Effects of element distortions on the performance of enriched quadrilateral elements

Abstract

It is well known that the accuracy of finite element solutions deteriorate in the presence of severe mesh distortions. But distortion is often unavoidable in mesh procedure involving complex geometry. Lee and Bathe (1993) studied the influence of mesh distortion on the serendipity and Lagrange quadrilateral elements. Lautersztajn and Samuelsson (2000) discussed the effects of geometric distortions on four-node isoparametric quadrilateral elements and concluded that the element performance can be rendered ?insensitive? to a particular type of mesh distortion by increasing the order of the interpolation functions for the displacement field. In order to overcome the influence of element distortions, unsymmetric 8-node (Rajendran and Liew 2003) and unsymmetric 20-node element (Ooi et al. 2004) are developed to reproducing any linear and quadratic displacement field under any admissible mesh distortions. However, they will produce an asymmetrical stiffness matrix, so these formulations require an asymmetrical solver to solve the resulting stiffness equations. The goal of this paper is to discuss the effects of element distortions on the accuracy and efficiency of enriched quadrilateral elements with bubble functions. A bubble function is defined as a function that vanishes along the element boundaries. Bubble functions have been introduced to construct plate element models (Auricchio and Taylor 1995, Cook et al. 2002, Hong et al. 2001). They are employed to solve advection-diffusion problems by Brezzi, Franca and Farhat (Brezzi et al. 1992, Brezzi and Russo 1994, Franca and Farhat 1995). Furthermore, the limitation of bubble functions is discussed by Franca and Farhat (1994) and error analysis of residual-free bubbles is discussed by Brezzi et al. (1999) and Sangalli (2000).