GENERAL

We saw in Section 1.3 that finite element equations are obtained by the classical approximation theories such as variational or weighted residual methods. However, there are some basic differences in philosophy between the classical approximation theories and finite element methods. In the finite element methods, the global functional representations ofa variable consist ofan assembly oflocal functional representations so that the global boundary conditions can be implemented in local elements by modification of the assembled algebraic equations. The local interpolation (shape, basis, or trial) functions are chosen in such a manner that continuity between adjacent elements is maintained.

The finite element interpolations are characterized by the shape ofthe finite element and the order ofthe approximations. In general, the choice ofa finite element depends on the geometry ofthe global domain, the degree ofaccurac y desired in the solution, the ease ofintegration over the domain, etc.

In Figure 9.1.1, a two-dimensional domain is discretized by a series of triangular elements and quadrilateral elements. It is seen that the global domain consists ofmany subdomains (the finite elements). The global domain may be one-, two-, or three-dimensional. The corresponding geometries ofthe finite elements are shown in Figure 9.1.2. A one-dimensional element (as we have studied in Chapters 1 and 8) is simply a straight line, a two-dimensional element may be triangular, rectangular, or quadrilateral, and a three-dimensional element can be a tetrahedron, a regular hexahedron, an irregular hexahedron, etc. The three-dimensional domain with axisymmetric geometry and axisymmetric physical behavior can be represented by a two-dimensional element generated into a three-dimensional ring by integration around the circumference.