. We discuss the code of our implementation and present a comparison with the other implementations, discussing the various tradeoffs in terms of performance and flexibility. This work will enable developers that need batched linear solvers to choose whichever implementation is more appropriate to the features and the requirements of their applications, and even to implement dynamic switching approaches that can choose the best implementation depending on the input data.« less

Acceleration of FDTD mode solver by high-performance computing techniques.

PubMed

Han, Lin; Xi, Yanping; Huang, Wei-Ping

2010-06-21

A two-dimensional (2D) compact finite-difference time-domain (FDTD) mode solver is developed based on wave equation formalism in combination with the matrix pencil method (MPM). The method is validated for calculation of both real guided and complex leaky modes of typical optical waveguides against the bench-mark finite-difference (FD) eigen mode solver. By taking advantage of the inherent parallel nature of the FDTD algorithm, the mode solver is implemented on graphics processing units (GPUs) using the compute unified device architecture (CUDA). It is demonstrated that the high-performance computing technique leads to significant acceleration of the FDTD mode solver with more than 30 times improvement in computational efficiency in comparison with the conventional FDTD mode solver running on CPU of a standard desktop computer. The computational efficiency of the accelerated FDTD method is in the same order of magnitude of the standard finite-difference eigen mode solver and yet require much less memory (e.g., less than 10%). Therefore, the new method may serve as an efficient, accurate and robust tool for mode calculation of optical waveguides even when the conventional eigen value mode solvers are no longer applicable due to memory limitation.

MIBPB: a software package for electrostatic analysis.

PubMed

Chen, Duan; Chen, Zhan; Chen, Changjun; Geng, Weihua; Wei, Guo-Wei

2011-03-01

The Poisson-Boltzmann equation (PBE) is an established model for the electrostatic analysis of biomolecules. The development of advanced computational techniques for the solution of the PBE has been an important topic in the past two decades. This article presents a matched interface and boundary (MIB)-based PBE software package, the MIBPB solver, for electrostatic analysis. The MIBPB has a unique feature that it is the first interface technique-based PBE solver that rigorously enforces the solution and flux continuity conditions at the dielectric interface between the biomolecule and the solvent. For protein molecular surfaces, which may possess troublesome geometrical singularities, the MIB scheme makes the MIBPB by far the only existing PBE solver that is able to deliver the second-order convergence, that is, the accuracy increases four times when the mesh size is halved. The MIBPB method is also equipped with a Dirichlet-to-Neumann mapping technique that builds a Green's function approach to analytically resolve the singular charge distribution in biomolecules in order to obtain reliable solutions at meshes as coarse as 1 Å--whereas it usually takes other traditional PB solvers 0.25 Å to reach similar level of reliability. This work further accelerates the rate of convergence of linear equation systems resulting from the MIBPB by using the Krylov subspace (KS) techniques. Condition numbers of the MIBPB matrices are significantly reduced by using appropriate KS solver and preconditioner combinations. Both linear and nonlinear PBE solvers in the MIBPB package are tested by protein-solvent solvation energy calculations and analysis of salt effects on protein-protein binding energies, respectively. Copyright © 2010 Wiley Periodicals, Inc.

Development and Verification of the Charring Ablating Thermal Protection Implicit System Solver

NASA Technical Reports Server (NTRS)

Amar, Adam J.; Calvert, Nathan D.; Kirk, Benjamin S.

2010-01-01

The development and verification of the Charring Ablating Thermal Protection Implicit System Solver is presented. This work concentrates on the derivation and verification of the stationary grid terms in the equations that govern three-dimensional heat and mass transfer for charring thermal protection systems including pyrolysis gas flow through the porous char layer. The governing equations are discretized according to the Galerkin finite element method with first and second order implicit time integrators. The governing equations are fully coupled and are solved in parallel via Newton's method, while the fully implicit linear system is solved with the Generalized Minimal Residual method. Verification results from exact solutions and the Method of Manufactured Solutions are presented to show spatial and temporal orders of accuracy as well as nonlinear convergence rates.

An Approximate Axisymmetric Viscous Shock Layer Aeroheating Method for Three-Dimensional Bodies

NASA Technical Reports Server (NTRS)

Brykina, Irina G.; Scott, Carl D.

1998-01-01

A technique is implemented for computing hypersonic aeroheating, shear stress, and other flow properties on the windward side of a three-dimensional (3D) blunt body. The technique uses a 2D/axisymmetric flow solver modified by scale factors for a, corresponding equivalent axisymmetric body. Examples are given in which a 2D solver is used to calculate the flow at selected meridional planes on elliptic paraboloids in reentry flight. The report describes the equations and the codes used to convert the body surface parameters into input used to scale the 2D viscous shock layer equations in the axisymmetric viscous shock layer code. Very good agreement is obtained with solutions to finite rate chemistry 3D thin viscous shock layer equations for a finite rate catalytic body.

An Optimized Multicolor Point-Implicit Solver for Unstructured Grid Applications on Graphics Processing Units

NASA Technical Reports Server (NTRS)

Zubair, Mohammad; Nielsen, Eric; Luitjens, Justin; Hammond, Dana

2016-01-01

In the field of computational fluid dynamics, the Navier-Stokes equations are often solved using an unstructuredgrid approach to accommodate geometric complexity. Implicit solution methodologies for such spatial discretizations generally require frequent solution of large tightly-coupled systems of block-sparse linear equations. The multicolor point-implicit solver used in the current work typically requires a significant fraction of the overall application run time. In this work, an efficient implementation of the solver for graphics processing units is proposed. Several factors present unique challenges to achieving an efficient implementation in this environment. These include the variable amount of parallelism available in different kernel calls, indirect memory access patterns, low arithmetic intensity, and the requirement to support variable block sizes. In this work, the solver is reformulated to use standard sparse and dense Basic Linear Algebra Subprograms (BLAS) functions. However, numerical experiments show that the performance of the BLAS functions available in existing CUDA libraries is suboptimal for matrices representative of those encountered in actual simulations. Instead, optimized versions of these functions are developed. Depending on block size, the new implementations show performance gains of up to 7x over the existing CUDA library functions.

Viscous Driven-Cavity Solver: User's Manual

NASA Technical Reports Server (NTRS)

Wood, William A.

1997-01-01

The viscous driven-cavity problem is solved using a stream-function and vorticity formulation for the incompressible Navier-Stokes equations. This report provides the user's manual and FORTRAN code for the set of governing equations presented in NASA TM-110262.

Conjugate Compressible Fluid Flow and Heat Transfer in Ducts

NASA Technical Reports Server (NTRS)

Cross, M. F.

2011-01-01

A computational approach to modeling transient, compressible fluid flow with heat transfer in long, narrow ducts is presented. The primary application of the model is for analyzing fluid flow and heat transfer in solid propellant rocket motor nozzle joints during motor start-up, but the approach is relevant to a wide range of analyses involving rapid pressurization and filling of ducts. Fluid flow is modeled through solution of the spatially one-dimensional, transient Euler equations. Source terms are included in the governing equations to account for the effects of wall friction and heat transfer. The equation solver is fully-implicit, thus providing greater flexibility than an explicit solver. This approach allows for resolution of pressure wave effects on the flow as well as for fast calculation of the steady-state solution when a quasi-steady approach is sufficient. Solution of the one-dimensional Euler equations with source terms significantly reduces computational run times compared to general purpose computational fluid dynamics packages solving the Navier-Stokes equations with resolved boundary layers. In addition, conjugate heat transfer is more readily implemented using the approach described in this paper than with most general purpose computational fluid dynamics packages. The compressible flow code has been integrated with a transient heat transfer solver to analyze heat transfer between the fluid and surrounding structure. Conjugate fluid flow and heat transfer solutions are presented. The author is unaware of any previous work available in the open literature which uses the same approach described in this paper.

Gauss-Seidel Iterative Method as a Real-Time Pile-Up Solver of Scintillation Pulses

NASA Astrophysics Data System (ADS)

Novak, Roman; Vencelj, Matja¿

2009-12-01

The pile-up rejection in nuclear spectroscopy has been confronted recently by several pile-up correction schemes that compensate for distortions of the signal and subsequent energy spectra artifacts as the counting rate increases. We study here a real-time capability of the event-by-event correction method, which at the core translates to solving many sets of linear equations. Tight time limits and constrained front-end electronics resources make well-known direct solvers inappropriate. We propose a novel approach based on the Gauss-Seidel iterative method, which turns out to be a stable and cost-efficient solution to improve spectroscopic resolution in the front-end electronics. We show the method convergence properties for a class of matrices that emerge in calorimetric processing of scintillation detector signals and demonstrate the ability of the method to support the relevant resolutions. The sole iteration-based error component can be brought below the sliding window induced errors in a reasonable number of iteration steps, thus allowing real-time operation. An area-efficient hardware implementation is proposed that fully utilizes the method's inherent parallelism.

User's Guide for ENSAERO_FE Parallel Finite Element Solver

NASA Technical Reports Server (NTRS)

Eldred, Lloyd B.; Guruswamy, Guru P.

1999-01-01

A high fidelity parallel static structural analysis capability is created and interfaced to the multidisciplinary analysis package ENSAERO-MPI of Ames Research Center. This new module replaces ENSAERO's lower fidelity simple finite element and modal modules. Full aircraft structures may be more accurately modeled using the new finite element capability. Parallel computation is performed by breaking the full structure into multiple substructures. This approach is conceptually similar to ENSAERO's multizonal fluid analysis capability. The new substructure code is used to solve the structural finite element equations for each substructure in parallel. NASTRANKOSMIC is utilized as a front end for this code. Its full library of elements can be used to create an accurate and realistic aircraft model. It is used to create the stiffness matrices for each substructure. The new parallel code then uses an iterative preconditioned conjugate gradient method to solve the global structural equations for the substructure boundary nodes.

Conical Euler solution for a highly-swept delta wing undergoing wing-rock motion

NASA Technical Reports Server (NTRS)

Lee, Elizabeth M.; Batina, John T.

1990-01-01

Modifications to an unsteady conical Euler code for the free-to-roll analysis of highly-swept delta wings are described. The modifications involve the addition of the rolling rigid-body equation of motion for its simultaneous time-integration with the governing flow equations. The flow solver utilized in the Euler code includes a multistage Runge-Kutta time-stepping scheme which uses a finite-volume spatial discretization on an unstructured mesh made up of triangles. Steady and unsteady results are presented for a 75 deg swept delta wing at a freestream Mach number of 1.2 and an angle of attack of 30 deg. The unsteady results consist of forced harmonic and free-to-roll calculations. The free-to-roll case exhibits a wing rock response produced by unsteady aerodynamics consistent with the aerodynamics of the forced harmonic results. Similarities are shown with a wing-rock time history from a low-speed wind tunnel test.

On Riemann solvers and kinetic relations for isothermal two-phase flows with surface tension

NASA Astrophysics Data System (ADS)

Rohde, Christian; Zeiler, Christoph

2018-06-01

We consider a sharp interface approach for the inviscid isothermal dynamics of compressible two-phase flow that accounts for phase transition and surface tension effects. Kinetic relations are frequently used to fix the mass exchange and entropy dissipation rate across the interface. The complete unidirectional dynamics can then be understood by solving generalized two-phase Riemann problems. We present new well-posedness theorems for the Riemann problem and corresponding computable Riemann solvers that cover quite general equations of state, metastable input data and curvature effects. The new Riemann solver is used to validate different kinetic relations on physically relevant problems including a comparison with experimental data. Riemann solvers are building blocks for many numerical schemes that are used to track interfaces in two-phase flow. It is shown that the new Riemann solver enables reliable and efficient computations for physical situations that could not be treated before.

A semi-direct procedure using a local relaxation factor and its application to an internal flow problem

NASA Technical Reports Server (NTRS)

Chang, S. C.

1984-01-01

Generally, fast direct solvers are not directly applicable to a nonseparable elliptic partial differential equation. This limitation, however, is circumvented by a semi-direct procedure, i.e., an iterative procedure using fast direct solvers. An efficient semi-direct procedure which is easy to implement and applicable to a variety of boundary conditions is presented. The current procedure also possesses other highly desirable properties, i.e.: (1) the convergence rate does not decrease with an increase of grid cell aspect ratio, and (2) the convergence rate is estimated using the coefficients of the partial differential equation being solved.

Some fast elliptic solvers on parallel architectures and their complexities

NASA Technical Reports Server (NTRS)

Gallopoulos, E.; Saad, Y.

1989-01-01

The discretization of separable elliptic partial differential equations leads to linear systems with special block tridiagonal matrices. Several methods are known to solve these systems, the most general of which is the Block Cyclic Reduction (BCR) algorithm which handles equations with nonconstant coefficients. A method was recently proposed to parallelize and vectorize BCR. In this paper, the mapping of BCR on distributed memory architectures is discussed, and its complexity is compared with that of other approaches including the Alternating-Direction method. A fast parallel solver is also described, based on an explicit formula for the solution, which has parallel computational compelxity lower than that of parallel BCR.

Some fast elliptic solvers on parallel architectures and their complexities

NASA Technical Reports Server (NTRS)

Gallopoulos, E.; Saad, Youcef

1989-01-01

The discretization of separable elliptic partial differential equations leads to linear systems with special block triangular matrices. Several methods are known to solve these systems, the most general of which is the Block Cyclic Reduction (BCR) algorithm which handles equations with nonconsistant coefficients. A method was recently proposed to parallelize and vectorize BCR. Here, the mapping of BCR on distributed memory architectures is discussed, and its complexity is compared with that of other approaches, including the Alternating-Direction method. A fast parallel solver is also described, based on an explicit formula for the solution, which has parallel computational complexity lower than that of parallel BCR.

Evaluation of new techniques for the calculation of internal recirculating flows

NASA Technical Reports Server (NTRS)

Van Doormaal, J. P.; Turan, A.; Raithby, G. D.

1987-01-01

The performance of discrete methods for the prediction of fluid flows can be enhanced by improving the convergence rate of solvers and by increasing the accuracy of the discrete representation of the equations of motion. This paper evaluates the gains in solver performance that are available when various acceleration methods are applied. Various discretizations are also examined and two are recommended because of their accuracy and robustness. Insertion of the improved discretization and solver accelerator into a TEACH code, that has been widely applied to combustor flows, illustrates the substantial gains that can be achieved.

Seven Wonders of the Ancient and Modern Quadratic World.

ERIC Educational Resources Information Center

Taylor, Sharon E.; Mittag, Kathleen Cage

2001-01-01

Presents four methods for solving a quadratic equation using graphing calculator technology: (1) graphing with the CALC feature; (2) quadratic formula program; (3) table; and (4) solver. Includes a worksheet for a lab activity on factoring quadratic equations. (KHR)

Second-order Poisson Nernst-Planck solver for ion channel transport

PubMed Central

Zheng, Qiong; Chen, Duan; Wei, Guo-Wei

2010-01-01

The Poisson Nernst-Planck (PNP) theory is a simplified continuum model for a wide variety of chemical, physical and biological applications. Its ability of providing quantitative explanation and increasingly qualitative predictions of experimental measurements has earned itself much recognition in the research community. Numerous computational algorithms have been constructed for the solution of the PNP equations. However, in the realistic ion-channel context, no second order convergent PNP algorithm has ever been reported in the literature, due to many numerical obstacles, including discontinuous coefficients, singular charges, geometric singularities, and nonlinear couplings. The present work introduces a number of numerical algorithms to overcome the abovementioned numerical challenges and constructs the first second-order convergent PNP solver in the ion-channel context. First, a Dirichlet to Neumann mapping (DNM) algorithm is designed to alleviate the charge singularity due to the protein structure. Additionally, the matched interface and boundary (MIB) method is reformulated for solving the PNP equations. The MIB method systematically enforces the interface jump conditions and achieves the second order accuracy in the presence of complex geometry and geometric singularities of molecular surfaces. Moreover, two iterative schemes are utilized to deal with the coupled nonlinear equations. Furthermore, extensive and rigorous numerical validations are carried out over a number of geometries, including a sphere, two proteins and an ion channel, to examine the numerical accuracy and convergence order of the present numerical algorithms. Finally, application is considered to a real transmembrane protein, the Gramicidin A channel protein. The performance of the proposed numerical techniques is tested against a number of factors, including mesh sizes, diffusion coefficient profiles, iterative schemes, ion concentrations, and applied voltages. Numerical predictions are compared with experimental measurements. PMID:21552336

A compressible Navier-Stokes solver with two-equation and Reynolds stress turbulence closure models

NASA Technical Reports Server (NTRS)

Morrison, Joseph H.

1992-01-01

This report outlines the development of a general purpose aerodynamic solver for compressible turbulent flows. Turbulent closure is achieved using either two equation or Reynolds stress transportation equations. The applicable equation set consists of Favre-averaged conservation equations for the mass, momentum and total energy, and transport equations for the turbulent stresses and turbulent dissipation rate. In order to develop a scheme with good shock capturing capabilities, good accuracy and general geometric capabilities, a multi-block cell centered finite volume approach is used. Viscous fluxes are discretized using a finite volume representation of a central difference operator and the source terms are treated as an integral over the control volume. The methodology is validated by testing the algorithm on both two and three dimensional flows. Both the two equation and Reynolds stress models are used on a two dimensional 10 degree compression ramp at Mach 3, and the two equation model is used on the three dimensional flow over a cone at angle of attack at Mach 3.5. With the development of this algorithm, it is now possible to compute complex, compressible high speed flow fields using both two equation and Reynolds stress turbulent closure models, with the capability of eventually evaluating their predictive performance.

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

PubMed Central

Wan, Xiaohai; Li, Zhilin

2012-01-01

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

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

PubMed

Wan, Xiaohai; Li, Zhilin

2012-06-01

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

Final Report, DE-FG01-06ER25718 Domain Decomposition and Parallel Computing

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

Widlund, Olof B.

2015-06-09

The goal of this project is to develop and improve domain decomposition algorithms for a variety of partial differential equations such as those of linear elasticity and electro-magnetics.These iterative methods are designed for massively parallel computing systems and allow the fast solution of the very large systems of algebraic equations that arise in large scale and complicated simulations. A special emphasis is placed on problems arising from Maxwell's equation. The approximate solvers, the preconditioners, are combined with the conjugate gradient method and must always include a solver of a coarse model in order to have a performance which is independentmore » of the number of processors used in the computer simulation. A recent development allows for an adaptive construction of this coarse component of the preconditioner.« less

Evaluation of a Multigrid Scheme for the Incompressible Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Swanson, R. C.

2004-01-01

A fast multigrid solver for the steady, incompressible Navier-Stokes equations is presented. The multigrid solver is based upon a factorizable discrete scheme for the velocity-pressure form of the Navier-Stokes equations. This scheme correctly distinguishes between the advection-diffusion and elliptic parts of the operator, allowing efficient smoothers to be constructed. To evaluate the multigrid algorithm, solutions are computed for flow over a flat plate, parabola, and a Karman-Trefftz airfoil. Both nonlifting and lifting airfoil flows are considered, with a Reynolds number range of 200 to 800. Convergence and accuracy of the algorithm are discussed. Using Gauss-Seidel line relaxation in alternating directions, multigrid convergence behavior approaching that of O(N) methods is achieved. The computational efficiency of the numerical scheme is compared with that of Runge-Kutta and implicit upwind based multigrid methods.

Numerical Simulations for Landing Gear Noise Generation and Radiation

NASA Technical Reports Server (NTRS)

Morris, Philip J.; Long, Lyle N.

2002-01-01

Aerodynamic noise from a landing gear in a uniform flow is computed using the Ffowcs Williams -Hawkings (FW-H) equation. The time accurate flow data on the surface is obtained using a finite volume flow solver on an unstructured and. The Ffowcs Williams-Hawkings equation is solved using surface integrals over the landing gear surface and over a permeable surface away from the landing gear. Two geometric configurations are tested in order to assess the impact of two lateral struts on the sound level and directivity in the far-field. Predictions from the Ffowcs Williams-Hawkings code are compared with direct calculations by the flow solver at several observer locations inside the computational domain. The permeable Ffowcs Williams-Hawkings surface predictions match those of the flow solver in the near-field. Far-field noise calculations coincide for both integration surfaces. The increase in drag observed between the two landing gear configurations is reflected in the sound pressure level and directivity mainly in the streamwise direction.

Matrix decomposition graphics processing unit solver for Poisson image editing

NASA Astrophysics Data System (ADS)

Lei, Zhao; Wei, Li

2012-10-01

In recent years, gradient-domain methods have been widely discussed in the image processing field, including seamless cloning and image stitching. These algorithms are commonly carried out by solving a large sparse linear system: the Poisson equation. However, solving the Poisson equation is a computational and memory intensive task which makes it not suitable for real-time image editing. A new matrix decomposition graphics processing unit (GPU) solver (MDGS) is proposed to settle the problem. A matrix decomposition method is used to distribute the work among GPU threads, so that MDGS will take full advantage of the computing power of current GPUs. Additionally, MDGS is a hybrid solver (combines both the direct and iterative techniques) and has two-level architecture. These enable MDGS to generate identical solutions with those of the common Poisson methods and achieve high convergence rate in most cases. This approach is advantageous in terms of parallelizability, enabling real-time image processing, low memory-taken and extensive applications.

Parallel-vector computation for linear structural analysis and non-linear unconstrained optimization problems

NASA Technical Reports Server (NTRS)

Nguyen, D. T.; Al-Nasra, M.; Zhang, Y.; Baddourah, M. A.; Agarwal, T. K.; Storaasli, O. O.; Carmona, E. A.

1991-01-01

Several parallel-vector computational improvements to the unconstrained optimization procedure are described which speed up the structural analysis-synthesis process. A fast parallel-vector Choleski-based equation solver, pvsolve, is incorporated into the well-known SAP-4 general-purpose finite-element code. The new code, denoted PV-SAP, is tested for static structural analysis. Initial results on a four processor CRAY 2 show that using pvsolve reduces the equation solution time by a factor of 14-16 over the original SAP-4 code. In addition, parallel-vector procedures for the Golden Block Search technique and the BFGS method are developed and tested for nonlinear unconstrained optimization. A parallel version of an iterative solver and the pvsolve direct solver are incorporated into the BFGS method. Preliminary results on nonlinear unconstrained optimization test problems, using pvsolve in the analysis, show excellent parallel-vector performance indicating that these parallel-vector algorithms can be used in a new generation of finite-element based structural design/analysis-synthesis codes.

Solving lattice QCD systems of equations using mixed precision solvers on GPUs

NASA Astrophysics Data System (ADS)

Clark, M. A.; Babich, R.; Barros, K.; Brower, R. C.; Rebbi, C.

2010-09-01

Modern graphics hardware is designed for highly parallel numerical tasks and promises significant cost and performance benefits for many scientific applications. One such application is lattice quantum chromodynamics (lattice QCD), where the main computational challenge is to efficiently solve the discretized Dirac equation in the presence of an SU(3) gauge field. Using NVIDIA's CUDA platform we have implemented a Wilson-Dirac sparse matrix-vector product that performs at up to 40, 135 and 212 Gflops for double, single and half precision respectively on NVIDIA's GeForce GTX 280 GPU. We have developed a new mixed precision approach for Krylov solvers using reliable updates which allows for full double precision accuracy while using only single or half precision arithmetic for the bulk of the computation. The resulting BiCGstab and CG solvers run in excess of 100 Gflops and, in terms of iterations until convergence, perform better than the usual defect-correction approach for mixed precision.

Efficiency optimization of a fast Poisson solver in beam dynamics simulation

NASA Astrophysics Data System (ADS)

Zheng, Dawei; Pöplau, Gisela; van Rienen, Ursula

2016-01-01

Calculating the solution of Poisson's equation relating to space charge force is still the major time consumption in beam dynamics simulations and calls for further improvement. In this paper, we summarize a classical fast Poisson solver in beam dynamics simulations: the integrated Green's function method. We introduce three optimization steps of the classical Poisson solver routine: using the reduced integrated Green's function instead of the integrated Green's function; using the discrete cosine transform instead of discrete Fourier transform for the Green's function; using a novel fast convolution routine instead of an explicitly zero-padded convolution. The new Poisson solver routine preserves the advantages of fast computation and high accuracy. This provides a fast routine for high performance calculation of the space charge effect in accelerators.

Distributed Memory Parallel Computing with SEAWAT

NASA Astrophysics Data System (ADS)

Verkaik, J.; Huizer, S.; van Engelen, J.; Oude Essink, G.; Ram, R.; Vuik, K.

2017-12-01

Fresh groundwater reserves in coastal aquifers are threatened by sea-level rise, extreme weather conditions, increasing urbanization and associated groundwater extraction rates. To counteract these threats, accurate high-resolution numerical models are required to optimize the management of these precious reserves. The major model drawbacks are long run times and large memory requirements, limiting the predictive power of these models. Distributed memory parallel computing is an efficient technique for reducing run times and memory requirements, where the problem is divided over multiple processor cores. A new Parallel Krylov Solver (PKS) for SEAWAT is presented. PKS has recently been applied to MODFLOW and includes Conjugate Gradient (CG) and Biconjugate Gradient Stabilized (BiCGSTAB) linear accelerators. Both accelerators are preconditioned by an overlapping additive Schwarz preconditioner in a way that: a) subdomains are partitioned using Recursive Coordinate Bisection (RCB) load balancing, b) each subdomain uses local memory only and communicates with other subdomains by Message Passing Interface (MPI) within the linear accelerator, c) it is fully integrated in SEAWAT. Within SEAWAT, the PKS-CG solver replaces the Preconditioned Conjugate Gradient (PCG) solver for solving the variable-density groundwater flow equation and the PKS-BiCGSTAB solver replaces the Generalized Conjugate Gradient (GCG) solver for solving the advection-diffusion equation. PKS supports the third-order Total Variation Diminishing (TVD) scheme for computing advection. Benchmarks were performed on the Dutch national supercomputer (https://userinfo.surfsara.nl/systems/cartesius) using up to 128 cores, for a synthetic 3D Henry model (100 million cells) and the real-life Sand Engine model ( 10 million cells). The Sand Engine model was used to investigate the potential effect of the long-term morphological evolution of a large sand replenishment and climate change on fresh groundwater resources. Speed-ups up to 40 were obtained with the new PKS solver.

Final Report for "Implimentation and Evaluation of Multigrid Linear Solvers into Extended Magnetohydrodynamic Codes for Petascale Computing"

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

Srinath Vadlamani; Scott Kruger; Travis Austin

Extended magnetohydrodynamic (MHD) codes are used to model the large, slow-growing instabilities that are projected to limit the performance of International Thermonuclear Experimental Reactor (ITER). The multiscale nature of the extended MHD equations requires an implicit approach. The current linear solvers needed for the implicit algorithm scale poorly because the resultant matrices are so ill-conditioned. A new solver is needed, especially one that scales to the petascale. The most successful scalable parallel processor solvers to date are multigrid solvers. Applying multigrid techniques to a set of equations whose fundamental modes are dispersive waves is a promising solution to CEMM problems.more » For the Phase 1, we implemented multigrid preconditioners from the HYPRE project of the Center for Applied Scientific Computing at LLNL via PETSc of the DOE SciDAC TOPS for the real matrix systems of the extended MHD code NIMROD which is a one of the primary modeling codes of the OFES-funded Center for Extended Magnetohydrodynamic Modeling (CEMM) SciDAC. We implemented the multigrid solvers on the fusion test problem that allows for real matrix systems with success, and in the process learned about the details of NIMROD data structures and the difficulties of inverting NIMROD operators. The further success of this project will allow for efficient usage of future petascale computers at the National Leadership Facilities: Oak Ridge National Laboratory, Argonne National Laboratory, and National Energy Research Scientific Computing Center. The project will be a collaborative effort between computational plasma physicists and applied mathematicians at Tech-X Corporation, applied mathematicians Front Range Scientific Computations, Inc. (who are collaborators on the HYPRE project), and other computational plasma physicists involved with the CEMM project.« less

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.

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

Regularization and computational methods for precise solution of perturbed orbit transfer problems

NASA Astrophysics Data System (ADS)

Woollands, Robyn Michele

The author has developed a suite of algorithms for solving the perturbed Lambert's problem in celestial mechanics. These algorithms have been implemented as a parallel computation tool that has broad applicability. This tool is composed of four component algorithms and each provides unique benefits for solving a particular type of orbit transfer problem. The first one utilizes a Keplerian solver (a-iteration) for solving the unperturbed Lambert's problem. This algorithm not only provides a "warm start" for solving the perturbed problem but is also used to identify which of several perturbed solvers is best suited for the job. The second algorithm solves the perturbed Lambert's problem using a variant of the modified Chebyshev-Picard iteration initial value solver that solves two-point boundary value problems. This method converges over about one third of an orbit and does not require a Newton-type shooting method and thus no state transition matrix needs to be computed. The third algorithm makes use of regularization of the differential equations through the Kustaanheimo-Stiefel transformation and extends the domain of convergence over which the modified Chebyshev-Picard iteration two-point boundary value solver will converge, from about one third of an orbit to almost a full orbit. This algorithm also does not require a Newton-type shooting method. The fourth algorithm uses the method of particular solutions and the modified Chebyshev-Picard iteration initial value solver to solve the perturbed two-impulse Lambert problem over multiple revolutions. The method of particular solutions is a shooting method but differs from the Newton-type shooting methods in that it does not require integration of the state transition matrix. The mathematical developments that underlie these four algorithms are derived in the chapters of this dissertation. For each of the algorithms, some orbit transfer test cases are included to provide insight on accuracy and efficiency of these individual algorithms. Following this discussion, the combined parallel algorithm, known as the unified Lambert tool, is presented and an explanation is given as to how it automatically selects which of the three perturbed solvers to compute the perturbed solution for a particular orbit transfer. The unified Lambert tool may be used to determine a single orbit transfer or for generating of an extremal field map. A case study is presented for a mission that is required to rendezvous with two pieces of orbit debris (spent rocket boosters). The unified Lambert tool software developed in this dissertation is already being utilized by several industrial partners and we are confident that it will play a significant role in practical applications, including solution of Lambert problems that arise in the current applications focused on enhanced space situational awareness.

Pycellerator: an arrow-based reaction-like modelling language for biological simulations.

PubMed

Shapiro, Bruce E; Mjolsness, Eric

2016-02-15

We introduce Pycellerator, a Python library for reading Cellerator arrow notation from standard text files, conversion to differential equations, generating stand-alone Python solvers, and optionally running and plotting the solutions. All of the original Cellerator arrows, which represent reactions ranging from mass action, Michales-Menten-Henri (MMH) and Gene-Regulation (GRN) to Monod-Wyman-Changeaux (MWC), user defined reactions and enzymatic expansions (KMech), were previously represented with the Mathematica extended character set. These are now typed as reaction-like commands in ASCII text files that are read by Pycellerator, which includes a Python command line interface (CLI), a Python application programming interface (API) and an iPython notebook interface. Cellerator reaction arrows are now input in text files. The arrows are parsed by Pycellerator and translated into differential equations in Python, and Python code is automatically generated to solve the system. Time courses are produced by executing the auto-generated Python code. Users have full freedom to modify the solver and utilize the complete set of standard Python tools. The new libraries are completely independent of the old Cellerator software and do not require Mathematica. All software is available (GPL) from the github repository at https://github.com/biomathman/pycellerator/releases. Details, including installation instructions and a glossary of acronyms and terms, are given in the Supplementary information. © The Author 2015. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oup.com.

Air-structure coupling features analysis of mining contra-rotating axial flow fan cascade

NASA Astrophysics Data System (ADS)

Chen, Q. G.; Sun, W.; Li, F.; Zhang, Y. J.

2013-12-01

The interaction between contra-rotating axial flow fan blade and working gas has been studied by means of establishing air-structure coupling control equation and combining Computational Fluid Dynamics (CFD) and Computational solid mechanics (CSM). Based on the single flow channel model, the Finite Volume Method was used to make the field discrete. Additionally, the SIMPLE algorithm, the Standard k-ε model and the Arbitrary Lagrangian-Eulerian dynamic grids technology were utilized to get the airflow motion by solving the discrete governing equations. At the same time, the Finite Element Method was used to make the field discrete to solve dynamic response characteristics of blade. Based on weak coupling method, data exchange from the fluid solver and the solid solver was processed on the coupling interface. Then interpolation was used to obtain the coupling characteristics. The results showed that the blade's maximum amplitude was on the tip of the last-stage blade and aerodynamic force signal could reflect the blade working conditions to some extent. By analyzing the flow regime in contra-rotating axial flow fan, it could be found that the vortex core region was mainly in the blade surface, the hub and the blade clearance. In those regions, the turbulence intensity was very high. The last-stage blade's operating life is shorter than that of the pre-stage blade due to the fatigue fracture occurs much more easily on the last-stage blade which bears more stress.

Conjunction of 2D and 3D modified flow solvers for simulating spatio-temporal wind induced hydrodynamics in the Caspian Sea

NASA Astrophysics Data System (ADS)

Zounemat-Kermani, Mohammad; Sabbagh-Yazdi, Saeed-Reza

2010-06-01

The main objective of this study is the simulation of flow dynamics in the deep parts of the Caspian Sea, in which the southern and middle deep regions are surrounded by considerable areas of shallow zones. To simulate spatio-temporal wind induced hydrodynamics in deep waters, a conjunctive numerical model consisting of a 2D depth average model and a 3D pseudo compressible model is proposed. The 2D model is applied to determine time dependent free surface oscillations as well as the surface velocity patterns and is conjunct to the 3D flow solver for computing three-dimensional velocity and pressure fields which coverage to steady state for the top boundary condition. The modified 2D and 3D sets of equations are conjunct considering interface shear stresses. Both sets of 2D and 3D equations are solved on unstructured triangular and tetrahedral meshes using the Galerkin Finite Volume Method. The conjunctive model is utilized to investigate the deep currents affected by wind, Coriolis forces and the river inflow conditions of the Caspian Sea. In this study, the simulation of flow field due to major winds as well as transient winds in the Caspian Sea during a period of 6 hours in the winter season has been conducted and the numerical results for water surface level are then compared to the 2D numerical results.

Summer Proceedings 2016: The Center for Computing Research at Sandia National Laboratories

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

Carleton, James Brian; Parks, Michael L.

Solving sparse linear systems from the discretization of elliptic partial differential equations (PDEs) is an important building block in many engineering applications. Sparse direct solvers can solve general linear systems, but are usually slower and use much more memory than effective iterative solvers. To overcome these two disadvantages, a hierarchical solver (LoRaSp) based on H2-matrices was introduced in [22]. Here, we have developed a parallel version of the algorithm in LoRaSp to solve large sparse matrices on distributed memory machines. On a single processor, the factorization time of our parallel solver scales almost linearly with the problem size for three-dimensionalmore » problems, as opposed to the quadratic scalability of many existing sparse direct solvers. Moreover, our solver leads to almost constant numbers of iterations, when used as a preconditioner for Poisson problems. On more than one processor, our algorithm has significant speedups compared to sequential runs. With this parallel algorithm, we are able to solve large problems much faster than many existing packages as demonstrated by the numerical experiments.« less

A high-order relativistic two-fluid electrodynamic scheme with consistent reconstruction of electromagnetic fields and a multidimensional Riemann solver for electromagnetism

NASA Astrophysics Data System (ADS)

Balsara, Dinshaw S.; Amano, Takanobu; Garain, Sudip; Kim, Jinho

2016-08-01

In various astrophysics settings it is common to have a two-fluid relativistic plasma that interacts with the electromagnetic field. While it is common to ignore the displacement current in the ideal, classical magnetohydrodynamic limit, when the flows become relativistic this approximation is less than absolutely well-justified. In such a situation, it is more natural to consider a positively charged fluid made up of positrons or protons interacting with a negatively charged fluid made up of electrons. The two fluids interact collectively with the full set of Maxwell's equations. As a result, a solution strategy for that coupled system of equations is sought and found here. Our strategy extends to higher orders, providing increasing accuracy. The primary variables in the Maxwell solver are taken to be the facially-collocated components of the electric and magnetic fields. Consistent with such a collocation, three important innovations are reported here. The first two pertain to the Maxwell solver. In our first innovation, the magnetic field within each zone is reconstructed in a divergence-free fashion while the electric field within each zone is reconstructed in a form that is consistent with Gauss' law. In our second innovation, a multidimensionally upwinded strategy is presented which ensures that the magnetic field can be updated via a discrete interpretation of Faraday's law and the electric field can be updated via a discrete interpretation of the generalized Ampere's law. This multidimensional upwinding is achieved via a multidimensional Riemann solver. The multidimensional Riemann solver automatically provides edge-centered electric field components for the Stokes law-based update of the magnetic field. It also provides edge-centered magnetic field components for the Stokes law-based update of the electric field. The update strategy ensures that the electric field is always consistent with Gauss' law and the magnetic field is always divergence-free. This collocation also ensures that electromagnetic radiation that is propagating in a vacuum has both electric and magnetic fields that are exactly divergence-free. Coupled relativistic fluid dynamic equations are solved for the positively and negatively charged fluids. The fluids' numerical fluxes also provide a self-consistent current density for the update of the electric field. Our reconstruction strategy ensures that fluid velocities always remain sub-luminal. Our third innovation consists of an efficient design for several popular IMEX schemes so that they provide strong coupling between the finite-volume-based fluid solver and the electromagnetic fields at high order. This innovation makes it possible to efficiently utilize high order IMEX time update methods for stiff source terms in the update of high order finite-volume methods for hyperbolic conservation laws. We also show that this very general innovation should extend seamlessly to Runge-Kutta discontinuous Galerkin methods. The IMEX schemes enable us to use large CFL numbers even in the presence of stiff source terms. Several accuracy analyses are presented showing that our method meets its design accuracy in the MHD limit as well as in the limit of electromagnetic wave propagation. Several stringent test problems are also presented. We also present a relativistic version of the GEM problem, which shows that our algorithm can successfully adapt to challenging problems in high energy astrophysics.

A new relativistic viscous hydrodynamics code and its application to the Kelvin-Helmholtz instability in high-energy heavy-ion collisions

NASA Astrophysics Data System (ADS)

Okamoto, Kazuhisa; Nonaka, Chiho

2017-06-01

We construct a new relativistic viscous hydrodynamics code optimized in the Milne coordinates. We split the conservation equations into an ideal part and a viscous part, using the Strang spitting method. In the code a Riemann solver based on the two-shock approximation is utilized for the ideal part and the Piecewise Exact Solution (PES) method is applied for the viscous part. We check the validity of our numerical calculations by comparing analytical solutions, the viscous Bjorken's flow and the Israel-Stewart theory in Gubser flow regime. Using the code, we discuss possible development of the Kelvin-Helmholtz instability in high-energy heavy-ion collisions.

