An extended stochastic finite element method for solving stochastic partial differential equations on random domains

Abstract

SUMMARYAn eXtended Stochastic Finite Element Method has been recently proposed for the numerical solution of partial dierential equations dened on random domains. This method is based on a mariage between the eXtended Finite Element Method and spectral stochastic methods. In this paper, we propose an extension of this method for the numerical simulation of random multi-phased materials. The random geometry of material interfaces is described implicitly by using random level-set functions. A xed deterministic nite element mesh, which is not conforming the random interfaces, is then introduced in order to approximate the geometry and the solution. Classical spectral stochastic nite element approximation spaces are not able to capture the irregularities of the solution eld with respect to spatial and stochastic variables, which leads to a deterioration of the accuracy and convergence properties of the approximate solution. In order to recover optimal convergence properties of the approximation, we propose an extension of the partition of unity method to the spectral stochastic framework. This technique allows the enrichment of approximation spaces with suitable functions based on an a priori knowledge of the irregularities in the solution. Numerical examples illustrate the eciency of the proposed method and demonstrate the relevance of the enrichment procedure.

The stochastic finite element method: Past, present and future

“…From a mathematical point of view, SFEM is a powerful tool for the solution of stochastic partial differential equations (PDEs) and it has been treated as such in numerous publications where convergence and error estimation issues are examined in detail e.g. [10][11][12]30,41,58,113,124]. SFEM has been successfully applied in a wide variety of problems (e.g.…”

“…Recently proposed methods. The most recent developments in spectral-Galerkin-based SFEM include the stochastic reduced basis methods (SRBMs) introduced in [118,154,116], the non-intrusive approaches proposed in [13][14][15], the use of the method in a multi-scale setting [191] and the extension to the stochastic framework of the eXtended finite element method (X-FEM) [124]. The SRBMs constitute an efficient alternative which is also limited to the analysis of random linear systems (at least in its present formulation).…”

Approximation Methods to Solve Stochastic Problems in Computational Electromagnetics

“…It consists in searching the solution in a tensorial space W (D) ⊗ P K P with W (D) the standard finite element space used in the deterministic case and P K P the space of approximation of random variables spanned by the basis functions (Ψ i [p(θ )]) 1≤i≤P introduced previously (see (10)). In magnetostatics, the vector potential is sought in a space generated by the basis function Ψ j (p(θ ))w i (x).…”

“…In a postprocessing step, quantities of interest (energy, flux,) can be also expressed using (10). Among the method proposed in the literature to determine these coefficients, some are called non-intrusive since they encapsulate a deterministic model in an environment of stochastic procedures.…”

Identification of random shapes from images through polynomial chaos expansion of random level set functions

“…The number of numerical strategies proposed for this kind of problems is limited [3,4]. Recently, the eXtended Stochastic Finite Element Method (X-SFEM) [5,6] has been proposed. This approach, which is an extension to the stochastic framework of the X-FEM method [7,8,9,10,11], relies on the implicit representation of complex geometries using random level-set functions and on the use of a Galerkin approximation at both stochastic and deterministic levels.…”