Partitioned versus global Krylov subspace iterative methods for FE solution of 3-D Biot's problem

Computer Methods in Applied Mechanics and Engineering

Volumn 196, Issue 25-28, 2007, Pages 2737-2750

  Chen, X ^a   Phoon, K K ^b   Toh, K C ^b  

^a Geosoft Pte Ltd   (Singapore)

^b NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Biot's consolidation equations; Generalized Jacobi preconditioner; Modified SSOR preconditioner; Symmetric indefinite linear system; Symmetric quasi minimal residual method

Indexed keywords

CONSOLIDATION; FINITE ELEMENT METHOD; ITERATIVE METHODS; JACOBIAN MATRICES; LINEAR SYSTEMS;

BIOT'S CONSOLIDATION EQUATIONS; GENERALIZED JACOBI PRECONDITIONER; SYMMETRIC INDEFINITE LINEAR SYSTEM; SYMMETRIC QUASI-MINIMAL RESIDUAL METHOD;

SOIL STRUCTURE INTERACTIONS;

CONSOLIDATION; FINITE ELEMENT METHOD; ITERATIVE METHODS; JACOBIAN MATRICES; LINEAR SYSTEMS; SOIL STRUCTURE INTERACTIONS;

EID: 34247101437     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2007.02.003     Document Type: Article

References (48)

