Robust methods for fluid-structure interaction with stabilised finite elements [Elektronische Ressource] / von Christiane Förster
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Robust methods for fluid-structure interaction with stabilised finite elements [Elektronische Ressource] / von Christiane Förster

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! Robust methods for fluid-structureinteraction with stabilised finite elementsvonChristiane F¨orsterBericht Nr. 51 (2007)Institut fur¨ Baustatik und Baudynamik der Universitat¨ StuttgartProfessor Dr.-Ing. M. BischoffStuttgart 2007c Christiane F¨orsterBerichte k¨onnen bezogen werden ub¨ er: / Reports are distributed by:Institut fur¨ Baustatik und BaudynamikUniversit¨at StuttgartPfaffenwaldring 7D-70550 StuttgartTel.: ++49(0)711/685 66123Fax: ++49(0)711/685 66130http://www.ibb.uni-stuttgart.de/¨Alle Rechte, insbesondere das der Ubersetzung in andere Sprachen, vorbehalten. Ohne Genehmi-gungdesAutorsistesnichtgestattet, diesenBerichtganzoderteilweiseaufphotomechanischem,elektronischem oder sonstigem Wege zu kommerziellen Zwecken zu vervielfaltigen.¨All rights reserved. In particular, the right to translate the text of this thesis into another lan-guage is reserved. No part of the material protected by this copyright notice may be reproduced orutilized in any form or by any means, electronic or mechanical, including photocopying, record-ing or by any other information storage and retrieval system, without written permission of theauthor.

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!

Robustmethodsforfluid-structure
interactionwithstabilisedfiniteelements

nov

ChristianeF¨orster

BerichtNr.51(2007)
Institutf¨urBaustatikundBaudynamikderUniversit¨atStuttgart
ProfessorStuttDr.-Ing.gart200M.7Bischoff

cChristianeF¨orster
Berichtek¨onnenbezogenwerden¨uber:/Reportsaredistributedby:
Institutf¨urBaustatikundBaudynamik
Universit¨atStuttgart
D-70Pfaffen550waldrStuttgingart7
Tel.:++49(0)711/68566123
Fax:++49(0)711/68566130
http://www.ibb.uni-stuttgart.de/

AlleRechte,insbesonderedasder¨UbersetzunginandereSprachen,vorbehalten.OhneGenehmi-
gungdesAutorsistesnichtgestattet,diesenBerichtganzoderteilweiseaufphotomechanischem,
elektronischemodersonstigemWegezukommerziellenZweckenzuvervielf¨altigen.
Allrightsreserved.Inparticular,therighttotranslatethetextofthisthesisintoanotherlan-
guageisreserved.Nopartofthematerialprotectedbythiscopyrightnoticemaybereproducedor
utilizedinanyformorbyanymeans,electronicormechanical,includingphotocopying,record-
ingorbyanyotherinformationstorageandretrievalsystem,withoutwrittenpermissionofthe
author.

D93-DissertationanderUniversit¨atStuttgart
ISBN978-3-00-022267-2

Robustmethodsforfluid-structure
interactionwithstabilisedfiniteelements

VonderFakult¨atBau-undUmweltingenieurwissenschaften
derUniversit¨atStuttgartzurErlangungderW¨urdeeinesDoktorsder
Ingenieurwissenschaften(Dr.-Ing.)genehmigteAbhandlung

vorgelegtvon

ChristianeF¨orster

Hauptberichter:Prof.
Dr.-Ing.1.Mitberichter:
2.Mitberichter:
Tagderm¨undlichenPr¨ufung:

aukwicZaus

Prof.Dr.-Ing.Dr.-Ing.E.h.Dr.h.c.EkkehardRamm
Prof.Dr.RamonCodina
Prof.Dr.-Ing.WolfgangA.Wall
Pr¨ufung:13.Juli2007

