Shape optimization for composite materials in linear elasticity

This article is devoted to the optimal design of the microstructure in composite materials, which are governed by the equations of linear elasticity. To this end, we combine homogenization with shape optimization. In particular, we determine the sensitivity of the homogenized coefficients of the elasticity tensor with respect to the shape of the periodic microstructure also in case of spatially varying material coefficients. We compute the respective Hadamard shape gradient and demonstrate the applicability and feasibility of our approach by numerical experiments for different problem settings.