An effective finite element method for 3-D eddy current problems

摘要

Until now we have introduced mixed finite element methods for three-dimensional eddy current problems, and have shown the effectiveness of mixed methods; see Kanayama et al. [5] and Kanayama and Kikuchi. However, the resultant linear systems derived from the mixed methods are indefinite, which cause difficulties in choosing the iterative solver, and the Lagrange multipliers turn out to be zero. This is why it is quite desirable to eliminate these extra quantities. Therefore, we have considered two iterative procedures for the eddy current problems, and have shown some numerical examples; see Kanayama et al. [7,8]. In this paper, we focus our attention on the time-harmonic electromagnetic field for which the magnetic vector potential is adopted as only one unknown complex function. This formulation is different from our previous methods; for example the electric scalar potential is set to be zero, which is sometimes used in three-dimensional eddy current problems. See [1] for a related formulation. The magnetic vector potential is discretized by the Nedelec element of simplex type; see Nedelec. Then, as in [7] and [8], we introduce a finite element method without the Lagrange multiplier, which is an extension of an iterative one derived from a perturbation problem of the magnetostatic problem in Kikuchi and Fukuhara. Moreover, to consider the continuity of an excitation current density, we also propose a correction method. It is well known that we need to consider the continuity of the excitation current density when the magnetic vector potential is an unknown function; see, for example, Fujiwara et al. [3] systems, and the iterative procedure converges for a rather wide range of the perturbation prameter.