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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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All sourcesNASA NTRSDOE OSTI
NASA NTRS2001

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

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.

Watson, W. R.↗
NASA NTRS1987

Solution of the two-dimensional Navier-Stokes equations using sparse matrix solvers

The use of direct sparse matrix solvers in the solution of the Navier-Stokes equations is investigated. The Yale Sparse Matrix Package and its implementation in the solution algorithm is described. The streamfunction-vorticity form of the Navier-Stokes equations are discretized and linearized and the resulting system of equations are solved using this package. Several viscous flow problems are investigated, including flow in a cavity and flow around a NACA0012 airfoil. Massively separated flow around a sine wave airfoil is investigated and high Reynolds number solutions are obtained. A solution of the unsteady flow around a Joukowski airfoil at high angle of attack is presented.

Bender, Erich E.↗
NASA NTRS1995

Nonlinear structural analysis on distributed-memory computers

A computational strategy is presented for the nonlinear static and postbuckling analyses of large complex structures on massively parallel computers. The strategy is designed for distributed-memory, message-passing parallel computer systems. The key elements of the proposed strategy are: (1) a multiple-parameter reduced basis technique; (2) a nested dissection (or multilevel substructuring) ordering scheme; (3) parallel assembly of global matrices; and (4) a parallel sparse equation solver. The effectiveness of the strategy is assessed by applying it to thermo-mechanical postbuckling analyses of stiffened composite panels with cutouts, and nonlinear large-deflection analyses of HSCT models on Intel Paragon XP/S computers. The numerical studies presented demonstrate the advantages of nested dissection-based solvers over traditional skyline-based solvers on distributed memory machines.

Watson, Brian C.↗
NASA NTRS1995

Postbuckling and large-deflection nonlinear analyses on distributed-memory computers

A computational strategy is presented for postbuckling and nonlinear static analyses of large complex structures on distributed-memory parallel computers. The strategy is designed for message-passing parallel computer systems. The key elements of the proposed strategy are: (1) a multiple-parameter reduced basis technique; (2) a nested dissection (or multilevel substructuring) ordering scheme; (3) parallel assembly of global matrices; and (4) a parallel sparse equation solver. The effectiveness of the strategy is assessed by performing thermomechanical postbuckling analyses of stiffened composite panels with cutouts, and nonlinear large-deflection analyses of High Speed Civil Transport models on three distributed-memory computers. The numerical studies presented demonstrate the advantages of nested dissection-based solvers over traditional skyline-based solvers on distributed-memory machines.

Watson, Brian C.↗
NASA NTRS1995

Sensitivity analysis for large-deflection and postbuckling responses on distributed-memory computers

A computational strategy is presented for calculating sensitivity coefficients for the nonlinear large-deflection and postbuckling responses of laminated composite structures on distributed-memory parallel computers. The strategy is applicable to any message-passing distributed computational environment. The key elements of the proposed strategy are: (1) a multiple-parameter reduced basis technique; (2) a parallel sparse equation solver based on a nested dissection (or multilevel substructuring) node ordering scheme; and (3) a multilevel parallel procedure for evaluating hierarchical sensitivity coefficients. The hierarchical sensitivity coefficients measure the sensitivity of the composite structure response to variations in three sets of interrelated parameters; namely, laminate, layer and micromechanical (fiber, matrix, and interface/interphase) parameters. The effectiveness of the strategy is assessed by performing hierarchical sensitivity analysis for the large-deflection and postbuckling responses of stiffened composite panels with cutouts on three distributed-memory computers. The panels are subjected to combined mechanical and thermal loads. The numerical studies presented demonstrate the advantages of the reduced basis technique for hierarchical sensitivity analysis on distributed-memory machines.

