Homogenization and topological optimization for wave propagation in periodic media

We are interested in waves in two-phase periodic materials, either occupying the whole 2D domain, or a thin interface between homogeneous media. The phase distribution in these media is to be optimized to obtain specific dispersive or transmission properties.

In these two cases, a first homogenization step is used to describe the effective wave propagation. The two-scale asymptotic homogenization procedure leads to an "enriched" dispersive wave equation in the first case, and effective transmission conditions across an equivalent interface in the second one. In both cases, the effective coefficients of the model are defined through elementary "cell" or "band" problems that we address with FFT-based solvers.

A topological optimization procedure is then presented. First, relevant propagation indicators are extracted from the effective models, and serve to define cost functionals to be minimized to achieve certain goals. The sensitivity of the functional to a localized phase change in the representative cell, also called topological derivative (TD), is computed and indicates optimal locations where to perform these phase changes. A TD-based level-set algorithm is finally used to optain optimal microstructures.

Applications of the method will finally be presented: maximizing the dispersion in given directions in the full space, or enhancing the transmitted wave in a specific direction.