Institutf¨urBaustatikundBaudynamikderUniversit¨atStuttgart
7200rtgaStutt

Zusenfassungamm

InnahezuallenBereichendesIngenieurwesenstretenMehrfeldproblemeauf,zudenen
auchFluid-Struktur-Interaktionen(FSI)zuz¨ahlensind.DieseArbeittr¨agtzurEntwick-
lungeinesstabilenundrobustennumerischenVerfahrenszurL¨osungsolcherFSI-Pro-
blemebei.Hierwerdenspeziellzwei-unddreidimensionaleStrukturenbetrachtet,diein
WechselwirkungmitinkompressiblenFl¨ussigkeitentreten.DabeiistdasStrukturverhal-
tendurchdienichtlinearenGleichungenderElastodynamikbestimmt.DieDynamikdes
FluidswirddurchdieinkompressiblenNavier-Stokes-Gleichungenbeschrieben.Beide
FelderwerdenmitHilfefiniterElementeimRaumundmittelsDifferenzenverfahreninder
Zeitdiskretisiert.UmdasgekoppelteProblemzul¨osen,kommteiniterativgestaffeltes
partitioniertesKopplungsverfahrenmitRelaxationzumEinsatz.
DerSchwerpunktdieserArbeitliegtaufmethodischenAspekten.Insbesonderesollen
dietheoretischenGrundlagendesnumerischenVerfahrensverbessertwerden.Dabeiist
dasZielsicherzustellen,daßdasVerfahrenstabill¨auftundf¨ureinenweitenParameterbe-
reichErgebnissevonverl¨aßlicherGenauigkeitliefert.BesondereAufmerksamkeitgiltdem
Fluidl¨oser,derinArbitraryLagrangeanEulerian“(ALE)Betrachtungsweiseformu-
liertist.DasVerha”ltendesFluidswirdalsoinBezugaufeinbewegtesKoordinatensystem
beschrieben.Dahergilteshier,nebendenklassischenErhaltungss¨atzenf¨urMasse,Impuls
undEnergieauchdiegeometrischeErhalungzubeachten.DerZusammenhangzwischen
denverschiedenenErhaltungss¨atzenundderStabilit¨atdesnumerischenVerfahrenswird
untersuchtundesk¨onnenStabilit¨atsgrenzeninFormvonmaximalenZeitschrittweiten
f¨urverschiedeneVerfahrenangegebenwerden.Weiterhinkanngezeigtwerden,wieein
unbedingtstabilesALEVerfahrenformuliertwerdenmuß.Einn¨achstesSchwerpunkt-
themaistdasstabilisierteFinite-Element-VerfahrenaufdembewegtenGebiet.Eswird
eineVersiondesStabilisierungsverfahrenshergeleitet,derenStabilit¨atvonderNetzbewe-
gungnahezuunber¨uhrtbleibt.WeitereUntersuchungenbetreffendieEmpfindlichkeitdes
VerfahrensinBezugaufkritischeParameterwiesehrkleineZeitschritte,steileGradien-
tenoderauchstarkverzerrteNetze.F¨urElementeh¨ohererOrdnungistdasstabilisierte
Verfahrenvollst¨andigkonsistent.Eswirdgezeigt,daßmitsolchenElementenauchauf
deutlichverzerrtenNeztensehrgenaueErgebnisseerzieltwerdenk¨onnen.
BesonderesAugenmerkwirdauchaufdieFluid-Struktur-KopplungimRahmenparti-
tionierterVerfahrengelegt.IndiesemZusammenhangbetriffteineersteFragedenAus-
tauschgenauerundmethodischkonsistenterKopplungsinformationanderGrenz߬ache
zwischenFluidundStruktur.Weiterhinwirddersogenannteartificialaddedmassef-
fect“analysiert.DieserEffektbezeichnetdieinherenteInstabilit”¨at,diebeisequentiell
gestaffeltenVerfahrenauftritt,wennleichteStrukturenmitinkompressiblenFluidenge-
koppeltwerden.DabeiistletztendlichdieInkompressibilit¨atdaf¨urverantwortlich,daß
einfachesequentiellgestaffelteVerfahrennichterfolgreichverwendetwerdenk¨onnen.Die
mathematischeAnalyse,dieimRahmendieserArbeitvorgenommenwird,zeigt,warum
dieInstabilit¨atnichtnurvomMassenverh¨altnisderbeteiligtenKontinua,sondernauch
vonderZeitdiskretisierungderFelderabh¨angt.Eswirddeutlich,warumgenauerezeitliche
Diskretisierungsans¨atzeeinfr¨uheresEintretenderInstabilit¨atzurFolgehaben.
DietheoretischenErgebnissewerdendurchbegleitendekleineBeispielrechnungenver-
anschaulicht.Einigegr¨oßereAnwendungendesVerfahrenswerdenamSchlußderArbeit
t.tierasen¨pr

