The role of preconditioning in the solution to FE coupled consolidation equations by Krylov subspace methods

The repeated solution in time of the linear system arising from the finite element integration of coupled consolidation equations is a major computational effort. This system can be written in either a symmetric or an unsymmetric form, thus calling for the implementation of different preconditioners and Krylov subspace solvers. The present paper aims at investigating when either a symmetric or an unsymmetric approach should be better used. The results from a number of representative numerical experiments indicate that a major role in selecting either form is played by the preconditioner rather than by the Krylov subspace method itself. Two other important issues addressed are the size of the time integration step and the possible lumping of the flow capacity matrix. It appears that ad hoc block constrained preconditioners provide the most robust algorithm independently of the time step size, lumping, and symmetry. Copyright © 2008 John Wiley & Sons, Ltd.

[1]  M. Biot.  General Theory of Three‐Dimensional Consolidation , 1941 .

[2]  Kim-Chuan Toh,et al.  Partitioned versus global Krylov subspace iterative methods for FE solution of 3-D Biot's problem , 2007 .

[3]  Graham F. Carey,et al.  Parallel conjugate gradient performance for least-squares finite elements and transport problems , 1998 .

[4]  J. Marchal,et al.  Preconditioned conjugate gradients for solving the transient Boussinesq equations in three-dimensional geometries , 1988 .

[5]  Valeria Simoncini,et al.  Krylov Subspace Methods for Saddle Point Problems with Indefinite Preconditioning , 2002, SIAM J. Matrix Anal. Appl..

[6]  Giuseppe Gambolati,et al.  Ill-conditioning of finite element poroelasticity equations , 2001 .

[7]  Youcef Saad,et al.  A Basic Tool Kit for Sparse Matrix Computations , 1990 .

[8]  X. Chen,et al.  A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations , 2006 .

[9]  C. Lanczos.  Solution of Systems of Linear Equations by Minimized Iterations1 , 1952 .

[10]  Michele Benzi,et al.  Robust Approximate Inverse Preconditioning for the Conjugate Gradient Method , 2000, SIAM J. Sci. Comput..

[11]  Luca Bergamaschi,et al.  Novel preconditioners for the iterative solution to FE-discretized coupled consolidation equations , 2007 .

[12]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[13]  P. Forsyth,et al.  Preconditioned conjugate gradient methods for three-dimensional linear elasticity , 1994 .

[14]  P. Sonneveld.  CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems , 1989 .

[15]  Jean-Herve Prevost,et al.  PARTITIONED SOLUTION PROCEDURE FOR SIMULTANEOUS INTEGRATION OF COUPLED-FIELD PROBLEMS , 1997 .

[16]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[17]  R. Freund,et al.  A new Krylov-subspace method for symmetric indefinite linear systems , 1994 .

[18]  K. Toh,et al.  An efficient diagonal preconditioner for finite element solution of Biot's consolidation equations , 2002 .

[19]  Kok-Kwang Phoon,et al.  A modified Jacobi preconditioner for solving ill‐conditioned Biot's consolidation equations using symmetric quasi‐minimal residual method , 2001 .

[20]  Gene H. Golub,et al.  SOR-like Methods for Augmented Systems , 2001 .

[21]  X. Chen,et al.  Block constrained versus generalized Jacobi preconditioners for iterative solution of large-scale Biot’s FEM equations , 2004 .

[22]  R. Freund,et al.  QMR: a quasi-minimal residual method for non-Hermitian linear systems , 1991 .

[23]  K. Toh,et al.  Block preconditioners for symmetric indefinite linear systems , 2004 .

[24]  Zhi-Hao Cao.  Fast uzawa algorithm for generalized saddle point problems , 2003 .

[25]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

[26]  K. Phoon,et al.  Performance of Jacobi preconditioning in Krylov subspace solution of finite element equations , 2002 .

[27]  Robert Beauwens,et al.  Problem-dependent preconditioners for iterative solvers in FE elastostatics , 1999 .

[28]  Ahmed Sameh,et al.  Efficient iterative solvers for structural dynamics problems , 2004 .

[29]  Luca Bergamaschi,et al.  Mixed Constraint Preconditioners for the iterative solution of FE coupled consolidation equations , 2008, J. Comput. Phys..

[30]  Michele Benzi,et al.  A Sparse Approximate Inverse Preconditioner for Nonsymmetric Linear Systems , 1998, SIAM J. Sci. Comput..

[31]  Yousef Saad,et al.  ILUT: A dual threshold incomplete LU factorization , 1994, Numer. Linear Algebra Appl..

[32]  Edmond Chow,et al.  Crout Versions of ILU for General Sparse Matrices , 2003, SIAM J. Sci. Comput..

[33]  Roland W. Freund,et al.  A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems , 1993, SIAM J. Sci. Comput..

[34]  Isam Shahrour,et al.  Use of sparse iterative methods for the resolution of three‐dimensional soil/structure interaction problems , 1999 .

[35]  M. Benzi.  Preconditioning techniques for large linear systems: a survey , 2002 .

[36]  Giuseppe Gambolati,et al.  Numerical performance of projection methods in finite element consolidation models , 2001 .

[37]  J. Reid,et al.  On the Automatic Scaling of Matrices for Gaussian Elimination , 1972 .

[38]  J. C. Small,et al.  An investigation of the stability of numerical solutions of Biot's equations of consolidation , 1975 .

[39]  G. Gambolati,et al.  Direct, partitioned and projected solution to finite element consolidation models , 2002 .

[40]  Giuseppe Gambolati,et al.  Scaling improves stability of preconditioned CG‐like solvers for FE consolidation equations , 2003 .