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Partitioned versus global Krylov subspace iterative methods for FE solution of 3-D Biot’s problem. (English) Zbl 1173.74405
Summary: Finite element analysis of 3-D Biot’s consolidation problem needs fast solution of discretized large \(2\times 2\) block symmetric indefinite linear systems. In this paper, partitioned iterative methods and global Krylov subspace iterative methods are investigated and compared. The partitioned iterative methods considered include stationary partitioned iteration and non-stationary Prevost’s PCG procedure. The global Krylov subspace methods considered include MINRES and Symmetric QMR (SQMR). Two efficient preconditioners are proposed for global methods. Numerical experiments based on a pile-group problem and simple footing problems with varied soil profiles are carried out. Numerical results show that when used in conjunction with suitable preconditioners, global Krylov subspace iterative methods are more promising for large-scale computations, and further improvement could be possible if significant differences in the solid material properties are addressed in these preconditioned iterative methods.

74S05 Finite element methods applied to problems in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76S05 Flows in porous media; filtration; seepage
65F10 Iterative numerical methods for linear systems
Full Text: DOI
[1] Gambolati, G.; Pini, G.; Ferronato, M., Direct, partitioned and projected solution to finite element consolidation models, Int. J. numer. anal. methods, 26, 1371-1383, (2002) · Zbl 1062.74612
[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.; Smith, D.W., Finite element formulation and algorithms for unsaturated soils. part I: theory, Int. J. numer. anal. methods, 27, 745-765, (2003) · Zbl 1085.74515
[4] Hestenes, M.R.; Stiefel, E., Methods of conjugate gradients for solving linear systems, J. res. nat. bur. stand., 49, 409-436, (1952) · Zbl 0048.09901
[5] Phoon, K.K.; Chan, S.H.; Toh, K.C.; Lee, F.H., Fast iterative solution of large undrained soil – structure interaction problems, Int. J. numer. anal. methods, 27, 159-181, (2003) · Zbl 1106.74413
[6] Paige, C.C.; Saunders, M.A., Solution of sparse indefinite systems of linear equations, SIAM J. numer. anal., 12, 617-629, (1975) · Zbl 0319.65025
[7] Barrett, R.; Berry, M.; Chan, T.; Demmel, J.; Donato, J.; Dongarra, J.; Eijkhout, V.; Pozo, R.; Romine, C.; 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.; Griffiths, D.V., Programming the finite element method, (1997), John Wiley · Zbl 0933.74002
[9] Chan, S.H.; Phoon, K.K.; 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, 1001-1025, (2001) · Zbl 1065.74605
[10] Lee, F.H.; Phoon, K.K.; Lim, K.C.; Chan, S.H., Performance of Jacobi preconditioning in Krylov subspace solution of finite element equations, Int. J. numer. anal. methods, 26, 341-372, (2002) · Zbl 1168.74457
[11] Phoon, K.K.; Toh, K.C.; Chan, S.H.; Lee, F.H., An efficient diagonal preconditioner for finite element solution of biot’s consolidation equations, Int. J. numer. methods engrg., 55, 377-400, (2002) · Zbl 1076.74558
[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] Rozložnı´k, M.; Simoncini, V., Krylov subspace methods for saddle point problems with indefinite preconditioning, SIAM J. matrix anal. A., 24, 368-391, (2002) · Zbl 1021.65016
[14] Toh, K.C., Solving large scale semidefinite programs via an iterative solver on the augmented systems, SIAM J. optim., 14, 670-698, (2003) · Zbl 1071.90026
[15] Toh, K.C.; Phoon, K.K.; Chan, S.H., Block preconditioners for symmetric indefinite linear systems, Int. J. numer. methods engrg., 60, 1361-1381, (2004) · Zbl 1065.65064
[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.; Vlček, J., Indefinitely preconditioned inexact Newton method for large sparse equality constrained non-linear programming problems, Numer. linear algebra appl., 5, 219-247, (1998) · Zbl 0937.65066
[18] Smith, I.M., A general purpose system for finite element analyses in parallel, Engrg. comput., 17, 75-91, (2000) · Zbl 0955.65082
