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NASA NTRS1990

On the parallel efficiency of the Frederickson-McBryan multigrid

To take full advantage of the parallelism in a standard multigrid algorithm requires as many processors as points. However, since coarse grids contain fewer points, most processors are idle during the coarse grid iterations. Frederickson and McBryan claim that retaining all points on all grid levels (using all processors) can lead to a superconvergent algorithm. The purpose of this work is to show that the parellel superconvergent multigrid (PSMG) algorithm of Frederickson and McBryan, though it achieves perfect processor utilization, is no more efficient than a parallel implementation of standard multigrid methods. PSMG is simply a new and perhaps simpler way of achieving the same results.

Decker, Naomi H.↗
NASA NTRS1999

Large-Scale Parallel Viscous Flow Computations using an Unstructured Multigrid Algorithm

The development and testing of a parallel unstructured agglomeration multigrid algorithm for steady-state aerodynamic flows is discussed. The agglomeration multigrid strategy uses a graph algorithm to construct the coarse multigrid levels from the given fine grid, similar to an algebraic multigrid approach, but operates directly on the non-linear system using the FAS (Full Approximation Scheme) approach. The scalability and convergence rate of the multigrid algorithm are examined on the SGI Origin 2000 and the Cray T3E. An argument is given which indicates that the asymptotic scalability of the multigrid algorithm should be similar to that of its underlying single grid smoothing scheme. For medium size problems involving several million grid points, near perfect scalability is obtained for the single grid algorithm, while only a slight drop-off in parallel efficiency is observed for the multigrid V- and W-cycles, using up to 128 processors on the SGI Origin 2000, and up to 512 processors on the Cray T3E. For a large problem using 25 million grid points, good scalability is observed for the multigrid algorithm using up to 1450 processors on a Cray T3E, even when the coarsest grid level contains fewer points than the total number of processors.

Mavriplis, Dimitri J.↗
NASA NTRS1994

A Parallel Incompressible Navier-Stokes Solver with Multigrid Iterations

We developed a parallel, numerically accurate and stable, and computationally efficient finate-difference incompressible Navier-Stokes (N-S) fluid flow solver. The solver runs on both sequential and massively parallel computers. The numerical method used here is a second-order projection method on a staggered grid. The code is highly modular and it can be used either as a stand-alone flow solver and or a template code which can be adapted or expanded to a specific application. Numerical results and parallel performances of our code on Intel Delta and Paragon are reported.

Navier-Stokes solver↗
NASA NTRS1988

Totally parallel multilevel algorithms

Four totally parallel algorithms for the solution of a sparse linear system have common characteristics which become quite apparent when they are implemented on a highly parallel hypercube such as the CM2. These four algorithms are Parallel Superconvergent Multigrid (PSMG) of Frederickson and McBryan, Robust Multigrid (RMG) of Hackbusch, the FFT based Spectral Algorithm, and Parallel Cyclic Reduction. In fact, all four can be formulated as particular cases of the same totally parallel multilevel algorithm, which are referred to as TPMA. In certain cases the spectral radius of TPMA is zero, and it is recognized to be a direct algorithm. In many other cases the spectral radius, although not zero, is small enough that a single iteration per timestep keeps the local error within the required tolerance.

Frederickson, Paul O.↗
NASA NTRS1989

Totally parallel multilevel algorithms for sparse elliptic systems

The fastest known algorithms for the solution of a large elliptic boundary value problem on a massively parallel hypercube all require O(log(n)) floating point operations and O(log(n)) distance-1 communications, if massively parallel is defined to mean a number of processors proportional to the size n of the problem. The Totally Parallel Multilevel Algorithm (TPMA) that has, as special cases, four of these fast algorithms is described. These four algorithms are Parallel Superconvergent Multigrid (PSMG), Robust Multigrid, the Fast Fourier Transformation (FFT) based Spectral Algorithm, and Parallel Cyclic Reduction. The algorithm TPMA, when described recursively, has four steps: (1) project to a collection of interlaced, coarser problems at the next lower level; (2) apply TPMA, recursively, to each of these lower level problems, solving directly at the lowest level; (3) interpolate these approximate solutions to the finer grid, and to verage them to form an approximate solution on this grid; and (4) refine this approximate solution with a defect-correction step, using a local approximate inverse. Choice of the projection operator (P), the interpolation operator (Q), and the smoother (S) determines the class of problems on which TPMA is most effective. There are special cases in which the first three steps produce an exact solution, and the smoother is not needed (e.g., constant coefficient operators).

