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DLA模型,Diffusion-limited Aggregation(DLA)扩散限制凝聚  

2013-01-31 13:02:06|  分类: 分形启示录 |  标签: |举报 |字号 订阅

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目录

摘要    2

1    Mandelbrot 与分形    4

1.1 Mandelbrot与其的研究课题    4

1.2 分形基础    7

1.2.1基本概念    7

1.2.2 维数    9

2    传统分形    12

2.1 构造分形专用函数包    12

2.2 Cantor    13

2.2.1 Cantor集的构造    13

2.2.2 Cantor集的实现    14

2.3 Kohn曲线    17

2.3.1 Kohn曲线与随机Kohn曲线    17

2.3.2 递归分形中的生成元    20

2.4 Sierpinski    27

2.4.1 Sierpinski三角形    27

2.4.2 Sierpinski地毯    29

3    分形模型和系统    31

3.1 L系统    31

3.1.1 L系统描述方法及算法    31

3.1.2 L系统的植物模拟    33

3.2 迭代函数系统    38

3.2.1 仿射变换与拼贴定理( collage theorm)    38

3.2.2 IFS自然景物模拟    43

3.3 DLA模型    44

3.3.1 分形凝聚体与DLA模型    44

3.3.2自然生长过程模拟及其维数的计算:    46

3.3.3 大尺度下的DLA模型改造    47

4    分形之应用    48

4.1高分子分形产生    49

4.1.1 具有生命活力的高分子分形分析    49

4.1.2 芳香烃族生成元的分形模拟    52

4.2    IFS系统与L系统的综合模拟树、花枝    54

4.3    宇宙中的分形综述    56

结束语    58

参考文献    59

附录    59

摘要

    自从B.Mandelbrot提出分形之后,由于它极其接近于大自然,在很多领域了人们一直进行相关的研究。在本文中,通过计算机编程语言Turbo C模拟出传统分形以及几个具有代表性的模型或者系统产生的分形图形。首先,我们介绍了被称为分形之父的Mandelbrot研究的课题,然后,稍微描述关于分形的定义以及维数方面的概念。其次,根据对分形的粗略理解,我们构建了Cantor集、Kohn曲线和Sierpinski集的生成算法,并绘制出它们的有限步图形。在分析Kohn曲线的过程中,分析构造出一种生成一类分形的方法:等长生成元和可变生成元。利用这个方法构造出各种奇异的线条以及根据生成元线条几何关系计算了部分图形的维数。再次,基于不同的目的,这里更加注重于构造分形的三个模型或系统:L系统,迭代函数系统(IFS)和DLA模型。然后构造出这些模型和系统的算法,同时在生成传统分形的基础上,用这些算法及其相应的程序,对大量的自然事物进行模拟,例如:柳条,枫树叶,树枝和凝聚物的生长过程等等。其间,在L系统模拟中,采用基本几何多边形来细化自然树枝在IFS中利用matlab计算压缩变换的系数;在DLA模型中用matlab计算了凝聚体的盒维数,提出了它在大尺度下的模型改造。最后,利用上面的分形模型算法和程序,我们分别在微观分子尺度、常规尺度和宇宙宏观大尺度上分别考虑并且探讨了分形的应用。从这三个尺度上,先是利用生成元方法分析了像蛋白质这样的高分子分形结构,并且模拟出一些苯的衍生物;随后综合运用L系统和IFS系统模拟出带分形叶子和花朵的枝条;最终从宇宙的观点,探讨了宇宙的分形特征并采用分形观点解释了一些现象例如木星的大红、黑斑,月亮的环形山,并推测出一些内在的结果,如陨星的大小分布。

 

关键字:维数;生成元;传统分形;L系统;迭代函数系统;DLA模型;高分子;宇宙

  

Abstract

Since fractal was advanced by B.Mandelbrot, researches on it in many fields have being con-

ducted, due to its approaching nature extremely. In this paper, some traditional fractals and some, created by several representative models or systems, are simulated by computer in a computer language Turbo C. Firstly, regarded as the father of fractal, Mandelbrot is introduced along with his research subjects. Then, there are certain conceptions about the definitions of fractal and its dimension described ratherish. Secondly, according to glancing comprehension, we construct arithmetics of the form about Cantor set, Kohn curve and Sierpinski set, and plot their finite figures. During the analysis of Kohn curve, a method, named createElement method, is analyzed and conformed in two aspects: equal and variable length. Moreover, some fantastic fractal lines are traced using the method above, and we calculate some dimensions of them by the geometrical relation of the element. Thirdly, dependent on different purposes, we attach more importance to three of fractal models or systems: L-system, IFS system and DLA model. Thereby, the arithmetics about aforementioned methods or systems are built. At the same time, including the traditional fractals, a mass of natural objects are painted by the programs to arithmetics, such as wicker, maple leaf, branch, the growth of agglutinating matters. Meanwhile, in L-system, braches are fine-drawed with leaves made up of polygons; in IFS, the coefficients of constringent transforms are computed by matlab; in DLA model, besides calculating the box-dimension of clots, we rebuild DLA model in the case of large scale. Lastly, making use of the arithmetics and corresponding programs of those fractal methods, we consider and probe into the application of fractal respectively in three scales: small scale of microcosmic molecule, routine scale and large scale of universe. Through these scales, we adopt createElement method to analyze the structure of some macromolecules such as protein, and simulate some ramifications of benzene; thereafter, synthetically utilize L-system and IFS method to create branches with leaves and flowers; in the view of large scale, we discuss the characters of universal fractal, and apply fractal viewpoint to explain some phenomena, for example large red or black spot on Jupiter, ring mountains on moon, and to speculate on connotative results such as the size distribution of meteorites.

Key Word: dimension; createElemet; traditional fractal ; L-system; IFS system; DLA model; macromolecule; universe

 

 

 全文阅读:http://www.360doc.com/showWeb/0/0/263389330.aspx

 

 

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