Showing papers on "Fractal dimension published in 1994"
TL;DR: An efficient differential box-counting approach to estimate fractal dimension is proposed and by comparison with four other methods, it has been shown that the method is both efficient and accurate.
Abstract: Fractal dimension is an interesting feature proposed to characterize roughness and self-similarity in a picture. This feature has been used in texture segmentation and classification, shape analysis and other problems. An efficient differential box-counting approach to estimate fractal dimension is proposed in this note. By comparison with four other methods, it has been shown that the authors, method is both efficient and accurate. Practical results on artificial and natural textured images are presented. >
767 citations
TL;DR: In this paper, the problem of the size effects on tensile strength and fracture energy of brittle and disordered materials (concrete, rocks, ceramics, etc.) is reconsidered under a new and unifying light cast on by fractal geometry.
354 citations
Book•
01 Jun 1994
TL;DR: Fractal Landscapes in Physiology & Medicine: Long-Range Correlations in DNA Sequences and Heart Rate Intervals, and how it can be Tuned and Measured.
Abstract: Fractal Structures in Biological Systems.- Mandelbrot's Fractals and the Geometry of Life: A Tribute to Benoit Mandelbrot on his 80th Birthday.- Gas Diffusion through the Fractal Landscape of the Lung: How Deep Does Oxygen Enter the Alveolar System?.- Is the Lung an Optimal Gas Exchanger?.- 3D Hydrodynamics in the Upper Human Bronchial Tree: Interplay between Geometry and Flow Distribution.- Fractal Aspects of Three-Dimensional Vascular Constructive Optimization.- Cognition Network Technology: Object Orientation and Fractal Topology in Biomedical Image Analysis. Method and Applications.- The Use of Fractal Analysis for the Quantification of Oocyte Cytoplasm Morphology.- Fractal Structures in Neurosciences.- Fractal Analysis: Pitfalls and Revelations in Neuroscience.- Ongoing Hippocampal Neuronal Activity in Human: Is it Noise or Correlated Fractal Process?.- Do Mental and Social Processes have a Self-similar Structure? The Hypothesis of Fractal Affect-Logic.- Scaling Properties of Cerebral Hemodynamics.- A Multifractal Dynamical Model of Human Gait.- Dual Antagonistic Autonomic Control Necessary for 1/f Scaling in Heart Rate.- Fractal Structures in Tumours and Diseases.- Tissue Architecture and Cell Morphology of Squamous Cell Carcinomas Compared to Granular Cell Tumours' Pseudo-epitheliomatous Hyperplasia and to Normal Oral Mucosae.- Statistical Shape Analysis Applied to Automatic Recognition of Tumor Cells.- Fractal Analysis of Monolayer Cell Nuclei from Two Different Prognostic Classes of Early Ovarian Cancer.- Fractal Analysis of Vascular Network Pattern in Human Diseases.- Quantification of Local Architecture Changes Associated with Neoplastic Progression in Oral Epithelium using Graph Theory.- Fractal Analysis of Canine Trichoblastoma.- Fractal Dimension as a Novel Clinical Parameter in Evaluation of the Urodynamic Curves.- Nonlinear Dynamics in Uterine Contractions Analysis.- Computer-Aided Estimate and Modelling of the Geometrical Complexity of the Corneal Stroma.- The Fractal Paradigm.- Complex-Dynamical Extension of the Fractal Paradigm and its Applications in Life Sciences.- Fractal-like Features of Dinosaur Eggshells.- Evolution and Regulation of Metabolic Networks.- Cytoskeleton as a Fractal Percolation Cluster: Some Biological Remarks.- A Mystery of the Gompertz Function.- Fractional Calculus and Symbolic Solution of Fractional Differential Equations.- Fox-Function Representation of a Generalized Arrhenius Law and Applications.
349 citations
24 May 1994
TL;DR: It is believed that the fractal dimension will help replace the uniformity and independence assumptions, allowing more accurate analysis for any spatial access method, as well as better estimates for query optimization on multi-attribute queries.
