Table Of ContentLecture Notes of the Unione Matematica Italiana
Michel Bercovier
Tanya Matskewich
Smooth Bézier
Surfaces over
Unstructured
Meshes
Lecture Notes of 22
the Unione Matematica Italiana
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Michel Bercovier • Tanya Matskewich
Smooth Bézier Surfaces over
Unstructured Quadrilateral
Meshes
123
MichelBercovier TanyaMatskewich
RachelandBeninSchoolofComputer MicrosoftCorporation
ScienceandEngineering Redmond,Washington
HebrewUniversityofJerusalem USA
Jerusalem,Israel
ISSN1862-9113 ISSN1862-9121 (electronic)
LectureNotesoftheUnioneMatematicaItaliana
ISBN978-3-319-63840-9 ISBN978-3-319-63841-6 (eBook)
DOI10.1007/978-3-319-63841-6
LibraryofCongressControlNumber:2017948657
MathematicsSubjectClassification(2010):65D17(principal),65M60(secondary)
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Foreword
ImetMichelBercovieralongtimeago,in1976tobeprecise,atthefamoussecond
conferenceonFiniteElementMethodsinFlowProblems,heldinSantaMargherita
Ligure,Italy.Hewasthenarecentdoctoralgraduateandfull-fledgedmemberofthe
“French school” of numerical analysis that brought a deep analytical perspective
to the study of the Finite Element Method. That school emanated from a single
figure,JacquesLouisLions,amathematicianofraregifts,andanindividualwhose
character,humanityandleadershipwere atthe veryleast the equalof hissingular
mathematicaltalent. Michel Bercovier was a PhD studentof Jacques LouisLions
anda worthydisciple.Michelwastalentedandenthusiastic;hepossessed thejoie
devivreofhiscultureandmademanyimportantresearchcontributionstotheFinite
ElementMethod.Fromtimetotimeweinteracted,atconferencesandoftenthrough
connectionswith industrialorganisationsthatwereutilising and/ordevelopingthe
Finite Element Method, as Michel was very actively engaged as a consultant to
industry.ThenonedayInoticedhestartedpublishingpapersaboutComputerAided
Design (CAD). I wonderedwhy. My reaction was “Who cares aboutthat?” I was
focused on the Finite Element Method, then and forever,or at least that is what I
thoughtatthetime.
The origins of the Finite Element Method are diffuse. One can point to many
seminalcontributionsinmathematicsandengineeringthatallseemtohavepointed
in the same direction. Let us say the wind was blowing in a direction and it
was felt by many with diverse backgroundsand perspectives, and they developed
theirideasthatevolvedintothebuildingblocksofwhattodaywerightlyconsider
an edifice: The Finite Element Method. As time has gone on, the paradigm
of the Finite Element Method has been refined through literally hundreds of
thousands of research papers and through implementation in numerous industrial
and commercially available computer programs. Within the paradigm, what more
canweexpect?Afterall,theprocessofdevelopmentandrefinementhasbeengoing
onforabout60years.Optimisingperformanceofcomputerprogramscanonlytake
onesofarwhentheparadigmisstagnant.
It is quite instructive to step back and adopt the perspective of the overall
process of product development. The fundamental steps are design, analysis and
v
vi Foreword
manufacturing. Design and analysis are linked in an iterative loop. One informs
the other until the design convergesand is ready for manufacture. It all seems so
simple,untilyoulookwherethetimegoes.Itisnotdominatedbydesign,anditis
notdominatedbyanalysis,butratherbytheprocessofconvertingcomputeraided
designfilestofiniteelementanalysisfiles.Extraordinaryasitseems,about90%of
overalltimeisdevotedtothistask.Itmaymaketheanalystshappyif,forexample,
theirlinearmatrixequationsolverisspeededupabit,butithaslittleeffectreducing
thetime oftheoveralldesign-through-analysisprocess.Theproblemisthechasm
thathasemergedbetweencomputerdesignandanalysistechnologies.Isogeometric
Analysis was invented to address this problem, but before Isogeometric Analysis
wasinvented,MichelBercovierwasworkingontheproblem,andmakingnumerous
contributionsalongtheway,convincedthat,inhiswords,“CAD andFEMshould
befriends.”
