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Comparative Analysis of High Performance Solvers for 3D Elasticity Problems

  • Ivan Lirkov
  • Yavor Vutov
  • Maria Ganzha
  • Marcin Paprzycki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5434)

Abstract

We consider the numerical solution of 3D linear elasticity equations. The investigated problem is described by a coupled system of second order elliptic partial differential equations. This system is then discretized by conforming or nonconforming finite elements. After applying the Finite Element Method (FEM) based discretization, a system of linear algebraic equations has to be solved. In this system the stiffness matrix is large, sparse and symmetric positive definite. In the solution process we utilize a well-known fact that the preconditioned conjugate gradient method is the best tool for efficient solution of large-scale symmetric systems with sparse positive definite matrices. In this context, the displacement decomposition (DD) technique is applied at the first step to construct a preconditioner that is based on a decoupled block diagonal part of the original matrix. Then two preconditioners, namely the Modified Incomplete Cholesky factorization MIC(0) and the Circulant Block-Factorization (CBF) preconditioning, are used to precondition thus obtained block diagonal matrix.

As far as the parallel implementation of the proposed solution methods is concerned, we utilize the Message Passing Interface (MPI) communication libraries. The aim of our work is to compare the performance of the two proposed preconditioners: the DD MIC(0) and the DD CBF. The presented comparative analysis is based on the execution times of actual codes run on modern parallel computers. Performed numerical tests demonstrate the level of parallel efficiency and robustness of the proposed algorithms. Furthermore, we discuss the number of iterations resulting from utilization of both preconditioners.

Keywords

Message Passing Interface Elasticity Problem Preconditioned Conjugate Gradient Method Linear Elasticity Problem Iterative Solution Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Axelsson, O.: Iterative solution methods. Cambridge Univ. Press, Cambridge (1994)CrossRefzbMATHGoogle Scholar
  2. 2.
    Axelsson, O., Gustafsson, I.: Iterative methods for the solution of the Navier equations of elasticity. Comp. Meth. Appl. Mech. Eng. 15, 241–258 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blaheta, R.: Displacement decomposition-incomplete factorization preconditioning techniques for linear elasticity problems. Num. Lin. Alg. Appl. 1, 107–128 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Georgiev, A., Baltov, A., Margenov, S.: Hipergeos benchmark problems related to bridge engineering applications. REPORT HG CP 94–0820–MOST–4Google Scholar
  5. 5.
    Lirkov, I.: MPI solver for 3D elasticity problems. Math. and computers in simulation 61(3–6), 509–516 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lirkov, I., Margenov, S.: MPI parallel implementation of CBF preconditioning for 3D elasticity problems. Math. and computers in simulation 50(1–4), 247–254 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Vutov, Y.: Parallel DD-MIC(0) Preconditioning of Nonconforming Rotated Trilinear FEM Elasticity Systems. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2007. LNCS, vol. 4818, pp. 745–752. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Snir, M., Otto, S., Huss-Lederman, S., Walker, D., Dongara, J.: MPI: The Complete Reference. Scientific and engineering computation series. The MIT Press, Cambridge (1997) (second printing)Google Scholar
  9. 9.
    Walker, D., Dongara, J.: MPI: a standard Message Passing Interface. Supercomputer 63, 56–68 (1996)Google Scholar
  10. 10.
  11. 11.
  12. 12.

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Ivan Lirkov
    • 1
  • Yavor Vutov
    • 1
  • Maria Ganzha
    • 2
  • Marcin Paprzycki
    • 2
  1. 1.Institute for Parallel ProcessingBulgarian Academy of SciencesSofiaBulgaria
  2. 2.Systems Research InstitutePolish Academy of SciencesWarsawPoland

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