Table Of ContentMeshing, Geometric Modeling and Numerical Simulation 1
For Anahita, Amitisse and Hilda
Geometric Modeling and Applications Set
coordinated by
Marc Daniel
Volume 1
Meshing, Geometric
Modeling and Numerical
Simulation 1
Form Functions, Triangulations and
Geometric Modeling
Houman Borouchaki
Paul Louis George
First published 2017 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 2017
The rights of Houman Borouchaki and Paul Louis George 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: 2017948827
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-038-6
Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter1.FiniteElementsandShapeFunctions . . . . . . . . . . . . . . . . . . . . . 15
1.1.Basicconcepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.Shapefunctions,completeelements . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.1.Genericexpressionofshapefunctions . . . . . . . . . . . . . . . . . . . . . . 18
1.2.2.Explicitexpressionfordegrees1–3 . . . . . . . . . . . . . . . . . . . . . . . 22
1.3.Shapefunctions,reducedelements . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.1.Simplices,trianglesandtetrahedra . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.2.Tensorelements,quadrilateralandhexahedralelements . . . . . . . . . . . . 31
1.3.3.Otherelements,prismsandpyramids . . . . . . . . . . . . . . . . . . . . . . 48
1.4.Shapefunctions,rationalelements . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.4.1.Rationaltrianglewithadegreeof2orarbitrarydegree . . . . . . . . . . . . 49
1.4.2.Rationalquadrilateralofanarbitrarydegree. . . . . . . . . . . . . . . . . . . 50
1.4.3.Generalcase,B-splinesorNurbselements . . . . . . . . . . . . . . . . . . . 50
Chapter2.LagrangeandBézierInterpolants . . . . . . . . . . . . . . . . . . . . . . . 53
2.1.Lagrange–Bézieranalogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.2.LagrangefunctionsexpressedinBézierforms . . . . . . . . . . . . . . . . . . . . 55
2.2.1.Thecaseoftensors,naturalcoordinates . . . . . . . . . . . . . . . . . . . . . 55
2.2.2.Simplicialcase,barycentriccoordinates . . . . . . . . . . . . . . . . . . . . . 63
2.3.BézierpolynomialsexpressedinLagrangianform . . . . . . . . . . . . . . . . . . 66
2.4.Applicationtocurves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.4.1.BézierexpressionforaLagrangecurve . . . . . . . . . . . . . . . . . . . . . 67
2.4.2.LagrangianexpressionforaBéziercurve . . . . . . . . . . . . . . . . . . . . 70
2.5.Applicationtopatches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.5.1.BézierexpressionforapatchinLagrangianform . . . . . . . . . . . . . . . 71
2.5.2.LagrangianexpressionforapatchinBézierform . . . . . . . . . . . . . . . 73
2.6.Reducedelements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6 Meshing,GeometricModelingandNumericalSimulation1
2.6.1.Thetensorcase,BézierexpressionforareducedLagrangianpatch. . . . . . 74
2.6.2.Thetensorcase,definitionofreducedBézierpatches . . . . . . . . . . . . . 82
2.6.3.Thetensorcase,LagrangianexpressionofareducedBézierpatch . . . . . . 90
2.6.4.Thecaseofsimplices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Chapter3.GeometricElementsandGeometricValidity . . . . . . . . . . . . . . . . . 95
3.1.Two-dimensionalelements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.2.Surfaceelements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.3.Volumetricelements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.4.Controlpointsbasedonnodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.5.Reducedelements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.5.1.Simplices,trianglesandtetrahedra . . . . . . . . . . . . . . . . . . . . . . . . 115
3.5.2.Tensorelements,quadrilateralsandhexahedra . . . . . . . . . . . . . . . . . 116
3.5.3.Otherelements,prismsandpyramids . . . . . . . . . . . . . . . . . . . . . . 120
3.6.Rationalelements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.6.1.ShiftfromLagrangerationalstoBézierrationals . . . . . . . . . . . . . . . . 121
3.6.2.Degree2,workingonthe(arcofa)circle . . . . . . . . . . . . . . . . . . . . 121
3.6.3.Applicationtotheanalysisofrationalelements. . . . . . . . . . . . . . . . . 123
3.6.4.Ontheuseofrationalelementsormore . . . . . . . . . . . . . . . . . . . . . 138
Chapter4.Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.1.Triangulation,definitions,basicconceptsandnaturalentities . . . . . . . . . . . . 142
4.1.1.Definitionsandbasicconcepts . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.1.2.Naturalentities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.1.3.Aball(topological)ofavertex . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.1.4.Ashellofak-face . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.1.5.Theringofak-face . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.2.Topologyandlocaltopologicalmodifications . . . . . . . . . . . . . . . . . . . . . 146
4.2.1.Flippinganedgeintwodimensions . . . . . . . . . . . . . . . . . . . . . . . 148
4.2.2.Flippingafaceinthreedimensions. . . . . . . . . . . . . . . . . . . . . . . . 148
4.2.3.Flippinganedgeinthreedimensions . . . . . . . . . . . . . . . . . . . . . . 148
