Homogenization and approximation of solutions to multiscale problems

Abstract

This project explores how to model steady-state diffusion in media that vary on very small spatial scales. These types of problems often arise in materials composed of repeating structures, and resolving all the microscopic details can be difficult. To address this, we use homogenization theory to approximate the behavior of such media by an effective equation that captures the macroscopic response. Starting from a periodic framework, we derive the homogenized equation using asymptotic expansions and construct the homogenized matrix that governs large-scale behavior. We then study how to estimate this matrix using both direct matrix methods and variational principles, leading to classical Voigt–Reiss inequalities. To improve on these estimates, we introduce the Hashin–Shtrikman bounds, which take into account additional structure like isotropy. The theory is applied to examples involving composite layers and skew-symmetric flows, showing how the bounds perform in practice. Overall, the report highlights how homogenization provides a useful tool for connecting fine-scale structure to effective material properties.