Watson, Brian C.↗
NASA NTRS2005

A Quasi-3-D Theory for Impedance Eduction in Uniform Grazing Flow

A 2-D impedance eduction methodology is extended to quasi-3-D sound fields in uniform or shearing mean flow. We introduce a nonlocal, nonreflecting boundary condition to terminate the duct and then educe the impedance by minimizing an objective function. The introduction of a parallel, sparse, equation solver significantly reduces the wall clock time for educing the impedance when compared to that of the sequential band solver used in the 2-D methodology. The accuracy, efficiency, and robustness of the methodology is demonstrated using two examples. In the first example, we show that the method reproduces the known impedance of a ceramic tubular test liner. In the second example, we illustrate that the approach educes the impedance of a four-segment liner where the first, second, and fourth segments consist of a perforated face sheet bonded to honeycomb, and the third segment is a cut from the ceramic tubular test liner. The ability of the method to educe the impedances of multisegmented liners has the potential to significantly reduce the amount of time and cost required to determine the impedance of several uniform liners by allowing them to be placed in series in the test section and to educe the impedance of each segment using a single numerical experiment. Finally, we probe the objective function in great detail and show that it contains a single minimum. Thus, our objective function is ideal for use with local, inexpensive, gradient-based optimizers.

Watson, W. R.↗
NASA NTRS2009

A Finite Element Theory for Predicting the Attenuation of Extended-Reacting Liners

A non-modal finite element theory for predicting the attenuation of an extended-reacting liner containing a porous facesheet and located in a no-flow duct is presented. The mathematical approach is to solve separate wave equations in the liner and duct airway and to couple these two solutions by invoking kinematic constraints at the facesheet that are consistent with a continuum theory of fluid motion. Given the liner intrinsic properties, a weak Galerkin finite element formulation with cubic polynomial basis functions is used as the basis for generating a discrete system of acoustic equations that are solved to obtain the coupled acoustic field. A state-of-the-art, asymmetric, parallel, sparse equation solver is implemented that allows tens of thousands of grid points to be analyzed. A grid refinement study is presented to show that the predicted attenuation converges. Excellent comparison of the numerically predicted attenuation to that of a mode theory (using a Haynes 25 metal foam liner) is used to validate the computational approach. Simulations are also presented for fifteen porous plate, extended-reacting liners. The construction of some of the porous plate liners suggest that they should behave as resonant liners while the construction of others suggest that they should behave as broadband attenuators. In each case the finite element theory is observed to predict the proper attenuation trend.

Watson, W. R.↗
NASA NTRS1988

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

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.

Poole, Eugene L.↗
DOE OSTI2021

Sparse Approximate Multifrontal Factorization with Butterfly Compression for High-Frequency Wave Equations

In this work, we present a fast and approximate multifrontal solver for large-scale sparse linear systems arising from finite-difference, finite-volume or finite-element discretization of high-frequency wave equations. The proposed solver leverages the butterfly algorithm and its hierarchical matrix extension for compressing and factorizing large frontal matrices via graph-distance guided entry evaluation or randomized matrix-vector multiplication-based schemes. Complexity analysis and numerical experiments demonstrate $\mathcal{O}(N\log^2 N)$ computation and $\mathcal{O}(N)$ memory complexity when applied to an $N\times N$ sparse system arising from 3D high-frequency Helmholtz and Maxwell problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Combinatorial Algorithms in Scientific Computing

We provide the final report for this grant, detailing the publications, software produced, students trained who have joined the DOE workforce, and the impact our work has had on computational mathematics and related disciplines.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1990

A Shifted Block Lanczos Algorithm 1: The Block Recurrence

In this paper we describe a block Lanczos algorithm that is used as the key building block of a software package for the extraction of eigenvalues and eigenvectors of large sparse symmetric generalized eigenproblems. The software package comprises: a version of the block Lanczos algorithm specialized for spectrally transformed eigenproblems; an adaptive strategy for choosing shifts, and efficient codes for factoring large sparse symmetric indefinite matrices. This paper describes the algorithmic details of our block Lanczos recurrence. This uses a novel combination of block generalizations of several features that have only been investigated independently in the past. In particular new forms of partial reorthogonalization, selective reorthogonalization and local reorthogonalization are used, as is a new algorithm for obtaining the M-orthogonal factorization of a matrix. The heuristic shifting strategy, the integration with sparse linear equation solvers and numerical experience with the code are described in a companion paper.

