Homogenized harmonic balance finite element method for nonlinear eddy current simulations of fast corrector magnets

TL;DR

The paper introduces HomHBFEM, a framework that marries laminated-yoke homogenization with harmonic balance finite element analysis to enable nonlinear eddy-current simulations of fast corrector magnets without resolving lamination thickness or skin depth in the FE mesh. By using frequency-dependent surrogate tensors for conductivity and reluctivity and a multi-harmonic solution, the method avoids costly time stepping and greatly reduces meshsize, achieving accurate predictions up to tens of kilohertz and validating against full 3D transient FE references. The authors demonstrate substantial computational efficiency gains (hours instead of days/weeks) while capturing key nonlinear effects on energy, flux densities, and eddy-current losses, including the impact of saturation and skin-depth phenomena. Application to PETRA IV FC magnets shows nonlinear effects can be mitigated by thinner laminations or materials with milder Rayleigh regions, making HomHBFEM a valuable tool for design and optimization of fast accelerator magnets and related laminated cores.

Abstract

This paper develops a homogenized harmonic balance finite element method (HomHBFEM) to predict the dynamic behavior of magnets with fast excitation cycles, including eddy current and skin effects. A homogenization technique for laminated yokes avoids resolving the individual laminates and the skin depth in the finite element (FE) mesh. Instead, the yoke is represented by a bulk surrogate material with frequency-dependent parameters. The ferromagnetic saturation of the yoke at higher excitation currents is tackled by a harmonic balance method, which accounts for a coupled set of frequency components. Thereby, a computationally expensive time-stepping of the eddy-current field problem and a convolution of the homogenized yoke model are avoided. The HomHBFEM enables, for the first time, to conduct nonlinear simulations of fast corrector magnets, which are embedded in a fast orbit feedback system to counteract orbit disturbances over a broad frequency spectrum, and thus guarantee a stable light-source operation. The results show the impact of the nonlinearity on the phase lag and the field attenuation as well as the eddy current losses at frequencies up to 65 kHz. The numerical validation for a C-dipole magnet example shows that the HomHBFEM achieves a sufficient accuracy at an affordable computational effort, with simulation times of a few hours. In comparison, standard 3D transient FE simulations need to resolve the lamination thickness and the skin depth in space and the largest relevant frequency in time, which leads to a two to three orders of magnitude larger mesh and prohibitive computational effort, with simulation times of a few weeks on a contemporary computer server.

Figures (25)

• Figure 1:
• Figure 2: Flow chart of the iterative procedure for the HomHBFEM. The index $i$ indicates the harmonic order and the superscript $k$ the iteration number. Blue boxes contain steps implemented in Python, red boxes steps in GetDP.
• Figure 3: 3D model of the laminated inductor.
• Figure 5: Magnetic energies in the ferromagnetic core computed with the HomHBFEM-1 at four fundamental frequencies of the source current, compared to the transient reference results.
• Figure 6: Magnetic energies in the ferromagnetic core computed with the HomHBFEM-2 at two fundamental frequencies of the source current, compared to the transient reference results.