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Journal articles on the topic 'Fractal modelling'

Author: Grafiati

Published: 4 June 2021

Last updated: 2 November 2024

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1

Bolshakov, V. I., V. M. Volchuk, M. A. Kotov, and D. P. Fisunenko. "Aspects of fractal modelling application." Physical Metallurgy and Heat Treatment of Metals 2, no. 2 (97) (2022): 7–18. http://dx.doi.org/10.30838/j.pmhtm.2413.050722.7.858.

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Purpose of research. More than 40 years ago, the theory of fractals applied to model of materials structure and properties firstly. During this time in many publications, the connection between the fractal (fractional) dimension of various materials structural elements and their physical and mechanical properties have confirmed. But unified approach to the organisation of fractal modelling not defined. This article analyses some steps of fractal modelling in order to assess their application to specific cases of predicting quality criteria for metals and concretes. Results. One of the fractal modelling algorithms used in materials science is considered. The algorithm consist of: calculation of the fractal dimension D for the research object according to F. Hausdorff's formula; definition of object self-similarity (invariance with reference to the representation scale); model investigation for compliance with the conditions corresponding to the sensitivity index; choice of a objective function (quality criterion), variables (fractal dimensions of structural elements) and reference points; formalization of the obtained results (selection of an adequate model describing the connection between the fractal structure of the material and its properties); estimation of the fractal object heterogeneity degree according to Rainier's formula for belonging to multifractals; interpretation of the obtained results. Examples of implementation for each step of the fractal modelling algorithm are given. The expediency of supplementing the considered algorithm due to the possibility of applying fractal formalism in the quality criteria ranking by the metal and concrete example is considered. The application of such a systematic approach in fractal modelling allows to improve the investigated material properties prediction based on the analysis of their structure and macrostructure. In turn, this leads to the finding of new structure-property regularities. Conclusions. Variants for supplementing the algorithm for fractal modelling of the structure and properties for metals (steel and cast iron) and concretes are proposed. The application of these algorithms allow the correlation and sensitivity estimation between the fractal dimension of the structure and the properties, as well as the ranking of the quality criteria for the materials based on the analysis of the working range of their values.

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2

Chen, Yanguang. "Characterizing Growth and Form of Fractal Cities with Allometric Scaling Exponents." Discrete Dynamics in Nature and Society 2010 (2010): 1–22. http://dx.doi.org/10.1155/2010/194715.

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Fractal growth is a kind of allometric growth, and the allometric scaling exponents can be employed to describe growing fractal phenomena such as cities. The spatial features of the regular fractals can be characterized by fractal dimension. However, for the real systems with statistical fractality, it is incomplete to measure the structure of scaling invariance only by fractal dimension. Sometimes, we need to know the ratio of different dimensions rather than the fractal dimensions themselves. A fractal-dimension ratio can make an allometric scaling exponent (ASE). As compared with fractal dimension, ASEs have three advantages. First, the values of ASEs are easy to be estimated in practice; second, ASEs can reflect the dynamical characters of system's evolution; third, the analysis of ASEs can be made through prefractal structure with limited scale. Therefore, the ASEs based on fractal dimensions are more functional than fractal dimensions for real fractal systems. In this paper, the definition and calculation method of ASEs are illustrated by starting from mathematical fractals, and, then, China's cities are taken as examples to show how to apply ASEs to depiction of growth and form of fractal cities.

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LAI, PENG-JEN. "HOW TO MAKE FRACTAL TILINGS AND FRACTAL REPTILES." Fractals 17, no. 04 (2009): 493–504. http://dx.doi.org/10.1142/s0218348x09004533.

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Intensive research on fractals began around 1980 and many new discoveries have been made. However, the connection between fractals, tilings and reptiles has not been thoroughly explored. This paper shows that a method, similar to that used to construct irregular tilings in ℜ2 can be employed to construct fractal tilings. Five main methods, including methods in Escher style paintings and the Conway criterion are used to create the fractal tilings. Also an algorithm is presented to generate fractal reptiles. These methods provide a more geometric way to understand fractal tilings and fractal reptiles and complements iteration methods.

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4

LAPIDUS, MICHEL L. "FRACTALS AND VIBRATIONS: CAN YOU HEAR THE SHAPE OF A FRACTAL DRUM?" Fractals 03, no. 04 (1995): 725–36. http://dx.doi.org/10.1142/s0218348x95000643.

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We study various aspects of the question “Can one hear the shape of a fractal drum?”, both for “drums with fractal boundary” (or “surface fractals”) and for “drums with fractal membrane” (or “mass fractals”).

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5

CHEN, YAN-GUANG. "FRACTAL TEXTURE AND STRUCTURE OF CENTRAL PLACE SYSTEMS." Fractals 28, no. 01 (2020): 2050008. http://dx.doi.org/10.1142/s0218348x20500085.

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The boundaries of central place models proved to be fractal lines, which compose fractal texture of central place networks. However, the fractal texture cannot be verified by empirical analyses based on observed data. On the other hand, fractal structure of central place systems in the real world can be empirically confirmed by positive studies, but there are no corresponding models. The spatial structure of classic central place models bears Euclidean dimension [Formula: see text] rather than fractal dimensions [Formula: see text]. This paper is devoted to deriving structural fractals of central place models from the textural fractals. The method is theoretical deduction based on the dimension rules of fractal sets. The main results and findings are as follows. First, the central place fractals were formulated by the [Formula: see text] numbers and [Formula: see text] numbers. Second, three structural fractal models were constructed for central place systems according to the corresponding fractal dimensions. Third, the classic central place models proved to comprise Koch snowflake curve, Sierpinski space filling curve, and Gosper snowflake curve. Moreover, the traffic principle plays a leading role in urban and rural settlements evolution. A conclusion was reached that the textural fractal dimensions of central place models can be converted into the structural fractal dimensions and vice versa, and the structural dimensions can be directly used to appraise human settlement distributions in reality. Thus, the textural fractals can be indirectly employed to characterize the systems of human settlements.