tractAbs

Variousmultifieldproblemsandamongthemfluid-structureinteractionapplicationsarise
innearlyallfieldsofengineering.Thepresentworkcontributestothedevelopmentof
astableandrobustapproachforthenumericalsimulationoffluid-structureinteraction
problems.Inparticulartwo-dimensionalandthree-dimensionalelasticstructuesinter-
actingwithincompressibleflowareconsidered.Thestructuralfieldisgovernedbythe
nonlinearelastodynamicequationswhilethedynamicsofthefluidfieldaredescribedby
theincompressibleNavier-Stokesequations.Bothfieldsarediscretisedbyfiniteele-
mentsinspaceandfinitedifferencemethodsintime.Aniterativelystaggeredpartitioned
couplingprocedurewithrelaxationisappliedtoobtaintheoverallcoupledsolution.
Thisworkfocusesonmethodologicalaspectsandcontributestoadeeperunderstanding
ofthetheoreticalfoundationsoftheapproach.Thisisnecessarytoensurethatthefor-
mulationisstableandoffersreliableresultsforawiderangeofparameters.Inparticular
theflowsolverformulatedinanarbitraryLagrangeanEulerianapproachisconsid-
ered.Inadditiontotheclassicalconservationlawsofmass,linearmomentumandenergy
geometricconservationhastobeconsidered.Thisisaconsequenceoftheformulationof
theflowequationswithrespecttoamovingframeofreference.Therelationshipofthese
conservationlawsandthestabilityofthenumericalschemeisinvestigatedandstability
limitsintermsofmaximaltimestepsizesfordifferentformulationsareestablished.It
isfurthershownhowanunconditionallystableALEformulationhastobeconstructed.
Anotherkeyissueisthestabilisedfiniteelementmethodemployedonthefluiddomain.
Thederivationofthemethodfromavirtualbubbleapproachisrevisitedwhilespecial
attentionisturnedtothefactthatthedomainismoving.Aversionofthestabilisation
isderivedwhichisnearlyunaffectedbythemotionoftheframeofreference.Furtherthe
sensitivityofthestabilisedformulationwithrespecttocriticalparameterssuchasvery
smalltimesteps,steepgradientsanddistortedmeshesisassessed.Atleastforhigher
orderelementswherefullconsistencyoftheformulationisassuredveryaccurateresults
canbeobtainedonhighlydistortedmeshes.
Asanothermainissuethecouplingoffluidandstructurewithinapartitionedscheme
isconsidered.Afirstconcerninthiscontextistheexchangeofpropercouplingdata
attheinterfacewhichiscrucialfortheconsistencyoftheoverallscheme.Subsequently
theso-calledartificialaddedmasseffectisanalysed.Thiseffectisresponsibleforan
inherentinstabilityofsequentiallystaggeredcouplingschemesappliedtothecouplingof
lightweightstructuresandincompressibleflow.Itisessentiallytheinfluenceoftheincom-
pressibiltywhichexcludesthesuccessfuluseofsimplestaggeredschemes.Theanalysis
derivedinthecourseofthisworkrevealswhytheartificialaddedmassinstabilityde-
pendsuponthemassratiobutfurtheronthespecifictimediscretisationusedonthefluid
andstructuralfield.Inparticularitisshownwhymoreaccuratetemporaldiscretisation
resultsinanearlieronsetoftheinstability.
Whilethetheoreticalconsiderationsareaccompaniedbysmallnumericalexamples
highlightingparticularaspectssomelargerapplicationsofthemethodarefinallypre-
ted.sen

