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

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