Almost-$C^1$ splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems

DOI10.1016/J.CMA.2022.115640arXiv2201.11491MaRDI QIDQ6389373

Publication date: 27 January 2022

Abstract: 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 B'ezier-extraction-based framework that can be easily implemented. Numerical tests suggest that the basis is well-conditioned and exhibits optimal approximation behavior.

Mathematics Subject Classification ID

Numerical computation using splines (65D07) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Computer-aided design (modeling of curves and surfaces) (65D17)

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