一个关于分形的教学实验 --对大学物理实验课程创新的探索

设计了一个分形（粘性指进）的教学实验，让学生通过动手操作和计算机模拟得出各种粘性指进分形图形，并测量分形维数．从而使他们对目前活跃、有特色的科学概念——分形有所认识，有所学习．     

分形多孔介质的粒子扩散特点(Ⅱ)

本文进一步讨论了粒子在分形结构中的扩散,首先计算测定了各分形结构的谱维数,然后对其布朗运动特点进 行了分析讨论,最后模拟得到了粒子扩散概率分布,它们表现出很多有趣的特征:不均匀性,内含空洞和标度性等.这些 结果可以更好地了解气体在分形结构中的扩散。     

一维凝聚生长的计算机模拟

在一维随机成核生长模型的基础上，考虑粒子的扩散和粒子数密度的影响，在最近邻及次近邻条件下模拟得到一些聚集生长的图形，并分析了其分形特征。     

一维随机成核生长模型的边界效应

通过计算机模拟，研究周期边界与固定边界对一维随机成生长模型的聚集生长图形形状及相应的分形维数的影响。     

分形图形与激光全息防伪技术

利用分形图形具有无限可分和比例尺度上的自相似的特征以及可以产生无限变量重复码的特点，通过计算机控制激光双光束干涉光刻成点阵全息光栅，从而形成数字像素全息图．构造出分形图形的迭代函数系统框架，形成丰富多彩，变化多端的分形图形，以此对点阵全息光栅编码，光刻成分形数字像素全息图．     

基于迭代法对分形物体转动惯量的研究

研究具有分形形态的物体绕定轴的转动惯量,以科赫雪花曲线所围图形为例,给出其绕过形心的某固定轴转动惯量的迭代算法和公式.所得结果与已有文献的精确结果完全一致.所提方法步骤明确,便于编程,且适合于一些推广的分形图形.     

分形分维

 分形理论是非线性科学的三个组成部分(混沌、孤立子和分形)之一。世界就其本质是非线性的,线性只不过是非线性的一个特例。而大多数非线性的几何表现和几何表示是分形理论的研究对象,分形是自然界的几何学。欧几里德几何研究的是规则的光滑图形。而分形研究不规则的不光滑图形。     

分形图形周界和面积的关系

 本文讨论分形图形周界和面积的关系；指出：在现实存在的有限层次自相似结构的近似分形图形中，Mandelbrot的周界-面积分维关系不适用。这可能是一些用此方法测出的分维值随码尺大小变化的根本原因。     

分形海面的微波电磁散射计算模型

利用2维Weierstrass带限函数建立了模拟粗糙海面形状的模型,讨论了分形维数、频率幅度尺度因子等分形参数对海面形状的影响。以粗糙海面形状模型为基础,针对模拟的分形海面形状,从亥姆霍兹积分出发,利用基尔霍夫近似推导出2维分形海面的电磁散射系数,并进行了数值模拟。对微波电磁散射特性随分形维数、频率幅度尺度因子、入射波入射角变化的规律做了进一步讨论分析。在低掠射角时电波会受海面的遮挡,用遮挡函数对散射系数进行修正。研究表明:随着分形维数的增大,散射峰分布变均匀。频率幅度尺度因子越大,散射也越分散。随着入射角的增大,后向散射也逐渐增强,而前向散射逐渐减弱。     

分形多孔介质中流体流动的LBM模拟

基于分形理论构造了规则Sierpinski地毯结构来模拟多孔介质孔隙结构,采用Lattice Boltzmann Method(LBM)分别对不同孔隙率与进出口压差下的分形多孔介质中的流体流动进行了数值模拟.结果表明,分形多孔介质的某些流场结构也具有分形特征;体积流量和流场进出口压差呈线性关系,符合Darcy定律;流体穿过分形多孔介质的体积流量与孔隙率成指数函数关系.     

FRACTAL PATTERN GROWTH OF METAL ATOM CLUSTERS IN ION IMPLANTED POLYMERS

The fractal and multi-fractal patterns of metal atoms are observed in the surface layer and cross section of a metal ion implanted polymer using TEM and SEM for the first time. The surface structure in the metal ion implanted polyethylene terephthalane (PET) is the random fractal. Certain average quantities of the random geometric patterns contain self-similarity. Some growth origins appeared in the fractal pattern which has a dimension of 1.67. The network structure of the fractal patterns is formed in cross section, having a fractal dimension of 1.87. So it can be seen that the fractal pattern is three-dimensional space fractal. We also find the collision cascade fractal in the cross section of implanted nylon, which is similar to the collision cascade pattern in transverse view calculated by the TRIM computer program. Finally, the mechanism for the formation and growth of the fractal patterns during ion implantation is discussed.     