SU-E-T-22: A Deterministic Solver of the Boltzmann-Fokker-Planck Equation for Dose Calculation

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

Hong, X; Gao, H; Paganetti, H

2015-06-15

Purpose: The Boltzmann-Fokker-Planck equation (BFPE) accurately models the migration of photons/charged particles in tissues. While the Monte Carlo (MC) method is popular for solving BFPE in a statistical manner, we aim to develop a deterministic BFPE solver based on various state-of-art numerical acceleration techniques for rapid and accurate dose calculation. Methods: Our BFPE solver is based on the structured grid that is maximally parallelizable, with the discretization in energy, angle and space, and its cross section coefficients are derived or directly imported from the Geant4 database. The physical processes that are taken into account are Compton scattering, photoelectric effect, pairmore » production for photons, and elastic scattering, ionization and bremsstrahlung for charged particles.While the spatial discretization is based on the diamond scheme, the angular discretization synergizes finite element method (FEM) and spherical harmonics (SH). Thus, SH is used to globally expand the scattering kernel and FFM is used to locally discretize the angular sphere. As a Result, this hybrid method (FEM-SH) is both accurate in dealing with forward-peaking scattering via FEM, and efficient for multi-energy-group computation via SH. In addition, FEM-SH enables the analytical integration in energy variable of delta scattering kernel for elastic scattering with reduced truncation error from the numerical integration based on the classic SH-based multi-energy-group method. Results: The accuracy of the proposed BFPE solver was benchmarked against Geant4 for photon dose calculation. In particular, FEM-SH had improved accuracy compared to FEM, while both were within 2% of the results obtained with Geant4. Conclusion: A deterministic solver of the Boltzmann-Fokker-Planck equation is developed for dose calculation, and benchmarked against Geant4. Xiang Hong and Hao Gao were partially supported by the NSFC (#11405105), the 973 Program (#2015CB856000) and the Shanghai Pujiang Talent Program (#14PJ1404500)« less

A low diffusive Lagrange-remap scheme for the simulation of violent air-water free-surface flows

NASA Astrophysics Data System (ADS)

Bernard-Champmartin, Aude; De Vuyst, Florian

2014-10-01

In 2002, Després and Lagoutière [17] proposed a low-diffusive advection scheme for pure transport equation problems, which is particularly accurate for step-shaped solutions, and thus suited for interface tracking procedure by a color function. This has been extended by Kokh and Lagoutière [28] in the context of compressible multifluid flows using a five-equation model. In this paper, we explore a simplified variant approach for gas-liquid three-equation models. The Eulerian numerical scheme has two ingredients: a robust remapped Lagrange solver for the solution of the volume-averaged equations, and a low diffusive compressive scheme for the advection of the gas mass fraction. Numerical experiments show the performance of the computational approach on various flow reference problems: dam break, sloshing of a tank filled with water, water-water impact and finally a case of Rayleigh-Taylor instability. One of the advantages of the present interface capturing solver is its natural implementation on parallel processors or computers.

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

NASA Technical Reports Server (NTRS)

Kim, S.-W.

1991-01-01

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

Constrained hierarchical least square nonlinear equation solvers. [for indefinite stiffness and large structural deformations

NASA Technical Reports Server (NTRS)

Padovan, J.; Lackney, J.

1986-01-01

The current paper develops a constrained hierarchical least square nonlinear equation solver. The procedure can handle the response behavior of systems which possess indefinite tangent stiffness characteristics. Due to the generality of the scheme, this can be achieved at various hierarchical application levels. For instance, in the case of finite element simulations, various combinations of either degree of freedom, nodal, elemental, substructural, and global level iterations are possible. Overall, this enables a solution methodology which is highly stable and storage efficient. To demonstrate the capability of the constrained hierarchical least square methodology, benchmarking examples are presented which treat structure exhibiting highly nonlinear pre- and postbuckling behavior wherein several indefinite stiffness transitions occur.

Modeling and Simulation of Swirl Stabilized Turbulent Non-Premixed Flames

NASA Astrophysics Data System (ADS)

Badillo-Rios, Salvador; Karagozian, Ann

2017-11-01

Flame stabilization is an important design criterion for many combustion chambers, especially at lean conditions and/or high power output, where insufficient stabilization can result in dangerous oscillations and noisy or damaged combustors. At high flow rates, swirling flow can offer a suitable stabilization mechanism, although understanding the dynamics of swirl-stabilized turbulent flames remains a significant challenge. Utilizing the General Equation and Mesh Solver (GEMS) code, which solves the Navier-Stokes equations along with the energy equation and five species equations, 2D axisymmetric and full 3D parametric studies and simulations are performed to guide the design and development of an experimental swirl combustor configuration and to study the effects of swirl on statistically stationary combustion. Results show that as the momentum of air is directed into the inner air inlet rather than the outer inlet of the swirl combustor, the central recirculating region becomes stronger and more unsteady, improving mixing and burning efficiency in that region. A high temperature region is found to occur as a result of burning of the trapped fuel from the central toroidal vortex. The effects of other parameters on flowfield and flame-stabilization dynamics are explored. Supported by ERC, Inc. (PS150006) and AFOSR (Dr. Chiping Li).

High-Maneuverability Airframe: Initial Investigation of Configuration’s Aft End for Increased Stability, Range, and Maneuverability

DTIC Science & Technology

2013-09-01

including the interaction effects between the fins and canards. 2. Solution Technique 2.1 Computational Aerodynamics The double-precision solver of a...and overset grids (unified-grid). • Total variation diminishing discretization based on a new multidimensional interpolation framework. • Riemann ... solvers to provide proper signal propagation physics including versions for preconditioned forms of the governing equations. • Consistent and

A level-set method for two-phase flows with moving contact line and insoluble surfactant

NASA Astrophysics Data System (ADS)

Xu, Jian-Jun; Ren, Weiqing

2014-04-01

A level-set method for two-phase flows with moving contact line and insoluble surfactant is presented. The mathematical model consists of the Navier-Stokes equation for the flow field, a convection-diffusion equation for the surfactant concentration, together with the Navier boundary condition and a condition for the dynamic contact angle derived by Ren et al. (2010) [37]. The numerical method is based on the level-set continuum surface force method for two-phase flows with surfactant developed by Xu et al. (2012) [54] with some cautious treatment for the boundary conditions. The numerical method consists of three components: a flow solver for the velocity field, a solver for the surfactant concentration, and a solver for the level-set function. In the flow solver, the surface force is dealt with using the continuum surface force model. The unbalanced Young stress at the moving contact line is incorporated into the Navier boundary condition. A convergence study of the numerical method and a parametric study are presented. The influence of surfactant on the dynamics of the moving contact line is illustrated using examples. The capability of the level-set method to handle complex geometries is demonstrated by simulating a pendant drop detaching from a wall under gravity.

Three-Dimensional Nacelle Aeroacoustics Code With Application to Impedance Education

NASA Technical Reports Server (NTRS)

Watson, Willie R.

2000-01-01

A three-dimensional nacelle acoustics code that accounts for uniform mean flow and variable surface impedance liners is developed. The code is linked to a commercial version of the NASA-developed General Purpose Solver (for solution of linear systems of equations) in order to obtain the capability to study high frequency waves that may require millions of grid points for resolution. Detailed, single-processor statistics for the performance of the solver in rigid and soft-wall ducts are presented. Over the range of frequencies of current interest in nacelle liner research, noise attenuation levels predicted from the code were in excellent agreement with those predicted from mode theory. The equation solver is memory efficient, requiring only a small fraction of the memory available on modern computers. As an application, the code is combined with an optimization algorithm and used to reduce the impedance spectrum of a ceramic liner. The primary problem with using the code to perform optimization studies at frequencies above I1kHz is the excessive CPU time (a major portion of which is matrix assembly). The research recommends that research be directed toward development of a rapid sparse assembler and exploitation of the multiprocessor capability of the solver to further reduce CPU time.

Chance-Constrained AC Optimal Power Flow for Distribution Systems With Renewables

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

DallAnese, Emiliano; Baker, Kyri; Summers, Tyler

This paper focuses on distribution systems featuring renewable energy sources (RESs) and energy storage systems, and presents an AC optimal power flow (OPF) approach to optimize system-level performance objectives while coping with uncertainty in both RES generation and loads. The proposed method hinges on a chance-constrained AC OPF formulation where probabilistic constraints are utilized to enforce voltage regulation with prescribed probability. A computationally more affordable convex reformulation is developed by resorting to suitable linear approximations of the AC power-flow equations as well as convex approximations of the chance constraints. The approximate chance constraints provide conservative bounds that hold for arbitrarymore » distributions of the forecasting errors. An adaptive strategy is then obtained by embedding the proposed AC OPF task into a model predictive control framework. Finally, a distributed solver is developed to strategically distribute the solution of the optimization problems across utility and customers.« less

Full Wave Parallel Code for Modeling RF Fields in Hot Plasmas

NASA Astrophysics Data System (ADS)

Spencer, Joseph; Svidzinski, Vladimir; Evstatiev, Evstati; Galkin, Sergei; Kim, Jin-Soo

2015-11-01

FAR-TECH, Inc. is developing a suite of full wave RF codes in hot plasmas. It is based on a formulation in configuration space with grid adaptation capability. The conductivity kernel (which includes a nonlocal dielectric response) is calculated by integrating the linearized Vlasov equation along unperturbed test particle orbits. For Tokamak applications a 2-D version of the code is being developed. Progress of this work will be reported. This suite of codes has the following advantages over existing spectral codes: 1) It utilizes the localized nature of plasma dielectric response to the RF field and calculates this response numerically without approximations. 2) It uses an adaptive grid to better resolve resonances in plasma and antenna structures. 3) It uses an efficient sparse matrix solver to solve the formulated linear equations. The linear wave equation is formulated using two approaches: for cold plasmas the local cold plasma dielectric tensor is used (resolving resonances by particle collisions), while for hot plasmas the conductivity kernel is calculated. Work is supported by the U.S. DOE SBIR program.

A Comparison of Ffowcs Williams-Hawkings Solvers for Airframe Noise Applications

NASA Technical Reports Server (NTRS)

Lockard, David P.

2002-01-01

This paper presents a comparison between two implementations of the Ffowcs Williams and Hawkings equation for airframe noise applications. Airframe systems are generally moving at constant speed and not rotating, so these conditions are used in the current investigation. Efficient and easily implemented forms of the equations applicable to subsonic, rectilinear motion of all acoustic sources are used. The assumptions allow the derivation of a simple form of the equations in the frequency-domain, and the time-domain method uses the restrictions on the motion to reduce the work required to find the emission time. The comparison between the frequency domain method and the retarded time formulation reveals some of the advantages of the different approaches. Both methods are still capable of predicting the far-field noise from nonlinear near-field flow quantities. Because of the large input data sets and potentially large numbers of observer positions of interest in three-dimensional problems, both codes utilize the message passing interface to divide the problem among different processors. Example problems are used to demonstrate the usefulness and efficiency of the two schemes.

Core-Collapse Supernovae Explored by Multi-D Boltzmann Hydrodynamic Simulations

NASA Astrophysics Data System (ADS)

Sumiyoshi, Kohsuke; Nagakura, Hiroki; Iwakami, Wakana; Furusawa, Shun; Matsufuru, Hideo; Imakura, Akira; Yamada, Shoichi

We report the latest results of numerical simulations of core-collapse supernovae by solving multi-D neutrino-radiation hydrodynamics with Boltzmann equations. One of the longstanding issues of the explosion mechanism of supernovae has been uncertainty in the approximations of the neutrino transfer in multi-D such as the diffusion approximation and ray-by-ray method. The neutrino transfer is essential, together with 2D/3D hydrodynamical instabilities, to evaluate the neutrino heating behind the shock wave for successful explosions and to predict the neutrino burst signals. We tackled this difficult problem by utilizing our solver of the 6D Boltzmann equation for neutrinos in 3D space and 3D neutrino momentum space coupled with multi-D hydrodynamics adding special and general relativistic extensions. We have performed a set of 2D core-collapse simulations from 11M ⊙ and 15M ⊙ stars on K-computer in Japan by following long-term evolution over 400 ms after bounce to reveal the outcome from the full Boltzmann hydrodynamic simulations with a sophisticated equation of state with multi-nuclear species and updated rates for electron captures on nuclei.

Fast Euler solver for transonic airfoils. I - Theory. II - Applications

NASA Technical Reports Server (NTRS)

Dadone, Andrea; Moretti, Gino

1988-01-01

Equations written in terms of generalized Riemann variables are presently integrated by inverting six bidiagonal matrices and two tridiagonal matrices, using an implicit Euler solver that is based on the lambda-formulation. The solution is found on a C-grid whose boundaries are very close to the airfoil. The fast solver is then applied to the computation of several flowfields on a NACA 0012 airfoil at various Mach number and alpha values, yielding results that are primarily concerned with transonic flows. The effects of grid fineness and boundary distances are analyzed; the code is found to be robust and accurate, as well as fast.

A narrow-band k-distribution model with single mixture gas assumption for radiative flows

NASA Astrophysics Data System (ADS)

Jo, Sung Min; Kim, Jae Won; Kwon, Oh Joon

2018-06-01

In the present study, the narrow-band k-distribution (NBK) model parameters for mixtures of H2O, CO2, and CO are proposed by utilizing the line-by-line (LBL) calculations with a single mixture gas assumption. For the application of the NBK model to radiative flows, a radiative transfer equation (RTE) solver based on a finite-volume method on unstructured meshes was developed. The NBK model and the RTE solver were verified by solving two benchmark problems including the spectral radiance distribution emitted from one-dimensional slabs and the radiative heat transfer in a truncated conical enclosure. It was shown that the results are accurate and physically reliable by comparing with available data. To examine the applicability of the methods to realistic multi-dimensional problems in non-isothermal and non-homogeneous conditions, radiation in an axisymmetric combustion chamber was analyzed, and then the infrared signature emitted from an aircraft exhaust plume was predicted. For modeling the plume flow involving radiative cooling, a flow-radiation coupled procedure was devised in a loosely coupled manner by adopting a Navier-Stokes flow solver based on unstructured meshes. It was shown that the predicted radiative cooling for the combustion chamber is physically more accurate than other predictions, and is as accurate as that by the LBL calculations. It was found that the infrared signature of aircraft exhaust plume can also be obtained accurately, equivalent to the LBL calculations, by using the present narrow-band approach with a much improved numerical efficiency.

Towards Real-Time Pilot-in-the-Loop Simulation of Rotorcraft With Fully-Coupled CFD Solutions of Rotor / Terrain Interactions

NASA Astrophysics Data System (ADS)

Oruc, Ilker

This thesis presents the development of computationally efficient coupling of Navier-Stokes CFD with a helicopter flight dynamics model, with the ultimate goal of real-time simulation of fully coupled aerodynamic interactions between rotor flow and the surrounding terrain. A particular focus of the research is on coupled airwake effects in the helicopter / ship dynamic interface. A computationally efficient coupling interface was developed between the helicopter flight dynamics model, GENHEL-PSU and the Navier-Stokes solvers, CRUNCH/CRAFT-CFD using both FORTRAN and C/C++ programming languages. In order to achieve real-time execution speeds, the main rotor was modeled with a simplified actuator disk using unsteady momentum sources, instead of resolving the full blade geometry in the CFD. All the airframe components, including the fuselage are represented by single aerodynamic control points in the CFD calculations. The rotor downwash influence on the fuselage and empennage are calculated by using the CFD predicted local flow velocities at these aerodynamic control points defined on the helicopter airframe. In the coupled simulations, the flight dynamics model is free to move within a computational domain, where the main rotor forces are translated into source terms in the momentum equations of the Navier-Stokes equations. Simultaneously, the CFD calculates induced velocities those are fed back to the simulation and affect the aerodynamic loads in the flight dynamics. The CFD solver models the inflow, ground effect, and interactional aerodynamics in the flight dynamics simulation, and these calculations can be coupled with solution of the external flow (e.g. ship airwake effects). The developed framework was utilized for various investigations of hovering, forward flight and helicopter/terrain interaction simulations including standard ground effect, partial ground effect, sloped terrain, and acceleration in ground effect; and results compared with different flight and experimental data. In near ground cases, the fully-coupled flight dynamics and CFD simulations predicted roll oscillations due to interactions of the rotor downwash, ground plane, and the feedback controller, which are not predicted by the conventional simulation models. Fully coupled simulations of a helicopter accelerating near ground predicted flow formations similar to the recirculation and ground vortex flow regimes observed in experiments. The predictions of hover power reductions due to ground effect compared well to a recent experimental data and the results showed 22% power reduction for a hover flight z/R=0.55 above ground level. Fully coupled simulations performed for a helicopter hovering over and approaching to a ship flight deck and results compared with the standalone GENHEL-PSU simulations without ship airwake and one-way coupled simulations. The fully-coupled simulations showed higher pilot workload compared to the other two cases. In order to increase the execution speeds of the CFD calculations, several improvements were made on the CFD solver. First, the initial coupling approach File I/O was replaced with a more efficient method called Multiple Program Multiple Data MPI framework, where the two executables communicate with each other by MPI calls. Next, the unstructured solver (CRUNCH CFD), which is 2nd-order accurate in space, was replaced with the faster running structured solver (CRAFT CFD) that is 5th-order accurate in space. Other improvements including a more efficient k-d tree search algorithm and the bounding of the source term search space within a small region of the grid surrounding the rotor were made on the CFD solver. The final improvement was to parallelize the search task with the CFD solver tasks within the solver. To quantify the speed-up of the improvements to the coupling interface described above, a study was performed to demonstrate the speedup achieved from each of the interface improvements. The improvements made on the CFD solver showed more than 40 times speedup from the baseline file I/O and unstructured solver CRUNCH CFD. Using a structured CFD solver with 5th-order spacial accuracy provided the largest reductions in execution times. Disregarding the solver numeric, the total speedup of all of the interface improvements including the MPMD rotor point exchange, k-d tree search algorithm, bounded search space, and paralleled search task, was approximately 231%, more than a factor of 2. All these improvements provided the necessary speedup for approach real-time CFD. (Abstract shortened by ProQuest.).

pyomo.dae: a modeling and automatic discretization framework for optimization with differential and algebraic equations

DOE PAGES

Nicholson, Bethany; Siirola, John D.; Watson, Jean-Paul; ...

2017-12-20

We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differentialmore » equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.« less

pyomo.dae: a modeling and automatic discretization framework for optimization with differential and algebraic equations

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

Nicholson, Bethany; Siirola, John D.; Watson, Jean-Paul

We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differentialmore » equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.« less

A fast solver for the Helmholtz equation based on the generalized multiscale finite-element method

NASA Astrophysics Data System (ADS)

Fu, Shubin; Gao, Kai

2017-11-01

Conventional finite-element methods for solving the acoustic-wave Helmholtz equation in highly heterogeneous media usually require finely discretized mesh to represent the medium property variations with sufficient accuracy. Computational costs for solving the Helmholtz equation can therefore be considerably expensive for complicated and large geological models. Based on the generalized multiscale finite-element theory, we develop a novel continuous Galerkin method to solve the Helmholtz equation in acoustic media with spatially variable velocity and mass density. Instead of using conventional polynomial basis functions, we use multiscale basis functions to form the approximation space on the coarse mesh. The multiscale basis functions are obtained from multiplying the eigenfunctions of a carefully designed local spectral problem with an appropriate multiscale partition of unity. These multiscale basis functions can effectively incorporate the characteristics of heterogeneous media's fine-scale variations, thus enable us to obtain accurate solution to the Helmholtz equation without directly solving the large discrete system formed on the fine mesh. Numerical results show that our new solver can significantly reduce the dimension of the discrete Helmholtz equation system, and can also obviously reduce the computational time.

The U.S. Geological Survey Modular Ground-Water Model - PCGN: A Preconditioned Conjugate Gradient Solver with Improved Nonlinear Control

USGS Publications Warehouse

Naff, Richard L.; Banta, Edward R.

2008-01-01

The preconditioned conjugate gradient with improved nonlinear control (PCGN) package provides addi-tional means by which the solution of nonlinear ground-water flow problems can be controlled as compared to existing solver packages for MODFLOW. Picard iteration is used to solve nonlinear ground-water flow equations by iteratively solving a linear approximation of the nonlinear equations. The linear solution is provided by means of the preconditioned conjugate gradient algorithm where preconditioning is provided by the modi-fied incomplete Cholesky algorithm. The incomplete Cholesky scheme incorporates two levels of fill, 0 and 1, in which the pivots can be modified so that the row sums of the preconditioning matrix and the original matrix are approximately equal. A relaxation factor is used to implement the modified pivots, which determines the degree of modification allowed. The effects of fill level and degree of pivot modification are briefly explored by means of a synthetic, heterogeneous finite-difference matrix; results are reported in the final section of this report. The preconditioned conjugate gradient method is coupled with Picard iteration so as to efficiently solve the nonlinear equations associated with many ground-water flow problems. The description of this coupling of the linear solver with Picard iteration is a primary concern of this document.

Performance of uncertainty quantification methodologies and linear solvers in cardiovascular simulations

NASA Astrophysics Data System (ADS)

Seo, Jongmin; Schiavazzi, Daniele; Marsden, Alison

2017-11-01

Cardiovascular simulations are increasingly used in clinical decision making, surgical planning, and disease diagnostics. Patient-specific modeling and simulation typically proceeds through a pipeline from anatomic model construction using medical image data to blood flow simulation and analysis. To provide confidence intervals on simulation predictions, we use an uncertainty quantification (UQ) framework to analyze the effects of numerous uncertainties that stem from clinical data acquisition, modeling, material properties, and boundary condition selection. However, UQ poses a computational challenge requiring multiple evaluations of the Navier-Stokes equations in complex 3-D models. To achieve efficiency in UQ problems with many function evaluations, we implement and compare a range of iterative linear solver and preconditioning techniques in our flow solver. We then discuss applications to patient-specific cardiovascular simulation and how the problem/boundary condition formulation in the solver affects the selection of the most efficient linear solver. Finally, we discuss performance improvements in the context of uncertainty propagation. Support from National Institute of Health (R01 EB018302) is greatly appreciated.

A new solver for granular avalanche simulation: Indoor experiment verification and field scale case study

NASA Astrophysics Data System (ADS)

Wang, XiaoLiang; Li, JiaChun

2017-12-01

A new solver based on the high-resolution scheme with novel treatments of source terms and interface capture for the Savage-Hutter model is developed to simulate granular avalanche flows. The capability to simulate flow spread and deposit processes is verified through indoor experiments of a two-dimensional granular avalanche. Parameter studies show that reduction in bed friction enhances runout efficiency, and that lower earth pressure restraints enlarge the deposit spread. The April 9, 2000, Yigong avalanche in Tibet, China, is simulated as a case study by this new solver. The predicted results, including evolution process, deposit spread, and hazard impacts, generally agree with site observations. It is concluded that the new solver for the Savage-Hutter equation provides a comprehensive software platform for granular avalanche simulation at both experimental and field scales. In particular, the solver can be a valuable tool for providing necessary information for hazard forecasts, disaster mitigation, and countermeasure decisions in mountainous areas.

An Implicit Solver on A Parallel Block-Structured Adaptive Mesh Grid for FLASH

NASA Astrophysics Data System (ADS)

Lee, D.; Gopal, S.; Mohapatra, P.

2012-07-01

We introduce a fully implicit solver for FLASH based on a Jacobian-Free Newton-Krylov (JFNK) approach with an appropriate preconditioner. The main goal of developing this JFNK-type implicit solver is to provide efficient high-order numerical algorithms and methodology for simulating stiff systems of differential equations on large-scale parallel computer architectures. A large number of natural problems in nonlinear physics involve a wide range of spatial and time scales of interest. A system that encompasses such a wide magnitude of scales is described as "stiff." A stiff system can arise in many different fields of physics, including fluid dynamics/aerodynamics, laboratory/space plasma physics, low Mach number flows, reactive flows, radiation hydrodynamics, and geophysical flows. One of the big challenges in solving such a stiff system using current-day computational resources lies in resolving time and length scales varying by several orders of magnitude. We introduce FLASH's preliminary implementation of a time-accurate JFNK-based implicit solver in the framework of FLASH's unsplit hydro solver.

Description and use of LSODE, the Livermore Solver for Ordinary Differential Equations

NASA Technical Reports Server (NTRS)

Radhakrishnan, Krishnan; Hindmarsh, Alan C.

1993-01-01

LSODE, the Livermore Solver for Ordinary Differential Equations, is a package of FORTRAN subroutines designed for the numerical solution of the initial value problem for a system of ordinary differential equations. It is particularly well suited for 'stiff' differential systems, for which the backward differentiation formula method of orders 1 to 5 is provided. The code includes the Adams-Moulton method of orders 1 to 12, so it can be used for nonstiff problems as well. In addition, the user can easily switch methods to increase computational efficiency for problems that change character. For both methods a variety of corrector iteration techniques is included in the code. Also, to minimize computational work, both the step size and method order are varied dynamically. This report presents complete descriptions of the code and integration methods, including their implementation. It also provides a detailed guide to the use of the code, as well as an illustrative example problem.

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

PubMed Central

Wang, Wansheng; Chen, Long; Zhou, Jie

2015-01-01

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

A FEniCS-based programming framework for modeling turbulent flow by the Reynolds-averaged Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Mortensen, Mikael; Langtangen, Hans Petter; Wells, Garth N.

2011-09-01

Finding an appropriate turbulence model for a given flow case usually calls for extensive experimentation with both models and numerical solution methods. This work presents the design and implementation of a flexible, programmable software framework for assisting with numerical experiments in computational turbulence. The framework targets Reynolds-averaged Navier-Stokes models, discretized by finite element methods. The novel implementation makes use of Python and the FEniCS package, the combination of which leads to compact and reusable code, where model- and solver-specific code resemble closely the mathematical formulation of equations and algorithms. The presented ideas and programming techniques are also applicable to other fields that involve systems of nonlinear partial differential equations. We demonstrate the framework in two applications and investigate the impact of various linearizations on the convergence properties of nonlinear solvers for a Reynolds-averaged Navier-Stokes model.

Gas-Kinetic Theory Based Flux Splitting Method for Ideal Magnetohydrodynamics

NASA Technical Reports Server (NTRS)

Xu, Kun

1998-01-01

A gas-kinetic solver is developed for the ideal magnetohydrodynamics (MHD) equations. The new scheme is based on the direct splitting of the flux function of the MHD equations with the inclusion of "particle" collisions in the transport process. Consequently, the artificial dissipation in the new scheme is much reduced in comparison with the MHD Flux Vector Splitting Scheme. At the same time, the new scheme is compared with the well-developed Roe-type MHD solver. It is concluded that the kinetic MHD scheme is more robust and efficient than the Roe- type method, and the accuracy is competitive. In this paper the general principle of splitting the macroscopic flux function based on the gas-kinetic theory is presented. The flux construction strategy may shed some light on the possible modification of AUSM- and CUSP-type schemes for the compressible Euler equations, as well as to the development of new schemes for a non-strictly hyperbolic system.

WARP3D-Release 10.8: Dynamic Nonlinear Analysis of Solids using a Preconditioned Conjugate Gradient Software Architecture

NASA Technical Reports Server (NTRS)

Koppenhoefer, Kyle C.; Gullerud, Arne S.; Ruggieri, Claudio; Dodds, Robert H., Jr.; Healy, Brian E.

1998-01-01

This report describes theoretical background material and commands necessary to use the WARP3D finite element code. WARP3D is under continuing development as a research code for the solution of very large-scale, 3-D solid models subjected to static and dynamic loads. Specific features in the code oriented toward the investigation of ductile fracture in metals include a robust finite strain formulation, a general J-integral computation facility (with inertia, face loading), an element extinction facility to model crack growth, nonlinear material models including viscoplastic effects, and the Gurson-Tver-gaard dilatant plasticity model for void growth. The nonlinear, dynamic equilibrium equations are solved using an incremental-iterative, implicit formulation with full Newton iterations to eliminate residual nodal forces. The history integration of the nonlinear equations of motion is accomplished with Newmarks Beta method. A central feature of WARP3D involves the use of a linear-preconditioned conjugate gradient (LPCG) solver implemented in an element-by-element format to replace a conventional direct linear equation solver. This software architecture dramatically reduces both the memory requirements and CPU time for very large, nonlinear solid models since formation of the assembled (dynamic) stiffness matrix is avoided. Analyses thus exhibit the numerical stability for large time (load) steps provided by the implicit formulation coupled with the low memory requirements characteristic of an explicit code. In addition to the much lower memory requirements of the LPCG solver, the CPU time required for solution of the linear equations during each Newton iteration is generally one-half or less of the CPU time required for a traditional direct solver. All other computational aspects of the code (element stiffnesses, element strains, stress updating, element internal forces) are implemented in the element-by- element, blocked architecture. This greatly improves vectorization of the code on uni-processor hardware and enables straightforward parallel-vector processing of element blocks on multi-processor hardware.

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

NASA Astrophysics Data System (ADS)

Angelidis, Dionysios; Chawdhary, Saurabh; Sotiropoulos, Fotis

2016-11-01

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

Heat Transfer Effects on a Fully Premixed Methane Impinging Flame

DTIC Science & Technology

2014-10-30

Houzeaux et al., 2009). The GM- RES solver is also employed to solve for the enthalpy and species mass fractions. The Gauss - Seidel iterative method is...the system is therefore split to solve the mo- mentum and continuity equations independently. This is achieved by applying an iterative strategy...the momentum equation twice and the continuity equation once. The momentum equation is solved using the GMRES or BICGSTAB method (diagonal and Gauss

Effect of Coannular Flow on Linearized Euler Equation Predictions of Jet Noise

NASA Technical Reports Server (NTRS)

Hixon, R.; Shih, S.-H.; Mankbadi, Reda R.

1997-01-01

An improved version of a previously validated linearized Euler equation solver is used to compute the noise generated by coannular supersonic jets. Results for a single supersonic jet are compared to the results from both a normal velocity profile and an inverted velocity profile supersonic jet.

Getting the Best out of Excel

ERIC Educational Resources Information Center

Heys, Chris

2008-01-01

Excel, Microsoft's spreadsheet program, offers several tools which have proven useful in solving some optimization problems that arise in operations research. We will look at two such tools, the Excel modules called Solver and Goal Seek--this after deriving an equation, called the "cash accumulation equation", to be used in conjunction with them.

Nonlinear Conservation Laws and Finite Volume Methods

NASA Astrophysics Data System (ADS)

Leveque, Randall J.

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

A mixture-energy-consistent six-equation two-phase numerical model for fluids with interfaces, cavitation and evaporation waves

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

Pelanti, Marica, E-mail: marica.pelanti@ensta-paristech.fr; Shyue, Keh-Ming, E-mail: shyue@ntu.edu.tw

2014-02-15

We model liquid–gas flows with cavitation by a variant of the six-equation single-velocity two-phase model with stiff mechanical relaxation of Saurel–Petitpas–Berry (Saurel et al., 2009) [9]. In our approach we employ phasic total energy equations instead of the phasic internal energy equations of the classical six-equation system. This alternative formulation allows us to easily design a simple numerical method that ensures consistency with mixture total energy conservation at the discrete level and agreement of the relaxed pressure at equilibrium with the correct mixture equation of state. Temperature and Gibbs free energy exchange terms are included in the equations as relaxationmore » terms to model heat and mass transfer and hence liquid–vapor transition. The algorithm uses a high-resolution wave propagation method for the numerical approximation of the homogeneous hyperbolic portion of the model. In two dimensions a fully-discretized scheme based on a hybrid HLLC/Roe Riemann solver is employed. Thermo-chemical terms are handled numerically via a stiff relaxation solver that forces thermodynamic equilibrium at liquid–vapor interfaces under metastable conditions. We present numerical results of sample tests in one and two space dimensions that show the ability of the proposed model to describe cavitation mechanisms and evaporation wave dynamics.« less

Asymptotically and exactly energy balanced augmented flux-ADER schemes with application to hyperbolic conservation laws with geometric source terms

NASA Astrophysics Data System (ADS)

Navas-Montilla, A.; Murillo, J.

2016-07-01

In this work, an arbitrary order HLL-type numerical scheme is constructed using the flux-ADER methodology. The proposed scheme is based on an augmented Derivative Riemann solver that was used for the first time in Navas-Montilla and Murillo (2015) [1]. Such solver, hereafter referred to as Flux-Source (FS) solver, was conceived as a high order extension of the augmented Roe solver and led to the generation of a novel numerical scheme called AR-ADER scheme. Here, we provide a general definition of the FS solver independently of the Riemann solver used in it. Moreover, a simplified version of the solver, referred to as Linearized-Flux-Source (LFS) solver, is presented. This novel version of the FS solver allows to compute the solution without requiring reconstruction of derivatives of the fluxes, nevertheless some drawbacks are evidenced. In contrast to other previously defined Derivative Riemann solvers, the proposed FS and LFS solvers take into account the presence of the source term in the resolution of the Derivative Riemann Problem (DRP), which is of particular interest when dealing with geometric source terms. When applied to the shallow water equations, the proposed HLLS-ADER and AR-ADER schemes can be constructed to fulfill the exactly well-balanced property, showing that an arbitrary quadrature of the integral of the source inside the cell does not ensure energy balanced solutions. As a result of this work, energy balanced flux-ADER schemes that provide the exact solution for steady cases and that converge to the exact solution with arbitrary order for transient cases are constructed.

A LAGRANGIAN GAUSS-NEWTON-KRYLOV SOLVER FOR MASS- AND INTENSITY-PRESERVING DIFFEOMORPHIC IMAGE REGISTRATION.

PubMed

Mang, Andreas; Ruthotto, Lars

2017-01-01

We present an efficient solver for diffeomorphic image registration problems in the framework of Large Deformations Diffeomorphic Metric Mappings (LDDMM). We use an optimal control formulation, in which the velocity field of a hyperbolic PDE needs to be found such that the distance between the final state of the system (the transformed/transported template image) and the observation (the reference image) is minimized. Our solver supports both stationary and non-stationary (i.e., transient or time-dependent) velocity fields. As transformation models, we consider both the transport equation (assuming intensities are preserved during the deformation) and the continuity equation (assuming mass-preservation). We consider the reduced form of the optimal control problem and solve the resulting unconstrained optimization problem using a discretize-then-optimize approach. A key contribution is the elimination of the PDE constraint using a Lagrangian hyperbolic PDE solver. Lagrangian methods rely on the concept of characteristic curves. We approximate these curves using a fourth-order Runge-Kutta method. We also present an efficient algorithm for computing the derivatives of the final state of the system with respect to the velocity field. This allows us to use fast Gauss-Newton based methods. We present quickly converging iterative linear solvers using spectral preconditioners that render the overall optimization efficient and scalable. Our method is embedded into the image registration framework FAIR and, thus, supports the most commonly used similarity measures and regularization functionals. We demonstrate the potential of our new approach using several synthetic and real world test problems with up to 14.7 million degrees of freedom.

Oasis: A high-level/high-performance open source Navier-Stokes solver

NASA Astrophysics Data System (ADS)

Mortensen, Mikael; Valen-Sendstad, Kristian

2015-03-01

Oasis is a high-level/high-performance finite element Navier-Stokes solver written from scratch in Python using building blocks from the FEniCS project (fenicsproject.org). The solver is unstructured and targets large-scale applications in complex geometries on massively parallel clusters. Oasis utilizes MPI and interfaces, through FEniCS, to the linear algebra backend PETSc. Oasis advocates a high-level, programmable user interface through the creation of highly flexible Python modules for new problems. Through the high-level Python interface the user is placed in complete control of every aspect of the solver. A version of the solver, that is using piecewise linear elements for both velocity and pressure, is shown to reproduce very well the classical, spectral, turbulent channel simulations of Moser et al. (1999). The computational speed is strongly dominated by the iterative solvers provided by the linear algebra backend, which is arguably the best performance any similar implicit solver using PETSc may hope for. Higher order accuracy is also demonstrated and new solvers may be easily added within the same framework.

Representing Sudden Shifts in Intensive Dyadic Interaction Data Using Differential Equation Models with Regime Switching.

PubMed

Chow, Sy-Miin; Ou, Lu; Ciptadi, Arridhana; Prince, Emily B; You, Dongjun; Hunter, Michael D; Rehg, James M; Rozga, Agata; Messinger, Daniel S

2018-06-01

A growing number of social scientists have turned to differential equations as a tool for capturing the dynamic interdependence among a system of variables. Current tools for fitting differential equation models do not provide a straightforward mechanism for diagnosing evidence for qualitative shifts in dynamics, nor do they provide ways of identifying the timing and possible determinants of such shifts. In this paper, we discuss regime-switching differential equation models, a novel modeling framework for representing abrupt changes in a system of differential equation models. Estimation was performed by combining the Kim filter (Kim and Nelson State-space models with regime switching: classical and Gibbs-sampling approaches with applications, MIT Press, Cambridge, 1999) and a numerical differential equation solver that can handle both ordinary and stochastic differential equations. The proposed approach was motivated by the need to represent discrete shifts in the movement dynamics of [Formula: see text] mother-infant dyads during the Strange Situation Procedure (SSP), a behavioral assessment where the infant is separated from and reunited with the mother twice. We illustrate the utility of a novel regime-switching differential equation model in representing children's tendency to exhibit shifts between the goal of staying close to their mothers and intermittent interest in moving away from their mothers to explore the room during the SSP. Results from empirical model fitting were supplemented with a Monte Carlo simulation study to evaluate the use of information criterion measures to diagnose sudden shifts in dynamics.

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 obtaining a robust and efficient high-order accurate flow solver. A goal-oriented error estimation technique has been developed to estimate the discretization error of simulation outputs. For high-order discretizations, it is shown that functional output error super-convergence can be obtained, provided the discretization satisfies a property known as dual consistency. The dual consistency of the DG methods developed in this work is shown via mathematical analysis and numerical experimentation. Goal-oriented error estimation is also used to drive an hp-adaptive mesh refinement strategy, where a combination of mesh or h-refinement, and order or p-enrichment, is employed based on the smoothness of the solution. The results demonstrate that the combination of goal-oriented error estimation and hp-adaptation yield superior accuracy, as well as enhanced robustness and efficiency for a variety of aerodynamic flows including flows with strong shock waves. This work demonstrates that DG discretizations can be the basis of an accurate, efficient, and robust CFD solver. Furthermore, enhancing the robustness of DG methods does not adversely impact the accuracy or efficiency of the solver for challenging and complex flow problems. In particular, when considering the computation of shocked flows, this work demonstrates that the available shock capturing techniques are sufficiently accurate and robust, particularly when used in conjunction with adaptive mesh refinement . This work also demonstrates that robust solutions of the Reynolds Averaged Navier-Stokes (RANS) and turbulence model equations can be obtained for complex and challenging aerodynamic flows. In this context, the most robust strategy was determined to be a low-order turbulence model discretization coupled to a high-order discretization of the RANS equations. Although RANS solutions using high-order accurate discretizations of the turbulence model were obtained, the behavior of current-day RANS turbulence models discretized to high-order was found to be problematic, leading to solver robustness issues. This suggests that future work is warranted in the area of turbulence model formulation for use with high-order discretizations. Alternately, the use of Large-Eddy Simulation (LES) subgrid scale models with high-order DG methods offers the potential to leverage the high accuracy of these methods for very high fidelity turbulent simulations. This thesis has developed the algorithmic improvements that will lay the foundation for the development of a three-dimensional high-order flow solution strategy that can be used as the basis for future LES simulations.

