Bloch Wave Homogenization Relative to a Microstructure

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.