The Finite Element Method in Electromagnetics, 3rd Edition

CHAPTER 2INTRODUCTION TO THE FINITE ELEMENT METHOD

The finite element method is a numerical technique for obtaining approximate solutions to boundary-value problems of mathematical physics. The method has a history of more than 60 years. It was first proposed in the 1940s by Courant to solve problems of equilibrium and vibration [1], and its use began in the 1950s for aircraft design. Thereafter, the method was developed and applied extensively to problems of elasticity and structural analysis and increasingly to problems in other fields such as fluid dynamics and electromagnetics. Today, the finite element method has become recognized as a general method widely applicable to engineering and mathematical problems. Many books have been written on the subject (see, for example, Refs. 2–16). In this chapter we first review two classical methods for solving boundary-value problems (both containing the roots of the finite element method) and then introduce the finite element method with the aid of a simple example. Finally, we describe the basic steps of the finite element analysis without reference to any specific problems so that the reader can obtain a clear idea about the basic principle of the method.

2.1 CLASSICAL METHODS FOR BOUNDARY-VALUE PROBLEMS

In this section we first define boundary-value problems and then review two classical methods for their solution. One is the Ritz variational method, the other is Galerkin’s method, and these two methods form the basis of the ...