Optimality Conditions with Topological Derivatives

In classical theory of shape optimization the first order necessary optimality conditions account for boundary variations of an optimal domain. On the other hand the relaxed formulation based on homogenization technique is used (Allaire et al, Numer Math 76(1):27–68, 1997, [5], Bendsoe, Optimization of structural topology, shape, and material, 1995, [46], Lewinski and Telega, Plates, laminates and shells, 2000, [173] in the topology optimization of energy functionals, the so called compliance in structural optimization. For such a formulation the coefficients of an elliptic operator are selected in an optimal way and the resulting optimal design takes the form of a composite microstructure rather than any geometrical domain.