分形生长的计算机模拟研究
摘要
本文首先介绍了一门新兴学科——分形,从分形的起源、发展等几个方面,简要回顾了分形的发展历史,着重介绍了分形的基本理论和基本要素,并在分形的维数测量方法上做了详细介绍。本文第二部分介绍了经典的扩散限制凝聚(Diffusion-Limited Aggregation)模型(简称DLA模型)的构建、原理以及分形分析方法。并进一步介绍了近年来对DLA模型的改进和研究现状。在保持DLA模型的固有性质(如扩散、凝聚概率等)不变的基础上,我们对DLA模型进行了改进,使凝聚区域半径随着形成分形结构的变化而变化,在最近邻和最近邻-次近邻两种不同粘合方式下我们分别得到了形状各异的分形,并对其形成机理进行了分析和讨论,测得其分维数比经典的DLA模型要小,我们对生长方向受限制的情形也做了理论分析,并得到了与实验结果相似的形貌。
本课题的研究意义是对经典的模型进行改进,得出了新颖的分形体,并对其形成过程进行了初步分析,为实验室膜诱导生长而得到的分形结构的形成机理提供了一些理论依据。
In this work, we first introduced a new subject——fractal. from the origin, basic principle, development etc, we reviewed the history of the development of fractal briefly, summarized the basic theory and the basic elements of fractal, made a particular description on the method of the fractal dimension measurement. In the second part, We introduced the classics model of Diffusion-Limited Aggregation (DLA) and the principles of fractal analysis, and also introduced the improvement the DLA model in recent years. At keeping the most inherent natures (such as diffusion mode, agglutinate probability, etc.) of the DLA model, we improved the DLA model, by changing the radius of the aggregation along with the framework of the formed fractal. The nearest-neighbor (NN) and the nearest-neighbor and the nest nearest-neighbor (NNN) and the limited growth direction of them were all researched, and we obtained variety of fractals, the dimension of the changed model is less than the one of DLA even as parts of them were similar to the experimental result.
The significance of this topic research is that we improved the classics model, had carried on the preliminary analysis to its forming process, which is possibly helpful to the experiment of our laboratory.
本课题的研究意义是对经典的模型进行改进,得出了新颖的分形体,并对其形成过程进行了初步分析,为实验室膜诱导生长而得到的分形结构的形成机理提供了一些理论依据。
In this work, we first introduced a new subject——fractal. from the origin, basic principle, development etc, we reviewed the history of the development of fractal briefly, summarized the basic theory and the basic elements of fractal, made a particular description on the method of the fractal dimension measurement. In the second part, We introduced the classics model of Diffusion-Limited Aggregation (DLA) and the principles of fractal analysis, and also introduced the improvement the DLA model in recent years. At keeping the most inherent natures (such as diffusion mode, agglutinate probability, etc.) of the DLA model, we improved the DLA model, by changing the radius of the aggregation along with the framework of the formed fractal. The nearest-neighbor (NN) and the nearest-neighbor and the nest nearest-neighbor (NNN) and the limited growth direction of them were all researched, and we obtained variety of fractals, the dimension of the changed model is less than the one of DLA even as parts of them were similar to the experimental result.
The significance of this topic research is that we improved the classics model, had carried on the preliminary analysis to its forming process, which is possibly helpful to the experiment of our laboratory.
引文
[1] Mandelbrot B B The Fractal Geometry of Nature NewYork : W.H.Freeman , 1982 p56
[2]孙霞 吴自勤 黄畇 编著 分形原理及其应用 2006 中国科学技术大学出版社 p25
[3]孙霞 吴自勤 黄畇 编著 分形原理及其应用 2006 中国科学技术大学出版社 p26
[4]Mandelbrot B B. Science. 156(1967) 636
[5]李后强,黄立基. 分形漫谈. 科学.1990 第 42 卷(2)p95
[6] B B Mandelbrot, D E Passoja and A J Parllay 1984 Nature, 308 (1984)721.
[7] H J Wiesmann, H R Zeller J. Appl. Phys. 60 (1986)1770.
[8] L V Meisel, M Johnson, P J Cote 1992 Phys. Rev. A. 45(1992) 6989.
[9] Wang M, Zhong S, Yin X B, Zhu J M, Peng R W, Wang Y, Zhang K Q, Ming N B Phys. Rev. Lett. 23 (2001)3827.
[10] J M Williams, T P Beebe J. Phys. Chem. 97 (1993)6249.
[11] B J West Physica D 86(1995)12
[12] S Lovejoy Science. 216(1982)185.
[13] J G Hou, Z Q Wu Phys. Rev. B 40 (1989)1008.
[14] T H Fang, S R Jian and D S Chuu J. Phys. D: Appl. Phys. 36 (2003)878
[15]李水根编 分形 高等教育出版社 2004 p8
[16] Mandelbrot B B. Science 156(1967) 636
[17] Falconer K. Fractal Geometry. 分形几何. 曾文曲,刘世耀译. 沈阳东北工学院出版社. 1991:p42
[18]李水根编 分形 高等教育出版社 2004 p6
[19]邓东皋,孙小礼,张祖贵编. 数学与文化. 北京大学出版社.1990:p338
[20]高安秀树著 分数维 沈步明 常子文译 北京 地震出版社 1989 p14
[21]董连科,王晓伟,王克钢等. 高压物理学报. 4(1990)259
[22]辛厚文 分形理论及其应用(三)合肥:中国科技大学出版社. 1993:p284
[23] T A Witten, L M Sander Phys. Rev. Lett 47 (1981)1400.