Frederickson, Paul O.↗
NASA NTRS1992

Parallel computing strategies for block multigrid implicit solution of the Euler equations

A multigrid diagonal implicit algorithm has been developed to solve the three-dimensional Euler equations of inviscid compressible flow on block-structured grids. An improved method of advancing the multigrid cycle has been examined with respect to convergence rates, accuracy, and efficiency. In this method, the multigrid cycle is advanced independently in each of the blocks, and the information exchange between the blocks is done using buffer arrays, allowing for the asynchronous updating of interface boundary conditions. This updating scheme is used to eliminate the convergence problems found in a previous implementation of the algorithm while retaining its potential for efficient parallel execution. Results are computed for transonic flows past wings and include pressure distributions to verify the accuracy of the scheme and convergence histories to demonstrate the efficiency of the method. Efficiencies that were obtained using a modest number of processors in parallel are also presented and discussed.

Yadlin, Yoram↗
NASA NTRS1993

A new conformal absorbing boundary condition for finite element meshes and parallelization of FEMATS

Some of the progress toward the development and parallelization of an improved version of the finite element code FEMATS is described. This is a finite element code for computing the scattering by arbitrarily shaped three dimensional surfaces composite scatterers. The following tasks were worked on during the report period: (1) new absorbing boundary conditions (ABC's) for truncating the finite element mesh; (2) mixed mesh termination schemes; (3) hierarchical elements and multigridding; (4) parallelization; and (5) various modeling enhancements (antenna feeds, anisotropy, and higher order GIBC).

Chatterjee, A.↗
NASA NTRS1993

Distributed parallel processing applied to an implicit multigrid Euler/Navier-Stokes algorithm

An implicit multigrid algorithm for the solution of the Euler and Navier-Stokes equations has been implemented within the framework of multiple block-structured grids in which the physical domain is spatially decomposed into several blocks and the solution is advanced in parallel on each block. Utilities have been developed to implement such a scheme in a distributed computing environment. The multi-block algorithm is designed so that the explicit residual calculation is identical to that of single-block scheme, and therefore converged solutions for both schemes must be the same. To accelerate convergence, synchronous and asynchronous multigrid strategies are implemented. Significant speedups have been achieved in a multiple processor environment, while convergence rates similar to those of the single-block scheme are observed.

Tysinger, T. L.↗
NASA NTRS1993

The multigrid preconditioned conjugate gradient method

A multigrid preconditioned conjugate gradient method (MGCG method), which uses the multigrid method as a preconditioner of the PCG method, is proposed. The multigrid method has inherent high parallelism and improves convergence of long wavelength components, which is important in iterative methods. By using this method as a preconditioner of the PCG method, an efficient method with high parallelism and fast convergence is obtained. First, it is considered a necessary condition of the multigrid preconditioner in order to satisfy requirements of a preconditioner of the PCG method. Next numerical experiments show a behavior of the MGCG method and that the MGCG method is superior to both the ICCG method and the multigrid method in point of fast convergence and high parallelism. This fast convergence is understood in terms of the eigenvalue analysis of the preconditioned matrix. From this observation of the multigrid preconditioner, it is realized that the MGCG method converges in very few iterations and the multigrid preconditioner is a desirable preconditioner of the conjugate gradient method.

