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

Several finite element formulations in terms of various potentials are reviewed for the calculation of transient eddy currents in nonlinear media. A fast and simple iterative technique is used to solve the nonlinear algebraic equations arising. The different formulations are compared by solving some large three dimensional problems by various methods. >

Various FEM formulations for the calculation of transient 3D eddy currents in nonlinear media

Some improvements to the finite element computation of static magnetic fields in three dimensions using a reduced magnetic scalar potential are presented. Methods are described for obtaining an edge element representation of the rotational part of the magnetic field from a given source current distribution. When the current distribution is not known in advance, a boundary value problem is set up in terms of a current vector potential. An edge element representation of the solution can be directly used in the subsequent magnetostatic calculation. The magnetic field in a DC arc furnace is calculated by first determining the current distribution in terms of a current vector potential. A 3-D problem involving a permanent magnet as well as a coil is solved, and the magnetic field in some points is compared with measurement results. >

Computation of 3-D magnetostatic fields using a reduced scalar potential

An advanced A-V method employing edge-based finite elements for the vector potential A and nodal shape functions for the scalar potential V is proposed. Both gauged and ungauged formulations are considered. The novel scheme is particularly well suited for efficient iterative solvers such as the preconditioned conjugate gradient method, since it leads to significantly faster numerical convergence rates than pure edge element schemes. In contrast to nodal finite element implementations, spurious solutions do not occur and the inherent singularities of the electromagnetic fields in the vicinity of perfectly conducting edges and corners are handled correctly. Several numerical examples are presented to verify the suggested approach.

A joint vector and scalar potential formulation for driven high frequency problems using hybrid edge and nodal finite elements

The application of evolution strategies to the optimal design of electromagnetic devices is investigated The corresponding field analysis is performed by the finite element method The strategies involve a simplified simulation of biological evolution They are especially advantageous in the case of strongly nonlinear optimization problems The three strategies used are the (1+1), the

FEM and evolution strategies in the optimal design of electromagnetic devices

Publisher Summary Computer-aided design (CAD) software packages for the field analysis methods of three-dimensional models are less widespread. This chapter attempts to develop robust and reliable numerical field analysis methods for three-dimensional models. A largely unified field analysis approach is proposed for various types of problems: static fields, eddy current fields, and general electromagnetic fields. The uniformity is attained by using similar potential functions to describe the field in each particular case. This allows for a great degree of generality with respect to material properties, because the continuity of the potentials is sufficient to ensure the satisfaction of the interface conditions on surfaces where the material characteristics change abruptly. The Coulomb gauge is invariably applied to ensure the uniqueness of the vector potentials, which are necessary in three-dimensional analysis. This results in great numerical stability and in the lack of any spurious solutions when the finite element method is employed. The robustness of the methods is shown by some illustrative examples in the chapter. When induced conductive currents are considered with the neglected displacement currents, the equations of eddy current fields are obtained. This case of the quasi-stationary limit is also discussed in the chapter. The chapter also presents the application of a special form of nodal finite elements by means of several examples of computer-aided analysis.

Oszkar Biro

Papers

Various FEM formulations for the calculation of transient 3D eddy currents in nonlinear media

Computation of 3-D magnetostatic fields using a reduced scalar potential

A joint vector and scalar potential formulation for driven high frequency problems using hybrid edge and nodal finite elements

FEM and evolution strategies in the optimal design of electromagnetic devices

CAD in Electromagnetism