Abstract: We propose the concept of fractal dimension of a set of points, in order to quantify the deviation from the uniformity distribution. Using measurements on real data sets (road intersections of U.S. counties, star coordinates from NASA's Infrared-Ultraviolet Explorer etc.) we provide evidence that real data indeed are skewed, and, moreover, we show that they behave as mathematical fractals, with a measurable, non-integer fractal dimension.Armed with this tool, we then show its practical use in predicting the performance of spatial access methods, and specifically of the R-trees. We provide the first analysis of R-trees for skewed distributions of points: We develop a formula that estimates the number of disk accesses for range queries, given only the fractal dimension of the point set, and its count. Experiments on real data sets show that the formula is very accurate: the relative error is usually below 5%, and it rarely exceeds 10%.We believe that the fractal dimension will help replace the uniformity and independence assumptions, allowing more accurate analysis for any spatial access method, as well as better estimates for query optimization on multi-attribute queries.
274 citations
Book•
01 Jan 1994
TL;DR: Fractal Function Wavelet theory as mentioned in this paper is a well-known extension of the basic wavelet theory and has been applied to the construction of Fractal Sets as Fractal Functions and Fractal Surfaces.
Abstract: (Subchapter Titles): I. Foundations. Mathematical Preliminaries: Analysis and Topology. Probability Theory. Algebra. Construction of Fractal Sets: Classical Fractal Sets. Iterated Function Systems. Recurrent Sets. Graph Directed Fractal Constructions. Dimension Theory: Topological Dimensions. Metric Dimensions. Probabilistic Dimensions. Dimension Results for Self-Affine Fractals. The Box Dimension of Projections. Dynamical Systems and Dimension. II. Fractal Functions and Fractal Surfaces: Fractal Function Construction: The Read-BajraktarevicOperator. Recurrent Sets as Fractal Functions. Iterative Interpolation Functions. Recurrent Fractal Functions. Hidden Variable Fractal Functions. Properties of Fractal Functions. Peano Curves. Fractal Functions of Class C gt gt . Dimension of Fractal Functions: gt Dimension Calculations. Function Spaces and Dimension. Fractal Functions and Wavelets: gt Basic Wavelet Theory. Fractal Function Wavelets. Fractal Surfaces: gt Tensor Product Fractal Surfaces. Affine Fractal Surfaces in R gt n+M gt . Properties of Fractal Surfaces. Fractal Surfaces of Class Ck gt . Fractal Surfaces and Wavelets in R gt n gt : gt Brief Review of Coxeter Groups. Fractal Functions on Foldable Figures. Interpolation on Foldable Figures. Dilation and W gt Invariant Spaces. Multiresolution Analyses. List of Symbols. Bibliography. Author Index. Subject Index.
239 citations
TL;DR: In this article, a numerical model of fracture closure is used to investigate the effects of different profile characteristics (D, A and sample size) on the nature of dilation and contact area, using the natural profiles and synthetic fractional Brownian motion profiles.
Abstract: Thirteen natural rock profiles (Barton and Choubey, 1977) are analyzed for their fractal properties. Most of the profiles were found to approximate fractal curves but some also showed features of specific wavelengths and amplitudes superimposed on fractal characteristics. The profiles showed fractal dimensions from 1.1 to 1.5 covering a range of selfsimilar and self-affine curves. The analysis results suggest a negative correlation between fractal dimension,D, and amplitude,A. Joint roughness coefficients (JRC) show a positive correlation with amplitude,A, and a negative correlation with fractal dimension,D. A numerical model of fracture closure is used to investigate the effects of different profile characteristics (D, A and sample size) on the nature of dilation and contact area, using the natural profiles and synthetic fractional Brownian motion profiles. Smooth profiles (low JRC, highD, lowA) display many small contact regions whereas rough fractures (high JRC, lowD, highA) display few large contact areas. The agreement with published experimental data supports the suggested correlations between JRC and the fractal parameters,A andD. It is suggested that observed scale effects in JRC and joint dilation can be explained by small differential strain discontinuities across fractures, which originate at the time of fracture formation.
211 citations
TL;DR: An in-depth review of the more commonly applied methods used in the determination of the fractal dimension of one-dimensional curves is presented in this article, where many often conflicting opinions about the different methods have been collected and are contrasted with each other.
Abstract: An in-depth review of the more commonly applied methods used in the determination of the fractal dimension of one-dimensional curves is presented. Many often conflicting opinions about the different methods have been collected and are contrasted with each other. In addition, several little known but potentially useful techniques are also reviewed. General recommendations which should be considered whenever applying any method are made.
205 citations
TL;DR: In this article, the specific extinction of carbonaceous aggregates (smoke) formed by combustion sources has been computed based on fractal concepts and specific extinction depends upon the primary particle size, the structure of the aggregate as represented by fractal dimension, the fractal prefactor, and the real and imaginary components of the refractive index of the particle material.
