Multiscale Finite Element Modeling of Nonlinear Quasistatic Electromagnetic Problems

The effective use of composite materials in the technology industry requires the development of accurate models. Typical such materials in electrotechnical applications are lamination stacks and soft magnetic composites, used in the so-called magnetoquasistatic (low frequency) regime.
Current homogenization models (e.g. the classical homogenization method, mean field homogenization, ...) fail to handle all the difficulties raised by the modeling of these materials, particularly taking into account the complexity of their microstructure and their nonlinear/hysteretic behaviour. In this thesis we develop a multiscale computational method which allows to effectively solve multiscale magnetoquasistatic problems.
The technique is inspired by the HMM (heterogeneous multiscale method), which involves the resolution of two types of problems: a macroscale problem that captures slow variations of the overall solution, and many mesoscale problems that allow to determine the constitutive laws at the macroscale and to construct accurate local fields. Macroscale and mesoscale weak, b-conform and h-conform formulations, are derived starting from the two-scale convergence and the periodic unfolding methods. We also use the asymptotic homogenization method for deriving the homogenized linear material laws and, in the end, we derive scale transitions for bridging the scales.
Numerical tests carried out in the two-dimensional case allow to validate the models. In the case of b-conform formulations, it is shown that the macroscale solution approximates well the average of the reference solution and that the resolution of the mesoscale problems allows to reconstruct accurate local fields and to compute accurate Joule losses and this, for materials with (non)linear and hysteretic behavior. Similar findings were obtained for the h-conform formulations.
In both cases, the deterioration of the accuracy for mesoscale problems located near the boundary of the computational domain could be treated by defining suit- able mesoscale problems near such boundaries. The extension of the model to three-dimensional problems, to multiphysical problems and the inclusion of the mesoscale domains with a stochastic distribution of phases are also some of the possible prospects for improving this work.