Grimes, Roger G.↗
DOE OSTI2021

Linear solvers for power grid optimization problems: A review of GPU-accelerated linear solvers

The linear equations that arise in interior methods for constrained optimization are sparse symmetric indefinite, and they become extremely ill-conditioned as the interior method converges. These linear systems present a challenge for existing solver frameworks based on sparse LU or LDL T decompositions. Here, we benchmark five well known direct linear solver packages on CPU- and GPU-based hardware, using matrices extracted from power grid optimization problems. The achieved solution accuracy varies greatly among the packages. None of the tested packages delivers significant GPU acceleration for our test cases. For completeness of the comparison we include results for MA57, which is one of the most efficient and reliable CPU solvers for this class of problem.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Impact of Reordering on the LU Factorization Performance of Bordered Block-Diagonal Sparse Matrix

Power engineers rely on computer-based simulation tools to assess grid performance and ensure security. At the core of these tools are solvers for sparse linear equations. When transformed into a bordered block-diagonal (BBD) structure, part of the sparse linear equation solving can be parallelized. This work focuses on using the Schur-complement-based method for LU factorization on BBD matrices, specifically, Jacobian matrices from large-scale systems. Our findings show that the natural ordering method outperforms the default ordering method in computational performance for each block of the BBD matrix. This observation is validated using synthetic 25k-bus and 70k-bus cases, showing a speedup of up to 38% when using natural ordering without permutation. Additionally, the impact of the number of partitions is studied, and the result shows that computational performance improves with more, smaller partitions in the BBD matrices.

BBD matrix↗
NASA NTRS1987

Methods for design and evaluation of integrated hardware/software systems for concurrent computation

Two testbed programming environments to support the evaluation of a large range of parallel architectures have been implemented under the program Parallel Implementation of Scientific Computing Environments (PISCES). The PISCES 1 environment was applied to two areas of aerospace interest: a sparse matrix iterative equation solver and a dynamic scene analysis system. Currently, the NICE/SPAR testbed system for structural analysis is being modified for parallel operation under PISCES 2; the PISCES 1 applications are also being adapted for PISCES 2. A new formal model of concurrent computation has been developed, based on the mathematical system known as H graph semantics together with a timed Petri net model of the parallel aspects of a system.

Pratt, Terrence W.↗
NASA NTRS2000

User's Manual for PCSMS (Parallel Complex Sparse Matrix Solver)

PCSMS (Parallel Complex Sparse Matrix Solver) is a computer code written to make use of the existing real sparse direct solvers to solve complex, sparse matrix linear equations. PCSMS converts complex matrices into real matrices and use real, sparse direct matrix solvers to factor and solve the real matrices. The solution vector is reconverted to complex numbers. Though, this utility is written for Silicon Graphics (SGI) real sparse matrix solution routines, it is general in nature and can be easily modified to work with any real sparse matrix solver. The User's Manual is written to make the user acquainted with the installation and operation of the code. Driver routines are given to aid the users to integrate PCSMS routines in their own codes.

Reddy, C. J.↗
DOE OSTI2023

symPACK: A GPU-Capable Fan-Out Sparse Cholesky Solver

Sparse symmetric positive definite systems of equations are ubiquitous in scientific workloads and applications. Parallel sparse Cholesky factorization is the method of choice for solving such linear systems. Therefore, the development of parallel sparse Cholesky codes that can efficiently run on today’s large-scale heterogeneous distributed-memory platforms is of vital importance. Modern supercomputers offer nodes that contain a mix of CPUs and GPUs. To fully utilize the computing power of these nodes, scientific codes must be adapted to offload expensive computations to GPUs. We present symPACK, a GPU-capable parallel sparse Cholesky solver that uses one-sided communication primitives and remote procedure calls provided by the UPC++ library. We also utilize the UPC++ "memory kinds" feature to enable efficient communication of GPU-resident data. We show that on a number of large problems, symPACK outperforms comparable state-of-the-art GPU-capable Cholesky factorization codes by up to 14x on the NERSC Perlmutter supercomputer.

Bellavita, Julian↗
NASA NTRS1993

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

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.

Whitaker, David L.↗
NASA NTRS2016

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

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.

Zubair, Mohammad↗