Summary
In this paper we consider the interrelation of numerical methods and algorithms with new parallel and vector processor capabilities. A brief historical note on the development of computer processors establishes the perspective of this study and leads into an examination of two proposed new processors: (1) an array processor and (2) a vectorized processor with some parallelism. Next we examine typical linear, nonlinear and transient finite element computations, noting where parallelism and vectorization are possible. Substructuring and splitting are taken as special examples, and we summarize some important features concerning direct and iterative methods for linear systems. Finally, the extension to nonlinear systems and the implication for a specific iteration scheme are considered.
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References
Matisoo, J.: The Superconducting Computer, Scientific American, 242, 5, 50–65, 1980.
Petersen, V. L. et al.: Future Computer Requirements for Computational Aerodynamics, Workshop Proceedings, NASA Conference Publication 2032, Feb. 1978.
Yanenko, N. N.: The Method of Fractional Steps, (trans M. Holt), Springer, Berlin, 1971.
MacCormack, R. W.: An Efficient Numerical Method for Solving the Time-Dependent, Compressible, Navier-Stokes Equations at High Reynolds Number, NASA TN X - 73, 1976.
Carey, G. F. and R. Krishnan: Navier Stokes Problems: Methods, Solution and Computation, TICOM Report 79–11, Texas Institute for Computational Mechanics, University of Texas at Austin, Austin, Texas, Sept. 1979.
Hayes, L. J.: Galerkin Alternating-Direction Methods for Non-Rectangular Regions Using Patch Approximations (to appear).
Noor, A. K. and J. J. Lambiotte, Jr.: Finite Element Dynamic Analysis on CDC Star-100 Computer, Computers and Structures, 10, 7–19, 1979.
Young, D.: Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.
Hayes, L. J.: Timing Analysis of Standard Iterative Methods on a Pipeline Computer, CNA Report 136, CNA, Univ. of Texas, April 1978.
Perrone, N. and R. Kao: A General Nonlinear Relaxation Technique for Solving Nonlinear Problems in Mechanics, J. Appl. Mech., 38, 371–376, 1971.
Carey, G. F. and R. Krishnan: On a Nonlinear Iterative Method in Applied Mechanics, Part I and II,. CMAME, (to appear).
Additional References
Birkhoff, G. and A. George: Elimination by Nested Dissection, Complexity of Sequential and Parallel Numerical Algorithms, J. F. Traub, ed., Academic Press, New York, 1973, pp. 221–269.
Buzbee, B. L.:A Fast Poisson Solver Amenable to Parallel Computation, IEEE Trans. Comput. C-22 (1973), pp. 793–796.
Calahan, D. A.: Complexity of Vectorized Solution of 2Dimensional Finite Element Grids, Report 91, SEL, Univ. of Michigan, 1975.
Calahan, D. A., W. G. Ames and E. J. Sesek: A Collection of Equation-Solving Codes for the CRAY 1, SEL Report No. 133, University of Michigan, Ann Arbor, Michigan, 1979.
Csanky, L.: Fast Parallel Matrix Inversion Algorithsm, SIAM J. Comput., 5 (1976), pp. 618–623.
George, A., W. G. Poole, Jr. and R. G. Voigt: Analysis of Dissection Algorithms for Vector Computers, ICASE, Hampton, Virginia, 1976.
Graham, W. R.: The Parallel and Pipeline Computers, Datamâtion, 68–71, April, 1970.
Heller, D.: A Survey of Parallel Algorithms in Numerical Linear Algebra, SIAM Review, 20, 4, 740–777, 1978.
Heller, D., D. K. Stevenson and J. F. Traub: Accelerated Iterative Methods for the Solution of Tridiagonal Linear Systems on Parallel Computers, J. Assoc. Comput. Mach., 23 (1976), pp. 636–654.
Lambiotte, J. J.,Jr. and R. G. Voigt: The Solution of Tri-diagonal Linear Systems on the CDC STAR-100 Computer, ACM Trans. Math. Software, 1 (1975), pp. 308–329.
Lazarus, R. B.: Computer Architecture and Very Large Problems, AFIPS Conference Proceedings, 40, 45–49, 1972.
Madsen, N. K., G. H. Rodrigue and J. I. Karush: Matrix Multiplication by Diagonals on a Vector/Parallel Processor, Information Processing Lett., 5 (1976), pp. 41–45.
Miranker, W. L.: A Survey of Parallelism in Numerical Analysis, SIAM Review, 13, 524–545, 1971.
Morice, Ph.: Calcul Parallele et Décomposition dans la Resolution d’Equations aux Derivees Partielles de Type Elliptique, IRIA, Rocquencourt, France, 1972.
Ortega, J. M. and W. C. Rheinboldt: Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
Pease, M. C.: Matrix Inversion Using Parallel Processing, J. Assoc. Comput. Mach., 14 (1967), pp. 757–764.
Poole, W. G., Jr. and R. G. Voigt: Numerical Algorithms for Parallel and Vector Computers: An Annotated Bibliography, CR, 15 (1974), pp. 379–388.
Sameh, A. H.: On Jacobi and Jacobi-like Algorithms for a Parallel Computer, Dept. of Computer Sci., Univ. of Illinois, Urbana, 1975.
Sameh, A. H., S. C. Chen and D. J. Kuck: Parallel Poisson and Biharmonic Solvers, Computing, 17 (1976) pp. 219–230.
Stone, H. S.: Parallel Tridiagonal Equation Solvers, ACM Trans. Math. Software, 1 (1975), pp. 289–307.
Traub, J. F.: Iterative Solution of Tridiagonal Systems on Parallel and Vector Computers, Complexity of Sequential and Parallel Numerical Algorithms, J. F. Traub, ed., Academic Press, New York, 1973, pp. 49–824
Watson, W. J.: The TI ASC, a Highly Modular and Flexible Super-Computer Architecture, AFIPS Fall 1972, AFIPS Press, Montvale, New Jersy, vol. 41, pt. 1, pp. 221–229.
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Carey, G.F. (1981). High-Speed Processors and Implication for Algorithms and Methods. In: Wunderlich, W., Stein, E., Bathe, KJ. (eds) Nonlinear Finite Element Analysis in Structural Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-81589-8_38
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DOI: https://doi.org/10.1007/978-3-642-81589-8_38
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