模糊分形及其在级配骨料研究中的应用

结合模糊集理论和分形理论提出了模糊分形集的概念,将分形论的应用范围从清晰现象推广到了模糊现象领域。模糊分形系统大量存在于自然界中,例如海岸线、混凝土的断裂表面、级配骨料的粒度分布等。级配骨料的分形特征具有模糊性,模糊分维是其粒度分布复杂程度的度量。文中给出模糊分形集在级配骨料研究中的应用实例。     

Non-linear elastic behavior of filler aggregates

A filled rubber network consists of polymer chains which are suspended between filler aggregates. In this contribution, the nonlinear elastic behavior of the aggregated filler particles inside the rubber matrix is investigated. Previously, by using scaling theory, the influence of initial length and fractal dimension of aggregates on the elastic response of aggregated structures was studied. Here we additionally take into account a deformation induced evolution of the aggregate structure. To this end, the directional topology of the aggregate structure is represented by its backbone chain. Thus, the analytical approach proposed describes not only the geometrically but also the physically non-linear behavior of aggregates. Our solution can further be generalized for colloidal structures as for example granular materials or suspended solid structures. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

岩石裂隙网络的分形特征

分形特征与分形维数广泛应用于岩石裂隙网络的量化,及与工程参数的关系模型建立.然而,严格的分形维数的极限定义形式难以直接应用,工程应用中多用近似分形维数值代替,近似的结果在建立量化关系模型时会产生蝴蝶效应,在量化及预测过程中产生巨大偏差.本文回顾了分形研究一系列的发展过程,并基于最新的分形定义提出了一种新的分形维数计算方法.通过对于十个岩石裂隙网络分形维数的计算,证明该方法能够准确有效的计算出图形的复杂度,避免了以往计算分形维数所产生的问题.     

On the arithmetic of fractal dimension using hyperhelices

A hyperhelix is a fractal curve generated by coiling a helix around a rect line, then another helix around the first one, a third around the second… an infinite number of times. A way to generate hyperhelices with any desired fractal dimension is presented, leading to the result that they have embedded an algebraic structure that allows making arithmetic with fractal dimensions and to the idea of an infinitesimal of fractal dimension.     

Fractal dimension evolution and spatial replacement dynamics of urban growth

This paper presents a new perspective of looking at the relation between fractals and chaos by means of cities. Especially, a principle of space filling and spatial replacement is proposed to interpret the fractal dimension of urban form. The fractal dimension evolution of urban growth can be empirically modeled with Boltzmann’s equation. For the normalized data, Boltzmann’s equation is just equivalent to the logistic function. The logistic equation can be transformed into the well-known 1-dimensional logistic map, which is based on a 2-dimensional map suggesting spatial replacement dynamics of city development. The 2-dimensional recurrence relations can be employed to generate the nonlinear dynamical behaviors such as bifurcation and chaos. A discovery is thus made in this article that, for the fractal dimension growth following the logistic curve, the normalized dimension value is the ratio of space filling. If the rate of spatial replacement (urban growth) is too high, the periodic oscillations and chaos will arise. The spatial replacement dynamics can be extended to general replacement dynamics, and bifurcation and chaos mirror a process of complex replacement.     

Fractal dimension of zeolite surfaces by calculation

A theoretical method for the estimation of the fractal dimensions of the pore surfaces of zeolites is proposed. The method is an analogy to the commonly employed box-counting method and uses imaginary meshes of various sizes (s) to trace the pore surfaces determined by the frameworks of crystalline zeolites. The surfaces formed by the geometrical shapes of the secondary building units of zeolites are taken into account for the calculations performed. The characteristics of the framework structures of the zeolites 13X, 5A and silicalite are determined by the help of the solid models of these zeolites and the total numbers of grid boxes intersecting the surfaces are estimated by using equations proposed in this study. As a result, the fractal dimension values of the zeolites 13X, 5A and silicalite are generally observed to vary in significant amounts with the range of mesh size used, especially for the relatively larger mesh sizes that are close to the sizes of real adsorbates. For these relatively larger mesh sizes, the fractal dimension of silicalite falls below 1.60 while the fractal dimension values of zeolite 13X and 5A tend to rise above 2. The fractal dimension values obtained by the proposed method seem to be consistent with those determined by using experimental adsorption data in their relative magnitudes while the absolute magnitudes may differ due to the different size ranges employed. The results of this study show that fractal dimension values much different from 2 (both higher and lower than 2) may be obtained for crystalline adsorbents, such as zeolites, in ranges of size that are close to those of real adsorbates.     

