IsoGeometric Analysis: A New Paradigm in the Numerical Approximation of PDEs: Cetraro, Italy 2012

203 Pages·2016·6.2 MB·English

by  Annalisa Buffa

Lecture Notes in Mathematics 2161
CIME Foundation Subseries
Annalisa Buffa
Giancarlo Sangalli Editors
IsoGeometric Analysis:
A New Paradigm
in the Numerical
Approximation of
PDEs
Cetraro, Italy 2012
Lecture Notes in Mathematics 2161
Editors-in-Chief:
J.-M.Morel,Cachan
B.Teissier,Paris
AdvisoryBoard:
CamilloDeLellis,Zurich
MariodiBernardo,Bristol
AlessioFigalli,Zurich
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Athens
GaborLugosi,Barcelona
MarkPodolskij,Aarhus
SylviaSerfaty,ParisandNewYork
CatharinaStroppel,Bonn
AnnaWienhard,Heidelberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/304

Annalisa Buffa • Giancarlo Sangalli
Editors
IsoGeometric Analysis:
A New Paradigm in the
Numerical Approximation
of PDEs
Cetraro, Italy 2012
123
Editors
AnnalisaBuffa GiancarloSangalli
IMATI-CNR DipartimentodiMatematica
Pavia,Italy UniversitàdiPavia
Pavia,Italy
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-319-42308-1 ISBN978-3-319-42309-8 (eBook)
DOI10.1007/978-3-319-42309-8
LibraryofCongressControlNumber:2016954372
MathematicsSubjectClassification(2010):65N30
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Preface
Isogeometric analysis (IGA) refers to a collection of methods, first introduced by
T.J.R.Hughesandcollaboratorsintheseminalpaper[3],thatusesplines,orsome
oftheirgeneralisationssuchasNURBS(non-uniformrationalB-splines),T-splines
andhierarchicalsplines,asfunctionstobuildapproximationspaceswhicharethen
usedtonumericallysolvepartialdifferentialequations(PDEs).Indeed,splinesand
their extensionsare the basic mathematicalengine behind CAD systems, and one
of the main motivation for IGA was to design numerical methods able to avoid
remeshing steps by using the output of CAD systems directly. In the last decade,
isogeometricmethodshaveattractedagreatinterestofthescientificcommunity,as
itisclearlydocumentedbythenumberofpublicationsonthesubjectthatappeared
inthefieldsofgeometricmodelling,mechanicalengineeringandnumericalanalysis
ofPDEs.
Isogeometric methods have been tested and applied on a variety of problems
of engineering interest, and there is indeed a large engineeringliterature showing
the beneficial effects of higher regularity in several practical problems; see, for
instance, the referencesgiven in [2] and in the introductionof Beirão da Veiga et
al. [1]. Indeed, the large activity around this class of methods shows that splines
yielda powerfulapproachtoPDEsdiscretisationandthat,thankstotheregularity
oftrialandtest spaces,spline-basedmethodsenjoyfeaturesthatwouldbe hardto
achievewithclassicalfiniteelements.Examplesarethedesignofregularandfully
conservative discretisations of flow and electromagnetic problems or the natural
discretisationsofhigh-orderPDEs.
The aim of this C.I.M.E. summer school has been to give an introduction to
isogeometric methods with a focus on their mathematical foundations. The four
chapters of this book collect the main topics that have been presented during the
lectures.
Chapter“StandardandNon-standardCAGDToolsforIsogeometricAnalysis:A
Tutorial” authored by Carla Manni and Hendrik Speleers, is a tutorial on splines
and generalisations that are used in CAD parametrisations and form the building
blocksofisogeometricmethods.Chapter“ModelsforIsogeometricAnalysisfrom
CAD”,byVibekeSkyttandTorDokken,givesanoverviewofgeometricmodelling
v
vi Preface
techniquesthatcanbeusedwithisogeometricmethods.Chapter“AnIntroductionto
theNumericalAnalysisofIsogeometricMethods”,writtenbyLourencoBeirãoda
Veiga, Rafael Vázquez and ourselves, is an overview of mathematical properties
of isogeometric spaces, and finally in chapter “Isogeometric Compatible Dis-
cretizationsforViscousIncompressibleFlow”,byJohnA.EvansandThomasJ.R
Hughes,givesanexampleapplicationoffundamentalimportance,theisogeometric
