Topological derivative for the nonlinear magnetostatic problem

Abstract

The topological derivative represents the sensitivity of a domain-dependent functional with respect to a local perturbation of the domain and is a valuable tool in topology optimization. Motivated by an application from electrical engineering, we derive the topological derivative for an optimization problem which is constrained by the quasilinear equation of two-dimensional magnetostatics. Here, the main ingredient is to establish a sufficiently fast decay of the variation of the direct state at scale 1 as |x| → ∞. In order to apply the method in a bi-directional topology optimization algorithm, we derive both the sensitivity for introducing air inside ferromagnetic material and the sensitivity for introducing material inside an air region. We explicitly compute the arising polarization matrices and introduce a way to efficiently evaluate the obtained formulas. Finally, we employ the derived formulas in a level-set based topology optimization algorithm and apply it to the design optimization of an electric motor.

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

“…In this section, we show a way to approximately evaluate formula (4.2) by first precomputing certain values in an offline phase and looking them up and interpolating them during the online phase of the optimization algorithm. We proceed in an analogous way to [4], Section 7.…”

“…The magnetization is supported in the permanent magnets Ω 2 ⊂ D ∖ Ω. In this particular application, which was also treated in a two-dimensional setting in [4,14], we assume the currents to be switched off, i.e., = 0 and therefore treat Ω 1 as air.…”

“…For 1 , items (i) and (ii) of Assumption A follow immediately from items (i) and (ii) of Assumption C, respecitively (see e.g., [25]). Moreover, it is shown in [4], Lemma 3.7 that Assumption C(iii) implies that 2 is twice continuously differentiable, which is sufficient for Assumption A(iii).…”

“…Concerning quasi-linear problems, in which the topological perturbation enters in the main part of the nonlinearity, even less work has been done. Here we mention [3] where the authors consider a regularised version of the -Poisson equation and also [4] where the topological derivative for the quasi-linear equation of 2D magnetostatics was derived. More recently, in [13] the topological derivative for a class of quasi-linear equations under fairly general assumptions in an 1 setting was presented.…”

“…The topological sensitivity of 2D nonlinear magnetostatics, which is a simplification of Maxwell's equation in 3D, was treated in [4]. The topological sensitivity of three dimensional linear Maxwell's equations has been studied in [22] and is based on asymptotics derived in [1].…”