Robust Topology Optimization of Electric Machines using Topological Derivatives

TL;DR

The paper addresses robust design of electric machines under material and operating uncertainties by marrying PDE‑constrained topology optimization with a level‑set/topological derivative framework. It solves a min–max problem using Clarke’s generalized gradient to handle inner worst‑case maximization and employs adjoint‑based sensitivities for efficient outer optimization. Theoretical results are provided for linear PDE constraints, and a practical algorithm with offline TD sampling is demonstrated on a two‑material PMSM, showing improved worst‑case torque with modest nominal performance trade‑offs. The approach promises more reliable, manufacturable machine designs under uncertainty and can extend to other PDE‑driven design problems with robust requirements.

Abstract

Designing high-performance electric machines that maintain their efficiency and reliability under uncertain material and operating conditions is crucial for industrial applications. In this paper, we present a novel framework for robust topology optimization with partial differential equation constraints to address this challenge. The robust optimization problem is formulated as a min-max optimization problem, where the inner maximization is the worst case with respect to predefined uncertainties, while the outer minimization aims to find an optimal topology that remains robust under these uncertainties using the topological derivative. The shape of the domain is represented by a level set function, which allows for arbitrary perturbation of the domain. The robust optimization problem is solved using a theorem of Clarke to compute subgradients of the worst case function. This allows the min-max problem to be solved efficiently and ensures that we find a design that performs well even in the presence of uncertainty. Finally, numerical results for a two-material permanent magnet synchronous machine demonstrate both the effectiveness of the method and the improved performance of robust designs under uncertain conditions.

Figures (14)

• Figure 1: Example of a function $g(x,q)$ with worst case $\max_{q\in U_q}$ in black (left); worst case w.r.t. $q$ (black) and w.r.t $(\delta x,q)$, i.e. $f(x)=\max_{(\delta x,q)\in (U_x,U_q)}g(x+\delta x,q)$ (right).
• Figure 2: Left: 2d cross section of permanent magnet synchronous machine: Rotor and stator iron in red, coils in yellow with phase distribution (A: red, B: blue, C: green), permanent magnets with magnetization direction in light blue and light green, airgap, airpockets and shaft in blue. Right: One pole of the machine, the computational domain $D_\mathrm{all}$. Design domain $D$ (dashed area) consisting of rotor iron $\Omega_f\subseteq D$ (red) and rotor air $\Omega_a\subseteq D$ (blue). Rotor $D_R$ consisting of shaft $D_\mathrm{shaft},$ design domain $D$ and permanent magnets $D_\mathrm{magnet1}, D_\mathrm{magnet2}$ with outer boundary $\Gamma_R.$ Stator $D_S$ consisting of airgap $D_\mathrm{airgap}$, iron $D_\mathrm{stator}$ and coils $D_{A^+},D_{B^-},D_{C^+}$ with outer boundary $\Gamma_S$. Boundaries of pole $\Gamma_1,\Gamma_2$.
• Figure 3: Some design domain $D$ consisting of two materials air in $\Omega_a$ and iron in $\Omega_f$. Perturbation of $\Omega_f$ by putting air in $\omega_\epsilon(z)$.
• Figure 4: Final design of nominal optimization $\Omega_\mathrm{NOM}$, $\mathcal{J}(\Omega_\mathrm{NOM})=-835\textrm{Nm}$.
• Figure 5: Final design $\Omega_\mathrm{ANG}$ (left), difference from nominal result $\Omega_\mathrm{NOM}$ (right) considering an uncertain load angle, see Section \ref{['subsec:ANG']}.

Theorems & Definitions (35)

• Remark 1
• Remark 2
• Remark 3
• Definition 1
• Remark 4
• Remark 5
• Remark 6
• Lemma 1
• proof
• Remark 7