An implicit G1-conforming bi-cubic interpolation for the analysis of smooth and folded Kirchhoff–Love shell assemblies

A quadrilateral bi-cubic G1-conforming finite element for the analysis of Kirchhoff–Love shell assemblies based on the rational Gregory interpolation is presented. The Gregory interpolation removes the symmetry of the second cross derivative at the corners of the element that allows an independent control of the side rotations of the boundaries of the element. In this way G1-conformity of the deformation can be implicitly obtained for any mesh of quadrilateral elements, also for not G1-continuous parametrizations. The interpolation is defined by means of the kinematics of the boundary ribbons. The ribbon is the differential set generated by the tangents at the boundary of the element. A new set of degrees of freedom is introduced in order to control the deformation of the boundary, and the non-linear map between this new set of degrees of freedom and the control points of the Gregory interpolation is derived. Due to the presence of rational terms, the interpolation is not consistent, so that, in order to recover consistency it is necessary to enforce the vanishing of the discontinuities of the second derivatives with additional constraints. The proposed G1-conforming shell element results accurate and robust as shown by several numerical investigations on benchmark problems.