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6

SUZUKI, MASUO. "FRACTAL FORM ANALYSIS." Fractals 04, no. 03 (1996): 237–39. http://dx.doi.org/10.1142/s0218348x96000327.

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7

Gospodinova, Evgeniya. "Methods and Algorithms for Simulation Modelling of Fractal Processes." Innovative STEM Education 1, no. 1 (2019): 48–58. http://dx.doi.org/10.55630/stem.2019.0107.

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The report presents methods and algorithms for simulation modelling of fractal processes. Fractal processes based on fractal Brownian motion, fractal Gaussian noise and, fractal Gaussian noise-wavelet transformation are simulated. Based on the performed comparative analysis of the algorithms for simulation modelling of fractal processes with respect to the accuracy parameter, it follows that the algorithms based on the models of fractal Gaussian noise and fractal Gaussian noise-wavelet transformation have the smallest relative error with respect to the Hurst parameter. The value of the Hurst parameter is one of the most important characteristics determining the degree of self-similarity of fractal processes. The considered algorithms based on these two models can be applied for modelling of physiological data, including cardiological data, because they have fractal properties.

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8

Avery, I., F. R. Hall, and C. E. N. Sturgess. "Fractal modelling of materials." Journal of Materials Processing Technology 80-81 (August 1998): 565–71. http://dx.doi.org/10.1016/s0924-0136(98)00124-1.

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9

Semkow, Thomas M. "Neighborhood Volume for Bounded, Locally Self-Similar Fractals." Fractals 05, no. 01 (1997): 23–33. http://dx.doi.org/10.1142/s0218348x97000048.

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We derive the formulas for neighborhood volume (Minkowski volume in d-dimensions) for fractals which have a curvature bias and are thus bounded. Both local surface fractal dimension and local mass fractal dimension are included as well as a radius of the neighborhood volume comparable with the size of the fractal. We consider two types of the neighborhood volumes: simplified and generalized, as well as the volumes below and above the fractal boundary. The formulas derived are generalizations of the equations for isotropic unbounded fractals. Based on the simplified-volume concept, we establish the procedure for calculating a distribution of physical quantities on bounded fractals and apply it to the distribution of trace elements in soil particles. Using the concept of the generalized volume, we show how an expectation value of a physical process can be calculated on bounded fractals, and apply it to the radon emanation from solid particles.

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10

Coppens, Marc-Olivier, and Gilbert F. Froment. "The Effectiveness of Mass Fractal Catalysts." Fractals 05, no. 03 (1997): 493–505. http://dx.doi.org/10.1142/s0218348x97000395.

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Many porous catalysts have a fractal surface, but only rarely do they have a fractal volume, the main exceptions being extremely porous aerogels. It has been suggested that a fractal shape of their volume would be ideal, because it has an infinite area per unit mass that is easily accessible by the reactants. This paper investigates the efficiency of mass fractals by comparing them with nonfractal catalysts. It is found that the specific surface areas of comparable nonfractal catalysts are of the same order of magnitude, if not higher than those of mass fractals. Despite the high effectiveness factor of mass fractals due to the exceptionally easy accessibility of their active sites, production in a nonfractal catalyst is often higher than in a mass fractal, because of the high porosity of the latter. For some strongly diffusion limited reactions, especially in mesoporous catalysts, an added mass fractal macroporosity, with a finite scaling regime, would increase the yields beyond what is possible with a nonfractal catalyst. Nonetheless, when transport through viscous flow in macropores is very rapid the effective reaction rates in classical bimodal catalysts are higher than in fractal catalysts with their high macroporosity.

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11

PE, JOSEPH L. "ANA'S GOLDEN FRACTAL." Fractals 11, no. 04 (2003): 309–13. http://dx.doi.org/10.1142/s0218348x03002269.

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In his fascinating book Wonders of Numbers, Clifford Pickover introduces the Ana sequence and fractal, two self-referential constructions arising from the use of language. This paper answers Pickover's questions on the relative composition of sequence terms and the dimension of the fractal. In the process, it introduces a novel way of obtaining fractals from iterative set operations. Also, it presents a beautiful variant of the Ana constructions involving the golden ratio. In conclusion, it suggests ways of constructing similar fractals for the Morse-Thue and "Look and Say" sequences.

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TATOM, FRANK B. "THE RELATIONSHIP BETWEEN FRACTIONAL CALCULUS AND FRACTALS." Fractals 03, no. 01 (1995): 217–29. http://dx.doi.org/10.1142/s0218348x95000175.

