An Optimal Design Problem for Two-dimensional Composite Materials. A Constructive Approach

Abstract

We consider an optimal design problem, when it is necessary to locate circular inclusions of fixed size having a given conductivity in a material of fixed shape and a given conductivity in such a way that with prescribed concentration of the inclusions and prescribed external field the whole composite material has extremal conductivity in a given direction. The problem is reformulated in terms of a boundary value problem for analytic functions with unknown geometrical parameters and constraints. The boundary value problem in special cases is solved in closed form and the optimization problem is reduced to the classical problem of the extrema of a continuous function on a compact set.