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 one-dimensional finite element magnetodynamic problem across half the lamination thickness. This 1D model is then able to deliver mesoscocpic information to be used, after appropriate homogenization, in the macroscopic modelling of an electrical machine or transformer. As each evaluation of such a homogenised model implies a finite element simulation at the mesoscale, a monolithic implementation of this method can become very time-consuming. This paper proposes an alternative methodology, assuming a periodic excitation of the system, where the homogenized material law is implemented with techniques of machine learning. The identified law is then used as a conventional constitutive relationship in the 2D or 3D modelling of an electrical machine or a transformer.