Almost-C¹ splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems

Isogeometric Analysis generalizes classical finite element analysis and intends to integrate it with the field of Computer-Aided Design. A central problem in achieving this objective is the reconstruction of analysis-suitable models from Computer-Aided Design models, which is in general a non-trivial and time-consuming task. In this article, we present a novel spline construction, that enables model reconstruction as well as simulation of high-order PDEs on the reconstructed models. The proposed almost-C¹ splines are biquadratic splines on fully unstructured quadrilateral meshes (without restrictions on placements or number of extraordinary vertices). They are C¹ smooth at all regular and extraordinary vertices. Moreover, they are C¹ smooth across all edges between regular vertices and C⁰ smooth across all edges that are adjacent to an extraordinary vertex. The splines thus form H²-nonconforming analysis-suitable discretization spaces. This is the lowest-degree unstructured spline construction that can be used to solve fourth-order problems. The associated spline basis is non-singular and has several B-spline-like properties (e.g., partition of unity, non-negativity, local support), the almost-C¹ splines are described in an explicit Bézier-extraction-based framework that can be easily implemented. Numerical tests suggest that the basis is well-conditioned and exhibits optimal approximation behaviour.