1. Gambolati G., Pini G., and Ferronato M. Direct, Partitioned and projected solution to finite element consolidation models. Int. J. Numer. Anal. Methods 26 (2002) 1371-1383
2. X. Chen, K.K. Phoon, K.C. Toh, Symmetric indefinite preconditioners for FE solution of Biot's consolidation problem, in: ASCE Geocongress, Atlanta, 2006.
3. Sheng D., Sloan S.W., Gens A., and Smith D.W. Finite element formulation and algorithms for unsaturated soils. Part I: Theory. Int. J. Numer. Anal. Methods 27 (2003) 745-765
4. Hestenes M.R., and Stiefel E. Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49 (1952) 409-436
5. Phoon K.K., Chan S.H., Toh K.C., and Lee F.H. Fast iterative solution of large undrained soil-structure interaction problems. Int. J. Numer. Anal. Methods 27 (2003) 159-181
6. Paige C.C., and Saunders M.A. Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12 (1975) 617-629
7. Barrett R., Berry M., Chan T., Demmel J., Donato J., Dongarra J., Eijkhout V., Pozo R., Romine C., and van der Vorst H.A. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (1994), SIAM Press, Philadelphia
8. Smith I.M., and Griffiths D.V. Programming the Finite Element Method. third ed. (1997), John Wiley
9. Chan S.H., Phoon K.K., and Lee F.H. A modified Jacobi preconditioner for solving ill-conditioned Biot's consolidation equations using symmetric quasi-minimal residual method. Int. J. Numer. Anal. Methods 25 (2001) 1001-1025
10. Lee F.H., Phoon K.K., Lim K.C., and Chan S.H. Performance of Jacobi preconditioning in Krylov subspace solution of finite element equations. Int. J. Numer. Anal. Methods 26 (2002) 341-372
11. Phoon K.K., Toh K.C., Chan S.H., and Lee F.H. An efficient diagonal preconditioner for finite element solution of Biot's consolidation equations. Int. J. Numer. Methods Engrg. 55 (2002) 377-400
12. K.K. Phoon, Iterative solution of large-scale consolidation and constrained finite element equations for 3-D problems, in: Proceedings, International e-Conference on Modern Trends in Foundation Engineering: Geotechnical Challenges and Solutions, IIT Madras, India, January 26-30, 2004.
13. Rozložni{dotless}́k M., and Simoncini V. Krylov subspace methods for saddle point problems with indefinite preconditioning. SIAM J. Matrix Anal. A. 24 (2002) 368-391
14. Toh K.C. Solving large scale semidefinite programs via an iterative solver on the augmented systems. SIAM J. Optim. 14 (2003) 670-698
15. Toh K.C., Phoon K.K., and Chan S.H. Block preconditioners for symmetric indefinite linear systems. Int. J. Numer. Methods Engrg. 60 (2004) 1361-1381
16. R.W. Freund, N.M. Nachtigal, A new Krylov-subspace method for symmetric indefinite linear system, in: Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics; Atlanta, USA, 1994, pp. 1253-1256.
17. Lukšan L., and Vlček J. Indefinitely preconditioned inexact Newton method for large sparse equality constrained non-linear programming problems. Numer. Linear Algebra Appl. 5 (1998) 219-247
18. Smith I.M. A general purpose system for finite element analyses in parallel. Engrg. Comput. 17 (2000) 75-91
19. Benzi M., Meyer C.D., and Tůma M. A sparse approximate inverse preconditioner for the conjugate gradient method. SIAM J. Sci. Comput. 17 (1996) 1135-1149
20. Grote M.J., and Huckle T. Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. 18 (1997) 838-853
21. Chow E., and Saad Y. Approximate inverse preconditioners via sparse-sparse iterations. SIAM J. Sci. Comput. 19 (1998) 995-1023
22. Gould N.I.M., and Scott J.A. Sparse approximate-inverse preconditioners using norm-minimization techniques. SIAM J. Sci. Comput. 19 (1998) 605-625
23. Saad Y. ILUT: a dual threshold incomplete LU factorization. Numer. Linear Algebra 1 (1994) 387-402
24. Saad Y. Iterative Methods for Sparse Linear Systems (1996), PWS Publishing Company, Boston
25. Meijerink J.A., and van der Vorst H.A. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comput. 31 (1977) 148-162
26. Benzi M. Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182 (2002) 418-477
27. Phoon K.K., Toh K.C., and Chen X. Block constrained versus generalized Jacobi preconditioners for iterative solution of large-scale Biot's FEM equations. Comput. Struct. 82 (2004) 2401-2411
28. Chen X., KC T., and KK P. A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations. Int. J. Numer. Methods Engrg. 65 (2006) 785-807
29. Eisenstat S.C. eEfficient implementation of A class of preconditioned conjugate gradient methods. SIAM J. Sci. Statist. Comput. 2 (1981) 1-4
30. M. Benzi, G.H. Golub, J. Liesen, Numerical solution of saddle point problems, Acta Numerica, 14 (2005) 1-137.
31. Biot M.A. General theory of three-dimensional consolidation. J. Appl. Phys. 12 (1941) 155-164
32. A.J. Abbo, Finite element algorithms for elastoplasticity and consolidation, Ph.D. Thesis, University of Newcastle, 1997.
33. Zienkiewicz O.C., and Taylor R.L. The Finite Element Method. 5th ed. (2000), Butterworth Heinemann, Oxford
34. Borja R.I. Composite Newton-PCG and quasi-Newton iterations for nonlinear consolidation. Comput. Method Appl. M. 86 (1991) 27-60
35. J.W. Demmel, Applied Numerical Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, c1997.
36. Golub G.H., Wu X., and Yuan J.Y. SOR-like methods for augmented systems. BIT 41 (2001) 071-085
37. Cao Z.H. Fast Uzawa algorithm for generalized saddle point problems. Appl. Numer. Math. 46 (2003) 157-171
38. Cui M.R. Analysis of iterative algorithms of Uzawa type for saddle point problems. Appl. Numer. Math. 50 (2004) 133-146
39. Hu Q., and Zou J. An iterative method with variable relaxation parameters for saddle-point problems. SIAM J. Matrix Anal. A. 23 (2001) 317-338
40. Prevost J.H. Partitioned solution procedure for simultaneous integration of coupled-field problems. Commun. Numer. Methods Engrg. 13 (1997) 239-247
41. Meurant G.A. Computer Solution of Large Linear Systems (1999), North-Holland, Elsevier, Amsterdam, New York
42. Greenbaum A. Iterative Methods for Solving Linear Systems (1997), SIAM, Philadelphia
43. Freund R.W., and Nachtigal N.M. Software for simplified Lanczos and QMR algorithms. Appl. Numer. Math. 19 (1995) 319-341
44. Y.Q. Wang, Preconditioning for the mixed formulation of linear plane elasticity, Ph.D. Thesis, Texas A&M University, 2004.
45. Dayde M.J., L'Excellent J.Y., and Gould N.I.M. Element-by-element preconditioners for large partially separable optimization problems. SIAM J. Sci. Comput. 18 (1997) 1767-1787
46. Cryer C.W. Comparison of the three-dimensional conclusion theories of Biot and Terzaghi. Quart. J. Mech. Appl. Math. 16 (1963) 401-412
47. Coussy O. Poromechanics (2004), John Wiley, Chichester
48. van der Vorst H.A. Iterative Krylov Methods for Large Linear Systems (2003), Cambridge University Press, New York