Overview of Finite Element Method

Chapter Outline

1.1 Basic Concept

The basic idea in the finite element method is to find the solution of a complicated problem by replacing it by a simpler one. Since the actual problem is replaced by a simpler one in finding the solution, we will be able to find only an approximate solution rather than the exact solution. The existing mathematical tools will not be sufficient to find the exact solution (and sometimes, even an approximate solution) of most of the practical problems. Thus, in the absence of any other convenient method to find even the approximate solution of a given problem, we have to prefer the finite element method. Moreover, in the finite element method, it will often be possible to improve or refine the approximate solution by spending more computational effort.

In the finite element method, the solution region is considered as built up of many small, interconnected subregions called finite elements. As an example of how a finite element model might be used to represent a complex geometrical shape, consider the milling machine structure shown in Figure 1.1(a). Since it is very difficult to find the exact response (like stresses and displacements) of the machine under any specified cutting (loading) condition, this structure is approximated as composed of several pieces as shown in Figure 1.1(b) in the finite element method. In each piece or element, a convenient approximate solution is assumed and the conditions of overall equilibrium of the structure are derived. The satisfaction of these conditions will yield an approximate solution for the displacements and stresses. Figure 1.2 shows the finite element idealization of a fighter aircraft.

1.2 Historical Background

Although the name of the finite element method was given recently, the concept dates back for several centuries. For example, ancient mathematicians found the circumference of a circle by approximating it by the perimeter of a polygon as shown in Figure 1.3. In terms of the present-day notation, each side of the polygon can be called a “finite element.” By considering the approximating polygon inscribed or circumscribed, one can obtain a lower bound S(l) or an upper bound S(u) for the true circumference S. Furthermore, as the number of sides of the polygon is increased, the approximate values converge to the true value. These characteristics, as will be seen later, will hold true in any general finite element application.

To find the differential equation of a surface of minimum area bounded by a specified closed curve, Schellback discretized the surface into several triangles and used a finite difference expression to find the total discretized area in 1851 [1.37]. In the current finite element method, a differential equation is solved by replacing it by a set of algebraic equations. Since the early 1900s, the behavior of structural frameworks, composed of several bars arranged in a regular pattern, has been approximated by that of an isotropic elastic body [1.38]. In 1943, Courant presented a method of determining the torsional rigidity of a hollow shaft by dividing the cross section into several triangles and using a linear variation of the stress function ϕ over each triangle in terms of the values of ϕ at net points (called nodes in the present day finite element terminology) [1.1]. This work is considered by some to be the origin of the present-day finite element method. Since mid-1950s, engineers in aircraft industry have worked on developing approximate methods for the prediction of stresses induced in aircraft wings. In 1956, Turner, Cough, Martin, and Topp [1.2] presented a method for modeling the wing skin using three-node triangles. At about the same time, Argyris and Kelsey presented several papers outlining matrix procedures, which contained some of the finite element ideas, for the solution of structural analysis problems [1.3]. Reference [1.2] is considered as one of the key contributions in the development of the finite element method.

The name finite element was coined, for the first time, by Clough in 1960 [1.4]. Although the finite element method was originally developed mostly based on intuition and physical argument, the method was recognized as a form of the classical Rayleigh-Ritz method in the early 1960s. Once the mathematical basis of the method was recognized, the developments of new finite elements for different types of problems and the popularity of the method started to grow almost exponentially [1.39–1.41]. The digital computer provided a rapid means of performing the many calculations involved in the finite element analysis and made the method practically viable. Along with the development of high-speed digital computers, the application of the finite element method also progressed at a very impressive rate. Zienkiewicz and Cheung [1.6] presented the broad interpretation of the method and its applicability to any general field problem. The book by Przemieniecki [1.5] presents the finite element method as applied to the solution of stress analysis problems.

With this broad interpretation of the finite element method, it has been found that the finite element equations can also be derived by using a weighted residual method such as Galerkin method or the least squares approach. This led to widespread interest among applied mathematicians in applying the finite element method for the solution of linear and nonlinear differential equations. It is to be noted that traditionally, mathematicians developed techniques such as matrix theory and solution methods for differential equations, and engineers used those methods to solve engineering analysis problems. Only in the case of finite element method, engineers developed and perfected the technique and applied mathematicians use the method for the solution of complex ordinary and partial differential equations. Today, it has become an industry standard to solve practical engineering problems using the finite element method. Millions of degrees of freedom (dof) are being used in the solution of some important practical problems.

A brief history of the beginning of the finite element method was presented by Gupta and Meek [1.7]. Books that deal with the basic theory, mathematical foundations, mechanical design, structural, fluid flow, heat transfer, electromagnetics and manufacturing applications, and computer programming aspects are given at the end of the chapter [1.10–1.32]. The rapid progress of the finite element method can be seen by noting that, annually about 3800 papers were being published with a total of about 56,000 papers and 380 books and 400 conference proceedings published as estimated in 1995 [1.42]. With all the progress, today the finite element method is considered one of the well-established and convenient analysis tools by engineers and applied scientists.