Multi-Scale FEM for the Eddy Current Problem in Ferromagnetic Laminated Media

The simulation of eddy currents in laminated iron cores by the finite element method (FEM) is of great interest in the design of electrical machines and transformers. The iron core is made of ferromagnetic grain oriented laminates. The material properties are anisotropic and exhibit a magnetic hysteresis. The scales vary from the meter range for the iron core to the thickness of single laminates (typically in the range of 0.2-0.3mm). Clearly, modeling each laminate individually is not a feasible solution. Many finite elements (FEs) have to be used in such a model leading to extremely large nonlinear systems of equations. That`s why an accurate simulation of eddy currents and the iron losses in laminated ferromagnetic cores with reasonable computer resources is by far not solved satisfactorily. It is still one of the major challenges in computational electromagnetics. Laminated cores represent a periodic microstructure and therefore are well suited for FEM with homogenization. Simulations with FEM and homogenization show a boundary layer quite similar to that which occurs in corresponding brute force models of such cores with anisotropic material properties. An accurate approximation of the boundary layer is essential for an exact evaluation of the iron losses. However, many FE layers are required, which considerably increases the total number of FEs in the model. The periodic nature of the lamination is interrupted by step lap joints or ventilation ducts or disturbed by skewing leading to complex geometries which are costly in the FE modeling on its own. An accurate approximation by the FEM with standard polynomials also in case of equations with rough coefficients, for instance materials with a microstructure, and problems with a boundary layer, requires extremely fine meshes. Therefore, we will develop new multiscale finite element methods (MSFEM) to cope with the microstructure, where the standard polynomial basis is augmented by special functions incorporating a priori information into the ansatz space to avoid fine FE meshes. Then, the MSFEM will be combined with the harmonic balance method to reduce the computational costs furthermore. To provide a comprehensive solution for the present topic, approaches for the boundary layer and for the above geometrical difficulties will be designed and integrated into MSFEM. Hysteresis will be considered by a Preisach model. Fast adapted numerical integration methods, a very important issue for an efficient MSFEM, will be developed which do not affect the accuracy of the approximation. All approaches will be developed for the time and frequency domain and for both potential formulations, the magnetic and the current vector potential. All new MSFEM approaches will be incorporated into the open source hp-FEM code Netgen/NGSolve. A benchmark to provide measured data and the supercomputer VSC to compute very expensive reference solutions will ensure an optimal development of the new MSFEM approaches. The aim is to create highly accurate numerical solutions consuming minimal computer resources to run on personal computers without any difficulty.

An accurate simulation of electrical machines and transformers in the design phase is chal- lenging, because their iron cores are composed of a large number of very thin iron sheets. The finite element method (FEM) is used for the simulation. Due to many sheets and the large dimensions of electrical devices, the FEM leads to extremely large equation systems, which can be solved by a super computer at the best. The aim of the project was to improve the standard FEM such that these problems can be solved on a personal computer. Due to the many sheets, the iron core is effectively a fine periodic structure. The developed multiscale FEMs use this property to drastically reduce the extremely large equation systems so that they can be solved on a personal computer. An optimal design of electrical machines and transformers means less iron losses in terms of heat and a smaller material consumption in the production, leading to a more environmentally friendly production, transport and conversion of electrical energy.