simulationofaviscousincompressibleflow.
We express our deepestgratitude to all the people who have contributedto the
successofthisC.I.M.E.summerschool:thelecturers,thespeakers,theauthorswho
have contributedto this C.I.M.E. FoundationSubseries book and the participants.
WethankFedericoMariniforhishelpinthepreparationofthisvolume.Lastbutnot
least, we thank C.I.M.E., in particular Elvira Mascolo (currentC.I.M.E. Director)
and Pietro Zecca (former C.I.M.E. Director) for their continuous support in the
organisationoftheschool.
Pavia,Italy AnnalisaBuffa
Pavia,Italy GiancarloSangalli
January2016
Acknowledgements
C.I.M.E. activity is carried out with the collaboration and financial support of:
INdAM(IstitutoNazionalediAltaMatematica),MIUR(Ministerodell’Istruzione,
dell’Universitàe della Ricerca),andEnte Cassa diRisparmio diFirenze.Partially
supportedbyERCStGGeoPDEs,grantn.20500.
References
1.L. Beirão da Veiga, A. Buffa, G. Sangalli, R. Vázquez, Mathematical analysis of variationl
isogeometricmethods. ActaNumer.23,157–287(2014)
2.J.A.Cottrell,T.J.R.Hughes,Y.Bazilevs, Isogeometric Analysis:TowardIntegrationofCAD
andFEA (Wiley,NewYork,2009)
3.T.J.R.Hughes,J.A.Cottrell,Y.Bazilevs,Isogeometricanalysis:CAD,finiteelements,NURBS,
exactgeometryandmeshrefinement. Comput.MethodsAppl.Mech.Eng.194(39–41),4135–
4195(2005)
vii
Contents
StandardandNon-standardCAGDToolsforIsogeometric
Analysis:ATutorial.............................................................. 1
CarlaManniandHendrikSpeleers
ModelsforIsogeometricAnalysisfromCAD................................. 71
VibekeSkyttandTorDokken
AnIntroductiontotheNumericalAnalysisofIsogeometricMethods..... 87
L.BeirãodaVeiga,A.Buffa,G.Sangalli,andR.Vázquez
Isogeometric Compatible Discretizations for Viscous
IncompressibleFlow............................................................. 155
JohnA.EvansandThomasJ.R.Hughes
ix
Standard and Non-standard CAGD Tools
for Isogeometric Analysis: A Tutorial
CarlaManniandHendrikSpeleers
Abstract We present a detailed summary of the main CAGD tools of interest in
IGA: Bernstein polynomials and B-splines. Besides their well-known algebraic
and geometric properties, we give a deeper insight into why the corresponding
representationsareso popularandefficientbyprovingthattheyare optimalbases
forthecorrespondingfunctionspaces.Moreover,wereviewsomegeneralizationsof
theB-splinestructureinfunctionspaceswhichextendclassicalpolynomialspaces.
Extensionsto the bivariate setting beyondthe straightforwardtensor-productcase
arediscussedaswell.Inparticular,wefocusonthetriangularsetting.
1 Introduction
Computeraidedgeometricdesign(CAGD)isawell-establishedareaofmathemat-
icsdevotedtotherepresentationandmanipulationofcurvesandsurfaces.Recently,
thisareahasattractedaconsiderableinterestfromresearchersworkinginnumerical
simulationduetothesuccessoftheemergingfieldofisogeometricanalysis(IGA),
see[20].
Theaimofthischapteristoprovideanextendedsummaryofthemostpopular
tools in CAGD, namely Bernstein polynomials and B-splines, and some possible
generalizationsforthem.
There exists a large amountof literature on the abovetopics, includingseveral
well-establishedbooks[8,26,35,58,63]andclassicalreviewpapers[6].However,
ourgoalistwofold.First,wewanttopresentthematerialfromtheIGAperspective.
Second, we aim to provide a deeper insight into the well-known algebraic and
geometric properties of Bernstein polynomials and B-splines, in order to explain
C.Manni((cid:2))
DipartimentodiMatematica,UniversitàdegliStudidiRoma“TorVergata”,Roma,Italy
e-mail:[email protected]
H.Speleers
DipartimentodiMatematica,UniversitàdegliStudidiRoma“TorVergata”,Roma,Italy
DepartementComputerwetenschappen,KatholiekeUniversiteitLeuven,Leuven,Belgium
e-mail:[email protected];[email protected]
©SpringerInternationalPublishingSwitzerland2016 1
A.Buffa,G.Sangalli(eds.),IsoGeometricAnalysis:ANewParadigm
intheNumericalApproximationofPDEs,LectureNotesinMathematics2161,
DOI10.1007/978-3-319-42309-8_1