Multiphysical Shape Optimization of Electrical Machines

Electrical machines are an integral part of our everyday life. Depending on the application, the machine should be designed in such a way that certain criteria such as a smooth rotation or a high average torque, which can be encoded into an objecitve functional, are satisfied as well as possible. These criteria depend on the magnetic field inside the motor, which also depends on the geometry of the motor. The goal of this project is to find a shape of certain parts of an electric motor which is optimal with respect to a given objective function in an efficient way while considering additional aspects. The magnetic field inside the motor is created, on the one hand, by means of the electric current which is induced in the coil areas of the motor, and on the other hand by permanent magnets inside the motor. The properties of these magnets depend on the current temperature. Conversely, the magnetic field (the so-called eddy currents) serve as a heat source. Moreover, also the mechanical behavior of the machine is of utter importance. Mechanical deformations are caused by two kinds of forces: on the one hand, by the thermal expansion due to the mentioned heating, and, on the other hand, by the rotation of the machine. If these forces are too high, the machine might break. Thus, there is a coupling between electromagnetic, thermal and mechanical fields which can be described mathematically by means of a coupled system of time-dependent partial differential equations. The goal of the project is to find a design for a given electrical machine that is optimal with respect to a given objective functional. Further criteria can be treated within the optimization problem at hand in the form of constraints. It is very important that the maximal occurring mechanical stresses do not exceed a given value. Taking care of this issue is a mathematically particularly delicate challenge since the corresponding objective functional is not differentiable. In the course of an optimization algorithm, based on analytically computed sensitivities, a given initial geometry is successively improved until a locally optimal design has been reached. Thereby, the time-dependent system of partial differential equations must be solved many times, thus making an efficient solution strategy indispensable. The idea of space-time methods for partial differential equations is to consider the time variable as an additional space variable, turning a two-dimensional time-dependent problem into a three-dimensional static problem. This way, it is possible to perform parallelization not only with respect to the space variables, but also with respect to the time variable. Using sufficiently many cores on a parallel computer, this yields a tremendous reduction in computational time.

Electric machines-including motors and generators-play a vital role in modern society. They convert electrical energy into mechanical energy, or vice versa, and account for over half of the world's electricity consumption. Given their significant role, enhancing the efficiency of electric machines is essential for advancing global climate objectives. Conventional design practices for electric machines involve selecting an initial configuration and modifying specific parameters such as radii, angles, or widths. However, such approaches restrict the range of achievable designs. In this project, more advanced mathematical methods, based on free-form shape optimization, were developed to allow machine shapes to evolve much more flexibly. This approach enables the discovery of highly efficient designs that may not have been found through conventional methods. We developed new mathematical theory and numerical techniques to optimize the shape of electric machines, with a focus on time-dependent phenomena like eddy currents. These currents can cause unwanted heating inside the machine, potentially leading to material damage. The machines were modeled in two spatial dimensions over time, forming a three-dimensional "space-time cylinder" on which the electromagnetic equations were solved using a space-time finite element method. Because electric machines typically operate with rotating parts, the methods were extended to handle moving geometries as well. A central element of the optimization process was the calculation of the shape derivative, which measures how performance changes when the machine's shape is slightly deformed. We adapted the computation of this derivative as well as the numerical optimization procedure to the particular time-dependent setting under movement, which can also be transferred to other engineering applications. In addition to electromagnetic and thermal efficiency, mechanical stability is crucial. Machines rotating at several thousand revolutions per minute are subject to significant mechanical stresses, and designs optimized solely for efficiency or thermal behavior may become structurally unstable. To address this challenge, the project treated the minimization of maximum mechanical stresses within a structure. This posed a particular challenge because the maximum stress is a non-smooth function and requires specialized mathematical techniques beyond classical optimization. Finally, methods were developed to not only identify designs that perform optimally with respect to a single cost function, but to find a set of designs, known as Pareto sets, representing the best possible trade-offs between several criteria such as efficiency, cost or mechanical stability. In contrast to conventional industrial workflows that involve evaluating a large number of candidate designs, derivative-based techniques were investigated that can explore these sets with significantly less computational effort. The outcomes of this project may contribute to the development of electric machines that are more efficient, durable, and environmentally sustainable.