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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

Array-based, parallel hierarchical mesh refinement algorithms for unstructured meshes

In this paper, we describe an array-based hierarchical mesh refinement capability through uniform refinement of unstructured meshes for efficient solution of PDE's using finite element methods and multigrid solvers. A multi-degree, multi-dimensional and multi-level framework is designed to generate the nested hierarchies from an initial coarse mesh that can be used for a variety of purposes such as in multigrid solvers/preconditioners, to do solution convergence and verification studies and to improve overall parallel efficiency by decreasing I/O bandwidth requirements (by loading smaller meshes and in memory refinement). We also describe a high-order boundary reconstruction capability that can be used to project the new points after refinement using high-order approximations instead of linear projection in order to minimize and provide more control on geometrical errors introduced by curved boundaries.The capability is developed under the parallel unstructured mesh framework "Mesh Oriented dAtaBase" (MOAB Tautges et al. (2004)). We describe the underlying data structures and algorithms to generate such hierarchies in parallel and present numerical results for computational efficiency and effect on mesh quality. Furthermore, we also present results to demonstrate the applicability of the developed capability to study convergence properties of different point projection schemes for various mesh hierarchies and to a multigrid finite-element solver for elliptic problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Array-based Hierarchical Mesh Generation in Parallel

In this paper, we describe an array-based hierarchical mesh generation capability through uniform refinement of unstructured meshes for efficient solution of PDE's using finite element methods and multigrid solvers. A multi-degree, multi-dimensional and multi-level framework is designed to generate the nested hierarchies from an initial mesh that can be used for a number of purposes such as multi-level methods to generating large meshes. The capability is developed under the parallel mesh framework “Mesh Oriented dAtaBase” a.k.a MOAB. We describe the underlying data structures and algorithms to generate such hierarchies and present numerical results for computational efficiency and mesh quality. In conclusion, we also present results to demonstrate the applicability of the developed capability to a multigrid finite-element solver.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Composite matrix construction for structured grid adaptive mesh refinement

Structured-grid adaptive mesh refinement (SAMR) is an approach to mesh generation that supports structured access to data and adaptive mesh refinement for discretized partial differential equations (PDEs). Solution algorithms often require that an inverse of an operator be applied, a system of algebraic equations must be solved, and this process is often the primary computational cost in an application. SAMR is well suited to geometric multigrid solvers, which can be effective, but often do not adapt well to complex geometry including material coefficients. Algebraic multigrid (AMG) is more robust in the face of complex geometry, in both boundary conditions and internal material interfaces. AMG requires a stored matrix linearization of the operator. We discuss an approach, and an implementation in the Chombo block-structured AMR framework, for constructing composite grid matrices from a SAMR hierarchy of grids for use in linear solvers in the PETSc numerical library. We consider a case study with the Chombo-based BISICLES ice sheet modeling application.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2000

Parallel performance investigations of an unstructured mesh Navier-Stokes solver

A Reynolds-averaged Navier-Stokes solver based on unstructured mesh techniques for analysis of high-lift configurations is described. The method makes use of an agglomeration multigrid solver for convergence acceleration. Implicit line-smoothing is employed to relieve the stiffness associated with highly stretched meshes. A GMRES technique is also implemented to speed convergence at the expense of additional memory usage. The solver is cache efficient and fully vectorizable, and is parallelized using a two-level hybrid MPI-OpenMP implementation suitable for shared and/or distributed memory architectures, as well as clusters of shared memory machines. Convergence and scalability results are illustrated for various high-lift cases.

Mavriplis, Dimitri J.↗
NASA NTRS1993

A mixed method Poisson solver for three-dimensional self-gravitating astrophysical fluid dynamical systems

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.

Duncan, Comer↗
DOE OSTI2015

Multilayer shallow shelf approximation: Minimisation formulation, finite element solvers and applications

In this paper, a multilayer generalisation of the Shallow Shelf Approximation (SSA) is considered. In this recent hybrid ice flow model, the ice thickness is divided into thin layers, which can spread out, contract and slide over each other in such a way that the velocity profile is layer-wise constant. Like the SSA (1-layer model), the multilayer model can be reformulated as a minimisation problem. However, unlike the SSA, the functional to be minimised involves a new penalisation term for the interlayer jumps of the velocity, which represents the vertical shear stresses induced by interlayer sliding. Taking advantage of this reformulation, numerical solvers developed for the SSA can be naturally extended layer-wise or column-wise. Numerical results show that the column-wise extension of a Newton multigrid solver proves to be robust in the sense that its convergence is barely influenced by the number of layers and the type of ice flow. In addition, the multilayer formulation appears to be naturally better conditioned than the one of the first-order approximation to face the anisotropic conditions of the sliding-dominant ice flow of ISMIP-HOM experiments.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Summary Report: Multigrid for Systems of Elliptic PDEs

We are interested in determining if multigrid can be effectively applied to the system. The conclusion that I seem to be drawn to is that it is impossible to develop a blackbox multigrid solver for these general systems. Analysis of the system of PDEs must be conducted first to determine pre-processing procedures on the continuous problem before applying a multigrid method. Determining this pre-processing is currently not incorporated in black-box multigrid strategies. Nevertheless, we characterize some system features that will make the system more amenable to multigrid approaches, techniques that may lead to more amenable systems, and multigrid procedures that are generally more appropriate for these systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Block smoothers and generalized ideal interpolation in AMG (Final Report)

The Pennsylvania State University (“Subcontractor”) worked on developing new parallel algebraic multilevel methods suitable for solving PDEs. Specifically, work on the design of multigrid solvers for coupled systems of partial differential equations arising in numerical modeling of various applications was completed. A main emphasis was on the design of new ideal algebraic multigrid interpolation for problems such as Maxwell’s equations where block smoothers are needed and the standard form of ideal interpolation is not an effective choice.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

