Improved four-node Hellinger-Reissner elements based on skew coordinates

摘要

Mixed four-node elements based on the Hellinger-Reissner (HR) functional are developed for stress representations in various coordinates, including the skew, natural and Cartesian ones. The two-field HR functional is used in the classical form and in the incremental form suitable for non-linear materials. We argue that the skew coordinates, not the natural ones, should be associated with the natural basis at the element's center. If 5- and 7-parameter stress representations are assumed in these coordinates, then, for a linear elastic case, the homogenous equilibrium equations and the stress form of compatibility equation are satisfied point-wise. Two mixed four-node elements are developed and tested: 1. An assumed stress element (HR5-S) is developed from the non-enhanced HR functional, for a 5-parameter representation of stresses, formally identical as the one used, for example, in Pian and Sumihara [Int. J. Numer Meth. Engng 1984; 20:1685-1695], but in terms of skew coordinates. This element is very simple and uses a smaller number of parameters, but is equally accurate as the elements by Yuan et al. [Int. J. Numer. Meth. Engng 1993; 36:1747-1763] and by Piltner and Taylor [Int. J. Numer Meth. Engng 1995; 38:1783-1808]. 2. An assumed stress/enhanced strain element (HR9) is developed from the enhanced HR functional, for a 7-parameter representation of stress and a 2-parameter enhanced assumed displacement gradient or enhanced assumed strain enhancement. Various forms of 7-parameter representations appearing in the literature are reviewed, and we prove that they are linked by a linear onto transformation. The choice of coordinates for the stress and the enhancement turns out to be the crucial factor, and four combinations of coordinates for which the element performs the best are identified. Both elements are based on the Green strain, and several numerical tests show their good accuracy, in particular, their robustness to shape distortions for coarse meshes. Two update schemes for the multipliers of modes and the incremental constitutive procedure accounting for the plane stress condition for non-linear materials are tested for large deformation problems. Copyright (C) 2008 John Wiley & Sons, Ltd.