Tatebe, Osamu↗
NASA NTRS1991

The improved robustness of multigrid elliptic solvers based on multiple semicoarsened grids

Multigrid convergence rates degenerate on problems with stretched grids or anisotropic operators, unless one uses line or plane relaxation. For 3-D problems, only plane relaxation suffices, in general. While line and plane relaxation algorithms are efficient on sequential machines, they are quite awkward and inefficient on parallel machines. A new multigrid algorithm is presented based on the use of multiple coarse grids, that eliminates the need for line or plane relaxation in anisotropic problems. This algorithm was developed and the standard multigrid theory was extended to establish rapid convergence for this class of algorithms. The new algorithm uses only point relaxation, allowing easy and efficient parallel implementation, yet achieves robustness and convergence rates comparable to line and plane relaxation multigrid algorithms. The algorithm described is a variant of Mulder's multigrid algorithm for hyperbolic problems. The latter uses multiple coarse grids to achieve robustness, but is unsuitable for elliptic problems, since its V-cycle convergence rate goes to one as the number of levels increases. The new algorithm combines the contributions from the multiple coarse grid via a local switch, based on the strength of the discrete operator in each coordinate direction.

Naik, Naomi H.↗
NASA NTRS1996

Multiple Coarse Grid Multigrid Methods for Solving Elliptic Problems

In this paper we describe some classes of multigrid methods for solving large linear systems arising in the solution by finite difference methods of certain boundary value problems involving Poisson's equation on rectangular regions. If parallel computing systems are used, then with standard multigrid methods many of the processors will be idle when one is working at the coarsest grid levels. We describe the use of Multiple Coarse Grid MultiGrid (MCGMG) methods. Here one first constructs a periodic set of equations corresponding to the given system. One then constructs a set of coarse grids such that for each grid corresponding to the grid size h there are four grids corresponding to the grid size 2*h. Multigrid operations such as restriction of residuals and interpolation of corrections are done in parallel at each grid level. For suitable choices of the multigrid operators the MCGMG method is equivalent to the Parallel Superconvergent MultiGrid (PSMG) method of Frederickson and McBryan. The convergence properties of MCGMG methods can be accurately analyzed using spectral methods.

Xiao, Shengyou↗
NASA NTRS1991

Parallel Methods on Large-Scale Structural Analysis and Physics Applications; Symposium, Hampton, VA, Feb. 5, 6, 1991, Selected Papers

Recent advances in parallel methods and algorithms integrated into large-scale codes are presented. Consideration is given to problem decomposition (substructuring), efficient matrix solution algorithms for shared memory architectures, dynamic and transient analysis algorithms for shared memory architectures, and algorithms for distributed and massively parallel architectures. Particular attention is given to partitioning of unstructured problems for parallel processing, parallel-vector computation for linear-structural analysis and nonlinear unconstraint optimization problems, a parallel-vector equation solver for unsymmetric matrices on supercomputers, parallel nonlinear finite element dynamic response, multigrid algorithms for solving structural mechanics problems on supercomputers, structural analysis on massively parallel computers, explicit finite element methods with contact-impact on SIMD computers, and the impact of mapping and sparsity on parallelized finite element method modules.

Storaasli, Olaf O.↗
NASA NTRS1998

Robust Multigrid Smoothers for Three Dimensional Elliptic Equations with Strong Anisotropies

We discuss the behavior of several plane relaxation methods as multigrid smoothers for the solution of a discrete anisotropic elliptic model problem on cell-centered grids. The methods compared are plane Jacobi with damping, plane Jacobi with partial damping, plane Gauss-Seidel, plane zebra Gauss-Seidel, and line Gauss-Seidel. Based on numerical experiments and local mode analysis, we compare the smoothing factor of the different methods in the presence of strong anisotropies. A four-color Gauss-Seidel method is found to have the best numerical and architectural properties of the methods considered in the present work. Although alternating direction plane relaxation schemes are simpler and more robust than other approaches, they are not currently used in industrial and production codes because they require the solution of a two-dimensional problem for each plane in each direction. We verify the theoretical predictions of Thole and Trottenberg that an exact solution of each plane is not necessary and that a single two-dimensional multigrid cycle gives the same result as an exact solution, in much less execution time. Parallelization of the two-dimensional multigrid cycles, the kernel of the three-dimensional implicit solver, is also discussed. Alternating-plane smoothers are found to be highly efficient multigrid smoothers for anisotropic elliptic problems.