170 citations
TL;DR: In this article, a weak correlation between fracture properties and the fractal dimensions is reported, as well as the relationship between a fractal analysis and the size effect law, where the fracture surfaces are fractal over the measured range of scales.
Abstract: Correlation between fractal dimensions and fracture properties in cementitious material is reported. Wedge splitting tests of specimens as much as 1.52m (5 ft) long and having 7.62cm (3-in.) maximum size aggregate were first performed to determine fracture toughness \IK\dI\dc\N and fracture energy \IG\dF\N. Subsequently, one of the split parts was mapped using a profilometer to provide detailed one-dimensional profiles. Finally, the fractal dimension of the profiles was determined by a specially developed computer program. The fracture surfaces are fractal over the measured range of scales, and the fractal dimension is independent of crack trajectory. There is a weak correlation between the fracture properties and the fractal dimensions. The implications of the fractal nature of the cracked surfaces on the fracture energy \IG\dF\N are discussed, as is the relationship between a fractal analysis and the size effect law.
119 citations
TL;DR: The results suggest that one should not place much reliance in the absolute value of a fractal estimate, but that the estimates do vary monotonically with D and might be useful descriptors in tasks such as image segmentation and description.
114 citations
TL;DR: In this article, a computer aided (CA) particle image processing system was used to analyze images which were obtained under different superficial gas velocities and solid circulating rates at different radial positions.
TL;DR: In this article, the fractal dimension of the Apollonian sphere packing has been computed numerically up to six trusty decimal digits, with an estimate of 2.4739465, where the last digit is questionable.
Abstract: The fractal dimension of the Apollonian sphere packing has been computed numerically up to six trusty decimal digits. Based on the 31 944 875 541 924 spheres of radius greater than 2−19 contained in the Apollonian packing of the unit sphere, we obtained an estimate of 2.4739465, where the last digit is questionable. Two fundamentally different algorithms have been employed. Outlines of both algorithms are given.
TL;DR: This review discusses the theory of fractal geometry using the classic examples of the Von Koch curve, the Cantor set and the Sierpinski gasket to discuss the problems of describing the dimensions of these objects.
TL;DR: In this article, the authors proposed a method for estimating effective fractal dimension of stationary Gaussian surfaces based on the variogram, which is based on log-linear regression and can be applied to non-Gaussian surfaces as well.
Abstract: The fractal dimension D of stationary Gaussian surfaces may be expressed very simply in terms of behaviour of the covariance function near the origin. Indeed, only the covariance of line transect samples is required, and that fact makes practical estimation of D relatively straightforward. The case of non-Gaussian surfaces is more poorly understood, but we might define'effective fractal dimension'in terms of the covariance function, as though the surface were Gaussian. In the present paper we suggest simple practical methods for estimating effective fractal dimension, based on the variogram. Like techniques proposed recently by Taylor and Taylor, ours are founded on log-linear regression
TL;DR: In this paper, it was shown that D = 2-H for self-affine profiles and that the roughness can be defined as the dispersion around a chosen fit to f(x) in an epsilon scale.
Abstract: One-dimensional profiles f(x) can be characterized by a Minkowski-Bouligand dimension D and by a scale-dependent generalized roughness W(f, epsilon ). This roughness can be defined as the dispersion around a chosen fit to f(x) in an epsilon -scale. It is shown that D=limepsilon to 0(2-In W(f, epsilon )/In epsilon ) holds for profiles nowhere differentiable. This establishes a close connection between the roughness and the fractal dimension and proves that D=2-H for self-affine profiles (H is the roughness or Hurst exponent). Two numerical algorithms based on the roughness, one around the local average (f(x))epsilon (usual roughness) and the other around the local RMS straight line (a generalized roughness), are discussed. The estimates of D for standard self-affine profiles are reliable and robust, especially for the last method.
TL;DR: In this article, the authors applied fractal analysis to soil bulk density data measured by X ray computed tomography (CT), a relatively new tool for nondestructively measuring macropore-scale density in soil cores.
