5.1 Introduction

Solving Maxwell's equations numerically by using the finite element method makes it possible to take into account material behaviors a priori unspecified (nonlinear, anisotropic, with or without hysteresis). For that purpose, adequate behavior models and implementing resolution algorithms for nonlinear problems are needed.

It is to be noted that the electromagnetic behavior of materials in the field of electrical engineering remains only very approximately represented in software available on the market at the present time. This is particularly the case for magnetic materials. Therefore, we will specifically deal with these materials in this chapter. In addition, the behavior of superconductors will also be covered.

The lack of behavior models actually used within the framework of the finite element method is partly explained by the fact that the microscopic phenomena at the origin of the macroscopic behavior of magnetic materials are complex and difficult to model. The transition from microscopic to macroscopic is not yet well understood. There is a real difficulty in finding models of magnetic behavior that achieve a good trade-off between accuracy and numerical simplicity for an effective integration in a software tool based on the finite element method. The "phenomenological" models traditionally suggested are in general quite far away from physical reality and the use of more realistic models of behaviors is still an object of research.

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The chapter starts by presenting the behavioral characteristics of magnetic materials: nonlinearity, anisotropy and hysteresis. Then two methods dealing with solving nonlinear problems are shown within...