Asymptotic analysis and topological derivative for 3D quasi-linear magnetostatics

In this paper we study the asymptotic behaviour of the quasilinear curl-curl equation of 3D magnetostatics with respect to a singular perturbation of the differential operator and prove the existence of the topological derivative using a Lagrangian approach. We follow the strategy proposed in Gangl and Sturm (ESAIM: COCV 26 (2020) 106) where a systematic and concise way for the derivation of topological derivatives for quasi-linear elliptic problems in H1 is introduced. In order to prove the asymptotics for the state equation we make use of an appropriate Helmholtz decomposition. The evaluation of the topological derivative at any spatial point requires the solution of a nonlinear transmission problem. We discuss an efficient way for the numerical evaluation of the topological derivative in the whole design domain using precomputation in an offline stage. This allows us to use the topological derivative for the design optimization of an electrical machine.