Group implicit concurrent algorithms in nonlinear structural dynamics

During the 70's and 80's, considerable effort was devoted to developing efficient and reliable time stepping procedures for transient structural analysis. Mathematically, the equations governing this type of problems are generally stiff, i.e., they exhibit a wide spectrum in the linear range. The algorithms best suited to this type of applications are those which accurately integrate the low frequency content of the response without necessitating the resolution of the high frequency modes. This means that the algorithms must be unconditionally stable, which in turn rules out explicit integration. The most exciting possibility in the algorithms development area in recent years has been the advent of parallel computers with multiprocessing capabilities. So, this work is mainly concerned with the development of parallel algorithms in the area of structural dynamics. A primary objective is to devise unconditionally stable and accurate time stepping procedures which lend themselves to an efficient implementation in concurrent machines. Some features of the new computer architecture are summarized. A brief survey of current efforts in the area is presented. A new class of concurrent procedures, or Group Implicit algorithms is introduced and analyzed. The numerical simulation shows that GI algorithms hold considerable promise for application in coarse grain as well as medium grain parallel computers

Implicit schemes and parallel computing in unstructured grid CFD

The development of implicit schemes for obtaining steady state solutions to the Euler and Navier-Stokes equations on unstructured grids is outlined. Applications are presented that compare the convergence characteristics of various implicit methods. Next, the development of explicit and implicit schemes to compute unsteady flows on unstructured grids is discussed. Next, the issues involved in parallelizing finite volume schemes on unstructured meshes in an MIMD (multiple instruction/multiple data stream) fashion are outlined. Techniques for partitioning unstructured grids among processors and for extracting parallelism in explicit and implicit solvers are discussed. Finally, some dynamic load balancing ideas, which are useful in adaptive transient computations, are presented

Parallel Adams methods

AbstractIn the literature, various types of parallel methods for integrating nonstiff initial-value problems for first-order ordinary differential equation have been proposed. The greater part of them are based on an implicit multistage method in which the implicit relations are solved by the predictor-corrector (or fixed point iteration) method. In the predictor-corrector approach the computation of the components of the stage vector iterate can be distributed over s processors, where s is the number of implicit stages of the corrector method. However, the fact that after each iteration the processors have to exchange their just computed results is often mentioned as a drawback, because it implies frequent communication between the processors. Particularly on distributed memory computers, such a fine grain parallelism is not attractive.An alternative approach is based on implicit multistage methods which are such that the implicit stages are already parallel, so that they can be solved independently of each other. This means that only after completion of a step, the processors need to exchange their results. The purpose of this paper is the design of a class of parallel methods for solving nonstiff IVPs. We shall construct explicit methods of order k + 1 with k parallel stages where each stage equation is of Adams-Bashforth type and implicit methods of order k + 2 with k parallel stages which are of Adams-Moulton type. The abscissae in both families of methods are proved to be the Lobatto points, so that the Adams-Bashforth type method can be used as a predictor for the Adams-Moulton-type corrector

Solving Vertical Transport and Chemistry in Air Pollution Models.

For the time integration of stiff transport-chemistry problems from air pollution modelling, standard ODE solvers are not feasible due to the large number of species and the 3D nature. The popular alternative, standard operator splitting, introduces artificial transients for short-lived species. This complicates the chemistry solution, easily causing large errors for such species. In the framework of an operational global air pollution model, we focus on the problem formed by chemistry and vertical transport, which is based on diffusion, cloud-related vertical winds, and wet deposition. Its specific nature leads to full Jacobian matrices, ruling out standard implicit integration. We compare Strang operator splitting with two alternatives: source splitting and an (unsplit) Rosenbrock method with approximate matrix factorization, all having equal computational cost. The comparison is performed with real data. All methods are applied with half-hour time steps, and give good accuracies. Rosenbrock is the most accurate, and source splitting is more accurate than Strang splitting. Splitting errors concentrate in short-lived species sensitive to solar radiation and species with strong emissions and depositions

Parallel methods for nonstiff VIDEs

We consider numerical methods for nonstiff initial-value problems for Volterra integro-differential equations. Such problems may be considered as initial-value problems for ordinary differential equations with expensive righthand side functions because each righthand side evaluation requires the application of a quadrature formula. The often considerable costs suggest the use of methods that require only one righthand side evaluation per step. One option is a conventional linear multistep method. However, if a parallel computer system is available, then one might also look for methods with more righthand sides per step, but such that they can all be evaluated in parallel. In this paper, we construct such parallel methods and we show that on parallel computers they are by far superior to the conventional linear multistep methods which do not have scope for parallelism. Moreover, the (real) stability interval is considerably larger

