Transactions of the Korean Society of Mechanical Engineers B (대한기계학회논문집B)

The Korean Society of Mechanical Engineers (대한기계학회)
권호 표지
DOI QR코드

DOI QR Code

Parallel Preconditioner for the Domain Decomposition Method of the Discretized Navier-Stokes Equation

이산화된 Navier-Stokes 방정식의 영역분할법을 위한 병렬 예조건화

  • 최형권 (서울산업대학교 기계공학과) ;
  • 유정열 (서울대학교 기계항공공학부) ;
  • 강성우 (서울대학교 대학원 기계항공공학부)
  • Published : 2003.06.01
Download PDF
⟨ Previous Next ⟩

Abstract

A finite element code for the numerical solution of the Navier-Stokes equation is parallelized by vertex-oriented domain decomposition. To accelerate the convergence of iterative solvers like conjugate gradient method, parallel block ILU, iterative block ILU, and distributed ILU methods are tested as parallel preconditioners. The effectiveness of the algorithms has been investigated when P1P1 finite element discretization is used for the parallel solution of the Navier-Stokes equation. Two-dimensional and three-dimensional Laplace equations are calculated to estimate the speedup of the preconditioners. Calculation domain is partitioned by one- and multi-dimensional partitioning methods in structured grid and by METIS library in unstructured grid. For the domain-decomposed parallel computation of the Navier-Stokes equation, we have solved three-dimensional lid-driven cavity and natural convection problems in a cube as benchmark problems using a parallelized fractional 4-step finite element method. The speedup for each parallel preconditioning method is to be compared using upto 64 processors.

Keywords

References

  1. Basermann, A., Reichel, B. and Schelthoff, C., 1997, 'Preconditioned CG methods for Sparse Matrices on Massively Parallel Machines,' Parallel Computing, Vol. 23, pp. 381-398 https://doi.org/10.1016/S0167-8191(97)00005-7
  2. Issman, E. and Degrez, G., 1997, 'Non-overlapping Preconditioners for a Parallel Implicit Navier-Stokes Solver,' Future Generation Comput. Syst., Vol. 13, pp. 303-313 https://doi.org/10.1016/S0167-739X(97)00032-0
  3. Wissink, A. M., Lyrintzis, A. S. and Chronopoulos, A. T., 1996, 'Efficient Iterative Methods Applied to the Solution of Transonic Flows,' J. Comput. Phys., Vol. 123, pp. 379-393 https://doi.org/10.1006/jcph.1996.0031
  4. Chronopoulos, A. T. and Wang, G., 1997, 'Parallel Solution of a Traffic Flow Simulation Problem,' Parallel Computing, Vol. 22, pp. 1965-1983 https://doi.org/10.1016/S0167-8191(97)00070-7
  5. Magolu monga Made, M. and van der Vorst, H. A., 2001, 'A Generalized Domain Decomposition Paradigm for Parallel Incomplete LU Factorization Preconditionings,' Future Generation Comput. Syst. Vol. 17, pp. 925-932 https://doi.org/10.1016/S0167-739X(01)00034-6
  6. Magolu monga Made, M. and van der Vorst, H. A., 2001,'Parallel Incomplete Factorizations with Pseudo-overlapped Subdomains,' Parallel Computing, Vol. 27, pp. 989-1008 https://doi.org/10.1016/S0167-8191(01)00082-5
  7. Saad, Y. and Sosonkina, M., 1999, 'Distributed Schur Complement Techniques for General Sparse Linear Systems,' SIAM J. Sci. Comput., Vol. 21, pp. 1337-1356 https://doi.org/10.1137/S1064827597328996
  8. Choi, H. G., Choi, H. and Yoo, J. Y., 1997, 'A Fractional Four-step Finite Element Formulation of the Unsteady Incompressible Navier-Stokes Equations Using SUPG and Linear Equal-order Element Methods,' Comput. Methods Appl. Mech. Eng., Vol. 143, pp. 333-348 https://doi.org/10.1016/S0045-7825(96)01156-5
  9. Radicati di Brozolo, G. and Robert, Y., 1989, 'Parallel Conjugate Gradient-like Algorithms for Solving Sparse Nonsymmetric Linear Systems on a Vector Multiprocessor ,' Parallel Computing, Vol. 11, pp. 223-239 https://doi.org/10.1016/0167-8191(89)90030-6
  10. Saad, Y., 1996, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, Boston, pp. 374-376
  11. Van der Vorst, H. A., 1992, 'Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Non-symmetric Linear Systems,' SIAM J. Sci. Statist. Comput., Vol. 12, pp. 631-634 https://doi.org/10.1137/0913035
  12. Snir, M., Otto, S., Huss-Lederman, S., Walker, D., Dongarra, J., 1996, MPI: The complete reference, The MIT Press, London
  13. http://www-users.cs.umn.edu/~karypis/metis
  14. Carey, G. F., Shen, Y. and McLay, R. T., 1998, 'Parallel Conjuate Gradient Performance for Least-Squares Finite Elements and Transport Problems,' Int. J. Numer. Meth. Fluids, Vol. 28, pp. 1421-1440 https://doi.org/10.1002/(SICI)1097-0363(19981230)28:10<1421::AID-FLD767>3.0.CO;2-F
  15. Kershaw, D. S., 1978, 'The Incomplete Cholesky-Conjugate Gradient Method for the Iterative Solution of Systems of Linear Equations,' J. Comput. Phys., Vol. 26, pp. 43-65 https://doi.org/10.1016/0021-9991(78)90098-0
  16. Jiang, B. N, Lin, T. L. and Povinelli, L. A., 1994, 'Large-scale Computation of Incompressible Viscous Flow by Least-squares Finite Element Method,' Comput. Methods Appl. Mech. Eng., Vol. 114, pp. 213-231 https://doi.org/10.1016/0045-7825(94)90172-4
  17. Fusegi, T., Hyun, J. M., Kuwahara, K. and Farouk, B., 1991, 'A Numerical Study of Three-dimensional Natural Convection in a Differentially Heated Cubical Enclosure,' Int. J. Heat Mass Transfer, Vol. 34, No.6, pp. 1543-1557 https://doi.org/10.1016/0017-9310(91)90295-P

Cited by

  1. PERFORMANCE ANALYSIS OF THE PARALLEL CUPID CODE IN DISTRIBUTED MEMORY SYSTEM BASED ETHERNET AND INFINIBAND NETWORK vol.19, pp.2, 2014, https://doi.org/10.6112/kscfe.2014.19.2.024
  2. Performance Analysis of the Parallel CUPID Code for Various Parallel Programming Models in Symmetric Multi-Processing System vol.38, pp.1, 2014, https://doi.org/10.3795/KSME-B.2014.38.1.071