论文标题
基于几何的放松算法，用于在二维中平衡三价多边形网络及其含义
A geometry-based relaxation algorithm for equilibrating a trivalent polygonal network in two dimensions and its implications
论文作者
论文摘要
在二维（2D）中，三价多边形网络的平衡本质上是一种普遍现象，但是基本的数学机制尚不清楚。在这项研究中，开发了基于简单的几何规则的弛豫算法来模拟平衡。提出的算法用Python语言实施。模拟的松弛将Voronoi网络的多边形细胞从椭圆形的多边形转向椭圆的最大铭刻多边形。同时，描述细胞之间相邻关系的Aboav-Weaire定律仍然统计。模拟的成功强烈支持椭圆堆积假设，该假设被提出，以解释三价结构的动态行为。模拟结果还表明，大细胞的边缘往往比小细胞的边缘短，反之亦然。另外，松弛增加了大细胞的面积和边缘长度，并降低了小细胞的面积和边缘长度。由于放松而引起的不同边缘细胞区域变化的模式几乎与von-neumann-mullins定律所描述的生长模式相同。这项工作中提出的结果可以帮助理解三价2D结构的动态行为的数学机制。
The equilibration of a trivalent polygonal network in two dimensions (2D) is a universal phenomenon in nature, but the underlying mathematical mechanism remains unclear. In this study, a relaxation algorithm based on a simple geometrical rule was developed to simulate the equilibration. The proposed algorithm was implemented in Python language. The simulated relaxation changed the polygonal cell of the Voronoi network from an ellipse's inscribed polygon toward the ellipse's maximal inscribed polygon. Meanwhile, the Aboav-Weaire's law, which describes the neighboring relationship between cells, still holds statistically. The succeed of simulation strongly supports the ellipse packing hypothesis that was proposed to explain the dynamic behaviors of a trivalent 2D structure. The simulation results also showed that the edge of large cells tends to be shorter than edges of small cells, and vice versa. In addition, the relaxation increases the area and edge length of large cells, and it decreases the area and edge length of small cells. The pattern of changes in the area of different-edged cells due to relaxation is almost the same as the growth pattern described by the von-Neumann-Mullins law. The results presented in this work can help to understand the mathematical mechanisms of the dynamic behaviors of trivalent 2D structures.
