Homogenization of Quasi-static Maxwell's Equations

Abstract

This paper studies the homogenization of quasi-static and nonlinear Maxwell's equations in grain-oriented (GO) silicon steel laminations. GO silicon steel laminations have multiple scales, and the ratio of the largest scale to the smallest scale can be up to $10^6$. Direct solution of three-dimensional nonlinear Maxwell's equations is very challenging and unrealistic for large electromagnetic devices. Based on the magnetic vector potential and the magnetic field, respectively, we propose two macroscale models for the quasi-static Maxwell's equations. We prove that microscale solutions converge to the solutions of the macroscale models weakly in $\boldsymbol{H}(\mathbf{curl},\Omega)$ and strongly in $\boldsymbol{L}^2(\Omega)$ as the thickness of lamination tends to zero. The well-posedness of the homogenized model is established by using weighted norms. Numerical experiments are carried out for a benchmark problem from the International Compumag Society, TEAM Workshop Problem $21^c$-M1 [Z. Cheng, N. Takaha...