Level set method with topological derivatives in shape optimization

A class of shape optimization problems is solved numerically by the level set method combined with the topological derivatives for topology optimization. Actually, the topology variations are introduced on the basis of asymptotic analysis, by an evaluation of extremal points (local maxima for the specific problem) of the so-called topological derivatives introduced by Sokolowski and Zochowski [J. Sokolowski and A. Zochowski, On the topological derivative in shape optimization. SIAM J. Control Optim. 37(4) (1999), pp. 1251-1272] for elliptic boundary value problems. Topological derivatives are given for energy functionals of linear boundary value problems. We present results, including numerical examples, which confirm that the application of topological derivatives in the framework of the level set method really improves the efficiency of the method. Examples show that the level set method combined with the asymptotic analysis is robust for the shape optimization problems, and it allows us to identify the better solution compared to the pure level set method exclusively based on the boundary variation technique.