A topic of recent interest in Isogeometric Analysis is the smooth merging of
tensor-product spline patches. It is obvious that smoothly joined spline patches
are a natural platform for the structural analysis of shells, and there are several
otherimportantapplicationsas well. The technicalitiesare greaterthanone might
imagine and it is a significant challenge. In 2014, in Pontignano, Tuscany, at the
FirstInternationalConferenceonSubdivision,GeometricandAlgebraicMethods,
Isogeometric Analysis and Reliability, I was very surprised to hear from Michel
aboutthe workthathe haddoneon thistopic about15yearsearlier. Atthattime,
apparently, there was not much of an interested audience but, with the rise of
IsogeometricAnalysis, there is now. I encouragedMichel to resurrectand further
develop his ideas and publish them, and this monograph is the product of that
endeavour.Better late thannever.Its publicationnowis an importantcontribution
toaproblemofconsiderablecontemporaryinterest.
Austin,TX,USA TomHughes
August2016
Preface
Thecoreofthepresentmonographdetailsandextendsresultsthatwereoriginally
developedfrom1999to2001andpresentedatseveralconferences(Dagshtul1999,
CurvesandSurfaces,StMalo1999,GeometricDesignandComputing,Sacramento
2001,andmore)butwerenotsubmittedforpublication.Thereasonwasthatthese
results did not seem to have an echo: on the one hand the actors in the field of
ComputerAidedDesignforGeometry(CAGD)didnotseeanyplusinusingFinite
Element (like) Methods (FEM) on quadrilaterals;besides, why be so rigid on the
mesh requirements? Just subdivide! On the other hand, mechanical engineers did
notseewhyoneshoulduseBéziercontrolpointsinsteadoftheclassicnodalones:
whatdoesthe“velocity”ofacontrolpointmean?
Our motivations are based on a long industrial experience of Computer Aided
Design as well as Finite Elements: computational domains deduced industrial
objects can be quite complex and are often defined by a collection of patches. A
naturalquestionisthen:canwebuildapproximationspacesonmulti-patcheswhile
keeping a high order of smoothness across the patches? What are the conditions
for such a solution to exist? These questions are the main motivating ones of the
present monograph. Considering the case of shells approximated by quadrilateral
planar or curved elements in the Finite Element Method, we want to build a free
formsurface,atleastC1.Thisconstructioncanbetheresultofashellequilibrium
problem(minimisation)oraninterpolationbasedontheunderlyingquadrilaterals,
andourquestionis:whenissuchaconstructionpossible?
Thisleadstothefollowingproblem:givenapolynomialorderofapproximation,
n, and the corresponding piecewise Bézier graph patches over an unstructured
quadrilateral mesh made of convex quadrilaterals with vertices of any valence,
definetheconditionsofexistenceofaC1 graphsurface.Inthe interpolationcase,
for instance, constraints defining regularity conditions across patches have to be
satisfied. The resulting number of free degrees of freedom must be such that the
interpolation problem has a solution! This is similar to studying the existence
and dimension of a minimal determining set (MDS) for such a G1 continuity
construction.
vii
viii Preface
Our positiveanswer wouldhavestayed a singularcuriosity,butfor the success
oftherecentlydevelopedIsogeometricAnalysis(IgA).ItstronglylinksComputer
Aided GeometryDesign and Analysis, with the aim to unify the underlyingbasis
functionsofbothfieldsandavoidingthecumbersomestepsofgeometrytomeshand
back. IgA relies on smooth shape functions, Cp(cid:2)1 continuous, splines or rational
splines of polynomial order p. However computational domains deducted from
industrial objects can be quite complex and are often defined by a collection of
patches.