Computational cost of two alternative formulations of Cahn-Hilliard equations

NASA Astrophysics Data System (ADS)

Paszyński, Maciej; Gurgul, Grzegorz; Łoś, Marcin; Szeliga, Danuta

2018-05-01

In this paper we propose two formulations of Cahn-Hilliard equations, which have several applications in cancer growth modeling and material science phase-field simulations. The first formulation uses one C4 partial differential equations (PDEs) the second one uses two C2 PDEs. Finally, we compare the computational costs of direct solvers for both formulations, using the refined isogeometric analysis (rIGA) approach.

The a(4) Scheme-A High Order Neutrally Stable CESE Solver

NASA Technical Reports Server (NTRS)

Chang, Sin-Chung

2009-01-01

The CESE development is driven by a belief that a solver should (i) enforce conservation laws in both space and time, and (ii) be built from a nondissipative (i.e., neutrally stable) core scheme so that the numerical dissipation can be controlled effectively. To provide a solid foundation for a systematic CESE development of high order schemes, in this paper we describe a new high order (4-5th order) and neutrally stable CESE solver of a 1D advection equation with a constant advection speed a. The space-time stencil of this two-level explicit scheme is formed by one point at the upper time level and two points at the lower time level. Because it is associated with four independent mesh variables (the numerical analogues of the dependent variable and its first, second, and third-order spatial derivatives) and four equations per mesh point, the new scheme is referred to as the a(4) scheme. As in the case of other similar CESE neutrally stable solvers, the a(4) scheme enforces conservation laws in space-time locally and globally, and it has the basic, forward marching, and backward marching forms. Except for a singular case, these forms are equivalent and satisfy a space-time inversion (STI) invariant property which is shared by the advection equation. Based on the concept of STI invariance, a set of algebraic relations is developed and used to prove the a(4) scheme must be neutrally stable when it is stable. Numerically, it has been established that the scheme is stable if the value of the Courant number is less than 1/3

Use of a residual distribution Euler solver to study the occurrence of transonic flow in Wells turbine rotor blades

NASA Astrophysics Data System (ADS)

Henriques, J. C. C.; Gato, L. M. C.

The aim of the present study is to investigate the occurrence of transonic flow in several cascade geometries and blade sections that have been considered in the design of Wells turbine rotor blades. The calculations were performed using an implicit Euler solver for two-dimensional flow. The numerical method uses a multi-dimensional upwind matrix residual distribution scheme formulated on a new symmetrized form of the Euler equations, both in time and in space, that decouples the entropy and the enthalpy equations. Second-order accurate steady-state solutions where obtained using a compact three-point stencil. The results show that unwanted transonic flow may occur in the turbine rotor at relatively low mean-flow Mach numbers.

GlobiPack v. 1.0

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

Bartlett, Roscoe

2010-03-31

GlobiPack contains a small collection of optimization globalization algorithms. These algorithms are used by optimization and various nonlinear equation solver algorithms.Used as the line-search procedure with Newton and Quasi-Newton optimization and nonlinear equation solver methods. These are standard published 1-D line search algorithms such as are described in the book Nocedal and Wright Numerical Optimization: 2nd edition, 2006. One set of algorithms were copied and refactored from the existing open-source Trilinos package MOOCHO where the linear search code is used to globalize SQP methods. This software is generic to any mathematical optimization problem where smooth derivatives exist. There is nomore » specific connection or mention whatsoever to any specific application, period. You cannot find more general mathematical software.« less

Three-dimensional unstructured grid Euler computations using a fully-implicit, upwind method

NASA Technical Reports Server (NTRS)

Whitaker, David L.

1993-01-01

A method has been developed to solve the Euler equations on a three-dimensional unstructured grid composed of tetrahedra. The method uses an upwind flow solver with a linearized, backward-Euler time integration scheme. Each time step results in a sparse linear system of equations which is solved by an iterative, sparse matrix solver. Local-time stepping, switched evolution relaxation (SER), preconditioning and reuse of the Jacobian are employed to accelerate the convergence rate. Implicit boundary conditions were found to be extremely important for fast convergence. Numerical experiments have shown that convergence rates comparable to that of a multigrid, central-difference scheme are achievable on the same mesh. Results are presented for several grids about an ONERA M6 wing.

Fast immersed interface Poisson solver for 3D unbounded problems around arbitrary geometries

NASA Astrophysics Data System (ADS)

Gillis, T.; Winckelmans, G.; Chatelain, P.

2018-02-01

We present a fast and efficient Fourier-based solver for the Poisson problem around an arbitrary geometry in an unbounded 3D domain. This solver merges two rewarding approaches, the lattice Green's function method and the immersed interface method, using the Sherman-Morrison-Woodbury decomposition formula. The method is intended to be second order up to the boundary. This is verified on two potential flow benchmarks. We also further analyse the iterative process and the convergence behavior of the proposed algorithm. The method is applicable to a wide range of problems involving a Poisson equation around inner bodies, which goes well beyond the present validation on potential flows.

FELIX-1.0: A finite element solver for the time dependent generator coordinate method with the Gaussian overlap approximation

NASA Astrophysics Data System (ADS)

Regnier, D.; Verrière, M.; Dubray, N.; Schunck, N.

2016-03-01

We describe the software package FELIX that solves the equations of the time-dependent generator coordinate method (TDGCM) in N-dimensions (N ≥ 1) under the Gaussian overlap approximation. The numerical resolution is based on the Galerkin finite element discretization of the collective space and the Crank-Nicolson scheme for time integration. The TDGCM solver is implemented entirely in C++. Several additional tools written in C++, Python or bash scripting language are also included for convenience. In this paper, the solver is tested with a series of benchmarks calculations. We also demonstrate the ability of our code to handle a realistic calculation of fission dynamics.

Fast Laplace solver approach to pore-scale permeability

NASA Astrophysics Data System (ADS)

Arns, C. H.; Adler, P. M.

2018-02-01

We introduce a powerful and easily implemented method to calculate the permeability of porous media at the pore scale using an approximation based on the Poiseulle equation to calculate permeability to fluid flow with a Laplace solver. The method consists of calculating the Euclidean distance map of the fluid phase to assign local conductivities and lends itself naturally to the treatment of multiscale problems. We compare with analytical solutions as well as experimental measurements and lattice Boltzmann calculations of permeability for Fontainebleau sandstone. The solver is significantly more stable than the lattice Boltzmann approach, uses less memory, and is significantly faster. Permeabilities are in excellent agreement over a wide range of porosities.

Use of general purpose graphics processing units with MODFLOW

USGS Publications Warehouse

Hughes, Joseph D.; White, Jeremy T.

2013-01-01

To evaluate the use of general-purpose graphics processing units (GPGPUs) to improve the performance of MODFLOW, an unstructured preconditioned conjugate gradient (UPCG) solver has been developed. The UPCG solver uses a compressed sparse row storage scheme and includes Jacobi, zero fill-in incomplete, and modified-incomplete lower-upper (LU) factorization, and generalized least-squares polynomial preconditioners. The UPCG solver also includes options for sequential and parallel solution on the central processing unit (CPU) using OpenMP. For simulations utilizing the GPGPU, all basic linear algebra operations are performed on the GPGPU; memory copies between the central processing unit CPU and GPCPU occur prior to the first iteration of the UPCG solver and after satisfying head and flow criteria or exceeding a maximum number of iterations. The efficiency of the UPCG solver for GPGPU and CPU solutions is benchmarked using simulations of a synthetic, heterogeneous unconfined aquifer with tens of thousands to millions of active grid cells. Testing indicates GPGPU speedups on the order of 2 to 8, relative to the standard MODFLOW preconditioned conjugate gradient (PCG) solver, can be achieved when (1) memory copies between the CPU and GPGPU are optimized, (2) the percentage of time performing memory copies between the CPU and GPGPU is small relative to the calculation time, (3) high-performance GPGPU cards are utilized, and (4) CPU-GPGPU combinations are used to execute sequential operations that are difficult to parallelize. Furthermore, UPCG solver testing indicates GPGPU speedups exceed parallel CPU speedups achieved using OpenMP on multicore CPUs for preconditioners that can be easily parallelized.

Computerised curve deconvolution of TL/OSL curves using a popular spreadsheet program.

PubMed

Afouxenidis, D; Polymeris, G S; Tsirliganis, N C; Kitis, G

2012-05-01

This paper exploits the possibility of using commercial software for thermoluminescence and optically stimulated luminescence curve deconvolution analysis. The widely used software package Microsoft Excel, with the Solver utility has been used to perform deconvolution analysis to both experimental and reference glow curves resulted from the GLOw Curve ANalysis INtercomparison project. The simple interface of this programme combined with the powerful Solver utility, allows the analysis of complex stimulated luminescence curves into their components and the evaluation of the associated luminescence parameters.

A fast and robust computational method for the ionization cross sections of the driven Schrödinger equation using an O (N) multigrid-based scheme

NASA Astrophysics Data System (ADS)

Cools, S.; Vanroose, W.

2016-03-01

This paper improves the convergence and robustness of a multigrid-based solver for the cross sections of the driven Schrödinger equation. Adding a Coupled Channel Correction Step (CCCS) after each multigrid (MG) V-cycle efficiently removes the errors that remain after the V-cycle sweep. The combined iterative solution scheme (MG-CCCS) is shown to feature significantly improved convergence rates over the classical MG method at energies where bound states dominate the solution, resulting in a fast and scalable solution method for the complex-valued Schrödinger break-up problem for any energy regime. The proposed solver displays optimal scaling; a solution is found in a time that is linear in the number of unknowns. The method is validated on a 2D Temkin-Poet model problem, and convergence results both as a solver and preconditioner are provided to support the O (N) scalability of the method. This paper extends the applicability of the complex contour approach for far field map computation (Cools et al. (2014) [10]).

Linear solver performance in elastoplastic problem solution on GPU cluster

NASA Astrophysics Data System (ADS)

Khalevitsky, Yu. V.; Konovalov, A. V.; Burmasheva, N. V.; Partin, A. S.

2017-12-01

Applying the finite element method to severe plastic deformation problems involves solving linear equation systems. While the solution procedure is relatively hard to parallelize and computationally intensive by itself, a long series of large scale systems need to be solved for each problem. When dealing with fine computational meshes, such as in the simulations of three-dimensional metal matrix composite microvolume deformation, tens and hundreds of hours may be needed to complete the whole solution procedure, even using modern supercomputers. In general, one of the preconditioned Krylov subspace methods is used in a linear solver for such problems. The method convergence highly depends on the operator spectrum of a problem stiffness matrix. In order to choose the appropriate method, a series of computational experiments is used. Different methods may be preferable for different computational systems for the same problem. In this paper we present experimental data obtained by solving linear equation systems from an elastoplastic problem on a GPU cluster. The data can be used to substantiate the choice of the appropriate method for a linear solver to use in severe plastic deformation simulations.

Efficient and Robust Optimization for Building Energy Simulation

PubMed Central

Pourarian, Shokouh; Kearsley, Anthony; Wen, Jin; Pertzborn, Amanda

2016-01-01

Efficiently, robustly and accurately solving large sets of structured, non-linear algebraic and differential equations is one of the most computationally expensive steps in the dynamic simulation of building energy systems. Here, the efficiency, robustness and accuracy of two commonly employed solution methods are compared. The comparison is conducted using the HVACSIM+ software package, a component based building system simulation tool. The HVACSIM+ software presently employs Powell’s Hybrid method to solve systems of nonlinear algebraic equations that model the dynamics of energy states and interactions within buildings. It is shown here that the Powell’s method does not always converge to a solution. Since a myriad of other numerical methods are available, the question arises as to which method is most appropriate for building energy simulation. This paper finds considerable computational benefits result from replacing the Powell’s Hybrid method solver in HVACSIM+ with a solver more appropriate for the challenges particular to numerical simulations of buildings. Evidence is provided that a variant of the Levenberg-Marquardt solver has superior accuracy and robustness compared to the Powell’s Hybrid method presently used in HVACSIM+. PMID:27325907

Efficient and Robust Optimization for Building Energy Simulation.

PubMed

Pourarian, Shokouh; Kearsley, Anthony; Wen, Jin; Pertzborn, Amanda

2016-06-15

Efficiently, robustly and accurately solving large sets of structured, non-linear algebraic and differential equations is one of the most computationally expensive steps in the dynamic simulation of building energy systems. Here, the efficiency, robustness and accuracy of two commonly employed solution methods are compared. The comparison is conducted using the HVACSIM+ software package, a component based building system simulation tool. The HVACSIM+ software presently employs Powell's Hybrid method to solve systems of nonlinear algebraic equations that model the dynamics of energy states and interactions within buildings. It is shown here that the Powell's method does not always converge to a solution. Since a myriad of other numerical methods are available, the question arises as to which method is most appropriate for building energy simulation. This paper finds considerable computational benefits result from replacing the Powell's Hybrid method solver in HVACSIM+ with a solver more appropriate for the challenges particular to numerical simulations of buildings. Evidence is provided that a variant of the Levenberg-Marquardt solver has superior accuracy and robustness compared to the Powell's Hybrid method presently used in HVACSIM+.

Impact of state-specific flowfield modeling on atomic nitrogen radiation

NASA Astrophysics Data System (ADS)

Johnston, Christopher O.; Panesi, Marco

2018-01-01

A hypersonic flowfield model that treats electronic levels of the dominant afterbody radiator N as individual species is presented. This model allows electron-ion recombination rate and two-temperature modeling improvements, the latter which are shown to decrease afterbody radiative heating by up to 30%. This decrease is primarily due to the addition of the electron-impact excitation energy-exchange term to the energy equation governing the vibrational-electronic electron temperature. This model also allows the validity of the often applied quasi-steady-state (QSS) approximation to be assessed. The QSS approximation is shown to fail throughout most of the afterbody region for lower electronic states, although this impacts the radiative intensity reaching the surface by less than 15%. By computing the electronic-state populations of N within the flowfield solver, instead of through the QSS approximation in the radiation solver, the coupling of nonlocal radiative transition rates to the species continuity equations becomes feasible. Implementation of this higher-fidelity level of coupling between the flowfield and radiation solvers is shown to increase the afterbody radiation by up to 50% relative to the conventional model.

An Initial Investigation of the Effects of Turbulence Models on the Convergence of the RK/Implicit Scheme

NASA Technical Reports Server (NTRS)

Swanson, R. C.; Rossow, C.-C.

2008-01-01

A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to be an effective solver for the fluid dynamic equations. This scheme has been applied to both the compressible and essentially incompressible Reynolds-averaged Navier-Stokes (RANS) equations using the algebraic turbulence model of Baldwin and Lomax (BL). In this paper we focus on the convergence of the RK/implicit scheme when the effects of turbulence are represented by either the Spalart-Allmaras model or the Wilcox k-! model, which are frequently used models in practical fluid dynamic applications. Convergence behavior of the scheme with these turbulence models and the BL model are directly compared. For this initial investigation we solve the flow equations and the partial differential equations of the turbulence models indirectly coupled. With this approach we examine the convergence behavior of each system. Both point and line symmetric Gauss-Seidel are considered for approximating the inverse of the implicit operator of the flow solver. To solve the turbulence equations we use a diagonally dominant alternating direction implicit (DDADI) scheme. Computational results are presented for three airfoil flow cases and comparisons are made with experimental data. We demonstrate that the two-dimensional RANS equations and transport-type equations for turbulence modeling can be efficiently solved with an indirectly coupled algorithm that uses the RK/implicit scheme for the flow equations.

Transforming parts of a differential equations system to difference equations as a method for run-time savings in NONMEM.

PubMed

Petersson, K J F; Friberg, L E; Karlsson, M O

2010-10-01

Computer models of biological systems grow more complex as computing power increase. Often these models are defined as differential equations and no analytical solutions exist. Numerical integration is used to approximate the solution; this can be computationally intensive, time consuming and be a large proportion of the total computer runtime. The performance of different integration methods depend on the mathematical properties of the differential equations system at hand. In this paper we investigate the possibility of runtime gains by calculating parts of or the whole differential equations system at given time intervals, outside of the differential equations solver. This approach was tested on nine models defined as differential equations with the goal to reduce runtime while maintaining model fit, based on the objective function value. The software used was NONMEM. In four models the computational runtime was successfully reduced (by 59-96%). The differences in parameter estimates, compared to using only the differential equations solver were less than 12% for all fixed effects parameters. For the variance parameters, estimates were within 10% for the majority of the parameters. Population and individual predictions were similar and the differences in OFV were between 1 and -14 units. When computational runtime seriously affects the usefulness of a model we suggest evaluating this approach for repetitive elements of model building and evaluation such as covariate inclusions or bootstraps.

Optimized and parallelized implementation of the electronegativity equalization method and the atom-bond electronegativity equalization method.

PubMed

Vareková, R Svobodová; Koca, J

2006-02-01

The most common way to calculate charge distribution in a molecule is ab initio quantum mechanics (QM). Some faster alternatives to QM have also been developed, the so-called "equalization methods" EEM and ABEEM, which are based on DFT. We have implemented and optimized the EEM and ABEEM methods and created the EEM SOLVER and ABEEM SOLVER programs. It has been found that the most time-consuming part of equalization methods is the reduction of the matrix belonging to the equation system generated by the method. Therefore, for both methods this part was replaced by the parallel algorithm WIRS and implemented within the PVM environment. The parallelized versions of the programs EEM SOLVER and ABEEM SOLVER showed promising results, especially on a single computer with several processors (compact PVM). The implemented programs are available through the Web page http://ncbr.chemi.muni.cz/~n19n/eem_abeem.

Aerodynamic Shape Optimization Using A Real-Number-Encoded Genetic Algorithm

NASA Technical Reports Server (NTRS)

Holst, Terry L.; Pulliam, Thomas H.

2001-01-01

A new method for aerodynamic shape optimization using a genetic algorithm with real number encoding is presented. The algorithm is used to optimize three different problems, a simple hill climbing problem, a quasi-one-dimensional nozzle problem using an Euler equation solver and a three-dimensional transonic wing problem using a nonlinear potential solver. Results indicate that the genetic algorithm is easy to implement and extremely reliable, being relatively insensitive to design space noise.

Jacobian-free approximate solvers for hyperbolic systems: Application to relativistic magnetohydrodynamics

NASA Astrophysics Data System (ADS)

Castro, Manuel J.; Gallardo, José M.; Marquina, Antonio

2017-10-01

We present recent advances in PVM (Polynomial Viscosity Matrix) methods based on internal approximations to the absolute value function, and compare them with Chebyshev-based PVM solvers. These solvers only require a bound on the maximum wave speed, so no spectral decomposition is needed. Another important feature of the proposed methods is that they are suitable to be written in Jacobian-free form, in which only evaluations of the physical flux are used. This is particularly interesting when considering systems for which the Jacobians involve complex expressions, e.g., the relativistic magnetohydrodynamics (RMHD) equations. On the other hand, the proposed Jacobian-free solvers have also been extended to the case of approximate DOT (Dumbser-Osher-Toro) methods, which can be regarded as simple and efficient approximations to the classical Osher-Solomon method, sharing most of it interesting features and being applicable to general hyperbolic systems. To test the properties of our schemes a number of numerical experiments involving the RMHD equations are presented, both in one and two dimensions. The obtained results are in good agreement with those found in the literature and show that our schemes are robust and accurate, running stable under a satisfactory time step restriction. It is worth emphasizing that, although this work focuses on RMHD, the proposed schemes are suitable to be applied to general hyperbolic systems.

Finite difference elastic wave modeling with an irregular free surface using ADER scheme

NASA Astrophysics Data System (ADS)

Almuhaidib, Abdulaziz M.; Nafi Toksöz, M.

2015-06-01

In numerical modeling of seismic wave propagation in the earth, we encounter two important issues: the free surface and the topography of the surface (i.e. irregularities). In this study, we develop a 2D finite difference solver for the elastic wave equation that combines a 4th- order ADER scheme (Arbitrary high-order accuracy using DERivatives), which is widely used in aeroacoustics, with the characteristic variable method at the free surface boundary. The idea is to treat the free surface boundary explicitly by using ghost values of the solution for points beyond the free surface to impose the physical boundary condition. The method is based on the velocity-stress formulation. The ultimate goal is to develop a numerical solver for the elastic wave equation that is stable, accurate and computationally efficient. The solver treats smooth arbitrary-shaped boundaries as simple plane boundaries. The computational cost added by treating the topography is negligible compared to flat free surface because only a small number of grid points near the boundary need to be computed. In the presence of topography, using 10 grid points per shortest shear-wavelength, the solver yields accurate results. Benchmark numerical tests using several complex models that are solved by our method and other independent accurate methods show an excellent agreement, confirming the validity of the method for modeling elastic waves with an irregular free surface.

Cpp Utility - Version 1.0

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

III, FredOppel; Rigdon, J. Brian

2014-09-08

A collection of general Umbra modules that are reused by other Umbra libraries. These capabilities include line segments, file utilities, color utilities, string utilities (for std::string), list utilities (for std ::vector ), bounding box intersections, range limiters, simple filters, cubic roots solvers and a few other utilities.

Large-scale optimization-based non-negative computational framework for diffusion equations: Parallel implementation and performance studies

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

Chang, Justin; Karra, Satish; Nakshatrala, Kalyana B.

It is well-known that the standard Galerkin formulation, which is often the formulation of choice under the finite element method for solving self-adjoint diffusion equations, does not meet maximum principles and the non-negative constraint for anisotropic diffusion equations. Recently, optimization-based methodologies that satisfy maximum principles and the non-negative constraint for steady-state and transient diffusion-type equations have been proposed. To date, these methodologies have been tested only on small-scale academic problems. The purpose of this paper is to systematically study the performance of the non-negative methodology in the context of high performance computing (HPC). PETSc and TAO libraries are, respectively, usedmore » for the parallel environment and optimization solvers. For large-scale problems, it is important for computational scientists to understand the computational performance of current algorithms available in these scientific libraries. The numerical experiments are conducted on the state-of-the-art HPC systems, and a single-core performance model is used to better characterize the efficiency of the solvers. Furthermore, our studies indicate that the proposed non-negative computational framework for diffusion-type equations exhibits excellent strong scaling for real-world large-scale problems.« less

Large-scale optimization-based non-negative computational framework for diffusion equations: Parallel implementation and performance studies

DOE PAGES

Chang, Justin; Karra, Satish; Nakshatrala, Kalyana B.

2016-07-26

It is well-known that the standard Galerkin formulation, which is often the formulation of choice under the finite element method for solving self-adjoint diffusion equations, does not meet maximum principles and the non-negative constraint for anisotropic diffusion equations. Recently, optimization-based methodologies that satisfy maximum principles and the non-negative constraint for steady-state and transient diffusion-type equations have been proposed. To date, these methodologies have been tested only on small-scale academic problems. The purpose of this paper is to systematically study the performance of the non-negative methodology in the context of high performance computing (HPC). PETSc and TAO libraries are, respectively, usedmore » for the parallel environment and optimization solvers. For large-scale problems, it is important for computational scientists to understand the computational performance of current algorithms available in these scientific libraries. The numerical experiments are conducted on the state-of-the-art HPC systems, and a single-core performance model is used to better characterize the efficiency of the solvers. Furthermore, our studies indicate that the proposed non-negative computational framework for diffusion-type equations exhibits excellent strong scaling for real-world large-scale problems.« less

Numerical MHD codes for modeling astrophysical flows

NASA Astrophysics Data System (ADS)

Koldoba, A. V.; Ustyugova, G. V.; Lii, P. S.; Comins, M. L.; Dyda, S.; Romanova, M. M.; Lovelace, R. V. E.

2016-05-01

We describe a Godunov-type magnetohydrodynamic (MHD) code based on the Miyoshi and Kusano (2005) solver which can be used to solve various astrophysical hydrodynamic and MHD problems. The energy equation is in the form of entropy conservation. The code has been implemented on several different coordinate systems: 2.5D axisymmetric cylindrical coordinates, 2D Cartesian coordinates, 2D plane polar coordinates, and fully 3D cylindrical coordinates. Viscosity and diffusivity are implemented in the code to control the accretion rate in the disk and the rate of penetration of the disk matter through the magnetic field lines. The code has been utilized for the numerical investigations of a number of different astrophysical problems, several examples of which are shown.

Parallel Dynamics Simulation Using a Krylov-Schwarz Linear Solution Scheme

DOE PAGES

Abhyankar, Shrirang; Constantinescu, Emil M.; Smith, Barry F.; ...

2016-11-07

Fast dynamics simulation of large-scale power systems is a computational challenge because of the need to solve a large set of stiff, nonlinear differential-algebraic equations at every time step. The main bottleneck in dynamic simulations is the solution of a linear system during each nonlinear iteration of Newton’s method. In this paper, we present a parallel Krylov- Schwarz linear solution scheme that uses the Krylov subspacebased iterative linear solver GMRES with an overlapping restricted additive Schwarz preconditioner. As a result, performance tests of the proposed Krylov-Schwarz scheme for several large test cases ranging from 2,000 to 20,000 buses, including amore » real utility network, show good scalability on different computing architectures.« less

A new relativistic viscous hydrodynamics code and its application to the Kelvin–Helmholtz instability in high-energy heavy-ion collisions

DOE PAGES

Okamoto, Kazuhisa; Nonaka, Chiho

2017-06-09

Here, we construct a new relativistic viscous hydrodynamics code optimized in the Milne coordinates. We also split the conservation equations into an ideal part and a viscous part, using the Strang spitting method. In the code a Riemann solver based on the two-shock approximation is utilized for the ideal part and the Piecewise Exact Solution (PES) method is applied for the viscous part. Furthemore, we check the validity of our numerical calculations by comparing analytical solutions, the viscous Bjorken’s flow and the Israel–Stewart theory in Gubser flow regime. Using the code, we discuss possible development of the Kelvin–Helmholtz instability inmore » high-energy heavy-ion collisions.« less

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.

Parallel Dynamics Simulation Using a Krylov-Schwarz Linear Solution Scheme

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

Abhyankar, Shrirang; Constantinescu, Emil M.; Smith, Barry F.

Fast dynamics simulation of large-scale power systems is a computational challenge because of the need to solve a large set of stiff, nonlinear differential-algebraic equations at every time step. The main bottleneck in dynamic simulations is the solution of a linear system during each nonlinear iteration of Newton’s method. In this paper, we present a parallel Krylov- Schwarz linear solution scheme that uses the Krylov subspacebased iterative linear solver GMRES with an overlapping restricted additive Schwarz preconditioner. As a result, performance tests of the proposed Krylov-Schwarz scheme for several large test cases ranging from 2,000 to 20,000 buses, including amore » real utility network, show good scalability on different computing architectures.« less

Computational Challenges of 3D Radiative Transfer in Atmospheric Models

NASA Astrophysics Data System (ADS)

Jakub, Fabian; Bernhard, Mayer

2017-04-01

The computation of radiative heating and cooling rates is one of the most expensive components in todays atmospheric models. The high computational cost stems not only from the laborious integration over a wide range of the electromagnetic spectrum but also from the fact that solving the integro-differential radiative transfer equation for monochromatic light is already rather involved. This lead to the advent of numerous approximations and parameterizations to reduce the cost of the solver. One of the most prominent one is the so called independent pixel approximations (IPA) where horizontal energy transfer is neglected whatsoever and radiation may only propagate in the vertical direction (1D). Recent studies implicate that the IPA introduces significant errors in high resolution simulations and affects the evolution and development of convective systems. However, using fully 3D solvers such as for example MonteCarlo methods is not even on state of the art supercomputers feasible. The parallelization of atmospheric models is often realized by a horizontal domain decomposition, and hence, horizontal transfer of energy necessitates communication. E.g. a cloud's shadow at a low zenith angle will cast a long shadow and potentially needs to communication through a multitude of processors. Especially light in the solar spectral range may travel long distances through the atmosphere. Concerning highly parallel simulations, it is vital that 3D radiative transfer solvers put a special emphasis on parallel scalability. We will present an introduction to intricacies computing 3D radiative heating and cooling rates as well as report on the parallel performance of the TenStream solver. The TenStream is a 3D radiative transfer solver using the PETSc framework to iteratively solve a set of partial differential equation. We investigate two matrix preconditioners, (a) geometric algebraic multigrid preconditioning(MG+GAMG) and (b) block Jacobi incomplete LU (ILU) factorization. The TenStream solver is tested for up to 4096 cores and shows a parallel scaling efficiency of 80-90% on various supercomputers.

Large-scale 3-D EM modelling with a Block Low-Rank multifrontal direct solver

NASA Astrophysics Data System (ADS)

Shantsev, Daniil V.; Jaysaval, Piyoosh; de la Kethulle de Ryhove, Sébastien; Amestoy, Patrick R.; Buttari, Alfredo; L'Excellent, Jean-Yves; Mary, Theo

2017-06-01

We put forward the idea of using a Block Low-Rank (BLR) multifrontal direct solver to efficiently solve the linear systems of equations arising from a finite-difference discretization of the frequency-domain Maxwell equations for 3-D electromagnetic (EM) problems. The solver uses a low-rank representation for the off-diagonal blocks of the intermediate dense matrices arising in the multifrontal method to reduce the computational load. A numerical threshold, the so-called BLR threshold, controlling the accuracy of low-rank representations was optimized by balancing errors in the computed EM fields against savings in floating point operations (flops). Simulations were carried out over large-scale 3-D resistivity models representing typical scenarios for marine controlled-source EM surveys, and in particular the SEG SEAM model which contains an irregular salt body. The flop count, size of factor matrices and elapsed run time for matrix factorization are reduced dramatically by using BLR representations and can go down to, respectively, 10, 30 and 40 per cent of their full-rank values for our largest system with N = 20.6 million unknowns. The reductions are almost independent of the number of MPI tasks and threads at least up to 90 × 10 = 900 cores. The BLR savings increase for larger systems, which reduces the factorization flop complexity from O(N2) for the full-rank solver to O(Nm) with m = 1.4-1.6. The BLR savings are significantly larger for deep-water environments that exclude the highly resistive air layer from the computational domain. A study in a scenario where simulations are required at multiple source locations shows that the BLR solver can become competitive in comparison to iterative solvers as an engine for 3-D controlled-source electromagnetic Gauss-Newton inversion that requires forward modelling for a few thousand right-hand sides.

Unsteady Aerodynamic Modeling of A Maneuvering Aircraft Using Indicial Functions

DTIC Science & Technology

2016-03-30

indicial functions are directly calculated using the results of unsteady Reynolds-averaged Navier - Stokes simulation and a grid-movement tool. Results are...but meanwhile, the full-order model based on Unsteady Reynolds-averaged Navier - Stokes (URANS) equation is too computationally expensive to be used...The flow solver used in this study solves the unsteady, three-dimensional and compressible Navier - Stokes equations. The equations in terms of

Solution of matrix equations using sparse techniques

NASA Technical Reports Server (NTRS)

Baddourah, Majdi

1994-01-01

The solution of large systems of matrix equations is key to the solution of a large number of scientific and engineering problems. This talk describes the sparse matrix solver developed at Langley which can routinely solve in excess of 263,000 equations in 40 seconds on one Cray C-90 processor. It appears that for large scale structural analysis applications, sparse matrix methods have a significant performance advantage over other methods.

Is it Code Imperfection or 'garbage in Garbage Out'? Outline of Experiences from a Comprehensive Adr Code Verification

NASA Astrophysics Data System (ADS)

Zamani, K.; Bombardelli, F. A.

2013-12-01

ADR equation describes many physical phenomena of interest in the field of water quality in natural streams and groundwater. In many cases such as: density driven flow, multiphase reactive transport, and sediment transport, either one or a number of terms in the ADR equation may become nonlinear. For that reason, numerical tools are the only practical choice to solve these PDEs. All numerical solvers developed for transport equation need to undergo code verification procedure before they are put in to practice. Code verification is a mathematical activity to uncover failures and check for rigorous discretization of PDEs and implementation of initial/boundary conditions. In the context computational PDE verification is not a well-defined procedure on a clear path. Thus, verification tests should be designed and implemented with in-depth knowledge of numerical algorithms and physics of the phenomena as well as mathematical behavior of the solution. Even test results need to be mathematically analyzed to distinguish between an inherent limitation of algorithm and a coding error. Therefore, it is well known that code verification is a state of the art, in which innovative methods and case-based tricks are very common. This study presents full verification of a general transport code. To that end, a complete test suite is designed to probe the ADR solver comprehensively and discover all possible imperfections. In this study we convey our experiences in finding several errors which were not detectable with routine verification techniques. We developed a test suit including hundreds of unit tests and system tests. The test package has gradual increment in complexity such that tests start from simple and increase to the most sophisticated level. Appropriate verification metrics are defined for the required capabilities of the solver as follows: mass conservation, convergence order, capabilities in handling stiff problems, nonnegative concentration, shape preservation, and spurious wiggles. Thereby, we provide objective, quantitative values as opposed to subjective qualitative descriptions as 'weak' or 'satisfactory' agreement with those metrics. We start testing from a simple case of unidirectional advection, then bidirectional advection and tidal flow and build up to nonlinear cases. We design tests to check nonlinearity in velocity, dispersivity and reactions. For all of the mentioned cases we conduct mesh convergence tests. These tests compare the results' order of accuracy versus the formal order of accuracy of discretization. The concealing effect of scales (Peclet and Damkohler numbers) on the mesh convergence study and appropriate remedies are also discussed. For the cases in which the appropriate benchmarks for mesh convergence study are not available we utilize Symmetry, Complete Richardson Extrapolation and Method of False Injection to uncover bugs. Detailed discussions of capabilities of the mentioned code verification techniques are given. Auxiliary subroutines for automation of the test suit and report generation are designed. All in all, the test package is not only a robust tool for code verification but also it provides comprehensive insight on the ADR solvers capabilities. Such information is essential for any rigorous computational modeling of ADR equation for surface/subsurface pollution transport.

A new fast direct solver for the boundary element method

NASA Astrophysics Data System (ADS)

Huang, S.; Liu, Y. J.

2017-09-01

A new fast direct linear equation solver for the boundary element method (BEM) is presented in this paper. The idea of the new fast direct solver stems from the concept of the hierarchical off-diagonal low-rank matrix. The hierarchical off-diagonal low-rank matrix can be decomposed into the multiplication of several diagonal block matrices. The inverse of the hierarchical off-diagonal low-rank matrix can be calculated efficiently with the Sherman-Morrison-Woodbury formula. In this paper, a more general and efficient approach to approximate the coefficient matrix of the BEM with the hierarchical off-diagonal low-rank matrix is proposed. Compared to the current fast direct solver based on the hierarchical off-diagonal low-rank matrix, the proposed method is suitable for solving general 3-D boundary element models. Several numerical examples of 3-D potential problems with the total number of unknowns up to above 200,000 are presented. The results show that the new fast direct solver can be applied to solve large 3-D BEM models accurately and with better efficiency compared with the conventional BEM.

libmpdata++ 1.0: a library of parallel MPDATA solvers for systems of generalised transport equations

NASA Astrophysics Data System (ADS)

Jaruga, A.; Arabas, S.; Jarecka, D.; Pawlowska, H.; Smolarkiewicz, P. K.; Waruszewski, M.

2015-04-01

This paper accompanies the first release of libmpdata++, a C++ library implementing the multi-dimensional positive-definite advection transport algorithm (MPDATA) on regular structured grid. The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitative representations of simulation results. Examples of applications include homogeneous transport in one, two and three dimensions in Cartesian and spherical domains; a shallow-water system compared with analytical solution (originally derived for a 2-D case); and a buoyant convection problem in an incompressible Boussinesq fluid with interfacial instability. All the examples are implemented out of the library tree. Regardless of the differences in the problem dimensionality, right-hand-side terms, boundary conditions and parallelisation approach, all the examples use the same unmodified library, which is a key goal of libmpdata++ design. The design, based on the principle of separation of concerns, prioritises the user and developer productivity. The libmpdata++ library is implemented in C++, making use of the Blitz++ multi-dimensional array containers, and is released as free/libre and open-source software.

libmpdata++ 0.1: a library of parallel MPDATA solvers for systems of generalised transport equations

NASA Astrophysics Data System (ADS)

Jaruga, A.; Arabas, S.; Jarecka, D.; Pawlowska, H.; Smolarkiewicz, P. K.; Waruszewski, M.

2014-11-01

This paper accompanies first release of libmpdata++, a C++ library implementing the Multidimensional Positive-Definite Advection Transport Algorithm (MPDATA). The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitative representations of simulation results. Examples of applications include: homogeneous transport in one, two and three dimensions in Cartesian and spherical domains; shallow-water system compared with analytical solution (originally derived for a 2-D case); and a buoyant convection problem in an incompressible Boussinesq fluid with interfacial instability. All the examples are implemented out of the library tree. Regardless of the differences in the problem dimensionality, right-hand-side terms, boundary conditions and parallelisation approach, all the examples use the same unmodified library, which is a key goal of libmpdata++ design. The design, based on the principle of separation of concerns, prioritises the user and developer productivity. The libmpdata++ library is implemented in C++, making use of the Blitz++ multi-dimensional array containers, and is released as free/libre and open-source software.

Thermal radiation heat transfer in participating media by finite volume discretization using collimated beam incidence

NASA Astrophysics Data System (ADS)

Harijishnu, R.; Jayakumar, J. S.

2017-09-01

The main objective of this paper is to study the heat transfer rate of thermal radiation in participating media. For that, a generated collimated beam has been passed through a two dimensional slab model of flint glass with a refractive index 2. Both Polar and azimuthal angle have been varied to generate such a beam. The Temperature of the slab and Snells law has been validated by Radiation Transfer Equation (RTE) in OpenFOAM (Open Field Operation and Manipulation), a CFD software which is the major computational tool used in Industry and research applications where the source code is modified in which radiation heat transfer equation is added to the case and different radiation heat transfer models are utilized. This work concentrates on the numerical strategies involving both transparent and participating media. Since Radiation Transfer Equation (RTE) is difficult to solve, the purpose of this paper is to use existing solver buoyantSimlpeFoam to solve radiation model in the participating media by compiling the source code to obtain the heat transfer rate inside the slab by varying the Intensity of radiation. The Finite Volume Method (FVM) is applied to solve the Radiation Transfer Equation (RTE) governing the above said physical phenomena.

