Preconditioners for Soil-Structure Interaction Problems with Significant Material Stiffness Contrast

Abstract

Three-dimensional finite element analysis of geotechnical problems usually involves a significant large number of variables (or unknowns) and non-uniformity of the materials. Recent advances on solution methods of linear systems show that Krylov subspace iterative methods in conjunction with appropriate preconditioning are potentially more effective than direct solution methods for large-scale systems. A preconditioner is the key for the success of iterative methods. For this reason, a number of publications have recently been devoted to propose effective preconditioners for the solution of large, often ill-conditioned coupled consolidation problems. Some of them may require a number of user-defined parameters, which may limit their practical use. Also, much of the work has been devoted on the ill-conditioning due to small time steps in the consolidation analysis. Little attention has been paid on the ill-conditioning due to significant contrasts in material properties such as stiffness and permeability. This significant difference in material properties may deteriorate the performance (Chen et al., 2007) of so called cheap and effective preconditioners such as generalized Jacobi (GJ) (Phoon et al., 2002) and modified symmetric successive over-relaxation (MSSOR) preconditioner (Chen et al., 2006). Similar degradation in performance was also observed for the standard Jacobi (Lee et al., 2002) and symmetric successive over-relaxation (SSOR) preconditioners (Mroueh and Shahrour, 1999) for the analysis of drained boundary value problems. On the other hand, pragmatic geotechnical problems often involve materials with highly varied material zones, such as in soil-structure interaction problems. Hence, the prime objective of the thesis was to propose a preconditioner that mitigates these adverse effects and yet remain practical for use. Firstly, the relative merits and demerits of MSSOR preconditioner was compared with ILU0 (incomplete LU factorization with zero fill-ins) for the Biot?s coupled consolidation equations. This is because the ILU-type preconditioners have also frequently been used for Biot?s problem (Gambolati et al., 2001, 2002, 2003). The comparison revealed that the ILU0 is occasionally unstable, but may be preferred over MSSOR if its instability problem is resolved and RAM constraint is not an issue. On the other hand, MSSOR and GJ were robust in solving even a severe ill-conditioned system. Secondly, the ill-conditioning due to the presence of different material zones with large relative differences in material stiffnesses was addressed by proposing block diagonal preconditioners. The effect of only stiffness contrasts was considered first (in Chapter 4) and stiffness/permeability contrasts in the consolidation analysis was studied next (in Chapter 5). The inexpensive block diagonal preconditioners for practical use were investigated numerically using preconditioned conjugate gradient (PCG) solver and symmetric quasi-minimal residual (SQMR) solver. Significant benefits in terms of CPU time in comparison to existing preconditioners were demonstrated with the help of a number of soil-structure interaction problems. Finally, the general applicability of the proposed block diagonal preconditioners for real-world problems was shown using two case history examples in Chapter 6.