Making sense of the quadrilateral area coordinate membrane elements

The introduction of the quadrilateral area coordinate (QAC) membrane elements has generated more heat than light on the subject of why distorted membrane elements perform very poorly. The canonical guidelines for construction of robust elements are continuity and completeness. It is known that isoparametric (or parametric) quadrilateral membrane elements satisfy the continuity requirements always. In rectangular form, they perform extremely well, except in cases where parasitic shear is involved (plane stress modeling of bending of thin beams). So their poor performance when the elements are used in general (i.e. distorted) quadrilateral form is now attributed to the failure to accommodate the completeness requirements. Completeness is understood to be maintained in physical (i.e. Cartesian, or metric) space and not in the natural (i.e. parametric) space. The QAC approach is a compromise that tries to use shape functions which are in physical space, but as these cannot ensure exact continuity, require a relaxed generalized continuity to be imposed. However, here, the patch test cannot be ensured in the strictest sense as a careful examination of the results from various QAC elements in this paper will reveal. Even this is not enough to remove locking in the simpler 4-node elements and additional displacements fields have to be introduced through internal parameters to ensure what Prathap calls the consistency requirement. These issues are carefully studied in this investigation.