A rotation-free quadrilateral thin shell subdivision finite element

We present a rotation-free, subdivision-based, isoparametric quadrilateral finite element for the analysis of doubly-curved thin shells. The element is a generalization of spline-based C¹ finite elements to curved domains, and general mesh connectivity. It uses semi-local basis functions that span the two-neighbourhood of a node. The traditional difficulty with this class of elements comes from the fact that these basis functions depend on the valence of the nodes of the element. For elements with 4-valence nodes the basis functions are bicubic polynomials, but there are no simple closed-form expressions for them when the nodes have general connectivity patterns. In the paper, we show how the basis functions for the general case may be efficiently evaluated as linear combinations of translation and scaling transformations of the bicubic polynomials. The performance of the resulting element is illustrated on a hemispherical shell model problem.