Topological sensitivity analysis in heterogeneous anisotropic elasticity problem. Theoretical and computational aspects

The topological sensitivity analysis for the heterogeneous and anisotropic elasticity problem in two-dimensions is performed in this work. The main result of the paper is an analytical closed-form of the topological derivative for the total potential energy of the problem. This derivative displays the sensitivity of the cost functional (the energy in this case) when a small singular perturbation is introduced in an arbitrary point of the domain. In this case, we consider a small disc with a completely different elastic material. Full mathematical justification for the derived formula, and derivations of precise estimates for the remainders of the topological asymptotic expansion are provided. Finally, the influence of the heterogeneity and anisotropy is shown through some numerical examples of structural topology optimization.