Showing papers on "Fractal dimension published in 1990"

Fractal Geometry: Mathematical Foundations and Applications

Abstract: Part I Foundations: mathematical background Hausdorff measure and dimension alternative definitions of dimension techniques for calculating dimensions local structure of fractals projections of fractals products of fractals intersections of fractals. Part II Applications and examples: fractals defined by transformations examples from number theory graphs of functions examples from pure mathematics dynamical systems iteration of complex functions-Julia sets random fractals Brownian motion and Brownian surfaces multifractal measures physical applications.

Estimating fractal dimension

Abstract: Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools for a variety of scientific disciplines, among which is chaos. Chaotic dynamical systems exhibit trajectories in their phase space that converge to a strange attractor. The fractal dimension of this attractor counts the effective number of degrees of freedom in the dynamical system and thus quantifies its complexity. In recent years, numerical methods have been developed for estimating the dimension directly from the observed behavior of the physical system. The purpose of this paper is to survey briefly the kinds of fractals that appear in scientific research, to discuss the application of fractals to nonlinear dynamical systems, and finally to review more comprehensively the state of the art in numerical methods for estimating the fractal dimension of a strange attractor.

Fractals in physics

Abstract: We give a brief overview of the impact of fractal geometry on physical sciences. In particular we will describe the prototype of fractal growth models and the recent developments in the direction of the formulation of an analytical theory that allows to understand why natural phenomena often give rise to fractal structures. Finally we will show that the large scale properties of the matter distribution in the universe need a radical revision on the of light of these new concepts.

Universal diffusion-limited colloid aggregation

Abstract: The authors study the process of diffusion-limited colloid aggregation (DLCA) using both static and dynamic light scattering. Static light scattering is used to measure the fractal dimension of the clusters as well as their structure factor, which is found to be in good agreement with that obtained from calculation using computer-generated clusters. Dynamic light scattering is used to probe both translational and rotational diffusion motion of the clusters. A method to separate their respective contributions is developed, allowing a quantitative determination of the average hydrodynamic radius. In addition, they determine the ratio of the hydrodynamic radius to the radius of gyration for individual aggregates, and find beta =0.93. A method is developed to scale all the dynamic light scattering data onto a single master curve, whose shape is sensitive to key features of the DLCA process. Good agreement is found between the prediction of the shape of the master curve and that obtained from experiments. Using several completely different colloids, they find that the shape of their master curves are identical, their fractal dimensions are identical and their aggregation kinetics are identical. This provides strong evidence of the universality of the DLCA regime of colloid aggregation.

Fractal processes in soil water retention

Abstract: Numerous empirical models exist for soil water retention and unsaturated hydraulic conductivity data. It has generally been recognized that the empirical fitting coefficients in these models are somehow related to soil texture. However, the fact that they are empirical means that elaborate laboratory experiments must be performed for each soil to obtain values for the parameters. Moreover, empirical models do not shed insight into the fundamental physical principles that govern the processes of unsaturated flow and drainage. We propose a physical conceptual model for soil texture and pore structure that is based on the concept of fractal geometry. The motivation for a fractal model of soil texture is that some particle size distributions in granular soils have already been shown to display self-similar scaling that is typical of fractal objects. Hence it is reasonable to expect that pore size distributions may also display fractal scaling properties. The paradigm that we use for the soil pore size distribution is the Sierpinski carpet, which is a fractal that contains self similar “holes” (or pores) over a wide range of scales. We evaluate the water retention properties of regular and random Sierpinski carpets and relate these properties directly to the Brooks and Corey (or Campbell) empirical water retention model. We relate the water retention curves directly to the fractal dimension of the Sierpinski carpet and show that the fractal dimension strongly controls the water retention properties of the Sierpinski carpet “soil”. Higher fractal dimensions are shown to mimic clay-type soils, with very slow dewatering characteristics and relatively low fractal dimensions are shown to mimic a sandy soil with relatively rapid dewatering characteristics. Our fractal model of soil water retention removes the empirical fitting parameters from the soil water retention models and provides parameters (fractal dimension) which are intrinsic to the nature of the fractal porous structure. The relative permeability functions of Burdine and Mualem are also shown to be fractal directly from fractal water retention results.

Fractal Dimension of Fractured Surfaces: A Universal Value?

