Multiscale FEM for the Linear 2-D/1-D Problem of Eddy Currents in Thin Iron Sheets

Abstract

A novel 2-D/1-D approach to simulate the eddy currents in a single thin iron sheet is presented. The introduced method utilizes ideas of the multiscale finite-element method by reducing the 3-D problem to a 2-D one. This is achieved via a decomposition of the solution with respect to its dependence on the coordinate directions. The decomposition is approximated using an expansion into polynomial shape functions. Integration over the respective coordinate gives an explicit 2-D problem. This eliminates the need to iteratively solve two coupled problems, as it is done in conventional 2-D/1-D methods. As another important advantage, the presented method allows for the incorporation of air gaps between the steel sheets. This can be done “for free,” i.e., without the introduction of additional unknowns and without changing the complexity of the resulting problem. The derivation is shown in detail for both the magnetic vector potential and the current vector potential formulation, giving explicit formulas for low polynomial orders of the used shape functions and illustrating how to proceed for shape functions of arbitrary degree. Finally, the developed method is tested in two numerical examples utilizing different geometries for the iron sheet and the computational efficiency is studied via comparison with a suitable standard finite-element model.