Topology optimization method with respect to the insertion of small coated inclusion

Article type: Research Article

Authors: Belhachmi, Z.^a | Ben Abda, A.^b | Meftahi, B.^b | Meftahi, H.^{c; *}

Affiliations: [a] Mathematics, Information Technology and Applications Laboratory (LMIA, EA 3993), 4 rue des Frères Lumière, 68093 Mulhouse, Cedex, France. E-mail: zakaria.belhachmi@uha.fr | [b] Laboratoire de Modélisation Mathématiques et Numériques dans les Sciences de L’ingénieurs, Université Tunis El-Manar, Tunis. E-mails: amel.benabda@enit.rnu.tn, belhassenmeftahi@hotmail.fr | [c] Technical University of Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany. E-mail: meftahi@math.tu-berlin.de

Correspondence: [*] Corresponding author.E-mail: meftahi@math.tu-berlin.de.

Abstract: We consider the two-dimensional steady-state heat conduction on a bounded domain Ω containing a heat source at an unknown location ω⊂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 L2 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

DOI: 10.3233/ASY-171441

Journal: Asymptotic Analysis, vol. 106, no. 2, pp. 99-119, 2018