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.

翻译：本文致力于电机内部结构的形状优化，具体而言，通过优化转子内部空气与铁磁材料的布局以提升电机转矩。控制物理问题为时变非线性准静态磁场的麦克斯韦方程组。该多物态问题可在真实三维圆柱构型的二维截面上重新表述；然而，由于电机旋转，各材料物态（铁磁材料、永磁体、空气等）的几何形状在时间周期内遵循预定运动。这一独特设定引出了若干问题。在理论层面，我们证明了这一具有移动几何特征的非线性演化问题的适定性。随后，我们通过相应的准静态磁位计算了依赖于铁磁相形状的性能准则的形状导数。解决该问题的数值框架基于形状梯度算法。在时域内采用牛顿-拉夫森法求解准静态磁位的非线性时间周期演化问题。离散化采用时空有限元方法，应用于所关注时空区域的精确网格化表示，该区域在时间周期的所有阶段均包含电机各材料物态的贴体表示。在通过学术问题评估数值框架效率后，我们展示了电机转子铁磁相优化设计的实际案例。