wroVort

DievorliegendeArbeitentstandinderZeitvonEnde2002bisAnfang2007w¨ahrendmei-
nerT¨atigkeitzun¨achstamInstitutf¨urBaustatikundnachderUmbenennungimSommer
2006amInstitutf¨urBaustatikundBaudynamikderUniversit¨atStuttgart.IndieserZeit
genoßichdasPrivilegalswissenschaftlicherMitarbeiterandiesemrenomiertenInstitut
undgleichzeitigalsMitglieddesSonderforschungsbereiches404”Mehrfeldproblemeinder
Kontinuumsmechanik“arbeitenzud¨urfen.
MeinemDoktorvaterHerrnProf.Dr.-Ing.Dr.-Ing.E.h.Dr.h.c.EkkehardRamm
m¨ochteichandieserStelleaußerordentlichherzlichf¨urdieM¨oglichkeitendanken,die
ermiranseinemInstituter¨offnethat.ErhateineAtmosph¨aregeschaffen,inderichmich
wohlgef¨uhltundsehrgerngearbeitethabe.VondemVertrauenunddemR¨uckhalt,dener
meinerArbeitgegebenhat,habeichnichtnurwissenschaftlichprofitierenk¨onnen.Auch
f¨urprivateProblemehatteerstetseinoffenesOhr.VielenDankdaf¨ur!
AuchHerrnProf.Dr.-Ing.habil.ManfredBischoff,derdasInstitutkurzvordemEnde
dieserArbeit¨ubernommenhat,dankeichf¨urdiegroßz¨ugigefachlicheundpers¨onliche
Unterst¨utzung,diemiralsgleichsamadoptiertemDoktorandenzuteilwurde.Erhatdas
beispiellosguteArbeitsumfeldamInstituterhaltenundmitneuemElangef¨ullt.Jeder
TagderZusammenarbeitmitihmwareineechteBereicherung.
HerrnProf.Dr.CodinavomDepartamentdeResistenciadeMaterialsiEstructu-
resal’Enginyeria“derUniversita”tPolitecnicadeCatalunya“inBarcelonadankeich
f¨urdie¨Ubernahmedes”Mitberichtes,diegr¨undlicheDurchsichtdesManuskriptessowie
KommentareundVerbesserungsvorschl¨age.SeinInteresseanmeinerArbeitistmireine
unverdientgroßeEhreundhatmichsehrgefreut.
OhnedasWirkenvonProf.Dr.-IngWolfgangA.WallvomLehrstuhlf¨urnumeri-
scheMechanikderTUM¨unchenw¨aredieseArbeitsonichtzustandegekommen.Seine
herrvorragendeDissertationhattedasThemaFluid-Struktur-InteraktionamInstitutf¨ur
BaustatikinStuttgarteinstetabliertundistdamitWegbereiterundBasisdervorliegen-
denArbeit.IchdankeProf.Wallf¨urseinefachlicheUnterst¨utzungindenletztenJahren,
diemirtrotzseinesstets¨ubervollenTerminkalenderszuteilwurde.Auchf¨ursomanchen
privatenZuspruchdankeichihmsehrherzlich.
Bedankenm¨ochteichmichauchbeidenKollegenamInstitutf¨urBaustatikundBau-
dynamikf¨urdieguteZusammenarbeit,gegenseitigesVertrauenundfreundschaftlichen
Umgangmiteinander.BesondererDankgiltHerrnMaltevonScheven,dermitseinem
selbstlosenBem¨uhenumdiePflegeundBetreuungdesinstitutseigenenQuellcodeszum
GelingendieserArbeitbeigetragenhat.
WertvolleUnterst¨utzungundHilfehabeichauchvonmeinemBruderUlrichK¨uttler
erfahren.Herzlichdankeichihmf¨urdieDurchsichtmeinerschriftlichenArbeit,dasstete
InteresseanmeinemVorankommenundmeinenErgebnissenundseinenvielfachenguten
RatinComputerfragen.
EinwesentlicherTeildervorligendenArbeitwurdeammathematischenDepartment
der”KungligaTekniskaH¨ogskolan“(KTH)inStockholmzusammengeschrieben.Ichbe-
dankemichbeimgesamtenDepartmentundganzbesondersbeiProf.Dr.AriLaptevf¨ur
dieherzlicheAufnahme,dieichdorterfahrenhabe.
GanzherzlicherDankgiltandieserStellemeinemMannClemens.ErhatmeineAr-

8

beitfachlichbereichertundmitbewundernswerterGeduldundAusdauerbedingungslos
unterst¨utzt.Ichdankeihmf¨urseinunersch¨utterlichesliebevollesVerst¨andnis.Herzlichen
Dankm¨ochteichauchmeinenEltern,GeschwisternundGroßelternsagen.Unterst¨utzung
undR¨uckhaltausderFamiliewarennichtnurf¨urdieseArbeitvonunsch¨atzbaremWert.
SchließlichdankeichderdeutschenForschungsgemeinschaftf¨urdieFinanzierungmei-
nerArbeitimRahmeneinesherausforderndenSonderforschungsbereiches.

