Topological sensitivity analysis of inclusion in two-dimensional linear elasticity

The topological derivative gives the sensitivity of the problem when the domain under consideration is perturbed by the introduction of a hole. Alternatively, this same concept can also be used to calculate the sensitivity of the problem when, instead of a hole, a small inclusion is introduced at a point in the domain. In the present paper we apply the Topological-Shape Sensitivity Method to obtain the topological derivative of inclusion in two-dimensional linear elasticity, adopting the total potential energy as the cost function and the equilibrium equation as a constraint. For the sake of completeness, initially we present a brief description of the Topological-Shape Sensitivity Method. Then, we calculate the topological derivative for the problem under consideration in two steps: firstly we perform the shape derivative and next we calculate the limit when the perturbation vanishes using classical asymptotic analysis around a circular inclusion. In addition, we use this information as a descent direction in a topology design algorithm which allows to simultaneously remove and insert material. Finally, we explore this feature showing some numerical experiments of structural topology design within the context of two-dimensional linear elasticity problem.