Regular Domain Perturbation

This chapter deals with the topological derivative of the so-called compliance shape functional associated with a modified Helmholtz problem, with respect to the nucleation of a small circular inclusion with different material property from the background. By taking into account the boundary value problem we are dealing with, three different cases are considered: (1) perturbation on its right-hand side, (2) perturbation on the lower order term, and (3) perturbation on the higher order term. The existence of the associated topological derivatives is ensured by using simple arguments from the analysis. Then, we derive their explicit forms which are useful for numerical methods in shape/topology optimization. Finally, a priori estimates for the remainders left in the topological asymptotic expansions are rigorously obtained, which are used to justify the obtained results.