Formal Solutions for Polarized Radiative Transfer. II. High-order Methods

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

Janett, Gioele; Steiner, Oskar; Belluzzi, Luca, E-mail: gioele.janett@irsol.ch

When integrating the radiative transfer equation for polarized light, the necessity of high-order numerical methods is well known. In fact, well-performing high-order formal solvers enable higher accuracy and the use of coarser spatial grids. Aiming to provide a clear comparison between formal solvers, this work presents different high-order numerical schemes and applies the systematic analysis proposed by Janett et al., emphasizing their advantages and drawbacks in terms of order of accuracy, stability, and computational cost.

Implementation of a Pseudo-Bending Seismic Travel-Time Calculator in a Distributed Parallel Computing Environment

DTIC Science & Technology

2008-09-01

algorithms that have been proposed to accomplish it fall into three broad categories. Eikonal solvers (e.g., Vidale, 1988, 1990; Podvin and Lecomte, 1991...difference eikonal solvers, the FMM algorithm works by following a wavefront as it moves across a volume of grid points, updating the travel times in...the grid according to the eikonal differential equation, using a second-order finite-difference scheme. We chose to use FMM for our comparison because

An unstructured shock-fitting solver for hypersonic plasma flows in chemical non-equilibrium

NASA Astrophysics Data System (ADS)

Pepe, R.; Bonfiglioli, A.; D'Angola, A.; Colonna, G.; Paciorri, R.

2015-11-01

A CFD solver, using Residual Distribution Schemes on unstructured grids, has been extended to deal with inviscid chemical non-equilibrium flows. The conservative equations have been coupled with a kinetic model for argon plasma which includes the argon metastable state as independent species, taking into account electron-atom and atom-atom processes. Results in the case of an hypersonic flow around an infinite cylinder, obtained by using both shock-capturing and shock-fitting approaches, show higher accuracy of the shock-fitting approach.

Notes on the ExactPack Implementation of the DSD Explosive Arc Solver

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

Kaul, Ann; Doebling, Scott William

It has been shown above that the discretization scheme implemented in the ExactPack solver for the DSD Explosive Arc equation is consistent with the Explosive Arc PDE. In addition, a stability analysis has provided a CFL condition for a stable time step. Together, consistency and stability imply convergence of the scheme, which is expected to be close to first-order in time and second-order in space. It is understood that the nonlinearity of the underlying PDE will affect this rate somewhat.

User's Manual for FOMOCO Utilities-Force and Moment Computation Tools for Overset Grids

NASA Technical Reports Server (NTRS)

Chan, William M.; Buning, Pieter G.

1996-01-01

In the numerical computations of flows around complex configurations, accurate calculations of force and moment coefficients for aerodynamic surfaces are required. When overset grid methods are used, the surfaces on which force and moment coefficients are sought typically consist of a collection of overlapping surface grids. Direct integration of flow quantities on the overlapping grids would result in the overlapped regions being counted more than once. The FOMOCO Utilities is a software package for computing flow coefficients (force, moment, and mass flow rate) on a collection of overset surfaces with accurate accounting of the overlapped zones. FOMOCO Utilities can be used in stand-alone mode or in conjunction with the Chimera overset grid compressible Navier-Stokes flow solver OVERFLOW. The software package consists of two modules corresponding to a two-step procedure: (1) hybrid surface grid generation (MIXSUR module), and (2) flow quantities integration (OVERINT module). Instructions on how to use this software package are described in this user's manual. Equations used in the flow coefficients calculation are given in Appendix A.

FELIX-1.0: A finite element solver for the time dependent generator coordinate method with the Gaussian overlap approximation

DOE PAGES

Regnier, D.; Verriere, M.; Dubray, N.; ...

2015-11-30

In this study, we describe the software package FELIX that solves the equations of the time-dependent generator coordinate method (TDGCM) in NN-dimensions (N ≥ 1) under the Gaussian overlap approximation. The numerical resolution is based on the Galerkin finite element discretization of the collective space and the Crank–Nicolson scheme for time integration. The TDGCM solver is implemented entirely in C++. Several additional tools written in C++, Python or bash scripting language are also included for convenience. In this paper, the solver is tested with a series of benchmarks calculations. We also demonstrate the ability of our code to handle amore » realistic calculation of fission dynamics.« less

A generalized Poisson solver for first-principles device simulations

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

Bani-Hashemian, Mohammad Hossein; VandeVondele, Joost, E-mail: joost.vandevondele@mat.ethz.ch; Brück, Sascha

2016-01-28

Electronic structure calculations of atomistic systems based on density functional theory involve solving the Poisson equation. In this paper, we present a plane-wave based algorithm for solving the generalized Poisson equation subject to periodic or homogeneous Neumann conditions on the boundaries of the simulation cell and Dirichlet type conditions imposed at arbitrary subdomains. In this way, source, drain, and gate voltages can be imposed across atomistic models of electronic devices. Dirichlet conditions are enforced as constraints in a variational framework giving rise to a saddle point problem. The resulting system of equations is then solved using a stationary iterative methodmore » in which the generalized Poisson operator is preconditioned with the standard Laplace operator. The solver can make use of any sufficiently smooth function modelling the dielectric constant, including density dependent dielectric continuum models. For all the boundary conditions, consistent derivatives are available and molecular dynamics simulations can be performed. The convergence behaviour of the scheme is investigated and its capabilities are demonstrated.« less

Ideal GLM-MHD: About the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations

NASA Astrophysics Data System (ADS)

Derigs, Dominik; Winters, Andrew R.; Gassner, Gregor J.; Walch, Stefanie; Bohm, Marvin

2018-07-01

The paper presents two contributions in the context of the numerical simulation of magnetized fluid dynamics. First, we show how to extend the ideal magnetohydrodynamics (MHD) equations with an inbuilt magnetic field divergence cleaning mechanism in such a way that the resulting model is consistent with the second law of thermodynamics. As a byproduct of these derivations, we show that not all of the commonly used divergence cleaning extensions of the ideal MHD equations are thermodynamically consistent. Secondly, we present a numerical scheme obtained by constructing a specific finite volume discretization that is consistent with the discrete thermodynamic entropy. It includes a mechanism to control the discrete divergence error of the magnetic field by construction and is Galilean invariant. We implement the new high-order MHD solver in the adaptive mesh refinement code FLASH where we compare the divergence cleaning efficiency to the constrained transport solver available in FLASH (unsplit staggered mesh scheme).

Function-Space-Based Solution Scheme for the Size-Modified Poisson-Boltzmann Equation in Full-Potential DFT.

PubMed

Ringe, Stefan; Oberhofer, Harald; Hille, Christoph; Matera, Sebastian; Reuter, Karsten

2016-08-09

The size-modified Poisson-Boltzmann (MPB) equation is an efficient implicit solvation model which also captures electrolytic solvent effects. It combines an account of the dielectric solvent response with a mean-field description of solvated finite-sized ions. We present a general solution scheme for the MPB equation based on a fast function-space-oriented Newton method and a Green's function preconditioned iterative linear solver. In contrast to popular multigrid solvers, this approach allows us to fully exploit specialized integration grids and optimized integration schemes. We describe a corresponding numerically efficient implementation for the full-potential density-functional theory (DFT) code FHI-aims. We show that together with an additional Stern layer correction the DFT+MPB approach can describe the mean activity coefficient of a KCl aqueous solution over a wide range of concentrations. The high sensitivity of the calculated activity coefficient on the employed ionic parameters thereby suggests to use extensively tabulated experimental activity coefficients of salt solutions for a systematic parametrization protocol.

Higher Order Time Integration Schemes for the Unsteady Navier-Stokes Equations on Unstructured Meshes

NASA Technical Reports Server (NTRS)

Jothiprasad, Giridhar; Mavriplis, Dimitri J.; Caughey, David A.; Bushnell, Dennis M. (Technical Monitor)

2002-01-01

The efficiency gains obtained using higher-order implicit Runge-Kutta schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier-Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each timestep are presented. The first algorithm (NMG) is a pseudo-time-stepping scheme which employs a non-linear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on Inexact Newton's methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson's iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the Generalized Minimal Residual method. Results demonstrating the relative superiority of these Newton's methods based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes with the more efficient nonlinear solvers.

One shot methods for optimal control of distributed parameter systems 1: Finite dimensional control

NASA Technical Reports Server (NTRS)

Taasan, Shlomo

1991-01-01

The efficient numerical treatment of optimal control problems governed by elliptic partial differential equations (PDEs) and systems of elliptic PDEs, where the control is finite dimensional is discussed. Distributed control as well as boundary control cases are discussed. The main characteristic of the new methods is that they are designed to solve the full optimization problem directly, rather than accelerating a descent method by an efficient multigrid solver for the equations involved. The methods use the adjoint state in order to achieve efficient smoother and a robust coarsening strategy. The main idea is the treatment of the control variables on appropriate scales, i.e., control variables that correspond to smooth functions are solved for on coarse grids depending on the smoothness of these functions. Solution of the control problems is achieved with the cost of solving the constraint equations about two to three times (by a multigrid solver). Numerical examples demonstrate the effectiveness of the method proposed in distributed control case, pointwise control and boundary control problems.

Solving Partial Differential Equations in a data-driven multiprocessor environment

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

Gaudiot, J.L.; Lin, C.M.; Hosseiniyar, M.

1988-12-31

Partial differential equations can be found in a host of engineering and scientific problems. The emergence of new parallel architectures has spurred research in the definition of parallel PDE solvers. Concurrently, highly programmable systems such as data-how architectures have been proposed for the exploitation of large scale parallelism. The implementation of some Partial Differential Equation solvers (such as the Jacobi method) on a tagged token data-flow graph is demonstrated here. Asynchronous methods (chaotic relaxation) are studied and new scheduling approaches (the Token No-Labeling scheme) are introduced in order to support the implementation of the asychronous methods in a data-driven environment.more » New high-level data-flow language program constructs are introduced in order to handle chaotic operations. Finally, the performance of the program graphs is demonstrated by a deterministic simulation of a message passing data-flow multiprocessor. An analysis of the overhead in the data-flow graphs is undertaken to demonstrate the limits of parallel operations in dataflow PDE program graphs.« less

Parallel-vector solution of large-scale structural analysis problems on supercomputers

NASA Technical Reports Server (NTRS)

Storaasli, Olaf O.; Nguyen, Duc T.; Agarwal, Tarun K.

1989-01-01

A direct linear equation solution method based on the Choleski factorization procedure is presented which exploits both parallel and vector features of supercomputers. The new equation solver is described, and its performance is evaluated by solving structural analysis problems on three high-performance computers. The method has been implemented using Force, a generic parallel FORTRAN language.

Novel Methods for Electromagnetic Simulation and Design

DTIC Science & Technology

2016-08-03

The resulting discretized integral equations are compatible with fast multipoleaccelerated solvers and will form the basis for high fidelity...expansion”) which are high-order, efficient and easy to use on arbitrarily triangulated surfaces. The resulting discretized integral equations are...created a user interface compatible with both low and high order discretizations , and implemented the generalized Debye approach of [4]. The

Highly efficient and exact method for parallelization of grid-based algorithms and its implementation in DelPhi

PubMed Central

Li, Chuan; Li, Lin; Zhang, Jie; Alexov, Emil

2012-01-01

The Gauss-Seidel method is a standard iterative numerical method widely used to solve a system of equations and, in general, is more efficient comparing to other iterative methods, such as the Jacobi method. However, standard implementation of the Gauss-Seidel method restricts its utilization in parallel computing due to its requirement of using updated neighboring values (i.e., in current iteration) as soon as they are available. Here we report an efficient and exact (not requiring assumptions) method to parallelize iterations and to reduce the computational time as a linear/nearly linear function of the number of CPUs. In contrast to other existing solutions, our method does not require any assumptions and is equally applicable for solving linear and nonlinear equations. This approach is implemented in the DelPhi program, which is a finite difference Poisson-Boltzmann equation solver to model electrostatics in molecular biology. This development makes the iterative procedure on obtaining the electrostatic potential distribution in the parallelized DelPhi several folds faster than that in the serial code. Further we demonstrate the advantages of the new parallelized DelPhi by computing the electrostatic potential and the corresponding energies of large supramolecular structures. PMID:22674480

Stochastic Partial Differential Equation Solver for Hydroacoustic Modeling: Improvements to Paracousti Sound Propagation Solver

NASA Astrophysics Data System (ADS)

Preston, L. A.

2017-12-01

Marine hydrokinetic (MHK) devices offer a clean, renewable alternative energy source for the future. Responsible utilization of MHK devices, however, requires that the effects of acoustic noise produced by these devices on marine life and marine-related human activities be well understood. Paracousti is a 3-D full waveform acoustic modeling suite that can accurately propagate MHK noise signals in the complex bathymetry found in the near-shore to open ocean environment and considers real properties of the seabed, water column, and air-surface interface. However, this is a deterministic simulation that assumes the environment and source are exactly known. In reality, environmental and source characteristics are often only known in a statistical sense. Thus, to fully characterize the expected noise levels within the marine environment, this uncertainty in environmental and source factors should be incorporated into the acoustic simulations. One method is to use Monte Carlo (MC) techniques where simulation results from a large number of deterministic solutions are aggregated to provide statistical properties of the output signal. However, MC methods can be computationally prohibitive since they can require tens of thousands or more simulations to build up an accurate representation of those statistical properties. An alternative method, using the technique of stochastic partial differential equations (SPDE), allows computation of the statistical properties of output signals at a small fraction of the computational cost of MC. We are developing a SPDE solver for the 3-D acoustic wave propagation problem called Paracousti-UQ to help regulators and operators assess the statistical properties of environmental noise produced by MHK devices. In this presentation, we present the SPDE method and compare statistical distributions of simulated acoustic signals in simple models to MC simulations to show the accuracy and efficiency of the SPDE method. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525.

A CFD Heterogeneous Parallel Solver Based on Collaborating CPU and GPU

NASA Astrophysics Data System (ADS)

Lai, Jianqi; Tian, Zhengyu; Li, Hua; Pan, Sha

2018-03-01

Since Graphic Processing Unit (GPU) has a strong ability of floating-point computation and memory bandwidth for data parallelism, it has been widely used in the areas of common computing such as molecular dynamics (MD), computational fluid dynamics (CFD) and so on. The emergence of compute unified device architecture (CUDA), which reduces the complexity of compiling program, brings the great opportunities to CFD. There are three different modes for parallel solution of NS equations: parallel solver based on CPU, parallel solver based on GPU and heterogeneous parallel solver based on collaborating CPU and GPU. As we can see, GPUs are relatively rich in compute capacity but poor in memory capacity and the CPUs do the opposite. We need to make full use of the GPUs and CPUs, so a CFD heterogeneous parallel solver based on collaborating CPU and GPU has been established. Three cases are presented to analyse the solver’s computational accuracy and heterogeneous parallel efficiency. The numerical results agree well with experiment results, which demonstrate that the heterogeneous parallel solver has high computational precision. The speedup on a single GPU is more than 40 for laminar flow, it decreases for turbulent flow, but it still can reach more than 20. What’s more, the speedup increases as the grid size becomes larger.

Extension of the Time-Spectral Approach to Overset Solvers for Arbitrary Motion

NASA Technical Reports Server (NTRS)

Leffell, Joshua Isaac; Murman, Scott M.; Pulliam, Thomas H.

2012-01-01

Forced periodic flows arise in a broad range of aerodynamic applications such as rotorcraft, turbomachinery, and flapping wing configurations. Standard practice involves solving the unsteady flow equations forward in time until the initial transient exits the domain and a statistically stationary flow is achieved. It is often required to simulate through several periods to remove the initial transient making unsteady design optimization prohibitively expensive for most realistic problems. An effort to reduce the computational cost of these calculations led to the development of the Harmonic Balance method [1, 2] which capitalizes on the periodic nature of the solution. The approach exploits the fact that forced temporally periodic flow, while varying in the time domain, is invariant in the frequency domain. Expanding the temporal variation at each spatial node into a Fourier series transforms the unsteady governing equations into a steady set of equations in integer harmonics that can be tackled with the acceleration techniques afforded to steady-state flow solvers. Other similar approaches, such as the Nonlinear Frequency Domain [3,4,5], Reduced Frequency [6] and Time-Spectral [7, 8, 9] methods, were developed shortly thereafter. Additionally, adjoint-based optimization techniques can be applied [10, 11] as well as frequency-adaptive methods [12, 13, 14] to provide even more flexibility to the method. The Fourier temporal basis functions imply spectral convergence as the number of harmonic modes, and correspondingly number of time samples, N, is increased. Some elect to solve the equations in the frequency domain directly, while others choose to transform the equations back into the time domain to simplify the process of adding this capability to existing solvers, but each harnesses the underlying steady solution in the frequency domain. These temporal projection methods will herein be collectively referred to as Time-Spectral methods. Time-Spectral methods have demonstrated marked success in reducing the computational costs associated with simulating periodic forced flows, but have yet to be fully applied to overset or Cartesian solvers for arbitrary motion with dynamic hole-cutting. Overset and Cartesian grid methodologies are versatile techniques capable of handling complex geometry configurations in practical engineering applications, and the combination of the Time-Spectral approach with this general capability potentially provides an enabling new design and analysis tool. In an arbitrary moving-body scenario for these approaches, a Lagrangian body moves through a fixed Eulerian mesh and mesh points in the Eulerian mesh interior to the solid body are removed (cut or blanked), leaving a hole in the Eulerian mesh. During the dynamic motion some gridpoints in the domain are blanked and do not have a complete set of time-samples preventing a direct implementation of the Time-Spectral method. Murman[6] demonstrated the Time-Spectral approach for a Cartesian solver with a rigid domain motion, wherein the hole cutting remains constant. Similarly, Custer et al. [15, 16] used the NASA overset OVERFLOW solver and limited the amount of relative motion to ensure static hole-cutting and interpolation. Recently, Mavriplis and Mundis[17] demonstrated a qualitative method for applying the Time-Spectral approach to an unstructured overset solver for arbitrary motion. The goal of the current work is to develop a robust and general method for handling arbitrary motion with the Time-Spectral approach within an overset or Cartesian mesh method, while still approaching the spectral convergence rate of the original Time-Spectral approach. The viscous OVERFLOW solver will be augmented with the new Time-Spectral algorithm and the capability of the method for benchmark problems in rotorcraft and turbomachinery will be demonstrated. This abstract begins with a brief synopsis of the Time-Spectral approach for overset grids and provides details of e current approach to allow for arbitrary motion. Model problem results in one and two dimensions are included to demonstrate the viability of the method and the convergence properties. Section IV briefly outlines the implementation into the OVERFLOW solver, and the abstract closes with a description of the benchmark test cases which will be included in the final paper.

Multi-GPU three dimensional Stokes solver for simulating glacier flow

NASA Astrophysics Data System (ADS)

Licul, Aleksandar; Herman, Frédéric; Podladchikov, Yuri; Räss, Ludovic; Omlin, Samuel

2016-04-01

Here we present how we have recently developed a three-dimensional Stokes solver on the GPUs and apply it to a glacier flow. We numerically solve the Stokes momentum balance equations together with the incompressibility equation, while also taking into account strong nonlinearities for ice rheology. We have developed a fully three-dimensional numerical MATLAB application based on an iterative finite difference scheme with preconditioning of residuals. Differential equations are discretized on a regular staggered grid. We have ported it to C-CUDA to run it on GPU's in parallel, using MPI. We demonstrate the accuracy and efficiency of our developed model by manufactured analytical solution test for three-dimensional Stokes ice sheet models (Leng et al.,2013) and by comparison with other well-established ice sheet models on diagnostic ISMIP-HOM benchmark experiments (Pattyn et al., 2008). The results show that our developed model is capable to accurately and efficiently solve Stokes system of equations in a variety of different test scenarios, while preserving good parallel efficiency on up to 80 GPU's. For example, in 3D test scenarios with 250000 grid points our solver converges in around 3 minutes for single precision computations and around 10 minutes for double precision computations. We have also optimized the developed code to efficiently run on our newly acquired state-of-the-art GPU cluster octopus. This allows us to solve our problem on more than 20 million grid points, by just increasing the number of GPU used, while keeping the computation time the same. In future work we will apply our solver to real world applications and implement the free surface evolution capabilities. REFERENCES Leng,W.,Ju,L.,Gunzburger,M. & Price,S., 2013. Manufactured solutions and the verification of three-dimensional stokes ice-sheet models. Cryosphere 7,19-29. Pattyn, F., Perichon, L., Aschwanden, A., Breuer, B., de Smedt, B., Gagliardini, O., Gudmundsson,G.H., Hindmarsh, R.C.A., Hubbard, A., Johnson, J.V., Kleiner, T., Konovalov,Y., Martin, C., Payne, A.J., Pollard, D., Price, S., Rckamp, M., Saito, F., Souk, O.,Sugiyama, S. & Zwinger, T., 2008. Benchmark experiments for higher-order and full-stokes ice sheet models (ismiphom). The Cryosphere 2, 95-108.

TransCut: interactive rendering of translucent cutouts.

PubMed

Li, Dongping; Sun, Xin; Ren, Zhong; Lin, Stephen; Tong, Yiying; Guo, Baining; Zhou, Kun

2013-03-01

We present TransCut, a technique for interactive rendering of translucent objects undergoing fracturing and cutting operations. As the object is fractured or cut open, the user can directly examine and intuitively understand the complex translucent interior, as well as edit material properties through painting on cross sections and recombining the broken pieces—all with immediate and realistic visual feedback. This new mode of interaction with translucent volumes is made possible with two technical contributions. The first is a novel solver for the diffusion equation (DE) over a tetrahedral mesh that produces high-quality results comparable to the state-of-art finite element method (FEM) of Arbree et al. but at substantially higher speeds. This accuracy and efficiency is obtained by computing the discrete divergences of the diffusion equation and constructing the DE matrix using analytic formulas derived for linear finite elements. The second contribution is a multiresolution algorithm to significantly accelerate our DE solver while adapting to the frequent changes in topological structure of dynamic objects. The entire multiresolution DE solver is highly parallel and easily implemented on the GPU. We believe TransCut provides a novel visual effect for heterogeneous translucent objects undergoing fracturing and cutting operations.

An Empirical Temperature Variance Source Model in Heated Jets

NASA Technical Reports Server (NTRS)

Khavaran, Abbas; Bridges, James

2012-01-01

An acoustic analogy approach is implemented that models the sources of jet noise in heated jets. The equivalent sources of turbulent mixing noise are recognized as the differences between the fluctuating and Favre-averaged Reynolds stresses and enthalpy fluxes. While in a conventional acoustic analogy only Reynolds stress components are scrutinized for their noise generation properties, it is now accepted that a comprehensive source model should include the additional entropy source term. Following Goldstein s generalized acoustic analogy, the set of Euler equations are divided into two sets of equations that govern a non-radiating base flow plus its residual components. When the base flow is considered as a locally parallel mean flow, the residual equations may be rearranged to form an inhomogeneous third-order wave equation. A general solution is written subsequently using a Green s function method while all non-linear terms are treated as the equivalent sources of aerodynamic sound and are modeled accordingly. In a previous study, a specialized Reynolds-averaged Navier-Stokes (RANS) solver was implemented to compute the variance of thermal fluctuations that determine the enthalpy flux source strength. The main objective here is to present an empirical model capable of providing a reasonable estimate of the stagnation temperature variance in a jet. Such a model is parameterized as a function of the mean stagnation temperature gradient in the jet, and is evaluated using commonly available RANS solvers. The ensuing thermal source distribution is compared with measurements as well as computational result from a dedicated RANS solver that employs an enthalpy variance and dissipation rate model. Turbulent mixing noise predictions are presented for a wide range of jet temperature ratios from 1.0 to 3.20.

Wind-US Users Guide Version 3.0

NASA Technical Reports Server (NTRS)

Yoder, Dennis A.

2016-01-01

Wind-US is a computational platform which may be used to numerically solve various sets of equations governing physical phenomena. Currently, the code supports the solution of the Euler and Navier-Stokes equations of fluid mechanics, along with supporting equation sets governing turbulent and chemically reacting flows. Wind-US is a product of the NPARC Alliance, a partnership between the NASA Glenn Research Center (GRC) and the Arnold Engineering Development Complex (AEDC) dedicated to the establishment of a national, applications-oriented flow simulation capability. The Boeing Company has also been closely associated with the Alliance since its inception, and represents the interests of the NPARC User's Association. The "Wind-US User's Guide" describes the operation and use of Wind-US, including: a basic tutorial; the physical and numerical models that are used; the boundary conditions; monitoring convergence; the files that are read and/or written; parallel execution; and a complete list of input keywords and test options. For current information about Wind-US and the NPARC Alliance, please see the Wind-US home page at http://www.grc.nasa.gov/WWW/winddocs/ and the NPARC Alliance home page at http://www.grc.nasa.gov/WWW/wind/. This manual describes the operation and use of Wind-US, a computational platform which may be used to numerically solve various sets of equations governing physical phenomena. Wind-US represents a merger of the capabilities of four CFD codes - NASTD (a structured grid flow solver developed at McDonnell Douglas, now part of Boeing), NPARC (the original NPARC Alliance structured grid flow solver), NXAIR (an AEDC structured grid code used primarily for store separation analysis), and ICAT (an unstructured grid flow solver developed at the Rockwell Science Center and Boeing).

ML 3.0 smoothed aggregation user's guide.

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

Sala, Marzio; Hu, Jonathan Joseph; Tuminaro, Raymond Stephen

2004-05-01

ML is a multigrid preconditioning package intended to solve linear systems of equations Az = b where A is a user supplied n x n sparse matrix, b is a user supplied vector of length n and x is a vector of length n to be computed. ML should be used on large sparse linear systems arising from partial differential equation (PDE) discretizations. While technically any linear system can be considered, ML should be used on linear systems that correspond to things that work well with multigrid methods (e.g. elliptic PDEs). ML can be used as a stand-alone package ormore » to generate preconditioners for a traditional iterative solver package (e.g. Krylov methods). We have supplied support for working with the AZTEC 2.1 and AZTECOO iterative package [15]. However, other solvers can be used by supplying a few functions. This document describes one specific algebraic multigrid approach: smoothed aggregation. This approach is used within several specialized multigrid methods: one for the eddy current formulation for Maxwell's equations, and a multilevel and domain decomposition method for symmetric and non-symmetric systems of equations (like elliptic equations, or compressible and incompressible fluid dynamics problems). Other methods exist within ML but are not described in this document. Examples are given illustrating the problem definition and exercising multigrid options.« less

ML 3.1 smoothed aggregation user's guide.

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

Sala, Marzio; Hu, Jonathan Joseph; Tuminaro, Raymond Stephen

2004-10-01

ML is a multigrid preconditioning package intended to solve linear systems of equations Ax = b where A is a user supplied n x n sparse matrix, b is a user supplied vector of length n and x is a vector of length n to be computed. ML should be used on large sparse linear systems arising from partial differential equation (PDE) discretizations. While technically any linear system can be considered, ML should be used on linear systems that correspond to things that work well with multigrid methods (e.g. elliptic PDEs). ML can be used as a stand-alone package ormore » to generate preconditioners for a traditional iterative solver package (e.g. Krylov methods). We have supplied support for working with the Aztec 2.1 and AztecOO iterative package [16]. However, other solvers can be used by supplying a few functions. This document describes one specific algebraic multigrid approach: smoothed aggregation. This approach is used within several specialized multigrid methods: one for the eddy current formulation for Maxwell's equations, and a multilevel and domain decomposition method for symmetric and nonsymmetric systems of equations (like elliptic equations, or compressible and incompressible fluid dynamics problems). Other methods exist within ML but are not described in this document. Examples are given illustrating the problem definition and exercising multigrid options.« less

A Stabilized Finite Element Method for Modified Poisson-Nernst-Planck Equations to Determine Ion Flow Through a Nanopore

PubMed Central

Chaudhry, Jehanzeb Hameed; Comer, Jeffrey; Aksimentiev, Aleksei; Olson, Luke N.

2013-01-01

The conventional Poisson-Nernst-Planck equations do not account for the finite size of ions explicitly. This leads to solutions featuring unrealistically high ionic concentrations in the regions subject to external potentials, in particular, near highly charged surfaces. A modified form of the Poisson-Nernst-Planck equations accounts for steric effects and results in solutions with finite ion concentrations. Here, we evaluate numerical methods for solving the modified Poisson-Nernst-Planck equations by modeling electric field-driven transport of ions through a nanopore. We describe a novel, robust finite element solver that combines the applications of the Newton's method to the nonlinear Galerkin form of the equations, augmented with stabilization terms to appropriately handle the drift-diffusion processes. To make direct comparison with particle-based simulations possible, our method is specifically designed to produce solutions under periodic boundary conditions and to conserve the number of ions in the solution domain. We test our finite element solver on a set of challenging numerical experiments that include calculations of the ion distribution in a volume confined between two charged plates, calculations of the ionic current though a nanopore subject to an external electric field, and modeling the effect of a DNA molecule on the ion concentration and nanopore current. PMID:24363784

Transport Equation Based Wall Distance Computations Aimed at Flows With Time-Dependent Geometry

NASA Technical Reports Server (NTRS)

Tucker, Paul G.; Rumsey, Christopher L.; Bartels, Robert E.; Biedron, Robert T.

2003-01-01

Eikonal, Hamilton-Jacobi and Poisson equations can be used for economical nearest wall distance computation and modification. Economical computations may be especially useful for aeroelastic and adaptive grid problems for which the grid deforms, and the nearest wall distance needs to be repeatedly computed. Modifications are directed at remedying turbulence model defects. For complex grid structures, implementation of the Eikonal and Hamilton-Jacobi approaches is not straightforward. This prohibits their use in industrial CFD solvers. However, both the Eikonal and Hamilton-Jacobi equations can be written in advection and advection-diffusion forms, respectively. These, like the Poisson s Laplacian, are commonly occurring industrial CFD solver elements. Use of the NASA CFL3D code to solve the Eikonal and Hamilton-Jacobi equations in advective-based forms is explored. The advection-based distance equations are found to have robust convergence. Geometries studied include single and two element airfoils, wing body and double delta configurations along with a complex electronics system. It is shown that for Eikonal accuracy, upwind metric differences are required. The Poisson approach is found effective and, since it does not require offset metric evaluations, easiest to implement. The sensitivity of flow solutions to wall distance assumptions is explored. Generally, results are not greatly affected by wall distance traits.

Transport Equation Based Wall Distance Computations Aimed at Flows With Time-Dependent Geometry

NASA Technical Reports Server (NTRS)

Tucker, Paul G.; Rumsey, Christopher L.; Bartels, Robert E.; Biedron, Robert T.

2003-01-01

Eikonal, Hamilton-Jacobi and Poisson equations can be used for economical nearest wall distance computation and modification. Economical computations may be especially useful for aeroelastic and adaptive grid problems for which the grid deforms, and the nearest wall distance needs to be repeatedly computed. Modifications are directed at remedying turbulence model defects. For complex grid structures, implementation of the Eikonal and Hamilton-Jacobi approaches is not straightforward. This prohibits their use in industrial CFD solvers. However, both the Eikonal and Hamilton-Jacobi equations can be written in advection and advection-diffusion forms, respectively. These, like the Poisson's Laplacian, are commonly occurring industrial CFD solver elements. Use of the NASA CFL3D code to solve the Eikonal and Hamilton-Jacobi equations in advective-based forms is explored. The advection-based distance equations are found to have robust convergence. Geometries studied include single and two element airfoils, wing body and double delta configurations along with a complex electronics system. It is shown that for Eikonal accuracy, upwind metric differences are required. The Poisson approach is found effective and, since it does not require offset metric evaluations, easiest to implement. The sensitivity of flow solutions to wall distance assumptions is explored. Generally, results are not greatly affected by wall distance traits.

Parallel Solver for Diffuse Optical Tomography on Realistic Head Models With Scattering and Clear Regions.

PubMed

Placati, Silvio; Guermandi, Marco; Samore, Andrea; Scarselli, Eleonora Franchi; Guerrieri, Roberto

2016-09-01

Diffuse optical tomography is an imaging technique, based on evaluation of how light propagates within the human head to obtain the functional information about the brain. Precision in reconstructing such an optical properties map is highly affected by the accuracy of the light propagation model implemented, which needs to take into account the presence of clear and scattering tissues. We present a numerical solver based on the radiosity-diffusion model, integrating the anatomical information provided by a structural MRI. The solver is designed to run on parallel heterogeneous platforms based on multiple GPUs and CPUs. We demonstrate how the solver provides a 7 times speed-up over an isotropic-scattered parallel Monte Carlo engine based on a radiative transport equation for a domain composed of 2 million voxels, along with a significant improvement in accuracy. The speed-up greatly increases for larger domains, allowing us to compute the light distribution of a full human head ( ≈ 3 million voxels) in 116 s for the platform used.

A semi-implicit augmented IIM for Navier–Stokes equations with open, traction, or free boundary conditions

PubMed Central

Li, Zhilin; Xiao, Li; Cai, Qin; Zhao, Hongkai; Luo, Ray

2016-01-01

In this paper, a new Navier–Stokes solver based on a finite difference approximation is proposed to solve incompressible flows on irregular domains with open, traction, and free boundary conditions, which can be applied to simulations of fluid structure interaction, implicit solvent model for biomolecular applications and other free boundary or interface problems. For some problems of this type, the projection method and the augmented immersed interface method (IIM) do not work well or does not work at all. The proposed new Navier–Stokes solver is based on the local pressure boundary method, and a semi-implicit augmented IIM. A fast Poisson solver can be used in our algorithm which gives us the potential for developing fast overall solvers in the future. The time discretization is based on a second order multi-step method. Numerical tests with exact solutions are presented to validate the accuracy of the method. Application to fluid structure interaction between an incompressible fluid and a compressible gas bubble is also presented. PMID:27087702

A semi-implicit augmented IIM for Navier-Stokes equations with open, traction, or free boundary conditions.

PubMed

Li, Zhilin; Xiao, Li; Cai, Qin; Zhao, Hongkai; Luo, Ray

2015-08-15

In this paper, a new Navier-Stokes solver based on a finite difference approximation is proposed to solve incompressible flows on irregular domains with open, traction, and free boundary conditions, which can be applied to simulations of fluid structure interaction, implicit solvent model for biomolecular applications and other free boundary or interface problems. For some problems of this type, the projection method and the augmented immersed interface method (IIM) do not work well or does not work at all. The proposed new Navier-Stokes solver is based on the local pressure boundary method, and a semi-implicit augmented IIM. A fast Poisson solver can be used in our algorithm which gives us the potential for developing fast overall solvers in the future. The time discretization is based on a second order multi-step method. Numerical tests with exact solutions are presented to validate the accuracy of the method. Application to fluid structure interaction between an incompressible fluid and a compressible gas bubble is also presented.

Solving a discrete model of the lac operon using Z3

NASA Astrophysics Data System (ADS)

Gutierrez, Natalia A.

2014-05-01

A discrete model for the Lcac Operon is solved using the SMT-solver Z3. Traditionally the Lac Operon is formulated in a continuous math model. This model is a system of ordinary differential equations. Here, it was considerated as a discrete model, based on a Boolean red. The biological problem of Lac Operon is enunciated as a problem of Boolean satisfiability, and it is solved using an STM-solver named Z3. Z3 is a powerful solver that allows understanding the basic dynamic of the Lac Operon in an easier and more efficient way. The multi-stability of the Lac Operon can be easily computed with Z3. The code that solves the Boolean red can be written in Python language or SMT-Lib language. Both languages were used in local version of the program as online version of Z3. For future investigations it is proposed to solve the Boolean red of Lac Operon using others SMT-solvers as cvc4, alt-ergo, mathsat and yices.

PUFoam : A novel open-source CFD solver for the simulation of polyurethane foams

NASA Astrophysics Data System (ADS)

Karimi, M.; Droghetti, H.; Marchisio, D. L.

2017-08-01

In this work a transient three-dimensional mathematical model is formulated and validated for the simulation of polyurethane (PU) foams. The model is based on computational fluid dynamics (CFD) and is coupled with a population balance equation (PBE) to describe the evolution of the gas bubbles/cells within the PU foam. The front face of the expanding foam is monitored on the basis of the volume-of-fluid (VOF) method using a compressible solver available in OpenFOAM version 3.0.1. The solver is additionally supplemented to include the PBE, solved with the quadrature method of moments (QMOM), the polymerization kinetics, an adequate rheological model and a simple model for the foam thermal conductivity. The new solver is labelled as PUFoam and is, for the first time in this work, validated for 12 different mixing-cup experiments. Comparison of the time evolution of the predicted and experimentally measured density and temperature of the PU foam shows the potentials and limitations of the approach.

Six Dimensional Trajectory Solver for Autonomous Proximity Operations

DTIC Science & Technology

1990-05-01

Clohessy - Wiltshire equations for relative position and quaternions for relative attitude are used to define a state space relationship between the initial...0 (2.23) y + 2nX = 0 (2.24) 2+ n2 z = 0 (2.25) which are commonly referred to as the Clohessy - Wiltshire equations. Although 11 the equations are...attributed to W. Clohessy and R. Wiltshire for their paper in the September 1960 issue of the Journal of Aerospace Science, another author developed the

Using high-order polynomial basis in 3-D EM forward modeling based on volume integral equation method

NASA Astrophysics Data System (ADS)

Kruglyakov, Mikhail; Kuvshinov, Alexey

2018-05-01

3-D interpretation of electromagnetic (EM) data of different origin and scale becomes a common practice worldwide. However, 3-D EM numerical simulations (modeling)—a key part of any 3-D EM data analysis—with realistic levels of complexity, accuracy and spatial detail still remains challenging from the computational point of view. We present a novel, efficient 3-D numerical solver based on a volume integral equation (IE) method. The efficiency is achieved by using a high-order polynomial (HOP) basis instead of the zero-order (piecewise constant) basis that is invoked in all routinely used IE-based solvers. We demonstrate that usage of the HOP basis allows us to decrease substantially the number of unknowns (preserving the same accuracy), with corresponding speed increase and memory saving.

A solver for General Unilateral Polynomial Matrix Equation with Second-Order Matrices Over Prime Finite Fields

NASA Astrophysics Data System (ADS)

Burtyka, Filipp

2018-03-01

The paper firstly considers the problem of finding solvents for arbitrary unilateral polynomial matrix equations with second-order matrices over prime finite fields from the practical point of view: we implement the solver for this problem. The solver’s algorithm has two step: the first is finding solvents, having Jordan Normal Form (JNF), the second is finding solvents among the rest matrices. The first step reduces to the finding roots of usual polynomials over finite fields, the second is essentially exhaustive search. The first step’s algorithms essentially use the polynomial matrices theory. We estimate the practical duration of computations using our software implementation (for example that one can’t construct unilateral matrix polynomial over finite field, having any predefined number of solvents) and answer some theoretically-valued questions.

Consistent initial conditions for the Saint-Venant equations in river network modeling

NASA Astrophysics Data System (ADS)

Yu, Cheng-Wei; Liu, Frank; Hodges, Ben R.