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The general relationship between fractional calculus and fractals is explored. Based on prior investigations dealing with random fractal processes, the fractal dimension of the function is shown to be a linear function of the order of fractional integro-differentiation. Emphasis is placed on the proper application of fractional calculus to the function of the random fractal, as opposed to the trail. For fractional Brownian motion, the basic relations between the spectral decay exponent, Hurst exponent, fractal dimension of the function and the trail, and the order of the fractional integro-differentiation are developed. Based on an understanding of fractional calculus applied to random fractal functions, consideration is given to an analogous application to deterministic or nonrandom fractals. The concept of expressing each coordinate of a deterministic fractal curve as a “pseudo-time” series is investigated. Fractional integro-differentiation of such series is numerically carried out for the case of quadric Koch curves. The resulting time series produces self-similar patterns with fractal dimensions which are linear functions of the order of the fractional integro-differentiation. These curves are assigned the name, fractional Koch curves. The general conclusion is reached that fractional calculus can be used to precisely change or control the fractal dimension of any random or deterministic fractal with coordinates which can be expressed as functions of one independent variable, which is typically time (or pseudo-time).

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13

Zhao, Shanrong, Jin Tan, Jiyang Wang, Xiaohong Xu, and Hong Liu. "A Dendrite with "Sierpinski Gasket" Fractal Morphology in Matt Glaze of LiAlSiO4-SiO2 System." Fractals 11, no. 03 (2003): 271–76. http://dx.doi.org/10.1142/s0218348x03001525.

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In this paper, we introduce a dendritic crystal, formed in matt glaze of LiAlSiO 4- SiO 2, having "Sierpinski gasket" fractal morphology. The crystal structure of this "Sierpinski gasket" dendrite is β-quartz. β-quartz can grow two kinds of fractal patterns: snow-shaped dendrite and "Sierpinski gasket" dendrite, depending on different supercooling conditions. These two kinds of fractals can develop together in one dendritic crystal. The evolution of the boundary morphologies between these two kinds of fractal dendrites can be described by another fractal — Koch curve. The "Sierpinski gasket" dendrite is a rather new fractal growth pattern which can introduce new opportunities to fractal growth research of nonlinear sciences.

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14

JAMPOUR, MAHDI, MAHDI YAGHOOBI, MARYAM ASHOURZADEH, and ADEL SOLEIMANI. "A NEW FAST TECHNIQUE FOR FINGERPRINT IDENTIFICATION WITH FRACTAL AND CHAOS GAME THEORY." Fractals 18, no. 03 (2010): 293–300. http://dx.doi.org/10.1142/s0218348x10005020.

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Fingerprints are one of the simplest and most reliable human biometric features for identification. Geometry of the fingerprint is fractal and we can classify a fingerprint database with fractal dimension, but one can't identify a fingerprint with fractal dimension uniquely. In this paper we present a new approach for identifying fingerprint uniquely; for this purpose a new fractal is initially made from a fingerprint by using Fractal theory and Chaos Game theory. While making the new fractal, five parameters that can be used in identification process can be achieved. Finally a fractal is made for each fingerprint, and then by analyzing the new fractal and parameters obtained by Chaos Game, fingerprint identification can be performed. We called this method Fingerprint Fractal Identification System (FFIS). The presented method besides having features of fractals such as stability against turning, magnifying, deleting a part of image, etc. also has a desirable speed.

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15

MILLÁN, H., M. AGUILAR, J. DOMÌNGUEZ, L. CÈSPEDES, E. VELASCO, and M. GÒNZALEZ. "A NOTE ON THE PHYSICS OF SOIL WATER RETENTION THROUGH FRACTAL PARAMETERS." Fractals 14, no. 02 (2006): 143–48. http://dx.doi.org/10.1142/s0218348x06003131.

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Fractals are important for studying the physics of water transport in soils. Many authors have assumed a mass fractal structure while others consider a fractal surface approach. Each model needs comparisons on the same data set in terms of goodness-of-fit and physical interpretation of parameters. In this note, it is shown, with some representative data sets, that a pore-solid interface fractal model could fit soil water retention data better than a mass fractal model. In addition to the interfacial fractal dimension, this model predicts the tension at dryness. This value is very close to 106 kPa as theoretically predicted.

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MEI, MAOFEI, BOMING YU, JIANCHAO CAI, and LIANG LUO. "A HIERARCHICAL MODEL FOR MULTI-PHASE FRACTAL MEDIA." Fractals 18, no. 01 (2010): 53–64. http://dx.doi.org/10.1142/s0218348x1000466x.

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The size distributions of solid particles and pores in porous media are approximately hierarchical and statistically fractals. In this paper, a model for single-phase fractal media is constructed, and the analytical expressions for area, fractal dimension and distribution function for solid particles are derived. The distribution function of solid particles obtained from the proposed model is in good agreement with available experimental data. Then, a model for approximate two-phase fractal media is developed. Good agreement is found between the predicted fractal dimensions for pore space from the two-phase fractal medium model and the existing measured data. A model for approximate three-phase fractal media is also presented by extending the obtained two-phase model.

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17

IGNATOWICH, MICHAEL J., DANIEL J. KELLEHER, CATHERINE E. MALONEY, DAVID J. MILLER, and KHRYSTYNA SERHIYENKO. "RESISTANCE SCALING FACTOR OF THE PILLOW AND FRACTALINA FRACTALS." Fractals 23, no. 02 (2015): 1550018. http://dx.doi.org/10.1142/s0218348x15500188.

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Much is known in the analysis of a finitely ramified self-similar fractal when the fractal has a harmonic structure: a Dirichlet form which respects the self-similarity of a fractal. What is still an open question is when such a structure exists in general. In this paper, we introduce two fractals, the fractalina and the pillow, and compute their resistance scaling factor. This is the factor which dictates how the Dirichlet form scales with the self-similarity of the fractal. By knowing this factor one can compute the harmonic structure on the fractal. The fractalina has scaling factor [Formula: see text], and the pillow fractal has scaling factor [Formula: see text].

