Homogenization of Quasi-static Maxwell's Equations

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 106. 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 macro-scale models for the quasi-static Maxwell’s equations. We prove that the microscale solutions converge to the solutions of the macro-scale models weakly in H(curl, Ω) and strongly in L(Ω) as the thickness of lamination tends to zero. The wellposedness 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 21c–M1. The numerical results show good agreements with the experimental data and validate the homogenized model.