最大二面角平分算法不能保持四面体剖分的网格正则性
Largest-dihedral-angle bisection algorithm does not preserve mesh regularity for tetrahedral partitions
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中文总结 AI 辅助
本文构造了一个非退化四面体的反例，证明最大二面角平分算法在每一步选择唯一最大二面角时，其嵌套分支可坍缩至单位边，导致元素退化，从而表明该算法不能保持网格正则性。
中文摘要 AI 辅助
我们给出一个固定的非退化四面体，其嵌套的最大二面角平分分支坍缩到一条单位边上。在每一步平分中，所选二面角都是唯一最大的。该分支元素的某些面角趋于零，相应的直径与内切球半径之比发散，且产生的元素直径不趋于零，尽管每个二面角始终一致远离0和π，且所有面角和二面角满足一致的最大角界。证明基于精确的不变区间论证。此例表明最大二面角平分算法可能产生退化的四面体剖分。
英文摘要
We present a fixed nondegenerate tetrahedron with a nested largest-dihedral-anglebisection branch that collapses onto a unit edge. The selected dihedral angles to bisect are uniquely largest ones at every bisection step. Some face angles of the branch elements tend to zero, the corresponding diameter-to-inradius ratio diverges, and the diameters of the elements produced do not tend to zero, although every dihedral angle stays uniformly away from both 0 and π and all face and dihedral angles satisfy a uniform maximum-angle bound. The proof is based on an exact invariant-range argument. This example shows that the largest-dihedral-angle bisection algorithm can produce degenerating tetrahedral partitions.
