A quadrilateral [formula omitted]-conforming finite element for the Kirchhoff plate model

Abstract

A quadrilateral bi-cubic G1-conforming finite element for the analysis of Kirchhoff plates is presented. The rational version of the Gregory patch proposed by Loop et al. (2009) is the starting point of our formulation. This version of the Gregory patch consists in rational enhancement of the bi-cubic Bézier interpolation representing a suitable tool for designing G1-conforming quadrilateral element on C0-conforming un-structured meshes. The element includes as additional degrees of freedom the edge rotations like in the Loof-formulations but is only displacement based.Because of the presence of the rational functions, the second derivatives of the interpolation present a finite discontinuity at the corners of the element, that prevent the element from passing the bending patch test. The element so formulated does not present optimal rate of convergence under h-refinement operation. The formulation is enhanced enforcing these discontinuities to be zero by means of Lagrange multipliers. It is shown that with these constraints the element passes the patch test and presents optimal rate of convergence for unstructured mesh. In this way the rational conforming approximation collapses into a conforming re-arrangement of the complete bi-cubic Bézier interpolation.Some examples and benchmarks are presented in order to test the performance of the element for the Kirchhoff plate model.