High precision finite elements

Abstract

The finite elements method has established itself as a synonym for Structural Analysis over the years. Nevertheless, the accuracy of FE-results continues to be unsatisfactory in many cases. The error is related to the fact that finite elements only approximate the partial differential equations describing the structure, where the quality of the approximation depends on the elementation. In this paper a new type of finite element is presented, which is based on exact solutions of the PDEs. This element is in fact the dynamic stiffness matrix of a continuous substructure that has not been discretized. Its elements are transcendental functions of the frequency. Special tools have to be used to find the eigensolutions. On the other hand, the matrix accurately describes the behaviour of a possibly very large structural element up to high frequencies. The elementation of the structure is solely determined by its geometry and not by precision constraints. In the past, only one-dimensional continuous FEs have been available under different names including “continuous elements” and “Exact Finite Elements”. Recently, the method has been modified to include two-dimensional elements. The modification does introduce an error, but still the performance compares favourably with the traditional FEM. Accordingly, the technique is termed high precision finite elements method.