Abstract: Different rupture modes and different fracture toughness values were obtained on an aluminium alloy subjected to four different heat treatments. The correlation functions of the bidimensional cuts of the four fractured surfaces were determined. It was found that these surfaces were fractal, and that their fractal dimensions were identical, within experimental error.

Fractal Physiology and Chaos in Medicine

Abstract: Introduction Physiology in Fractal Dimensions Dynamics in Fractal Dimensions Statistics in Fractal Dimensions Applications of Chaotic Attractors Physiologic Networks: The Final Chapter?.

Fractal properties of pulmonary blood flow: characterization of spatial heterogeneity.

Abstract: The heterogeneity of pulmonary blood flow was examined using a fractal analytic procedure, and the results were compared with the traditional gravitational model of flow distribution. 99mTc-labeled macroaggregate was injected intravenously at functional residual capacity in six supine anesthetized dogs. The lungs were fixed in situ and sliced in transverse sections. The slices were imaged on a planar gamma camera, and a three-dimensional array of blood flow measurements was reconstructed for each lung. Fractal analysis was used to examine the spatial heterogeneity or RDs (relative dispersion = SD/mean) as a function of the number of pieces into which the flow array was subdivided. RDs was fractal and could be characterized by a fractal dimension (Ds) of 1.09 +/- 0.02, where a Ds of 1.0 reflects homogeneous flow and 1.5 indicates a random flow distribution. The data fit the fractal model exceptionally well with an average r = 0.98. RDs was examined in gravitational and isogravitational planes and as expected was greatest in the gravitational direction. However, the difference was small, suggesting that gravitation plays a secondary role to an underlying process producing heterogeneity. Within the limits of resolution attained by this study (piece volumes greater than 0.25 cm3), the heterogeneity of pulmonary blood flow is well characterized by a fractal model.

The fractal properties of retinal vessels: embryological and clinical implications.

Abstract: The branching patterns of retinal arterial and venous systems have characteristics of a fractal, a geometrical pattern whose parts resemble the whole. Fluorescein angiogram collages were digitised and analysed, demonstrating that retinal arterial and venous patterns have fractal dimensions of 1.63 +/- 0.05 and 1.71 +/- 0.07, respectively, consistent with the 1.68 +/- 0.05 dimension of diffusion limited aggregation. This finding prompts speculation that factors controlling retinal angiogenesis may obey Laplace's equation, with fluctuations in the distribution of embryonic cell-free spaces providing the randomness needed for fractal behaviour and for the uniqueness of each individual's retinal vascular pattern. Since fractal dimensions characterise how completely vascular patterns span the retina, they can provide insight into the relationship between vascular patterns and retinal disease. Fractal geometry offers a more accurate description of ocular anatomy and pathology than classical geometry, and provides a new language for posing questions about the complex geometrical patterns that are seen in ophthalmic practice.

Fractal geometry of marine snow and other biological aggregates

Abstract: Fractal dimensions of aggregates can potentially be used to classify aggregate morphology as well as to identify coagulation mechanisms. Microbial aggregates of Zoogloea ramigera have a cluster fractal dimension of 1.8±0.3 (± SD), suggesting that these aggregates are formed through cluster-cluster coagulation. An analysis of size-porosity correlations for two types of marine snow aggregates yielded fractal dimensions of 1.39±0.06 and 1.52±0.19, which were lower than values describing inorganic colloidal aggregation.

A simple method to estimate the fractal dimension of a self‐affine series

Abstract: Time/space series of natural variables (e.g., surface topography) are often self-affine, i.e., measurements taken at different resolutions have the same statistical characteristics when rescaled by factors that are generally different for the horizontal and vertical coordinates. Self-affinity implies that the standard deviation measured on a sample spanning a length w is proportional to wH = w2 − D, where H is the Hurst exponent and D is the fractal dimension (1 ≤ D ≤2 for a fractal series). In this paper, a “roughness-length” method based on this property of self-affine series is presented. In practice, the root-mean-square roughness is computed in a number of windows of varying length w, and H is measured from the slope of a log-log plot of roughness versus w. Montecarlo simulations show that the fractal dimension as measured by the roughness-length method is approximately the same as that defined by the power spectrum. The roughness-length method is closely related to the grid fractal dimension, is simple to implement, and can be applied to non-uniformly spaced series.

