Multiscale model reduction for stochastic elasticity problems using ensemble variable-separated method

Abstract

Elasticity is a fundamental model in mechanics and material sciences. In the article, we present an ensemble variable-separated multiscale method for elasticity problems in random media. A variable-separated method is utilized to provide a low-rank approximation for random fields. The variable-separated method is integrated with the Generalized Multiscale Finite Element Method (GMsFEM) to obtain an efficient reduced model. To this end, a few local problems are solved for snapshots to construct the multiscale basis functions in GMsFEM. An ensemble method is used to construct stochastic basis functions shared by a series of elasticity equations, and a residual decomposition is adopted to calculate the deterministic physical components for all ensemble members. The ensemble variable-separation is used to solve the local problems of the multiscale basis functions and achieves efficient online-computation for the basis. Then, under the Galerkin projection of the generalized finite element method, the solutions of stochastic elasticity problems are projected onto the low-dimensional space spanned by the online multiscale basis functions. This renders an effective coarse model. The ensemble variable-separated multiscale method is applied to steady-state elastic deformation and elastic wave propagation in random media. A few numerical results are carried out to demonstrate the accuracy and efficiency of the presented method.