Sparse Direct Solver Research Articles

Exploiting large grain parallelism in a sparse direct linear system solver

MCSPARSE is a parallel solver based on large grain parallelism, combined with medium and fine grain parallelism. For the multiple CPU Cray-systems, the large grain parallelism can be exploited using Cray's macro-tasking while Cray's micro-tasking facilities can be used to implement the medium grain parallelism of MCSPARSE. The fine grain parallelism can be mapped on the parallel fine grain Cray processor. In this paper, this design together with its performance results on the Cray Y-MP 4/464 are presented. More specifically, we study the impact of an elaborate reordering scheme H ∗ on the resulting efficiency of MCSPARSE.

Preconditioned conjugate gradient methods for three‐dimensional linear elasticity

AbstractFinite element modelling of three‐dimensional elasticity problems give rise to large sparse matrices. Various preconditioning methods are developed for use in preconditioned conjugate gradient iterative solution techniques. Incomplete factorizations based on levels of fill, drop tolerance, and a two‐level hierarchical basis are developed. Various techniques for ensuring that the incomplete factors have positive pivots are presented. Computational tests are carried out for problems generated using unstructured tetrahedral meshes. Quadratic basis functions are used. The performance of the iterative methods is compared to a standard direct sparse matrix solver. Problems with up to 70 000 degrees of freedom and small (≪1) element aspect ratio are considered.

A Flux-Split Solution Procedure for Unsteady Inlet Flow Calculations

The solution of the reduced Navier Stokes (RNS) equations is considered using a flux-split procedure. Unsteady flow in a two dimensional engine inlet is computed. The problems of unstart and restart are investigated. A sparse matrix direct solver combined with a domain decomposition strategy is used to compute the unsteady flow field. Strong shock-boundary layer interaction, time varying shocks, and time varying recirculation regions are efficiently captured.

Transient behavior of supersonic flow through inlets

A solution technique to compute inlet flow behavior is presented. The phenomena of inlet unstart and restart are investigated using a flux-split procedure applied to the Euler and Reduced Navier Stokes (RNS) equations. A time consistent direct sparse matrix solver in conjunction with a domain decomposition strategy is applied to compute the transient flow behavior both internal and external to the inlet. Time varying shocks and time varying recirculation regions are efficiently analyzed. The code is quite general and is suitable for the computation of flow for a wide variety of geometries and over a wide range of Mach and Reynolds numbers.

Derivative-Free Restricted Maximum Likelihood Estimation in Animal Models with a Sparse Matrix Solver

Estimation of (co)variance components by derivative-free REML requires repeated evaluation of the log-likelihood function of the data. Gaussian elimination of the augmented mixed model coefficient matrix is often used to evaluate the likelihood function, but it can be costly for animal models with large coefficient matrices. This study investigated the use of a direct sparse matrix solver to obtain the log-likelihood function. The sparse matrix package SPARSPAK was used to reorder the mixed model equations once and then repeatedly to solve the equations by Cholesky factorization to generate the terms required to calculate the likelihood. The animal model used for comparison contained 19 fixed levels, 470 maternal permanent environmental effects, and 1586 direct and 1586 maternal genetic effects, resulting in a coefficient matrix of order 3661 with .3% nonzero elements after including numerator relationships. Compared with estimation via Gaussian elimination of the unordered system, utilization of SPARSPAK required 605 and 240 times less central processing unit time on mainframes and personal computers, respectively. The SPARSPAK package also required less memory and provided solutions for all effects in the model.

Thetranair rectangular grid approach to solving the nonlinear full-potential equation about complex configurations

This paper describestranair, a numerical method that uses a rectangular grid to solve the nonlinear full-potential equation about general complex configurations. The grid is locally refined to resolve the high velocity gradients arising from leading edge expansions or shock waves. The grid penetrates the boundary (described by networks of quadrilateral panels) and is generated automatically. Discrete operators are constructed using the finite element method. The system of nonlinear discrete equations is solved iteratively using an orthogonal conjugate gradient method preconditioned by an exterior Poisson solver and a direct sparse solver. The primary emphasis of this work is to provide design engineers with an aerodynamics analysis tool (thetranair code) which is as accurate, reliable, economical, and flexible to use as panel methods. Results obtained by usingtranair to analyse several interesting configurations are presented.

