中文摘要：

采用Potts模型Monte Carlo方法研究了晶粒棱长、尺寸与拓扑学特征之间的统计关系．结果表明，晶粒棱长与晶粒面数之间呈线性统计关系，并且平均Ⅳ面体晶粒模型和Poisson．Voronoi组织均支持该结论．不同时刻的晶粒长大仿真数据表明，在准稳态晶粒长大阶段晶粒棱长的分布具有自相似性．个体晶粒的平均棱长随晶粒面数（或晶粒尺寸）的增加而逐渐增大，这说明一些理论模型中采用的“不同面数的晶粒平均棱长均相等”的假设具有局限性．仿真数据和纯铁实验数据均表明，晶粒尺寸与晶粒面数之间的统计关系表现为一条单调递增的凸曲线．

英文摘要：

The relation among grain edge length, grain size and topology is studied by large-scale Potts model-Monte Carlo simulation. The results show that the grain edge length is related to the number of grain faces by a linear relation, which is also observed to hold in Poisson-Voronoi structure and average N-bedra. The distribution of the grain edge lengths maintains self-similarity in the period of quasi-stationary grain growth. The average length of an edge segment in individual grains varies with grain size （or the number of grain faces）, thereby indicating that assuming the lengths of edge segments in all individual grains to be equal averagely as does in some models is subject to some limitations. The data of Monte Carlo simulated grains and pure iron show that the grain size is related to the face number by a curve convex upward.