An algebraic multigrid method with interpolation reproducing rigid body modes for semi-definite problems in two-dimensional linear elasticity

Journal of Computational and Applied Mathematics

Volumn 200, Issue 2, 2007, Pages 637-652

  Xiao, Ying Xiong ^a   Zhang, Ping ^a   Shu, Shi ^a  

^a XIANGTAN UNIVERSITY   (China)

Author keywords

Algebraic multigrid; Linear elasticity; Orthogonal decomposition; Rigid body modes; Semi definite problems

Indexed keywords

BOUNDARY CONDITIONS; ELASTICITY; INTERPOLATION; LINEAR EQUATIONS; MATHEMATICAL OPERATORS; PROBLEM SOLVING;

ALGEBRAIC MULTIGRID; LINEAR ELASTICITY; ORTHOGONAL DECOMPOSITION; RIGID BODY MODES; SEMI-DEFINITE PROBLEMS;

NUMERICAL METHODS;

EID: 33846081866     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2006.01.021     Document Type: Article

References (43)

1. Berman A.B., and Plemmons R.J. Nonnegative Matrices in the Mathematical Sciences, SIAM Classics in Applied Mathematics (1994), SIAM, Philadelphia, PA
2. Bitzarakis S., Papadrakakis M., and Kotsopoulos A. Parallel solution techniques in computational structural mechanics. Comput. Methods Appl. Mech. Eng. 148 (1997) 75-104
3. Bramble J.H., and Pasciak J.E. Iterative techniques for time dependent stocks problems. Comput. Math. Appl. 33 (1997) 13-30
4. Bramble J.H., Pasciak J.E., Wang J., and Xu J. Convergence estimates for multigrid algorithms without regularity assumptions. Math. Comput. 57 (1999) 23-45
5. Brezina M., Cleary A.J., Falgout R.D., Henson V.E., Jones J.E., Manteuffel T.A., McCormick S.F., and Ruge J.W. Algebraic multigrid based on element interpolation (AMGe). SIAM J. Sci. Comput. 22 (2000) 1570-1592 (electronic)
6. R.E. Caflisch, C. Connell, Atomistic theory of elasticity for thin epitaxial films, J. Elasticity, 2001, submitted for publication.
7. Chan T.F., Xu J., and Zikatanov L. An agglomeration multigrid method for unstructured grids. Contemporary Mathematics, AMS 218 (1998) 67-81
8. Q. Chang, W. Sun, Multigrid methods for singular system, 1999, preprint.
9. Chang Q., Wong Y., and Fu H. On algebraic multigrid method. J. Comput. Phys. 125 (1996) 279-292
10. Chen L., and Xu J. On a schur complement operator arising from Navier-Stokes equations and its preconditioning. In: Charles A.M., Chen Z., Li Y., and Xu Y. (Eds). Advances in Computational Mathematics, Proceedings of the Guangzhou International Symposium vol. 202 (1998), Marcel Dekker, New York
11. Farhat C., Chen P.S., and Mandel J. A scalable Lagrange multiplier based on domain decomposition method for implicit time-dependent problems. Int. J. Numer. Mech. Eng. 38 (1995) 3831-3854
12. Farhat C., Pierson K., and Lesoinne M. The second generation FETI methods and their application to parallel solution of large-scale linear and geometrically non-linear analysis problems. Comput. Methods Appl. Mech. Eng. 184 (2000) 333-374
13. Farhat C., and Roux F.-X. A method of finite element and interconnecting and its parallel solution algorithm. Int. J. Numer. Mech. Eng. 32 (1991) 1205-1227
14. Felippa C.A., and Park K.C. A direct flexibility method. Comput. Methods Appl. Mech. Eng. 149 (1997) 319-337
15. Felippa C.A., Park K.C., and Justino Filho M.R. The construction of free-free flexibility matrices as generalized stiffness inverses. Comput. Struct. 68 (1998) 411-418
16. Geradin M., and Rixen D. Mechanical Vibrations, Theory and Application to Structural Dynamics. second ed. (1997), Wiley, Chichester
17. Golub G.H., and Van Loan C.F. Matrix Computations (1990), John Hopkins University Press, Baltimore, MD
18. Griebel M., Oeltz D., and Schweitzer M.A. An algebraic multigrid method for linear elasticity. SIAM J. Sci. Comput. 25 (2003) 385-407