Esistberechtigt,wennhierderEindruckentsteht,daßichnahezuausschließlichmit
tenfachlicdurhfte.heraDieusragZeitenindenStuttgundartmenscundhlicUmghegroßbungartigwarenheraKousllegenfordernd,undChefsspannend,zusammenainteresrbsaeni-t
undbereichernd.Allen,diedazubeigetragenhaben,seiganzherzlichgedankt.

ChristianeF¨orster

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Allzumenscheshlic

Contents

Contentsi
ListofFiguresv
ListofTablesvii
ixureenclatNom1Introduction1
1.1Motivation....................................1
1.2Scopeandobjective...............................3
1.3Overview.....................................5
2Governingequationsandmodelproblems7
2.1Systemsofreference..............................7
2.2Structure....................................7
2.2.1Modelling................................8
2.2.2Kinematics...............................8
2.2.3Constitutiveequation..........................9
2.2.4Balanceoflinearmomentum......................10
2.2.5Initialconditionsandboundaryconditions..............10
2.2.6Weakformulation............................11
2.2.7Discretisationinspaceandtime....................12
2.3Fluid.......................................14
2.3.1Modelling................................14
2.3.2Kinematics...............................14
2.3.3Constitutiveequation..........................14
2.3.4Conservationofmass..........................15
2.3.5Conservationoflinearmomentum...................16
2.3.6Conservationofenergy.........................17
2.3.7PropertiesoftheincompressibleNavier-Stokesequations......18
2.3.8Initialconditionsandboundaryconditions..............18
2.4Couplingconditions..............................20
2.5Modelproblems.................................20
2.5.1Stokesproblem............................21
2.5.2Transientadvection-diffusionequation................21
2.5.3Singulardiffusion............................22

i

ii

3

4

5

tsentCon

Flowsolveronmovingmeshes23
3.1Introduction...................................23
3.2ALEformsoftheNavier-Stokesequations..................24
3.2.1TheALEsystemofreference.....................24
3.2.2Convectiveformulation.........................26
3.2.3Divergenceformulation.........................26
3.3Discretisation..................................27
3.3.1Discretisationintime..........................27
3.3.2Discretisationinspace.........................30
3.4Conservationlawsondeformingdomains...................33
3.4.1Conservationoflinearmomentum...................34
3.4.2Geometricconservation.........................34
3.4.3Conservationofenergy.........................38
3.5StabilityofALEformulations.........................40
3.5.1StabilityoftheconvectiveALEformulationoftheadvection-diffusion
equation.................................41
3.5.2StabilityofthedivergenceALEformulationoftheadvection-diffustion
equation.................................44
3.5.3Anunconditionallystableformulationondeformingdomains....46
3.6Summary....................................47

Stabilisedfiniteelementmethodsforincompressibleflow49
4.1Wigglesandtheneedforstabilisation.....................49
4.1.1Convectiondominatedproblems....................50
4.1.2SpuriouspressureoscillationsandtheLBBcondition........51
4.2Stabilisation–omnipresentinflowsolvers..................52
4.2.1Convectionstabilisation........................52
4.2.2CircumventingtheLBBcondition-pressurestabilisation......54
4.3Residualbasedstabilisationmethods.....................58
4.3.1Virtualbubbles.............................59
4.3.2Alookbeyond–thevariationalmultiscalemethod.........62
4.3.3Generalisationandrelatives......................63
4.3.4Acommentonconservationinstabilisedproblems..........66
4.3.5Stabilisedequalorderelementsanddiscretelydivergencefreefunc-
tions...................................66
4.3.6Stabilisedformulationinmatrixnotation...............68
4.3.7Stabilisationparameter.........................70
4.3.8Elementlengthdefinitions.......................72
4.3.9Residualbasedstabilisationandlinearelements...........73
4.4Stabilityofastabilisedmethodonamovingmesh..............74
4.4.1CoercivityofGLSstabilisedALEformulation............75
4.4.2CoercivityofUSFEMstabilisedALEformulation..........76
4.5Summary....................................80

Stabilisedfiniteelementmethodsatcriticalparameters81
5.1Introduction...................................81
5.2Residualbasedstabilisationatsmalltimesteps...............82

tsentCon

6

7

iii

5.2.1Dominatingzerothorderterms....................84
5.2.2Acloserlookatstabilisationfordominatingzerothorderterms..87
5.2.3Coercivityanalysisofadvection-diffusion-reactionmodelproblem.88
5.2.4Exampleatsmalltimesteps......................94
5.3Residualbasedstabilisationondistortedmeshes...............99
5.3.1SensitivityofUSFEMstabilisationvariant..............99
5.3.2Kim-Moinflow.............................100
5.3.3Flowpastcylinder...........................107
5.4Summary....................................112

