The development of technologies for the additive manufacturing, in particular of metallic materials, is offering the possibility of producing parts with complex geometries. This opens up to the possibility of using topological optimization methods for the design of electromagnetic devices. Hence, a wide variety of approaches, originally developed for solid mechanics, have recently become attractive also in the field of electromagnetics. The general distinction between gradient-based and gradient-free methods drives the structure of the paper, with the latter becoming particularly attractive in the last years due to the concepts of artificial neural networks. The aim of this paper is twofold. On one hand, the paper aims at summarizing and describing the state-of-art on topology optimization techniques while on the other it aims at showing how the latter methodologies developed in non-electromagnetic framework (e.g., solid mechanics field) can be applied for the optimization of electromagnetic devices. Discussions and comparisons are both supported by theoretical aspects and numerical results.

A fast and general partial element equivalent circuit (PEEC) method based on the fast Fourier transform (FFT) is proposed for the first time. The numerical tool only requires common CAD data input files (e.g., $.\text{stl}$ format), then the discretization process is performed automatically by means of a fast voxelization technique based on ray intersection, thus, drastically reducing the human effort required to setup the model. The method allows for considering at the same time inductive and capacitive effects, and is focused on power electronics applications where propagation effects can be neglected, whereas all the other electromagnetic phenomena are considered. Specifically, the proposed method is particularly suited for problems where both electric and magnetic fields are equally important and, therefore, quasistatic approximations do not apply. An ad-hoc preconditioner which significantly speeds-up the solver is also proposed and, thanks to the FFT, both memory and computation time are significantly reduced, without the need of applying data compression. Both linear and nonlinear materials are considered by the proposed FFT-PEEC method. Sample implementation of the method is made publicly available.

Topology optimization for the design of electromagnetic devices has recently garnered significant interest, thanks to advancements in additive manufacturing techniques that enable the fabrication of intricate geometries. This has opened up new possibilities for utilizing integral equation methods to solve electromagnetic (EM) problems, particularly in the analysis of Inductive Power Transfer (IPT) devices. In this context, we introduce a novel topology optimization tool called TopIE, which is built upon the Integral Equation method for EM problem solutions. TopIE adopts the logic of Topology Optimization of Binary Structures (TOBS), a sensitivity-based approach that leverages binary design variables to clearly differentiate the material properties within the design domain. The tool is specifically employed for optimizing IPT devices, and sample implementations of the method are made publicly available. By combining the power of topology optimization and integral equation methods, TopIE offers a promising avenue for enhancing the design and performance of EM devices.

A volume integral (VI) equation method based on the fast Fourier transform (FFT) is proposed for the electromagnetic simulation of complex systems in large magnetic confinement fusion (MCF) devices. Such FFT-VI allows for dramatically reducing the time and the human effort required to adapt the CAD input files to models suited for electromagnetic simulations. Moreover, with FFT-VI the mesh generation is performed automatically and is very efficient and robust. The adoption of the FFT allows for significantly reducing the computation time and the memory requirements, therefore making simulations with several millions of unknowns feasible. Numerical results show that, in terms of numerical performances, the FFT-VI method applied to challenging electromagnetic problems in representative MCF devices is markedly superior with respect to the state of the art VI techniques. A sample implementation of FFT-VI is made publicly available.

This article presents an integral equation method based on the current–potential ( $\mathbf {J}-\varphi _{e}$ ) formulation, for the solution of the eddy currents problem in superconducting structures. Two efficient methods for storing and manipulating the fully populated matrices resulting from the discretization of the integral equation are proposed in this work. These approaches leverage hierarchical ( $\mathcal {H}$ ) matrices and the fast Fourier transform technique, tailored for voxelized structures. These techniques drastically reduce the computational cost of the numerical simulations. The proposed approach is compared to existing methods, as the $\mathbf {H}-\varphi _{m}$ formulation, and is efficiently applied to calculate the power losses for a realistic high-temperature superconductor coil.