2017-09-01

Initial conditions for flows and depths (cross-sectional areas) throughout a river network are required for any time-marching (unsteady) solution of the one-dimensional (1-D) hydrodynamic Saint-Venant equations. For a river network modeled with several Strahler orders of tributaries, comprehensive and consistent synoptic data are typically lacking and synthetic starting conditions are needed. Because of underlying nonlinearity, poorly defined or inconsistent initial conditions can lead to convergence problems and long spin-up times in an unsteady solver. Two new approaches are defined and demonstrated herein for computing flows and cross-sectional areas (or depths). These methods can produce an initial condition data set that is consistent with modeled landscape runoff and river geometry boundary conditions at the initial time. These new methods are (1) the pseudo time-marching method (PTM) that iterates toward a steady-state initial condition using an unsteady Saint-Venant solver and (2) the steady-solution method (SSM) that makes use of graph theory for initial flow rates and solution of a steady-state 1-D momentum equation for the channel cross-sectional areas. The PTM is shown to be adequate for short river reaches but is significantly slower and has occasional non-convergent behavior for large river networks. The SSM approach is shown to provide a rapid solution of consistent initial conditions for both small and large networks, albeit with the requirement that additional code must be written rather than applying an existing unsteady Saint-Venant solver.

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.

Euler equation computations for the flow over a hovering helicopter rotor

NASA Technical Reports Server (NTRS)

Roberts, Thomas Wesley

1988-01-01

A numerical solution technique is developed for computing the flow field around an isolated helicopter rotor in hover. The flow is governed by the compressible Euler equations which are integrated using a finite volume approach. The Euler equations are coupled to a free wake model of the rotary wing vortical wake. This wake model is incorporated into the finite volume solver using a prescribed flow, or perturbation, technique which eliminates the numerical diffusion of vorticity due to the artificial viscosity of the scheme. The work is divided into three major parts: (1) comparisons of Euler solutions to experimental data for the flow around isolated wings show good agreement with the surface pressures, but poor agreement with the vortical wake structure; (2) the perturbation method is developed and used to compute the interaction of a streamwise vortex with a semispan wing. The rapid diffusion of the vortex when only the basic Euler solver is used is illustrated, and excellent agreement with experimental section lift coefficients is demonstrated when using the perturbation approach; and (3) the free wake solution technique is described and the coupling of the wake to the Euler solver for an isolated rotor is presented. Comparisons with experimental blade load data for several cases show good agreement, with discrepancies largely attributable to the neglect of viscous effects. The computed wake geometries agree less well with experiment, the primary difference being that too rapid a wake contraction is predicted for all the cases.

A new iterative scheme for solving the discrete Smoluchowski equation

NASA Astrophysics Data System (ADS)

Smith, Alastair J.; Wells, Clive G.; Kraft, Markus

2018-01-01

This paper introduces a new iterative scheme for solving the discrete Smoluchowski equation and explores the numerical convergence properties of the method for a range of kernels admitting analytical solutions, in addition to some more physically realistic kernels typically used in kinetics applications. The solver is extended to spatially dependent problems with non-uniform velocities and its performance investigated in detail.

Entropy Analysis of Kinetic Flux Vector Splitting Schemes for the Compressible Euler Equations

NASA Technical Reports Server (NTRS)

Shiuhong, Lui; Xu, Jun

1999-01-01

Flux Vector Splitting (FVS) scheme is one group of approximate Riemann solvers for the compressible Euler equations. In this paper, the discretized entropy condition of the Kinetic Flux Vector Splitting (KFVS) scheme based on the gas-kinetic theory is proved. The proof of the entropy condition involves the entropy definition difference between the distinguishable and indistinguishable particles.

Iterative Methods to Solve Linear RF Fields in Hot Plasma

NASA Astrophysics Data System (ADS)

Spencer, Joseph; Svidzinski, Vladimir; Evstatiev, Evstati; Galkin, Sergei; Kim, Jin-Soo

2014-10-01

Most magnetic plasma confinement devices use radio frequency (RF) waves for current drive and/or heating. Numerical modeling of RF fields is an important part of performance analysis of such devices and a predictive tool aiding design and development of future devices. Prior attempts at this modeling have mostly used direct solvers to solve the formulated linear equations. Full wave modeling of RF fields in hot plasma with 3D nonuniformities is mostly prohibited, with memory demands of a direct solver placing a significant limitation on spatial resolution. Iterative methods can significantly increase spatial resolution. We explore the feasibility of using iterative methods in 3D full wave modeling. The linear wave equation is formulated using two approaches: for cold plasmas the local cold plasma dielectric tensor is used (resolving resonances by particle collisions), while for hot plasmas the conductivity kernel (which includes a nonlocal dielectric response) is calculated by integrating along test particle orbits. The wave equation is discretized using a finite difference approach. The initial guess is important in iterative methods, and we examine different initial guesses including the solution to the cold plasma wave equation. Work is supported by the U.S. DOE SBIR program.

Helmholtz and parabolic equation solutions to a benchmark problem in ocean acoustics.

PubMed

Larsson, Elisabeth; Abrahamsson, Leif

2003-05-01

The Helmholtz equation (HE) describes wave propagation in applications such as acoustics and electromagnetics. For realistic problems, solving the HE is often too expensive. Instead, approximations like the parabolic wave equation (PE) are used. For low-frequency shallow-water environments, one persistent problem is to assess the accuracy of the PE model. In this work, a recently developed HE solver that can handle a smoothly varying bathymetry, variable material properties, and layered materials, is used for an investigation of the errors in PE solutions. In the HE solver, a preconditioned Krylov subspace method is applied to the discretized equations. The preconditioner combines domain decomposition and fast transform techniques. A benchmark problem with upslope-downslope propagation over a penetrable lossy seamount is solved. The numerical experiments show that, for the same bathymetry, a soft and slow bottom gives very similar HE and PE solutions, whereas the PE model is far from accurate for a hard and fast bottom. A first attempt to estimate the error is made by computing the relative deviation from the energy balance for the PE solution. This measure gives an indication of the magnitude of the error, but cannot be used as a strict error bound.

Turbulent Flow Validation in the Helios Strand Solver

DTIC Science & Technology

2014-01-07

usual (̄) notation is omitted for simplicity). The pressure is obtained from the ideal gas equation of state given as: P = (γ−1) [ Et − 1 2 ρ ( u2 + v2...2. SA-RANS System The state vector and flux vectors including those of the SA model equation for three-dimensional flow are explicitly given as: u...number, PrT is the turbulent Prandtl number, and T is the temperature. The ideal gas equation of state , p = ρRT is used to close the equations . IV.A

Numerical Approach to Spatial Deterministic-Stochastic Models Arising in Cell Biology.

PubMed

Schaff, James C; Gao, Fei; Li, Ye; Novak, Igor L; Slepchenko, Boris M

2016-12-01

Hybrid deterministic-stochastic methods provide an efficient alternative to a fully stochastic treatment of models which include components with disparate levels of stochasticity. However, general-purpose hybrid solvers for spatially resolved simulations of reaction-diffusion systems are not widely available. Here we describe fundamentals of a general-purpose spatial hybrid method. The method generates realizations of a spatially inhomogeneous hybrid system by appropriately integrating capabilities of a deterministic partial differential equation solver with a popular particle-based stochastic simulator, Smoldyn. Rigorous validation of the algorithm is detailed, using a simple model of calcium 'sparks' as a testbed. The solver is then applied to a deterministic-stochastic model of spontaneous emergence of cell polarity. The approach is general enough to be implemented within biologist-friendly software frameworks such as Virtual Cell.

EUPDF-II: An Eulerian Joint Scalar Monte Carlo PDF Module : User's Manual

NASA Technical Reports Server (NTRS)

Raju, M. S.; Liu, Nan-Suey (Technical Monitor)

2004-01-01

EUPDF-II provides the solution for the species and temperature fields based on an evolution equation for PDF (Probability Density Function) and it is developed mainly for application with sprays, combustion, parallel computing, and unstructured grids. It is designed to be massively parallel and could easily be coupled with any existing gas-phase CFD and spray solvers. The solver accommodates the use of an unstructured mesh with mixed elements of either triangular, quadrilateral, and/or tetrahedral type. The manual provides the user with an understanding of the various models involved in the PDF formulation, its code structure and solution algorithm, and various other issues related to parallelization and its coupling with other solvers. The source code of EUPDF-II will be available with National Combustion Code (NCC) as a complete package.

An efficient direct solver for rarefied gas flows with arbitrary statistics

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

Diaz, Manuel A., E-mail: f99543083@ntu.edu.tw; Yang, Jaw-Yen, E-mail: yangjy@iam.ntu.edu.tw; Center of Advanced Study in Theoretical Science, National Taiwan University, Taipei 10167, Taiwan

2016-01-15

A new numerical methodology associated with a unified treatment is presented to solve the Boltzmann–BGK equation of gas dynamics for the classical and quantum gases described by the Bose–Einstein and Fermi–Dirac statistics. Utilizing a class of globally-stiffly-accurate implicit–explicit Runge–Kutta scheme for the temporal evolution, associated with the discrete ordinate method for the quadratures in the momentum space and the weighted essentially non-oscillatory method for the spatial discretization, the proposed scheme is asymptotic-preserving and imposes no non-linear solver or requires the knowledge of fugacity and temperature to capture the flow structures in the hydrodynamic (Euler) limit. The proposed treatment overcomes themore » limitations found in the work by Yang and Muljadi (2011) [33] due to the non-linear nature of quantum relations, and can be applied in studying the dynamics of a gas with internal degrees of freedom with correct values of the ratio of specific heat for the flow regimes for all Knudsen numbers and energy wave lengths. The present methodology is numerically validated with the unified treatment by the one-dimensional shock tube problem and the two-dimensional Riemann problems for gases of arbitrary statistics. Descriptions of ideal quantum gases including rotational degrees of freedom have been successfully achieved under the proposed methodology.« less

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

Vay, Jean-Luc, E-mail: jlvay@lbl.gov; Haber, Irving; Godfrey, Brendan B.

Pseudo-spectral electromagnetic solvers (i.e. representing the fields in Fourier space) have extraordinary precision. In particular, Haber et al. presented in 1973 a pseudo-spectral solver that integrates analytically the solution over a finite time step, under the usual assumption that the source is constant over that time step. Yet, pseudo-spectral solvers have not been widely used, due in part to the difficulty for efficient parallelization owing to global communications associated with global FFTs on the entire computational domains. A method for the parallelization of electromagnetic pseudo-spectral solvers is proposed and tested on single electromagnetic pulses, and on Particle-In-Cell simulations of themore » wakefield formation in a laser plasma accelerator. The method takes advantage of the properties of the Discrete Fourier Transform, the linearity of Maxwell’s equations and the finite speed of light for limiting the communications of data within guard regions between neighboring computational domains. Although this requires a small approximation, test results show that no significant error is made on the test cases that have been presented. The proposed method opens the way to solvers combining the favorable parallel scaling of standard finite-difference methods with the accuracy advantages of pseudo-spectral methods.« less

Divergence-Free SPH for Incompressible and Viscous Fluids.

PubMed

Bender, Jan; Koschier, Dan

2017-03-01

In this paper we present a novel Smoothed Particle Hydrodynamics (SPH) method for the efficient and stable simulation of incompressible fluids. The most efficient SPH-based approaches enforce incompressibility either on position or velocity level. However, the continuity equation for incompressible flow demands to maintain a constant density and a divergence-free velocity field. We propose a combination of two novel implicit pressure solvers enforcing both a low volume compression as well as a divergence-free velocity field. While a compression-free fluid is essential for realistic physical behavior, a divergence-free velocity field drastically reduces the number of required solver iterations and increases the stability of the simulation significantly. Thanks to the improved stability, our method can handle larger time steps than previous approaches. This results in a substantial performance gain since the computationally expensive neighborhood search has to be performed less frequently. Moreover, we introduce a third optional implicit solver to simulate highly viscous fluids which seamlessly integrates into our solver framework. Our implicit viscosity solver produces realistic results while introducing almost no numerical damping. We demonstrate the efficiency, robustness and scalability of our method in a variety of complex simulations including scenarios with millions of turbulent particles or highly viscous materials.

Simulations of pattern dynamics for reaction-diffusion systems via SIMULINK

PubMed Central

2014-01-01

Background Investigation of the nonlinear pattern dynamics of a reaction-diffusion system almost always requires numerical solution of the system’s set of defining differential equations. Traditionally, this would be done by selecting an appropriate differential equation solver from a library of such solvers, then writing computer codes (in a programming language such as C or Matlab) to access the selected solver and display the integrated results as a function of space and time. This “code-based” approach is flexible and powerful, but requires a certain level of programming sophistication. A modern alternative is to use a graphical programming interface such as Simulink to construct a data-flow diagram by assembling and linking appropriate code blocks drawn from a library. The result is a visual representation of the inter-relationships between the state variables whose output can be made completely equivalent to the code-based solution. Results As a tutorial introduction, we first demonstrate application of the Simulink data-flow technique to the classical van der Pol nonlinear oscillator, and compare Matlab and Simulink coding approaches to solving the van der Pol ordinary differential equations. We then show how to introduce space (in one and two dimensions) by solving numerically the partial differential equations for two different reaction-diffusion systems: the well-known Brusselator chemical reactor, and a continuum model for a two-dimensional sheet of human cortex whose neurons are linked by both chemical and electrical (diffusive) synapses. We compare the relative performances of the Matlab and Simulink implementations. Conclusions The pattern simulations by Simulink are in good agreement with theoretical predictions. Compared with traditional coding approaches, the Simulink block-diagram paradigm reduces the time and programming burden required to implement a solution for reaction-diffusion systems of equations. Construction of the block-diagram does not require high-level programming skills, and the graphical interface lends itself to easy modification and use by non-experts. PMID:24725437

Simulations of pattern dynamics for reaction-diffusion systems via SIMULINK.

PubMed

Wang, Kaier; Steyn-Ross, Moira L; Steyn-Ross, D Alistair; Wilson, Marcus T; Sleigh, Jamie W; Shiraishi, Yoichi

2014-04-11

Investigation of the nonlinear pattern dynamics of a reaction-diffusion system almost always requires numerical solution of the system's set of defining differential equations. Traditionally, this would be done by selecting an appropriate differential equation solver from a library of such solvers, then writing computer codes (in a programming language such as C or Matlab) to access the selected solver and display the integrated results as a function of space and time. This "code-based" approach is flexible and powerful, but requires a certain level of programming sophistication. A modern alternative is to use a graphical programming interface such as Simulink to construct a data-flow diagram by assembling and linking appropriate code blocks drawn from a library. The result is a visual representation of the inter-relationships between the state variables whose output can be made completely equivalent to the code-based solution. As a tutorial introduction, we first demonstrate application of the Simulink data-flow technique to the classical van der Pol nonlinear oscillator, and compare Matlab and Simulink coding approaches to solving the van der Pol ordinary differential equations. We then show how to introduce space (in one and two dimensions) by solving numerically the partial differential equations for two different reaction-diffusion systems: the well-known Brusselator chemical reactor, and a continuum model for a two-dimensional sheet of human cortex whose neurons are linked by both chemical and electrical (diffusive) synapses. We compare the relative performances of the Matlab and Simulink implementations. The pattern simulations by Simulink are in good agreement with theoretical predictions. Compared with traditional coding approaches, the Simulink block-diagram paradigm reduces the time and programming burden required to implement a solution for reaction-diffusion systems of equations. Construction of the block-diagram does not require high-level programming skills, and the graphical interface lends itself to easy modification and use by non-experts.

SC-GRAPPA: Self-constraint noniterative GRAPPA reconstruction with closed-form solution.

PubMed

Ding, Yu; Xue, Hui; Ahmad, Rizwan; Ting, Samuel T; Simonetti, Orlando P

2012-12-01

Parallel MRI (pMRI) reconstruction techniques are commonly used to reduce scan time by undersampling the k-space data. GRAPPA, a k-space based pMRI technique, is widely used clinically because of its robustness. In GRAPPA, the missing k-space data are estimated by solving a set of linear equations; however, this set of equations does not take advantage of the correlations within the missing k-space data. All k-space data in a neighborhood acquired from a phased-array coil are correlated. The correlation can be estimated easily as a self-constraint condition, and formulated as an extra set of linear equations to improve the performance of GRAPPA. The authors propose a modified k-space based pMRI technique called self-constraint GRAPPA (SC-GRAPPA) which combines the linear equations of GRAPPA with these extra equations to solve for the missing k-space data. Since SC-GRAPPA utilizes a least-squares solution of the linear equations, it has a closed-form solution that does not require an iterative solver. The SC-GRAPPA equation was derived by incorporating GRAPPA as a prior estimate. SC-GRAPPA was tested in a uniform phantom and two normal volunteers. MR real-time cardiac cine images with acceleration rate 5 and 6 were reconstructed using GRAPPA and SC-GRAPPA. SC-GRAPPA showed a significantly lower artifact level, and a greater than 10% overall signal-to-noise ratio (SNR) gain over GRAPPA, with more significant SNR gain observed in low-SNR regions of the images. SC-GRAPPA offers improved pMRI reconstruction, and is expected to benefit clinical imaging applications in the future.

Input-output-controlled nonlinear equation solvers

NASA Technical Reports Server (NTRS)

Padovan, Joseph

1988-01-01

To upgrade the efficiency and stability of the successive substitution (SS) and Newton-Raphson (NR) schemes, the concept of input-output-controlled solvers (IOCS) is introduced. By employing the formal properties of the constrained version of the SS and NR schemes, the IOCS algorithm can handle indefiniteness of the system Jacobian, can maintain iterate monotonicity, and provide for separate control of load incrementation and iterate excursions, as well as having other features. To illustrate the algorithmic properties, the results for several benchmark examples are presented. These define the associated numerical efficiency and stability of the IOCS.

Object-Oriented Design for Sparse Direct Solvers

NASA Technical Reports Server (NTRS)

Dobrian, Florin; Kumfert, Gary; Pothen, Alex

1999-01-01

We discuss the object-oriented design of a software package for solving sparse, symmetric systems of equations (positive definite and indefinite) by direct methods. At the highest layers, we decouple data structure classes from algorithmic classes for flexibility. We describe the important structural and algorithmic classes in our design, and discuss the trade-offs we made for high performance. The kernels at the lower layers were optimized by hand. Our results show no performance loss from our object-oriented design, while providing flexibility, case of use, and extensibility over solvers using procedural design.

A User’s Guide to the SEVP (Stabilized Error Vector Propagation) Solver: An Efficient Direct Solver for Elliptic Partial Differential Equations

DTIC Science & Technology

1989-04-13

19 5.3 The Solution, BSM2 , BSM3 . ...................................... 21 6. Description of test example...are modified for the boundary conditions. The sections on the preprocessor subroutine BSM1 and the solution subroutines BSM2 , BSM3 may be skipped by...interior row j = N-1 to the solution error C5 on the second row j = IE(2) of the last block, so that P3 = C5 R31 (5.18) 20 5.3 The Solution. BSM2

Treating convection in sequential solvers

NASA Technical Reports Server (NTRS)

Shyy, Wei; Thakur, Siddharth

1992-01-01

The treatment of the convection terms in the sequential solver, a standard procedure found in virtually all pressure based algorithms, to compute the flow problems with sharp gradients and source terms is investigated. Both scalar model problems and one-dimensional gas dynamics equations have been used to study the various issues involved. Different approaches including the use of nonlinear filtering techniques and adoption of TVD type schemes have been investigated. Special treatments of the source terms such as pressure gradients and heat release have also been devised, yielding insight and improved accuracy of the numerical procedure adopted.

Numerical simulation of the hydrodynamic instabilities of Richtmyer-Meshkov and Rayleigh-Taylor

NASA Astrophysics Data System (ADS)

Fortova, S. V.; Shepelev, V. V.; Troshkin, O. V.; Kozlov, S. A.

2017-09-01

The paper presents the results of numerical simulation of the development of hydrodynamic instabilities of Richtmyer-Meshkov and Rayleigh-Taylor encountered in experiments [1-3]. For the numerical solution used the TPS software package (Turbulence Problem Solver) that implements a generalized approach to constructing computer programs for a wide range of problems of hydrodynamics, described by the system of equations of hyperbolic type. As numerical methods are used the method of large particles and ENO-scheme of the second order with Roe solver for the approximate solution of the Riemann problem.

Mathematics of the total alkalinity-pH equation - pathway to robust and universal solution algorithms: the SolveSAPHE package v1.0.1

NASA Astrophysics Data System (ADS)

Munhoven, G.

2013-08-01

The total alkalinity-pH equation, which relates total alkalinity and pH for a given set of total concentrations of the acid-base systems that contribute to total alkalinity in a given water sample, is reviewed and its mathematical properties established. We prove that the equation function is strictly monotone and always has exactly one positive root. Different commonly used approximations are discussed and compared. An original method to derive appropriate initial values for the iterative solution of the cubic polynomial equation based upon carbonate-borate-alkalinity is presented. We then review different methods that have been used to solve the total alkalinity-pH equation, with a main focus on biogeochemical models. The shortcomings and limitations of these methods are made out and discussed. We then present two variants of a new, robust and universally convergent algorithm to solve the total alkalinity-pH equation. This algorithm does not require any a priori knowledge of the solution. SolveSAPHE (Solver Suite for Alkalinity-PH Equations) provides reference implementations of several variants of the new algorithm in Fortran 90, together with new implementations of other, previously published solvers. The new iterative procedure is shown to converge from any starting value to the physical solution. The extra computational cost for the convergence security is only 10-15% compared to the fastest algorithm in our test series.

Coupled electromagnetic-thermodynamic simulations of microwave heating problems using the FDTD algorithm.

PubMed

Kopyt, Paweł; Celuch, Małgorzata

2007-01-01

A practical implementation of a hybrid simulation system capable of modeling coupled electromagnetic-thermodynamic problems typical in microwave heating is described. The paper presents two approaches to modeling such problems. Both are based on an FDTD-based commercial electromagnetic solver coupled to an external thermodynamic analysis tool required for calculations of heat diffusion. The first approach utilizes a simple FDTD-based thermal solver while in the second it is replaced by a universal commercial CFD solver. The accuracy of the two modeling systems is verified against the original experimental data as well as the measurement results available in literature.

Shock-driven fluid-structure interaction for civil design

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

Wood, Stephen L; Deiterding, Ralf

The multiphysics fluid-structure interaction simulation of shock-loaded structures requires the dynamic coupling of a shock-capturing flow solver to a solid mechanics solver for large deformations. The Virtual Test Facility combines a Cartesian embedded boundary approach with dynamic mesh adaptation in a generic software framework of flow solvers using hydrodynamic finite volume upwind schemes that are coupled to various explicit finite element solid dynamics solvers (Deiterding et al., 2006). This paper gives a brief overview of the computational approach and presents first simulations that utilize the general purpose solid dynamics code DYNA3D for complex 3D structures of interest in civil engineering.more » Results from simulations of a reinforced column, highway bridge, multistory building, and nuclear reactor building are presented.« less

The novel high-performance 3-D MT inverse solver

NASA Astrophysics Data System (ADS)

Kruglyakov, Mikhail; Geraskin, Alexey; Kuvshinov, Alexey

2016-04-01

We present novel, robust, scalable, and fast 3-D magnetotelluric (MT) inverse solver. The solver is written in multi-language paradigm to make it as efficient, readable and maintainable as possible. Separation of concerns and single responsibility concepts go through implementation of the solver. As a forward modelling engine a modern scalable solver extrEMe, based on contracting integral equation approach, is used. Iterative gradient-type (quasi-Newton) optimization scheme is invoked to search for (regularized) inverse problem solution, and adjoint source approach is used to calculate efficiently the gradient of the misfit. The inverse solver is able to deal with highly detailed and contrasting models, allows for working (separately or jointly) with any type of MT responses, and supports massive parallelization. Moreover, different parallelization strategies implemented in the code allow optimal usage of available computational resources for a given problem statement. To parameterize an inverse domain the so-called mask parameterization is implemented, which means that one can merge any subset of forward modelling cells in order to account for (usually) irregular distribution of observation sites. We report results of 3-D numerical experiments aimed at analysing the robustness, performance and scalability of the code. In particular, our computational experiments carried out at different platforms ranging from modern laptops to HPC Piz Daint (6th supercomputer in the world) demonstrate practically linear scalability of the code up to thousands of nodes.

Numerical solution of the two-dimensional time-dependent incompressible Euler equations

NASA Technical Reports Server (NTRS)

Whitfield, David L.; Taylor, Lafayette K.

1994-01-01

A numerical method is presented for solving the artificial compressibility form of the 2D time-dependent incompressible Euler equations. The approach is based on using an approximate Riemann solver for the cell face numerical flux of a finite volume discretization. Characteristic variable boundary conditions are developed and presented for all boundaries and in-flow out-flow situations. The system of algebraic equations is solved using the discretized Newton-relaxation (DNR) implicit method. Numerical results are presented for both steady and unsteady flow.

Navier-Stokes computation of compressible turbulent flows with a second order closure

NASA Technical Reports Server (NTRS)

Dingus, C.; Kollmann, W.

1991-01-01

The objective was the development of a complete second order closure for wall bounded flows, including all components of the dissipation rate tensor and a numerical solution procedure for the resulting system of equations. The main topics discussed are the closure of the pressure correlations and the viscous destruction terms in the dissipation rate equations and the numerical solution scheme based on a block-tridiagonal solver for the nine equations required for the prediction of plane or axisymmetric flows.

Methods for the computation of the multivalued Painlevé transcendents on their Riemann surfaces

NASA Astrophysics Data System (ADS)

Fasondini, Marco; Fornberg, Bengt; Weideman, J. A. C.

2017-09-01

We extend the numerical pole field solver (Fornberg and Weideman (2011) [12]) to enable the computation of the multivalued Painlevé transcendents, which are the solutions to the third, fifth and sixth Painlevé equations, on their Riemann surfaces. We display, for the first time, solutions to these equations on multiple Riemann sheets. We also provide numerical evidence for the existence of solutions to the sixth Painlevé equation that have pole-free sectors, known as tronquée solutions.

Modeling hemodynamics in intracranial aneurysms: Comparing accuracy of CFD solvers based on finite element and finite volume schemes.

PubMed

Botti, Lorenzo; Paliwal, Nikhil; Conti, Pierangelo; Antiga, Luca; Meng, Hui

2018-06-01

Image-based computational fluid dynamics (CFD) has shown potential to aid in the clinical management of intracranial aneurysms (IAs) but its adoption in the clinical practice has been missing, partially due to lack of accuracy assessment and sensitivity analysis. To numerically solve the flow-governing equations CFD solvers generally rely on two spatial discretization schemes: Finite Volume (FV) and Finite Element (FE). Since increasingly accurate numerical solutions are obtained by different means, accuracies and computational costs of FV and FE formulations cannot be compared directly. To this end, in this study we benchmark two representative CFD solvers in simulating flow in a patient-specific IA model: (1) ANSYS Fluent, a commercial FV-based solver and (2) VMTKLab multidGetto, a discontinuous Galerkin (dG) FE-based solver. The FV solver's accuracy is improved by increasing the spatial mesh resolution (134k, 1.1m, 8.6m and 68.5m tetrahedral element meshes). The dGFE solver accuracy is increased by increasing the degree of polynomials (first, second, third and fourth degree) on the base 134k tetrahedral element mesh. Solutions from best FV and dGFE approximations are used as baseline for error quantification. On average, velocity errors for second-best approximations are approximately 1cm/s for a [0,125]cm/s velocity magnitude field. Results show that high-order dGFE provide better accuracy per degree of freedom but worse accuracy per Jacobian non-zero entry as compared to FV. Cross-comparison of velocity errors demonstrates asymptotic convergence of both solvers to the same numerical solution. Nevertheless, the discrepancy between under-resolved velocity fields suggests that mesh independence is reached following different paths. This article is protected by copyright. All rights reserved.

A FAST ITERATIVE METHOD FOR SOLVING THE EIKONAL EQUATION ON TRIANGULATED SURFACES*

PubMed Central

Fu, Zhisong; Jeong, Won-Ki; Pan, Yongsheng; Kirby, Robert M.; Whitaker, Ross T.

2012-01-01

This paper presents an efficient, fine-grained parallel algorithm for solving the Eikonal equation on triangular meshes. The Eikonal equation, and the broader class of Hamilton–Jacobi equations to which it belongs, have a wide range of applications from geometric optics and seismology to biological modeling and analysis of geometry and images. The ability to solve such equations accurately and efficiently provides new capabilities for exploring and visualizing parameter spaces and for solving inverse problems that rely on such equations in the forward model. Efficient solvers on state-of-the-art, parallel architectures require new algorithms that are not, in many cases, optimal, but are better suited to synchronous updates of the solution. In previous work [W. K. Jeong and R. T. Whitaker, SIAM J. Sci. Comput., 30 (2008), pp. 2512–2534], the authors proposed the fast iterative method (FIM) to efficiently solve the Eikonal equation on regular grids. In this paper we extend the fast iterative method to solve Eikonal equations efficiently on triangulated domains on the CPU and on parallel architectures, including graphics processors. We propose a new local update scheme that provides solutions of first-order accuracy for both architectures. We also propose a novel triangle-based update scheme and its corresponding data structure for efficient irregular data mapping to parallel single-instruction multiple-data (SIMD) processors. We provide detailed descriptions of the implementations on a single CPU, a multicore CPU with shared memory, and SIMD architectures with comparative results against state-of-the-art Eikonal solvers. PMID:22641200

Some Approaches Towards Constructing Optimally Efficient Multigrid Solvers for the Inviscid Flow Equations

NASA Technical Reports Server (NTRS)

Sidilkover, David

1997-01-01

Some important advances took place during the last several years in the development of genuinely multidimensional upwind schemes for the compressible Euler equations. In particular, a robust, high-resolution genuinely multidimensional scheme which can be used for any of the flow regimes computations was constructed. This paper summarizes briefly these developments and outlines the fundamental advantages of this approach.

Parallel SOR methods with a parabolic-diffusion acceleration technique for solving an unstructured-grid Poisson equation on 3D arbitrary geometries

NASA Astrophysics Data System (ADS)

Zapata, M. A. Uh; Van Bang, D. Pham; Nguyen, K. D.

2016-05-01

This paper presents a parallel algorithm for the finite-volume discretisation of the Poisson equation on three-dimensional arbitrary geometries. The proposed method is formulated by using a 2D horizontal block domain decomposition and interprocessor data communication techniques with message passing interface. The horizontal unstructured-grid cells are reordered according to the neighbouring relations and decomposed into blocks using a load-balanced distribution to give all processors an equal amount of elements. In this algorithm, two parallel successive over-relaxation methods are presented: a multi-colour ordering technique for unstructured grids based on distributed memory and a block method using reordering index following similar ideas of the partitioning for structured grids. In all cases, the parallel algorithms are implemented with a combination of an acceleration iterative solver. This solver is based on a parabolic-diffusion equation introduced to obtain faster solutions of the linear systems arising from the discretisation. Numerical results are given to evaluate the performances of the methods showing speedups better than linear.

Final Report - Subcontract B623760

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

Bank, R.

2017-11-17

During my visit to LLNL during July 17{27, 2017, I worked on linear system solvers. The two level hierarchical solver that initiated our study was developed to solve linear systems arising from hp adaptive finite element calculations, and is implemented in the PLTMG software package, version 12. This preconditioner typically requires 3-20% of the space used by the stiffness matrix for higher order elements. It has multigrid like convergence rates for a wide variety of PDEs (self-adjoint positive de nite elliptic equations, convection dominated convection-diffusion equations, and highly indefinite Helmholtz equations, among others). The convergence rate is not independent ofmore » the polynomial degree p as p ! 1, but but remains strong for p 9, which is the highest polynomial degree allowed in PLTMG, due to limitations of the numerical quadrature rules implemented in the software package. A more complete description of the method and some numerical experiments illustrating its effectiveness appear in. Like traditional geometric multilevel methods, this scheme relies on knowledge of the underlying finite element space in order to construct the smoother and the coarse grid correction.« less

On the Development of an Efficient Parallel Hybrid Solver with Application to Acoustically Treated Aero-Engine Nacelles

NASA Technical Reports Server (NTRS)

Watson, Willie R.; Nark, Douglas M.; Nguyen, Duc T.; Tungkahotara, Siroj

2006-01-01

A finite element solution to the convected Helmholtz equation in a nonuniform flow is used to model the noise field within 3-D acoustically treated aero-engine nacelles. Options to select linear or cubic Hermite polynomial basis functions and isoparametric elements are included. However, the key feature of the method is a domain decomposition procedure that is based upon the inter-mixing of an iterative and a direct solve strategy for solving the discrete finite element equations. This procedure is optimized to take full advantage of sparsity and exploit the increased memory and parallel processing capability of modern computer architectures. Example computations are presented for the Langley Flow Impedance Test facility and a rectangular mapping of a full scale, generic aero-engine nacelle. The accuracy and parallel performance of this new solver are tested on both model problems using a supercomputer that contains hundreds of central processing units. Results show that the method gives extremely accurate attenuation predictions, achieves super-linear speedup over hundreds of CPUs, and solves upward of 25 million complex equations in a quarter of an hour.

Crashworthiness of light aircraft fuselage structures: A numerical and experimental investigation

NASA Technical Reports Server (NTRS)

Nanyaro, A. P.; Tennyson, R. C.; Hansen, J. S.

1984-01-01

The dynamic behavior of aircraft fuselage structures subject to various impact conditions was investigated. An analytical model was developed based on a self-consistent finite element (CFE) formulation utilizing shell, curved beam, and stringer type elements. Equations of motion were formulated and linearized (i.e., for small displacements), although material nonlinearity was retained to treat local plastic deformation. The equations were solved using the implicit Newmark-Beta method with a frontal solver routine. Stiffened aluminum fuselage models were also tested in free flight using the UTIAS pendulum crash test facility. Data were obtained on dynamic strains, g-loads, and transient deformations (using high speed photography in the latter case) during the impact process. Correlations between tests and predicted results are presented, together with computer graphics, based on the CFE model. These results include level and oblique angle impacts as well as the free-flight crash test. Comparisons with a hybrid, lumped mass finite element computer model demonstrate that the CFE formulation provides the test overall agreement with impact test data for comparable computing costs.

Treatment of charge singularities in implicit solvent models.

PubMed

Geng, Weihua; Yu, Sining; Wei, Guowei

2007-09-21

This paper presents a novel method for solving the Poisson-Boltzmann (PB) equation based on a rigorous treatment of geometric singularities of the dielectric interface and a Green's function formulation of charge singularities. Geometric singularities, such as cusps and self-intersecting surfaces, in the dielectric interfaces are bottleneck in developing highly accurate PB solvers. Based on an advanced mathematical technique, the matched interface and boundary (MIB) method, we have recently developed a PB solver by rigorously enforcing the flux continuity conditions at the solvent-molecule interface where geometric singularities may occur. The resulting PB solver, denoted as MIBPB-II, is able to deliver second order accuracy for the molecular surfaces of proteins. However, when the mesh size approaches half of the van der Waals radius, the MIBPB-II cannot maintain its accuracy because the grid points that carry the interface information overlap with those that carry distributed singular charges. In the present Green's function formalism, the charge singularities are transformed into interface flux jump conditions, which are treated on an equal footing as the geometric singularities in our MIB framework. The resulting method, denoted as MIBPB-III, is able to provide highly accurate electrostatic potentials at a mesh as coarse as 1.2 A for proteins. Consequently, at a given level of accuracy, the MIBPB-III is about three times faster than the APBS, a recent multigrid PB solver. The MIBPB-III has been extensively validated by using analytically solvable problems, molecular surfaces of polyatomic systems, and 24 proteins. It provides reliable benchmark numerical solutions for the PB equation.

Treatment of charge singularities in implicit solvent models

NASA Astrophysics Data System (ADS)

Geng, Weihua; Yu, Sining; Wei, Guowei

2007-09-01

This paper presents a novel method for solving the Poisson-Boltzmann (PB) equation based on a rigorous treatment of geometric singularities of the dielectric interface and a Green's function formulation of charge singularities. Geometric singularities, such as cusps and self-intersecting surfaces, in the dielectric interfaces are bottleneck in developing highly accurate PB solvers. Based on an advanced mathematical technique, the matched interface and boundary (MIB) method, we have recently developed a PB solver by rigorously enforcing the flux continuity conditions at the solvent-molecule interface where geometric singularities may occur. The resulting PB solver, denoted as MIBPB-II, is able to deliver second order accuracy for the molecular surfaces of proteins. However, when the mesh size approaches half of the van der Waals radius, the MIBPB-II cannot maintain its accuracy because the grid points that carry the interface information overlap with those that carry distributed singular charges. In the present Green's function formalism, the charge singularities are transformed into interface flux jump conditions, which are treated on an equal footing as the geometric singularities in our MIB framework. The resulting method, denoted as MIBPB-III, is able to provide highly accurate electrostatic potentials at a mesh as coarse as 1.2Å for proteins. Consequently, at a given level of accuracy, the MIBPB-III is about three times faster than the APBS, a recent multigrid PB solver. The MIBPB-III has been extensively validated by using analytically solvable problems, molecular surfaces of polyatomic systems, and 24 proteins. It provides reliable benchmark numerical solutions for the PB equation.

Algorithmically scalable block preconditioner for fully implicit shallow-water equations in CAM-SE

DOE PAGES

Lott, P. Aaron; Woodward, Carol S.; Evans, Katherine J.

2014-10-19

Performing accurate and efficient numerical simulation of global atmospheric climate models is challenging due to the disparate length and time scales over which physical processes interact. Implicit solvers enable the physical system to be integrated with a time step commensurate with processes being studied. The dominant cost of an implicit time step is the ancillary linear system solves, so we have developed a preconditioner aimed at improving the efficiency of these linear system solves. Our preconditioner is based on an approximate block factorization of the linearized shallow-water equations and has been implemented within the spectral element dynamical core within themore » Community Atmospheric Model (CAM-SE). Furthermore, in this paper we discuss the development and scalability of the preconditioner for a suite of test cases with the implicit shallow-water solver within CAM-SE.« less

A stochastic-dynamic model for global atmospheric mass field statistics

NASA Technical Reports Server (NTRS)

Ghil, M.; Balgovind, R.; Kalnay-Rivas, E.

1981-01-01

A model that yields the spatial correlation structure of atmospheric mass field forecast errors was developed. The model is governed by the potential vorticity equation forced by random noise. Expansion in spherical harmonics and correlation function was computed analytically using the expansion coefficients. The finite difference equivalent was solved using a fast Poisson solver and the correlation function was computed using stratified sampling of the individual realization of F(omega) and hence of phi(omega). A higher order equation for gamma was derived and solved directly in finite differences by two successive applications of the fast Poisson solver. The methods were compared for accuracy and efficiency and the third method was chosen as clearly superior. The results agree well with the latitude dependence of observed atmospheric correlation data. The value of the parameter c sub o which gives the best fit to the data is close to the value expected from dynamical considerations.