The Fractal Nature of Premixed Turbulent Flames

Abstract: Two-dimensional flame structure measurements of a freely propagating premixed turbulent flame were examined for fractal character over a range of turbulent Reynolds numbers from 52 to 1431 and Damkohler numbers from 10 to 889. The fractal dimension was found to increase with u'/SL, from 2.13 at u'/SL of 0.25 to 2.32 at u'/SL of 11.9. An inner cutoff was not identified due to limited spatial resolution, however, it was shown that the inner cutoff was smaller than the Gibson scale for the conditions studied. The outer cutoff was found to occur at scales comparable to the integral scale, however, the cutoff was a gradual, rather than sharp, transition from fractal behavior. A heuristic relationship between the turbulent flame surface fractal dimension and u'/SL was developed which agrees well with the fractal dimension measurements made in this study, as well as those made in a number of different flame configurations.

Fractal measures in multiparticle production.

Abstract: A new set of moments {ital G}{sub {ital q}} of the multiplicity distribution in a narrow rapidity window is suggested for analysis so that the chaotic fluctuations of the rapidity distribution can be described in the formalism of multifractals. Phenomenological evidence is given to show that there is a fractal structure in the data of multiparticle production at high energy. The fractal dimension and its generalizations can be determined to provide a description of the spectrum of scaling indices. The implication on the emergence of a new approach to the study of multiparticle production in the framework of multifractals in the theory of chaos is discussed.

On the interpretation of the fractal character of fracture surfaces

Abstract: To examine the usefulness of the fractal concept in quantitative fractography, a series of classical fracture surfaces, namely transgranular cleavage, intergranular fracture, microvoid coalescence, quasicleavage and intergranular microvoid coalescence, are analyzed in terms of fractal geometry. Specifically, the five brittle and ductile fracture modes are studied, from three well characterized steels (a mild steel, a low-alloy steel and a 32 wt% Mn-steel) where the salient microstructural dimensions contributing to the final fracture morphology have been measured. Resulting plots of the mean angular deviation, and Richardson (fractal) plots of the lineal roughness, as a function of the measuring step size, are interpreted with the aid of computer-simulated fracture-surface profiles with known characteristics. It is found that the ranges of resolution, over which the fractal dimension is constant, correspond to the pertinent metallurgical dimensions on the fracture surface, and thus can be related to microstructural size-scales.

Efficient box-counting determination of generalized fractal dimensions

Abstract: An alternative algorithm for the numerical analysis of fractal structures and measures is presented, which consumes computer time and memory only quasiproportionally to the size of the input data set. This efficient tool is applied to various deterministic and random multifractals, in particular to the growth probability measures of diffusion-limited aggregation clusters in two- and three-dimensional embedding space.

Negative fractal dimensions and multifractals

Abstract: A new notion of fractal dimension is defined When it is positive, it effectively falls back on known definitions But its motivating virtue is that it can take negative values, which measure usefully the degree of emptiness of empty sets The main use concerns random multifractals for which f(α)

Microbial growth patterns described by fractal geometry.

Abstract: Fractal geometry has made important contributions to understanding the growth of inorganic systems in such processes as aggregation, cluster formation, and dendritic growth. In biology, fractal geometry was previously applied to describe, for instance, the branching system in the lung airways and the backbone structure of proteins as well as their surface irregularity. This investigation applies the fractal concept to the growth patterns of two microbial species, Streptomyces griseus and Ashbya gossypii. It is a first example showing fractal aggregates in biological systems, with a cell as the smallest aggregating unit and the colony as an aggregate. We find that the global structure of sufficiently branched mycelia can be described by a fractal dimension, D, which increases during growth up to 1.5. D is therefore a new growth parameter. Two different box-counting methods (one applied to the whole mass of the mycelium and the other applied to the surface of the system) enable us to evaluate fractal dimensions for the aggregates in this analysis in the region of D = 1.3 to 2. Comparison of both box-counting methods shows that the mycelial structure changes during growth from a mass fractal to a surface fractal.

