Enhanced nodal gradient finite elements with new numerical integration schemes for 2D and 3D geometrically nonlinear analysis

Abstract

The consecutive-interpolation procedure (CIP) has been recently proposed as an enhanced technique for traditional finite element method (FEM) with various desirable properties such as continuous nodal gradients and higher accuracy without increasing the total number of degrees of freedom (DOFs). It is common knowledge that linear finite elements, e.g., four-node quadrilateral (Q4) or eight-node hexahedral (HH8) elements, are not highly suitable for geometrically nonlinear analysis. The elements with quadratic interpolation functions have to be used instead. In this paper, the CIP-enhanced four-node quadrilateral element (CQ4), and the CIP-enhanced eight-node hexahedral element (CHH8), are for the first time extended to investigate geometrically nonlinear problems of two- (2D) and three-dimensional (3D) structures. To further enhance the efficiency of the present approaches, novel numerical integration schemes based on the concept of mid-point rules, namely element mid-points (EM) and element mid-edges (EE) are integrated into the present CQ4 element. For CHH8, the 3D-version of EM (namely 3D-EM) and the element mid-faces (EF) scheme are investigated. The accuracy and computational efficiency of the two novel quadrature schemes in both regular and irregular (distorted) meshes are analyzed. Numerical results indicate that the new integration approaches perform more efficiently than the well-known Gaussian quadrature while gaining equivalent accuracy. The performance of the CIP-enhanced elements, which is examined through numerical experiments, is found to be equivalent to that of quadratic Lagrangian finite element counterparts, while having the same discretization with that by the linear finite elements. In addition, we also apply the present CQ4 and CHH8 elements associated with different numerical integration techniques to nearly incompressible materials.