Abstract: The size, shape, and arrangement of structured voids 1–10 mm in size play an important role in the transport of water and solutes through soil. However, these characteristics are complex and difficult to quantify. Improved methods are needed to quantify the characteristics of these voids to better understand and predict the behavior of water and solutes passing through them. This study applied fractal analysis to soil bulk density data measured by X ray computed tomography (CT), a relatively new tool for nondestructively measuring macropore-scale density in soil cores. Studies were conducted using undisturbed soil cores (7.6 cm ID) from forested and cultivated sites in the A horizon of a Menfro silt loam soil containing macropores and using two groups of soil cores which were uniformly packed with Menfro aggregates from 1–2 mm in diameter for one group and <1 mm in diameter for the other group. Samples were probed using CT to produce a 512 by 512 digital matrix of CT pixel values corresponding to bulk density. Pixels above a specified CT “cutoff” value were designated as occupied. A box-counting method was used to find the fractal dimension of the perimeters between occupied and unoccupied pixels and of the areas formed by the unoccupied pixels. For length scales from 1 to 10 mm, perimeters and areas of these regions appeared to be fractal systems. Single degree of freedom orthogonal contrast tests determined from analysis of variance showed significant differences between the fractal dimension for (1) forest and cultivated cores versus uniformly packed cores, (2) two groups of uniformly packed cores made of different aggregate sizes, and (3) forest versus cultivated cores.
TL;DR: In this paper, the characteristics of four different methods of estimating the fractal dimension of profiles were compared using synthetic 1024-point profiles generated by three methods, and using two profiles derived from a gridded DEM and two profiles from a laser-scanned soil surface.
Abstract: This paper examines the characteristics of four different methods of estimating the fractal dimension of profiles. The semi-variogram, roughness-length, and two spectral methods are compared using synthetic 1024-point profiles generated by three methods, and using two profiles derived from a gridded DEM and two profiles from a laser-scanned soil surface. The analysis concentrates on the Hurst exponent H,which is linearly related to fractal dimension D,and considers both the accuracy and the variability of the estimates of H.The estimation methods are found to be quite consistent for Hnear 0.5, but the semivariogram method appears to be biased for Happroaching 0 and 1, and the roughness-length method for Happroaching 0. The roughness-length or the maximum entropy spectral methods are recommended as the most suitable methods for estimating the fractal dimension of topographic profiles. The fractal model fitted the soil surface data at fine scales but not at broad scales, and did not appear to fit the DEM profiles well at any scale.
TL;DR: A hierarchical cluster-cluster aggregation computer model is introduced in this paper, which allows one to build random fractal aggregates on a d-dimensional lattice with a fractal dimension fixed a priori.
Abstract: A hierarchical cluster-cluster aggregation computer model is introduced which allows one to build random fractal aggregates on a d-dimensional lattice with a fractal dimension fixed a priori. The algorithm works iteratively by sticking aggregates of the same number of particles at the correct centre-to-centre distance in order to recover the desired scaling. With the more efficient versions of the model, any fractal dimension ranging from 1 up to a d-dependent upper limit DM(d) can be obtained. One estimates DM(2) approximately=1.80+or-0.03 and DM(3) approximately=2.55+or-0.04. Calculations up to d=8 show that the ratio DM(d)/d decreases as d increases.
01 Aug 1994
TL;DR: In this article, in situ photographs of marine snow ranging in size from 1 to 60 mm were used to calculate one and two-dimensional fractal dimensions, D1 and D2, in order to characterize aggregate morphology with respect to aggregate perimeter and cross-sectional area.
Abstract: Seventy seven in situ photographs of marine snow ranging in size from 1 to 60 mm were used to calculate one- and two-dimensional fractal dimensions, D1 and D2, in order to characterize aggregate morphology with respect to aggregate perimeter and cross-sectional area. The lowest fractal dimension of D2 = 1.28 ± 0.11 was calculated for marine snow aggregates composed predominantly of a single type of particle (e.g. diatoms or fecal pellets) containing large amounts of miscellaneous debris. Marine snow formed by the aggregation of fecal pellets (D2 = 1.34 ± 0.16), non-identifiable particles (amorphous, D2 = 1.63 ± 0.72), and diatoms (D2 = 1.86 ± 0.13) had increasingly larger fractal dimensions. When combined into a single group, all marine snow aggregates had a fractal dimension of 1.72 ± 0.07. Larvacean houses, formed originally from a single, nearly spherical particle, were found to have a D2 value close to the Euclidean value of 2. Based on fractal geometrical relationships, D2 should have been equal to previous estimates of D3, a three-dimensional fractal dimension. Instead, the D2 value of 1.72 for the combined group was larger than previous estimates of D3 of 1.39 and 1.52, probably because of the dominant influence of the diatom aggregates on the combined group. Diatom aggregates had the highest fractal dimensions and covered the widest size range of all categories of particles examined.
