Showing papers on "Fractal dimension published in 1986"

Fractals and fragmentation

Abstract: The use of renormalization group techniques on fragmentation problems is examined. The equations which represent fractals and the size-frequency distributions of fragments are presented. Method for calculating the size distributions of asteriods and meteorites are described; the frequency-mass distribution for these interplanetary objects are due to fragmentation. The application of two renormalization group models to fragmentation is analyzed. It is observed that the models yield a fractal behavior for fragmentation; however, different values for the fractal dimension are produced . It is concluded that fragmentation is a scale invariant process and that the fractal dimension is a measure of the fragility of the fragmented material.

The fractal facets of turbulence

Abstract: Speculations abound that several facets of fully developed turbulent flows are fractals. Although the earlier leading work of Mandelbrot (1974, 1975) suggests that these speculations, initiated largely by himself, are plausible, no effort has yet been made to put them on firmer ground by, resorting to actual measurements in turbulent shear flows. This work is an attempt at filling this gap. In particular, we examine the following questions: (a) Is the turbulent/non-turbulent interface a self-similar fractal, and (if so) what is its fractal dimension ? Does this quantity differ from one class of flows to another? (b) Are constant-property surfaces (such as the iso-velocity and iso-concentration surfaces) in fully developed flows fractals? What are their fractal dimensions? (c) Do dissipative structures in fully developed turbulence form a fractal set? What is the fractal dimension of this set? Answers to these questions (and others to be less fully discussed here) are interesting because they bring the theory of fractals closer to application to turbulence and shed new light on some classical problems in turbulence - for example, the growth of material lines in a turbulent environment. The other feature of this work is that it tries to quantify the seemingly complicated geometric aspects of turbulent flows, a feature that has not received its proper share of attention. The overwhelming conclusion of this work is that several aspects of turbulence can be described roughly by fractals, and that their fractal dimensions can be measured. However, it is not clear how (or whether), given the dimensions for several of its facets, one can solve (up to a useful accuracy) the inverse problem of reconstructing the original set (that is, the turbulent flow itself).

Dynamics of fractal networks

Abstract: Random structures often exhibit fractal geometry, defined in terms of the mass scaling exponent, D, the fractal dimension. The vibrational dynamics of fractal networks are expressed in terms of the exponent d, the fracton dimensionality. The eigenstates on a fractal network are spatially localized for d less than or equal to 2. The implications of fractal geometry are discussed for thermal transport on fractal networks. The electron-fracton interaction is developed, with a brief outline given for the time dependence of the electronic relaxation on fractal networks. It is suggested that amorphous or glassy materials may exhibit fractal properties at short length scales or, equivalently, at high energies. The calculations of physical properties can be used to test the fractal character of the vibrational excitations in these materials.

Optics of Fractal Clusters Such as Smoke

Abstract: Scalar or vector light of wavelength 2~/k strikes N small refracting and absorbing spherules, each of radius a, which have coagulated into a sparse random cluster with fractal dimension D (for smok...

Surface roughening and the fractal nature of rocks.

Abstract: The microstructure of sedimentary rocks is studied by small-angle neutron scattering for length scales between 5 and 500 \AA{}. In limestones and dolomites, we find that the pore surfaces are effectively smooth above 50 \AA{}, but there is evidence for roughening on shorter length scales. In sandstones and shales, the pore surfaces show fractal character due to the presence of clay. The fractal dimension is nonuniversal. We attribute these observations to impurity effects, which can lower the surface tension and maximize the surface area.

Radial viscous fingers and diffusion-limited aggregation: Fractal dimension and growth sites.

Abstract: Etude de la formation de doigts fractaux visqueux dans une cellule de Hele-Shaw a symetrie radiale

Fractal sandstone pores: Automated measurements using scanning-electron-microscope images

Abstract: An automatic technique has been developed to measure precisely the fractal dimension of the microstructure of sandstones from scanning-electron-microscope (SEM) images of fracture surfaces. The technique involves digitizing the images, filtering, counting geometrical features as a function of feature size, and fitting feature histograms. The magnification of the SEM is changed to cover 2.5 orders of magnitude in feature sizes. A power-law model, which includes the resolution of the digital filter, accounts for the feature size distributions for all magnifications and the scaling from magnification to magnification. Results have been obtained for a dozen sandstones, and the fractal dimension is observed to range from 2.55 to 2.85. The precision for averaged images is \ifmmode\pm\else\textpm\fi{}0.01. In addition, a long-length limit to the fractal regime is defined and measured.

