Almost-C^1 splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems
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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-C^1 are biquadratic splines on fully unstructured quadrilateral meshes (without restrictions on placements or number of extraordinary vertices). They are C^1 smooth almost everywhere, that is, at all vertices and across most edges, and in addition almost (i.e. approximately) C^1 smooth across all other edges. Thus, the splines form H^2-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-C^1 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 behavior.
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