Chapter 7 - Numerical Solution of Finite Element Equations

Abstract

Numerical methods of solving different types of finite element equations are presented. For solving equilibrium equations, the Gaussian elimination method and Choleski method (for symmetric matrices) are presented. For solving the matrix eigenvalue problem, first the methods of converting a general eigenvalue problem into a standard eigenvalue problem are presented. Then some of the popular methods used for solving the eigenvalue problem, including the Jacobi method, power method, and Rayleigh–Ritz subspace iteration method, are presented. For solving the equations of propagation problems, first the equations are converted into a set of simultaneous first-order differential equations with appropriate boundary conditions. Then methods for solving the first-order differential equations, including the fourth-order Runge–Kutta numerical method and the direct integration methods (finite difference method and Newmark method) as well as the mode superposition method are presented. The computational details of most of the methods are illustrated with examples. Finally, the feasibility of using parallel processing in finite element analysis is indicated.