01 Jan 1994
TL;DR: In this paper, the voltage dependence of electrical tree propagation from both recessed voids and embedded needle electrodes is described in terms of a quantitatively expressed physical model, and voltage dependent changes in the fractal dimension of the trees are used to explain the nonmonotonic behavior observed in embedded needle geometry.
Abstract: The voltage dependence of electrical tree propagation from both recessed voids and embedded needle electrodes is described in terms of a quantitatively expressed physical model. Voltage dependent changes in the fractal dimension of the trees are used to explain the nonmonotonic behavior observed in the embedded needle geometry. The relationship of the model to inverse power law and ‘Z’ lifeline characteristics is briefly discussed.
TL;DR: The fractal dimensions of a habitat, coupled with energy requirements, may be one explanation for the species abundance/body size distribution of the associated animal assemblages.
TL;DR: In this paper, the authors show that the shear bands that appear during pure shear numerical simulations of rocks with a non-associated plastic flow rule form fractal networks, and that the system drives spontaneously into a state in which the length distribution of shear band follows a power law with exponent 2.07.
Abstract: We show that the shear bands that appear during the pure shear numerical simulations of rocks with a non-associated plastic flow rule form fractal networks. The system drives spontaneously into a state in which the length distribution of shear bands follows a power law (self-organized criticality) with exponent 2.07. The distribution of local gradients in deviatoric strain rate has different scaling exponents for each moment, in particular the geometrical fractal dimension is 1.7. Samples of granodiorite sheared under high confining pressure from the Pyrenees are analyzed and their properties compared with the numerical results.
TL;DR: In this article, a method of characterizing the geometry and statistical nature of vegetation patterns and for studying their fractal dimension is proposed, which utilizes the concept of multifractals, and is especially suited to the description of complex patterns.
Abstract: . A method of characterizing the geometry and statistical nature of vegetation patterns and for studying their fractal dimension is proposed. The method utilizes the concept of multifractals, and is especially suited to the description of complex patterns. The properties of multifractals and their role in detecting the scale of vegetation patterns are explained. We suggest an extension of the term multifractal for use in landscape ecology and coenology connected with patterns of many different kinds of points. Relationships between information-statistical functions and the fractal dimensions introduced are shown. A computer-simulated example demonstrates the use of statistical functions and illustrates its applicability in vegetation science.
TL;DR: In this article, the authors show that the fractal dimension of extensive chaos is determined by the average spatial disorder as measured by the spatial correlation length associated with the equal-time two-point correlation function, a measure of the correlations between different regions of the system.
Abstract: SUSTAINED nonequiibrium systems can be characterized by a fractal dimension D⩾0, which can be considered to be a measure of the number of independent degrees of freedom1. The dimension D is usually estimated from time series2 but the available algorithms are unreliable and difficult to apply when D is larger than about 5 (refs 3,4). Recent advances in experimental technique5–8 and in parallel computing have now made possible the study of big systems with large fractal dimensions, raising new questions about what physical properties determine D and whether these physical properties can be used in place of time-series to estimate large fractal dimensions. Numerical simulations9–11 suggest that sufficiently large homogeneous systems will generally be extensively chaotic12, which means that D increases linearly with the system volume V. Here we test an hypothesis that follows from this observation: that the fractal dimension of extensive chaos is determined by the average spatial disorder as measured by the spatial correlation length e associated with the equal-time two-point correlation function —a measure of the correlations between different regions of the system. We find that the hypothesis fails for a representative spatiotemporal chaotic system. Thus, if there is a length scale that characterizes homogeneous extensive chaos, it is not the characteristic length scale of spatial disorder.
TL;DR: This study shows that colorectal polyps have a fractal structure over a defined range of magnification and Euclidean morphometric measurements will be invalid outside precisely defined conditions of resolution and magnification.
