关于最长边n等分细分下四面体退化性的研究

中文摘要 AI 辅助

对于每个整数 $n \ge 2$，我们构造了显式的四面体反例，表明重复的最长边（LE）n等分细分通常不能保持形状正则性，这与长期以来的猜想预期相反。对于 $n = 2$，该构造否定了 Adler 于1983年提出的有限族猜想，以及 Rivara 和 Levin 于1992年提出的四面体最长边对分非退化猜想。它还表明，Stynes 在1983年基于其1980年平面有限相似性结果所提出的更高维直径衰减的更强结论，对于任意四面体不成立。对于 $n = 3$，它否定了 Suárez、Abellón、Abad 和 Plaza 于2011年基于数值实验提出的四面体最长边三分非退化猜想。由此产生的无限后代序列同时违反了最小和最大角条件。对于 $n \ge 3$，在每一步细分中，所选边始终是唯一的最长边。构造中使用的两步递推表现出两种不同的退化行为：对于 $2 \le n \le 5$ 是扁平四面体状态，对于 $n \ge 6$ 是细长四面体状态。退化序列也可以在由一致性 LE n等分算法生成的一致性剖分中实现。对于经典和一致性 LE n等分算法，无论选择哪条最长边，随着细分步数趋于无穷大，最大单元直径趋于零。因此，直径收敛甚至一致性并不能防止形状退化。这些算法在需要均匀形状正则性的应用中应谨慎使用。

英文摘要

For every integer $n \ge 2$, we construct explicit tetrahedral counterexamples showing that repeated longest-edge (LE) n-section refinement does not, in general, preserve shape regularity, contrary to long-standing conjectural expectations. For $n = 2$, the construction disproves the finite-family conjecture put forward by Adler in 1983 and the non-degeneracy conjecture for tetrahedral longest-edge bisection formulated by Rivara and Levin in 1992. It also shows that the sharper higher-dimensional diameter decay suggested by Stynes in 1983,following his 1980 planar finite-similarity result, cannot hold for arbitrary tetrahedra. For $n = 3$, it disproves the published non-degeneracy conjecture for tetrahedral longest-edge trisection formulated in 2011 by Suárez, Abellón, Abad and Plaza on the basis of numerical experiments. The resulting infinite descendant sequences violate both the minimum and maximum angle conditions. For $n \ge 3$, the selected edge is always the unique longest edge at every refinement step. The two-step recurrence used in the construction exhibits two distinct degeneration behaviors: a flat-tetrahedron regime for $2 \le n \le 5$ and a skinny-tetrahedron regime for $n \ge 6$. The degenerating sequences can also be realized within conforming partitions generated by the conforming LE n-section algorithm. For both the classical and the conforming LE n-section algorithms, the maximal element diameter tends to zero as the number of refinement steps tends to infinity, regardless of which longest edge is chosen. Thus diameter convergence, and even conformity, do not prevent shape degeneration. These algorithms should therefore be used with care in applications requiring uniform shape regularity.

发表机构

• Mälardalen University（马尔默达伦大学）

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