ABSTRACT
The Þnite element method is a powerful computational technique for the solution of differential and integral equations that arise in various Þelds of engineering and applied science. The method is a generalization of the classical variational (i.e., the Ritz) and weighted-residual (e.g., weak-form Galerkin, least-squares, collocation, etc.) methods, which are based on the idea that the solution u of a differential equation can be represented as a linear combination of unknown parameters cj and appropriately selected functions φj in the entire domain of the problem. The parameters cj are then determined such that the differential equation is satisÞed, often, in a weighted-integral sense. The functions φj , called the approximation functions, are selected such that they satisfy the boundary conditions of the problem. For additional details on the variational and weighted-residual methods, the reader may consult Mikhlin [1,2], Reddy [3-6], and Reddy and Rasmussen [7]. The traditional variational and weighted-residual methods suffer from one major
shortcoming: construction of the approximation functions that satisfy the boundary conditions of the problem to be solved. Most real-world problems are deÞned on regions that are geometrically complex, and therefore it is difficult to generate approximation functions that satisfy different types of boundary conditions on different portions of the boundary of the complex domain. However, if the domain can be represented as a collection of “simple subdomains” that permit construction of the approximation functions for any arbitrary but physically meaningful boundary conditions, then the traditional variational methods can be used to solve practical engineering problems. The basic idea of the Þnite element method is to view a given domain as an assemblage of simple geometric shapes, called Þnite elements, for which it is possible to systematically generate the approximation functions needed in the solution of differential equations by any of the variational and weightedresidual methods. The ability to represent domains with irregular geometries by a collection of Þnite elements makes the method a valuable practical tool for the solution of boundary, initial, and eigenvalue problems arising in various Þelds of engineering. The approximation functions are often constructed using ideas from interpolation theory, and hence they are also called interpolation functions. Thus, the Þnite element method is an element-wise application of the classical variational and weighted-residual methods. For a given differential equation, it is possible to develop different Þnite element models, depending on the choice of a particular variational or weighted-residual method. The Þnite element model is a set of algebraic relations among the unknown parameters of the approximation.