131ingcouplFSI6.1Introduction...................................113
6.2PartitionedFSIalgorithmanditsdetails...................114
6.2.1Ageneralpartitionedalgorithm....................114
6.2.2Structuralpredictor...........................115
6.2.3Fluidvelocityboundarycondition...................116
6.2.4Structuralforceboundarycondition..................117
6.3Artificialaddedmasseffect...........................118
6.3.1Introduction...............................118
6.3.2AddedmassoperatorforLBBstablefluidelements.........118
6.3.3Addedmassoperatorforstabilisedfiniteelements..........121
6.3.4Influenceofthediscretisationintime................124
6.3.5Consequencesoftheartificialaddedmasseffect...........129
6.3.6Generalinstability...........................130
6.3.7Furtherinfluencesontheartificialaddedmasseffect.........131
6.3.8Numericalinvestigation.........................131
6.4Stablepartitionedschemes...........................136
6.4.1Iterativelystaggeredschemesandtheneedforrelaxation......136
6.4.2Projectionbasedsemi-implicitscheme................137
6.5Summary....................................138

NumericalExamples139
7.1Introduction...................................139
7.1.1Generalalgorithmicinformationandmodelling...........139
7.1.2Afewcommentsoncomputationaltools...............139
7.2Bridgecrosssectioninlaminarflow......................140
7.2.1Geometryandmaterialdata......................140
7.2.2Modellinganddiscretisation......................141
7.2.3Results..................................142
7.3Channelwithbackwardfacingstepandmembrane..............145
7.3.1Geometryandmaterialdata......................145
7.3.2Discretisationandinitialmembranerelaxation............145
7.3.3Resultsontwomeshes.........................146
7.4VibratingU-pipe................................151
7.4.1Geometryandmaterialdata......................151
7.4.2Modellinganddiscretisation......................152

iv

tContsen

7.4.3Results..................................153

8SummaryandConclusions155
8.1Summary....................................155
8.2Prospectus....................................156

AAppendix159
A.1Thekinematicformulaeinadeformingframeofreference..........159
A.1.1Geometricconservationlaw......................159
A.1.2Reynoldstransporttheorem.....................160
A.2Somemathematicalbackground........................161
A.2.1Thescalarproduct...........................161
A.2.2Lax-Milgramlemma.........................161
A.2.3Someinequalities............................161
A.3ErrorsinKim-Moinflow............................162
A.4Flowaroundrigidcylinder...........................165

senceRefer

Index

691

821

esgurFiofList

1.1Samplefluid-structureinteractionproblems..................1

3.1SketchofALEmeasuringofthefluiddomain................25
3.2Drivencavityflowexample-problemdescription..............37
3.3Errorinhorizontalreactionforce.......................37
3.4Sketchofmeshvelocity.............................43

5.1Velocityoscillationsondrivencavityproblem................82
5.2Modelproblem,solutionsatdifferentlevelsofdiscretisation........85
5.3Problemsettingofdrivencavityexample..................95
5.4Temporalevolutionoftopforceatdrivencavityproblem..........96
5.5DrivencavitysolvedwithlinearelementsandfullUSFEMoperator....97
5.6DrivencavitysolvedwithlinearelementsandreducedUSFEMoperator..97
5.7DrivencavitysolvedwithquadraticelementsandfullUSFEMoperator..98
5.8DrivencavitysolvedwithquadraticelementsandreducedUSFEMoperator98
5.9Drivencavityproblemsolvedonreferencemesh...............98
5.10Kim-Moinflow-velocityvectorsonpressurefield..............101
5.11Distortedmeshesusedforerrorevaluation..................102
5.12VelocityerrorsinKim-Moinflow.......................103
5.13Velocityerrordependingonmeshdistortion.................103
5.14Pressureerrordependingonmeshdistortion.................104
5.15Mesheswithoffsetedgenodes.........................106
5.16EvolutionofnormalisederrorsforSUPGandGLS..............106
5.17EvolutionofnormalisederrorforUSFEM..................107
5.18Flowpastcylinder,geometryandmeshdata................108
5.19LiftcoefficientonmeshA4...........................109
5.20DragcoefficientonmeshA4..........................109
5.21PressureprofileonmeshA4..........................110
5.22Slightlyoscillatingpressureprofilewithclose-upviewobtainedonMesh
C9withminimalelementlengthvi......................110
5.23LiftanddragcoefficientsonmeshC9.....................111

