Wavelet-based multi-scale projection method in homogenization of heterogeneous media

Abstract

Standard homogenization of highly heterogeneous media often filters out the fine scale information, and as a consequence, it produces acceptable results only for certain type of periodic structures. In this work, a wavelet-based multi-scale homogenization is introduced for highly heterogeneous materials where the standard asymptotic technique cannot be effectively applied. A set of scaling and wavelet functions based on the linear hat function and its corresponding wavelet transformation matrix are constructed. The advantages of this wavelet transformation constructed by hat function compared to that formulated using Haar function are identified. The mirror image technique has been employed to preserve the accurate representation of boundary conditions and to avoid numerical oscillation near the boundaries. This wavelet-based multi-scale transformation hierarchically filters out the high-scale components of the solution, and thus provides an effective framework for the multi-scale selection of the most essential scales of the solution.