Optimization of dispersive coefficients in the homogenization of the wave equation in periodic structures

Grégoire Allaire (1) , T Yamada (2, 1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 Department of Mechanical Engineering and Science

We study dispersive effects of wave propagation in periodic media, which can be modelled by adding a fourth-order term in the homogenized equation. The corresponding fourth-order dispersive tensor is called Burnett tensor and we numerically optimize its values in order to minimize or maximize dispersion. More precisely, we consider the case of a two-phase composite medium with an 8-fold symmetry assumption of the periodicity cell in two space dimensions. We obtain upper and lower bound for the dispersive properties, along with optimal microgeometries.

Mots clés: periodic structure; homogenization; Bloch waves; dispersion; shape optimization

Domaines: Analyse numérique [math.NA]; Optimisation et contrôle [math.OC]

Grégoire Allaire, T Yamada. Optimization of dispersive coefficients in the homogenization of the wave equation in periodic structures. Numerische Mathematik, 2018, 140 (2), pp.265-326. ⟨hal-01341082⟩