ABSTRACT
This chapter introduces the basic concepts needed to develop approximations to the solution of differential equations that will ultimately lead to the numerical algorithm known as the finite element method. The theory of the finite element method is found in variational calculus, and its mathematical basis allowed it to be developed in a very short time and become the powerful tool for engineers. However, this also created the misconception that a strong mathematical background is essential to understand the finite element method. The underlying principle of the finite element method resides in the method of weighted residuals. The two most commonly used procedures are the Rayleigh–Ritz and Galerkin methods. In both of the methods, the dependent variable is expressed by means of a finite series approximation in which the “shape” of the solution is assumed known, and it depends on a finite number of parameters to be determined.