Design and analysis of fractal detector for high resolution radars

Recently, fractal geometry has been used as a tool for improving the detection of targets in radar systems. The fractal dimension is utilized as a feature to distinguish between target and clutter in fractal detectors. In this paper, a general model is proposed for the target and clutter signals in high resolution radar (HRR). The fractal dimensions of the clutter and the target plus clutter are evaluated. Performing statistical tests on the distribution of the fractal dimension, it is proved that a gaussian distribution can approximately model the distribution of the fractal dimension for HRR signals. The fractal detector is designed based on the gaussian distribution of the fractal dimension and its performance is compared with a semi-optimum detector. It is demonstrated that the fractal detector has great capabilities in the rejection of colored clutter. Moreover, we show that the fractal detector is CFAR, i.e., the probability of false alarm remains approximately constant in different interference powers.     

Structure function and spectral density of fractal profiles

Spectral density and structure function for fractal profile are analyzed. It is found that the fractal dimension obtained from spectral density is not exactly the same as that obtained from structure function. The fractal dimension of structure function is larger than that of spectral density for small fractal dimension, and is smaller than that of spectral density for larger fractal dimension. The fractal dimension of structure function strongly depends on the spectral density at low and high wave numbers. The spectral density at low wave number affects the structure function at long distance, especially for small fractal dimension. The spectral density at high wave number affects the structure function at short distance, especially for large fractal dimension. This problem is more serious for bifractal profiles. Therefore, in order to obtain a correct fractal dimension, both spectral density and structure function should be checked.     

On the complexity of simultaneous price-quantity adjustment processes

While the Walrasian price tâtonnement represents the traditional dynamic process in the general equilibrium context with and without production, Walras and other classics designed the process exclusively for pure exchange economies. In productive economies, the short-run output adjustment of existing firms and the entry/exit of firms should be modeled as well. So-called cross-dual processes which represent the classical approach to the dynamics of productive economies are discussed and extended. Complex motion can emerge in a discrete-time version of the original two-dimensional system when the aggregate demand function has a non-standard shape. A simultaneous process of price and short-run quantity adjustment with free entry and exit of competitive firms in a single market with a continuum of firms can generate closed orbits via a Hopf bifurcation when the slope of the demand function is positive at equilibrium. When the continuum economy is replaced by an economy with a finite number of firms, noisy limit cycles and complicated behavior can be observed.     

炼油污水处理的多元方差分析

提高炼油污水生化处理的达标率 ,气浮絮凝处理是关键。我们将某炼油厂目前使用的复合絮凝剂PAFC +HIHON与新研制的复合絮凝剂TIDI +HIHON投入工业运行对比试验。对炼油污水所含的石油类、CODcr、Ar OH等主要污染物进行监测并计算出它们的去除率。对三类指标的去除率运用多元方差分析 ,结果是应用新研制的絮凝剂去除率有显著提高 ,从而为新研制絮凝剂投入工业生产运行提供了可靠的依据     

The growth of fractal dimension of an interface evolution from the interaction of a shock wave with a rectangular block of SF6

The interface between air and a rectangular block of sulphur hexafluoride (SF6), impulsively accelerated by the passage of a planar shock wave, undergoes Richtmyer–Meshkov instability and the flow becomes turbulent. The evolution of the interface was previously simulated using a multi-component model based on a thermodynamically consistent and fully conservative formulation and results were validated against available experimental data (Bates et al. Richtmyer–Meshkov instability induced by the interaction of a shock wave with a rectangular block of SF6, Phys Fluids, 2007; 19:036101). In this study, the CFD results are analyzed using the fractal theory approach and the evolution of fractal dimension of the interface during the transition of the flow into fully developed turbulence is measured using the standard box-counting method. It is shown that as the Richtmyer–Meshkov instability on the interface develops and the flow becomes turbulent, the fractal dimension of the interface increases asymptotically toward a value close to 1.39, which agrees well to those measured for classical shear and fully developed turbulences.     

