Fractals in engineering: new trends in theory and applications PDF
Preview Fractals in engineering: new trends in theory and applications
Fractals in Engineering Jacques Lévy-Véhel and Evelyne Lutton (Eds.) Fractals in Engineering New Trends in Theory and Applications With106Figures 123 JacquesLévy-Véhel EvelyneLutton INRIA Rocquencourt DomainedeVoluceau-Rocquencourt B.P.105 78153LeChesnayCedex France BritishLibraryCataloguinginPublicationData Fractalsinengineering:newtrendsintheoryand applications 1.Engineeringmathematics2.Fractals I.Lévy-Véhel,Jacques,1960-II.Lutton,Evelyne,1962- 620’.001514742 ISBN-10:1846280478 LibraryofCongressControlNumber:2005927902 ISBN-10: 1-84628-047-8 e-ISBN:1-84628-048-6 Printedonacid-freepaper ISBN-13: 978-1-84628-047-4 ©Springer-VerlagLondonLimited2005 Apartfromanyfairdealingforthepurposesofresearchorprivatestudy,orcriticismorreview,as permittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers,orinthecaseofreprographicreproductioninaccordancewiththetermsoflicencesissued bytheCopyrightLicensingAgency.Enquiriesconcerningreproductionoutsidethosetermsshouldbe senttothepublishers. Theuseofregisterednames,trademarks,etc.inthispublicationdoesnotimply,evenintheabsenceof aspecificstatement,thatsuchnamesareexemptfromtherelevantlawsandregulationsandtherefore freeforgeneraluse. Thepublishermakesnorepresentation,expressorimplied,withregardtotheaccuracyoftheinfor- mationcontainedinthisbookandcannotacceptanylegalresponsibilityorliabilityforanyerrorsor omissionsthatmaybemade. PrintedinGermany 987654321 SpringerScience+BusinessMedia springeronline.com Foreword This volume is a sequel to the books Fractals: Theory and Applications in Engineering (Springer-Verlag, 1999) and Fractals in Engineering. From Theory to Industrial Applications (Springer-Verlag,1997),presentingsome of the most recent advances in the field. It is a fascinating exercise to follow the progressof knowledgein this interdisciplinaryarea,as witnessedby these three volumes. First,confirmingprevioustrendsobservedin1997and1999,appliedmath- ematicalresearchonfractalshasnowreacheda maturelevel,where beautiful theories are developed in direct contact with engineering concerns. The four papers in the Mathematical Aspects section constitute valuable additions to the set of tools needed by the engineer: Synthetic pictures modelling and rendering in computer graphics (Theory and Applications of Fractal Tops, by Michael Barnsley), curve approximation and ”fractal B-splines” (Splines, Fractal Functions, and Besov and Triebel-Lizorkin Spaces, by Peter Masso- pust), deep understanding of the Ho¨lderian properties of certain stochastic processes useful in a large number of applications (Ho¨lderian random func- tions, by Antoine Ayache et al.), and study of the invariant measure of a coupled discrete dynamical system (Fractal Stationary Density in Coupled Maps, by Ju¨rgen Jost et al.). The second section of the book describes novel physical applications as well as recent progress on more classicalones. The paper A Network of Frac- tal Force Chains and Their Effect in Granular Materials under Compression by Luis E.Vallejoet al. offersanexplanationto the well-knownexperimental fact that granular material develop fractal fragments as a result of compres- vi Foreword sion. In Percolation and permeability of three dimensional fracture networks with a power law size distribution, V.V. Mourzenko et al. provide a new and interesting addition to the large body of work devoted to fractal analysis of percolationinfracturenetworks.Theyperformathoroughnumericalstudyof percolationin polydisperse fracture networks,allowing to define anappropri- ate percolation parameter and to develop two heuristic analytical models. A newandverypromisingapplicationoffractalanalysistoacousticsintheframe ofurbanstructures is developedby Philippe WoloszyninAcoustic diffraction patterns from fractal to urban structures:Applications to the Sierpinski trian- gleandtoaneoclassicalurbanfacade.RolfBaderdevelopsanotherapplication to acoustics, proposing an interesting Turbulent k−(cid:1) model of flute-like mu- sical instrument sound production. ThesectiononChemical Engineeringfeaturestwopapers.InAsimple dis- crete stochastic model for laser-induced jet-chemical etching, Alejandro Mora et al. describe a discrete stochastic model for the description of laser-induced wet-chemical etching. This model enables one to describe the aspect of the surface depending on the velocity of the laser beam. A deep study of fluid mixing in two dimensions is made in Invariant structures and multifractal measures in 2d mixing systems by Massimiliano Giona al., through a con- nectionbetweengeometricinvariantstructuresandthe spatialdistributionof periodic points. Fractal modelling of financial time series has a long and rich history. The section onFinance focuses on the specific question oflong rangedependence, with two papers. In Long range dependence in financial markets, Rama Cont discusses the relevance of this property in financial modelling, and highlights possible economic mechanisms accounting for its presence in financial time series.PierreBertrandderivesinFinancial Modelling byMultiscale Fractional BrownianMotionthepriceofaEuropeanoptionforthismodelofstockprices. Application of fractalanalysis to Internet traffic, which is the topic of the fifth section, started in the 1990’s, and an extremely large number of studies have been devoted to this topic in recent years. The paper Limiting Frac- tal Random Processes in Heavy-Tailed Systems by Ingemar Kaj investigates the asymptotic behavior of stochastic processes build through aggregationof independent subsystems and simultaneous time rescaling. This behavior de- pends considerably on the relative speed of aggregationdegree and rescaling. Although primarily of interest in telecommunications, these results extend in higher dimensions (e.g. spatial Poisson point processes). The concept of crossing tree previously introduced by the authors