Multiharmonic multiscale modelling in 3-D nonlinear magnetoquasistatics: Composite material made of insulated particles

The use of the classical finite element method (FEM) to solve problems with magnetic composites leads to huge linear systems that are impossible to solve. Instead, homogenization and multiscale methods are often used with the composite material replaced by a homogeneous material with the homogenized constitutive law obtained by solving cell-problems representing the mesoscale material structure. For non-linear time-dependent problems, FEM is often used with a time-transient method (TTM) and the solution is obtained one time-step at a time. However, in cases where a steady-state solution is of interest, the multiharmonic method can be faster and more cost effective for the same accuracy of the time discretization. In addition, when solving magnetoquasistatic multiscale problems with TTM, the dynamic hysteresis in the homogenized fields can slow down or even impede the convergence of the macro-scale problem due to the possibly non-continuously differentiable homogenized material laws. This work presents a novel robust modelling approach for non-linear magnetoquasistatic problems combining multiharmonic method with the multiscale method.