[24] J. Nittmann, L. Pietronero and H. J. Wiesmann Phys. Rev. Lett 52(1984)1033
[25]H. Honjo J. Phys. Soc. Japan. 55(1986)2487
[26]吴自勤 王兵, 薄膜生长 科学技术出版社 2001 chapter 10
[27] T A Witten and L M Sander Phys. Rev. B 27(1983)5686
[28]孙霞 吴自勤 黄畇 编著 分形原理及其应用 中国科学技术大学出版社 2006 p120
[29]R. Ball, T A Witten, Phys. Rev.A 29(1984)2966
[30] Miyagawa Y. Honjo H.; Katsuragi H. Journal of Crystal Growth, 240(2002)1
[31]候建国 毕岭松.a—Ge/Au 双层膜分形过程的计算机模拟[J].物理学报, 39(1990)1183.
[32] C. Tang, Phys. Rev. A 31(1985)1977
[33] P Meakin J Kertesz and T Vicsek J. Phys. A: Math. Gen. 21 (1988)1271
[34]孙霞 吴自勤 黄畇 编著 分形原理及其应用 2006 中国科学技术大学出版社 p122
[35] S Miyashita, Y Saito and M Uwaha J. Crystal Growth. 283(2005)533
[2]孙霞 吴自勤 黄畇 编著 分形原理及其应用 2006 中国科学技术大学出版社 p25
[3]孙霞 吴自勤 黄畇 编著 分形原理及其应用 2006 中国科学技术大学出版社 p26
[4]Mandelbrot B B. Science. 156(1967) 636
[5]李后强,黄立基. 分形漫谈. 科学.1990 第 42 卷(2)p95
[6] B B Mandelbrot, D E Passoja and A J Parllay 1984 Nature, 308 (1984)721.
[7] H J Wiesmann, H R Zeller J. Appl. Phys. 60 (1986)1770.
[8] L V Meisel, M Johnson, P J Cote 1992 Phys. Rev. A. 45(1992) 6989.
[9] Wang M, Zhong S, Yin X B, Zhu J M, Peng R W, Wang Y, Zhang K Q, Ming N B Phys. Rev. Lett. 23 (2001)3827.
[10] J M Williams, T P Beebe J. Phys. Chem. 97 (1993)6249.
[11] B J West Physica D 86(1995)12
[12] S Lovejoy Science. 216(1982)185.
[13] J G Hou, Z Q Wu Phys. Rev. B 40 (1989)1008.
[14] T H Fang, S R Jian and D S Chuu J. Phys. D: Appl. Phys. 36 (2003)878
[15]李水根编 分形 高等教育出版社 2004 p8
[16] Mandelbrot B B. Science 156(1967) 636
[17] Falconer K. Fractal Geometry. 分形几何. 曾文曲,刘世耀译. 沈阳东北工学院出版社. 1991:p42
[18]李水根编 分形 高等教育出版社 2004 p6
[19]邓东皋,孙小礼,张祖贵编. 数学与文化. 北京大学出版社.1990:p338
[20]高安秀树著 分数维 沈步明 常子文译 北京 地震出版社 1989 p14
[21]董连科,王晓伟,王克钢等. 高压物理学报. 4(1990)259
[22]辛厚文 分形理论及其应用(三)合肥:中国科技大学出版社. 1993:p284
[23] T A Witten, L M Sander Phys. Rev. Lett 47 (1981)1400.
[24] J. Nittmann, L. Pietronero and H. J. Wiesmann Phys. Rev. Lett 52(1984)1033
[25]H. Honjo J. Phys. Soc. Japan. 55(1986)2487
[26]吴自勤 王兵, 薄膜生长 科学技术出版社 2001 chapter 10
[27] T A Witten and L M Sander Phys. Rev. B 27(1983)5686
[28]孙霞 吴自勤 黄畇 编著 分形原理及其应用 中国科学技术大学出版社 2006 p120
[29]R. Ball, T A Witten, Phys. Rev.A 29(1984)2966
[30] Miyagawa Y. Honjo H.; Katsuragi H. Journal of Crystal Growth, 240(2002)1
[31]候建国 毕岭松.a—Ge/Au 双层膜分形过程的计算机模拟[J].物理学报, 39(1990)1183.
[32] C. Tang, Phys. Rev. A 31(1985)1977
[33] P Meakin J Kertesz and T Vicsek J. Phys. A: Math. Gen. 21 (1988)1271
[34]孙霞 吴自勤 黄畇 编著 分形原理及其应用 2006 中国科学技术大学出版社 p122
[35] S Miyashita, Y Saito and M Uwaha J. Crystal Growth. 283(2005)533