基于分形多孔介质三维网络模型的非混溶两相流驱替数值模拟

提出了一个描述多孔介质孔隙尺寸分布的三维分形网络模型,利用该模型对多孔介质中的非混溶两相流驱替进行了数值模拟,研究了孔隙尺寸分布分维D_f和两相流黏滞比M对驱替前沿指进型的影响,结果表明指进型容量维数D_h随着孔隙尺寸分布分维D_f以及黏滞比M的增大而减少,并通过曲线拟合得到了它们之间的定量关系. 关键词： 多孔介质 三维网络 黏滞指进 非混溶两相流     

Diffraction Fields of Fractally Bounded Apertures

Experimental study is performed of optical diffractions by two-dimensional, self-similar mass and surface fractal apertures as well as mass fractal apertures bounded by surface fractals. Self-similar intensity distributions are observed in the Fraunhofer diffraction fields from surface and mass fractal apertures. Power law decays in average intensities of the diffraction patterns are also investigated for the mass, surface and combined fractal apertures in connection with the fractal dimensions of the objects.     

Computer simulation of viscous fingering in a lifting Hele-Shaw cell with grooved plates

We simulate viscous fingering generated by separating two plates with a constant force, in a lifting Hele-Shaw cell. Variation in the patterns for different fluid viscosity and lifting force is studied. Viscous fingering is strongly affected by anisotropy. We report a computer simulation study of fingering patterns, where circular or square grooves are etched on to the lower plate. Results are compared with experiments.     

Coarsening of fractal viscous fingering patterns

We have measured the coarsening due to surface tension of radially grown fractal viscous fingering patterns. The patterns at late times depend on the structural form at the onset of coarsening, providing information on the age of the fractal. The coarsening process is not dynamically scale invariant, exhibiting two dynamic length scales that grow as L1(t) approximately t(0.22+/-0.02) and L2(t) approximately t(0.31+/-0.02). The measured exponents are in agreement with the results of recent numerical studies of diffusion-controlled coarsening of a diffusion-limited aggregation fractal [Phys. Rev. E 65, 050501 (2002)]].     

Fractal Germanium Patterns: Annealing Strategies and Perspectives of Metal-Induced Crystallization

Semiconductor germanium (Ge) in contact with some metals, such as Al, Pd, and Au, etc., is a class of distinctive materials with non-integer dimensions (D) that differ from integer dimensional materials, such as nanoparticles (0D), nanowires/nanorods//nanotubes/nanoribbons (1D), and thin films (2D). In this article, we describe our efforts toward understanding the annealing strategies and perspectives of metal-induced crystallization for the amorphous Ge embedded in Al, Pd, and Au matrices prepared by high vacuum thermal evaporation techniques, highlighting contributions from our laboratory. First, we present the Al-induced crystallization of amorphous Ge and formation processes of fractal Ge patterns. In addition, the fractal Ge patterns induced by Pd nanoparticles with solid-state reactions will be summarized in detail. Temperature-dependent properties of resistance and fractal dimension in Pd/Ge bilayer films will be expounded. In particular, the nonlinear optical properties are discussed in detail. Finally, we will emphasize the in situ observations by transmission electron microscopy and multi-fractal analysis for the fractal Ge patterns induced by Au nanoparticles. Moreover, the polycondensation-type fractal Ge patterns with non-integer dimensions, thick branches and smooth edges, and metastable gamma-Au_0.6Ge_0.4 are further investigated. The computer simulation indicated that the experimental results are good agreement with the simulation patterns, which were carried out by a ripening mechanism of non uniform grains. This review may provide a novel insight to modulate their competent performance and promote rational design of micro/nanodevices.     

多孔介质热导率的数值计算

通过数值模拟得出面积分形维数和孔隙率相同,而孔隙结构不同的分形多孔介质的导热特性是不相同的.说明仅仅依靠孔隙率和分形维数两个参数无法有效确定多孔介质的热物理性质,寻找新的表征孔隙结构的参数是必要的.同时也说明了前人所得出的关于多孔介质的热物性的解析表达式具有一定的局限性和不确定性.     

稳恒磁场下金属电沉积枝晶生长的模拟

采用Monte-Carlo模拟方法,给出用于描述稳恒磁场作用下电沉积枝晶生长的模型.该模型综合考虑了外加磁场B,电解溶液浓度和离子在阴极发生还原反应的几率P_s等因素对电沉积过程的影响,模拟得到与实验结果一致的枝晶生长图形.模拟结果表明:团簇的形状和它们的分形维数都与外加磁场B的强弱,即体现在模型中离子的旋转角速度ω的大小有关;随着磁场强度的增加,沉积团簇会从分形向非分形转变;在相对强的外加磁场作用下,较高离子浓度时的沉积物是非分形的;离子在阴极的反应概率P_s越小,随着磁场强度的增加枝晶生长越易趋向非分形.     

Dynamics of conformal maps for a class of non-Laplacian growth phenomena

Time-dependent conformal maps are used to model a class of growth phenomena limited by coupled non-Laplacian transport processes, such as nonlinear diffusion, advection, and electromigration. Both continuous and stochastic dynamics are described by generalizing conformal-mapping techniques for viscous fingering and diffusion-limited aggregation, respectively. The theory is applied to simulations of advection-diffusion-limited aggregation in a background potential flow. A universal crossover in morphology is observed from diffusion-limited to advection-limited fractal patterns with an associated crossover in the growth rate, controlled by a time-dependent effective Péclet number. Remarkably, the fractal dimension is not affected by advection, in spite of dramatic increases in anisotropy and growth rate, due to the persistence of diffusion limitation at small scales.