4.2.4.Otherflips? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.3.Enricheddatastructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.3.1.Minimalstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.3.2.Enrichedstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.4.Constructionofnaturalentities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.5.Triangulation,constructionmethods . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.6.Theincrementalmethod,agenericmethod . . . . . . . . . . . . . . . . . . . . . . 159
4.6.1.Naivetriangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.6.2.Delaunaytriangulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Chapter5.DelaunayTriangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.1.History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.2.Definitionsandproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.3.TheincrementalmethodforDelaunay . . . . . . . . . . . . . . . . . . . . . . . . . 175
Contents 7
5.4.Othermethodsofconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.5.Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.6.Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Chapter6.TriangulationandConstraints . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.1.Triangulationofadomain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
6.1.1.Triangulationofadomainintwodimensions . . . . . . . . . . . . . . . . . . 195
6.1.2.Triangulationofadomaininthreedimensions . . . . . . . . . . . . . . . . . 202
6.2.DelaunayTriangulation“Delaunayadmissibility” . . . . . . . . . . . . . . . . . . 214
6.3.Triangulationofavariety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.4.Topologicalinvariants(trianglesandtetrahedra) . . . . . . . . . . . . . . . . . . . 222
Chapter7.GeometricModeling:Methods . . . . . . . . . . . . . . . . . . . . . . . . . 233
7.1.Implicitorexplicitform(CAD),startingfromananalyticaldefinition . . . . . . . 234
7.1.1.Modelinganimplicitcurve,continuous→discrete . . . . . . . . . . . . . . 234
7.1.2.Modelingaparametriccurve . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
7.1.3.Modelinganimplicitsurface . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
7.1.4.Modelingofaparametricsurface. . . . . . . . . . . . . . . . . . . . . . . . . 242
7.2.Startingfromadiscretizationortriangulation,discrete→continuous . . . . . . . 246
7.2.1.Caseofacurve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
7.2.2.Thecaseofasurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
7.3.Startingfromapointcloud,discrete→discrete . . . . . . . . . . . . . . . . . . . 278
7.3.1.Thecaseofacurveintwodimensions . . . . . . . . . . . . . . . . . . . . . . 278
7.3.2.Thecaseofasurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
7.4.Extractionofcharacteristicpointsandcharacteristiclines . . . . . . . . . . . . . . 302
Chapter8.GeometricModeling:Examples . . . . . . . . . . . . . . . . . . . . . . . . . 305
8.1.Geometricmodelingofparametricpatches . . . . . . . . . . . . . . . . . . . . . . 306
8.2.Characteristiclinesofadiscretesurface . . . . . . . . . . . . . . . . . . . . . . . . 311
8.3.Parametrizationofasurfacepatchthroughunfolding . . . . . . . . . . . . . . . . 311
8.4.Geometricsimplificationofasurfacetriangulation . . . . . . . . . . . . . . . . . . 324
8.5.Geometricsupportforadiscretesurface . . . . . . . . . . . . . . . . . . . . . . . . 325
8.6.Discretereconstructionofadigitizedobjectorenvironment. . . . . . . . . . . . . 330
Chapter9.AFewBasicAlgorithmsandFormulae . . . . . . . . . . . . . . . . . . . . 343
9.1.Subdivisionofanentity(DeCasteljau) . . . . . . . . . . . . . . . . . . . . . . . . 344
9.1.1.Subdivisionofacurve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
9.1.2.Subdivisionofapatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
9.2.Computingcontrolcoefficients(higherorderelements) . . . . . . . . . . . . . . . 348
9.3.Algorithmsfortheinsertionofapoint(Delaunay) . . . . . . . . . . . . . . . . . . 351
9.3.1.Classicalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
9.3.2.Modifiedalgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
9.4.Constructionofneighboringrelationships,ballsandshells . . . . . . . . . . . . . 357
9.4.1.Neighboringrelationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
8 Meshing,GeometricModelingandNumericalSimulation1
9.4.2.Constructionoftheballofavertex . . . . . . . . . . . . . . . . . . . . . . . . 359
9.4.3.Constructionoftheshellofanedge . . . . . . . . . . . . . . . . . . . . . . . 361
9.5.Localizationproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
9.5.1.Triangulationsorsimplicialmeshes . . . . . . . . . . . . . . . . . . . . . . . 363
9.5.2.Othermeshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
9.6.Someformulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
ConclusionsandPerspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
Foreword
The“GeometricModelingandApplications”seriesismadeupoffivevolumesofresearch
ongeometricmodeling. Theactofgeometricmodelingisanage-oldone. Tolookatafewkey
pointsinitshistory: geometristsinantiquitydevotedthemselvestoit,though,ofcourse,ithad
adifferentformfromthatwhichweknowtoday; Descartes, inthemid-17thCentury, worked
onpartitioningofplanes;andVoronoïestablishedconceptsrelatingtothistowardtheendofthe
19thCentury.
