A quasi-Newton method in shape optimization for a transmission problem

We consider optimal design problems in stationary diffusion for mixtures of two isotropic phases. The goal is to find an optimal distribution of the phases such that the energy functional is maximized. By following the identity perturbation method, we calculate the first- and second-order shape derivatives in the distributional representation under weak regularity assumptions. Ascent methods based on the distributed first- and second-order shape derivatives are implemented and tested in classes of problems for which the classical solutions exist and can be explicitly calculated from the optimality conditions. A proposed quasi-Newton method offers a better ascent vector compared to gradient methods, reaching the optimal design in half as many steps. The method applies well also for multiple state problems.