The stochastic finite element method: Past, present and future

Abstract

a b s t r a c tA powerful tool in computational stochastic mechanics is the stochastic finite element method (SFEM). SFEM is an extension of the classical deterministic FE approach to the stochastic framework i.e. to the solution of static and dynamic problems with stochastic mechanical, geometric and/or loading properties. The considerable attention that SFEM received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large-scale problems. This article aims at providing a state-of-the-art review of past and recent developments in the SFEM area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future.

Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients

“…A valid alternative choice that has been widely used in literature (see e.g. [7,20,21]) is given by the Total Degree polynomial space, that includes those polynomials whose total degree is lower than or equal to w: such space contains indeed only  N+w N  polynomials, which is much lower than (1 + w) N , and still has good approximation properties. A number of possible polynomial spaces has been listed and analyzed e.g.…”

“…Moreover, we see from (20) that to ensure convergence of the estimate we need ϵ M,n > 0, which enforces a constraint on M. Namely, taken any 0 < δ < ϵ max we require ϵ M,n > δ which implies…”

Efficient stochastic simulation approach for RTM process with random fibrous permeability

“…Since the zero residual is only ensured at the set of collocation points, they must be able to capture the high probability region in the sample space. In analogous to the Gaussian quadrature, the collocation points are usually chosen from the roots of the polynomials of one order higher than the PCE, and kept as close to the origin as possible in sample space [20,21]. For example, if the 2 nd -order PCM is employed, the optimal collocation points should have the coordinates (in sample space) 0 or 3 ± (as roots of ( )…”

A stochastic computational multiscale approach; Application to MEMS resonators

“…When considering a finite element analysis, the structural response variability can be predicted using the direct Monte Carlo method, which can lead to an overwhelming computation cost as it involves the finite-element discretization of the heterogeneities. In order to solve the problem of structural stochasticity at a reasonable computation cost, Stochastic Finite Element (SFE) analyzes were developed [1][2][3].…”