Table Of ContentMeshing, Geometric Modeling and Numerical Simulation 2
Geometric Modeling and Applications Set
coordinated by
Marc Daniel
Volume 2
Meshing, Geometric Modeling
and Numerical Simulation 2
Metrics, Meshes and
Mesh Adaptation
Paul Louis George
Houman Borouchaki
Frédéric Alauzet
Patrick Laug
Adrien Loseille
Loïc Maréchal
First published 2019 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
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ISTE Ltd John Wiley & Sons, Inc.
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© ISTE Ltd 2019
The rights of Paul Louis George, Houman Borouchaki, Frédéric Alauzet, Patrick Laug, Adrien Loseille
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Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter1.Metrics,DefinitionsandProperties . . . . . . . . . . . . . . . . . . . . . . 1
1.1.Definitionsandproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.Metricinterpolationandintersection . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1.Metricinterpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2.Metricintersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.Geometricmetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.1.Geometricmetricforacurve . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.2.Geometricmetricforasurface . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.3.Turninganymetricintoageometricmetric . . . . . . . . . . . . . . . . . . . 23
1.4.Meshingmetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5.Metricsgradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.6.Elementmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6.1.Metricofasimplicialelement . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6.2.Metricofanon-simplicialelement . . . . . . . . . . . . . . . . . . . . . . . . 37
1.6.3.Metricofanelementofarbitrarydegree . . . . . . . . . . . . . . . . . . . . . 38
1.7.Elementshapeandmetricquality. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.8.Practicalcomputationsinthepresenceofametric . . . . . . . . . . . . . . . . . . 46
1.8.1.Calculationofthelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.8.2.Thecalculationofanangle,areaorvolume . . . . . . . . . . . . . . . . . . . 49
Chapter2.InterpolationErrorsandMetrics . . . . . . . . . . . . . . . . . . . . . . . 53
2.1.Someproperties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.2.Interpolationerrorofaquadraticfunction . . . . . . . . . . . . . . . . . . . . . . . 55
2.3.Bézierformulationandinterpolationerror . . . . . . . . . . . . . . . . . . . . . . . 62
2.3.1.Foraquadraticfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.3.2.Foracubicfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.3.3.Forapolynomialfunctionofarbitrarydegree . . . . . . . . . . . . . . . . . . 80
vi Meshing,GeometricModelingandNumericalSimulation2
2.3.4.Errorthresholdormeshdensity. . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.4.Computationsofdiscretederivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.4.1.TheL2doubleprojectionmethod . . . . . . . . . . . . . . . . . . . . . . . . 86
2.4.2.Greenformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.4.3.LeastsquareandTaylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Chapter3.CurveMeshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.1.Parametriccurvemeshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.1.1.CurveinR3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.1.2.Aboutmetricsusedandcomputationsoflengths . . . . . . . . . . . . . . . . 99
3.1.3.Curveplottedonapatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.2.Discretecurvemeshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.3.Remeshingameshedcurve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Chapter4.SimplicialMeshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.1.Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2.Variety(surface)meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.2.1.Patch-basedmeshing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2.2.Discretesurfaceremeshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.2.3.Meshingusingavolumemesher . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.3.Themeshingofaplaneorofavolumedomain . . . . . . . . . . . . . . . . . . . . 122
4.3.1.Tree-basedmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.3.2.Front-basedmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.3.3.Delaunay-basedmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.3.4.Remeshingofamesheddomain . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.4.Othergenerationmethods? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Chapter5.Non-simplicialMeshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.1.Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.2.Varietymeshing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.3.Constructionmethodsformeshingaplanarorvolumedomain . . . . . . . . . . . 145
