Two new triangular G1-conforming finite elements with cubic edge rotation for the analysis of Kirchhoff plates

Two triangular G1-conforming elements, based on the triangular Gregory patch, suitable for the analysis of the Kirchhoff plate model are presented. Both have cubic normal derivative along the sides, so that they can be effectively used in combination with generalized Hermitian elements. The Gregory patch consists in a rational enhancement of the base-polynomial spaces useful to design G1-conforming elements on general C0-conforming unstructured meshes. Because of the presence of the rational functions, the second derivatives at the corners of the element present finite discontinuities, that prevent the elements from passing the bending patch test. The discontinuities are removed using a constrained version of the Gregory patch, with Lagrange multipliers. In this way, the rational conforming space collapses into a conforming rearrangement of the original polynomial interpolant spaces. The proposed formulation design elements that pass the bending patch test and present optimal rate of convergence on general unstructured meshes.