Structural dynamic analysis on a parallel computer: The finite element machine

The application of parallel computers to reservoir simulation is studied with the aid of two recently introduced Multiple Instruction, Multiple Data (MIMD) computers. The problems posed by reservoir simulators are shown to be particularly well suited for such parallel computers* Fundamental parallel programming techniques using Fortran are presented for the Hetrogeneous Element Processor (HEP) produced by Denelcor and the iPSC Hypercube produced by Intel. These techniques are then applied to two of the most time consuming tasks in reservoir simulation: forming the matrix coefficients and producing a sparse matrix solution. The problem of parallel formation of sparse matrix coefficients is addressed for the single and multiphase cases using both black oil and compositional fluid models. Parallel sparse matrix solution is addressed by considering D4 ordered Gaussian elimination, and the multigrid, conjugate gradient, and Successive Over-Relaxation (SOR) methods. Three parallel algorithms based on SOR are developed to illustrate the possible diversity of parallel algorithms. The red-black and multicolored SOR methods which are often used on vector machines are shown to be easily adaptable to parallel MIMD machine. Matrix partitioning and a new method of iteration pipelining are also presented as parallel SOR methods. The various SOR algorithms, in parallel form, are mapped onto the HEP to illustrate the speedups currently possible on an MIMD machine. Projections are made on the future importance of these types of machines to reservoir simulation.

The problem addressed is how to use transputer networks for image processing. As is well known, SIMD (single instruction/multiple data) or MIMD (multiple instruction/multiple data) architectures are very useful in parallel computing. The authors describe how to use transputers as a SIMD or MIMD machine to implement parallel image processing. Several image-processing algorithms, including Laplacian of Gaussian, edge detection, region labeling, generalized Hough transform, and thinning, have been tested. The authors maintain that transputer-based systems are suitable for real-time image processing. It is also pointed out that, compared with the other parallel machines, transputer systems can provide high performance and more flexibility with relatively low cost. >

We present fast and highly scalable parallel computations for a number of important and fundamental matrix problems on distributed memory systems (DMS). These problems include matrix multiplication, matrix chain product, and computing the powers, the inverse, the characteristic polynomial, the determinant, the rank, the Krylov matrix, and an LU- and a QR-factorization of a matrix, and solving linear systems of equations. Our highly scalable parallel computations for these problems are based on a highly scalable implementation of the fastest sequential matrix multiplication algorithm on DMS. We show that compared with the best known parallel time complexities on parallel random access machines (PRAM), the most powerful but unrealistic shared memory model of parallel computing, our parallel matrix computations achieve the same speeds on distributed memory parallel computers (DMPC), and have an extra polylog factor in the time complexities on DMS with hypercubic networks. Furthermore, our parallel matrix computations are fully scalable on DMPC and highly scalable over a wide range of system size on DMS with hypercubic networks. Such fast (in terms of parallel time complexity) and highly scalable (in terms of our definition of scalability) parallel matrix computations were rarely seen before on any distributed memory systems.

We present fast and scalable parallel computations for a number of important and fundamental matrix problems on distributed memory systems (DMS). These problems include computing the powers, the inverse, the characteristic polynomial, the determinant, the rank, the Krylov matrix, and an LU- and a QR-factorization of a matrix, and solving linear systems of equations. These parallel computations are based on efficient implementations of the fastest sequential matrix multiplication algorithm on DMS. We show that compared with the best known time complexities on PRAM, our parallel matrix computations achieve the same speeds on distributed memory parallel computers (DMPC), and have an extra polylog factor in the time complexities on DMS with hypercubic networks. Furthermore, our parallel matrix computations are fully scalable on DMPC and highly scalable over a wide range of system size on DMS with hypercubic networks. Such fast and scalable parallel matrix computations were not seen before on any distributed memory systems.

