Heaviside enriched extended stochastic FEM for problems with uncertain material interfaces

Abstract

This paper is concerned with the modeling of heterogeneous materials with uncertain inclusion geometry. The eXtended stochastic finite element method (X-SFEM) is a recently proposed approach for modeling stochastic partial differential equations defined on random domains. The X-SFEM combines the deterministic eXtended finite element method (XFEM) with a polynomial chaos expansion (PCE) in the stochastic domain. The X-SFEM has been studied for random inclusion problems with a $C^0$-continuous solution at the inclusion interface. This work proposes a new formulation of the X-SFEM using the Heaviside enrichment for modeling problems with either continuous or discontinuous solutions at the uncertain inclusion interface. The Heaviside enrichment formulation employs multiple enrichment levels for each material subdomain which allows more complex inclusion geometry to be accurately modeled. A PCE is applied in the stochastic domain, and a random level set function implicitly defines the uncertain interface geometry. The Heaviside enrichment leads to a discontinuous solution in the spatial and stochastic domains. Adjusting the support of the stochastic approximation according to the active stochastic subdomain for each degree of freedom is proposed. Numerical examples for heat diffusion and linear elasticity are studied to illustrate convergence and accuracy of the scheme under spatial and stochastic refinements. In addition to problems with discontinuous solutions, the Heaviside enrichment is applicable to problems with $C^0$-continuous solutions by enforcing continuity at the interface. A higher convergence rate is achieved using the proposed Heaviside enriched X-SFEM for $C^0$-continuous problems when compared to using a $C^0$-continuous enrichment.