Topological derivative for an anisotropic and heterogeneous heat diffusion problem

The topological derivative measures the sensitivity of a given shape functional with respect to an infinitesimal singular domain perturbation. According to the literature, the topological derivative has been fully developed for a wide range of physical phenomenon modeled by partial differential equations, considering homogeneous and isotropic constitutive behavior. In fact, only a few works dealing with heterogeneous and anisotropic material behavior can be found in the literature, and, in general, the derived formulas are given in an abstract form. In this work, we derive the topological derivative in its closed form for the total potential energy associated to an anisotropic and heterogeneous heat diffusion problem, when a small circular inclusion of the same nature of the bulk phase is introduced at an arbitrary point of the domain. In addition, we provide a full mathematical justification for the derived formula and develop precise estimates for the remainders of the topological asymptotic expansion. Finally, the influence of the heterogeneity and anisotropy are shown through some numerical examples of heat conductor topology optimization.