Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Aggregation multigrid method for schur complement system in FE analysis of continuum elements

  • Ko, Jin-Hwan (Department of Aerospace information Engineering, Konkuk University) ;
  • Lee, Byung Chai (Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology)
  • Received : 2007.10.15
  • Accepted : 2008.08.21
  • Published : 2008.11.10
⟨ Previous Next ⟩

Abstract

An aggregation multigrid method (AMM) is a leading iterative solver in solid mechanics. Recently, AMM is applied for solving Schur Complement system in the FE analysis of shell structures. In this work, an extended application of AMM for solving Schur Complement system in the FE analysis of continuum elements is presented. Further, the performance of the proposed AMM in multiple load cases, which is a challenging problem for an iterative solver, is studied. The proposed method is developed by combining the substructuring and the multigrid methods. The substructuring method avoids factorizing the full-size matrix of an original system and the multigrid method gives near-optimal convergence. This method is demonstrated for the FE analysis of several elastostatic problems. The numerical results show better performance by the proposed method as compared to the preconditioned conjugate gradient method. The smaller computational cost for the iterative procedure of the proposed method gives a good alternative to a direct solver in large systems with multiple load cases.

Keywords

References

  1. Adams, M. (2002), "Evaluation of three unstructured multigrid methods on 3D finite element problems in solid mechanics", Int. J. Numer. Meth. Eng., 55, 519-534 https://doi.org/10.1002/nme.506
  2. Ashcraft, C. and Grimes, R. (1999). "SPOOLES: An object-oriented sparse matrix library", 9th SIAM Conference on Parallel Processing for Scientific Computing, San Antonio, Texas. http://citeseer.ist.psu.edu/ ashcraft99spooles.html
  3. Brenner, S.C. (1999), "The condition number of the Schur complement in domain decomposition", Numer. Math., 83, 187-203 https://doi.org/10.1007/s002110050446
  4. Bulgakov, V.E. (1995), "High performance of multi-level iterative aggregation solver for large finite-element structural analysis problems", Int. J. Numer. Meth. Eng., 38, 3529-3544 https://doi.org/10.1002/nme.1620382010
  5. Bulgakov, V.E (1997), "The use of the multi-level iterative aggregation method in 3-d finite element analysis of solid, truss, frame and shell structure", Comp. Struct., 63(5), 927-938 https://doi.org/10.1016/S0045-7949(96)00388-4
  6. Carvalho, L.M., Giraud, L. and Le Tallec, P. (2001), "Algebraic two-level preconditioners for the Schur Complement method", SIAM J. Sci. Comput., 22(6), 1987-2005 https://doi.org/10.1137/S1064827598340809
  7. Fish, J. and Belsky, V. (1997), "Generalized aggregation multi-level solver", Int. J. Numer. Meth. Eng., 40, 4341-4361 https://doi.org/10.1002/(SICI)1097-0207(19971215)40:23<4341::AID-NME261>3.0.CO;2-C
  8. Karypis, G. and Kumar, V. (1995), METIS: unstructured graph partitioning and sparse matrix ordering system, Tech. rep., Department of Computer Science, University of Minnesota, available on the WWW at URL http:// www-users.cs.umn.edu/~karypis/metis/metis/index.html
  9. Ko. J.H. and Lee, B.C. (2006), "Preconditioning Schur complement matrices based on an aggregation multigrid method for shell structures", Comp. Struct., 84(29-30), 1853-1865 https://doi.org/10.1016/j.compstruc.2006.08.014
  10. Ko, J.H. (2004), "A preconditioner for schur complement matrices based on aggregation multigrid method in linear finite element analysis," Ph. D. dissertation, KAIST
  11. Saad, Y. and Sosonkina, M. (1999), "Distributed schur complement technique for general sparse linear systems", SIAM J. Sci. Comput., 21(4), 1337-1356 https://doi.org/10.1137/S1064827597328996
  12. Saad, Y. (1996), Iterative Methods for Sparse Linear System, New York: PWS publishing
  13. Saint-Georges, P., Warzee, G., Notay, Y. and Beauwens, R. (1999), "Problem-dependent preconditioners for iterative solvers in FE elastostatics", Comp. Struct., 73, 33-42 https://doi.org/10.1016/S0045-7949(98)00277-6
  14. Sonneveld, P. (1989), "CGS, a fast Lanczos-type solver for nonsymmetric linear system", SIAM J. Sci. Statistical Comput., 10(1), 36-52 https://doi.org/10.1137/0910004
  15. Vanek, P., Mandel, J. and Brezina, M. (1996), "Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems", Computing, 56, 179-196 https://doi.org/10.1007/BF02238511