Optimal fine-scale structures in composite materials

A classical problem consists in optimizing the structure of a composite material, for instance to achieve high rigidity against a prescribed mechanical loading. In the simplest case, the material is a composite of void and the elastic base material. The problem then reduces to finding the optimal topology and geometry of the structure. One typically aims to minimize a weighted sum of material volume, structure perimeter, and structure compliance (a measure of the inverse structure stiffness). This task is not only of interest for optimal material designs, but also represents a prototype problem to better understand biological structures. The high nonconvexity of the problem makes it impossible to find the globally optimal design; if in addition the weight of the perimeter is chosen small, very fine material structures are optimal that can hardly be resolved numerically. However, for certain geometries an energy scaling law can be proven that describes how the minimum of the objective functional scales with the model parameters and that provides near-optimal designs. The optimal design problem is strongly related to a few well-known pattern formation problems but has several distinct features.