Generalization of the twist-Kirchhoff theory of plate elements to arbitrary quadrilaterals and assessment of convergence

We generalize the recently introduced twist-Kirchhoff theory of rectangular plate elements to arbitrary quadrilateral elements. A key feature is the use of Raviart–Thomas vector-field approximations for rotations. To preserve continuity of the normal components of the rotation vector across mesh edges, we employ the Piola transformation to map the rotations from the parent domain to the physical domain. These elements possess a unique combination of efficiency and robustness in that minimal quadrature rules are sufficient to guarantee stability without rank deficiency. In particular, only one-point Gauss quadrature is required for the lowest-order element in the twist-Kirchhoff family. We numerically study the convergence and accuracy of the first two members of the twist-Kirchhoff family of quadrilateral elements on square, rhombic and circular plate problems.