Space-time shape optimization of rotating electric machines

TL;DR

The paper addresses shape optimization of the rotor's ferromagnetic domain in a rotating electric machine under a nonlinear magneto-quasi-static model with moving geometry. It develops a space-time finite element framework, proves well-posedness for the moving-geometry MQS problem, and derives Hadamard-type shape derivatives via an adjoint approach, including linear and nonlinear extensions. The numerical workflow solves the state and adjoint problems on a space-time mesh with Newton iterations and uses a Hilbert-space-based descent to update the rotor interface, enabling simultaneous optimization across time. Demonstrations on an academic 1D example and a realistic rotating-machine setting show torque improvements and provide insights into computational costs, convergence, and potential enhancements such as multiphysics coupling and topology methods.

Abstract

This article is devoted to the shape optimization of the internal structure of an electric motor, and more precisely of the arrangement of air and ferromagnetic material inside the rotor part with the aim to increase the torque of the machine. The governing physical problem is the time-dependent, non linear magneto-quasi-static version of Maxwell's equations. This multiphase problem can be reformulated on a 2d section of the real cylindrical 3d configuration; however, due to the rotation of the machine, the geometry of the various material phases at play (the ferromagnetic material, the permanent magnets, air, etc.) undergoes a prescribed motion over the considered time period. This original setting raises a number of issues. From the theoretical viewpoint, we prove the well-posedness of this unusual non linear evolution problem featuring a moving geometry. We then calculate the shape derivative of a performance criterion depending on the shape of the ferromagnetic phase via the corresponding magneto-quasi-static potential. Our numerical framework to address this problem is based on a shape gradient algorithm. The non linear time periodic evolution problems for the magneto-quasi-static potential is solved in the time domain, with a Newton-Raphson method. The discretization features a space-time finite element method, applied on a precise, meshed representation of the space-time region of interest, which encloses a body-fitted representation of the various material phases of the motor at all the considered stages of the time period. After appraising the efficiency of our numerical framework on an academic problem, we present a quite realistic example of optimal design of the ferromagnetic phase of the rotor of an electric machine.

Figures (9)

• Figure 1: Two-dimensional cross-section $D$ of an electric machine of the form considered in \ref{['sec.setting']}: the ferromagnetic material is depicted in red, air is in dark blue, the coils are in yellow, and the permanent magnets are in light blue.
• Figure 2: The variation $\Omega_\theta$ of $\Omega$ featured in the method of Hadamard is obtained by displacing its points according to the "small" vector field $\theta$.
• Figure 3: One deformation $\theta$ of the shape $\Omega$, with the associated deformation $\Theta$ of the space-time cylinder $Q_\Omega$.
• Figure 4: (a) Right-hand side $f(t,x)$; (b) Space-time cylinder $Q$ associated to the initial design $\Omega^0$ in the 1d academic example of \ref{['sec.1dex']}; (c) Potential $u_{\Omega^0}$ associated to $\Omega^0$.
• Figure 5: (a) Descent direction $\Theta^0$ from the initial design $\Omega^0$, after extension of $\theta^0$ to the space-time cylinder; (b) Space-time cylinder $Q_{\Omega^*}$ associated to the optimized design $\Omega^*$; (c) Potential $u_{\Omega^*}$ for the optimized design $\Omega^*$.

Theorems & Definitions (34)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 3.1
• Remark 3.2
• Lemma 3.1
• proof
• Remark 4.1
• Theorem 4.1
• Lemma 4.1