论文标题
几乎 - $ c^1 $花键：非结构化四边形网格上的双Quadratic花键及其在第四阶问题上的应用
Almost-$C^1$ splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems
论文作者
论文摘要
等几何分析概括了经典的有限元分析，并打算将其与计算机辅助设计领域集成。实现这一目标的一个核心问题是重建计算机辅助设计模型的分析模型，这通常是一项非平凡且耗时的任务。在本文中，我们提出了一种新型的样条结构，该构建可以使模型重建以及对重建模型上的高阶PDE的模拟。所提出的几乎$ c^1 $是完全非结构化的四边形网格上的双Quadratic花键（在放置或非凡顶点的数量上无限制）。它们几乎到处都是$ c^1 $，即在所有顶点和大多数边缘，而且几乎（即大约）$ c^1 $在所有其他边缘上平滑。因此，花键形成$ h^2 $ -NONONON-NONON-NONCONTING分析适合离散空间。这是可用于解决四阶问题的最低程度的非结构化样条结构。相关的样条基础是非单位的，并且具有几种类似B的属性（例如，统一分区，非负分区，局部支撑），在基于Bézier-actaction-actaction的显式框架中描述了几乎 - $ c^1 $的光谱，可以轻松实现。数值测试表明该基础是良好的条件，并且表现出最佳的近似行为。
Isogeometric Analysis generalizes classical finite element analysis and intends to integrate it with the field of Computer-Aided Design. A central problem in achieving this objective is the reconstruction of analysis-suitable models from Computer-Aided Design models, which is in general a non-trivial and time-consuming task. In this article, we present a novel spline construction, that enables model reconstruction as well as simulation of high-order PDEs on the reconstructed models. The proposed almost-$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.
