A FETI‐preconditioned conjugate gradient method for large‐scale stochastic finite element problems

Abstract

SUMMARYIn the spectral stochastic finite element method for analyzing an uncertain system, the uncertainty is represented by a set of random variables, and a quantity of interest such as the system response is considered as a function of these random variables. Consequently, the underlying Galerkin projection yields a block system of deterministic equations where the blocks are sparse but coupled. The solution of this algebraic system of equations becomes rapidly challenging when the size of the physical system and/or the level of uncertainty is increased. This paper addresses this challenge by presenting a preconditioned conjugate gradient method for such block systems where the preconditioning step is based on the dual-primal finite element tearing and interconnecting method equipped with a Krylov subspace reusage technique for accelerating the iterative solution of systems with multiple and repeated right-hand sides. Preliminary performance results on a Linux Cluster suggest that the proposed solution method is numerically scalable and demonstrate its potential for making the uncertainty quantification of realistic systems tractable.

Uncertain static plane stress analysis with interval fields

“…In the context of probabilistic analysis framework, the effects of spatial dependency of the uncertain parameters are modelled by the concept of random field [35][36][37]. Because of the well-established theoretical background, the random field has been incorporated into finite element method (FEM) to propose the stochastic FEM (SFEM) analysis framework [38][39][40][41][42], whose implantation has been extended into a wide range of engineering applications [43][44][45][46][47][48].Despite of various superiorities associated with the stochastic computational methods basing on the concept of random field, the applicability of such advanced stochastic analysis framework is also frustrated by the insufficiency of the information on uncertain system parameters, especially for situations where the information of distribution type, as well as the spatial dependency, of uncertain parameters are ambiguous. Consequently, the stochastic analysis basing on random field modelling is inapplicable for engineering situations suffering from information insufficiency.…”

“…In the context of probabilistic analysis framework, the effects of spatial dependency of the uncertain parameters are modelled by the concept of random field [35][36][37]. Because of the well-established theoretical background, the random field has been incorporated into finite element method (FEM) to propose the stochastic FEM (SFEM) analysis framework [38][39][40][41][42], whose implantation has been extended into a wide range of engineering applications [43][44][45][46][47][48].…”

On the Capabilities of the Polynomial Chaos Expansion Method within SFE Analysis—An Overview

“…Methods such as employing domain decomposition techniques [35,37,45], generalized spectral decomposition method [58], multiliniear algebra [24] and multigrid solvers [46,63] can be mentioned in this context.…”

Efficient stochastic structural analysis using Guyan reduction

“…In order to circumvent this drawback, collocation based P-C formulations are proposed, where the deterministic FE solver can be used in a black-box fashion [16,20,21,34,35]. Finally, solutions employing parallel computing and domain decomposition techniques have been also proposed [13,17,28].…”