Recently, use of parallel machines in reservoir simulation has received considerable attention from the petroleum industry. This paper presents parallelization of a 3D compositional, equation-of-state reservoir simulator on the CM-5 and KSR-1. To the best of our knowledge, this is the first time that the parallelization of a compositional reservoir simulator has been performed on both the CM-5 and KSR-1. For new users of the CM-5 machines, the software and hardware of CM-5 architecture is presented, as well as details of the parallelization techniques. For example, domain decomposition, I/O's, phase equilibrium computations, and well model are described. The parallelism techniques on the KSR-1 are presented with the emphasis on the porting of the phase equilibrium calculation. The performance of each machine is evaluated by showing the speedup on different sets of processing nodes. Two test problems were used to explore the capability of the parallelized version of the code; one is a waterflood problem and the other is a CO2 multiple contact miscible flood, both in a West Texas oil field. These field problems were run on 1, 2, 4, 8, 16, and 32 processors to get insight into the locations of communication bottlenecks, generally occurring in the programming with distributed memory machines. The problems of latency and bandwidth which are associated with communication efficiency of the CM-5 are also addressed.

A parallel algorithm for the simulation of steady‐state, two‐dimensional flows of integral viscoelastic fluids on Multiple Instruction Multiple DATA (MIMD) computers with distributed memory is proposed. The algorithm has been implemented within the commercial finite element package polyflow developed at Louvain‐la‐Neuve and widely used in polymer processing applications. Parallel performance results obtained on the Intel iPSC/860 hypercube are discussed for the flow of a K.B.K.Z. fluid through an abrupt contraction. It is found that satisfactory levels of parallel speed‐up can be obtained by using a rather simple allocation scheme to schedule workload to the available processors. The optimal use of the parallel computer requires, however, a dynamic load allocation scheme. An adaptive strategy that reallocates the workload to the processors has been proposed on the basis of the history of the nonlinear iterative process. The proposed adaptive allocation scheme is found to carry only marginal compute and communication overheads.

Department of Computing Science, Chalmers University of Technology, S-41296 G¨oteborg, Sweden,lindskog@cs.chalmers.seFinite element meshes and nodenumberingssuitable for parallel solution with equally loaded processors are pre-sented for linear orthotropic elliptic partial differential equations. These problems are of great importance for in-stance in oil and airfoil industry. The linear systems of equations are solved by the conjugate gradient methodpreconditionedby modified incompletefactorization, MIC.The basic method presented,is based on fronts of un-coupled nodes and unlike earlier methods it has the advantage of no requirement of a specific orientation of themesh. This method is howeverin generalrestricted to small degreeof anisotropyin the differential equation. An-other method, which does not suffer from this limitation, uses rotation of the differential equation and spectralequivalence.Therotation is madein such a way that in thenew coordinatesystem, thebasic methodis applicable.The spectral equivalence property is used for estimation of the condition number of the preconditioned system.Both methods are suitable for implementation on parallel computers. The computer architecture could be singleinstruction multiple data (SIMD) as well as multiple instruction multiple data (MIMD) with shared or distributedmemory.Implementation ofthe basicmethodon a sharedmemory parallelcomputershowsasignificantimprove-ment by use of the MIC method compared to the diagonalscaling preconditioningmethod.

When using MIMD (multiple instruction, multiple data) parallel computers, one is often confronted with solving a task composed of many independent subtasks where it is necessary to synchronize the processors after all the subtasks have been completed. This paper studies how the subtasks should be allocated to the processors in order to minimize the expected time it takes to finish all the subtasks (sometimes called the makespan). We assume that the running times of the subtasks are independent, identically distributed, increasing failure rate random variables, and that assigning one or more subtasks to a processor entails some overhead, or communication time, that is independent of the number of subtasks allocated. Our analyses, which use ideas from renewal theory, reliability theory, order statistics, and the theory of large deviations, are valid for a wide class of distributions. We show that allocating an equal number of subtasks to each processor all at once has good efficiency. This appears as a consequence of a rather general theorem which shows how some consequences of the central limit theorem hold even when we cannot prove that the central limit theorem applies.

As advances and simulation prediction design become more prevalent, the methods for solving rotor dynamics response are challenged to meet higher standards for precision, efficiency, and convergence. To address the limitations of current approaches like Newmark and Runge–Kutta in solving nonlinear responses of rotor dynamics in engineering applications, this paper introduces an improved differential-quadrature precision method. This method fits the Duhamel term with a nonequidistant differential-quadrature technique and uses Taylor and Lagrange formulas for self-starting predictions, balancing accuracy with computational efficiency. Furthermore, the paper presents an iterative strategy that effectively addresses the convergence issues of precise integration methods under large step sizes for nonlinear response solutions. Comparative analysis with real aeroengine gas turbine rotor models demonstrates the performance differences between the improved precise method and Newmark method. The result shows that in steady-state response solutions the improved precise method achieves greater precision at twice the step size of the Newmark method. In transient response solutions, the improved precise method outperforms Newmark in terms of computational efficiency, and the iterative strategy proves both reliable and effective.