PDF approach for compressible turbulent reacting flows

NASA Technical Reports Server (NTRS)

Hsu, A. T.; Tsai, Y.-L. P.; Raju, M. S.

1993-01-01

The objective of the present work is to develop a probability density function (pdf) turbulence model for compressible reacting flows for use with a CFD flow solver. The probability density function of the species mass fraction and enthalpy are obtained by solving a pdf evolution equation using a Monte Carlo scheme. The pdf solution procedure is coupled with a compressible CFD flow solver which provides the velocity and pressure fields. A modeled pdf equation for compressible flows, capable of capturing shock waves and suitable to the present coupling scheme, is proposed and tested. Convergence of the combined finite-volume Monte Carlo solution procedure is discussed, and an averaging procedure is developed to provide smooth Monte-Carlo solutions to ensure convergence. Two supersonic diffusion flames are studied using the proposed pdf model and the results are compared with experimental data; marked improvements over CFD solutions without pdf are observed. Preliminary applications of pdf to 3D flows are also reported.

Cosmic-ray propagation with DRAGON2: I. numerical solver and astrophysical ingredients

NASA Astrophysics Data System (ADS)

Evoli, Carmelo; Gaggero, Daniele; Vittino, Andrea; Di Bernardo, Giuseppe; Di Mauro, Mattia; Ligorini, Arianna; Ullio, Piero; Grasso, Dario

2017-02-01

We present version 2 of the DRAGON code designed for computing realistic predictions of the CR densities in the Galaxy. The code numerically solves the interstellar CR transport equation (including inhomogeneous and anisotropic diffusion, either in space and momentum, advective transport and energy losses), under realistic conditions. The new version includes an updated numerical solver and several models for the astrophysical ingredients involved in the transport equation. Improvements in the accuracy of the numerical solution are proved against analytical solutions and in reference diffusion scenarios. The novel features implemented in the code allow to simulate the diverse scenarios proposed to reproduce the most recent measurements of local and diffuse CR fluxes, going beyond the limitations of the homogeneous galactic transport paradigm. To this end, several applications using DRAGON2 are presented as well. This new version facilitates the users to include their own physical models by means of a modular C++ structure.

Effects of numerical tolerance levels on an atmospheric chemistry model for mercury

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

Ferris, D.C.; Burns, D.S.; Shuford, J.

1996-12-31

A Box Model was developed to investigate the atmospheric oxidation processes of mercury in the environment. Previous results indicated the most important influences on the atmospheric concentration of HgO(g) are (i) the flux of HgO(g) volatilization, which is related to the surface medium, extent of contamination, and temperature, and (ii) the presence of Cl{sub 2} in the atmosphere. The numerical solver which has been incorporated into the ORganic CHemistry Integrated Dispersion (ORCHID) model uses the Livermore Solver of Ordinary Differential Equations (LSODE). In the solution of the ODE`s, LSODE uses numerical tolerances. The tolerances effect computer run time, the relativemore » accuracy of ODE calculated species concentrations and whether or not LSODE converges to a solution using this system of equations. The effects of varying these tolerances on the solution of the box model and the ORCHID model will be discussed.« less

Partitioned coupling of advection-diffusion-reaction systems and Brinkman flows

NASA Astrophysics Data System (ADS)

Lenarda, Pietro; Paggi, Marco; Ruiz Baier, Ricardo

2017-09-01

We present a partitioned algorithm aimed at extending the capabilities of existing solvers for the simulation of coupled advection-diffusion-reaction systems and incompressible, viscous flow. The space discretisation of the governing equations is based on mixed finite element methods defined on unstructured meshes, whereas the time integration hinges on an operator splitting strategy that exploits the differences in scales between the reaction, advection, and diffusion processes, considering the global system as a number of sequentially linked sets of partial differential, and algebraic equations. The flow solver presents the advantage that all unknowns in the system (here vorticity, velocity, and pressure) can be fully decoupled and thus turn the overall scheme very attractive from the computational perspective. The robustness of the proposed method is illustrated with a series of numerical tests in 2D and 3D, relevant in the modelling of bacterial bioconvection and Boussinesq systems.

Novel numerical techniques for magma dynamics

NASA Astrophysics Data System (ADS)

Rhebergen, S.; Katz, R. F.; Wathen, A.; Alisic, L.; Rudge, J. F.; Wells, G.

2013-12-01

We discuss the development of finite element techniques and solvers for magma dynamics computations. These are implemented within the FEniCS framework. This approach allows for user-friendly, expressive, high-level code development, but also provides access to powerful, scalable numerical solvers and a large family of finite element discretisations. With the recent addition of dolfin-adjoint, FeniCS supports automated adjoint and tangent-linear models, enabling the rapid development of Generalised Stability Analysis. The ability to easily scale codes to three dimensions with large meshes, and/or to apply intricate adjoint calculations means that efficiency of the numerical algorithms is vital. We therefore describe our development and analysis of preconditioners designed specifically for finite element discretizations of equations governing magma dynamics. The preconditioners are based on Elman-Silvester-Wathen methods for the Stokes equation, and we extend these to flows with compaction. Our simulations are validated by comparison of results with laboratory experiments on partially molten aggregates.

Multigrid approaches to non-linear diffusion problems on unstructured meshes

NASA Technical Reports Server (NTRS)

Mavriplis, Dimitri J.; Bushnell, Dennis M. (Technical Monitor)

2001-01-01

The efficiency of three multigrid methods for solving highly non-linear diffusion problems on two-dimensional unstructured meshes is examined. The three multigrid methods differ mainly in the manner in which the nonlinearities of the governing equations are handled. These comprise a non-linear full approximation storage (FAS) multigrid method which is used to solve the non-linear equations directly, a linear multigrid method which is used to solve the linear system arising from a Newton linearization of the non-linear system, and a hybrid scheme which is based on a non-linear FAS multigrid scheme, but employs a linear solver on each level as a smoother. Results indicate that all methods are equally effective at converging the non-linear residual in a given number of grid sweeps, but that the linear solver is more efficient in cpu time due to the lower cost of linear versus non-linear grid sweeps.

A GPU accelerated and error-controlled solver for the unbounded Poisson equation in three dimensions

NASA Astrophysics Data System (ADS)

Exl, Lukas

2017-12-01

An efficient solver for the three dimensional free-space Poisson equation is presented. The underlying numerical method is based on finite Fourier series approximation. While the error of all involved approximations can be fully controlled, the overall computation error is driven by the convergence of the finite Fourier series of the density. For smooth and fast-decaying densities the proposed method will be spectrally accurate. The method scales with O(N log N) operations, where N is the total number of discretization points in the Cartesian grid. The majority of the computational costs come from fast Fourier transforms (FFT), which makes it ideal for GPU computation. Several numerical computations on CPU and GPU validate the method and show efficiency and convergence behavior. Tests are performed using the Vienna Scientific Cluster 3 (VSC3). A free MATLAB implementation for CPU and GPU is provided to the interested community.

Far-Field Turbulent Vortex-Wake/Exhaust Plume Interaction for Subsonic and HSCT Airplanes

NASA Technical Reports Server (NTRS)

Kandil, Osama A.; Adam, Ihab; Wong, Tin-Chee

1996-01-01

Computational study of the far-field turbulent vortex-wake/exhaust plume interaction for subsonic and high speed civil transport (HSCT) airplanes is carried out. The Reynolds-averaged Navier-Stokes (NS) equations are solved using the implicit, upwind, Roe-flux-differencing, finite-volume scheme. The two-equation shear stress transport model of Menter is implemented with the NS solver for turbulent-flow calculation. For the far-field study, the computations of vortex-wake interaction with the exhaust plume of a single engine of a Boeing 727 wing in a holding condition and two engines of an HSCT in a cruise condition are carried out using overlapping zonal method for several miles downstream. These results are obtained using the computer code FTNS3D. The results of the subsonic flow of this code are compared with those of a parabolized NS solver known as the UNIWAKE code.

Evaluation of the Lattice-Boltzmann Equation Solver PowerFLOW for Aerodynamic Applications

NASA Technical Reports Server (NTRS)

Lockard, David P.; Luo, Li-Shi; Singer, Bart A.; Bushnell, Dennis M. (Technical Monitor)

2000-01-01

A careful comparison of the performance of a commercially available Lattice-Boltzmann Equation solver (Power-FLOW) was made with a conventional, block-structured computational fluid-dynamics code (CFL3D) for the flow over a two-dimensional NACA-0012 airfoil. The results suggest that the version of PowerFLOW used in the investigation produced solutions with large errors in the computed flow field; these errors are attributed to inadequate resolution of the boundary layer for reasons related to grid resolution and primitive turbulence modeling. The requirement of square grid cells in the PowerFLOW calculations limited the number of points that could be used to span the boundary layer on the wing and still keep the computation size small enough to fit on the available computers. Although not discussed in detail, disappointing results were also obtained with PowerFLOW for a cavity flow and for the flow around a generic helicopter configuration.

Numerical Approach to Spatial Deterministic-Stochastic Models Arising in Cell Biology

PubMed Central

Gao, Fei; Li, Ye; Novak, Igor L.; Slepchenko, Boris M.

2016-01-01

Hybrid deterministic-stochastic methods provide an efficient alternative to a fully stochastic treatment of models which include components with disparate levels of stochasticity. However, general-purpose hybrid solvers for spatially resolved simulations of reaction-diffusion systems are not widely available. Here we describe fundamentals of a general-purpose spatial hybrid method. The method generates realizations of a spatially inhomogeneous hybrid system by appropriately integrating capabilities of a deterministic partial differential equation solver with a popular particle-based stochastic simulator, Smoldyn. Rigorous validation of the algorithm is detailed, using a simple model of calcium ‘sparks’ as a testbed. The solver is then applied to a deterministic-stochastic model of spontaneous emergence of cell polarity. The approach is general enough to be implemented within biologist-friendly software frameworks such as Virtual Cell. PMID:27959915

Application of a compressible flow solver and barotropic cavitation model for the evaluation of the suction head in a low specific speed centrifugal pump impeller channel

NASA Astrophysics Data System (ADS)

Limbach, P.; Müller, T.; Skoda, R.

2015-12-01

Commonly, for the simulation of cavitation in centrifugal pumps incompressible flow solvers with VOF kind cavitation models are applied. Since the source/sink terms of the void fraction transport equation are based on simplified bubble dynamics, empirical parameters may need to be adjusted to the particular pump operating point. In the present study a barotropic cavitation model, which is based solely on thermodynamic fluid properties and does not include any empirical parameters, is applied on a single flow channel of a pump impeller in combination with a time-explicit viscous compressible flow solver. The suction head curves (head drop) are compared to the results of an incompressible implicit standard industrial CFD tool and are predicted qualitatively correct by the barotropic model.

Aerodynamic Design Optimization on Unstructured Meshes Using the Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Nielsen, Eric J.; Anderson, W. Kyle

1998-01-01

A discrete adjoint method is developed and demonstrated for aerodynamic design optimization on unstructured grids. The governing equations are the three-dimensional Reynolds-averaged Navier-Stokes equations coupled with a one-equation turbulence model. A discussion of the numerical implementation of the flow and adjoint equations is presented. Both compressible and incompressible solvers are differentiated and the accuracy of the sensitivity derivatives is verified by comparing with gradients obtained using finite differences. Several simplifying approximations to the complete linearization of the residual are also presented, and the resulting accuracy of the derivatives is examined. Demonstration optimizations for both compressible and incompressible flows are given.

A high-order relativistic two-fluid electrodynamic scheme with consistent reconstruction of electromagnetic fields and a multidimensional Riemann solver for electromagnetism

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

Balsara, Dinshaw S., E-mail: dbalsara@nd.edu; Amano, Takanobu, E-mail: amano@eps.s.u-tokyo.ac.jp; Garain, Sudip, E-mail: sgarain@nd.edu

In various astrophysics settings it is common to have a two-fluid relativistic plasma that interacts with the electromagnetic field. While it is common to ignore the displacement current in the ideal, classical magnetohydrodynamic limit, when the flows become relativistic this approximation is less than absolutely well-justified. In such a situation, it is more natural to consider a positively charged fluid made up of positrons or protons interacting with a negatively charged fluid made up of electrons. The two fluids interact collectively with the full set of Maxwell's equations. As a result, a solution strategy for that coupled system of equationsmore » is sought and found here. Our strategy extends to higher orders, providing increasing accuracy. The primary variables in the Maxwell solver are taken to be the facially-collocated components of the electric and magnetic fields. Consistent with such a collocation, three important innovations are reported here. The first two pertain to the Maxwell solver. In our first innovation, the magnetic field within each zone is reconstructed in a divergence-free fashion while the electric field within each zone is reconstructed in a form that is consistent with Gauss' law. In our second innovation, a multidimensionally upwinded strategy is presented which ensures that the magnetic field can be updated via a discrete interpretation of Faraday's law and the electric field can be updated via a discrete interpretation of the generalized Ampere's law. This multidimensional upwinding is achieved via a multidimensional Riemann solver. The multidimensional Riemann solver automatically provides edge-centered electric field components for the Stokes law-based update of the magnetic field. It also provides edge-centered magnetic field components for the Stokes law-based update of the electric field. The update strategy ensures that the electric field is always consistent with Gauss' law and the magnetic field is always divergence-free. This collocation also ensures that electromagnetic radiation that is propagating in a vacuum has both electric and magnetic fields that are exactly divergence-free. Coupled relativistic fluid dynamic equations are solved for the positively and negatively charged fluids. The fluids' numerical fluxes also provide a self-consistent current density for the update of the electric field. Our reconstruction strategy ensures that fluid velocities always remain sub-luminal. Our third innovation consists of an efficient design for several popular IMEX schemes so that they provide strong coupling between the finite-volume-based fluid solver and the electromagnetic fields at high order. This innovation makes it possible to efficiently utilize high order IMEX time update methods for stiff source terms in the update of high order finite-volume methods for hyperbolic conservation laws. We also show that this very general innovation should extend seamlessly to Runge–Kutta discontinuous Galerkin methods. The IMEX schemes enable us to use large CFL numbers even in the presence of stiff source terms. Several accuracy analyses are presented showing that our method meets its design accuracy in the MHD limit as well as in the limit of electromagnetic wave propagation. Several stringent test problems are also presented. We also present a relativistic version of the GEM problem, which shows that our algorithm can successfully adapt to challenging problems in high energy astrophysics.« less

Robust parallel iterative solvers for linear and least-squares problems, Final Technical Report

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

Saad, Yousef

2014-01-16

The primary goal of this project is to study and develop robust iterative methods for solving linear systems of equations and least squares systems. The focus of the Minnesota team is on algorithms development, robustness issues, and on tests and validation of the methods on realistic problems. 1. The project begun with an investigation on how to practically update a preconditioner obtained from an ILU-type factorization, when the coefficient matrix changes. 2. We investigated strategies to improve robustness in parallel preconditioners in a specific case of a PDE with discontinuous coefficients. 3. We explored ways to adapt standard preconditioners formore » solving linear systems arising from the Helmholtz equation. These are often difficult linear systems to solve by iterative methods. 4. We have also worked on purely theoretical issues related to the analysis of Krylov subspace methods for linear systems. 5. We developed an effective strategy for performing ILU factorizations for the case when the matrix is highly indefinite. The strategy uses shifting in some optimal way. The method was extended to the solution of Helmholtz equations by using complex shifts, yielding very good results in many cases. 6. We addressed the difficult problem of preconditioning sparse systems of equations on GPUs. 7. A by-product of the above work is a software package consisting of an iterative solver library for GPUs based on CUDA. This was made publicly available. It was the first such library that offers complete iterative solvers for GPUs. 8. We considered another form of ILU which blends coarsening techniques from Multigrid with algebraic multilevel methods. 9. We have released a new version on our parallel solver - called pARMS [new version is version 3]. As part of this we have tested the code in complex settings - including the solution of Maxwell and Helmholtz equations and for a problem of crystal growth.10. As an application of polynomial preconditioning we considered the problem of evaluating f(A)v which arises in statistical sampling. 11. As an application to the methods we developed, we tackled the problem of computing the diagonal of the inverse of a matrix. This arises in statistical applications as well as in many applications in physics. We explored probing methods as well as domain-decomposition type methods. 12. A collaboration with researchers from Toulouse, France, considered the important problem of computing the Schur complement in a domain-decomposition approach. 13. We explored new ways of preconditioning linear systems, based on low-rank approximations.« less

Complex wet-environments in electronic-structure calculations

NASA Astrophysics Data System (ADS)

Fisicaro, Giuseppe; Genovese, Luigi; Andreussi, Oliviero; Marzari, Nicola; Goedecker, Stefan

The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of an applied electrochemical potentials, including complex electrostatic screening coming from the solvent. In the present work we present a solver to handle both the Generalized Poisson and the Poisson-Boltzmann equation. A preconditioned conjugate gradient (PCG) method has been implemented for the Generalized Poisson and the linear regime of the Poisson-Boltzmann, allowing to solve iteratively the minimization problem with some ten iterations. On the other hand, a self-consistent procedure enables us to solve the Poisson-Boltzmann problem. The algorithms take advantage of a preconditioning procedure based on the BigDFT Poisson solver for the standard Poisson equation. They exhibit very high accuracy and parallel efficiency, and allow different boundary conditions, including surfaces. The solver has been integrated into the BigDFT and Quantum-ESPRESSO electronic-structure packages and it will be released as a independent program, suitable for integration in other codes. We present test calculations for large proteins to demonstrate efficiency and performances. This work was done within the PASC and NCCR MARVEL projects. Computer resources were provided by the Swiss National Supercomputing Centre (CSCS) under Project ID s499. LG acknowledges also support from the EXTMOS EU project.

The a(3) Scheme--A Fourth-Order Space-Time Flux-Conserving and Neutrally Stable CESE Solver

NASA Technical Reports Server (NTRS)

Chang, Sin-Chung

2008-01-01

The CESE development is driven by a belief that a solver should (i) enforce conservation laws in both space and time, and (ii) be built from a non-dissipative (i.e., neutrally stable) core scheme so that the numerical dissipation can be controlled effectively. To initiate a systematic CESE development of high order schemes, in this paper we provide a thorough discussion on the structure, consistency, stability, phase error, and accuracy of a new 4th-order space-time flux-conserving and neutrally stable CESE solver of an 1D scalar advection equation. The space-time stencil of this two-level explicit scheme is formed by one point at the upper time level and three points at the lower time level. Because it is associated with three independent mesh variables (the numerical analogues of the dependent variable and its 1st-order and 2ndorder spatial derivatives, respectively) and three equations per mesh point, the new scheme is referred to as the a(3) scheme. Through the von Neumann analysis, it is shown that the a(3) scheme is stable if and only if the Courant number is less than 0.5. Moreover, it is established numerically that the a(3) scheme is 4th-order accurate.

Design of a Modular Monolithic Implicit Solver for Multi-Physics Applications

NASA Technical Reports Server (NTRS)

Carton De Wiart, Corentin; Diosady, Laslo T.; Garai, Anirban; Burgess, Nicholas; Blonigan, Patrick; Ekelschot, Dirk; Murman, Scott M.

2018-01-01

The design of a modular multi-physics high-order space-time finite-element framework is presented together with its extension to allow monolithic coupling of different physics. One of the main objectives of the framework is to perform efficient high- fidelity simulations of capsule/parachute systems. This problem requires simulating multiple physics including, but not limited to, the compressible Navier-Stokes equations, the dynamics of a moving body with mesh deformations and adaptation, the linear shell equations, non-re effective boundary conditions and wall modeling. The solver is based on high-order space-time - finite element methods. Continuous, discontinuous and C1-discontinuous Galerkin methods are implemented, allowing one to discretize various physical models. Tangent and adjoint sensitivity analysis are also targeted in order to conduct gradient-based optimization, error estimation, mesh adaptation, and flow control, adding another layer of complexity to the framework. The decisions made to tackle these challenges are presented. The discussion focuses first on the "single-physics" solver and later on its extension to the monolithic coupling of different physics. The implementation of different physics modules, relevant to the capsule/parachute system, are also presented. Finally, examples of coupled computations are presented, paving the way to the simulation of the full capsule/parachute system.

Examples of Linking Codes Within GeoFramework

NASA Astrophysics Data System (ADS)

Tan, E.; Choi, E.; Thoutireddy, P.; Aivazis, M.; Lavier, L.; Quenette, S.; Gurnis, M.

2003-12-01

Geological processes usually encompass a broad spectrum of length and time scales. Traditionally, a modeling code (solver) is written to solve a problem with specific length and time scales in mind. The utility of the solver beyond the designated purpose is usually limited. Furthermore, two distinct solvers, even if each can solve complementary parts of a new problem, are difficult to link together to solve the problem as a whole. For example, Lagrangian deformation model with visco-elastoplastic crust is used to study deformation near plate boundary. Ideally, the driving force of the deformation should be derived from underlying mantle convection, and it requires linking the Lagrangian deformation model with a Eulerian mantle convection model. As our understanding of geological processes evolves, the need of integrated modeling codes, which should reuse existing codes as much as possible, begins to surface. GeoFramework project addresses this need by developing a suite of reusable and re-combinable tools for the Earth science community. GeoFramework is based on and extends Pyre, a Python-based modeling framework, recently developed to link solid (Lagrangian) and fluid (Eulerian) models, as well as mesh generators, visualization packages, and databases, with one another for engineering applications. Under the framework, a solver is aware of the existence of other solvers and can interact with each other via exchanging information across adjacent boundary. A solver needs to conform a standard interface and provide its own implementation for exchanging boundary information. The framework also provides facilities to control the coordination between interacting solvers. We will show an example of linking two solvers within GeoFramework. CitcomS is a finite element code which solves for thermal convection within a 3D spherical shell. CitcomS can solve for problems either within a full spherical (global) domain or a restricted (regional) domain of a full sphere by using different meshers. We can embed a regional CitcomS solver within a global CitcomS solver. We not that linking instances of the same solver is conceptually equivalent to linking to different solvers. The global solver has a coarser grid and a longer stable time step than the regional solver. Therefore, a global-solver time step consists of several regional-solver time steps. The time-marching scheme is described below. First, the global solver is advanced one global-solver time step. Then, the regional solver is advanced for several regional-solver time steps until it catches up global solver. Within each regional-solver time step, the velocity field of the global solver is interpolated in time and then is imposed to the regional solver as boundary conditions. Finally, the temperature field of the regional solver is extrapolated in space and is fed back to the global. These two solvers are linked and synchronized by the time-marching scheme. An effort to embed a visco-elastoplastic representation of the crust within viscous mantle flow is underway.

An exterior Poisson solver using fast direct methods and boundary integral equations with applications to nonlinear potential flow

NASA Technical Reports Server (NTRS)

Young, D. P.; Woo, A. C.; Bussoletti, J. E.; Johnson, F. T.

1986-01-01

A general method is developed combining fast direct methods and boundary integral equation methods to solve Poisson's equation on irregular exterior regions. The method requires O(N log N) operations where N is the number of grid points. Error estimates are given that hold for regions with corners and other boundary irregularities. Computational results are given in the context of computational aerodynamics for a two-dimensional lifting airfoil. Solutions of boundary integral equations for lifting and nonlifting aerodynamic configurations using preconditioned conjugate gradient are examined for varying degrees of thinness.

A numerical framework for bubble transport in a subcooled fluid flow

NASA Astrophysics Data System (ADS)

Jareteg, Klas; Sasic, Srdjan; Vinai, Paolo; Demazière, Christophe

2017-09-01

In this paper we present a framework for the simulation of dispersed bubbly two-phase flows, with the specific aim of describing vapor-liquid systems with condensation. We formulate and implement a framework that consists of a population balance equation (PBE) for the bubble size distribution and an Eulerian-Eulerian two-fluid solver. The PBE is discretized using the Direct Quadrature Method of Moments (DQMOM) in which we include the condensation of the bubbles as an internal phase space convection. We investigate the robustness of the DQMOM formulation and the numerical issues arising from the rapid shrinkage of the vapor bubbles. In contrast to a PBE method based on the multiple-size-group (MUSIG) method, the DQMOM formulation allows us to compute a distribution with dynamic bubble sizes. Such a property is advantageous to capture the wide range of bubble sizes associated with the condensation process. Furthermore, we compare the computational performance of the DQMOM-based framework with the MUSIG method. The results demonstrate that DQMOM is able to retrieve the bubble size distribution with a good numerical precision in only a small fraction of the computational time required by MUSIG. For the two-fluid solver, we examine the implementation of the mass, momentum and enthalpy conservation equations in relation to the coupling to the PBE. In particular, we propose a formulation of the pressure and liquid continuity equations, that was shown to correctly preserve mass when computing the vapor fraction with DQMOM. In addition, the conservation of enthalpy was also proven. Therefore a consistent overall framework that couples the PBE and two-fluid solvers is achieved.

Geometric multigrid for an implicit-time immersed boundary method

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

Guy, Robert D.; Philip, Bobby; Griffith, Boyce E.

2014-10-12

The immersed boundary (IB) method is an approach to fluid-structure interaction that uses Lagrangian variables to describe the deformations and resulting forces of the structure and Eulerian variables to describe the motion and forces of the fluid. Explicit time stepping schemes for the IB method require solvers only for Eulerian equations, for which fast Cartesian grid solution methods are available. Such methods are relatively straightforward to develop and are widely used in practice but often require very small time steps to maintain stability. Implicit-time IB methods permit the stable use of large time steps, but efficient implementations of such methodsmore » require significantly more complex solvers that effectively treat both Lagrangian and Eulerian variables simultaneously. Moreover, several different approaches to solving the coupled Lagrangian-Eulerian equations have been proposed, but a complete understanding of this problem is still emerging. This paper presents a geometric multigrid method for an implicit-time discretization of the IB equations. This multigrid scheme uses a generalization of box relaxation that is shown to handle problems in which the physical stiffness of the structure is very large. Numerical examples are provided to illustrate the effectiveness and efficiency of the algorithms described herein. Finally, these tests show that using multigrid as a preconditioner for a Krylov method yields improvements in both robustness and efficiency as compared to using multigrid as a solver. They also demonstrate that with a time step 100–1000 times larger than that permitted by an explicit IB method, the multigrid-preconditioned implicit IB method is approximately 50–200 times more efficient than the explicit method.« less

Large liquid rocket engine transient performance simulation system

NASA Technical Reports Server (NTRS)

Mason, J. R.; Southwick, R. D.

1991-01-01

A simulation system, ROCETS, was designed and developed to allow cost-effective computer predictions of liquid rocket engine transient performance. The system allows a user to generate a simulation of any rocket engine configuration using component modules stored in a library through high-level input commands. The system library currently contains 24 component modules, 57 sub-modules and maps, and 33 system routines and utilities. FORTRAN models from other sources can be operated in the system upon inclusion of interface information on comment cards. Operation of the simulation is simplified for the user by run, execution, and output processors. The simulation system makes available steady-state trim balance, transient operation, and linear partial generation. The system utilizes a modern equation solver for efficient operation of the simulations. Transient integration methods include integral and differential forms for the trapezoidal, first order Gear, and second order Gear corrector equations. A detailed technology test bed engine (TTBE) model was generated to be used as the acceptance test of the simulation system. The general level of model detail was that reflected in the Space Shuttle Main Engine DTM. The model successfully obtained steady-state balance in main stage operation and simulated throttle transients, including engine starts and shutdown. A NASA FORTRAN control model was obtained, ROCETS interface installed in comment cards, and operated with the TTBE model in closed-loop transient mode.

Investigation of a Parabolic Iterative Solver for Three-dimensional Configurations

NASA Technical Reports Server (NTRS)

Nark, Douglas M.; Watson, Willie R.; Mani, Ramani

2007-01-01

A parabolic iterative solution procedure is investigated that seeks to extend the parabolic approximation used within the internal propagation module of the duct noise propagation and radiation code CDUCT-LaRC. The governing convected Helmholtz equation is split into a set of coupled equations governing propagation in the positive and negative directions. The proposed method utilizes an iterative procedure to solve the coupled equations in an attempt to account for possible reflections from internal bifurcations, impedance discontinuities, and duct terminations. A geometry consistent with the NASA Langley Curved Duct Test Rig is considered and the effects of acoustic treatment and non-anechoic termination are included. Two numerical implementations are studied and preliminary results indicate that improved accuracy in predicted amplitude and phase can be obtained for modes at a cut-off ratio of 1.7. Further predictions for modes at a cut-off ratio of 1.1 show improvement in predicted phase at the expense of increased amplitude error. Possible methods of improvement are suggested based on analytic and numerical analysis. It is hoped that coupling the parabolic iterative approach with less efficient, high fidelity finite element approaches will ultimately provide the capability to perform efficient, higher fidelity acoustic calculations within complex 3-D geometries for impedance eduction and noise propagation and radiation predictions.

Modelling atmospheric flows with adaptive moving meshes

NASA Astrophysics Data System (ADS)

Kühnlein, Christian; Smolarkiewicz, Piotr K.; Dörnbrack, Andreas

2012-04-01

An anelastic atmospheric flow solver has been developed that combines semi-implicit non-oscillatory forward-in-time numerics with a solution-adaptive mesh capability. A key feature of the solver is the unification of a mesh adaptation apparatus, based on moving mesh partial differential equations (PDEs), with the rigorous formulation of the governing anelastic PDEs in generalised time-dependent curvilinear coordinates. The solver development includes an enhancement of the flux-form multidimensional positive definite advection transport algorithm (MPDATA) - employed in the integration of the underlying anelastic PDEs - that ensures full compatibility with mass continuity under moving meshes. In addition, to satisfy the geometric conservation law (GCL) tensor identity under general moving meshes, a diagnostic approach is proposed based on the treatment of the GCL as an elliptic problem. The benefits of the solution-adaptive moving mesh technique for the simulation of multiscale atmospheric flows are demonstrated. The developed solver is verified for two idealised flow problems with distinct levels of complexity: passive scalar advection in a prescribed deformational flow, and the life cycle of a large-scale atmospheric baroclinic wave instability showing fine-scale phenomena of fronts and internal gravity waves.

Parallel O(N) Stokes’ solver towards scalable Brownian dynamics of hydrodynamically interacting objects in general geometries

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

Zhao, Xujun; Li, Jiyuan; Jiang, Xikai

An efficient parallel Stokes’s solver is developed towards the complete inclusion of hydrodynamic interactions of Brownian particles in any geometry. A Langevin description of the particle dynamics is adopted, where the long-range interactions are included using a Green’s function formalism. We present a scalable parallel computational approach, where the general geometry Stokeslet is calculated following a matrix-free algorithm using the General geometry Ewald-like method. Our approach employs a highly-efficient iterative finite element Stokes’ solver for the accurate treatment of long-range hydrodynamic interactions within arbitrary confined geometries. A combination of mid-point time integration of the Brownian stochastic differential equation, the parallelmore » Stokes’ solver, and a Chebyshev polynomial approximation for the fluctuation-dissipation theorem result in an O(N) parallel algorithm. We also illustrate the new algorithm in the context of the dynamics of confined polymer solutions in equilibrium and non-equilibrium conditions. Our method is extended to treat suspended finite size particles of arbitrary shape in any geometry using an Immersed Boundary approach.« less

Numerical aspects and implementation of a two-layer zonal wall model for LES of compressible turbulent flows on unstructured meshes

NASA Astrophysics Data System (ADS)

Park, George Ilhwan; Moin, Parviz

2016-01-01

This paper focuses on numerical and practical aspects associated with a parallel implementation of a two-layer zonal wall model for large-eddy simulation (LES) of compressible wall-bounded turbulent flows on unstructured meshes. A zonal wall model based on the solution of unsteady three-dimensional Reynolds-averaged Navier-Stokes (RANS) equations on a separate near-wall grid is implemented in an unstructured, cell-centered finite-volume LES solver. The main challenge in its implementation is to couple two parallel, unstructured flow solvers for efficient boundary data communication and simultaneous time integrations. A coupling strategy with good load balancing and low processors underutilization is identified. Face mapping and interpolation procedures at the coupling interface are explained in detail. The method of manufactured solution is used for verifying the correct implementation of solver coupling, and parallel performance of the combined wall-modeled LES (WMLES) solver is investigated. The method has successfully been applied to several attached and separated flows, including a transitional flow over a flat plate and a separated flow over an airfoil at an angle of attack.

Implicit integration methods for dislocation dynamics

DOE PAGES

Gardner, D. J.; Woodward, C. S.; Reynolds, D. R.; ...

2015-01-20

In dislocation dynamics simulations, strain hardening simulations require integrating stiff systems of ordinary differential equations in time with expensive force calculations, discontinuous topological events, and rapidly changing problem size. Current solvers in use often result in small time steps and long simulation times. Faster solvers may help dislocation dynamics simulations accumulate plastic strains at strain rates comparable to experimental observations. Here, this paper investigates the viability of high order implicit time integrators and robust nonlinear solvers to reduce simulation run times while maintaining the accuracy of the computed solution. In particular, implicit Runge-Kutta time integrators are explored as a waymore » of providing greater accuracy over a larger time step than is typically done with the standard second-order trapezoidal method. In addition, both accelerated fixed point and Newton's method are investigated to provide fast and effective solves for the nonlinear systems that must be resolved within each time step. Results show that integrators of third order are the most effective, while accelerated fixed point and Newton's method both improve solver performance over the standard fixed point method used for the solution of the nonlinear systems.« less

An installed nacelle design code using a multiblock Euler solver. Volume 1: Theory document

NASA Technical Reports Server (NTRS)

Chen, H. C.

1992-01-01

An efficient multiblock Euler design code was developed for designing a nacelle installed on geometrically complex airplane configurations. This approach employed a design driver based on a direct iterative surface curvature method developed at LaRC. A general multiblock Euler flow solver was used for computing flow around complex geometries. The flow solver used a finite-volume formulation with explicit time-stepping to solve the Euler Equations. It used a multiblock version of the multigrid method to accelerate the convergence of the calculations. The design driver successively updated the surface geometry to reduce the difference between the computed and target pressure distributions. In the flow solver, the change in surface geometry was simulated by applying surface transpiration boundary conditions to avoid repeated grid generation during design iterations. Smoothness of the designed surface was ensured by alternate application of streamwise and circumferential smoothings. The capability and efficiency of the code was demonstrated through the design of both an isolated nacelle and an installed nacelle at various flow conditions. Information on the execution of the computer program is provided in volume 2.

Parallel O(N) Stokes’ solver towards scalable Brownian dynamics of hydrodynamically interacting objects in general geometries

DOE PAGES

Zhao, Xujun; Li, Jiyuan; Jiang, Xikai; ...

2017-06-29

An efficient parallel Stokes’s solver is developed towards the complete inclusion of hydrodynamic interactions of Brownian particles in any geometry. A Langevin description of the particle dynamics is adopted, where the long-range interactions are included using a Green’s function formalism. We present a scalable parallel computational approach, where the general geometry Stokeslet is calculated following a matrix-free algorithm using the General geometry Ewald-like method. Our approach employs a highly-efficient iterative finite element Stokes’ solver for the accurate treatment of long-range hydrodynamic interactions within arbitrary confined geometries. A combination of mid-point time integration of the Brownian stochastic differential equation, the parallelmore » Stokes’ solver, and a Chebyshev polynomial approximation for the fluctuation-dissipation theorem result in an O(N) parallel algorithm. We also illustrate the new algorithm in the context of the dynamics of confined polymer solutions in equilibrium and non-equilibrium conditions. Our method is extended to treat suspended finite size particles of arbitrary shape in any geometry using an Immersed Boundary approach.« less

Notes on the ExactPack Implementation of the DSD Rate Stick Solver

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

Kaul, Ann

It has been shown above that the discretization scheme implemented in the ExactPack solver for the DSD Rate Stick equation is consistent with the Rate Stick PDE. In addition, a stability analysis has provided a CFL condition for a stable time step. Together, consistency and stability imply convergence of the scheme, which is expected to be close to first-order in time and second-order in space. It is understood that the nonlinearity of the underlying PDE will affect this rate somewhat. In the solver I implemented in ExactPack, I used the one-sided boundary condition described above at the outer boundary. Inmore » addition, I used 80% of the time step calculated in the stability analysis above. By making these two changes, I was able to implement a solver that calculates the solution without any arbitrary limits placed on the values of the curvature at the boundary. Thus, the calculation is driven directly by the conditions at the boundary as formulated in the DSD theory. The chosen scheme is completely coherent and defensible from a mathematical standpoint.« less

Novel Scalable 3-D MT Inverse Solver

NASA Astrophysics Data System (ADS)

Kuvshinov, A. V.; Kruglyakov, M.; Geraskin, A.

2016-12-01

We present a new, robust and fast, three-dimensional (3-D) magnetotelluric (MT) inverse solver. As a forward modelling engine a highly-scalable solver extrEMe [1] is used. The (regularized) inversion is based on an iterative gradient-type optimization (quasi-Newton method) and exploits adjoint sources approach for fast calculation of the gradient of the misfit. The inverse solver is able to deal with highly detailed and contrasting models, allows for working (separately or jointly) with any type of MT (single-site and/or inter-site) responses, and supports massive parallelization. Different parallelization strategies implemented in the code allow for optimal usage of available computational resources for a given problem set up. To parameterize an inverse domain a mask approach is implemented, which means that one can merge any subset of forward modelling cells in order to account for (usually) irregular distribution of observation sites. We report results of 3-D numerical experiments aimed at analysing the robustness, performance and scalability of the code. In particular, our computational experiments carried out at different platforms ranging from modern laptops to high-performance clusters demonstrate practically linear scalability of the code up to thousands of nodes. 1. Kruglyakov, M., A. Geraskin, A. Kuvshinov, 2016. Novel accurate and scalable 3-D MT forward solver based on a contracting integral equation method, Computers and Geosciences, in press.

Recent applications of the transonic wing analysis computer code, TWING

NASA Technical Reports Server (NTRS)

Subramanian, N. R.; Holst, T. L.; Thomas, S. D.

1982-01-01

An evaluation of the transonic-wing-analysis computer code TWING is given. TWING utilizes a fully implicit approximate factorization iteration scheme to solve the full potential equation in conservative form. A numerical elliptic-solver grid-generation scheme is used to generate the required finite-difference mesh. Several wing configurations were analyzed, and the limits of applicability of this code was evaluated. Comparisons of computed results were made with available experimental data. Results indicate that the code is robust, accurate (when significant viscous effects are not present), and efficient. TWING generally produces solutions an order of magnitude faster than other conservative full potential codes using successive-line overrelaxation. The present method is applicable to a wide range of isolated wing configurations including high-aspect-ratio transport wings and low-aspect-ratio, high-sweep, fighter configurations.