for estimating the Hurst index of self-similar processes is used as a tool for A non-parametric test for self-similarity and stationarity in network traffic, by Owen Jones et al.. The last section deals with applications in image processing. In Contin- uous evolution of functions and measures toward fixed points of contraction mappings, Jerry Bona et al. study a class of evolution equations associated with contraction mappings on a Banach space of functions. This enables one to performcontinuous,fractal-like,”touch-up” operationsonimages.Fahima Foreword vii Nekka et al. use the autocorrelation function, the regularization dimension as well as the Hausdorff measure spectrum function to analyze textures in Various Mathematical Approaches to Extract Information from Textures of Increasing Complexities. The celebrated inverse problem of fractal coding is the topic ofFractal Inverse Problem: Approximation Formulation and Differ- entialMethodsbyEricGu´erinetal.Usingananalyticalapproach,theyobtain interesting results both in one and two dimensions. While it is obviously impossible to cover the wealth of all applications of fractalanalysisinengineeringsciencesinasinglevolume,thisbook doespro- videanoverviewofsomeofthemoreprominentrecentadvances,whichshould be of interest to anyone willing to keep up with the fast pace of development in this field. Wewouldliketothankalltheauthorswhohavecontributedtothisbook. Thanks also to Nathalie Gaudechoux for her Latex skills. Finally, we are grateful to INRIA and our publisher Springer-Verlag for their support. Jacques L´evy V´ehel, Evelyne Lutton. Contents 1 MATHEMATICAL ASPECTS ................................................................ 1 Theory and Applications of Fractal Tops Michael Barnsley ................................................ 3 Splines, Fractal Functions, and Besov and Triebel-Lizorkin Spaces Peter Massopust ................................................. 21 Ho¨lderian random functions Antoine Ayache, Philippe Heinrich, Laurence Marsalle, Charles Suquet . 33 Fractal Stationary Density in Coupled Maps Ju¨rgen Jost, Kiran M. Kolwankar.................................. 57 2 PHYSICS ................................................................ 65 A Network of Fractal Force Chains and Their Effect in Granular Materials under Compression Luis E. Vallejo, Sebastian Lobo-Guerrero, Zamri Chik ................ 67 Percolation and permeability of three dimensional fracture networks with a power law size distribution V.V. Mourzenko, Jean-Franc¸ois Thovert, Pierre M. Adler............. 81 Acoustic diffraction patterns from fractal to urban structures: applications to the Sierpinski triangle and to a neoclassical urban facade Philippe Woloszyn................................................ 97 x Contents Turbulent k−(cid:1) model of flute-like musical instrument sound production Rolf Bader ......................................................109 3 CHEMICAL ENGINEERING ................................................................123 A simple discrete stochastic model for laser-induced jet-chemical etching Alejandro Mora, Thomas Rabbow, Bernd Lehle, Peter J. Plath, Maria Haase ..........................................................125 Invariant structures and multifractal measures in 2d mixing systems Massimiliano Giona, Stefano Cerbelli, and Alessandra Adrover ........141 4 FINANCE ................................................................157 Long range dependence in financial markets Rama Cont......................................................159 Financial Modelling by Multiscale Fractional Brownian Motion Pierre Bertrand..................................................181 5 INTERNET TRAFFIC ................................................................197 Limiting Fractal Random Processes in Heavy-Tailed Systems Ingemar Kaj ....................................................199 A non-parametric test for self-similarity and stationarity in network traffic Owen Dafydd Jones, Yuan Shen ...................................219 6 IMAGE PROCESSING ................................................................235 Continuous evolution of functions and measures toward fixed points of contraction mappings Jerry L. Bona, Edward R. Vrscay..................................237 Various Mathematical Approaches to Extract Information from Textures of Increasing Complexities Fahima Nekka, Jun Li............................................255 Contents xi Fractal Inverse Problem: Approximation Formulation and Differential Methods Eric Gu´erin, Eric Tosan ..........................................271 Index..........................................................287 List of Contributors Pierre Adler Pierre Bertrand IPGP Laboratoire de Math´ematiques - Tour 24 UMR CNRS 6620, 4 Place Jussieu Universit´e Blaise Pascal (Clermont- 75252 Paris Cedex 05, France Ferrand II), [email protected] 24 Avenue des Landais, 63117 Aubi`ere Cedex, France. Alessandra Adrover [email protected] Dipartimento di Ingegneria Chimica Facolta` di Ingegneria Universita` di Roma “La Sapienza” Jerry L. Bona via Eudossiana 18 Department of Mathematics, 00184 Roma, Italy Statistics and Computer Science [email protected] The University of Illinois at Chicago Chicago, Illinois, USA 60607-7045 Antoine Ayache [email protected] Laboratoire P. Painlev´e, CNRS UMR 8524,Universit´e Lille 1, Stefano Cerbelli 59650 Villeneuve d’Ascq cedex, Dipartimento di Ingegneria Chimica France Facolta` di Ingegneria [email protected] Universita` di Roma “La Sapienza” Rolf Bader via Eudossiana 18 University of Hamburg 00184 Roma, Italy Institute of Musicology [email protected] Neue Rabenstr. 13 20354 Hamburg, Germany Zamri Chik R [email protected] Department of Civil and Michael Barnsley Environmental Engineering Australian National University University of Pittsburgh Canberra, ACT 0200, Australia Pittsburgh, PA 15261,U.S.A. [email protected] [email protected]
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