AI 中文总结
研究基于最大角n等分的网格细化算法，将其与最长边n等分规则对比。证明最大角n等分不会产生退化，能满足最小角条件，且递归算法生成的三角剖分族中k级后代最大直径随k趋于无穷而趋于零。
AI 中文摘要
我们定义了一种基于将平面剖分中三角形单元的最大角分成n等份规则的网格细化算法，并分析了该技术生成的三角剖分的（几何）性质。将此最大角n等分规则与经典的最长边n等分规则进行比较，已知最长边二等分和三等分可产生非退化三角剖分，但n≥4时最长边n等分总会产生最小角趋于零的（无限）三角形序列。我们证明最大角n等分不会出现这种退化，对于n≥2，它能产生满足最小角条件的剖分，且递归算法生成的三角剖分族中，k级后代的最大直径随k趋于无穷而趋于零。
英文摘要
We define a mesh refinement algorithm based on the rule of dividing the largest angles of triangular elements of planar partitions in focus into $n$ equal parts, and analyse the (geometric) properties of triangulations generated by this technique. This largest-angle $n$-section rule is compared with the classical longest-edge $n$-section rule, where it is the longest edges which are split into $n$ equal parts. The longest-edge bisection and trisection are known to produce nondegenerate triangulations (possibly with hanging nodes), but the longest-edge $n$-sections with $n\geq 4$ always produce (infinite) sequences of triangles with minimum angles tending to zero (moreover, their relevant maximum angles tend to $π$), thus breaking the minimum and maximum angle conditions. We show that this degeneration effect is not a consequence of $n$-section itself. For every $n\geq 2$, the largest-angle $n$-sections produce partitions satisfying the minimum angle condition (and, therefore, the maximum angle condition). More precisely, if the initial triangle has its smallest angle $γ_0>0$, then all descendant triangles have angles bounded below by $m_n=\min\left\{γ_0,\fracπ{3n}\right\},$ and, correspondingly, bounded above by $π-2m_n<π$. We also show that the recursive largest-angle $n$-section algorithm always produces a family of triangular partitions, i.e. the maximum diameter of level-$k$ descendants tends to zero as $k \to \infty$.
Comments7 pages, 1 figure
