Topological derivative for steady-state orthotropic heat diffusion problem

Abstract

The aim of this work is to present the calculation of the topological derivative for the total potential energy associated to the steady-state orthotropic heat diffusion problem, when a circular inclusion is introduced at an arbitrary point of the domain. By a simple change of variables and using the first order Pólya-Szegö polarization tensor, we obtain a closed formula for the topological sensitivity. For the sake of completeness, the analytical expression for the topological derivative is checked numerically using the standard Finite Element Method. Finally, we present two numerical experiments showing the influency of the orthotropy in the topological derivative field and also one example concerning the optimal design of a heat conductor.