Suitability of three‐dimensional finite elements for modelling material incompressibility using exact integration

Abstract

AbstractThis paper examines the suitability of three‐dimensional finite elements to model accurately problems involving material incompressibility, using the displacement finite element method and exact numerical integration. The previously used method for classification of element suitability is presented and extended to the three‐dimensional case. However, an alternative approach for examining suitability, quantified in terms of free degrees of freedom (equal to the degrees of freedom minus the incompressibility constraints), is introduced. This is used to examine Lagrangian cubic, serendipity cubic and tetrahedral three‐dimensional elements configured in a regular cubic arrangement. The findings of this paper are substantiated by a number of three‐dimensional numerical experiments and comparison with a separate two‐dimensional study.All serendipity cubic elements are found to be unsuitable. The linear strain tetrahedron is on the borderline of suitability in a 6‐tetrahedra‐per‐cube arrangement, and is thought to be only suitable if the mesh boundary nodes are not over‐constrained. The same element in a 5‐tetrahedra‐per‐cube arrangement, and higher order tetrahedra (quadratic strain etc), are suitable. The Lagrangian cube elements of higher order than the 27‐node cube are suitable, but are probably not as efficient computationally as the tetrahedral elements of the same order.