v

vi

6.16.26.36.46.56.66.7

7.17.27.37.47.57.67.77.87.907.117.127.137.147.1

A.1A.2A.3A.4

ListFiguresof

Polynomialp(λi)accordingto(6.44)fordifferentvaluesofµi........126
Geometryandmaterialdataofdrivencavityexamplewithflexiblebottom132
EvolutionofcouplingforcewithdifferentstructuraldensitiesandBE...132
EvolutionofcouplingforcewithdifferentstructuraldensitiesandBDF2..133
Evolutionofcouplingforcewithdifferentpredictors.............134
Evolutionofcouplingforcewithdifferenttimestepsizes..........135
Evolutionofcouplingforceforconsistentandinconsistentstabilisation..136

Geometryofbridgecrosssectioninflowfield................141
Evolutionofverticalforceonbridgedeck...................142
Evolutionofangularmomentumonbridgedeck...............143
Streamlinesonpressurefield.........................143
Bridgedeckatdifferenttimeinstants.....................144
Initialgeometryofbackwardfacingstepproblem..............145
Membraneproblem:velocityonmeshoflinearelements...........148
Membraneproblem:velocityonmeshofquadraticelements........149
Temporalevolutionofverticaldisplacementofmembranemidpoint....150
Close-upviewofmembraneproblem.....................150
Geometryandmaterialofflowmetertube...................151
Boundaryconditionsofflowmeter.......................152
VerticaldisplacementsatAandB......................153
ResultsofU-pipesimulation..........................154

Lift,dragandpressureonmeshB4......................165
Lift,dragandpressureonmeshC4......................166
Lift,dragandpressureonmeshA9......................167
Lift,dragandpressureonmeshB9......................168

ListofTables

3.1

4.1

5.15.25.3

6.1

A.1A.2A.3A.4A.5A.6

ConservationpropertiesofALEforms.....................47

Notationusedtodistinguishdifferentversionsofthestabilisationoperator65

VelocityandpressureerrorinnormalisedH1seminormforlinearelements104
VelocityandpressureerrorinnormalisedH1seminormforquadraticelements105
Strouhalnumberforflowpastcylinder...................108

Instabilityconstants..............................127

L2erroronundistortedmeshoflinearelements...............163
L2erroronundistortedmeshofquadraticelements.............163
L2erroronmode1distortedmeshoflinearelements............163
L2erroronmode1distortedmeshofquadraticelements..........164
L2erroronmode2distortedmeshoflinearelements............164
L2erroronmode2distortedmeshofquadraticelements..........164

vii

ureenclatNom

nsiotaAbbreviALEBDF2BESCBCFDDGCLFEMFSIGCLGLSEODPSPGSUPGTRUSFEM

Spaces1(Ω)H2(Ω)LCR(ΩB)ehVB

ArbitraryLagrangeanEulerian
SecondorderBackwardDifferentiationFormula
BackwardEulertimeintegration
SplitBasedcteristic-CharaComputationalFluidDynamics
DiscreteGeometricConservationLaw
FiniteElementMethod
Fluid-StructureInteraction
GeometricConservationLaw
Galerkin/Least-Squares
OrdinaryDifferentialEquation
PressureStabilisedPetrov-Galerkin
StreamlineupwindPetrov-Galerkin
TruleezoidalaprUnusualStabilisedFiniteElementMethod

Sobolevspaceofsquare-integrablefunctionswithsquare-integra-
blederivatives
Sobolevspaceofsquare-integrablefunctions
ersbumncomplexersbumnrealbubblespaceonelemente
discretefiniteelementspaceenrichedbybubblefunctions

ix

xV0,SVD,ShV,S0hVD,ShVFhV0,FhPPm]P[m

nsmaiDoΓΩ0ΩF0ΩFΩGΩS0ΩSΩtΩeΩ+1n

Normsandderivatives
˙).(DtDGradΔDiv.L∞(Ω)
.Ω

turelamencNoweightingfunctionspaceforstructuraldisplacement
solutionfunctionspaceforstructuraldisplacement
discreteweightingfunctionspaceforstructuraldisplacement
discretesolutionfunctionspaceforstructuraldisplacement
discretesolutionfunctionspaceforfluidvelocity
discreteweightingfunctionspaceforfluidvelocity
discretefunctionspaceforfluidpressure
continuouspolynomialspaceoforderm
discontinuouspolynomialspaceoforderm

