Comput Mech (2015) 56:753–767 DOI 10.1007/s00466-015-1199-1 ORIGINAL PAPER Heaviside enriched extended stochastic FEM for problems with uncertain material interfaces Christapher Lang 1 · Ashesh Sharma 2 · Alireza Doostan 2 · Kurt Maute 2 Received: 23 April 2015 / Accepted: 21 August 2015 / Published online: 5 September 2015 © Springer-Verlag Berlin Heidelberg 2015 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 formu- lation of the X-SFEM using the Heaviside enrichment for modeling problems with either continuous or discontinuous solutions at the uncertain inclusion interface. The Heavi- side enrichment formulation employs multiple enrichment levels for each material subdomain which allows more com- plex 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 sto- chastic subdomain for each degree of freedom is proposed. Numerical examples for heat diffusion and linear elastic- ity are studied to illustrate convergence and accuracy of the scheme under spatial and stochastic refinements. In addi- tion 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 B Alireza Doostan doostan@colorado.edu 1 Structural Mechanics and Concepts Branch, NASA Langley Research Center, Hampton, VA, USA 2 Aerospace Engineering Sciences, University of Colorado, Boulder, CO, USA convergence rate is achieved using the proposed Heaviside enriched X-SFEM for C 0 -continuous problems when com- pared to using a C 0 -continuous enrichment. Keywords X-SFEM · Level set method · Heaviside enrichment · Polynomial chaos · Uncertainty quantification 1 Introduction Computational methods for the propagation of uncertainties through models governed by partial differential equations are powerful tools for the prediction of a system’s response, model validation, and engineering design. For heterogeneous composite materials, the material layout has uncertainty due to fabrication techniques. In order to relate the effective prop- erties to the material layout, the uncertainty in geometry requires methods that account for the random material inter- faces. This work proposes an approach to model problems with either a weak or a strong discontinuity across a random material interface. Examples from the first class of problems include perfectly bonded interfaces, while examples from the latter class of problems include imperfectly bonded inter- faces, crack analysis, and the phonon Boltzmann transport model for heat diffusion at the submicron scale. The proposed approach introduces the Heaviside enrichment function in the eXtended stochastic finite element method (X-SFEM) [12], which extends the eXtended finite element method (XFEM) [10] to the stochastic domain using a polynomial chaos expansion (PCE) [25] to approximate the degrees of freedom based on the random parameters characterizing the interface position. Following the work by Hansbo and Hansbo [4], the Heav- iside enriched XFEM is a deterministic approach for solving problems with strong discontinuities across an embedded 123