Three-dimensional electromagnetic vector field computation in lossy magnetic media by finite element method

Abstract

A three dimensional finite element method software package has been developed for solving electromagnetic vector fields in conducting, magnetic materials and has been applied in two dimensions to ferromagnetic filaments and in three dimensions to a sphere. The bulk of this dissertation describes the approach to formulating the problem, choosing a solution routine, developing a method of discretization, verifying the accuracy and characterizing the computational efficiency of the package. Spurious vector solutions, which arise in numerical approximations to three dimensional electromagnetic problems, were eliminated by using a node-based formulation, with modified vector wave equation to ensure that divergence free conditions are satisfied. Conjugate gradient, iterative quasi-minimal residual solver (QMR) with a non-zero matrix element storage scheme expedited computation and reduced memory requirements. An automatic mesh generator for hexahedral elements was developed for discretization. The two dimensional study continued earlier analytical and experimental work on induction heating of multi-filament ferromagnetic strands. The present results demonstrate that coupling between filaments does not occur in two dimensions and is, in fact, a three dimensional effect provided the filaments are not in electrical contact. Furthermore, the accuracy of the solution can be established quantitatively by a single parameter, the ratio of one side of the finite element to the electromagnetic skin (or penetration) depth. The three dimensional parametric study investigates the effects on power absorption patterns in the sphere as a function of conductivity and permeability. Primarily, this research demonstrates that these types of problems can be solved accurately. Finally, it is shown that while the discretization must extend completely throughout the sphere for non-magnetic, moderately lossy media (conductivity, σ∼ 1 S/m, relative permeability, μᵣ ∼1), it need only consist of a thin shell of about three skin depths thick for highly conducting magnetic materials. The core of this sphere can be replaced by a field free inner perfectly conducting sphere. While the problem of computing power absorption in ferromagnetic implants for hyperthermia, the motivation for this study, was not solved completely, the foundations have been laid. Dependence of power absorption upon size, shape, permeability and conductivity as well as interactions between filaments of finite length can be addressed with this beginning.