TOPOLOGICAL DERIVATIVE FOR STEADY-STATE ORTHOTROPIC HEAT DIFFUSION PROBLEM S.M. GIUSTI, A.A. NOVOTNY, AND J. SOKO LOWSKI 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 P´olya-Szeg¨o 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. 1. Introduction The topological sensitivity analysis gives the topological asymptotic expansion of a shape functional with respect to an infinitesimal singular domain perturbation, like the insertion of holes, inclusions, source-term or cracks. The main term of this expansion, called topological derivative ([12, 31, 10]), is now of common use in numerical procedures of resolution for topology optimization ([4, 22]), image processing ([20, 7, 21]) and inverse problems ([14, 6, 9]). Concerning the theoretical development of the topological asymptotic analysis, the reader may refer to [26], for instance. We refer the reader to [1, 28, 15] and [16], for the numerical methods of shape and topology optimization which include the topological derivatives in the numerical procedure of the levelset type. In order to introduce these concepts, let us consider an open bounded domain Ω ⊂ R 2 , which is submitted to a non-smooth perturbation in a small region ω ε ( x)= εω of size ε with center at an arbitrary point  x ∈ Ω. Thus, we assume that a given shape functional ψ admits the following topological asymptotic expansion ψ(Ω ε )= ψ(Ω) + f (ε)D T ( x)+ o(f (ε)) , (1.1) where Ω ε is the topologically perturbed domain and f (ε) is a positive function that decreases monotonically such that f (ε) → 0 when ε → 0. Then, the term D T ( x) is defined as the topological derivative of ψ. Therefore, this derivative can be seen as a first order correction on ψ(Ω) to estimate ψ(Ω ε ). In addition, from (1.1), we have that the classical definition of the topological derivative is given by D T ( x) = lim ε→0 ψ(Ω ε ) − ψ(Ω) f (ε) . (1.2) On the other hand, in the work of [31], the topological sensitivity associated to the nucleation of a hole in a domain characterized by an orthotropic material was calculated. In order to simplify the analysis, the domain was perturbed introducing an elliptical hole oriented in the directions of the orthotropy and with semi-axis proportional to the material properties coefficients in each orthogonal direction. In this paper, we extend the above result considering as perturbation a small circular inclusion of size ε of the same nature as the bulk material (see Fig.1), instead of an elliptical hole. In summary, we present the calculation of the topological derivative for the total potential energy associated to the steady-state orthotropic heat diffusion problem, considering the nucleation of a small circular inclusion. Key words and phrases. Topological asymptotic analysis, steady-state orthotropic heat diffusion, topological derivative, polarization tensor. 1