Fast iterative solution of large undrained soil‐structure interaction problems

Abstract

SUMMARYIn view of rapid developments in iterative solvers, it is timely to re-examine the merits of using mixed formulation for incompressible problems. This paper presents extensive numerical studies to compare the accuracy of undrained solutions resulting from the standard displacement formulation with a penalty term and the two-field mixed formulation. The standard displacement and two-field mixed formulations are solved using both direct and iterative approaches to assess if it is cost-effective to achieve more accurate solutions. Numerical studies of a simple footing problem show that the mixed formulation is able to solve the incompressible problem 'exactly', does not create pressure and stress instabilities, and obviate the need for an ad hoc penalty number. In addition, for large-scale problems where it is not possible to perform direct solutions entirely within available random access memory, it turns out that the larger system of equations from mixed formulation also can be solved much more efficiently than the smaller system of equations arising from standard formulation by using the symmetric quasi-minimal residual (SQMR) method with the generalized Jacobi (GJ) preconditioner. Iterative solution by SQMR with GJ preconditioning also is more elegant, faster, and more accurate than the popular Uzawa method.

Element‐based preconditioners for elasto‐plastic problems in geotechnical engineering

“…In this paper we focus on the standard displacement formulation (see, for example, [25]) and do not consider alternative methods such as mixed formulations (see, for example, [26]), which may be more effective for particular types of incompressible problems. For the displacement finite element method, the n × n (where n is the total number of degrees of freedom) structure stiffness matrix K in (1) is found by evaluating…”

Comparison between iterative solution of symmetric and non‐symmetric forms of Biot's FEM equations using the generalized Jacobi preconditioner

“…Physically, t → 0 produces the 'undrained' problem, whereas t →∞ produces the 'drained' problem. The former and latter problems (symmetric form) were studied in some details by Phoon et al [11] and Lee et al [12].…”

Effective block diagonal preconditioners for Biot's consolidation equations in piled‐raft foundations

“…The generalized Jacobi (GJ) preconditioner developed by Phoon et al . has been shown to be effective in mitigating the ill‐conditioning because of low permeability in Biot's consolidation equations . Observing the breakdown of conventional symmetric successive over‐relaxation (SSOR) preconditioner for consolidation problems, Chen et al .…”

“…The generalized Jacobi (GJ) preconditioner developed by Phoon et al [6] has been shown to be effective in mitigating the ill-conditioning because of low permeability in Biot's consolidation equations [16][17][18][19]. Observing the breakdown of conventional symmetric successive over-relaxation (SSOR) preconditioner for consolidation problems, Chen et al [7] proposed the modified SSOR (MSSOR) preconditioner with GJ as the main diagonal, which outperforms both the GJ and the block constrained preconditioners [12].…”