Abstract: Colorectal polyps have a subjectively self-similar structure which suggests that these structures may have fractal elements and that the fractal dimension may be a useful morphometric discriminant. The fractal dimensions of images from haematoxylin and eosin-stained sections of 359 colorectal polyps (214 tubulovillous adenomas, 41 'pure' tubular adenomas, 29 'pure' villous adenomas, 68 metaplastic polyps, and 7 inflammatory polyps) were measured using a box-counting method implemented on a microcomputer-based image analysis system. Results were assessed using polychotomous logistic regression, confusion matrices, and kappa statistics. All examined polyps were shown to have a fractal structure in the range of scales examined. The fractal dimension was significantly different between different diagnostic categories (P 0.05). This study shows that colorectal polyps have a fractal structure over a defined range of magnification and Euclidean morphometric measurements will be invalid outside precisely defined conditions of resolution and magnification. The fractal dimension is a better way of quantitating the polyp shape and is a useful morphometric discriminant between diagnostic categories.
TL;DR: The fractal dimension findings paralleled the manifest EEG abnormalities in a way that suggests it has potential clinical utility in metric studies on the EEG, especially when applied to the dementias.
TL;DR: In this article, a Monte Carlo simulation has been developed to describe the gas phase coagulation and sintering of nan-clusters, and the model is modified to include a finite interparticle binding energy.
Abstract: A Monte Carlo simulation has been developed to describe the gas phase coagulation and sintering of nan-oclusters. The cluster-cluster aggregation model is modified to include a finite interparticle binding energy. Particle restructuring and densification (sintering) are incorporated into the model by modifying Kadanoff's algorithm for random particle walks on the surface of the cluster. The effect of sintering on aggregate size distribution and fractal dimension has been investigated in simulations of two-dimensional clusters. The binding energy and the relative rates of aggregation and sintering are the primary variables affecting particle structure. In the initial stages, the sintering process results in aggregates which are compact on small length scales. As time progresses and the aggregates become larger, the sintering process slows down and the fractal dimension of the aggregates decreases. The model is able to track the effect of reactor residence time and temperature on the specific surface area a...
TL;DR: The results indicate that the box counting algorithm is not suitable for fractal analysis on images of cancellous bone and that the fractal appearance of the trabecular network reported previously is artifactual.
Abstract: Fractal analysis has recently been suggested [Med. Phys. 20, 1611-1619 (1993)] as a means to characterize the structure of cancellous bone by measuring the fractal dimension using a box counting algorithm. This work re-examines the possible fractal nature of such structures on nuclear magnetic resonance (NMR) images of cancellous bone by estimating the trabecular boundary length as a function of box size under various experimental conditions. On high-resolution images (pixel sizes on the order of 50 microns) and signal-to-noise ratios of 30, the trabecular boundary turns out to be a smooth surface relative to the achievable resolution and is thus nonfractal. The fractal dimension of the trabecular structure is undefined and can vary significantly as a function of image signal-to-noise ratio. The present work further indicates the "apparent" fractal dimension obtained by box counting to be a reflection of marrow pore size. In conclusion, the results indicate that, at the currently achievable resolution, the box counting algorithm is not suitable for fractal analysis on images of cancellous bone and that the fractal appearance of the trabecular network reported previously is artifactual.
TL;DR: In this paper, the configuration entropy is compared to the multifractal analysis on computer simulated morphologies, and it is shown that at the percolation threshold, the entropy undergoes a maximum and its optimum length a minimum.
Abstract: When the random morphology of ramified or percolating clusters exhibit local fluctuations, the framework of the theory of random percolation with its critical exponents and fractal dimension is still not enough to describe the disorder and the optical properties. We propose an alternative concept: the configuration entropy, that we compare to the multifractal analysis on computer simulated morphologies. At the percolation threshold, the entropy undergoes a maximum and its optimum length a minimum. In contrast with the multifractal analysis, the configuration entropy gives unambiguous results, relatively independent of the finite size of the image.
TL;DR: In this paper, a computer simulation model is developed which is capable of simulating the growth of thermal dendrites in two dimensions, where the thermal field governing the growth is determined by the non-steady state heat diffusion equation but is subject to random fluctuations whose level can be controlled.
Abstract: A computer simulation model is developed which is capable of simulating the growth of thermal dendrites in two dimensions. The thermal field governing the growth is determined by the non-steady state heat diffusion equation but is subject to random fluctuations whose level can be controlled. The anisotropy of growth velocity is shown up by simulating the growth on an appropriate finite grid and allowing growth only into the nearest neighbours. The growth can be followed in real time and the effect of material properties and process parameters can be determined. The morphology is quantified using the fractal dimension. The variation of the fractal dimension with material properties and process parameters is studied. A number of experimentally observed features are exhibited by the simulated results, including the transition to absolute stability at high velocities.