Homogenization of Thin Sheets: Static Magnetic Field Problems and Effective Materials

Abstract

The simulation of eddy currents in thin iron sheets is still a challenging task. The thickness of the sheets is in the range of less than a millimeter, while the overall dimensions are in the range of meters. Modeling each single sheet by standard finite element methods leads to an extremely large nonlinear system of equations, which cannot reasonably be solved with present computer capacities. Homogenization treats laminated materials as a bulk medium, and thus avoids the need to model the multiscale problem. This already reduces the computational effort drastically. To this end an effective material is determined based on solving proper cell problems at negligible computational costs. The effective material is complex-valued to cover the eddy current losses too. Instead of solving the basic eddy current problem, a related static magnetic field problem with the effective material is solved. In contrast to multiscale methods, the static magnetic field problem can in most cases be described with only a scalar potential, which significantly reduces the complexity of the implementation and also considerably reduces the required computational resources. Three different applications are discussed to show the methods wide applicability. It efficiently provides accurate fictitious field distributions and thus losses and reactive powers.