Multi-processing on supercomputers for computational aerodynamics

NASA Technical Reports Server (NTRS)

Yarrow, Maurice; Mehta, Unmeel B.

1990-01-01

The MIMD concept is applied, through multitasking, with relatively minor modifications to an existing code for a single processor. This approach maps the available memory to multiple processors, exploiting the C-FORTRAN-Unix interface. An existing single processor algorithm is mapped without the need for developing a new algorithm. The procedure of designing a code utilizing this approach is automated with the Unix stream editor. A Multiple Processor Multiple Grid (MPMG) code is developed as a demonstration of this approach. This code solves the three-dimensional, Reynolds-averaged, thin-layer and slender-layer Navier-Stokes equations with an implicit, approximately factored and diagonalized method. This solver is applied to a generic, oblique-wing aircraft problem on a four-processor computer using one process for data management and nonparallel computations and three processes for pseudotime advance on three different grid systems.

Finite Element Modeling of Coupled Flexible Multibody Dynamics and Liquid Sloshing

DTIC Science & Technology

2006-09-01

tanks is presented. The semi-discrete combined solid and fluid equations of motions are integrated using a time- accurate parallel explicit solver...Incompressible fluid flow in a moving/deforming container including accurate modeling of the free-surface, turbulence, and viscous effects ...paper, a single computational code which uses a time- accurate explicit solution procedure is used to solve both the solid and fluid equations of

LORENE: Spectral methods differential equations solver

NASA Astrophysics Data System (ADS)

Gourgoulhon, Eric; Grandclément, Philippe; Marck, Jean-Alain; Novak, Jérôme; Taniguchi, Keisuke

2016-08-01

LORENE (Langage Objet pour la RElativité NumériquE) solves various problems arising in numerical relativity, and more generally in computational astrophysics. It is a set of C++ classes and provides tools to solve partial differential equations by means of multi-domain spectral methods. LORENE classes implement basic structures such as arrays and matrices, but also abstract mathematical objects, such as tensors, and astrophysical objects, such as stars and black holes.

Simulation of nonlinear propagation of biomedical ultrasound using pzflex and the Khokhlov-Zabolotskaya-Kuznetsov Texas code

PubMed Central

Qiao, Shan; Jackson, Edward; Coussios, Constantin C.; Cleveland, Robin O.

2016-01-01

Nonlinear acoustics plays an important role in both diagnostic and therapeutic applications of biomedical ultrasound and a number of research and commercial software packages are available. In this manuscript, predictions of two solvers available in a commercial software package, pzflex, one using the finite-element-method (FEM) and the other a pseudo-spectral method, spectralflex, are compared with measurements and the Khokhlov-Zabolotskaya-Kuznetsov (KZK) Texas code (a finite-difference time-domain algorithm). The pzflex methods solve the continuity equation, momentum equation and equation of state where they account for nonlinearity to second order whereas the KZK code solves a nonlinear wave equation with a paraxial approximation for diffraction. Measurements of the field from a single element 3.3 MHz focused transducer were compared with the simulations and there was good agreement for the fundamental frequency and the harmonics; however the FEM pzflex solver incurred a high computational cost to achieve equivalent accuracy. In addition, pzflex results exhibited non-physical oscillations in the spatial distribution of harmonics when the amplitudes were relatively low. It was found that spectralflex was able to accurately capture the nonlinear fields at reasonable computational cost. These results emphasize the need to benchmark nonlinear simulations before using codes as predictive tools. PMID:27914432

Aerodynamic Shape Optimization of Complex Aircraft Configurations via an Adjoint Formulation

NASA Technical Reports Server (NTRS)

Reuther, James; Jameson, Antony; Farmer, James; Martinelli, Luigi; Saunders, David

1996-01-01

This work describes the implementation of optimization techniques based on control theory for complex aircraft configurations. Here control theory is employed to derive the adjoint differential equations, the solution of which allows for a drastic reduction in computational costs over previous design methods (13, 12, 43, 38). In our earlier studies (19, 20, 22, 23, 39, 25, 40, 41, 42) it was shown that this method could be used to devise effective optimization procedures for airfoils, wings and wing-bodies subject to either analytic or arbitrary meshes. Design formulations for both potential flows and flows governed by the Euler equations have been demonstrated, showing that such methods can be devised for various governing equations (39, 25). In our most recent works (40, 42) the method was extended to treat wing-body configurations with a large number of mesh points, verifying that significant computational savings can be gained for practical design problems. In this paper the method is extended for the Euler equations to treat complete aircraft configurations via a new multiblock implementation. New elements include a multiblock-multigrid flow solver, a multiblock-multigrid adjoint solver, and a multiblock mesh perturbation scheme. Two design examples are presented in which the new method is used for the wing redesign of a transonic business jet.

Simulation of nonlinear propagation of biomedical ultrasound using pzflex and the Khokhlov-Zabolotskaya-Kuznetsov Texas code.

PubMed

Qiao, Shan; Jackson, Edward; Coussios, Constantin C; Cleveland, Robin O

2016-09-01

Nonlinear acoustics plays an important role in both diagnostic and therapeutic applications of biomedical ultrasound and a number of research and commercial software packages are available. In this manuscript, predictions of two solvers available in a commercial software package, pzflex, one using the finite-element-method (FEM) and the other a pseudo-spectral method, spectralflex, are compared with measurements and the Khokhlov-Zabolotskaya-Kuznetsov (KZK) Texas code (a finite-difference time-domain algorithm). The pzflex methods solve the continuity equation, momentum equation and equation of state where they account for nonlinearity to second order whereas the KZK code solves a nonlinear wave equation with a paraxial approximation for diffraction. Measurements of the field from a single element 3.3 MHz focused transducer were compared with the simulations and there was good agreement for the fundamental frequency and the harmonics; however the FEM pzflex solver incurred a high computational cost to achieve equivalent accuracy. In addition, pzflex results exhibited non-physical oscillations in the spatial distribution of harmonics when the amplitudes were relatively low. It was found that spectralflex was able to accurately capture the nonlinear fields at reasonable computational cost. These results emphasize the need to benchmark nonlinear simulations before using codes as predictive tools.

Detailed analysis of the effects of stencil spatial variations with arbitrary high-order finite-difference Maxwell solver

DOE PAGES

Vincenti, H.; Vay, J. -L.

2015-11-22

Due to discretization effects and truncation to finite domains, many electromagnetic simulations present non-physical modifications of Maxwell's equations in space that may generate spurious signals affecting the overall accuracy of the result. Such modifications for instance occur when Perfectly Matched Layers (PMLs) are used at simulation domain boundaries to simulate open media. Another example is the use of arbitrary order Maxwell solver with domain decomposition technique that may under some condition involve stencil truncations at subdomain boundaries, resulting in small spurious errors that do eventually build up. In each case, a careful evaluation of the characteristics and magnitude of themore » errors resulting from these approximations, and their impact at any frequency and angle, requires detailed analytical and numerical studies. To this end, we present a general analytical approach that enables the evaluation of numerical discretization errors of fully three-dimensional arbitrary order finite-difference Maxwell solver, with arbitrary modification of the local stencil in the simulation domain. The analytical model is validated against simulations of domain decomposition technique and PMLs, when these are used with very high-order Maxwell solver, as well as in the infinite order limit of pseudo-spectral solvers. Results confirm that the new analytical approach enables exact predictions in each case. It also confirms that the domain decomposition technique can be used with very high-order Maxwell solver and a reasonably low number of guard cells with negligible effects on the whole accuracy of the simulation.« less

Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers

NASA Astrophysics Data System (ADS)

Balsara, Dinshaw S.; Dumbser, Michael

2015-10-01

Several advances have been reported in the recent literature on divergence-free finite volume schemes for Magnetohydrodynamics (MHD). Almost all of these advances are restricted to structured meshes. To retain full geometric versatility, however, it is also very important to make analogous advances in divergence-free schemes for MHD on unstructured meshes. Such schemes utilize a staggered Yee-type mesh, where all hydrodynamic quantities (mass, momentum and energy density) are cell-centered, while the magnetic fields are face-centered and the electric fields, which are so useful for the time update of the magnetic field, are centered at the edges. Three important advances are brought together in this paper in order to make it possible to have high order accurate finite volume schemes for the MHD equations on unstructured meshes. First, it is shown that a divergence-free WENO reconstruction of the magnetic field can be developed for unstructured meshes in two and three space dimensions using a classical cell-centered WENO algorithm, without the need to do a WENO reconstruction for the magnetic field on the faces. This is achieved via a novel constrained L2-projection operator that is used in each time step as a postprocessor of the cell-centered WENO reconstruction so that the magnetic field becomes locally and globally divergence free. Second, it is shown that recently-developed genuinely multidimensional Riemann solvers (called MuSIC Riemann solvers) can be used on unstructured meshes to obtain a multidimensionally upwinded representation of the electric field at each edge. Third, the above two innovations work well together with a high order accurate one-step ADER time stepping strategy, which requires the divergence-free nonlinear WENO reconstruction procedure to be carried out only once per time step. The resulting divergence-free ADER-WENO schemes with MuSIC Riemann solvers give us an efficient and easily-implemented strategy for divergence-free MHD on unstructured meshes. Several stringent two- and three-dimensional problems are shown to work well with the methods presented here.

Unstructured Mesh Methods for the Simulation of Hypersonic Flows

NASA Technical Reports Server (NTRS)

Peraire, Jaime; Bibb, K. L. (Technical Monitor)

2001-01-01

This report describes the research work undertaken at the Massachusetts Institute of Technology. The aim of this research is to identify effective algorithms and methodologies for the efficient and routine solution of hypersonic viscous flows about re-entry vehicles. For over ten years we have received support from NASA to develop unstructured mesh methods for Computational Fluid Dynamics. As a result of this effort a methodology based on the use, of unstructured adapted meshes of tetrahedra and finite volume flow solvers has been developed. A number of gridding algorithms flow solvers, and adaptive strategies have been proposed. The most successful algorithms developed from the basis of the unstructured mesh system FELISA. The FELISA system has been extensively for the analysis of transonic and hypersonic flows about complete vehicle configurations. The system is highly automatic and allows for the routine aerodynamic analysis of complex configurations starting from CAD data. The code has been parallelized and utilizes efficient solution algorithms. For hypersonic flows, a version of the, code which incorporates real gas effects, has been produced. One of the latest developments before the start of this grant was to extend the system to include viscous effects. This required the development of viscous generators, capable of generating the anisotropic grids required to represent boundary layers, and viscous flow solvers. In figures I and 2, we show some sample hypersonic viscous computations using the developed viscous generators and solvers. Although these initial results were encouraging, it became apparent that in order to develop a fully functional capability for viscous flows, several advances in gridding, solution accuracy, robustness and efficiency were required. As part of this research we have developed: 1) automatic meshing techniques and the corresponding computer codes have been delivered to NASA and implemented into the GridEx system, 2) a finite element algorithm for the solution of the viscous compressible flow equations which can solve flows all the way down to the incompressible limit and that can use higher order (quadratic) approximations leading to highly accurate answers, and 3) and iterative algebraic multigrid solution techniques.

Analytical results for post-buckling behaviour of plates in compression and in shear

NASA Technical Reports Server (NTRS)

Stein, M.

1985-01-01

The postbuckling behavior of long rectangular isotropic and orthotropic plates is determined. By assuming trigonometric functions in one direction, the nonlinear partial differential equations of von Karman large deflection plate theory are converted into nonlinear ordinary differential equations. The ordinary differential equations are solved numerically using an available boundary value problem solver which makes use of Newton's method. Results for longitudinal compression show different postbuckling behavior between isotropic and orthotropic plates. Results for shear show that change in inplane edge constraints can cause large change in postbuckling stiffness.

Efficient Solution of Three-Dimensional Problems of Acoustic and Electromagnetic Scattering by Open Surfaces

NASA Technical Reports Server (NTRS)

Turc, Catalin; Anand, Akash; Bruno, Oscar; Chaubell, Julian

2011-01-01

We present a computational methodology (a novel Nystrom approach based on use of a non-overlapping patch technique and Chebyshev discretizations) for efficient solution of problems of acoustic and electromagnetic scattering by open surfaces. Our integral equation formulations (1) Incorporate, as ansatz, the singular nature of open-surface integral-equation solutions, and (2) For the Electric Field Integral Equation (EFIE), use analytical regularizes that effectively reduce the number of iterations required by iterative linear-algebra solution based on Krylov-subspace iterative solvers.

Hybrid ODE/SSA methods and the cell cycle model

NASA Astrophysics Data System (ADS)

Wang, S.; Chen, M.; Cao, Y.

2017-07-01

Stochastic effect in cellular systems has been an important topic in systems biology. Stochastic modeling and simulation methods are important tools to study stochastic effect. Given the low efficiency of stochastic simulation algorithms, the hybrid method, which combines an ordinary differential equation (ODE) system with a stochastic chemically reacting system, shows its unique advantages in the modeling and simulation of biochemical systems. The efficiency of hybrid method is usually limited by reactions in the stochastic subsystem, which are modeled and simulated using Gillespie's framework and frequently interrupt the integration of the ODE subsystem. In this paper we develop an efficient implementation approach for the hybrid method coupled with traditional ODE solvers. We also compare the efficiency of hybrid methods with three widely used ODE solvers RADAU5, DASSL, and DLSODAR. Numerical experiments with three biochemical models are presented. A detailed discussion is presented for the performances of three ODE solvers.

A fast parallel 3D Poisson solver with longitudinal periodic and transverse open boundary conditions for space-charge simulations

NASA Astrophysics Data System (ADS)

Qiang, Ji

2017-10-01

A three-dimensional (3D) Poisson solver with longitudinal periodic and transverse open boundary conditions can have important applications in beam physics of particle accelerators. In this paper, we present a fast efficient method to solve the Poisson equation using a spectral finite-difference method. This method uses a computational domain that contains the charged particle beam only and has a computational complexity of O(Nu(logNmode)) , where Nu is the total number of unknowns and Nmode is the maximum number of longitudinal or azimuthal modes. This saves both the computational time and the memory usage of using an artificial boundary condition in a large extended computational domain. The new 3D Poisson solver is parallelized using a message passing interface (MPI) on multi-processor computers and shows a reasonable parallel performance up to hundreds of processor cores.

A spectral Poisson solver for kinetic plasma simulation

NASA Astrophysics Data System (ADS)

Szeremley, Daniel; Obberath, Jens; Brinkmann, Ralf

2011-10-01

Plasma resonance spectroscopy is a well established plasma diagnostic method, realized in several designs. One of these designs is the multipole resonance probe (MRP). In its idealized - geometrically simplified - version it consists of two dielectrically shielded, hemispherical electrodes to which an RF signal is applied. A numerical tool is under development which is capable of simulating the dynamics of the plasma surrounding the MRP in electrostatic approximation. In this contribution we concentrate on the specialized Poisson solver for that tool. The plasma is represented by an ensemble of point charges. By expanding both the charge density and the potential into spherical harmonics, a largely analytical solution of the Poisson problem can be employed. For a practical implementation, the expansion must be appropriately truncated. With this spectral solver we are able to efficiently solve the Poisson equation in a kinetic plasma simulation without the need of introducing a spatial discretization.

GENASIS Mathematics : Object-oriented manifolds, operations, and solvers for large-scale physics simulations

NASA Astrophysics Data System (ADS)

Cardall, Christian Y.; Budiardja, Reuben D.

2018-01-01

The large-scale computer simulation of a system of physical fields governed by partial differential equations requires some means of approximating the mathematical limit of continuity. For example, conservation laws are often treated with a 'finite-volume' approach in which space is partitioned into a large number of small 'cells,' with fluxes through cell faces providing an intuitive discretization modeled on the mathematical definition of the divergence operator. Here we describe and make available Fortran 2003 classes furnishing extensible object-oriented implementations of simple meshes and the evolution of generic conserved currents thereon, along with individual 'unit test' programs and larger example problems demonstrating their use. These classes inaugurate the Mathematics division of our developing astrophysics simulation code GENASIS (Gen eral A strophysical Si mulation S ystem), which will be expanded over time to include additional meshing options, mathematical operations, solver types, and solver variations appropriate for many multiphysics applications.

TADS: A CFD-based turbomachinery and analysis design system with GUI. Volume 2: User's manual

NASA Technical Reports Server (NTRS)

Myers, R. A.; Topp, D. A.; Delaney, R. A.

1995-01-01

The primary objective of this study was the development of a computational fluid dynamics (CFD) based turbomachinery airfoil analysis and design system, controlled by a graphical user interface (GUI). The computer codes resulting from this effort are referred to as the Turbomachinery Analysis and Design System (TADS). This document is intended to serve as a user's manual for the computer programs which comprise the TADS system. TADS couples a throughflow solver (ADPAC) with a quasi-3D blade-to-blade solver (RVCQ3D) in an interactive package. Throughflow analysis capability was developed in ADPAC through the addition of blade force and blockage terms to the governing equations. A GUI was developed to simplify user input and automate the many tasks required to perform turbomachinery analysis and design. The coupling of various programs was done in a way that alternative solvers or grid generators could be easily incorporated into the TADS framework.

Comparison of an algebraic multigrid algorithm to two iterative solvers used for modeling ground water flow and transport

USGS Publications Warehouse

Detwiler, R.L.; Mehl, S.; Rajaram, H.; Cheung, W.W.

2002-01-01

Numerical solution of large-scale ground water flow and transport problems is often constrained by the convergence behavior of the iterative solvers used to solve the resulting systems of equations. We demonstrate the ability of an algebraic multigrid algorithm (AMG) to efficiently solve the large, sparse systems of equations that result from computational models of ground water flow and transport in large and complex domains. Unlike geometric multigrid methods, this algorithm is applicable to problems in complex flow geometries, such as those encountered in pore-scale modeling of two-phase flow and transport. We integrated AMG into MODFLOW 2000 to compare two- and three-dimensional flow simulations using AMG to simulations using PCG2, a preconditioned conjugate gradient solver that uses the modified incomplete Cholesky preconditioner and is included with MODFLOW 2000. CPU times required for convergence with AMG were up to 140 times faster than those for PCG2. The cost of this increased speed was up to a nine-fold increase in required random access memory (RAM) for the three-dimensional problems and up to a four-fold increase in required RAM for the two-dimensional problems. We also compared two-dimensional numerical simulations of steady-state transport using AMG and the generalized minimum residual method with an incomplete LU-decomposition preconditioner. For these transport simulations, AMG yielded increased speeds of up to 17 times with only a 20% increase in required RAM. The ability of AMG to solve flow and transport problems in large, complex flow systems and its ready availability make it an ideal solver for use in both field-scale and pore-scale modeling.

Calm water resistance prediction of a bulk carrier using Reynolds averaged Navier-Stokes based solver

NASA Astrophysics Data System (ADS)

Rahaman, Md. Mashiur; Islam, Hafizul; Islam, Md. Tariqul; Khondoker, Md. Reaz Hasan

2017-12-01

Maneuverability and resistance prediction with suitable accuracy is essential for optimum ship design and propulsion power prediction. This paper aims at providing some of the maneuverability characteristics of a Japanese bulk carrier model, JBC in calm water using a computational fluid dynamics solver named SHIP Motion and OpenFOAM. The solvers are based on the Reynolds average Navier-Stokes method (RaNS) and solves structured grid using the Finite Volume Method (FVM). This paper comprises the numerical results of calm water test for the JBC model with available experimental results. The calm water test results include the total drag co-efficient, average sinkage, and trim data. Visualization data for pressure distribution on the hull surface and free water surface have also been included. The paper concludes that the presented solvers predict the resistance and maneuverability characteristics of the bulk carrier with reasonable accuracy utilizing minimum computational resources.

Using the Multiplicative Schwarz Alternating Algorithm (MSAA) for Solving the Large Linear System of Equations Related to Global Gravity Field Recovery up to Degree and Order 120

NASA Astrophysics Data System (ADS)

Safari, A.; Sharifi, M. A.; Amjadiparvar, B.

2010-05-01

The GRACE mission has substantiated the low-low satellite-to-satellite tracking (LL-SST) concept. The LL-SST configuration can be combined with the previously realized high-low SST concept in the CHAMP mission to provide a much higher accuracy. The line of sight (LOS) acceleration difference between the GRACE satellite pair is the mostly used observable for mapping the global gravity field of the Earth in terms of spherical harmonic coefficients. In this paper, mathematical formulae for LOS acceleration difference observations have been derived and the corresponding linear system of equations has been set up for spherical harmonic up to degree and order 120. The total number of unknowns is 14641. Such a linear equation system can be solved with iterative solvers or direct solvers. However, the runtime of direct methods or that of iterative solvers without a suitable preconditioner increases tremendously. This is the reason why we need a more sophisticated method to solve the linear system of problems with a large number of unknowns. Multiplicative variant of the Schwarz alternating algorithm is a domain decomposition method, which allows it to split the normal matrix of the system into several smaller overlaped submatrices. In each iteration step the multiplicative variant of the Schwarz alternating algorithm solves linear systems with the matrices obtained from the splitting successively. It reduces both runtime and memory requirements drastically. In this paper we propose the Multiplicative Schwarz Alternating Algorithm (MSAA) for solving the large linear system of gravity field recovery. The proposed algorithm has been tested on the International Association of Geodesy (IAG)-simulated data of the GRACE mission. The achieved results indicate the validity and efficiency of the proposed algorithm in solving the linear system of equations from accuracy and runtime points of view. Keywords: Gravity field recovery, Multiplicative Schwarz Alternating Algorithm, Low-Low Satellite-to-Satellite Tracking

Calculation of the recirculating compressible flow downstream a sudden axisymmetric expansion

NASA Technical Reports Server (NTRS)

Vandromme, D.; Haminh, H.; Brunet, H.

1988-01-01

Significant progress has been made during the last five years to adapt conventional Navier-Stokes solver for handling nonconservative equations. A primary type of application is to use transport equation turbulence models, but the extension is also possible for describing the transport of nonpassive scalars, such as in reactive media. Among others, combustion and gas dissociation phenomena are topics needing a considerable research effort. An implicit two step scheme based on the well-known MacCormack scheme has been modified to treat compressible turbulent flows on complex geometries. Implicit treatment of nonconservative equations (in the present case a two-equation turbulence model) opens the way to the coupled solution of thermochemical transport equations.

MHD stagnation-point flow over a nonlinearly shrinking sheet with suction effect

NASA Astrophysics Data System (ADS)

Awaludin, Izyan Syazana; Ahmad, Rokiah; Ishak, Anuar

2018-04-01

The stagnation point flow over a shrinking permeable sheet in the existence of magnetic field is numerically investigated in this paper. The system of partial differential equations are transformed to a nonlinear ordinary differential equation using similarity transformation and is solved numerically using the boundary value problem solver, bvp4c, in Matlab software. It is found that dual solutions exist for a certain range of the shrinking strength.

Mixed-Integer Conic Linear Programming: Challenges and Perspectives

DTIC Science & Technology

2013-10-01

The novel DCCs for MISOCO may be used in branch- and-cut algorithms when solving MISOCO problems. The experimental software CICLO was developed to...perform limited, but rigorous computational experiments. The CICLO solver utilizes continuous SOCO solvers, MOSEK, CPLES or SeDuMi, builds on the open...submitted Fall 2013. Software: 1. CICLO : Integer conic linear optimization package. Authors: J.C. Góez, T.K. Ralphs, Y. Fu, and T. Terlaky

An Adaptively-Refined, Cartesian, Cell-Based Scheme for the Euler and Navier-Stokes Equations. Ph.D. Thesis - Michigan Univ.

NASA Technical Reports Server (NTRS)

Coirier, William John

1994-01-01

A Cartesian, cell-based scheme for solving the Euler and Navier-Stokes equations in two dimensions is developed and tested. Grids about geometrically complicated bodies are generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, polygonal 'cut' cells are created. The geometry of the cut cells is computed using polygon-clipping algorithms. The grid is stored in a binary-tree data structure which provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite-volume formulation. The convective terms are upwinded, with a limited linear reconstruction of the primitive variables used to provide input states to an approximate Riemann solver for computing the fluxes between neighboring cells. A multi-stage time-stepping scheme is used to reach a steady-state solution. Validation of the Euler solver with benchmark numerical and exact solutions is presented. An assessment of the accuracy of the approach is made by uniform and adaptive grid refinements for a steady, transonic, exact solution to the Euler equations. The error of the approach is directly compared to a structured solver formulation. A non smooth flow is also assessed for grid convergence, comparing uniform and adaptively refined results. Several formulations of the viscous terms are assessed analytically, both for accuracy and positivity. The two best formulations are used to compute adaptively refined solutions of the Navier-Stokes equations. These solutions are compared to each other, to experimental results and/or theory for a series of low and moderate Reynolds numbers flow fields. The most suitable viscous discretization is demonstrated for geometrically-complicated internal flows. For flows at high Reynolds numbers, both an altered grid-generation procedure and a different formulation of the viscous terms are shown to be necessary. A hybrid Cartesian/body-fitted grid generation approach is demonstrated. In addition, a grid-generation procedure based on body-aligned cell cutting coupled with a viscous stensil-construction procedure based on quadratic programming is presented.

Researched applied to transonic compressors in numerical fluid mechanics of inviscid flow and viscous flow

NASA Technical Reports Server (NTRS)

Thompkins, W. T., Jr.

1985-01-01

A streamline Euler solver which combines high accuracy and good convergence rates with capabilities for inverse or direct mode solution modes and an analysis technique for finite difference models of hyperbolic partial difference equations were developed.

Some special solutions to the Hyperbolic NLS equation

NASA Astrophysics Data System (ADS)

Vuillon, Laurent; Dutykh, Denys; Fedele, Francesco

2018-04-01

The Hyperbolic Nonlinear SCHRöDINGER equation (HypNLS) arises as a model for the dynamics of three-dimensional narrow-band deep water gravity waves. In this study, the symmetries and conservation laws of this equation are computed. The PETVIASHVILI method is then exploited to numerically compute bi-periodic time-harmonic solutions of the HypNLS equation. In physical space they represent non-localized standing waves. Non-trivial spatial patterns are revealed and an attempt is made to describe them using symbolic dynamics and the language of substitutions. Finally, the dynamics of a slightly perturbed standing wave is numerically investigated by means a highly accurate FOURIER solver.

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

NASA Technical Reports Server (NTRS)

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

1995-01-01

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

An accurate and efficient acoustic eigensolver based on a fast multipole BEM and a contour integral method

NASA Astrophysics Data System (ADS)

Zheng, Chang-Jun; Gao, Hai-Feng; Du, Lei; Chen, Hai-Bo; Zhang, Chuanzeng

2016-01-01

An accurate numerical solver is developed in this paper for eigenproblems governed by the Helmholtz equation and formulated through the boundary element method. A contour integral method is used to convert the nonlinear eigenproblem into an ordinary eigenproblem, so that eigenvalues can be extracted accurately by solving a set of standard boundary element systems of equations. In order to accelerate the solution procedure, the parameters affecting the accuracy and efficiency of the method are studied and two contour paths are compared. Moreover, a wideband fast multipole method is implemented with a block IDR (s) solver to reduce the overall solution cost of the boundary element systems of equations with multiple right-hand sides. The Burton-Miller formulation is employed to identify the fictitious eigenfrequencies of the interior acoustic problems with multiply connected domains. The actual effect of the Burton-Miller formulation on tackling the fictitious eigenfrequency problem is investigated and the optimal choice of the coupling parameter as α = i / k is confirmed through exterior sphere examples. Furthermore, the numerical eigenvalues obtained by the developed method are compared with the results obtained by the finite element method to show the accuracy and efficiency of the developed method.

Incorporation of a Chemical Equilibrium Equation of State into LOCI-Chem

NASA Technical Reports Server (NTRS)

Cox, Carey F.

2005-01-01

Renewed interest in development of advanced high-speed transport, reentry vehicles and propulsion systems has led to a resurgence of research into high speed aerodynamics. As this flow regime is typically dominated by hot reacting gaseous flow, efficient models for the characteristic chemical activity are necessary for accurate and cost effective analysis and design of aerodynamic vehicles that transit this regime. The LOCI-Chem code recently developed by Ed Luke at Mississippi State University for NASA/MSFC and used by NASA/MSFC and SSC represents an important step in providing an accurate, efficient computational tool for the simulation of reacting flows through the use of finite-rate kinetics [3]. Finite rate chemistry however, requires the solution of an additional N-1 species mass conservation equations with source terms involving reaction kinetics that are not fully understood. In the equilibrium limit, where the reaction rates approach infinity, these equations become very stiff. Through the use of the assumption of local chemical equilibrium the set of governing equations is reduced back to the usual gas dynamic equations, and thus requires less computation, while still allowing for the inclusion of reacting flow phenomenology. The incorporation of a chemical equilibrium equation of state module into the LOCI-Chem code was the primary objective of the current research. The major goals of the project were: (1) the development of a chemical equilibrium composition solver, and (2) the incorporation of chemical equilibrium solver into LOCI-Chem. Due to time and resource constraints, code optimization was not considered unless it was important to the proper functioning of the code.

Three-dimensional forward modeling and inversion of marine CSEM data in anisotropic conductivity structures

NASA Astrophysics Data System (ADS)

Han, B.; Li, Y.

2016-12-01

We present a three-dimensional (3D) forward and inverse modeling code for marine controlled-source electromagnetic (CSEM) surveys in anisotropic media. The forward solution is based on a primary/secondary field approach, in which secondary fields are solved using a staggered finite-volume (FV) method and primary fields are solved for 1D isotropic background models analytically. It is shown that it is rather straightforward to extend the isotopic 3D FV algorithm to a triaxial anisotropic one, while additional coefficients are required to account for full tensor conductivity. To solve the linear system resulting from FV discretization of Maxwell' s equations, both iterative Krylov solvers (e.g. BiCGSTAB) and direct solvers (e.g. MUMPS) have been implemented, makes the code flexible for different computing platforms and different problems. For iterative soloutions, the linear system in terms of electromagnetic potentials (A-Phi) is used to precondition the original linear system, transforming the discretized Curl-Curl equations to discretized Laplace-like equations, thus much more favorable numerical properties can be obtained. Numerical experiments suggest that this A-Phi preconditioner can dramatically improve the convergence rate of an iterative solver and high accuracy can be achieved without divergence correction even for low frequencies. To efficiently calculate the sensitivities, i.e. the derivatives of CSEM data with respect to tensor conductivity, the adjoint method is employed. For inverse modeling, triaxial anisotropy is taken into account. Since the number of model parameters to be resolved of triaxial anisotropic medias is twice or thrice that of isotropic medias, the data-space version of the Gauss-Newton (GN) minimization method is preferred due to its lower computational cost compared with the traditional model-space GN method. We demonstrate the effectiveness of the code with synthetic examples.

An adjoint view on flux consistency and strong wall boundary conditions to the Navier–Stokes equations

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

Stück, Arthur, E-mail: arthur.stueck@dlr.de

2015-11-15

Inconsistent discrete expressions in the boundary treatment of Navier–Stokes solvers and in the definition of force objective functionals can lead to discrete-adjoint boundary treatments that are not a valid representation of the boundary conditions to the corresponding adjoint partial differential equations. The underlying problem is studied for an elementary 1D advection–diffusion problem first using a node-centred finite-volume discretisation. The defect of the boundary operators in the inconsistently defined discrete-adjoint problem leads to oscillations and becomes evident with the additional insight of the continuous-adjoint approach. A homogenisation of the discretisations for the primal boundary treatment and the force objective functional yieldsmore » second-order functional accuracy and eliminates the defect in the discrete-adjoint boundary treatment. Subsequently, the issue is studied for aerodynamic Reynolds-averaged Navier–Stokes problems in conjunction with a standard finite-volume discretisation on median-dual grids and a strong implementation of noslip walls, found in many unstructured general-purpose flow solvers. Going out from a base-line discretisation of force objective functionals which is independent of the boundary treatment in the flow solver, two improved flux-consistent schemes are presented; based on either body wall-defined or farfield-defined control-volumes they resolve the dual inconsistency. The behaviour of the schemes is investigated on a sequence of grids in 2D and 3D.« less

Aeroelastic Computations of a Compressor Stage Using the Harmonic Balance Method

NASA Technical Reports Server (NTRS)

Reddy, T. S. R.

2010-01-01

The aeroelastic characteristics of a compressor stage were analyzed using a computational fluid dynamic (CFD) solver that uses the harmonic balance method to solve the governing equations. The three dimensional solver models the unsteady flow field due to blade vibration using the Reynolds-Averaged Navier-Stokes equations. The formulation enables the study of the effect of blade row interaction through the inclusion of coupling modes between blade rows. It also enables the study of nonlinear effects of high amplitude blade vibration by the inclusion of higher harmonics of the fundamental blade vibration frequency. In the present work, the solver is applied to study in detail the aeroelastic characteristics of a transonic compressor stage. Various parameters were included in the study: number of coupling modes, blade row axial spacing, and operating speeds. Only the first vibration mode is considered with amplitude of oscillation in the linear range. Both aeroelastic stability (flutter) of rotor blade and unsteady loading on the stator are calculated. The study showed that for the stage considered, the rotor aerodynamic damping is not influenced by the presence of the stator even when the axial spacing is reduced by nearly 25 percent. However, the study showed that blade row interaction effects become important for the unsteady loading on the stator when the axial spacing is reduced by the same amount.

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

NASA Technical Reports Server (NTRS)

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

1994-01-01

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

Computational simulations of supersonic magnetohydrodynamic flow control, power and propulsion systems

NASA Astrophysics Data System (ADS)

Wan, Tian

This work is motivated by the lack of fully coupled computational tool that solves successfully the turbulent chemically reacting Navier-Stokes equation, the electron energy conservation equation and the electric current Poisson equation. In the present work, the abovementioned equations are solved in a fully coupled manner using fully implicit parallel GMRES methods. The system of Navier-Stokes equations are solved using a GMRES method with combined Schwarz and ILU(0) preconditioners. The electron energy equation and the electric current Poisson equation are solved using a GMRES method with combined SOR and Jacobi preconditioners. The fully coupled method has also been implemented successfully in an unstructured solver, US3D, and convergence test results were presented. This new method is shown two to five times faster than the original DPLR method. The Poisson solver is validated with analytic test problems. Then, four problems are selected; two of them are computed to explore the possibility of onboard MHD control and power generation, and the other two are simulation of experiments. First, the possibility of onboard reentry shock control by a magnetic field is explored. As part of a previous project, MHD power generation onboard a re-entry vehicle is also simulated. Then, the MHD acceleration experiments conducted at NASA Ames research center are simulated. Lastly, the MHD power generation experiments known as the HVEPS project are simulated. For code validation, the scramjet experiments at University of Queensland are simulated first. The generator section of the HVEPS test facility is computed then. The main conclusion is that the computational tool is accurate for different types of problems and flow conditions, and its accuracy and efficiency are necessary when the flow complexity increases.

Overview of the CHarring Ablator Response (CHAR) Code

NASA Technical Reports Server (NTRS)

Amar, Adam J.; Oliver, A. Brandon; Kirk, Benjamin S.; Salazar, Giovanni; Droba, Justin

2016-01-01

An overview of the capabilities of the CHarring Ablator Response (CHAR) code is presented. CHAR is a one-, two-, and three-dimensional unstructured continuous Galerkin finite-element heat conduction and ablation solver with both direct and inverse modes. Additionally, CHAR includes a coupled linear thermoelastic solver for determination of internal stresses induced from the temperature field and surface loading. Background on the development process, governing equations, material models, discretization techniques, and numerical methods is provided. Special focus is put on the available boundary conditions including thermochemical ablation and contact interfaces, and example simulations are included. Finally, a discussion of ongoing development efforts is presented.

Overview of the CHarring Ablator Response (CHAR) Code

NASA Technical Reports Server (NTRS)

Amar, Adam J.; Oliver, A. Brandon; Kirk, Benjamin S.; Salazar, Giovanni; Droba, Justin

2016-01-01

An overview of the capabilities of the CHarring Ablator Response (CHAR) code is presented. CHAR is a one-, two-, and three-dimensional unstructured continuous Galerkin finite-element heat conduction and ablation solver with both direct and inverse modes. Additionally, CHAR includes a coupled linear thermoelastic solver for determination of internal stresses induced from the temperature field and surface loading. Background on the development process, governing equations, material models, discretization techniques, and numerical methods is provided. Special focus is put on the available boundary conditions including thermochemical ablation, surface-to-surface radiation exchange, and flowfield coupling. Finally, a discussion of ongoing development efforts is presented.

MILAMIN 2 - Fast MATLAB FEM solver

NASA Astrophysics Data System (ADS)

Dabrowski, Marcin; Krotkiewski, Marcin; Schmid, Daniel W.

2013-04-01

MILAMIN is a free and efficient MATLAB-based two-dimensional FEM solver utilizing unstructured meshes [Dabrowski et al., G-cubed (2008)]. The code consists of steady-state thermal diffusion and incompressible Stokes flow solvers implemented in approximately 200 lines of native MATLAB code. The brevity makes the code easily customizable. An important quality of MILAMIN is speed - it can handle millions of nodes within minutes on one CPU core of a standard desktop computer, and is faster than many commercial solutions. The new MILAMIN 2 allows three-dimensional modeling. It is designed as a set of functional modules that can be used as building blocks for efficient FEM simulations using MATLAB. The utilities are largely implemented as native MATLAB functions. For performance critical parts we use MUTILS - a suite of compiled MEX functions optimized for shared memory multi-core computers. The most important features of MILAMIN 2 are: 1. Modular approach to defining, tracking, and discretizing the geometry of the model 2. Interfaces to external mesh generators (e.g., Triangle, Fade2d, T3D) and mesh utilities (e.g., element type conversion, fast point location, boundary extraction) 3. Efficient computation of the stiffness matrix for a wide range of element types, anisotropic materials and three-dimensional problems 4. Fast global matrix assembly using a dedicated MEX function 5. Automatic integration rules 6. Flexible prescription (spatial, temporal, and field functions) and efficient application of Dirichlet, Neuman, and periodic boundary conditions 7. Treatment of transient and non-linear problems 8. Various iterative and multi-level solution strategies 9. Post-processing tools (e.g., numerical integration) 10. Visualization primitives using MATLAB, and VTK export functions We provide a large number of examples that show how to implement a custom FEM solver using the MILAMIN 2 framework. The examples are MATLAB scripts of increasing complexity that address a given technical topic (e.g., creating meshes, reordering nodes, applying boundary conditions), a given numerical topic (e.g., using various solution strategies, non-linear iterations), or that present a fully-developed solver designed to address a scientific topic (e.g., performing Stokes flow simulations in synthetic porous medium). References: Dabrowski, M., M. Krotkiewski, and D. W. Schmid MILAMIN: MATLAB-based finite element method solver for large problems, Geochem. Geophys. Geosyst., 9, Q04030, 2008

Laplace-domain waveform modeling and inversion for the 3D acoustic-elastic coupled media

NASA Astrophysics Data System (ADS)

Shin, Jungkyun; Shin, Changsoo; Calandra, Henri

2016-06-01

Laplace-domain waveform inversion reconstructs long-wavelength subsurface models by using the zero-frequency component of damped seismic signals. Despite the computational advantages of Laplace-domain waveform inversion over conventional frequency-domain waveform inversion, an acoustic assumption and an iterative matrix solver have been used to invert 3D marine datasets to mitigate the intensive computing cost. In this study, we develop a Laplace-domain waveform modeling and inversion algorithm for 3D acoustic-elastic coupled media by using a parallel sparse direct solver library (MUltifrontal Massively Parallel Solver, MUMPS). We precisely simulate a real marine environment by coupling the 3D acoustic and elastic wave equations with the proper boundary condition at the fluid-solid interface. In addition, we can extract the elastic properties of the Earth below the sea bottom from the recorded acoustic pressure datasets. As a matrix solver, the parallel sparse direct solver is used to factorize the non-symmetric impedance matrix in a distributed memory architecture and rapidly solve the wave field for a number of shots by using the lower and upper matrix factors. Using both synthetic datasets and real datasets obtained by a 3D wide azimuth survey, the long-wavelength component of the P-wave and S-wave velocity models is reconstructed and the proposed modeling and inversion algorithm are verified. A cluster of 80 CPU cores is used for this study.

