Solving 1D non-linear magneto quasi-static Maxwell's equations using neural networks

Electromagnetics (EM) can be described, together with the constitutive laws, by four PDEs, called Maxwell's equations. "Quasi-static" approximations emerge from neglecting particular couplings of electric and magnetic field related quantities. In case of slowly time varying fields, if inductive and resistive effects have to be considered, whereas capacitive effects can be neglected, the magneto quasi-static (MQS) approximation applies. The solution of the MQS Maxwell's equations, traditionally obtained with finite differences and elements methods, is crucial in modelling EM devices. In this paper, the applicability of an unsupervised deep learning model is studied in order to solve MQS Maxwell's equations, in both frequency and time domain. In this framework, a straightforward way to model hysteretic and anhysteretic non-linearity is shown. The introduced technique is used for the field analysis in the place of the classical finite elements in two applications: on the one hand, the B-H curve inverse determination of AISI 4140, on the other, the simulation of an induction heating process. Finally, since many of the commercial FEM packages do not allow modelling hysteresis, it is shown how the present approach could be further adopted for the inverse magnetic properties identification of new magnetic flux concentrators for induction applications.