Dielectric breakdown in solids modeled by DBM and DLA

Using numerical simulation, two stochastic models of electrical treeing in solid dielectrics are compared. These are the diffusion-limited aggregation (DLA) model and the dielectric breakdown model (DBM or η-model). On a linear two-dimensional geometry, the relationship between both models, when the size of the structures is of the order of the experimental samples (the electrode gap is 100 times the length of the discharge channel), is explored by statistical methods. Although there is a one-to-one correspondence between DBM with η=1 and the DLA model when the structure size is very large, the case of rather smaller structures is not well known. From a fractal analysis, employing the method of the correlation function C(r), it follows that average fractal dimension of electrical trees, generated with the DLA or with the DBM (η=1), collapse (up to the numerical uncertainty), on a single curve that “universally” accounts for finite size effects. Even more, from this analysis we conclude that the two curves obtained for DLA and DBM (η=1) cannot be distinguished if one takes into account the error bars. This means that finite size effects in the fractal analysis of DLA and DBM (η=1) are quite the same (despite the differences in the algorithms respectively used to generate the electrical trees). To our knowledge no comparison has ever been made between the similarities and differences of the DBM and DLA approach on a geometry other than the open-planar geometry.     

Temporal variation of velocity components in a turbulent open channel flow: Identification of fractal dimensions

Fractals are objects which have similar appearances when viewed at different scales. Such objects have details at arbitrarily small scales, making them too complex to be represented by Euclidian space; hence, they are assigned a non-integer dimension. Some natural phenomena have been modeled as fractals with success; examples include geologic deposits, topographic surfaces and seismic activities. In particular, time series have been represented as a curve with fractal dimensions between one and two. There are different ways to define fractal dimension, most being equivalent in the continuous domain. However, when applied in practice to discrete data sets, different ways lead to different results. In this study, three methods for estimating fractal dimension are described and two standard algorithms, Hurst’s rescaled range analysis and box-counting method (BC), are compared with the recently introduced variation method (VM). It was confirmed that the last method offers a superior efficiency and accuracy, and hence may be recommended for fractal dimension calculations for time series data. All methods were applied to the measured temporal variation of velocity components in turbulent flows in an open channel in Shiraz University laboratory. The analyses were applied to 2500 measurements at different Reynold’s numbers and it was concluded that a certain degree of randomness may be associated with the velocity in all directions which is a unique character of the flow independent of the Reynold’s number. Results also suggest that the rigid lateral confinement of flow to the fixed channel width allows for designation of a more-or-less constant fractal dimension for the spanwise velocity component. On the contrary, in vertical and streamwise directions more freedom of movements for fluid particles sets more room for variation in fractal dimension at different Reynold’s numbers.     

Linear correlation between fractal dimension of surface EMG signal from Rectus Femoris and height of vertical jump

Fractal dimension was demonstrated to be able to characterize the complexity of biological signals. The EMG time series are well known to have a complex behavior and some other studies already tried to characterize these signals by their fractal dimension.This paper is aimed at studying the correlation between the fractal dimension of surface EMG signal recorded over Rectus Femoris muscles during a vertical jump and the height reached in that jump.Healthy subjects performed vertical jumps at different heights. Surface EMG from Rectus Femoris was recorded and the height of each jump was measured by an optoelectronic motion capture system.Fractal dimension of sEMG was computed and the correlation between fractal dimension and eight of the jump was studied.Linear regression analysis showed a very high correlation coefficient between the fractal dimension and the height of the jump for all the subjects.The results of this study show that the fractal dimension is able to characterize the EMG signal and it can be related to the performance of the jump. Fractal dimension is therefore an useful tool for EMG interpretation.     

The fractal dimension of pullback attractors for the 2D Navier–Stokes equations with delay

This paper is concerned with the bounded fractal and Hausdorff dimension of the pullback attractors for 2D nonautonomous incompressible Navier-Stokes equations with constant delay terms. Using the construction of trace formula with two bases for phase spaces of product flow, the upper boundedness of fractal dimension has been achieved.     

