Variational Procedures and Convergence of Finite-Element Methods

Abstract

There are several advantages to basing the finite-element method on variational principles. One is to put the method on a sound theoretical foundation. Another is to clarify the requirements for interelement continuity, as these requirements are explicit in the statement of the variational principles. The variational principles of linear structural mechanics are the principles of minimum potential energy and of minimum complementary energy. The principle of minimum complementary energy is concerned with stress fields that satisfy the conditions of equilibrium but not necessarily the requirements of compatibility. Among all statically admissible stress fields, the one that satisfies the stress–strain relations in the interior of the structure and the displacement boundary conditions makes the complementary energy an absolute minimum. Apart from the fundamental approximation errors, the standard finite element techniques introduce additional perturbation errors. Boundary conditions are not satisfied exactly, the original domain is perturbed by triangularization, various coefficients of the differential equations are approximated, and exact integration is often replaced by numerical quadrature.