5.3.1.Cylindricalgeometryandextrusionmethod . . . . . . . . . . . . . . . . . . . 147
5.3.2.Algebraicmethodsandblock-basedmethods . . . . . . . . . . . . . . . . . . 148
5.3.3.Tree-basedmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.3.4.Pairingmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.3.5.Polygonalorpolyhedralcellmeshing . . . . . . . . . . . . . . . . . . . . . . 176
5.3.6.Constructionofboundarylayers . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.4.Othergenerationmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.4.1.“Q-morphism”or“H-morphism”meshing . . . . . . . . . . . . . . . . . . . 182
5.4.2.Meshingusingareferenceframefield . . . . . . . . . . . . . . . . . . . . . . 183
5.5.Topologicalinvariants(quadrilateralsandhexahedra) . . . . . . . . . . . . . . . . 185
Chapter6.High-orderMeshConstruction . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.1.Straightmeshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Contents vii
6.1.1.Localnodenumbering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.1.2.Overallnodenumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.1.3.Nodepositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.1.4.Onfillingupmatricesaccordingtoelementdegrees . . . . . . . . . . . . . . 207
6.2.Constructionofcurvedmeshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
6.2.1.First-degreemesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6.2.2.Nodecreation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6.2.3.Deformationandvalidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.2.4.Generalscheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
6.3.Curvedmeshesonavariety,curveorsurface . . . . . . . . . . . . . . . . . . . . . 215
Chapter7.MeshOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.1.Towardadefinitionofquality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
7.2.Optimizationprocess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
7.2.1.Globalmethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
7.2.1.1.Optimizationofacostfunction . . . . . . . . . . . . . . . . . . . . . . . 233
7.2.1.2.Iterativerelaxationofthepositionofverticesbyduality(simplices) . . 234
7.2.1.3.Globaloptimizationofthepositionofvertices(quadrilateralsandhex-
ahedra). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
7.2.2.Localoperatorsandlocalmethods . . . . . . . . . . . . . . . . . . . . . . . . 236
7.2.2.1.Vertexmovesbybarycentering . . . . . . . . . . . . . . . . . . . . . . . 236
7.2.2.2.VertexmovesandLaplacianoperator . . . . . . . . . . . . . . . . . . . 237
7.2.2.3.Movingorremovingverticesandflipsbyinsertionorreinsertion . . . 241
7.2.2.4.Edgeflips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
7.2.2.5.Clusterofedgeflips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
7.2.2.6.Edgeorfaceflipbyreinsertion . . . . . . . . . . . . . . . . . . . . . . . 244
7.2.2.7.Edgeslicing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
7.2.2.8.Removalofanedgebymerging . . . . . . . . . . . . . . . . . . . . . . 245
7.2.2.9.Metricfieldupdate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.2.2.10.Topologicalandmetriccriteria . . . . . . . . . . . . . . . . . . . . . . 246
7.2.2.11.Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.3.Planarmesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
7.4.Surfacemesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
7.5.Volumemeshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
7.6.High-degreemeshing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Chapter8.MeshAdaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
8.1.Genericframeworkforadaptivecomputation,thecontinuousmesh . . . . . . . . 266
8.1.1.Dualitybetweendiscreteandcontinuousgeometricentities . . . . . . . . . . 267
8.1.2.Dualitybetweendiscreteandcontinuousinterpolationerror . . . . . . . . . 269
8.1.3.Discrete–continuousdualityinonediagram. . . . . . . . . . . . . . . . . . . 272
8.2.OptimalcontroloftheinterpolationerrorinLp-norm . . . . . . . . . . . . . . . . 272
8.3.Genericschemeofstationaryadaptation . . . . . . . . . . . . . . . . . . . . . . . . 279
8.3.1.Errorestimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
8.3.2.Interpolationofsolutionfields . . . . . . . . . . . . . . . . . . . . . . . . . . 287
viii Meshing,GeometricModelingandNumericalSimulation2
8.4.Unsteadyadaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
8.4.1.Space–timeerrorestimatorsbasedonthecharacteristicsofthesolution . . . 290
8.4.2. Extension of the error analysis for the fixed-point algorithm for unsteady
meshadaptation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
8.4.3.Meshadaptationforunsteadyproblems . . . . . . . . . . . . . . . . . . . . . 292
8.4.4.Unsteadymeshadaptationtargetedatafunctionofinterest . . . . . . . . . . 294
