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Abstract

We consider the two-dimensional steady-state heat conduction on a bounded domain Ω containing a heat source at an unknown location $ω \subset Ω$ . We are interested in determining the locations ω allowing a suitable thermal environment. The resulting shape optimization problem consists of a geometric control of either the maximum of the temperature or its $L^{2}$ mean oscillations in Ω. We derive the topological asymptotic expansion of the considered shape functionals with respect to the insertion of small circular coated inclusion characterized by a discontinuous thermal conductivity and their radius. We propose a reconstruction algorithm based on this topological gradient to identify the locations. Some numerical simulations are presented to show the efficiency of the algorithm.

Keywords

Topological optimization asymptotic analysis coated inclusion heat conduction

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