Numerical upscaling for the eddy-current model with stochastic magnetic materials

This paper deals with the upscaling of the time-harmonic Maxwell equations for heterogeneous media. We analyze the eddy-current approximation of Maxwell’s equations to describe the electric field for heterogeneous, isotropic magnetic materials. The magnetic permeability of the materials is assumed to have random heterogeneities described by a Gaussian random field. We apply the so-called Coarse Graining method to develop a numerical upscaling of the eddy-current model. The upscaling uses filtering and averaging procedures in Fourier space which results in a formulation of the eddy-current model on coarser resolution scales where the influence of sub-scale fluctuations is modeled by effective scale- and space-dependent reluctivity tensors. The effective reluctivity tensors can be obtained by solving local partial differential equations which contain a Laplacian as well as a curl–curl operator. We present a computational method how the equation of the combined operators can be discretized and solved numerically using an extended variational formulation compared to standard discretizations. We compare the results of the numerical upscaling of the eddy-current model with theoretical results of Eberhard [J.P. Eberhard, Upscaling for the time-harmonic Maxwell equations with heterogeneous magnetic materials, Physical Review E 72 (3), (2005)] and obtain a very good agreement.