Eigenfrequency optimization in optimal design

We maximize the first eigenfrequency, or a sum of the first ones, of a bounded domain occupied by two elastic materials with a volume constraint for the most rigid one. A relaxed formulation of this problem is introduced, which allows for composite materials as admissible designs. These composites are obtained by homogenization of fine mixtures of the two original materials. We prove a saddle-point theorem that permits to reduce the full (unknown) set of admissible composite designs to the smaller set of sequential laminates which is explicitly known. Although our relaxation theorem is valid only for two non-degenerate materials, we deduce from it a numerical algorithm for eigenfrequency optimization in the context of optimal shape design (i.e. when one of the two materials is void). As is the case with all homogenization methods, our algorithm can be seen as a topology optimizer. Numerical results are presented for various two- and three-dimensional problems.