Adaptive Strategy for Stochastic Homogenization and Multiscale Stochastic Stress Analysis

Abstract

This paper discusses the probabilistic analysis of a multiscale problem of heterogeneous materials, such as composite materials, for estimating the probabilistic characteristics of their homogenized equivalent elastic properties and their macroscopic and microscopic stress fields. For this purpose, a function approximation-based stochastic homogenization method or a perturbation-based multiscale stochastic analysis method is employed. When using these methods for the probabilistic analyses, an appropriate set of samples must be selected for the approximation and the approximation order must be appropriately determined. For this problem, to improve the accuracy of a lower-order approximation-based analysis, some adaptive strategies for the multiscale stochastic analysis are introduced. One is based on the approximation of a response function with the adaptive weighted least-squares method and the other is a piecewise linear approximation with the adaptive expansion of a response function. As a numerical example, a stochastic homogenization and multiscale stochastic stress analysis of a glass particle-reinforced composite material is solved. On the basis of the results, the effectiveness of the proposed approaches is discussed.