There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Introduction to Electromagnetic Compatibility

The finite-element method (FEM) is a full-wave numerical method that discretizes the variational of a functional. The evolution of this method within the scope of electromagnetics traces back to the solving of two classes of problems, namely, the eigenmode problems and the deterministic problems. If we try to use some examples to illustrate the most typical and the most complete techniques with the most complete solution, the eigenmode problem of a dielectrically loaded waveguide and the wave propagation in a three-dimensional (3D) discontinuous waveguide are good candidates representing the eigenmode and the closed-domain solutions, respectively. As for the open-domain scattering problem and radiating problems, the authors consider the essential and key parts are presented in solving the 3D scattering problems. For this reason, we will illustrate the theoretical basics, the critical solving techniques and the typical skills involved in FEM through solving of the above three specific problems. At the end of this chapter, we will also briefly review the FEM solution for some other problems.

Finite-Element Method

Various magnetic vector potential formulations for the eddy-current problem are reviewed. The uniqueness of the vector potential is given special attention. The aim is to develop a numerically stable finite-element scheme that performs well at low and high frequencies, does not require an unduly high number of degrees of freedom, and is capable of treating multiple connected conductors. >

On the use of the magnetic vector potential in the finite-element analysis of three-dimensional eddy currents

Numerical Methods for Scientists and Engineers

Edge element formulations of eddy current problems

An overview of various finite element techniques based on the magnetic vector potential for the solution of three-dimensional magnetostatic problems is presented. If nodal finite elements are used for the approximation of the vector potential, a lack of gauging results in an ill-conditioned system. The implicit enforcement of the Coulomb gauge dramatically improves the numerical stability, but the normal component of the vector potential must be allowed to be discontinuous on iron/air interfaces. If the vector potential is is interpolated with the aid of edge finite elements and no gauge is enforced, a singular system results. It can be solved efficiently by conjugate gradient methods, provided care is taken to ensure that the current density is divergence free. Finally, if a tree-cotree gauging of the vector potential is introduced, the numerical stability depends on how the tree is selected with no obvious optimal choice available.

On the use of the magnetic vector potential in the nodal and edge finite element analysis of 3D magnetostatic problems

A 3-D finite-element solution has been used to solve controlled-source electromagnetic (EM) induction problems in heterogeneous electrically conducting media. The solution is based on a weak formulation of the governing Maxwell equations using Coulomb-gauged EM potentials. The resulting sparse system of linear algebraic equations is solved efficiently using the quasi-minimal residual method with simple Jacobi scaling as a preconditioner. The main aspects of this work include the implementation of a 3-D cylindrical mesh generator with high-quality local mesh refinement and a formulation in terms of secondary EM potentials that eliminates singularities introduced by the source. These new aspects provide quantitative induction-log interpretation for petroleum exploration applications. Examples are given for 1-D, 2-D, and 3-D problems, and favorable comparisons are presented against other, previously published multidimensional EM induction codes. The method is general and can also be adapted for controlled-source EM modeling in mining, groundwater, and environmental geophysics in addition to fundamental studies of EM induction in heterogeneous media.

Finite-element analysis of controlled-source electromagnetic induction using Coulomb-gauged potentials

The authors review formulations of three-dimensional (3-D) eddy current problems in terms of various magnetic and electric potentials. The differential equations and boundary conditions are formulated to include the necessary gauging conditions and thus to ensure the uniqueness of the potentials. Different sets of potentials can be used in distinct subregions, thus facilitating an economic treatment of various types of problems. A novel technique for interfacing conducting regions with an electric vector and a magnetic scalar potential to eddy-current-free regions with a magnetic vector potential is described. Finite-element solutions to several large eddy-current problems are presented. >

Oszkar Biro

Papers

On the use of the magnetic vector potential in the finite-element analysis of three-dimensional eddy currents

Edge element formulations of eddy current problems

On the use of the magnetic vector potential in the nodal and edge finite element analysis of 3D magnetostatic problems

Finite-element analysis of controlled-source electromagnetic induction using Coulomb-gauged potentials

Finite element analysis of 3-D eddy currents