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18

XIANG, ZHIYANG, KAI-QING ZHOU, and YIBO GUO. "GAUSSIAN MIXTURE NOISED RANDOM FRACTALS WITH ADVERSARIAL LEARNING FOR AUTOMATED CREATION OF VISUAL OBJECTS." Fractals 28, no. 04 (2020): 2050068. http://dx.doi.org/10.1142/s0218348x20500681.

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Because of the self-similarity properties of nature, fractals are widely adopted as generators of natural object multimedia contents. Unfortunately, fractals are difficult to control due to their iterated function systems, and traditional researches on fractal generating visual objects focus on mathematical manipulations. In Generative Adversarial Nets (GANs), visual object generators can be automatically guided by a single image. In this work, we explore the problem of guiding fractal generators with GAN. We assume that the same category of fractal patterns is produced by a group of parameters of initial patterns, affine transformations and random noises. Connections between these fractal parameters and visual objects are modeled by a Gaussian mixture model (GMM). Generator trainings are performed as gradients on GMM instead of fractals, so that evaluation numbers of iterated function systems are minimized. The proposed model requires no mathematical expertise from the user because parameters are trained by automatic procedures of GMM and GAN. Experiments include one 2D demonstration and three 3D real-world applications, where high-resolution visual objects are generated, and a user study shows the effectiveness of artificial intelligence guidances on fractals.

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ZENG, QIUHUA, and HOUQIANG LI. "DIFFUSION EQUATION FOR DISORDERED FRACTAL MEDIA." Fractals 08, no. 01 (2000): 117–21. http://dx.doi.org/10.1142/s0218348x00000123.

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The movement of the fractal Brownian particle in isotropic and homogeneous two-dimensional assembling fractal spaces is studied by the standard diffusion equation on fractals, and we find that particle movement belongs to the anomalous diffusion. At the same time, by discussing the defectiveness of earlier proposed equations, a general form of analytic fractional diffusion equation is proposed for description of probability density of particles diffusing on fractal geometry at fractal time, and the solution connects with the ordinary solutions in the normal space time limit.

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STRUZIK, ZBIGNIEW R. "SOLVING THE TWO-DIMENSIONAL INVERSE FRACTAL PROBLEM WITH THE WAVELET TRANSFORM." Fractals 04, no. 04 (1996): 469–75. http://dx.doi.org/10.1142/s0218348x96000583.

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The methodology of the solution to the inverse fractal problem with the wavelet transform1,2 is extended to two-dimensional self-affine functions. Similar to the one-dimensional case, the two-dimensional wavelet maxima bifurcation representation used is derived from the continuous wavelet decomposition. It possesses translational and scale invariance necessary to reveal the invariance of the self-affine fractal. As many fractals are naturally defined on two-dimensions, this extension constitutes an important step towards solving the related inverse fractal problem for a variety of fractal types.

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21

Li, Baihua, and James Miller. "Fractal Cityscape." International Journal of Virtual Reality 9, no. 4 (2010): 7–12. http://dx.doi.org/10.20870/ijvr.2010.9.4.2785.

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There is an increasing demand for the simulation of large-scale "free-flow" urban environments in games or other virtual reality worlds. Currently most 3D cityscapes are generated by time consuming manually pre-scripted "room-to-room" approaches. Although the effectiveness of fractal modelling of rural landscape or architecture has been studied extensively, there is a relative dearth of research on fractal modelling of cityscapes. Inspired by modern fractal city theory, we intend to develop an automatic fractal-based method to address this problem. In particular, we will investigate how fractal theory can be used to mathematically simulate the multi-level hierarchical urban structure corresponding to various land use property probabilities of city regions. Tentative findings and results are presented in this paper

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22

JÁNOSI, IMRE M., and ANDRÁS CZIRÓK. "FRACTAL CLUSTERS AND SELF-ORGANIZED CRITICALITY." Fractals 02, no. 01 (1994): 153–68. http://dx.doi.org/10.1142/s0218348x94000156.

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Self-organized criticality (SOC) and fractals have been shown to be related in various ways. On the one hand, the original idea of SOC suggests that the common explanation of the origin of fractal shapes in nature may be based on self-organized processes. Thus different models exhibiting SOC result in relaxation clusters or avalanches whose geometrical characteristics could be described by fractals. On the other hand, there exist several models for fractal growth phenomena, such as viscous fingering, invasion percolation, dielectric breakdown, etc., and it is possible that the concept of SOC may help in finding the common feature of these models. In this paper we review the recent results on self-organized critical behaviour in various fractal growth models. Next we discuss the relation of fractals and self-organized criticality by concentrating on the geometrical properties of SOC clusters in 2–4 dimensions. A short analysis of the cluster growth processes is given as well.

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23

Giacomazzi, Eugenio, Claudio Bruno, and Bernardo Favini. "Fractal modelling of turbulent mixing." Combustion Theory and Modelling 3, no. 4 (1999): 637–55. http://dx.doi.org/10.1088/1364-7830/3/4/303.

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Giacomazzi, Eugenio, Claudio Bruno, and Bernardo Favini. "Fractal modelling of turbulent combustion." Combustion Theory and Modelling 4, no. 4 (2000): 391–412. http://dx.doi.org/10.1088/1364-7830/4/4/302.