Quantitative topographic analysis of fractal surfaces by scanning tunneling microscopy

Abstract: The applicability of models based on fractal geometry to length scales of nanometers is confirmed by Fourier analysis of scanning tunneling microscopy images of a sputter deposited gold film, a copper fatigue fracture surface, and a single crystal silicon fracture surface. Surfaces are characterized in terms of fractal geometry with a Fourier profile analysis, the calculations yielding fractal dimensions with high precision. Fractal models are shown to apply at length scales to 12 A, at which point the STM tip geometry influences the information. Directionality and spatial variation of the topographic structures are measured. For the directions investigated, the gold and silicon appeared isotropic, while the copper fracture surface exhibited large differences in structure. The influences of noise in the images and of intrinsic mathematical scatter in the calculations are tested with profiles generated from fractal Brownian motion and the Weierstrass-Mandelbrot function. Accurate estimates of the fractal dimension of surfaces from STM data result only when images consist of at least 1000–2000 points per line and 1/f-type noise has amplitudes two orders of magnitude lower than the image signal. Analysis of computer generated ideal profiles from the Weierstrass-Mandelbrot function and fractional Brownian motion also illustrates that the Fourier analysis is useful only in determining the local fractal dimension. This requirement of high spatial resolution (vertical information density) is met by STM data. The fact that fractal models can be used at lengths as small as nanometers implies that continued topographic structural analyses may be used to study atomistic processes such as those occurring during fracture of elastic solids.

Physiology in fractal dimensions: Error tolerance

Abstract: The natural variability in physiological form and function is herein related to the geometric concept of a fractal. The average dimensions of the branches in the tracheobronchial tree, long thought to be exponential, are shown to be an inverse power law of the generation number modulated by a harmonic variation. A similar functional form is found for the power spectrum of the QRS-complex of the healthy human heart. These results follow from the assumption that the bronchial tree and the cardiac conduction system are fractal forms. The fractal concept provides a mechanism for the morphogenesis of complex structures which are more stable than those generated by classical scaling (i.e., they are more error tolerant).

Stochastic modelling of electrical treeing: fractal and statistical characteristics

Abstract: A two-dimensional stochastic model of electrical treeing in solid dielectrics has been examined using 'fractal' analysis and statistical methods. It is found that simulated trees display remarkably similar behaviour to that found experimentally. A distribution of dimensions is found and simulated failure probability is well described by the two-parameter Weibull distribution. Several methods of assessing fractal dimension are critically compared and correlations are found between different methods. The influence of model parameters on tree growth behaviour is also examined.

Fast aggregation of colloidal silica

Abstract: The aggregation kinetics of colloidal silica is highly dependent on conditions such as the {ital p}H and salt concentration. In this paper we investigate the aggregation of colloidal silica under conditions that promote rapid growth, contrasting our findings with earlier investigations of the slow growth of silica. A number of interesting effects are observed, including power-law growth of the mean aggregate radius, dependence of the aggregation rate on concentration and the chemical nature of the salt used, a reduced aggregate fractal dimension, fragmentation of the fast aggregates under changing solution conditions, and shear-induced restructuring of aggregates. Finally, we present evidence that the fractal dimension of aggregates is not strongly universal, but depends weakly on such factors as the solution concentration. We conclude that although the diffusion-limited cluster-cluster aggregation model gives a good first-order description of rapid aggregation, real systems exhibit richer behavior that is not given to such a facile interpretation.

Growth of fractal fault patterns.

Abstract: Experiments on the formation of faults in a laboratory model of the Earth's crust are presented; they respect its vertical rheological stratification, a brittle layer on top of ductile layers. As a result of the competition between the different nature of the brittle and ductile-layer deformations, complex fractal patterns of faults are formed with fractal dimension 1.70\ifmmode\pm\else\textpm\fi{}0.05 independent of the fault densities associated with different brittle-ductile coupling. We propose a kinematic formulation of the mechanical problem which suggests an analogy with previously studied fractal-growth problems.

Stochastic analysis of a three-phase fluidized bed; Fractal approach

Abstract: This paper reports on three-phase fluidized beds that have played important roles in various areas of chemical and biochemical processing. The characteristics of such beds are highly stochastic due to the influence of a variety of phenomena, including the jetting and bubbling of the fluidizing medium and the motion of the fluidized particles. A novel approach based on the concept of fractals, has been adopted to analyze these complicated and stochastic characteristics. Specifically, pressure fluctuations in a gas-liquid-solid fluidized bed under different batch operating conditions have been analyzed in terms of Hurst's rescaled range (R/S) analysis, thus yielding the estimates for the so-called Hurst exponent, H. The time series of the pressure fluctuations has a local fractal dimension of d{sub FL} = 2 {minus} H. An H value of 1/2 signifies that the time series follows Brownian motion; otherwise, it follows fractional Brownian motion (FBM), which has been found to be the case for the three-phase fluidized bed investigated.