Extended stochastic FEM for diffusion problems with uncertain material interfaces.

This paper is concerned with the prediction of heat transfer in composite materials with uncertain inclusion geometry. To numerically solve the governing equation, which is defined on a random domain, an approach based on the combination of the Extended finite element method (X-FEM) and the spectral stochastic finite element method is studied. Two challenges of the extended stochastic finite element method (X-SFEM) are choosing an enrichment function and numerical integration over the probability domain. An enrichment function, which is based on knowledge of the interface location, captures the C-continuous solution in the spatial and probability domains without a conforming mesh. Standard enrichment functions and enrichment functions tailored to X-SFEM are analyzed and compared, and the basic elements of a successful enrichment function are identified. We introduce a partition approach for accurate integration over the probability domain. The X-FEM solution is studied as a function of the parameters describing the inclusion geometry and the different enrichment functions. The efficiency and accuracy of a spectral polynomial chaos expansion and a finite element approximation in the probability domain are compared. Numerical examples of a two-dimensional heat conduction problem with a random inclusion show the spectral PC approximation with a suitable choice of enrichment function is as accurate and more efficient than the finite element approach. Though focused on heat transfer in composite materials, the techniques and observations in this paper are also applicable to other types of problems with uncertain geometry. [ABSTRACT FROM AUTHOR]