Asymptotic analysis and topological derivatives for shape and topology optimization of elasticity problems in two spactial dimensions

Abstract

Topological derivatives for elasticity problems are used in shape and topology optimization in structural mechanics. We propose an approach to the asymptotic analysis of singular perturbations of geometrical domains. This approach can be used in order to determine the exact solutions of elasticity boundary value problems in domains with small holes, and determine the explicit asymptotic expansions of solutions with respect to small parameter which describes the radius of internal hole. The elastic potentials of Muskhelishvili gives us an explicite solution in the ring $C(\rho,R)=\{\rho < |x| < R \}$ in the form of complex valued series. The series depends on the small parameter, the radius $\rho$ of the ring, and we are interested in the behavior of the series for the passage $\rho\to 0$. Such analysis leads to the expansion of the elastic energy in the form $$\mathcal{E}(\rho,R)=\mathcal{E}(0,R)+\rho^2\mathcal{E}^1(R)+\rho^4\mathcal{E}^2(R)+\dots\ ,$$ where $\mathcal{E}^1(R)$ is used to determine the first order topological derivatives of shape functionals, and $\mathcal{E}^2(R)$ can be used to determine the second order topological derivatives of shape functionals. In the paper the first order term $\mathcal{E}^1(R)$ is given, however the method is general and can be used to determine the subsequent terms of the energy expansion and the topological derivatives of higher order