Topological Derivative of the Thermo-Electro-Mechanical Coupled Problem

Abstract

In this work, we perform a topological asymptotic analysis of a particular case of thermo-electro-mechanical problem known as the coupled Joule-heating with thermal expansion problem. The objective is to obtain a closed formula of the associated first-order topological derivative. This result is useful in topology optimization since it can be used to obtain optimum designs of thermo-electro-mechanical devices, such as Micro-Electro-Mechanical Systems (MEMS). The topological derivative is obtained by means of a Lagrangian technique for a particular class of cost functionals considering regular and circular perturbations of the material properties distribution. A numerical procedure for validating the analytical expression of the obtained topological derivative is performed. Good concordance between the numerical approximation and the analytical expression has been obtained. Finally, we provide a full mathematical justification for the derived expressions and develop precise estimates for the remainder of the topological asymptotic expansion.