Construction of fractal surfaces by recurrent fractal interpolation curves

A method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system (RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal surfaces by fractal curves and Lipschitz functions and calculate the box-counting dimension of the constructed surfaces. Finally, we combine both methods to have more flexible constructions of fractal surfaces.     

铜基复合材料组织形态分形特征的统计分析与研究

通过对铜基复合材料显微组织结构相图的分析和研究,根据分形理论,计算了不同实验条件下铜基复合材料横截面和平行压制力面的显微组织结构相图的分形维数,同时结合统计方法分析了铜基复合材料分形维数的一些统计特性,结果表明,分形维数反映了石墨在样品中的分布规律,分形维数越大,组织结构相图越复杂,石墨分布越不规则,故石墨分布的不规则性可用分形维数来刻画,分形维数可作为材料组织形态分析的一个表征参数,通过统计分析可知,铜基复合材料横截面和平行压制力面的组织结构相图的分形维数服从正态分布,且横截面和平行压制力面的分形维数随石墨含量变化的情况互不影响。     

界面弧形裂纹对混凝土开裂强度的影响研究

混凝土由于水分蒸发、干缩、泌水以及骨料与砂浆变形不一致等原因会导致骨料与砂浆的界面层中产生弧形裂纹,从而对混凝土开裂强度产生很大影响.从细观角度将混凝土视作由粗骨料和水泥砂浆组成的两相复合材料,并将界面层视为粗骨料与水泥砂浆的接触层进行分析.首先基于相互作用直推估计(interaction direct derivative, IDD)法,考虑混凝土中骨料颗粒的相互作用,将施加在混凝土表征体积元的远场外荷载等效为无限大基体中含单一骨料的等效外荷载.然后,将等效外荷载转化为最大和最小主应力,基于断裂力学理论得到界面层中弧形裂纹的应力强度因子,并根据复合型裂纹幂准则判断弧形裂纹是否发生开裂,进而来研究混凝土开裂强度的变化规律.通过与数值模拟结果的比较,验证了界面弧形裂纹应力强度因子解析解的有效性,参数分析结果表明,当裂纹与最大主应力垂直或与最小主应力呈45°夹角时,骨料周围弧形裂纹最易发生开裂破坏.随着裂纹长度增加,混凝土受拉和受压开裂强度先减小后增大,且均存在最不利的裂纹长度.混凝土开裂强度随着骨料体积分数的增加而增大,随着骨料粒径的增大而减小.在裂纹长度较小时,增大骨料的弹性模量有利于提高混凝土开裂强度.骨料周围承受同号应力可以提高混凝土的开裂强度,反之,异号应力会降低开裂强度.     

Fractal dimension for fractal structures: A Hausdorff approach revisited

In this paper, we use fractal structures to study a new approach to the Hausdorff dimension from both continuous and discrete points of view. We show that it is possible to generalize the Hausdorff dimension in the context of Euclidean spaces equipped with their natural fractal structure. To do this, we provide three definitions of fractal dimension for a fractal structure and study their relationships and mathematical properties.     

Convergence of trajectories in fractal interpolation of stochastic processes

The notion of fractal interpolation functions (FIFs) can be applied to stochastic processes. Such construction is especially useful for the class of α-self-similar processes with stationary increments and for the class of α-fractional Brownian motions. For these classes, convergence of the Minkowski dimension of the graphs in fractal interpolation of the Hausdorff dimension of the graph of original process was studied in [Herburt I, Małysz R. On convergence of box dimensions of fractal interpolation stochastic processes. Demonstratio Math 2000;4:873–88. [11]], [Małysz R. A generalization of fractal interpolation stochastic processes to higher dimension. Fractals 2001;9:415–28. [15]], and [Herburt I. Box dimension of interpolations of self-similar processes with stationary increments. Probab Math Statist 2001;21:171–8. [10]].We prove that trajectories of fractal interpolation stochastic processes converge to the trajectory of the original process. We also show that convergence of the trajectories in fractal interpolation of stochastic processes is equivalent to the convergence of trajectories in linear interpolation.