Therefore one recent research theme in IgA is the study of C1 assembly of
patches.The simplerBézier setup is a toolforthe analysisof thisproblem,hence
the interest of the presentwork: it providesthe algorithmsfor the constructionof
B-spline approximations on such general assemblies. The locality of the Bézier
patches simplifies the construction of inter patch C1 relations and the actual
explicit definition of basis functions. Based on the equivalence of G1 and C1 on
straight edges, we introduce a sufficient G1 condition that is better adapted to the
presentproblem.Boundaryconditionsarethenanalysedincludingnormalderivative
constraints (common in FEM but not in Computer Aided Design). The MDS is
constructed for both polygonal meshes and meshes with G1-smooth piecewise
Béziercubicglobalboundary.
Limitingourselvestographsurfacesovera piecewisebilinearparameterisation
andtoBézierpatchesinsteadofB-Splinesallowsustostudythestronginteraction
between possible solutions and the geometry of the underlying parameterisation.
ExtensiontoB-Splinesisthenstraightforward.Thusthisworkdevelopsimportant
toolsfortheanalysisofG1Isogeometricapproximationsoverunstructuredpatches
andprovidesanswerstothequestionoftheAnalysisSuitabilityofG1Isogeometric
spaces. We suppose that the reader has a good knowledge of the foundations of
CAGDasgivenin[24]or[15]andhassomenotionsofIgA,asintroducedin[18].
The publicationofthepresentwork owesmuchtoTom HugheswhoatSMART
2014 in Pontignano pointed to us the importance of our results and strongly
encouraged us to publish them. Special thanks also go to Annalisa Buffa for her
supportandtoBertJüttlerandhisstudents(especiallyMarioKapl)whoextended
our results and motivated us in detailing some examples. Special thanks also to
the anonymous reviewer whose remarks and comments greatly contributed to the
quality of the present version. Thanks also to friends and colleagues who helped
correct the manuscript. And most special thanks from the first author to Nicole
Poinsowhoenduredayearofrewriting.
Jerusalem,Israel MichelBercovier
Redmond,WA,USA TanyaMatskewich
January2017
Contents
1 Introduction .................................................................. 1
1.1 Motivation .............................................................. 1
1.2 ProblemDefinition .................................................... 4
1.2.1 BézierPatches.................................................. 4
1.2.2 DescriptionoftheProblem.................................... 6
1.2.3 BriefReviewofRelatedWorks............................... 9
1.2.4 ThePrincipalAimofthePresentWork....................... 12
1.2.5 DomainsofApplication....................................... 16
1.3 Notations................................................................ 17
1.3.1 PointsandVectors ............................................. 17
1.3.2 Polynomials.................................................... 18
1.3.3 PlanarMeshData.............................................. 19
1.3.4 PartialDerivativesofanIn-PlaneParameterisation.......... 20
1.3.5 WeightFunctions .............................................. 21
1.3.6 3DDataoftheResultingSurface............................. 21
1.3.7 DefinitionsofSpecialSets,SpacesandEquations........... 24
2 G1-SmoothSurfaces ......................................................... 25
2.1 BasicDefinitionsRelatedtoSmoothnessofGluedSurfaces......... 25
2.2 TheVertexEnclosureProblem ........................................ 27
2.2.1 GeneralFormulationoftheVertexEnclosureConstraint.... 27
2.3 LinearisationoftheSmoothnessConstraintsandMinimisation..... 30
2.3.1 LinearisationoftheSmoothnessCondition .................. 30
2.3.2 SimpleExampleofDegree3.................................. 34
2.3.3 LinearFormofAdditionalConstraints....................... 36
2.3.4 QuadraticFormoftheEnergyFunctional.................... 38
2.4 PrinciplesofConstructionofanMDS ................................ 39
2.4.1 SpecialSubsetsofControlPointsandTheir
Dimensionality................................................. 39
2.4.2 RelationBetweenMDSsandtheAdditionalConstraints.... 40
2.4.3 ThePrincipleofLocalityintheConstructionofMDSs ..... 41
ix