Inviscid steady/unsteady flow calculations

The solution of the Euler equations using a flux splitting procedure is considered for low subsonic to high supersonic flows. Steady and unsteady, internal and external flow fields, are computed. For transient flow, a direct sparse matrix solver is applied to compute the flow field at each instant of time. Oscillation free normal and oblique shocks are captured. Unstart and restart of a simplified two-dimensional inlet is investigated.

A review of reduced Navier-Stokes computations for compressible viscous flows

A reduced form of the Navier-Stokes equations, defined by a single composite of the Euler, boundary layer and triple deck approximations, is considered for the computation of viscous interacting flows. Primitive variable formulations based on pressure velocity flux-vector differencing, and composite velocity differencing, wherein rotational and pseudo-potential fields are combined in the spirit of matched expansions, are reviewed. Global pressure or pseudo-potential relaxation methods are integrated with coupled sparse matrix direct solvers or coupled strongly implicit ILU algorithms to efficiently capture sharp shocks and regions of recirculation. Solutions are obtained for a variety of two-and three-dimensional geometries and for Mach numbers ( M) spanning the range from incompressible ( M = 0) to supersonic ( M = 6) flow.

Three‐dimensional reduced Navier–Stokes solutions for subsonic separated and non‐separated flows, using a global pressure relaxation procedure

AbstractThe reduced Navier–Stokes and thin layer approximations to the Navier–Stokes equations are used to obtain solutions for viscous subsonic three‐dimensional flows. A spatial marching method is combined with a direct sparse matrix solver to obtain successive solutions in a global relaxation process. Results have been obtained for flow fields with and without regions of flow reversal.

Some aspects of parallel implementation of the finite-element method on message passing architectures

We discuss some aspects of implementing the finite-element method on parallel computers with local memory and message passing. In particular, we compare the costs of using high-order and low-order elements and of direct and iterative solvers for solving the linear systems that occur. Our model of parallel computation is a two-dimensional grid of processors chosen to be similar in shape to the underlying grid. Our main conclusions are that use of high-order methods is an effective way to achieve high accuracy for some problems, on both serial and parallel computers, and that such methods provide a natural way to achieve efficiency in parallel implementations. In addition, we show that sparse direct solvers generalize naturally to methods based on high-order elements, and that direct solvers are adequate for two-dimensional problems, especially for multiple load vectors.

Solving Sparse Linear Systems with Sparse Backward Error

When solving sparse linear systems, it is desirable to produce the solution of a nearby sparse problem with the same sparsity structure. This kind of backward stability helps guarantee, for example, that a problem with the same physical connectivity as the original has been solved. Theorems of Oettli, Prager [Numer Math., 6 (1964), pp. 405-409] and Skeel [Math. Comput., 35 (1980), pp. 817-832] show that one step of iterative refinement, even with single precision accumulation of residuals, guarantees such a small backward error if the final matrix is not too ill-conditioned and the solution components do not vary too much in magnitude. These results are incorporated into the stopping criterion of the iterative refinement step of a direct sparse matrix solver, and numerical experiments verify that the algorithm frequently stops after one step of iterative refinement with a componentwise relative backward error at the level of the machine precision. Furthermore, calculating this stopping criterion is very i...

Application of conjugate‐gradient‐like methods to a hyperbolic problem in porous‐media flow

AbstractThis paper presents the application of a preconditioned conjugate‐gradient‐like method to a non‐self‐adjoint problem of interest in underground flow simulation. The method furnishes a reliable iterative solution scheme for the non‐symmetric matrices arising at each iteration of the non‐linear time‐stepping scheme. The method employs a generalized conjugate residual scheme with nested factorization as a preconditioner. Model runs demonstrate significant computational savings over direct sparse matrix solvers.