A methodology is presented which is aimed at the development of efficient and portable software for general-purpose highly-parallel computers. The methodology has two major components: a programming language that allows the programmer to express the parallelism of an application at a high level and an abstract model of parallel computers that allows programs written for it to be mapped efficiently to different multiple instruction-multiple data (MIMD) parallel computers. >

Frequency response analysis often provides a great deal of information about the system response over the entire range of operation. This can be computationally expensive if time‐domain methods are used, especially for large structural models. Presence of non‐linearity in the system makes it difficult to employ standard frequency response analysis techniques, which are linear in nature. If the system contains mild‐nonlinearities and the response of the system can be assumed to be periodic, it is possible to obtain nonlinear frequency response of the system using harmonic balance techniques. This paper presents the application of the harmonic balance method for solving nonlinear structural dynamics problems. To improve robustness of the solution and capture unstable branches, the continuation procedure technique is included along with the harmonic balance method. The method developed has been implemented in MSC Nastran SOL 128.

Fuzzy reliability analysis can be implemented using two discrete optimization maps in the processes of reliability and fuzzy analysis. Actually, the efficiency and robustness of the iterative reliability methods are two main factors in the fuzzy-based reliability analysis due to the huge computational burdens and unstable results. In the structural fuzzy reliability analysis, the first-order reliability method (FORM) using discrete nonlinear map can provide a C membership function. In this paper, a discrete nonlinear conjugate map is proposed using a relaxed-finite step size method for fuzzy structural reliability analysis, namely Fuzzy conjugate relaxed-finite step size method fuzzy CRS. A discrete conjugate map is stabilized using two adaptive factors to compute the relaxed factor and step size in FORM. The framework of the proposed fuzzy structural reliability method is established using two linked iterative discrete maps as an outer loop, which constructs the membership function of the response using alpha level set optimization based on genetic operator, and the inner loop, implemented for reliability analysis using proposed conjugate relaxed-finite step size method. The fuzzy CRS and fuzzy HL-RF methods are compared to evaluate the membership functions of five structural problems with highly nonlinear limit state functions. Results demonstrated that the fuzzy CRS method is computationally more efficient and is strongly more robust than the HL-RF for fuzzy-based reliability analysis of the nonlinear structural reliability problems.

A Monte Carlo technique is presented which can effectively be used for nonlinear response analysis of a structure subjected to a random pressure field undergoing large deflections. The pressure field is idealized as a multidimensional Gaussian process with mean zero and homogeneous both in time and space. The response analysis is performed in the time domain by numerically simulating generalized forces rather than in the frequency domain. The solution satisfies the boundary conditions and the differential equation in a Galerkin sense. Two numerical examples involving large deflections of a string and a plate are worked out. The result indicates that the present method indeed provides a powerful tool in solving nonlinear structural response problems under random excitations.

A Monte Carlo technique is presented which can effectively be used for nonlinear response analysis of a structure subjected to a random pressure field undergoing large deflections. The pressure field is idealized as a multidimensional Gaussian process with mean zero and homogeneous both in time and space. The response analysis is performed in the time domain by numerically simulating generalized forces rather than in the frequency domain. The solution satisfies the boundary conditions and the differential equation in a Galerkin sense. Two numerical examples involving large deflections of a string and a plate are worked out. The result indicates that the present method indeed provides a powerful tool in solving nonlinear structural response problems under random excitations.

The parallel implicit time‐integration algorithm (PITA) is among a very limited number of time‐integrators that have been successfully applied to the time‐parallel solution of linear second‐order hyperbolic problems such as those encountered in structural dynamics. Time‐parallelism can be of paramount importance to fast computations, for example, when space‐parallelism is unfeasible as in problems with a relatively small number of degrees of freedom in general, and reduced‐order model applications in particular, or when reaching the fastest possible CPU time is desired and requires the exploitation of both space‐ and time‐parallelisms. This paper extends the previously developed PITA to the non‐linear case. It also demonstrates its application to the reduction of the time‐to‐solution on a Linux cluster of sample non‐linear structural dynamics problems. Copyright © 2008 John Wiley & Sons, Ltd.