Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
A multiscale finite element method for elliptic problems in composite materials and porous media
Journal of Computational Physics
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Uncertainty propagation using Wiener-Haar expansions
Journal of Computational Physics
Multi-resolution analysis of wiener-type uncertainty propagation schemes
Journal of Computational Physics
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
An adaptive multi-element generalized polynomial chaos method for stochastic differential equations
Journal of Computational Physics
Journal of Computational Physics
Numerical Methods for Differential Equations in Random Domains
SIAM Journal on Scientific Computing
Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
A stochastic variational multiscale method for diffusion in heterogeneous random media
Journal of Computational Physics
A fictitious domain approach to the numerical solution of PDEs in stochastic domains
Numerische Mathematik
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
Numerical Methods for Stochastic Computations: A Spectral Method Approach
Numerical Methods for Stochastic Computations: A Spectral Method Approach
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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 0-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.