A Material Law Based on Neural Networks and Homogenization for the Accurate Finite Element Simulation of Laminated Ferromagnetic Cores in the Periodic Regime

Electromagnetic fields and eddy currents in thin electrical steel laminations are governed by the laws of magnetodynamics with hysteresis. If the lamination is large with respect to its thickness, field and current distributions are accurately resolved by solving a 1-D finite element magnetodynamic problem with hysteresis across half the lamination thickness. This 1-D model is able to deliver mesoscopic information to be used, after appropriate homogenization, in the macroscopic modeling of an electrical machine or transformer. As each evaluation of such a homogenized model implies a finite element simulation at the mesoscale, a monolithic coupling might be very time-consuming. This article proposes an alternative approach, assuming a periodic excitation of the system, where the parameters of a parametric homogenized material law are determined in each finite element with a neural network. The local material law can then be used as a conventional constitutive relationship in a 2-D or 3-D modeling, with a massive speed-up with respect to the monolithic coupling.