Treatment of geometric singularities in implicit solvent models

NASA Astrophysics Data System (ADS)

Yu, Sining; Geng, Weihua; Wei, G. W.

2007-06-01

Geometric singularities, such as cusps and self-intersecting surfaces, are major obstacles to the accuracy, convergence, and stability of the numerical solution of the Poisson-Boltzmann (PB) equation. In earlier work, an interface technique based PB solver was developed using the matched interface and boundary (MIB) method, which explicitly enforces the flux jump condition at the solvent-solute interfaces and leads to highly accurate biomolecular electrostatics in continuum electric environments. However, such a PB solver, denoted as MIBPB-I, cannot maintain the designed second order convergence whenever there are geometric singularities, such as cusps and self-intersecting surfaces. Moreover, the matrix of the MIBPB-I is not optimally symmetrical, resulting in the convergence difficulty. The present work presents a new interface method based PB solver, denoted as MIBPB-II, to address the aforementioned problems. The present MIBPB-II solver is systematical and robust in treating geometric singularities and delivers second order convergence for arbitrarily complex molecular surfaces of proteins. A new procedure is introduced to make the MIBPB-II matrix optimally symmetrical and diagonally dominant. The MIBPB-II solver is extensively validated by the molecular surfaces of few-atom systems and a set of 24 proteins. Converged electrostatic potentials and solvation free energies are obtained at a coarse grid spacing of 0.5Å and are considerably more accurate than those obtained by the PBEQ and the APBS at finer grid spacings.

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

Vincenti, H.; Vay, J. -L.

Due to discretization effects and truncation to finite domains, many electromagnetic simulations present non-physical modifications of Maxwell's equations in space that may generate spurious signals affecting the overall accuracy of the result. Such modifications for instance occur when Perfectly Matched Layers (PMLs) are used at simulation domain boundaries to simulate open media. Another example is the use of arbitrary order Maxwell solver with domain decomposition technique that may under some condition involve stencil truncations at subdomain boundaries, resulting in small spurious errors that do eventually build up. In each case, a careful evaluation of the characteristics and magnitude of themore » errors resulting from these approximations, and their impact at any frequency and angle, requires detailed analytical and numerical studies. To this end, we present a general analytical approach that enables the evaluation of numerical discretization errors of fully three-dimensional arbitrary order finite-difference Maxwell solver, with arbitrary modification of the local stencil in the simulation domain. The analytical model is validated against simulations of domain decomposition technique and PMLs, when these are used with very high-order Maxwell solver, as well as in the infinite order limit of pseudo-spectral solvers. Results confirm that the new analytical approach enables exact predictions in each case. It also confirms that the domain decomposition technique can be used with very high-order Maxwell solver and a reasonably low number of guard cells with negligible effects on the whole accuracy of the simulation.« less

Extending fields in a level set method by solving a biharmonic equation

NASA Astrophysics Data System (ADS)

Moroney, Timothy J.; Lusmore, Dylan R.; McCue, Scott W.; McElwain, D. L. Sean

2017-08-01

We present an approach for computing extensions of velocities or other fields in level set methods by solving a biharmonic equation. The approach differs from other commonly used approaches to velocity extension because it deals with the interface fully implicitly through the level set function. No explicit properties of the interface, such as its location or the velocity on the interface, are required in computing the extension. These features lead to a particularly simple implementation using either a sparse direct solver or a matrix-free conjugate gradient solver. Furthermore, we propose a fast Poisson preconditioner that can be used to accelerate the convergence of the latter. We demonstrate the biharmonic extension on a number of test problems that serve to illustrate its effectiveness at producing smooth and accurate extensions near interfaces. A further feature of the method is the natural way in which it deals with symmetry and periodicity, ensuring through its construction that the extension field also respects these symmetries.

A mixed method Poisson solver for three-dimensional self-gravitating astrophysical fluid dynamical systems

NASA Technical Reports Server (NTRS)

Duncan, Comer; Jones, Jim

1993-01-01

A key ingredient in the simulation of self-gravitating astrophysical fluid dynamical systems is the gravitational potential and its gradient. This paper focuses on the development of a mixed method multigrid solver of the Poisson equation formulated so that both the potential and the Cartesian components of its gradient are self-consistently and accurately generated. The method achieves this goal by formulating the problem as a system of four equations for the gravitational potential and the three Cartesian components of the gradient and solves them using a distributed relaxation technique combined with conventional full multigrid V-cycles. The method is described, some tests are presented, and the accuracy of the method is assessed. We also describe how the method has been incorporated into our three-dimensional hydrodynamics code and give an example of an application to the collision of two stars. We end with some remarks about the future developments of the method and some of the applications in which it will be used in astrophysics.

TOUGH3 v1.0

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

PAU, GEORGE; JUNG, YOOJIN; FINSTERLE, STEFAN

2016-09-14

TOUGH3 V1.0 capabilities to simulate multi-dimensional, multi-phase, multi-component, non-isothermal flow and transport in fractured porous media, with applications geosciences and reservoir engineering and other application areas. TOUGH3 V1.0 supports a number of different combinations of fluids and components (updated equation-of-state (EOS) modules from previous versions of TOUGH, including EOS1, EOS2, EOS3, EOS4, EOS5, EOS7, EOS7R, EOS7C, EOS7CA, EOS8, EOS9, EWASG, TMVOC, ECO2N, and ECO2M). This upgrade includes (a) expanded list of updated equation-of-state (EOS) modules, (b) new hysteresis models, (c) new implementation of parallel and solver functionalities, (d) new linear solver options based on PETSc libraries, (e) new automatic buildmore » system that automatically downloads and builds third-party libraries and TOUGH3, (f) new printout in CSV format, (g) dynamic memory allocation, (h) various user features, and (i) bug fixes.« less

The importance of the boundary condition in the transport of intensity equation based phase measurement

NASA Astrophysics Data System (ADS)

Zhang, Jialin; Chen, Qian; Li, Jiaji; Zuo, Chao

2017-02-01

The transport of intensity equation (TIE) is a powerful tool for direct quantitative phase retrieval in microscopy imaging. However, there may be some problems when dealing with the boundary condition of the TIE. The previous work introduces a hard-edged aperture to the camera port of the traditional bright field microscope to generate the boundary signal for the TIE solver. Under this Neumann boundary condition, we can obtain the quantitative phase without any assumption or prior knowledge about the test object and the setup. In this paper, we will demonstrate the effectiveness of this method based on some experiments in practice. The micro lens array will be used for the comparison of two TIE solvers results based on introducing the aperture or not and this accurate quantitative phase imaging technique allows measuring cell dry mass which is used in biology to follow cell cycle, to investigate cell metabolism, or to address effects of drugs.

A multiblock multigrid three-dimensional Euler equation solver

NASA Technical Reports Server (NTRS)

Cannizzaro, Frank E.; Elmiligui, Alaa; Melson, N. Duane; Vonlavante, E.

1990-01-01

Current aerodynamic designs are often quite complex (geometrically). Flexible computational tools are needed for the analysis of a wide range of configurations with both internal and external flows. In the past, geometrically dissimilar configurations required different analysis codes with different grid topologies in each. The duplicity of codes can be avoided with the use of a general multiblock formulation which can handle any grid topology. Rather than hard wiring the grid topology into the program, it is instead dictated by input to the program. In this work, the compressible Euler equations, written in a body-fitted finite-volume formulation, are solved using a pseudo-time-marching approach. Two upwind methods (van Leer's flux-vector-splitting and Roe's flux-differencing) were investigated. Two types of explicit solvers (a two-step predictor-corrector and a modified multistage Runge-Kutta) were used with multigrid acceleration to enhance convergence. A multiblock strategy is used to allow greater geometric flexibility. A report on simple explicit upwind schemes for solving compressible flows is included.

Cosmic-ray propagation with DRAGON2: I. numerical solver and astrophysical ingredients

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

Evoli, Carmelo; Gaggero, Daniele; Vittino, Andrea

2017-02-01

We present version 2 of the DRAGON code designed for computing realistic predictions of the CR densities in the Galaxy. The code numerically solves the interstellar CR transport equation (including inhomogeneous and anisotropic diffusion, either in space and momentum, advective transport and energy losses), under realistic conditions. The new version includes an updated numerical solver and several models for the astrophysical ingredients involved in the transport equation. Improvements in the accuracy of the numerical solution are proved against analytical solutions and in reference diffusion scenarios. The novel features implemented in the code allow to simulate the diverse scenarios proposed tomore » reproduce the most recent measurements of local and diffuse CR fluxes, going beyond the limitations of the homogeneous galactic transport paradigm. To this end, several applications using DRAGON2 are presented as well. This new version facilitates the users to include their own physical models by means of a modular C++ structure.« less

Techniques and Software for Monolithic Preconditioning of Moderately-sized Geodynamic Stokes Flow Problems

NASA Astrophysics Data System (ADS)

Sanan, Patrick; May, Dave A.; Schenk, Olaf; Bollhöffer, Matthias

2017-04-01

Geodynamics simulations typically involve the repeated solution of saddle-point systems arising from the Stokes equations. These computations often dominate the time to solution. Direct solvers are known for their robustness and ``black box'' properties, yet exhibit superlinear memory requirements and time to solution. More complex multilevel-preconditioned iterative solvers have been very successful for large problems, yet their use can require more effort from the practitioner in terms of setting up a solver and choosing its parameters. We champion an intermediate approach, based on leveraging the power of modern incomplete factorization techniques for indefinite symmetric matrices. These provide an interesting alternative in situations in between the regimes where direct solvers are an obvious choice and those where complex, scalable, iterative solvers are an obvious choice. That is, much like their relatives for definite systems, ILU/ICC-preconditioned Krylov methods and ILU/ICC-smoothed multigrid methods, the approaches demonstrated here provide a useful addition to the solver toolkit. We present results with a simple, PETSc-based, open-source Q2-Q1 (Taylor-Hood) finite element discretization, in 2 and 3 dimensions, with the Stokes and Lamé (linear elasticity) saddle point systems. Attention is paid to cases in which full-operator incomplete factorization gives an improvement in time to solution over direct solution methods (which may not even be feasible due to memory limitations), without the complication of more complex (or at least, less-automatic) preconditioners or smoothers. As an important factor in the relevance of these tools is their availability in portable software, we also describe open-source PETSc interfaces to the factorization routines.

Collisionless Spectral Kinetic Simulation of Ideal Multipole Resonance Probe

NASA Astrophysics Data System (ADS)

Gong, Junbo; Wilczek, Sebastian; Szeremley, Daniel; Oberrath, Jens; Eremin, Denis; Dobrygin, Wladislaw; Schilling, Christian; Friedrichs, Michael; Brinkmann, Ralf Peter

2016-09-01

Active Plasma Resonance Spectroscopy denotes a class of industry-compatible plasma diagnostic methods which utilize the natural ability of plasmas to resonate on or near the electron plasma frequency ωpe. One particular realization of APRS with a high degree of geometric and electric symmetry is the Multipole Resonance Probe (MRP). The Ideal MRP(IMRP) is an even more symmetric idealization which is suited for theoretical investigations. In this work, a spectral kinetic scheme is presented to investigate the behavior of the IMRP in the low pressure regime. However, due to the velocity difference, electrons are treated as particles whereas ions are only considered as stationary background. In the scheme, the particle pusher integrates the equations of motion for the studied particles, the Poisson solver determines the electric field at each particle position. The proposed method overcomes the limitation of the cold plasma model and covers kinetic effects like collisionless damping.

ASIS v1.0: an adaptive solver for the simulation of atmospheric chemistry

NASA Astrophysics Data System (ADS)

Cariolle, Daniel; Moinat, Philippe; Teyssèdre, Hubert; Giraud, Luc; Josse, Béatrice; Lefèvre, Franck

2017-04-01

This article reports on the development and tests of the adaptive semi-implicit scheme (ASIS) solver for the simulation of atmospheric chemistry. To solve the ordinary differential equation systems associated with the time evolution of the species concentrations, ASIS adopts a one-step linearized implicit scheme with specific treatments of the Jacobian of the chemical fluxes. It conserves mass and has a time-stepping module to control the accuracy of the numerical solution. In idealized box-model simulations, ASIS gives results similar to the higher-order implicit schemes derived from the Rosenbrock's and Gear's methods and requires less computation and run time at the moderate precision required for atmospheric applications. When implemented in the MOCAGE chemical transport model and the Laboratoire de Météorologie Dynamique Mars general circulation model, the ASIS solver performs well and reveals weaknesses and limitations of the original semi-implicit solvers used by these two models. ASIS can be easily adapted to various chemical schemes and further developments are foreseen to increase its computational efficiency, and to include the computation of the concentrations of the species in aqueous-phase in addition to gas-phase chemistry.

New algorithms for field-theoretic block copolymer simulations: Progress on using adaptive-mesh refinement and sparse matrix solvers in SCFT calculations

NASA Astrophysics Data System (ADS)

Sides, Scott; Jamroz, Ben; Crockett, Robert; Pletzer, Alexander

2012-02-01

Self-consistent field theory (SCFT) for dense polymer melts has been highly successful in describing complex morphologies in block copolymers. Field-theoretic simulations such as these are able to access large length and time scales that are difficult or impossible for particle-based simulations such as molecular dynamics. The modified diffusion equations that arise as a consequence of the coarse-graining procedure in the SCF theory can be efficiently solved with a pseudo-spectral (PS) method that uses fast-Fourier transforms on uniform Cartesian grids. However, PS methods can be difficult to apply in many block copolymer SCFT simulations (eg. confinement, interface adsorption) in which small spatial regions might require finer resolution than most of the simulation grid. Progress on using new solver algorithms to address these problems will be presented. The Tech-X Chompst project aims at marrying the best of adaptive mesh refinement with linear matrix solver algorithms. The Tech-X code PolySwift++ is an SCFT simulation platform that leverages ongoing development in coupling Chombo, a package for solving PDEs via block-structured AMR calculations and embedded boundaries, with PETSc, a toolkit that includes a large assortment of sparse linear solvers.

Development of a Chemically Reacting Flow Solver on the Graphic Processing Units

DTIC Science & Technology

2011-05-10

been implemented on the GPU by Schive et al. (2010). The outcome of their work is the GAMER code for astrophysical simulation. Thibault and...Euler equations at each cell. For simplification, consider the Euler equations in one dimension with no source terms; the discretized form of the...is known to be more diffusive than the other fluxes due to the large bound of the numerical signal velocities: b+, b-. 3.4 Time Marching Methods

NASA-Ames three-dimensional potential flow analysis system (POTFAN) equation solver code (SOLN) version 1

NASA Technical Reports Server (NTRS)

Davis, J. E.; Bonnett, W. S.; Medan, R. T.

1976-01-01

A computer program known as SOLN was developed as an independent segment of the NASA-Ames three-dimensional potential flow analysis systems of linear algebraic equations. Methods used include: LU decomposition, Householder's method, a partitioning scheme, and a block successive relaxation method. Due to the independent modular nature of the program, it may be used by itself and not necessarily in conjunction with other segments of the POTFAN system.

Massively Parallel Solution of Poisson Equation on Coarse Grain MIMD Architectures

NASA Technical Reports Server (NTRS)

Fijany, A.; Weinberger, D.; Roosta, R.; Gulati, S.

1998-01-01

In this paper a new algorithm, designated as Fast Invariant Imbedding algorithm, for solution of Poisson equation on vector and massively parallel MIMD architectures is presented. This algorithm achieves the same optimal computational efficiency as other Fast Poisson solvers while offering a much better structure for vector and parallel implementation. Our implementation on the Intel Delta and Paragon shows that a speedup of over two orders of magnitude can be achieved even for moderate size problems.

Multiscale Numerical Methods for Non-Equilibrium Plasma

DTIC Science & Technology

2015-08-01

current paper reports on the implementation of a numerical solver on the Graphic Processing Units (GPUs) to model reactive gas mixtures with detailed...Governing equations The flow ismodeled as amixture of gas specieswhile neglecting viscous effects. The chemical reactions taken place between the gas ...components are to be modeled in great detail. The set of the Euler equations for a reactive gas mixture can be written as: ∂Q ∂t + ∇ · F̄ = Ω̇ (1) where Q

Solution algorithms for the two-dimensional Euler equations on unstructured meshes

NASA Technical Reports Server (NTRS)

Whitaker, D. L.; Slack, David C.; Walters, Robert W.

1990-01-01

The objective of the study was to analyze implicit techniques employed in structured grid algorithms for solving two-dimensional Euler equations and extend them to unstructured solvers in order to accelerate convergence rates. A comparison is made between nine different algorithms for both first-order and second-order accurate solutions. Higher-order accuracy is achieved by using multidimensional monotone linear reconstruction procedures. The discussion is illustrated by results for flow over a transonic circular arc.

A new stream function formulation for the Euler equations

NASA Technical Reports Server (NTRS)

Atkins, H. L.; Hassan, H. A.

1983-01-01

A new stream function formulation is developed for the solution of Euler's equations in the transonic flow region. The stream function and the density are the dependent variables in this method, while the governing equations for adiabatic flow are the momentum equations which are solved in the strong conservation law form. The application of this method does not require a knowledge of the vorticity. The algorithm is combined with the automatic grid solver (GRAPE) of Steger and Sorenson (1979) in order to study arbitrary geometries. Results of the application of this method are presented for the NACA 0012 airfoil at various Mach numbers and angles of attack, and cylinders. In addition, detailed comparisons are made with other solutions of the Euler equations.

Physics-Based Preconditioning of a Compressible Flow Solver for Large-Scale Simulations of Additive Manufacturing Processes

NASA Astrophysics Data System (ADS)

Weston, Brian; Nourgaliev, Robert; Delplanque, Jean-Pierre

2017-11-01

We present a new block-based Schur complement preconditioner for simulating all-speed compressible flow with phase change. The conservation equations are discretized with a reconstructed Discontinuous Galerkin method and integrated in time with fully implicit time discretization schemes. The resulting set of non-linear equations is converged using a robust Newton-Krylov framework. Due to the stiffness of the underlying physics associated with stiff acoustic waves and viscous material strength effects, we solve for the primitive-variables (pressure, velocity, and temperature). To enable convergence of the highly ill-conditioned linearized systems, we develop a physics-based preconditioner, utilizing approximate block factorization techniques to reduce the fully-coupled 3×3 system to a pair of reduced 2×2 systems. We demonstrate that our preconditioned Newton-Krylov framework converges on very stiff multi-physics problems, corresponding to large CFL and Fourier numbers, with excellent algorithmic and parallel scalability. Results are shown for the classic lid-driven cavity flow problem as well as for 3D laser-induced phase change. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Open Collaboration: A Problem Solving Strategy That Is Redefining NASA's Innovative Spirit

NASA Technical Reports Server (NTRS)

Rando, Cynthia M.; Fogarty, Jennifer A.; Richard, Elizabeth E.; Davis, Jeffrey R.

2011-01-01

In 2010, NASA?s Space Life Sciences Directorate announced the successful results from pilot experiments with open innovation methodologies. Specifically, utilization of internet based external crowd sourcing platforms to solve challenging problems in human health and performance related to the future of spaceflight. The follow-up to this success was an internal crowd sourcing pilot program entitled NASA@work, which was supported by the InnoCentive@work software platform. The objective of the NASA@work pilot was to connect the collective knowledge of individuals from all areas within the NASA organization via a private web based environment. The platform provided a venue for NASA Challenge Owners, those looking for solutions or new ideas, to pose challenges to internal solvers, those within NASA with the skill and desire to create solutions. The pilot was launched in 57 days, a record for InnoCentive and NASA, and ran for three months with a total of 20 challenges posted Agency wide. The NASA@work pilot attracted over 6000 participants throughout NASA with a total of 183 contributing solvers for the 20 challenges posted. At the time of the pilot?s closure, solvers provided viable solutions and ideas for 17 of the 20 posted challenges. The solver community provided feedback on the pilot describing it as a barrier breaking activity, conveying that there was a satisfaction associated with helping co-workers, that it was "fun" to think about problems outside normal work boundaries, and it was nice to learn what challenges others were facing across the agency. The results and the feedback from the solver community have demonstrated the power and utility of an internal collaboration tool, such as NASA@work.

Open Collaboration: A Problem Solving Strategy That is Redefining NASA's Innovative Spirit

NASA Technical Reports Server (NTRS)

Rando, Cynthia M.; Fogarty, Jennifer A.; Richard, E. E.; Davis, Jeffrey R.

2011-01-01

In 2010, NASA's Space Life Sciences Directorate announced the successful results from pilot experiments with open innovation methodologies. Specifically, utilization of internet based external crowdsourcing platforms to solve challenging problems in human health and performance related to the future of spaceflight. The follow-up to this success was an internal crowdsourcing pilot program entitled NASA@work, which was supported by the InnoCentive@work software platform. The objective of the NASA@work pilot was to connect the collective knowledge of individuals from all areas within the NASA organization via a private web based environment. The platform provided a venue for NASA Challenge Owners, those looking for solutions or new ideas, to pose challenges to internal solvers, those within NASA with the skill and desire to create solutions. The pilot was launched in 57 days, a record for InnoCentive and NASA, and ran for three months with a total of 20 challenges posted Agency wide. The NASA@work pilot attracted over 6,000 participants throughout NASA with a total of 183 contributing solvers for the 20 challenges posted. At the time of the pilot's closure, solvers provided viable solutions and ideas for 17 of the 20 posted challenges. The solver community provided feedback on the pilot describing it as a barrier breaking activity, conveying that there was a satisfaction associated with helping co-workers, that it was fun to think about problems outside normal work boundaries, and it was nice to learn what challenges others were facing across the agency. The results and the feedback from the solver community have demonstrated the power and utility of an internal collaboration tool, such as NASA@work.

Navier-Stokes Computations With One-Equation Turbulence Model for Flows Along Concave Wall Surfaces

NASA Technical Reports Server (NTRS)

Wang, Chi R.

2005-01-01

This report presents the use of a time-marching three-dimensional compressible Navier-Stokes equation numerical solver with a one-equation turbulence model to simulate the flow fields developed along concave wall surfaces without and with a downstream extension flat wall surface. The 3-D Navier- Stokes numerical solver came from the NASA Glenn-HT code. The one-equation turbulence model was derived from the Spalart and Allmaras model. The computational approach was first calibrated with the computations of the velocity and Reynolds shear stress profiles of a steady flat plate boundary layer flow. The computational approach was then used to simulate developing boundary layer flows along concave wall surfaces without and with a downstream extension wall. The author investigated the computational results of surface friction factors, near surface velocity components, near wall temperatures, and a turbulent shear stress component in terms of turbulence modeling, computational mesh configurations, inlet turbulence level, and time iteration step. The computational results were compared with existing measurements of skin friction factors, velocity components, and shear stresses of the developing boundary layer flows. With a fine computational mesh and a one-equation model, the computational approach could predict accurately the skin friction factors, near surface velocity and temperature, and shear stress within the flows. The computed velocity components and shear stresses also showed the vortices effect on the velocity variations over a concave wall. The computed eddy viscosities at the near wall locations were also compared with the results from a two equation turbulence modeling technique. The inlet turbulence length scale was found to have little effect on the eddy viscosities at locations near the concave wall surface. The eddy viscosities, from the one-equation and two-equation modeling, were comparable at most stream-wise stations. The present one-equation turbulence model is an effective approach for turbulence modeling in the near solid wall surface region of flow over a concave wall.

QuTiP 2: A Python framework for the dynamics of open quantum systems

NASA Astrophysics Data System (ADS)

Johansson, J. R.; Nation, P. D.; Nori, Franco

2013-04-01

We present version 2 of QuTiP, the Quantum Toolbox in Python. Compared to the preceding version [J.R. Johansson, P.D. Nation, F. Nori, Comput. Phys. Commun. 183 (2012) 1760.], we have introduced numerous new features, enhanced performance, and made changes in the Application Programming Interface (API) for improved functionality and consistency within the package, as well as increased compatibility with existing conventions used in other scientific software packages for Python. The most significant new features include efficient solvers for arbitrary time-dependent Hamiltonians and collapse operators, support for the Floquet formalism, and new solvers for Bloch-Redfield and Floquet-Markov master equations. Here we introduce these new features, demonstrate their use, and give a summary of the important backward-incompatible API changes introduced in this version. Catalog identifier: AEMB_v2_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEMB_v2_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU General Public License, version 3 No. of lines in distributed program, including test data, etc.: 33625 No. of bytes in distributed program, including test data, etc.: 410064 Distribution format: tar.gz Programming language: Python. Computer: i386, x86-64. Operating system: Linux, Mac OSX. RAM: 2+ Gigabytes Classification: 7. External routines: NumPy, SciPy, Matplotlib, Cython Catalog identifier of previous version: AEMB_v1_0 Journal reference of previous version: Comput. Phys. Comm. 183 (2012) 1760 Does the new version supercede the previous version?: Yes Nature of problem: Dynamics of open quantum systems Solution method: Numerical solutions to Lindblad, Floquet-Markov, and Bloch-Redfield master equations, as well as the Monte Carlo wave function method. Reasons for new version: Compared to the preceding version we have introduced numerous new features, enhanced performance, and made changes in the Application Programming Interface (API) for improved functionality and consistency within the package, as well as increased compatibility with existing conventions used in other scientific software packages for Python. The most significant new features include efficient solvers for arbitrary time-dependent Hamiltonians and collapse operators, support for the Floquet formalism, and new solvers for Bloch-Redfield and Floquet-Markov master equations. Restrictions: Problems must meet the criteria for using the master equation in Lindblad, Floquet-Markov, or Bloch-Redfield form. Running time: A few seconds up to several tens of hours, depending on size of the underlying Hilbert space.

A gradual update method for simulating the steady-state solution of stiff differential equations in metabolic circuits.

PubMed

Shiraishi, Emi; Maeda, Kazuhiro; Kurata, Hiroyuki

2009-02-01

Numerical simulation of differential equation systems plays a major role in the understanding of how metabolic network models generate particular cellular functions. On the other hand, the classical and technical problems for stiff differential equations still remain to be solved, while many elegant algorithms have been presented. To relax the stiffness problem, we propose new practical methods: the gradual update of differential-algebraic equations based on gradual application of the steady-state approximation to stiff differential equations, and the gradual update of the initial values in differential-algebraic equations. These empirical methods show a high efficiency for simulating the steady-state solutions for the stiff differential equations that existing solvers alone cannot solve. They are effective in extending the applicability of dynamic simulation to biochemical network models.

An algebraic multigrid method for Q2-Q1 mixed discretizations of the Navier-Stokes equations

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

Prokopenko, Andrey; Tuminaro, Raymond S.

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily co-located at mesh points. Speci cally, we investigate a Q 2-Q 1 mixed finite element discretization of the incompressible Navier-Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees-of-freedom (dofs) are defined at spatial locations where there are no corresponding pressure dofs. Thus, AMG approaches lever- aging this co-located structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocitymore » dof relationships of the Q 2-Q 1 discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity dofs resembles that on the nest grid. To define coefficients within the inter-grid transfers, an energy minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier-Stokes problems.« less

Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications

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

Du, Qiang, E-mail: jyanghkbu@gmail.com; Yang, Jiang, E-mail: qd2125@columbia.edu

This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simplemore » ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models.« less

An algebraic multigrid method for Q2-Q1 mixed discretizations of the Navier-Stokes equations

DOE PAGES

Prokopenko, Andrey; Tuminaro, Raymond S.

2016-07-01

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily co-located at mesh points. Speci cally, we investigate a Q 2-Q 1 mixed finite element discretization of the incompressible Navier-Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees-of-freedom (dofs) are defined at spatial locations where there are no corresponding pressure dofs. Thus, AMG approaches lever- aging this co-located structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocitymore » dof relationships of the Q 2-Q 1 discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity dofs resembles that on the nest grid. To define coefficients within the inter-grid transfers, an energy minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier-Stokes problems.« less

The simulation of shock- and impact-driven flows with Mie-Gruneisen equations of state

NASA Astrophysics Data System (ADS)

Ward, Geoffrey M.

An investigation of shock- and impact-driven flows with Mie-Gruneisen equation of state derived from a linear shock-particle speed Hugoniot relationship is presented. Cartesian mesh methods using structured adaptive refinement are applied to simulate several flows of interest in an Eulerian frame of reference. The flows central to the investigation include planar Richtmyer-Meshkov instability, the impact of a sphere with a plate, and an impact-driven Mach stem. First, for multicomponent shock-driven flows, a dimensionally unsplit, spatially high-order, hybrid, center-difference, limiter methodology is developed. Effective switching between center-difference and upwinding schemes is achieved by a set of robust tolerance and Lax-entropy-based criteria [49]. Oscillations that result from such a mixed stencil scheme are minimized by requiring that the upwinding method approaches the center-difference method in smooth regions. The solver is then applied to investigate planar Richtmyer-Meshkov instability in the context of an equation of state comparison. Comparisons of simulations with materials modeled by isotropic stress Mie-Gruneisen equations of state derived from a linear shock-particle speed Hugoniot relationship [36,52] to those of perfect gases are made with the intention of exposing the role of the equation of state. First, results for single- and triple-mode planar Richtmyer-Meshkov instability between mid-ocean ridge basalt (MORB) and molybdenum modeled by Mie-Gruneisen equations of state are presented for the case of a reflected shock. The single-mode case is explored for incident shock Mach numbers of 1.5 and 2.5. Additionally, examined is single-mode Richtmyer-Meshkov instability when a reflected expansion wave is present for incident Mach numbers of 1.5 and 2.5. Comparison to perfect gas solutions in such cases yields a higher degree of similarity in start-up time and growth rate oscillations. Vorticity distribution and corrugation centerline shortly after shock interaction is also examined. The formation of incipient weak shock waves in the heavy fluid driven by waves emanating from the perturbed transmitted shock is observed when an expansion wave is reflected. Next, the ghost fluid method [83] is explored for application to impact-driven flows with Mie-Gruneisen equations of state in a vacuum. Free surfaces are defined utilizing a level-set approach. The level-set is reinitialized to the signed distance function periodically by solution to a Hamilton-Jacobi differential equation in artificial time. Flux reconstruction along each Cartesian direction of the domain is performed by subdividing in a way that allows for robust treatment of grid-scale sized voids. Ghost cells in voided regions near the material-vacuum interface are determined from surface-normal Riemann problem solution. The method is then applied to several impact problems of interest. First, a one-dimensional impact problem is examined in Mie-Gruneisen aluminum with simple point erosion used to model separation by spallation under high tension. A similar three-dimensional axisymmetric simulation of two rods impacting is then performed without a model for spallation. Further results for three-dimensional axisymmetric simulation of a sphere hitting a plate are then presented. Finally, a brief investigation of the assumptions utilized in modeling solids as isotropic fluids is undertaken. An Eulerian solver approach to handling elastic and elastic-plastic solids is utilized for comparison to the simple fluid model assumption. First, in one dimension an impact problem is examined for elastic, elastic-plastic, and fluid equations of state for aluminum. The results demonstrate that in one dimension the fluid models the plastic shock structure of the flow well. Further investigation is made using a three-dimensional axisymmetric simulation of an impact problem involving a copper cylinder surrounded by aluminum. An aluminum slab impact drives a faster shock in the outer aluminum region yielding a Mach reflection in the copper. The results demonstrate similar plastic shock structures. Several differences are also notable that include a lack of roll-up instability at the material interface and slip-line emanating from the Mach stem's triple point. (Abstract shortened by UMI.)

Recent Advances in Modeling Hugoniots with Cheetah

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

Glaesemann, K R; Fried, L E

2005-07-26

We describe improvements to the Cheetah thermochemical-kinetics code's equilibrium solver to enable it to find a wider range of thermodynamic states. Cheetah supports a wide range of elements, condensed detonation products, and gas phase reactions. Therefore, Cheetah can be applied to a wide range of shock problems involving both energetic and non-energetic materials. An improve equation of state is also introduced. New experimental validations of Cheetah's equation of state methodology have been performed, including both reacted and unreacted Hugoniots.

Recent Advances in Modeling Hugoniots with Cheetah

NASA Astrophysics Data System (ADS)

Glaesemann, K. R.; Fried, L. E.

2006-07-01

We describe improvements to the Cheetah thermochemical-kinetics code's equilibrium solver to enable it to find a wider range of thermodynamic states. Cheetah supports a wide range of elements, condensed detonation products, and gas phase reactions. Therefore, Cheetah can be applied to a wide range of shock problems involving both energetic and non-energetic materials. An improve equation of state is also introduced. New experimental validations of Cheetah's equation of state methodology have been performed, including both reacted and unreacted Hugoniots.

Solution of elliptic partial differential equations by fast Poisson solvers using a local relaxation factor. 2: Two-step method

NASA Technical Reports Server (NTRS)

Chang, S. C.

1986-01-01

A two-step semidirect procedure is developed to accelerate the one-step procedure described in NASA TP-2529. For a set of constant coefficient model problems, the acceleration factor increases from 1 to 2 as the one-step procedure convergence rate decreases from + infinity to 0. It is also shown numerically that the two-step procedure can substantially accelerate the convergence of the numerical solution of many partial differential equations (PDE's) with variable coefficients.

A Numerical Method for Solving the 3D Unsteady Incompressible Navier-Stokes Equations in Curvilinear Domains with Complex Immersed Boundaries.

PubMed

Ge, Liang; Sotiropoulos, Fotis

2007-08-01

A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus.

A Numerical Method for Solving the 3D Unsteady Incompressible Navier-Stokes Equations in Curvilinear Domains with Complex Immersed Boundaries

PubMed Central

Ge, Liang; Sotiropoulos, Fotis

2008-01-01

A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus. PMID:19194533

Small Internal Combustion Engine Testing for a Hybrid-Electric Remotely-Piloted Aircraft

DTIC Science & Technology

2011-03-01

differential equations (ODEs) were formed and solved for numerically using various solvers in MATLAB . From these solutions, engine performance...program 5. □ Make sure eddy-current absorber and sprockets are free of debris and that no loose materials are close enough to become entangled

Diagnosis of Enzyme Inhibition Using Excel Solver: A Combined Dry and Wet Laboratory Exercise

ERIC Educational Resources Information Center

Dias, Albino A.; Pinto, Paula A.; Fraga, Irene; Bezerra, Rui M. F.

2014-01-01

In enzyme kinetic studies, linear transformations of the Michaelis-Menten equation, such as the Lineweaver-Burk double-reciprocal transformation, present some constraints. The linear transformation distorts the experimental error and the relationship between "x" and "y" axes; consequently, linear regression of transformed data…

Eric Kozubal | NREL

Science.gov Websites

researches new methods and technologies for energy-efficient air conditioning systems. He has tested more -6155 Eric joined NREL in 2002 and is a member of the Commercial Buildings Research Group. Eric recommendations. He uses tools such as CAD, Matlab, Engineer Equation Solver, Excel, and statistical software to

Application of a Modular Particle-Continuum Method to Partially Rarefied, Hypersonic Flow

NASA Astrophysics Data System (ADS)

Deschenes, Timothy R.; Boyd, Iain D.

2011-05-01

The Modular Particle-Continuum (MPC) method is used to simulate partially-rarefied, hypersonic flow over a sting-mounted planetary probe configuration. This hybrid method uses computational fluid dynamics (CFD) to solve the Navier-Stokes equations in regions that are continuum, while using direct simulation Monte Carlo (DSMC) in portions of the flow that are rarefied. The MPC method uses state-based coupling to pass information between the two flow solvers and decouples both time-step and mesh densities required by each solver. It is parallelized for distributed memory systems using dynamic domain decomposition and internal energy modes can be consistently modeled to be out of equilibrium with the translational mode in both solvers. The MPC results are compared to both full DSMC and CFD predictions and available experimental measurements. By using DSMC in only regions where the flow is nonequilibrium, the MPC method is able to reproduce full DSMC results down to the level of velocity and rotational energy probability density functions while requiring a fraction of the computational time.

Factorizable Upwind Schemes: The Triangular Unstructured Grid Formulation

NASA Technical Reports Server (NTRS)

Sidilkover, David; Nielsen, Eric J.

2001-01-01

The upwind factorizable schemes for the equations of fluid were introduced recently. They facilitate achieving the Textbook Multigrid Efficiency (TME) and are expected also to result in the solvers of unparalleled robustness. The approach itself is very general. Therefore, it may well become a general framework for the large-scale, Computational Fluid Dynamics. In this paper we outline the triangular grid formulation of the factorizable schemes. The derivation is based on the fact that the factorizable schemes can be expressed entirely using vector notation. without explicitly mentioning a particular coordinate frame. We, describe the resulting discrete scheme in detail and present some computational results verifying the basic properties of the scheme/solver.

Aerodynamic optimization studies on advanced architecture computers

NASA Technical Reports Server (NTRS)

Chawla, Kalpana

1995-01-01

The approach to carrying out multi-discipline aerospace design studies in the future, especially in massively parallel computing environments, comprises of choosing (1) suitable solvers to compute solutions to equations characterizing a discipline, and (2) efficient optimization methods. In addition, for aerodynamic optimization problems, (3) smart methodologies must be selected to modify the surface shape. In this research effort, a 'direct' optimization method is implemented on the Cray C-90 to improve aerodynamic design. It is coupled with an existing implicit Navier-Stokes solver, OVERFLOW, to compute flow solutions. The optimization method is chosen such that it can accomodate multi-discipline optimization in future computations. In the work , however, only single discipline aerodynamic optimization will be included.

A parallel finite-difference method for computational aerodynamics

NASA Technical Reports Server (NTRS)

Swisshelm, Julie M.

1989-01-01

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