Fractal structure of the interplanetary magnetic field

Abstract: Under some conditions, time series of the interplanetary magnetic field strength and components have the properties of fractal curves. Magnetic field measurements made near 8.5 AU by Voyager 2 from June 5 to August 24, 1981 were self-similar over time scales from approximately 20 sec to approximately 3 x 100,000 sec, and the fractal dimension of the time series of the strength and components of the magnetic field was D = 5/3, corresponding to a power spectrum P(f) approximately f sup -5/3. Since the Kolmogorov spectrum for homogeneous, isotropic, stationary turbulence is also f sup -5/3, the Voyager 2 measurements are consistent with the observation of an inertial range of turbulence extending over approximately four decades in frequency. Interaction regions probably contributed most of the power in this interval. As an example, one interaction region is discussed in which the magnetic field had a fractal dimension D = 5/3.

Structure of the solar granulation.

Abstract: The structure of the solar granulation has been analysed using computer-processed images of two very high resolution (0″.25) white-light pictures obtained at the Pic-du-Midi Observatory. The narrow dispersion in the distribution of granule sizes is not confirmed. On the contrary, it is found that the number of granules increases continuously toward smaller scales; this means that the solar granulation has no characteristic or mean scale. Nevertheless, the granules appear to have a critical scale of 1″.37, at which drastic changes in the properties of granules occur; in particular the fractal dimension changes at the critical scale. The granules smaller than this scale could be of turbulent origin.

Fractal characterization of inhomogeneous geophysical measuring networks

Abstract: The measuring stations of most in situ geophysical networks are spatially distributed in a highly inhomogeneous manner, being mainly concentrated on continents and population centres. When inhomogeneity occurs over a wide range of scales in a space of dimension E, it can be characterized by a fractal dimension Dm. For measuring networks, there is no reason to assume a priori that Dm equals E; it will usually be less than E. The world meteorological network studied here is an example on a surface for which E = 2 (the surface of the Earth) whereas, the network has an empirical dimension Dm ≍ 1.75. Whenever Dm < E, any sufficiently sparsely distributed phenomena (with dimension Dp

On the measurement of the fractal dimension of aggregated particles by electron microscopy : experimental method, corrections and comparison with numerical models

Abstract: Digital annular dark-field images of aggregated iron polydisperse particles are obtained using a computer-controlled STEM. Different methods are discussed on how to extract the fractal dimension of the aggregates, and the sources of error are analysed. The results are compared with computer simulations on a polydisperse version of the cluster-cluster aggregation model. Simulations show that polydispersity does not affect the fractal dimension. The experimental result for the fractal dimension (D=1.9±0.1) is consistent with the cluster-cluster model with linear trajectories Des images digitalisees en fond noir annulaire d'agregats polydisperses de billes de fer sont obtenues avec un microscope electronique pilote par ordinateur. On discute des differentes methodes capables d'extraire la dimension fractale des agregats et on analyse les differentes sources d'erreur. Les resultats sont compares avec des simulations a l'ordinateur utilisant une version polydisperse du modele d'agregation par collage d'amas. Les simulations montrent que la polydispersite n'affecte pas la valeur de la dimension fractale. Le resultat experimental (D=1,9±0,1) est consistant avec un modele d'agregation par collage d'amas avec trajectoires lineaires

Equations for the conductivity of macroscopic mixtures

Abstract: Two equations which describe the conductivity (resistivity) of a wide variety of binary macroscopic conducting mixtures as a function of the conductivity of the components, the volume fraction of each, the space dimension and a single parameter are presented. These equations are interpolations between Bruggeman's symmetric- and asymmetric-effective-media theories. The parameter, possibly a fractal dimension, is determined from a critical composition; for example, in a metal-perfect-insulator mixture it is the metal-insulator transition point. Good agreement is found with a wide range of experimental data.

Characterization and Measurement of Random Fractals

Abstract: Mandelbrot's fractal geometry provides both a description and a mathematical model for many of the seemingly complex shapes found in nature. Such shapes often possess a remarkable invariance under changes of magnification. This statistical self-similarity may be characterized by a fractal dimension D, a number that agrees with our intuitive notion of dimension but need not be an integer. A brief mathematical charactrization of random fractals is presented with emphasis on variations of Mandelbrot's fractional Brownian motion. The important concepts of fractal dimension and exact and statisical self-similarity and self-affinity will be reviewed. The various methods and difficulties of estimating the fractal dimension and lacunarity from experimental images or point sets are summarized.

