Rational approach to solid finite elements

Abstract

The finite element method today has become a powerful tool in engineering analysis and design. The ease of application, reliability of solutions, and its ability to model complex geometries seem to be the main reasons for its popularity. Yet the method does possess some inherent difficulties. Some of these are the problems of locking (shear locking, membrane locking, and incompressibility locking), poor stress predictions within the element domain, violent stress oscillations, poor convergence, etc. There have been many attempts to overcome these disadvantages. The techniques used include reduced/selective integration, assumed strain methods, use of incompatible bubble modes, mixed formulations, use of drilling degrees of freedom, assumed?stress hybrid formulations, etc. More often than not, these techniques lack suitable explanations for their success. Here we examine the formulation of hexahedral finite elements using interpolation functions which satisfy the governing differential equations of equilibrium a priori. Solutions (for example, the Papcovitch–Neuber solution) to these equations have been available in the form of a harmonic vector and a harmonic function for some time but have not been used in the formulation of finite elements. These solutions have been shrouded in controversy regarding whether four or three functions are required for three?dimensional solids. We take a fresh look at them to show that all four functions are required to represent the displacement field within a solid. We then proceed to use them to obtain hexahedral solid elements. We obtain a number of such elements, which are formulated without taking recourse to any technique extraneous to the variational procedure. We compute stresses at any point of interest within the element. We then subject the elements to a number of test problems proposed in the literature. The solutions obtained from these elements in single?element situations match with that of theory. The performance of the element is exact when subjected to bending moment loads, out?of?plane shear, and twist, even under distorted conditions. Therefore, we conclude that it is possible to formulate finite elements which predict both stresses and displacements accurately, simply by using interpolation functions based on the governing differential equations. Finally, we look at some aspects of the problem that require further investigation.