Classical optimal designs on annulus and numerical approximations

Abstract

We consider optimal design problems for stationary diffusion equation, seeking for the arrangement of two isotropic materials, with prescribed amounts, which maximizes the energy functional. The aim is to present some classes of problems on an annulus with classical solutions.The first class is a single state equation problem with a constant right-hand side and homogenous Dirichlet boundary condition. By analyzing the optimality conditions, we are able to show that there exists a unique (classical) solution. We prove that, depending on the amounts of given materials, only two optimal configurations in both two- and three-dimensional case are possible.The second class of problems deals with a two-state optimal design problem. A numerical method based on shape derivative is presented, showing good results when applied to described problems with classical solutions on annulus.