AN ADAPTIVE PARTICLE-MESH GRAVITY SOLVER FOR ENZO

We describe and implement an adaptive particle-mesh algorithm to solve the Poisson equation for grid-based hydrodynamics codes with nested grids. The algorithm is implemented and extensively tested within the astrophysical code Enzo against the multigrid solver available by default. We find that while both algorithms show similar accuracy for smooth mass distributions, the adaptive particle-mesh algorithm is more accurate for the case of point masses, and is generally less noisy. We also demonstrate that the two-body problem can be solved accurately in a configuration with nested grids. In addition, we discuss the effect of subcycling, and demonstrate that evolving all the levels with the same timestep yields even greater precision.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1992

A fast direct solver for a class of two-dimensional separable elliptic equations on the sphere

An efficient, direct, second-order solver for the discrete solution of two-dimensional separable elliptic equations on the sphere 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 wavenumber 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.

Moorthi, Shrinivas↗
NASA NTRS1996

On DGS Relaxation: The Stokes Problem

Multigrid methods have proven to be efficient methods for solving partial differential equations (especially those of elliptic type). There is also growing experience with multigrid solvers for fluids problems, e.g., the Stokes and Navier-Stokes equations (using both finite element and finite difference discretizations). It is also well known that at the heart of any multigrid method is the smoother. In this work we look at a smoother introduced by Brandt and Dinar (DGS relaxation), and we examine some of its properties and consider some possible modifications to it. It is well known that multigrid performance using DGS relaxation is sensitive to the treatment of boundaries; this issue is addressed.

Meir, A. J.↗
NASA NTRS1993

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

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.

Moorthi, Shrinivas↗
DOE OSTI2016

Hydrodynamics of suspensions of passive and active rigid particles: a rigid multiblob approach

We develop a rigid multiblob method for numerically solving the mobility problem for suspensions of passive and active rigid particles of complex shape in Stokes flow in unconfined, partially confined, and fully confined geometries. As in a number of existing methods, we discretize rigid bodies using a collection of minimally resolved spherical blobs constrained to move as a rigid body, to arrive at a potentially large linear system of equations for the unknown Lagrange multipliers and rigid-body motions. Here we develop a block-diagonal preconditioner for this linear system and show that a standard Krylov solver converges in a modest number of iterations that is essentially independent of the number of particles. Key to the efficiency of the method is a technique for fast computation of the product of the blob-blob mobility matrix and a vector. For unbounded suspensions, we rely on existing analytical expressions for the Rotne-Prager-Yamakawa tensor combined with a fast multipole method (FMM) to obtain linear scaling in the number of particles. For suspensions sedimented against a single no-slip boundary, we use a direct summation on a graphical processing unit (GPU), which gives quadratic asymptotic scaling with the number of particles. For fully confined domains, such as periodic suspensions or suspensions confined in slit and square channels, we extend a recently developed rigid-body immersed boundary method by B. Kallemov, A. P. S. Bhalla, B. E. Griffith, and A. Donev (Commun. Appl. Math. Comput. Sci. 11 (2016), no. 1, 79-141) to suspensions of freely moving passive or active rigid particles at zero Reynolds number. We demonstrate that the iterative solver for the coupled fluid and rigid-body equations converges in a bounded number of iterations regardless of the system size. In our approach, each iteration only requires a few cycles of a geometric multigrid solver for the Poisson equation, and an application of the block-diagonal preconditioner, leading to linear scaling with the number of particles. We optimize a number of parameters in the iterative solvers and apply our method to a variety of benchmark problems to carefully assess the accuracy of the rigid multiblob approach as a function of the resolution. We also model the dynamics of colloidal particles studied in recent experiments, such as passive boomerangs in a slit channel, as well as a pair of non-Brownian active nanorods sedimented against a wall.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
NASA NTRS2004

Textbook Multigrid Efficiency for the Steady Euler Equations

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.

Roberts, Thomas W.↗
NASA NTRS1983

A multigrid method for the Euler equations

A multigrid algorithm has been developed for the numerical solution of the steady two-dimensional Euler equations. Flux vector splitting and one-sided differencing are employed to define the spatial discretization. Newton's method is used to solve the nonlinear equations, and a multigrid solver is used on each linear problem. The relaxation scheme for the linear problems is symmetric Gauss-Seidel. Standard restriction and interpolation operators are employed. Local mode analysis is used to predict the convergence rate of the multigrid process on the linear problems. Computed results for transonic flows over airfoils are presented.

Jespersen, D. C.↗
NASA NTRS1985

Multigrid solutions to quasi-elliptic schemes

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

Brandt, A.↗
NASA NTRS1985

Multigrid solutions to quasi-elliptic schemes

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

Brandt, A.↗
DOE OSTI2018

A multigrid waveform relaxation method for solving the poroelasticity equations

In this work, a multigrid waveform relaxation method is proposed for solving a collocated finite difference discretization of the linear Biot’s model. This gives rise to the first space–time multigrid solver for poroelasticity equations in the literature. The waveform relaxation iteration is based on a point-wise Vanka smoother that couples the pressure variable at a grid-point with the displacements around it. A semi-algebraic mode analysis is proposed to theoretically analyze the convergence of the multigrid waveform relaxation algorithm. This analysis is novel since it combines the semi-algebraic analysis, suitable for parabolic problems, with the non-standard analysis for overlapping smoothers. The practical utility of the method is illustrated through several numerical experiments in one and two dimensions.

97 MATHEMATICS AND COMPUTING↗