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Saunders, Russell W., and John M. C. Plane. "Fractal growth modelling of nanoparticles." Journal of Aerosol Science 37, no. 12 (2006): 1737–49. http://dx.doi.org/10.1016/j.jaerosci.2006.08.007.

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ZHANG, XIN-MIN, L. RICHARD HITT, BIN WANG, and JIU DING. "SIERPIŃSKI PEDAL TRIANGLES." Fractals 16, no. 02 (2008): 141–50. http://dx.doi.org/10.1142/s0218348x08003934.

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We generalize the construction of the ordinary Sierpiński triangle to obtain a two-parameter family of fractals we call Sierpiński pedal triangles. These fractals are obtained from a given triangle by recursively deleting the associated pedal triangles in a manner analogous to the construction of the ordinary Sierpiński triangle, but their fractal dimensions depend on the choice of the initial triangles. In this paper, we discuss the fractal dimensions of the Sierpiński pedal triangles and the related area ratio problem, and provide some computer-generated graphs of the fractals.

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Zair, C. E., and E. Tosan. "Fractal Geometric Modeling in Computer Graphics." Fractals 05, supp02 (1997): 45–61. http://dx.doi.org/10.1142/s0218348x97000826.

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Free form techniques and fractals are complementary tools for modeling respectively man-made objects and complex irregular shapes. Fractal techniques, having the advantage of describing self-similar objects, suffer from the drawback of a lack of control of the fractal figures. In contrast, free form techniques provide a high flexibility with smooth figures. Our work focuses on the definition of an IFS-based model designed to inherit the advantages of fractals and free form techniques (control by a set of control points, convex hull) in order to manipulate fractal figures in the way as classical free form shapes (BÉZIER, spline). The work is essentially based on the study of the functional equation Φ(τ * t) = T Φ (t), where Φ is a continuous function, τ and T are both contractive affine operators. We prove that there is a strong relationship between this functional equation and IFS attractors. This relationship will be used for the construction of parametric fractal attractors.

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BAK, PER, and MAYA PACZUSKI. "THE DYNAMICS OF FRACTALS." Fractals 03, no. 03 (1995): 415–29. http://dx.doi.org/10.1142/s0218348x95000345.

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Fractals are formed by avalanches, driving the system toward a critical state. This critical state is a fractal in d spatial plus one temporal dimension. Long range spatial and temporal properties are described by different cuts in this fractal attractor. We unify the origin of fractals, 1/f noise, Hurst exponents, Levy flights, and punctuated equilibria in terms of avalanche dynamics, and elucidate their relationships through analytical and numerical studies of simple models.

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CAI, JIANCHAO, FERNANDO SAN JOSÉ MARTÍNEZ, MIGUEL ANGEL MARTÍN, and EDMUND PERFECT. "AN INTRODUCTION TO FLOW AND TRANSPORT IN FRACTAL MODELS OF POROUS MEDIA: PART I." Fractals 22, no. 03 (2014): 1402001. http://dx.doi.org/10.1142/s0218348x14020010.

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This special issue gathers together a number of recent papers on fractal geometry and its applications to the modeling of flow and transport in porous media. The aim is to provide a systematic approach for analyzing the statics and dynamics of fluids in fractal porous media by means of theory, modeling and experimentation. The topics covered include lacunarity analyses of multifractal and natural grayscale patterns, random packing's of self-similar pore/particle size distributions, Darcian and non-Darcian hydraulic flows, diffusion within fractals, models for the permeability and thermal conductivity of fractal porous media and hydrophobicity and surface erosion properties of fractal structures.

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WANG, LEI, and SHENGWEN TANG. "EDITORIAL: AN INTRODUCTION TO FRACTALS IN CONSTRUCTION MATERIALS." Fractals 29, no. 02 (2021): 2102001. http://dx.doi.org/10.1142/s0218348x21020011.

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In recent years, the application of fractal theory in construction materials has drawn tremendous attention worldwide. This special issue section containing seven papers publishes the recent advances in the investigation and application of fractal-based approaches implemented in construction materials. The topics covered in this introduction mainly include: (1) the fractal characterization of construction materials from nano- to micro-scales; (2) combining fractals methods with other theoretical, numerical and/or experimental methods to evaluate or predict the macroscopic behavior of construction materials; (3) the relationship of fractal dimension with the macro-properties (i.e. mechanical property, shrinkage behavior, permeability, frost resistance, abrasion resistance, etc.) of construction materials.

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VINOY, K. J., JOSE K. ABRAHAM, and V. K. VARADAN. "IMPACT OF FRACTAL DIMENSION IN THE DESIGN OF MULTI-RESONANT FRACTAL ANTENNAS." Fractals 12, no. 01 (2004): 55–66. http://dx.doi.org/10.1142/s0218348x04002288.

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During the last few decades, fractal geometries have found numerous applications in several fields of science and engineering such as geology, atmospheric sciences, forest sciences, physiology and electromagnetics. Although the very fractal nature of these geometries have been the impetus for their application in many of these areas, a direct quantifiable link between a fractal property such as dimension and antenna characteristics has been elusive thus far. In this paper, the variations in the input characteristics of multi-resonant antennas based on generalizations of Koch curves and fractal trees are examined by numerical simulations. Schemes for such generalizations of these geometries to vary their fractal dimensions are presented. These variations are found to have a direct influence on the primary resonant frequency, the input resistance at this resonance, and ratios resonant frequencies of these antennas. It is expected that these findings would further enhance the popularity of the study of fractals.

