Efficient evaluation of forward and inverse energy-based magnetic hysteresis operators

The energy-based vector hysteresis model of Francois-Lavet et al. establishes an implicit relation between magnetic fields and fluxes via internal magnetic polarizations which are determined by convex but non-smooth minimization problems. The systematic solution of these problems for every material point is a key ingredient for the efficient implementation of the model into standard magnetic field solvers. We propose to approximate the non-smooth terms via regularization which allows to employ standard Newton methods for the evaluation of the local material models while being in control of the error in this approximation. We further derive the inverse of the regularized hysteresis operator which amounts to a regularized version of the inverse hysteresis model. The magnetic polarizations in this model are again determined by local minimization problems which here are coupled across the different pinning forces. An efficient algorithm for solving the Newton systems is proposed which allows evaluation of the inverse hysteresis operator at the same cost as the forward model. Numerical tests on standard benchmark problems are presented for illustration of our results.