# Multigrid Solvers for 3D Nonlinear Multiharmonic Magnetic Field Computations

## Florian Bachinger

**Oct. 7, 2003, 11:45 a.m. T 711**

The efficient calculation of induction and eddy currents in electromagnetical problems is of significant interest for many industrial applications. A fast solver is required for the endeavor to reduce eddy current losses in electrical machines, for example, or contrariwise for the optimization of eddy current welding.

In this talk, we present a rigorous analysis of the underlying mathematical problem combined with a strategy for time-discretization which takes advantage of the periodicity of the solution.

Two main features of eddy current problems complicate the design of an efficient solver: First, the relation between magnetic field strength and induction is in general nonlinear. Secondly, the magnetic field and the thereby generated eddy currents hardly penetrate into conducting materials and thus form a small layer of strong induction at the boundaries of this material. This peculiarity leads to difficulties in computations, because the skin depth has to be considered in the discretization.

We have developped a powerful solver that handles the problem of the boundary layers by adaptive refinement and by an increase of the polynomial degree in the basis functions; the nonlinearity is dealt with by a Newton iteration. The capacities of our solver for three-dimensional eddy current problems are illustrated by several tests, even including the challenging real-life problem of eddy current welding.