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ELIA, MATTEO, and ROBERTO PEIRONE. "EIGENFORMS ON FRACTALS WITH CONNECTED INTERIOR AND THREE VERTICES." Fractals 26, no. 04 (2018): 1850082. http://dx.doi.org/10.1142/s0218348x18500822.

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An important problem in analysis on fractals is the existence and the determination of an eigenform on a given finitely ramified fractal. It is known that on every fractal either with three vertices or with connected interior, an eigenform exists for suitable weights on the cells. In this paper, we prove that if the fractal has three vertices and connected interior, the form having all coefficients equal to [Formula: see text] is an eigenform for suitable weights on the cells.

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HUYNH, HOAI NGUYEN, and LOCK YUE CHEW. "ARC-FRACTAL AND THE DYNAMICS OF COASTAL MORPHOLOGY." Fractals 19, no. 02 (2011): 141–62. http://dx.doi.org/10.1142/s0218348x11005178.

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In this paper, we present an idea of creating fractals by using the geometric arc as the basic element. This approach of generating fractals, through the tuning of just three parameters, gives a universal way to obtain many novel fractals including the classic ones. Although this arc-fractal system shares similar features with the well-known Lindenmayer system, such as the same set of invariant points and the ability to tile the space, they do have different properties. One of which is the generation of pseudo-random number, which is not available in the Lindenmayer system. Furthermore, by assuming that coastline formation is based purely on the processes of erosion and deposition, the arc-fractal system can also serve as a dynamical model of coastal morphology, with each level of its construction corresponds to the time evolution of the shape of the coastal features. Remarkably, our results indicate that the arc-fractal system can provide an explanation on the origin of fractality in real coastline.

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LI, TINGTING, KAN JIANG, and LIFENG XI. "AVERAGE DISTANCE OF SELF-SIMILAR FRACTAL TREES." Fractals 26, no. 01 (2018): 1850016. http://dx.doi.org/10.1142/s0218348x18500160.

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In this paper, we introduce a method which can generate a family of growing symmetrical tree networks. The networks are constructed by replacing each edge with a reduced-scale of the initial graph. Repeating this procedure, we obtain the fractal networks. In this paper, we define the average geodesic distance of fractal tree in terms of some integral, and calculate its accurate value. We find that the limit of the average geodesic distance of the finite networks tends to the average geodesic distance of the fractal tree. This result generalizes the paper [Z. Zhang, S. Zhou, L. Chen, M. Yin and J. Guan, Exact solution of mean geodesic distance for Vicsek fractals, J. Phys. A: Math. Gen. 41(48) (2008) 7199–7200] for which the mean geodesic distance of Vicsek fractals was considered.

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El-Basil, Sherif. "Fractal relations. Modelling fractals and analogy with Feigenbaum's period-doubling diagram." Journal of Molecular Structure: THEOCHEM 313, no. 2 (1994): 237–64. http://dx.doi.org/10.1016/0166-1280(94)85006-2.

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Hasegawa, M., J. C. Liu, K. Okuda, and M. Nunobiki. "A New Approach to Modelling Machined Surface Profiles." Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 210, no. 2 (1996): 177–82. http://dx.doi.org/10.1243/pime_proc_1996_210_103_02.

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A new approach to modelling machined surface profiles is presented in this paper. The fractal characteristics of the ARMA model are first investigated. It is found that the ARMA model behaves in a fractal manner. Then, based on this result, the relationships between the parameters of the ARMA model and its fractal dimensions are studied by use of the multiple regression analysis. In this way, the parameters of the ARMA model of a machined surface profile can be decided by calculating its fractal dimension. The way the fractal dimensions of the machined surface profiles are affected by the machining conditions is also studied. Therefore using these results, the ARMA model of the machined surface profile can be derived directly from the machining conditions. The AIC (Akaike information criterion) is used as the selection criterion for the model obtained using this approach. Finally, modelling of cut surface profiles is carried out.

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RAMÍREZ, JOSÉ L., GUSTAVO N. RUBIANO, and BORUT JURČIČ ZLOBEC. "GENERATING FRACTAL PATTERNS BY USING p-CIRCLE INVERSION." Fractals 23, no. 04 (2015): 1550047. http://dx.doi.org/10.1142/s0218348x15500474.

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In this paper, we introduce the [Formula: see text]-circle inversion which generalizes the classical inversion with respect to a circle ([Formula: see text]) and the taxicab inversion [Formula: see text]. We study some basic properties and we also show the inversive images of some basic curves. We apply this new transformation to well-known fractals such as Sierpinski triangle, Koch curve, dragon curve, Fibonacci fractal, among others. Then we obtain new fractal patterns. Moreover, we generalize the method called circle inversion fractal be means of the [Formula: see text]-circle inversion.

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Shalaev, V. M., E. Y. Poliakov, V. A. Markel, V. P. Safonov, and A. K. Sarychev. "Surface-Enhanced Optical Nonlinearities of Nanostructured Fractal Materials." Fractals 05, supp02 (1997): 63–82. http://dx.doi.org/10.1142/s0218348x97000838.

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Localization of optical excitations in fractal metal nanocomposites, such as colloidal aggregates, self-affine and semicontinuous films, results in strong enhancements of optical nonlinearities. The localized modes of fractals cover a broad spectral range, from the visible to the far-infrared. Results of both theoretical and experimental studies of enhanced nonlinear optical responses of fractal nanostructures are considered.