8.4.5.Conservativeinterpolationofsolutionfields. . . . . . . . . . . . . . . . . . . 295
8.5.Mobilegeometrywithorwithoutdeformation . . . . . . . . . . . . . . . . . . . . 297
8.5.1.Generalcontextoftheadaptationformobileand/ordeformablegeometries. 297
8.5.2.ALEcontinuousoptimalmeshminimizingtheinterpolationerrorinLp-norm 298
8.5.3.Space–timeerrorestimatorformovinggeometryproblems . . . . . . . . . . 300
Chapter9.MeshingandParallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
9.1.Renumberingviaafillingcurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
9.2.Parallelism:twomemoryparadigmsanddifferentstrategies . . . . . . . . . . . . 307
9.3.Algorithmparallelizationformeshconstruction . . . . . . . . . . . . . . . . . . . 312
9.4.Parallelizationofameshconstructionprocess,partitionthenmeshing . . . . . . . 324
9.5.Meshparallelization,meshingthenpartition . . . . . . . . . . . . . . . . . . . . . 326
Chapter10.Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
10.1.Surfacemeshing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
10.2.Incomputationalfluiddynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
10.3.Computationalsolidmechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
10.4.Computationalelectromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
10.5.Renumberingandparallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
10.6.Othermoreexoticapplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Chapter11.SomeAlgorithmsandFormulas . . . . . . . . . . . . . . . . . . . . . . . . 353
11.1.Localnumberingofnodesofhigh-orderelements. . . . . . . . . . . . . . . . . . 354
11.2.Lengthcomputationsetc.,inthepresenceofametricfield . . . . . . . . . . . . . 364
11.3.Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
ConclusionsandPerspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Foreword
The“GeometricModelingandApplications”seriesismadeupoffivevolumesofresearch
ongeometricmodeling. Theactofgeometricmodelingisanage-oldone. Tolookatafewkey
pointsinitshistory: geometristsinantiquitydevotedthemselvestoit,though,ofcourse,ithad
adifferentformfromthatwhichweknowtoday; Descartes, inthemid-17thCentury, worked
onpartitioningofplanes;andVoronoiestablishedconceptsrelatingtothistowardtheendofthe
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,foranyobjectorentitythatistobemodeled,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.
x Meshing,GeometricModelingandNumericalSimulation2
•“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
Introduction
Triangulations and, more precisely, meshes, with the subtle difference that separates these
twoentities, lieattheheartofanynumberofproblemsthatarisefromvariedscientificdisci-
plines. Atriangulationorameshisadiscreterepresentation,usingsimplegeometricelements
(triangle,quadrilateral,tetrahedron,etc.,anyarbitrarypolygonorpolyhedron),ofadomainthat
maybeanobjectoraregionofwhichwewantadiscrete,spatialdescription. Thereare,thus,
manyapplications,includingnumericalsimulationsofanykindofphysicalproblem,thoughnot
restrictedtothese. Inparticular,adiscreterepresentationofa(volume)objectorasurfacemay
simplybeseenasageometricmodelingproblemasis.Thisbookadoptsadoublepointofview,
asindicatedbyitstitle,andwewilllookbothattheuseofmeshesinnumericalsimulations(the
finiteelementmethod,especially)with,ofcourse,theunderlyingconstraints,aswellastheuse
ofthesemeshesforthe(discrete)modelingofgeometry.
The literature on triangulations and meshes may be classified in two chief categories: one
morepurelymathematicalandgeometric,theothermoreorientedtowardindustrialapplications
(numericalsimulations)with,ofcourse,thoughnotalways,relationsbetweenthesecategories.
The first point of view is covered by the computation geometry community, which stud-
ies (among others) Delaunay triangulations in all cuts, definitions, properties, construction al-
gorithms and the complexity of these algorithms. They also study some applications of these
triangulations. Nonetheless,relativelyrecently,meshgenerationproblemsarealsobeingstud-
ied,butunderamoretheoreticalangle,generallyrelatingtosituationsthatallowfortheuseof
Delaunaytriangulationsforwhichrobustconstructionmethodsaswellasinterestinggeometric
propertieshavebeenknownforalongtime.This(somewhatmonotheistic)philosophynecessar-
ilyimposeslimitsonthenatureoftheproblemsworked(workable).Thefirstreferencebookon
thesesubjectsisthatofPreparataandShamos[Preparata,Shamos-1985]publishedin1985.This
wasfollowedbyseveralothers,amongwhichwecitetwobyEdelsbrunner[Edelsbrunner-1987]
and [Edelsbrunner-2001], that of Yvinnec and Boissonnat [Boissonnat,Yvinec-1997], by Dey
[Dey-2007]andwindingupthelistwiththebookbyChengetal. [Chengetal. 2012],which
waspublishedin2012. Withafewexceptions,theorientationchosenbythesereferencesisnot
alwaysguidedbythepreoccupationsofmathematicians,engineersandtheworldofnumerical
simulationingeneral.
