Table Of Content

Geometric Modeling of Fractal Forms for CAD Geometric Modeling and Applications Set coordinated by Marc Daniel Volume 5 Geometric Modeling of Fractal Forms for CAD Christian Gentil Gilles Gouaty Dmitry Sokolov First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com © ISTE Ltd 2021 The rights of Christian Gentil, Gilles Gouaty and Dmitry Sokolov to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2021932086 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-040-9 Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter1.TheBC-IFSModel . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.Self-similarityandIFS . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1.Mathematicaldefinitionsandreminders. . . . . . . . . . . . . . . . 4 1.1.2.Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3.Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.4.Iteratedfunctionsystems . . . . . . . . . . . . . . . . . . . . . . . . 15 1.1.5.Visualizationandapproximation . . . . . . . . . . . . . . . . . . . . 18 1.1.6.Computerimplementation . . . . . . . . . . . . . . . . . . . . . . . 19 1.1.7.Notionsofaddressandparameterization . . . . . . . . . . . . . . . 23 1.1.8.Finiteaddress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.1.9.IFSmorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.1.10.Exampleofacurve. . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.1.11.TheexampleoftheSierpinskitriangle . . . . . . . . . . . . . . . . 28 1.2.Controllediteratedfunctionsystem . . . . . . . . . . . . . . . . . . . . 30 1.2.1.Formalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.2.2.Visualizationandapproximation . . . . . . . . . . . . . . . . . . . . 36 1.2.3.Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.3.Boundarycontrollediteratedfunctionsystem . . . . . . . . . . . . . . 38 1.3.1.Attractorsdefinedfromasetofcontrolpoints . . . . . . . . . . . . 43 1.3.2.Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.3.3.Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1.3.4.Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.3.5.Topologycontrol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.3.6.Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 1.3.7.Evaluationtreeandquotientgraph. . . . . . . . . . . . . . . . . . . 62 vi GeometricModelingofFractalFormsforCAD 1.3.8.Constraintresolution . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1.3.9.Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 1.3.10.Controlofthelocaltopology . . . . . . . . . . . . . . . . . . . . . 77 1.3.11.Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 1.3.12.Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 1.3.13.Choiceofthedisplayprimitive . . . . . . . . . . . . . . . . . . . . 82 1.3.14.Choiceofdimension . . . . . . . . . . . . . . . . . . . . . . . . . . 83 1.3.15.Choiceofgeometry . . . . . . . . . . . . . . . . . . . . . . . . . . 83 1.3.16.Primitiveproperty . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 1.3.17.BC-IFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Chapter2.DesignExamples . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.1.Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.2.Wiredstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.3.Surfacesandlaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.3.1.Quadrangularsubdivisions . . . . . . . . . . . . . . . . . . . . . . . 98 2.3.2.Triangularsubdivisions . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.3.3.Otherregularpolygonalsubdivisions . . . . . . . . . . . . . . . . . 109 2.3.4.TheSierpinskitriangle . . . . . . . . . . . . . . . . . . . . . . . . . 109 2.3.5.Penrosetilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2.4.Volumesandlacunarobjects . . . . . . . . . . . . . . . . . . . . . . . . 116 2.5.Treestructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 2.5.1.Basicexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 2.5.2.Basicexamplewithanundividedtrunk . . . . . . . . . . . . . . . . 120 2.5.3.Examplebytilingthespace . . . . . . . . . . . . . . . . . . . . . . 122 2.6.Formassembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Chapter3.SurfaceNURBS,SubdivisionSurfacesandBC-IFS . . . 129 3.1.Beziercurvesandsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.2.UniformB-splinecurvesandsurfaces . . . . . . . . . . . . . . . . . . . 132 3.3.Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.3.1.Polynomialcurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.3.2.Rationalcurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.4.NURBScurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.4.1.Knot-doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.4.2.Curvesubdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 3.4.3.Surfacesubdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 3.5.Subdivisioncurvesandsurfaces . . . . . . . . . . . . . . . . . . . . . . 