Chapter 5 - Derivation of Element Matrices and Vectors

Abstract

Three methods can be used to derive the element matrices and vectors: the direct method, the variational method, and the weighted residual method. Since the direct method is not used commonly, the variational and weighted residual methods are presented. For any physical problem that can be formulated as a variational problem, the application of the conditions for the stationary value of the functional yields the governing differential equation and the natural and geometric boundary conditions. The derivations and solutions of equilibrium, eigenvalue, and propagation problems using the variational method, also known as the Rayleigh–Ritz method, are presented. The weighted residual approach, which does not require a functional, is considered to derive the finite element equations. The point collocation, subdomain collocation, Galerkin, and least squares methods are outlined along with illustrative examples. The application of the weighted residual method for the solution of eigenvalue and propagation problems is also indicated. The derivations of finite element equations using the Galerkin and least squares methods are given along with numerical examples. The strong and weak form formulations of the finite element equations are discussed and the advantages of the weak form formulations are indicated.