PH2T样条，第一部分：一个合理的网格假设

中文摘要 AI 辅助

本文是关于在分层T网格上构造具有最高阶光滑度的多项式样条（称为$\PHtwoT$-样条）的三部分系列论文中的第一篇。对于双次数$(d,d)$的样条，我们研究了后续基构造所需的合适细化条件，这要求底层样条空间的维数稳定性。我们给出了两组例子，分别考虑无限制的分层细化和没有可消失T$l$-边的细化。第一种设置允许新边而不增加自由度。在第二组中，每个细化级别排除可消失的T$l$-边并增加维数。然而，两组中的维数都不稳定。然后，我们引入模板平移来描述每个细化区域为单元索引网格中固定模板的平移并集。结合在$(d-1)\times(d-1)$模板细化下的已知稳定性结果，这些例子证明该条件是后续$\PHtwoT$-样条构造的一个合理网格假设。

英文摘要

This paper is the first in a three-part series on the construction of polynomial splines with the highest order of smoothness over hierarchical T-meshes, referred to as $\PHtwoT$-splines. For splines of bi-degree $(d,d)$, we study suitable refinement conditions for the subsequent basis construction, which requires dimensional stability of the underlying spline space. We present two groups of examples, considering unrestricted hierarchical refinement and refinement without vanishable T $l$-edges, respectively. The first setting permits new edges without additional degrees of freedom. In the second group, every refinement level excludes vanishable T $l$-edges and increases the dimension. Nevertheless, the dimension is unstable in both groups. We then introduce template translations to describe each refined region as a union of translates of a fixed template in the cell-index grid. Together with the known stability result under $(d-1)\times(d-1)$ template refinement, these examples justify this condition as a reasonable mesh assumption for the subsequent $\PHtwoT$-spline construction.

发表机构

• University of Science and Technology of China（中国科学技术大学）

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