A fractal approach to the relationship between ore grade and tonnage

Abstract: It is now recognized that a variety of natural processes exhibit scale invariance over a wide range of scales. Scale invariant processes often exhibit a fractal behavior. There are various ways to define fractal behavior; one is to relate the frequency of occurrence to size. Tonnage and grade relations for economic ore deposits exhibit a fractal behavior if the tonnage of ore with a mean grade is proportional to the mean grade raised to a power. We show that such a relation is a good approximation for mercury, copper, and uranium deposits in the United States; the respective fractal dimensions are 2.01, 1.16, and 1.48. The renormalization group approach is a quantitative method for studying scale-invariant processes. If it is assumed that the concentration of elements in ores is statistically scale invariant, the renormalization group approach can be used to derive a fractal relationship between tonnage and mean grade.

Fractal behavior in classical collisional energy transfer

Abstract: A plot of final vibrational action vs initial vibrational phase for a two degree of freedom model for He + I2 collisions is examined and shown to exhibit a chattering region. The chattering region is shown to exhibit a very intricate structure, including an infinite number of regions of regularity which we term icicles. Structure is evident on all scales of examination and a fractal dimension close to 2 is obtained for some parts of the chattering region. The survival probability associated with the complexes shows an initial fast decay, due to the icicles, but can be roughly characterized over a longer and wider time range by a more slowly decaying exponential.

Self-affine fractal sets, i: the basic fractal dimensions

Abstract: The notion of -fractal dimension is explored -for various -fractal curves or dusts that are not self -similar, but are diagonally self - affine. A diagonal self -affinity stretches the coordinates in different ratios. It is showed that, in contrast to the unique -fractal dimension of strictly self-similar sets, one needs in general several distinct notions. Most important are the concepts of dimension obtained via the mass in a sphere and via covering by uniform boxes. One -finds it does not matter which definition is taken, but it matters greatly whether one interpolates or extrapolates. Thus, one obtains two sharply distinct dimensions: a local one, valid on scales well below, and a global one, valid on scales well above, a certain crossover scale.

Scattering exponents for polydisperse surface and mass fractals

Abstract: Scattering exponents are discussed for systems of power-law polydisperse surface and mass fractals. It is found that the scattering exponent for power-law polydisperse mass fractals lies on the interval [0, d] and always depends on the mass fractal dimension. The scattering exponent for power-law polydisperse surface fractals lies on [0, d + 1] and is independent of the surface fractal dimension for a large range of the polydispersity exponent.

Experimental Methods for Studying Fractal Aggregates

Abstract: Porous materials, aggregates and ramified structures frequently display self-similarity, ie, their structure is associated with power-law density-density correlation functions In a real physical system, such behavior has two natural limits: the particle or pore size and the cluster size Within these limits, the system is well described by a fractal or Hausdorff dimension For the experimentalist, it is important to find methods which can isolate this behavior, ie, to distinguish it from the particle and cluster contributions, in order to determine the fractal dimension unambiguously Depending on the characteristic size of the fractal object, different experimental methods can be used We discuss the use of microscopic photographs and give the theory of scattering applied to fractal objects and fractal surfaces We present, in particular, the method of separating the particle and the cluster size contributions Finally we give a short introduction to the dynamical aspects of fractals and to the interpretation of low frequency vibrational spectra of fractal systems in terms of fractons

Fractal dimensions and geometries of caves

Abstract: Lengths of all caves in a region have been observed previously to be distributed hyperbolically, like self-similar geomorphic phenomena identified by Mandetbrot as exhibiting fractal geometra:. Proper cave lengths exhibit a fractal dimension of about 1.4. These concepts are extended to other self-similar geometric properties of caves with the /bllowing consequences. Length of a cave is defined as the sum of sizes of passage-filling, linked modular elements larger than the cave-defining modulus. If total length of all caves in a region is a self-similar fractal, it has a fractal dimension between 2 and 3; and the total number of linked modular elements in a region is a self-similar fractal of the same dimension. Cave volume in any modular element size range may be calculated from the distribution. The expected conditional distribution of modular element sizes in a cave, given length and modulus, also is distributed hyperbolically. Data from Little Brush Creek Cave (Utah) agree and yield a fractal dimension of about 2.8 (like the Menger Sponge). The expected number of modular elements in a cave equals approximately the 0.9 power of length of the cave divided by modulus. This result yields an intriguing ' 'parlor trick. "An algorithm for estimating modular element sizes from survey data provides a means for further analysis of cave surveys.

