Improved 4-node Hu–Washizu elements based on skew coordinates

Mixed 4-node elements based on the Hu–Washizu (HW) functional are developed for stress and strain representations in various coordinates, including the skew, natural and Cartesian ones. The HW functional is used in incremental form, suitable for non-linear materials. The key features of our approach are as follows.(1)The representations of stress and strain are assumed in skew coordinates associated with the natural basis at the element’s center, which implies that, for a linear elastic case, the homogenous equilibrium equations and the compatibility condition are satisfied point-wise. For stress, the same 5- and 7-parameter representations as for the Hellinger–Reissner (HR) elements by Wisniewski and Turska [Wisniewski K, Turska E. Improved four-node Hellinger–Reissner elements based on skew coordinates. Int J Numer Methods Eng 2008;76:798–836] are used. For strain, a 9-parameter linear representation is selected.(2)A mixed element HW14-S using a 5-parameter representation of stresses assumed in skew coordinates is developed from the non-enhanced HW functional. This element is equally accurate as our HR5-S element of Wisniewski and Turska (1998), the HR element by Yuan et al. [Yuan K-Y, Huang Y-S, Pian THH. New strategy for assumed stress for 4-node hybrid stress membrane element. Int J Numer Methods Eng 1993;36:1747–63], and the HW elements by Piltner and Taylor [Piltner R, Taylor RL. A quadrilateral mixed finite element with two enhanced strain modes. Int J Numer Methods Eng 1995;38:1783–808; Piltner R, Taylor RL. A systematic construction of B-bar functions for linear and non-linear mixed-enhanced finite elements for plane elasticity problems. Int J Numer Methods Eng 1999;44:615–39], and Piltner [Piltner R. An implementation of mixed enhanced finite elements with strains assumed in Cartesian and natural element coordinates using sparse B¯-matrices. Eng Comput 2000;17(8):933–49]. Compared to these HW elements, our element uses a smaller number of parameters.(3)A mixed/enhanced element HW18 using a 7-parameter representation of stress is developed from the enhanced HW functional. For the elements based on this stress representation, the strain representation has to be enriched; we use a 2-parameter EADG enhancement. Various combinations of the natural, skew and Cartesian coordinates are tested, and these for which this element performs best are selected.(4)A specific modification of the FTFFTF product, consisting of the expansion of FF and the selection of meaningful terms in the product, was applied to selected elements. With this modification, the element HW14-S performs better for coarse distorted meshes than the HW elements described in the literature.The developed elements are based on the Green strain, and are tested for linear and non-linear constitutive laws modified by the zero normal stress condition, because they will be used as a membrane part of a shell element. Several numerical tests show their performance, in particular, their robustness to the element’s shape distortion for coarse meshes.