[19] Benzi, M.; Meyer, C.D.; Tůma, M., A sparse approximate inverse preconditioner for the conjugate gradient method, SIAM J. sci. comput., 17, 1135-1149, (1996) · Zbl 0856.65019
[20] Grote, M.J.; Huckle, T., Parallel preconditioning with sparse approximate inverses, SIAM J. sci. comput., 18, 838-853, (1997) · Zbl 0872.65031
[21] Chow, E.; Saad, Y., Approximate inverse preconditioners via sparse – sparse iterations, SIAM J. sci. comput., 19, 995-1023, (1998) · Zbl 0922.65034
[22] Gould, N.I.M.; Scott, J.A., Sparse approximate-inverse preconditioners using norm-minimization techniques, SIAM J. sci. comput., 19, 605-625, (1998) · Zbl 0911.65037
[23] Saad, Y., ILUT: a dual threshold incomplete LU factorization, Numer. linear algebra, 1, 387-402, (1994) · Zbl 0838.65026
[24] Saad, Y., Iterative methods for sparse linear systems, (1996), PWS Publishing Company Boston · Zbl 1002.65042
[25] Meijerink, J.A.; 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, 148-162, (1977) · Zbl 0349.65020
[26] Benzi, M., Preconditioning techniques for large linear systems: a survey, J. comput. phys., 182, 418-477, (2002) · Zbl 1015.65018
[27] Phoon, K.K.; Toh, K.C.; Chen, X., Block constrained versus generalized Jacobi preconditioners for iterative solution of large-scale biot’s FEM equations, Comput. struct., 82, 2401-2411, (2004)
[28] Chen, X.; KC, Toh; KK, Phoon, A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations, Int. J. numer. methods engrg., 65, 785-807, (2006) · Zbl 1114.74056
[29] Eisenstat, S.C., Eefficient implementation of A class of preconditioned conjugate gradient methods, SIAM J. sci. statist. comput., 2, 1-4, (1981) · Zbl 0474.65020
[30] M. Benzi, G.H. Golub, J. Liesen, Numerical solution of saddle point problems, Acta Numerica, 14 (2005) 1-137. · Zbl 1115.65034
[31] Biot, M.A., General theory of three-dimensional consolidation, J. appl. phys., 12, 155-164, (1941) · JFM 67.0837.01
[32] A.J. Abbo, Finite element algorithms for elastoplasticity and consolidation, Ph.D. Thesis, University of Newcastle, 1997.
[33] Zienkiewicz, O.C.; Taylor, R.L., The finite element method, (2000), Butterworth Heinemann Oxford · Zbl 0991.74002
[34] Borja, R.I., Composite Newton-PCG and quasi-Newton iterations for nonlinear consolidation, Comput. method appl. M., 86, 27-60, (1991) · Zbl 0788.65061
[35] J.W. Demmel, Applied Numerical Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, c1997. · Zbl 0665.65021
[36] Golub, G.H.; Wu, X.; Yuan, J.Y., SOR-like methods for augmented systems, Bit, 41, 071-085, (2001)
[37] Cao, Z.H., Fast Uzawa algorithm for generalized saddle point problems, Appl. numer. math., 46, 157-171, (2003) · Zbl 1032.65029
[38] Cui, M.R., Analysis of iterative algorithms of Uzawa type for saddle point problems, Appl. numer. math., 50, 133-146, (2004) · Zbl 1056.65026
[39] Hu, Q.; Zou, J., An iterative method with variable relaxation parameters for saddle-point problems, SIAM J. matrix anal. A., 23, 317-338, (2001) · Zbl 1007.65019
[40] Prevost, J.H., Partitioned solution procedure for simultaneous integration of coupled-field problems, Commun. numer. methods engrg., 13, 239-247, (1997) · Zbl 0878.73073
[41] Meurant, G.A., Computer solution of large linear systems, (1999), North-Holland, Elsevier Amsterdam, New York · Zbl 0934.65032
[42] Greenbaum, A., Iterative methods for solving linear systems, (1997), SIAM Philadelphia · Zbl 0883.65022
[43] Freund, R.W.; Nachtigal, N.M., Software for simplified Lanczos and QMR algorithms, Appl. numer. math., 19, 319-341, (1995) · Zbl 0853.65041
[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.; Gould, N.I.M., Element-by-element preconditioners for large partially separable optimization problems, SIAM J. sci. comput., 18, 1767-1787, (1997) · Zbl 0897.65045
[46] Cryer, C.W., Comparison of the three-dimensional conclusion theories of Biot and terzaghi, Quart. J. mech. appl. math., 16, 401-412, (1963) · Zbl 0121.21502
[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 · Zbl 1023.65027
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