Parallel methods for nonstiff VIDEs

We consider numerical methods for nonstiff initial-value problems for Volterra integro-differential equations. Such problems may be considered as initial-value problems for ordinary differential equations with expensive righthand side functions because each righthand side evaluation requires the application of a quadrature formula. The often considerable costs suggest the use of methods that require only one righthand side evaluation per step. One option is a conventional linear multistep method. However, if a parallel computer system is available, then one might also look for methods with more righthand sides per step, but such that they can all be evaluated in parallel. In this paper, we construct such parallel methods and we show that on parallel computers they are by far superior to the conventional linear multistep methods which do not have scope for parallelism. Moreover, the (real) stability interval is considerably larger

Krylov's methods in function space for waveform relaxation.

by Wai-Shing Luk.Thesis (Ph.D.)--Chinese University of Hong Kong, 1996.Includes bibliographical references (leaves 104-113).Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Functional Extension of Iterative Methods --- p.2Chapter 1.2 --- Applications in Circuit Simulation --- p.2Chapter 1.3 --- Multigrid Acceleration --- p.3Chapter 1.4 --- Why Hilbert Space? --- p.4Chapter 1.5 --- Parallel Implementation --- p.5Chapter 1.6 --- Domain Decomposition --- p.5Chapter 1.7 --- Contributions of This Thesis --- p.6Chapter 1.8 --- Outlines of the Thesis --- p.7Chapter 2 --- Waveform Relaxation Methods --- p.9Chapter 2.1 --- Basic Idea --- p.10Chapter 2.2 --- Linear Operators between Banach Spaces --- p.14Chapter 2.3 --- Waveform Relaxation Operators for ODE's --- p.16Chapter 2.4 --- Convergence Analysis --- p.19Chapter 2.4.1 --- Continuous-time Convergence Analysis --- p.20Chapter 2.4.2 --- Discrete-time Convergence Analysis --- p.21Chapter 2.5 --- Further references --- p.24Chapter 3 --- Waveform Krylov Subspace Methods --- p.25Chapter 3.1 --- Overview of Krylov Subspace Methods --- p.26Chapter 3.2 --- Krylov Subspace methods in Hilbert Space --- p.30Chapter 3.3 --- Waveform Krylov Subspace Methods --- p.31Chapter 3.4 --- Adjoint Operator for WBiCG and WQMR --- p.33Chapter 3.5 --- Numerical Experiments --- p.35Chapter 3.5.1 --- Test Circuits --- p.36Chapter 3.5.2 --- Unstructured Grid Problem --- p.39Chapter 4 --- Parallel Implementation Issues --- p.50Chapter 4.1 --- DECmpp 12000/Sx Computer and HPF --- p.50Chapter 4.2 --- Data Mapping Strategy --- p.55Chapter 4.3 --- Sparse Matrix Format --- p.55Chapter 4.4 --- Graph Coloring for Unstructured Grid Problems --- p.57Chapter 5 --- The Use of Inexact ODE Solver in Waveform Methods --- p.61Chapter 5.1 --- Inexact ODE Solver for Waveform Relaxation --- p.62Chapter 5.1.1 --- Convergence Analysis --- p.63Chapter 5.2 --- Inexact ODE Solver for Waveform Krylov Subspace Methods --- p.65Chapter 5.3 --- Experimental Results --- p.68Chapter 5.4 --- Concluding Remarks --- p.72Chapter 6 --- Domain Decomposition Technique --- p.80Chapter 6.1 --- Introduction --- p.80Chapter 6.2 --- Overlapped Schwarz Methods --- p.81Chapter 6.3 --- Numerical Experiments --- p.83Chapter 6.3.1 --- Delay Circuit --- p.83Chapter 6.3.2 --- Unstructured Grid Problem --- p.86Chapter 7 --- Conclusions --- p.90Chapter 7.1 --- Summary --- p.90Chapter 7.2 --- Future Works --- p.92Chapter A --- Pseudo Codes for Waveform Krylov Subspace Methods --- p.94Chapter B --- Overview of Recursive Spectral Bisection Method --- p.101Bibliography --- p.10