Bloch Wave Homogenization Relative to a Microstructure

TL;DR

This paper introduces a new approach to Bloch wave homogenization that accounts for interactions between two microstructures, providing a spectral representation and homogenization results.

Contribution

It develops a novel homogenization framework incorporating microstructure interactions via a new macro quantity $B^{#}$ and its spectral representation.

Findings

Introduction of the macro quantity $B^{#}$ capturing microstructure interactions

Spectral representation of $B^{#}$ in the Bloch wave context

Homogenization results relating $B^{#}$ to microstructure convergence

Abstract

In this work, we study the aspect of Bloch wave homogenization of the new notion of convergence of microstructures represented by matrices $B^\epsilon$ related to the classical $H$-convergence of $A^\epsilon$ introduced in \cite{TG-MV-1}. The new macro quantity $B^{#}$ appears to incorporate the interaction between the two microstructures $A^\epsilon,B^\epsilon$. Here we present its Bloch spectral representation along with the homogenization result.