Complete topological asymptotic expansion for $L_2$ and $H^1$ tracking-type cost functionals in dimension two and three

In this paper, we study the topological asymptotic expansion of a topology optimisation problem that is constrained by the Poisson equation with the design/shape variable entering through the right hand side. Using an averaged adjoint approach, we give explicit formulas for topological derivatives of arbitrary order for both an $L_2$ and $H^1$ tracking-type cost function in both dimension two and three and thereby derive the complete asymptotic expansion. As the asymptotic behaviour of the fundamental solution of the Laplacian differs in dimension two and three, also the derivation of the topological expansion significantly differs in dimension two and three. The complete expansion for the $H^1$ cost functional directly follows from the analysis of the variation of the state equation. However, the proof of the asymptotics of the $L_2$ tracking-type cost functional is significantly more involved and, surprisingly, the asymptotic behaviour of the bi-harmonic equation plays a crucial role in our proof.