Optimal Design in Conductivity

Abstract

This chapter is concerned with optimal design problems in the conductivity setting that we shall treat by the homogenization method. There is a huge variety of optimal design problems, but we focus only on two-phase optimization problems. Such problems are defined as one in which we seek the optimal distribution of two components in a given domain that minimizes a criterion (also called an objective function), computed through the solution of a partial differential equation modeling the conductivity of the whole domain (the state equation). This type of problem includes, as a limit case, shape optimization problems. These are defined as ones in which we seek the shape of a domain (filled with a single material) that minimizes a criterion, computed again through the solution of a state equation. Indeed, if in a two-phase problem the conductivity of one of the components is allowed to go to zero, then, in the limit, this weak component mimics void or holes in the domain, supporting homogeneous Neumann boundary conditions. Therefore, a two-phase problem yields a shape optimization problem, where the boundaries, as well as the topology of the holes (i.e., their number and connectivity), are subject to optimization. In other words, the weak phase with very small conductivity properties mimics holes with current-free boundary conditions. From a physical point of view, this limit procedure is clear, but its mathematical justification is delicate and not always known.