Introduction

First-order triangular finite elements made their initial appearance in electrical engineering applications in 1968. They were then used for the solution of comparatively simple waveguide problems, but have since been employed in many areas where two-dimensional scalar potentials or wave functions need to be determined. Because of their relatively low accuracy, first-order elements have been supplanted in many applications by elements of higher orders. However, they continue to find wide use in problems where material nonlinearities or complicated geometric shapes are encountered; for example, in analysing the magnetic fields of electric machines, or the charge and current distributions in semiconductor devices.

The first-order methods using triangular elements may be regarded as two-dimensional generalizations of piecewise-linear approximation, a tool widely used in virtually all areas of electrical engineering. The mathematics required in defining such elements is easily mastered, and computer programming at a very simple level can produce many useful results. There are few methods in electromagnetic field theory for which such sweeping claims can be made, and indeed it is surprising that finite elements have not penetrated into electrical engineering applications even more deeply.

In this chapter, simple triangular finite element methods will be developed for solving two-dimensional scalar potential problems. The construction of these simple elements is useful in its own right; but perhaps more importantly, it will also illustrate by way of example many of the principles involved in all finite element methods.