An implementation of mixed enhanced finite elements with strains assumed in Cartesian and natural element coordinates using sparse B̄-matrices

The use of enhanced strains leads to an improved performance of low order finite elements. A modified Hu-Washizu variational formulation with orthogonal stress and strain functions is considered. The use of orthogonal functions leads to a formulation with B̄-strain matrices which avoids numerical inversion of matrices. Depending on the choice of the stress and strain functions in Cartesian or natural element coordinates one can recover, for example, the hybrid stress element P-S of Pian-Sumihara or the Trefftz-type element QE2 of Piltner and Taylor. With the mixed formulation discussed in this paper a simple extension of the high precision elements P-S and QE2 to general non-linear problems is possible, since the final computer implementation of the mixed element is very similar to the implementation of a displacement element. Instead of sparse B-matrices, sparse B̄-matrices are used and the typical matrix inversions of hybrid and mixed methods can be avoided. The two most efficient four-node B-elements for plane strain and plane stress in this study are denoted B̄(x, y)-QE4 and B̄(ξ, η)-QE4.