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Amir, Shahizat, Mohamed Nor Sabirin, and Siti Aishah Hashim Ali. "Using Polymer Electrolyte Membranes as Media to Culture Fractals: A Simulation Study." Advanced Materials Research 93-94 (January 2010): 35–38. http://dx.doi.org/10.4028/www.scientific.net/amr.93-94.35.

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In the authors' laboratory, fern-like fractals have been cultured in polymer electrolyte membranes of polyethylene oxide (PEO) doped with ammonium iodide (NH4I). The simulation study was then carried out utilizing the Diffusion Limited Aggregation (DLA) based on random motion of aggregating particles modelling technique. The fractal dimension values and the forms of the simulated fractals are comparable to those observed in the PEO polymer membranes. These indicate that the simulation using the DLA model done in this study has resulted outputs that are in abidance with the original fractals cultured in the polymer membranes.

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Sidorenko, A. S. "Superconducting Fractal Multilayers." Fractals 05, supp02 (1997): 101–17. http://dx.doi.org/10.1142/s0218348x97000851.

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The influence of fractal geometry on superconductivity has been studied for layered superconductors. Superconducting multilayers consisting of alternating Nb and Cu layers with fractal stacking sequence and fractal dimension Df=0.63 including the two limiting cases Df= 0 (single superconducting film) and Df=1 (periodic multilayers) were prepared by electron-beam evaporation in ultrahigh vacuum. The layers of Nb and Cu were put down alternately via computer control of the target shutter. The structure of the samples has been checked with in situ reflection high-energy electron diffraction (RHEED) and Auger depth profiling, confirmed the prescribed layering geometry. Superconductivity was investigated by measurements of the critical temperature of superconducting transition Tc, and of the temperature and of the angular dependence of the upper critical magnetic fields Bc2. The observed dependences of Tc on the parameters of fractal samples are in a good qualitative agreement with the proximity effect theory developed for layered superconductors with a self-similar fractal structure. The behavior of the upper critical magnetic field is directly related to the type of the layering. At low temperatures, all samples show the same two-dimensional behavior essentially governed by the topological dimension of the individual superconducting layers, independent of the fractal dimensionality Df of the samples, whereas for temperatures near Tc the type of layering determines the dimensionality, resulting in a multicrossover behavior of fractal samples. The angular dependence of the upper critical magnetic field Bc2(θ) of fractals corresponds to the theory for a two-dimensional superconductor at all temperatures, reflecting the multicrossover behavior of the fractal multilayers, as long as the temperature-dependent coherence length is comparable with a certain scale of fractal.

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LI, WEN XIA. "THE DIMENSION OF SETS DETERMINED BY THEIR CODE BEHAVIOR." Fractals 11, no. 04 (2003): 345–52. http://dx.doi.org/10.1142/s0218348x0300218x.

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By prescribing their code run behavior, we consider some subsets of Moran fractals. Fractal dimensions of these subsets are exactly obtained. Meanwhile, an interesting decomposition of Moran fractals is given.

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Pereira, Luis M. "Fractal Pharmacokinetics." Computational and Mathematical Methods in Medicine 11, no. 2 (2010): 161–84. http://dx.doi.org/10.1080/17486700903029280.

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Pharmacokinetics (PK) has been traditionally dealt with under the homogeneity assumption. However, biological systems are nowadays comprehensively understood as being inherently fractal. Specifically, the microenvironments where drug molecules interact with membrane interfaces, metabolic enzymes or pharmacological receptors, are unanimously recognized as unstirred, space-restricted, heterogeneous and geometrically fractal. Therefore, classical Fickean diffusion and the notion of the compartment as a homogeneous kinetic space must be revisited. Diffusion in fractal spaces has been studied for a long time making use of fractional calculus and expanding on the notion of dimension. Combining this new paradigm with the need to describe and explain experimental data results in defining time-dependent rate constants with a characteristic fractal exponent. Under the one-compartment simplification this strategy is straightforward. However, precisely due to the heterogeneity of the underlying biology, often at least a two-compartment model is required to address macroscopic data such as drug concentrations. This simple modelling step-up implies significant analytical and numerical complications. However, a few methods are available that make possible the original desideratum. In fact, exploring the full range of parametric possibilities and looking at different drugs and respective biological concentrations, it may be concluded that all PK modelling approaches are indeed particular cases of the fractal PK theory.

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Manousopoulos, Polychronis, Vasileios Drakopoulos, and Efstathios Polyzos. "Financial Time Series Modelling Using Fractal Interpolation Functions." AppliedMath 3, no. 3 (2023): 510–24. http://dx.doi.org/10.3390/appliedmath3030027.

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Time series of financial data are both frequent and important in everyday practice. Numerous applications are based, for example, on time series of asset prices or market indices. In this article, the application of fractal interpolation functions in modelling financial time series is examined. Our motivation stems from the fact that financial time series often present fluctuations or abrupt changes which the fractal interpolants can inherently model. The results indicate that the use of fractal interpolation in financial applications is promising.

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BODRI, L. "SOME LONG-RUN PROPERTIES OF CLIMATIC RECORDS." Fractals 01, no. 03 (1993): 601–5. http://dx.doi.org/10.1142/s0218348x93000630.

