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通过两点相关函数对非均质材料进行建模:基本原理

Modeling heterogeneous materials via two-point correlation functions: basic principles.

作者信息

Jiao Y, Stillinger F H, Torquato S

机构信息

Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA.

出版信息

Phys Rev E Stat Nonlin Soft Matter Phys. 2007 Sep;76(3 Pt 1):031110. doi: 10.1103/PhysRevE.76.031110. Epub 2007 Sep 11.

Abstract

Heterogeneous materials abound in nature and man-made situations. Examples include porous media, biological materials, and composite materials. Diverse and interesting properties exhibited by these materials result from their complex microstructures, which also make it difficult to model the materials. Yeong and Torquato [Phys. Rev. E 57, 495 (1998)] introduced a stochastic optimization technique that enables one to generate realizations of heterogeneous materials from a prescribed set of correlation functions. In this first part of a series of two papers, we collect the known necessary conditions on the standard two-point correlation function S2(r) and formulate a conjecture. In particular, we argue that given a complete two-point correlation function space, S2(r) of any statistically homogeneous material can be expressed through a map on a selected set of bases of the function space. We provide examples of realizable two-point correlation functions and suggest a set of analytical basis functions. We also discuss an exact mathematical formulation of the (re)construction problem and prove that S2(r) cannot completely specify a two-phase heterogeneous material alone. Moreover, we devise an efficient and isotropy-preserving construction algorithm, namely, the lattice-point algorithm to generate realizations of materials from their two-point correlation functions based on the Yeong-Torquato technique. Subsequent analysis can be performed on the generated images to obtain desired macroscopic properties. These developments are integrated here into a general scheme that enables one to model and categorize heterogeneous materials via two-point correlation functions. We will mainly focus on basic principles in this paper. The algorithmic details and applications of the general scheme are given in the second part of this series of two papers.

摘要

非均质材料在自然界和人造环境中比比皆是。例子包括多孔介质、生物材料和复合材料。这些材料所展现出的多样且有趣的特性源于其复杂的微观结构,这也使得对这些材料进行建模变得困难。杨和托尔夸托[《物理评论E》57, 495 (1998)]引入了一种随机优化技术,该技术能使人们从一组规定的相关函数生成非均质材料的实现。在这两篇系列论文的第一部分中,我们收集了关于标准两点相关函数S2(r)的已知必要条件并提出一个猜想。特别地,我们认为给定一个完整的两点相关函数空间,任何统计均匀材料的S2(r)都可以通过函数空间选定基集上的一个映射来表示。我们给出了可实现的两点相关函数的例子并提出了一组解析基函数。我们还讨论了(重)构建问题的精确数学表述,并证明S2(r)不能单独完全指定一种两相非均质材料。此外,我们设计了一种高效且保持各向同性的构建算法,即格点算法,用于基于杨 - 托尔夸托技术从两点相关函数生成材料的实现。随后可以对生成的图像进行分析以获得所需的宏观性质。这些进展在此被整合到一个通用方案中,该方案能使人们通过两点相关函数对非均质材料进行建模和分类。在本文中我们将主要关注基本原理。该通用方案的算法细节和应用将在这两篇系列论文的第二部分给出。

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