151 3.5.1.Subdivisioncurves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.5.2.Subdivisionsurfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Contents vii Chapter4.BuildingOperations,AssistancetoDesign andApplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4.1.Topologicalconsistencyandsymmetryconstraints . . . . . . . . . . . 168 4.1.1.Orientationconstraints . . . . . . . . . . . . . . . . . . . . . . . . . 168 4.1.2.Permutationoperatorsandsymmetryconstraints. . . . . . . . . . . 170 4.2.Topologicalcombination . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.2.1.Topologicaltensorproduct . . . . . . . . . . . . . . . . . . . . . . . 179 4.2.2.Exampleofsurfacegenerationfromcurves. . . . . . . . . . . . . . 184 4.2.3.Volumegenerationsfromcurves . . . . . . . . . . . . . . . . . . . . 188 4.2.4.Generationofthetreestructure . . . . . . . . . . . . . . . . . . . . 188 4.3.Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 4.3.1.Interactionbetweenrepresentationmodels . . . . . . . . . . . . . . 193 4.3.2.Structureoptimization. . . . . . . . . . . . . . . . . . . . . . . . . . 195 4.3.3.Roughnessandspacerefilling . . . . . . . . . . . . . . . . . . . . . 208 4.3.4.Thedesignofaheatexchanger. . . . . . . . . . . . . . . . . . . . . 210 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Preface This work introduces a model of geometric representation for describing and manipulating complex non-standard shapes such as rough surfaces or porous volumes. It is aimed at students in scientific education (mathematicians, computer scientists, physicists, etc.), engineers, researchers or anyone familiar with the mathematical concepts addressed at early stages of the graduate level. However, many parts are accessible to all, in particular, all introductory sections that present ideas with examples. People with no prior background, whether they are artists, designers or simply curious, will be able to understand the philosophy of our approach,anddiscoveranewuniverseofunsuspectedandexcitingforms. Geometric representation models are mathematical tools integrated into computer-aided geometric design (CAGD) software. They make the production of numerical representations of forms possible. By means of graphical interfaces or programming tools, users can draw and/or manipulate these shapes. They can also testorevaluatetheirphysicalproperties(mechanical,electro-magnetic,acoustic,etc.) by communicating geometric descriptions to further specific numerical simulation software. The geometric representation model we present here is based on the fractal geometry paradigm. The principle behind this consists of studying the properties (signal, geometry, phenomena, etc.) at different scales and identifying the invariants fromthere.Theobjectsaredescribedasself-referentialbetweentwoscales: eachof the object features (namely, the lower scale level) is described as a reference to the object itself (namely, the higher scale). This approach is not conventional and often confusingatfirst.Wecometoperceiveitsrichnessandpowerveryquickly,however. The universe of forms that can possibly be created is infinite and has still only partiallybeenexplored. In this book, we present the mathematical foundations, so that the reader can accessalltheinformationtounderstand,testandmakeuseofthismodel.Properties, x GeometricModelingofFractalFormsforCAD theoremsandconstructionmethodsaresupplementedwithalgorithmsandnumerous examples and illustrations. Concerning the formalization, we have chosen to use precise and rigorous mathematical notations to remove any ambiguity and make understandingeasier. Readersunwillingtobeconcernedwithmathematicalformalismscangettogrips with the philosophy of our approach by focusing on the sections found at the beginnings of the chapters, in which ideas and principles are intuitively presented, basedonexamples. This book is the result of 25 years of research carried out mainly in the LIRIS laboratories of the University of Lyon I and LIB of the University of Burgundy Franche-Comté. This research was initiated by Eric Tosan, who was instrumental at theoriginofthisformalismandtowhomwededicatethiswork. ChristianGENTIL GillesGOUATY DmitrySOKOLOV February2021 Introduction I.1. Fractalsforindustry: whatfor? This book shows our first steps toward the fundamental and applied aspects of geometric modeling. This area of research addresses the acquisition, analysis and optimizationofthenumericalrepresentationof3Dobjects. a) 3Dmodel b) 3Dprinting FigureI.1.3Dtreebuiltbyiterativemodeling(source: projectMODITEREno.ANR-09-COSI-014) Figure I.1 shows an example of a structure that admits high vertical loads, while minimizing the transfer of heat between the top and bottom of the part. Additive manufacturing (3D printing) allows, for the first time, the creation of such complex objects, even in metal (here with a high-end laser printer EOS M270). This type of technologywillhaveahighsocietalandeconomicimpact,enablingbettersystemsto be created (engines, cars, airplanes, etc.), designed and adapted numerically for optimalfunctionality,thusconsuminglessrawmaterial,fortheirmanufacturing,and energy,whenused.