Multiscale Problems in Materials Science: A Mathematical Approach to the Role of Uncertainty.

摘要

One approach to uncertainty quantification in materials science problems is to use dedicated numerical approaches in a multiscale framework, such as the Multiscale Finite Element Method (MsFEM). In this report, we first consider a variant of stochastic homogenization, well suited to model materials that are periodic up to a random deformation. This variant admits a homogenized limit. However, the homogenized matrix is expensive to compute, as is often the case in stochastic homogenization. We propose here an efficient MsFEM-type approach dedicated to that setting. We next turn to studying the robustness of the MsFEM approach to perturbations of the equation coefficients that are non- oscillatory. Our idea is that MsFEM approaches are devoted to capturing the highly oscillatory modes of the solution, which are poorly captured by a standard FEM approach using a limited number of degrees of freedom. When the coefficient in the equation is modified by a non-oscillatory component, the high frequencies are not modified, and the MsFEM approach can be expected to be robust with respect to these perturbations. This is exactly the question we considered in the final portion of this three-year study.