Butgeometricmodeling,asweunderstanditnow,canbeconsideredtobelinkedtotheuseof
computers,especiallysincethistoolhasbecomeubiquitous.Wearetemptedtofixthescientific
recognitionofthisactivitytothedateofthe“FirstConferenceonComputerAidedGeometric
Design”,whichtookplaceinMarch1974attheUniversityofUtah. ThiswasorganizedbyR.
Barhnill and R. Riesenfeld, whose presentations may be found in the book “Computer Aided
GeometricDesign, AcademicPress, 1974”, whichclearlyassociatesgeometricmodelingwith
thecomputer.
Sincethen,manydetailedandhigh-qualitybookshavebeenpublishedinthisfield. Oneof
thepioneeringworkswas,undoubtedly,thatauthoredbyD.FauxandM.Pratt,“Computational
Geometry for Design and Manufacture, Ellis Horwood Publishers, 1979”. Some books went
throughseveraleditions,takingnoteoftheadvancesmadeinresearchinthefield.Inthiscontext,
bringingoutanewgeneralbookonthesubjectwouldmarknosignificantscientificprogress.
Moreover,foranyobjetorentitythatistobemodeled,thereisnosinglegeometricmodel.
Instead, thereareseveralgeometricmodels, eachbeingadaptedtotheprocessestobecarried
out. For example, the model to finely analyze the shape of an object is necessarily different
frommodelsthatareusedformechanicalcomputationsonthissameobject.Wehave,therefore,
chosentofocusonspecificpointsofresearchinfieldsthatwefindparticularlyimportant.Thus,
thefivereferencesgivenhere,writtenbyspecialists,bringyouuptodateonadvancesinresearch
infivedomains:
•“Constraintsandgeometricmodeling”, byDominiqueMichelucci, PascalSchreck and
PascalMathis,looksatgeometricproblemsthatcanbeapproachedbydescribingtheconstraints,
geometricormorecomplex,thatasolutionmustverify,ratherthandirectlygivingthegeometric
definitionofsuchasolution.
10 Meshing,GeometricModelingandNumericalSimulation1
•“GeometricmodelingoffractalformsCAD”,byChristianGentil,DmitrySokolovand
GillesGouaty,discussesthedefinitionoffractalforms:thisformalismmakesitpossibletoreuse
smoothformsofgeometricmodelsthatareclassictodaytodefineformsthathavenotyetbeen
exploredtillpresent.
•“Meshing,geometricmodelingandnumericalsimulation”,intwovolumes,byHouman
Borouchaki and Paul-Louis George, details methodologies advanced for the construction of
meshes and geometric modeling for numerial simulation with applications, especially in me-
chanics.
•“Geometric and Topological Mesh Feature Extraction for 3D Shape Analysis”, by
Jean-Luc Mari, Franck Hétroy-Wheeler and Gérard Subsol, studies the analysis of three-
dimensional-meshedshapesviatheextractionofcharacteristics,bothgeometricandtopological.
Thefieldsofapplicationarealsodetailedtoillustratethetransdisciplinarynatureofthiswork.
•“RelevantTriangulationsandSamplingsfor3DShapes”,byRaphaëlleChaineandJulie
Digne,discussessampling,meshingandcompressionoffreeformsfromalargequantityofreal
datafromavirtual,interactivedeformationprocess.
Thesearebooksthatpresentrecentresearch. Thereadercanfind, inthegeneralbookson
geometricmodelingthatwerementionedbefore,elementstofillanygapstheymayhaveandto
enablethemtoreadthebooksinthisseries.
Happyreading.
MarcDANIEL
