Although there are now many research papers in the literature that describe the application of finite element methods to problems of electromagnetics, no textbook has appeared to date in this area. This is surprising, since the first papers on finite element solution of electrical engineering problems appeared in 1968, about the same time as the first textbook on finite element applications in civil engineering.

The authors have both taught courses in finite elements to a variety of electrical engineering audiences, and have sorely felt the lack of a suitable background book. The present work is intended to be suitable for advanced undergraduate students, as well as for engineers in professional practice. It is undemanding mathematically, and stresses applications in established areas, rather than attempting to delineate the research forefront.

Both authors wish to thank the many people who have helped shape this book – especially their students.

Introduction

Many of the problems of classical electrophysics that interest electrical engineers are inherently nonlinear. In fact it may be said that nonlinearity is an essential ingredient in almost all practical device analysis; nonlinearities determine the values over which field variables may range, and thereby fix the limits of device performance. Many useful devices could not function at all were they linear; mixers, modulators, and microwave circulators spring to mind as examples, but others can be found easily at any power level or in any part of the frequency spectrum.

Nonlinear finite element methods found early use in the analysis of power-frequency magnetic devices, most of which were not easily amenable to any other analytic technique, and a large part of the finite element literature in electrical engineering still deals with nonlinear magnetics. The nonlinearities encountered in such problems can often be assumed single-valued and monotonic so that comparatively simple methods can be made computationally efficient and reliable. For this reason, the present chapter will outline the major nonlinear methods used in magnetic field analysis. Other areas, such as problems of plasma electrochemistry or semiconductor devices, are generally amenable to similar mathematical techniques, though the electromagnetic formulations are often much more complicated.

Functional minimization for magnetic fields

The basic principles of linear finite element analysis carry over to nonlinear problems almost without modification. As always, a stationary functional is constructed and discretized over finite elements.

Introduction

This chapter introduces the finite element method by showing how a simple one-dimensional problem can be solved. Subsequent chapters of this book then generalize that elementary technique to cover more complicated situations.

The illustrative problem considered in this beginning chapter is that of a lossy direct-current transmission line, a problem not only physically simple but analytically solvable that permits direct comparison of the finite element approximation with known exact solutions. Two- and three-dimensional cases are dealt with in later chapters, where both the mathematical formulation of finite elements and the relevant numerical solution techniques are dealt with.

A direct-current transmission line

Suppose a pair of buried pipes is used for transmission of electrical signals, as in Fig. 1.1(a). Such a situation may arise, for example, if gas pipelines are used to carry signals for flow measurement, flaw detection, or other communications. It will be assumed that only very low frequencies are of interest, but that the pipes are resistive so there will be some longitudinal voltage drop as well as current leakage through the earth between pipes. The transmission line formed by the pair of pipes is assumed to be driven by a voltage source (e.g., an amplifier connected to thermocouples); the receiving end is taken to be very lightly loaded, practically an open circuit. The problem is to determine the distribution of signal voltage along the line, and the voltage at the receiving end.

Introduction

This Appendix lists several generally useful subprograms for matrix manipulation. It also clarifies some points of programming style as followed throughout this book. Most of the subprograms have no particularly close association with the material of any one chapter, but they are called upon to perform necessary services by programs listed in the several chapters. It therefore seems appropriate to group them in a single listing.

Input-data arrangements

Nearly all the programs in this book accept input data in roughly similar formats. Typically, a list of nodes is followed by a list of elements, which is in turn followed by a list of constraints. The lists are separated by lines blank except for a slant character / in the leading character position. This technique is based on the special meaning Fortran 77 assigns to the / character in list-directed input-output. It leads to program constructs a little different from C, Pascal, or many other languages, and therefore merits a brief discussion.

List-directed input–output is free-format. Numbers may be placed anywhere on the line, only their sequence need be correct. When input is requested, as many full lines are read in as is necessary to ensure that the input list has been fully satisfied, even if that means reading several lines. Since Fortran read statements always read full lines, a blank line will not be visible at all.

Introduction

Any polygon, no matter how irregular, can be represented exactly as a union of triangles, and any polyhedron can be represented as a union of tetradehra. It is thus reasonable to employ the triangle as the fundamental element shape in two dimensions, and to extend a similar treatment to three-dimensional problems by using tetrahedra.

The solution accuracy obtained with simple elements may be satisfactory in some problems, but it can be raised markedly by using piecewise polynomials instead of piecewise-linear functions on each element. If desired, this increased accuracy can be traded for computing cost, by using high-order approximations on each element but choosing much larger elements than in the first-order method. Indeed, both theory and experience indicate that for many two-dimensional problems, it is best to subdivide the problem region into the smallest possible number of large triangles, and to achieve the desired solution accuracy by the use of highorder polynomial approximants on this very coarse mesh.

In the following, details will be given for the construction of simplicial elements – triangles and tetrahedra – for the inhomogeneous scalar Helmholtz equation. This equation is particularly valuable because of its generality; a formulation valid for the inhomogeneous Helmholtz equation allows problems in Laplace's equation, Poisson's equation, or the homogeneous Helmholtz equation to be solved by merely dropping terms from this general equation. Scalar and quasi-scalar problems will be considered throughout, while all materials will be assumed locally linear and isotropic.

Introduction

Triangular and tetrahedral elements as described in the foregoing chapters are widely used because any polygonal object can be decomposed into a set of simplexes without approximation. In fact, such decomposition can be carried out by computer programs without human intervention, since there do exist mathematical techniques guaranteed to produce correct decomposition into simplexes. Unfortunately, simplex elements also have some serious shortcomings. They do not lend themselves well to the modelling of curved shapes, so that the intrinsically high accuracy of high-order elements may be lost in rather rough geometric approximation. They use polynomial approximation functions throughout, so that fields containing very rapid variations, or even singularities, cannot be well approximated. A third disadvantage of scalar simplex elements is precisely that they do model scalar quantities; they are not well suited to describing vector fields. Finally, they are unable to model large (or infinite) regions economically; yet many field problems of electromagnetics are ‘open’, in the sense that the region of principal interest is embedded in an infinitely extending homogeneous exterior space.

Alternative element shapes and alternative types of approximating functions can avoid most of the problems inherent in the simplex elements. The use of so-called isoparametric elements, which have curved sides, can often alleviate the problems encountered in geometric modelling, by shaping the elements to fit the real geometry. Vector fields require vector approximating functions, and while their theory is neither so simple, nor quite so completely developed as for scalars, useful vectorvalued functions do exist.

Introduction

Finite element methods have been successful in electromagnetics largely because the conventional field equations permit numerous different reformulations. These bring the electromagnetic field within the scope of numerical methods that rely on high-order local approximations while permitting comparative laxity with respect to boundary and interface conditions. After a very brief review of electromagnetic theory as it is relevant to finite elements, this chapter describes the projective and variational reformulations that lead directly to finite element methods, the distinctions between various types of boundary conditions, and the sometimes confusing terminology attached to them.

Maxwell's equations

Electromagnetic field problems occupy a relatively favourable position in engineering and physics in that their governing laws can be expressed very concisely by a single set of four equations. These evolved through the efforts of several well-known scientists, mainly during the nineteenth century, and were cast in their now accepted form as differential equations by Maxwell. There also exists an equivalent integral form.

Differential relations

The variables that the Maxwell equations relate are the following set of five vectors and one scalar:

Each of these may be a function of three space coordinates x, y, z and the time t. The four Maxwell equations in differential form are usually written as follows:

To these differential relations are added the constitutive relations

describing the macroscopic properties of the medium being dealt with in terms of its permittivity ∈, permeability µ and conductivity σ.