Magnetic Field Conforming Multiscale Formulations for Locally-Confined Nonlinear Eddy Current Problems Using the FE-HMM Method

Abstract

Magnetic composites used for the conversion of electrical energy often incorporate ferromagnetic inclusions insulated from each other to mitigate eddy current losses. Numerical models for these composites must be robust enough to address potential convergence issues arising from the presence of nonlinear magnetic inclusions and presence of significant confined eddy currents in the cell. This paper introduces an $\mathbf{h}$-conforming multiscale formulation for magnetic composites in a periodic framework. The proposed method uses the Heterogeneous Multiscale Method (HMM) with two mesoscale problems: a magnetoquasistatic problem for upscaling the homogenized magnetic flux density $\mathbf{B}_M$ and a magnetostatic problem for upscaling the macroscale incremental reluctivity $(\partial \mathbf{B}_M/\partial \mathbf{H}_M)$. Additionally, the method uses relaxed Newton--Raphson schemes at both macro and mesoscale levels to mitigate the well-known NR convergence issue linked to nonlinear BH constitutive laws in $\mathbf{h}$-conforming formulations. The accuracy and performance of the formulation are evaluated using 2D and 3D idealized periodic soft magnetic composites with linear and nonlinear BH curves. Furthermore, the paper demonstrates that the magnetoquasistatic mesoscale problem can be replaced by a magnet a