Multiscale finite element method for a locally nonperiodic heterogeneous medium

Abstract

A generalization of the mathematical homogenization theory to account for locally nonperiodic solutions is presented. Such nonperiodicity may arise either due to the rapidly varying microstructure (e.g.: graded materials, microcracks) or because the macroscopic solution is not smooth and may have significant variation within a microstructure. In the portion of the problem domain where the material is formed by a spatial repetition of the base cell and the macroscopic solution is smooth, a double scale asymptotic expansion and solution periodicity are assumed, and consequently, mathematical homogenization theory is employed to uncouple the microscopic problem from the global solution. For the rest of the problem domain it is assumed that the periodic solution does not exist (cutouts, cracks, free edges in composites, etc.) and the approximation space is decomposed into macroscopic and microscopic fields. Compatibility between the two regions is explicitly enforced. The proposed method is applied to resolve the structure of the microscopic fields in the single ply composite plates with a centered hole and with a centered crack and in the [0/90] _s laminated plate. Numerical results are compared to the reference solution, an engineering global-local approach, and the direct extraction from the mathematical homogenization method.