Fractal geometry of vapor-phase aggregates.

Abstract: Static and dynamic light scattering and small-angle x-ray scattering measurements are reported for vapor-phase aggregates of silica. From the static scattering data the fractal dimension is found to be approx.1.84 +- 0.08, in agreement with the prediction of the cluster-cluster aggregation model. The Rayleigh linewidth is found to vary as the 2.7 power of the momuntum transfer for large momentum transfer.

Fractal viscous fingering in clay slurries

Abstract: The fractal growth of viscous fingers on injection of a low-viscosity fluid into a high-viscosity fluid represents a type of hydrodynamic instability which was first demonstrated by Nittman et al.1,2, by pushing water through an aqueous polymer solution. The conditions for the occurrence of the fractal morphology (in place of classical smooth fingering of the Saffman–Taylor3 type) are a combination of high viscosity contrast and low interfacial tension, which leads to a high capillary number and, most probably, non-newtonian rheological properties2,4. In fact, these are conditions which are easily fulfilled on injecting a solvent into a concentrated suspension of colloidal particles in the same solvent. In particular, we show here that fractal viscous fingering generally occurs when water flows through clay slurries in a radial Hele–Shaw cell, but the fractal dimension of the pattern is strongly dependent on the concentration of the clay slurry. Given the importance of clay/water systems in nature, this result has far-reaching consequences, which we discuss.

Gelation and percolation: swelling effect

Abstract: Small angle neutron scattering experiments were performed on Polyurethane branched polymers synthetized near the gelation threshold. The exponent τ of the size distribution function is measured, on a system cross-linked by polycondensation. We find τ = 2.2 ± 0.04. This leads to a fractal dimension for the polymers in the reaction bath Dp = 2.5 ± 0.09, in agreement with percolation model. In the dilute state, the fractal dimension is D = 1.98 ± 0.03, in agreement with recent predictions. Thus the largest branched polymers are swollen by dilution, and behave as lattice animals.

Dimension and Dynamics for Fractal Recurrent Sets

Abstract: The fractal 'recurrent sets' defined by F. M. Dekking are analysed using subshifts of finite type. We show how Dekking's method is related to a construction due to J. Hutchinson, and prove a conjecture of Dekking concerning conditions under which the best general upper bound for the Hausdorff dimension for recurrent sets is actually equal to the Hausdorff dimension. Introduction Since the publication of Mandelbrot's book [9] there has been wide interest in fractals. The appearence of [9] stimulated several mathematical papers, including one by Hutchinson [8] which analysed strictly self-similar sets using constructions closely related to the full shift spaces and Markov partitions used in the study of dynamical systems. A different construction of fractals was given by Dekking [5]. Here we show how Dekking's method is related to that of Hutchinson. There is a best general upper estimate of Hausdorff dimension for recurrent sets, and we are interested in finding conditions under which this general estimate is actually equal to the Hausdorff dimension. When the scaling map is a similitude, this depends on how the different pieces of the fine structure of a recurrent set intersect. Dekking [7] gives a condition called 'resolvability' which ensures that such intersection is only slight. We prove his conjecture, that resolvability holds if and only if the general estimate is equal to the Hausdorff dimension, by using techniques from dynamical systems. Dekking's formalism is useful for constructing mathematical objects because it gives a large degree of control over the geometric properties of the fractal to be generated. In another paper [2] we use recurrent sets to construct invariant sets and 'canonical' Markov partitions for hyperbolic automorphisms of the 3-torus. I should like to thank F. M. Dekking whose work stimulated my own and C. Series who supervised part of this work. I should also like to thank Noel Lloyd and the referee for their comments. This work was supported by the SERC and the King's College Research Centre. Background and definitions A quantity which gives a notion of the 'rarity' or 'thickness' of a set is Hausdorff dimension. Given Y c U the a-dimensional Hausdorff measure of Y is defined as HMa(Y) = lim infix |diamQ|°: {JC^Y £-»o u i Y Received 1 March 1985. 1980 Mathematics Subject Classification 28A75. J. London Math. Soc. (2) 33 (1986) 89-100