Llorente, Ignacio M.↗
NASA NTRS1992

Parallel computational fluid dynamics - Implementations and results

The present volume on parallel CFD discusses implementations on parallel machines, numerical algorithms for parallel CFD, and performance evaluation and computer science issues. Attention is given to a parallel algorithm for compressible flows through rotor-stator combinations, a massively parallel Euler solver for unstructured grids, a fast scheme to analyze 3D disk airflow on a parallel computer, and a block implicit multigrid solution of the Euler equations. Topics addressed include a 3D ADI algorithm on distributed memory multiprocessors, clustered element-by-element computations for fluid flow, hypercube FFT and the Fourier pseudospectral method, and an investigation of parallel iterative algorithms for CFD. Also discussed are fluid dynamics using interface methods on parallel processors, sorting for particle flow simulation on the connection machine, a large grain mapping method, and efforts toward a Teraflops capability for CFD.

Simon, Horst D.↗
NASA NTRS1999

Large-scale Parallel Unstructured Mesh Computations for 3D High-lift Analysis

A complete "geometry to drag-polar" analysis capability for the three-dimensional high-lift configurations is described. The approach is based on the use of unstructured meshes in order to enable rapid turnaround for complicated geometries that arise in high-lift configurations. Special attention is devoted to creating a capability for enabling analyses on highly resolved grids. Unstructured meshes of several million vertices are initially generated on a work-station, and subsequently refined on a supercomputer. The flow is solved on these refined meshes on large parallel computers using an unstructured agglomeration multigrid algorithm. Good prediction of lift and drag throughout the range of incidences is demonstrated on a transport take-off configuration using up to 24.7 million grid points. The feasibility of using this approach in a production environment on existing parallel machines is demonstrated, as well as the scalability of the solver on machines using up to 1450 processors.

Mavriplis, Dimitri J.↗
NASA NTRS1999

Large-Scale Parallel Unstructured Mesh Computations for 3D High-Lift Analysis

A complete "geometry to drag-polar" analysis capability for three-dimensional high-lift configurations is described. The approach is based on the use of unstructured meshes in order to enable rapid turnaround for complicated geometries which arise in high-lift configurations. Special attention is devoted to creating a capability for enabling analyses on highly resolved grids. Unstructured meshes of several million vertices are initially generated on a work-station, and subsequently refined on a supercomputer. The flow is solved on these refined meshes on large parallel computers using an unstructured agglomeration multigrid algorithm. Good prediction of lift and drag throughout the range of incidences is demonstrated on a transport take-off configuration using up to 24.7 million grid points. The feasibility of using this approach in a production environment on existing parallel machines is demonstrated, as well as the scalability of the solver on machines using up to 1450 processors.

Mavriplis, D. J.↗
NASA NTRS1999

Large-Scale Parallel Unstructured Mesh Computations for 3D High-Lift Analysis

A complete "geometry to drag-polar" analysis capability for three-dimensional high-lift configurations is described. The approach is based on the use of unstructured meshes in order to enable rapid turnaround for complicated geometries which arise in high-lift con gurations. Special attention is devoted to creating a capability for enabling analyses on highly resolved grids. Unstructured meshes of several million vertices are initially generated on a work-station, and subsequently refined on a supercomputer. The flow is solved on these refined meshes on large parallel computers using an unstructured agglomeration multigrid algorithm. Good prediction of lift and drag throughout the range of incidences is demonstrated on a transport take-off configuration using up to 24.7 million grid points. The feasibility of using this approach in a production environment on existing parallel machines is demonstrated, as well as the scalability of the solver on machines using up to 1450 processors.

Mavriplis, D. J.↗