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Fractal analysis allows for a mathematical comparison of climatic changes obtained from a variety of observations and recorded at different time scales. Analysis of the oxygen isotope curve of Pacific core V28–239 indicated fractal geometry of the oxygen isotope record for time scales 5000 to 2000000 years, with a fractal dimension of 1.22. On a time scale of some hundred years, tree ring index values yield an average fractal dimension of 1.32±0.02. For annual precipitation records from between 1817 and 1963 at nine major cities in the United States, a mean fractal dimension of 1.26±0.03 was found. Analysis of the annual mean surface air temperatures at seven meteorological stations in Hungary for the period of 1901–1991, indicates that the considered temperatures are fractals with a mean fractal dimension of 1.23±0.01. For the global surface air temperature change estimated from meteorological station records from 1880 to 1985, we derive in the present study a fractal dimension of 1.21. It is reasonable therefore to assume that climatic changes are characterized by one general fractal dimension over the spectral range 10 to 106 years. Records with such values of fractal dimension have some long-term persistence: even observations sufficiently distant from each other are not completely independent. If these conclusions become confirmed through analysis of a wider set of climatic records, long-run climatic prediction (in statistical sense) on different time scales will appear feasible.

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Sheng, L. Q., J. H. Pei, and X. Y. Liu. "Self-affine Fractal Modelling of Aircraft Echoes from Low-resolution Radars." Defence Science Journal 66, no. 2 (2016): 151. http://dx.doi.org/10.14429/dsj.66.8423.

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<p>For complex targets, the non rigid vibration of an aircraft as well as its attitude changes and the rotation of its rotating parts will induce complex nonlinear modulation on its echo from low-resolution radars. If one performs the fractal analysis of measures on an aircraft echo, it may offer a fine description of the dynamic characteristics which induce the echo structure. On basis of introducing self-affine fractal theory, the paper models real recorded aircraft echo data from a low-resolution radar using the self-affine fractal representation, and investigates the application of echo self-affine fractal characteristics in aircraft target classification. Results analysis shows that aircraft echoes from low-resolution radars can be modelled by using the self-affine fractal method, and the self-affine fractal features can be effectively applied to target classification and recognition.</p><p> </p>

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BARNSLEY, MICHAEL, JOHN HUTCHINSON, and ÖRJAN STENFLO. "A FRACTAL VALUED RANDOM ITERATION ALGORITHM AND FRACTAL HIERARCHY." Fractals 13, no. 02 (2005): 111–46. http://dx.doi.org/10.1142/s0218348x05002799.

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We describe new families of random fractals, referred to as "V-variable", which are intermediate between the notions of deterministic and of standard random fractals. The parameter V describes the degree of "variability": at each magnification level any V-variable fractals has at most V key "forms" or "shapes". V-variable random fractals have the surprising property that they can be computed using a forward process. More precisely, a version of the usual Random Iteration Algorithm, operating on sets (or measures) rather than points, can be used to sample each family. To present this theory, we review relevant results on fractals (and fractal measures), both deterministic and random. Then our new results are obtained by constructing an iterated function system (a super IFS) from a collection of standard IFSs together with a corresponding set of probabilities. The attractor of the super IFS is called a superfractal; it is a collection of V-variable random fractals (sets or measures) together with an associated probability distribution on this collection. When the underlying space is for example ℝ2, and the transformations are computationally straightforward (such as affine transformations), the superfractal can be sampled by means of the algorithm, which is highly efficient in terms of memory usage. The algorithm is illustrated by some computed examples. Some variants, special cases, generalizations of the framework, and potential applications are mentioned.

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Silberschmidt, V. V. "Fractal approach in modelling of earthquakes." Geologische Rundschau 85, no. 1 (1996): 116. http://dx.doi.org/10.1007/s005310050060.

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Beer, T. "Modelling rainfall as a fractal process." Mathematics and Computers in Simulation 32, no. 1-2 (1990): 119–24. http://dx.doi.org/10.1016/0378-4754(90)90225-8.

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Dennis, T. J., and N. G. Dessipris. "Fractal modelling in image texture analysis." IEE Proceedings F Radar and Signal Processing 136, no. 5 (1989): 227. http://dx.doi.org/10.1049/ip-f-2.1989.0036.

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SENGUPTA, KAUSHIK, and K. J. VINOY. "A NEW MEASURE OF LACUNARITY FOR GENERALIZED FRACTALS AND ITS IMPACT IN THE ELECTROMAGNETIC BEHAVIOR OF KOCH DIPOLE ANTENNAS." Fractals 14, no. 04 (2006): 271–82. http://dx.doi.org/10.1142/s0218348x06003313.

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In recent years, fractal geometries have been explored in various branches of science and engineering. In antenna engineering several of these geometries have been studied due to their purported potential of realizing multi-resonant antennas. Although due to the complex nature of fractals most of these previous studies were experimental, there have been some analytical investigations on the performance of the antennas using them. One such analytical attempt was aimed at quantitatively relating fractal dimension with antenna characteristics within a single fractal set. It is however desirable to have all fractal geometries covered under one framework for antenna design and other similar applications. With this objective as the final goal, we strive in this paper to extend an earlier approach to more generalized situations, by incorporating the lacunarity of fractal geometries as a measure of its spatial distribution. Since the available measure of lacunarity was found to be inconsistent, in this paper we propose to use a new measure to quantize the fractal lacunarity. We also demonstrate the use of this new measure in uniquely explaining the behavior of dipole antennas made of generalized Koch curves and go on to show how fundamental lacunarity is in influencing electromagnetic behavior of fractal antennas. It is expected that this averaged measure of lacunarity may find applications in areas beyond antennas.

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