19. D.C. Haarbis, M. Papadrakakis, Approximate Dirichlet preconditioners for the FETI method, in: Fourth International Collaboration on Computation of Shell and Spatial Structures (IASS-IACN 2000), Book of Abstract, 2000.
20. Hackbusch W. Multi-Grid Methods and Applications (1985), Springer, New York
21. W. Hackbusch (Ed.), Robust Multigrid Methods, Friedr. Vieweg 8 Sohn, 1988.
22. W. Hackbusch, U. Trottenberg (Eds.) Multigrid Methods II, Lecture Notes in Mathematics, vol. 1228, Springer, Berlin, 1982.
23. Henson V.E., and Vassilevski P.S. Element-free AMGe: general algorithms for computing interpolation weights in AMG. SIAM J. Sci. Comput. 23 (2001) 629-650 (electronic)
24. Keller H.B. On the solution of singular and semidefinite linear systems by iteration. J. Soc. Indust. Appl. Math. B Numer. Anal. 2 (1965) 281-290
25. Y.J. Lee, J.B. Wu, J.C. Xu, L. Zikatanov, A sharp convergence estimate of the method of subspace corrections for singular system of equations, Math. Comput., 2005, to appear.
26. Lutz E., Ye W.J., and Mukherjee S. Elimination of rigid body modes from discreted boundary integral equations. lnt J. Solids Structures 35 (1998) 4427-4436
27. Mandel J. Balancing domain decomposition. Comm. Appl. Numer. Mech. 9 (1993) 233-241
28. Mandel J., Brezina M., and Vanek P. Energy optimization of algebraic multigrid bases. Computing 62 (1999) 205-228
29. Mandel J., McCormick S., and Ruge J. An algebraic theory for multigrid methods for variational problems. SIAM J. Numer. Anal. 25 (1988) 91-110
30. L. Marek, D.B. Szyld, Algebraic Schwarz methods for the numerical solution of Markov chains, 2003, preprint.
31. Papadrakakis M., and Fragakis Y. An integrated geometric-algebraic method for solving semi-definite problems in structural mechanics. Comput. Methods Appl. Mech. Eng. 190 (2001) 6513-6532
32. Papadrakakis M., and Tsopanakis Y. Domain decomposition methods for parallel solution of shape sensitivity analysis problems. Int. J. Numer. Mech. Eng. 44 (1999) 281-303
33. Rixen D. Dual Schur complement method for semi-definite problems. Contemp. Math. 18 (1998) 341-348
34. Ruge J., and Stuben K. Algebraic multigrid. In: McCormick S.F. (Ed). Multigrid Methods vol. 4 (1987), SIAM, Philadelphia, PA 77-130
35. Saad Y. Iterative Methods for Sparse Linear Systems. second ed. (2003), SIAM, Philadelphia, PA
36. Schindler A.C., Gyure M.F., Simms G.D., Vvedensky D.D., Caflisch R.E., Connell C., and Luo E. Theory of strain relaxation in heteroepitaxial systems. Phys. Rev. B 67 (2003)
37. Shu S., Xu J., Xiao Y., and Zikatanov L. Algebraic multigrid methods for lattice block materials. In: Chan T.F., et al. (Ed). Recent Progress in Computational and Applied PDEs vol. 22 (2002), Boston/London 287-307
38. Vanek P., Brezina M., and Mandel J. Convergence of algebraic multigrid based on smoothed aggregation. Numer. Math. 88 (2001) 559-579
39. Vassilevski P.S., and Jones J.E. AMGe based element agglomeration. SIAM J. Sci. Comput. 23 (2001) 109-133 (electronic)
40. P.S. Vassilevski, L.T. Zikatanov, Multiple vector preserving interpolation mappings in algebraic multigrid, SIAM J. Matrix Anal. Appl., 2004, submitted for publication. Also available as Lawrence Livermore National Laboratory Technical Report UCRL-JRNL-208036, November 2004.
41. Xiao Y., Shu S., Zhang P., Mo Z., and Xu J. A kind of semi-coarsing AMG method for two dimensional energy equations with three temperatures. J. Numer. Methods Comput. Appl. 24 (2003) 293-303
42. Xiao Y., Zhang P., and Shu S. Algebraic multigrid method for a multiscale elastic model of thin epitaxial films. Natural Sci. J. Xiangtan Univ. 25 (2003) 75-82
43. Xu J. Iterative methods by space decomposition and subspace correction. SIAM Rev. 34 (1992) 581-613