terfaceinructuretfluid-sinitialandreferencedomainofmodelproblem
indomafluidfluidreferencedomain
maindomeshdomainalstructurdomainerencerefalstructurtimedependentspatialdomainofadvection-diffusionproblem
indomatalelemenspatialdomainofadvection-diffusionproblemattimeleveln+1

materialtimederivativeof(.)
materialtimederivative
gradientwithrespecttothematerialsystemX
Laplaceoperatorwithrespecttothespatialsystemx
gradientwithrespecttospatialsystemx
divergencewithrespecttomaterialsystemX
∞Ωon-normL2Ωin-normL

tureNomenclax||rsatoerOpbstaLMRMCLCRΠBPRBStr

Euclidiannormofthevectorx,i.e.itslength

xi

stabilisationoperatorformomentumequation
residualoftemporallydiscretisedbalanceoflinearmomentumin
fluidthestabilisationoperatorforcontinuityequation
residualoffluidcontinuityequation
projectionoperatorontothespace[Pm]
projectionoperatoronbubblespace
Residualoftheadvection-diffusion-problem
solutionoperatorinbubblespace
traceoperator

Discretematricesandvectors
MAaddedmassoperator
MAl,MAstabaddedmassoperatorbasedonlumpedmassmatricesandonsta-
ionsulatrmfobilisedd¨vectorofnodalstructuralaccelerations
d˙vectorofnodalstructuralvelocities
dvectorofnodalstructuraldisplacements
DSstructuraldampingmatrix
fbFrighthandsidevectorofdiscretefluidmomentumbalanceobtained
fromhistoryvaluesandbodyforces
fhFrighthandsidevectorofdiscretefluidmomentumbalanceobtained
fromtractionforces
FSextexternalstructuralforces
FSintinternalstructuralforces
fΓnodalforcevectoratinterfacecontainingphysicalcouplingforce
aluesvGdiscretegradientoperator
KFfluidstiffnessmatrixobtainedfromviscousterm
MFfluidmassmatrix

xiiSMN)u(Npu

turelamencNo

trixmamassalstructurmatrixofshapefunctions
fluidcoefficientmatrixstemmingfromconvectiveterm
vectorofnodalvaluesoffluidpressure
vectorofnodalvaluesoffluidvelocity

Subscriptsandsuperscripts
(.)nvalueattimeleveln
(.)0,(.)+stabilisationoperatororweakformincludingzerothorderweight-
functioning(.)Breferringtobubblespace
(.)Creferringtocontinuityequation
(.)dreferringtodivergenceformulation
(.)ereferringtoelemente
(.)Freferringtofluidfield
(.)GLS,(.)USFEMGLSorUSFEMversionofstabilisationoperatororstabilisedweak
form(.)galreferringtoGalerkinweakform
(.)Greferringtomeshfield
(.)modreferringtomodelproblem
(.)Mreferringtomomentumbalanceequation
(.)Sreferringtostructuralfield
(.)|XevaluateatfixedmaterialcoordinateX
(.)|xevaluateatfixedspatialcoordinatex
(.)Γinterfacedegreesoffreedomofmatrix(.)
(.)|χevaluateatfixedALEcoordinateχ
lsobSymAeelementalarea
bFspecificfluidbodyforces
bSspecificstructuralbodyforcevector
Bmodbilinearformofmodelproblem

aeartalelemenspecificfluidbodyforces
specificstructuralbodyforcevector
bilinearformofmodelproblem

tureNomencla(4)CCCedEEeEkinFhdhheJtkLmenp¯ePeePpr,rθ,rBDF2
rd,rd,θ,rd,BDF2
ReeRerGStST

xiii

elasticitytensor
rightCauchy-Greentensor
constantofanelementalinverseinequality
numberofspatialdimensions
Green-Lagrangeanstraintensor
Young’smodulusofthestructuralmaterial
elementcounter
gyrenekineticdeformationgradienttensor
totalmomentumflux
fluidtractionvector
characteristicelementlength
timedependentJacobiandeterminantoftheALEmapping
polynomialorderorderoffiniteelementspace
characteristiclengthofaflowproblem
elementalpolynomialinterpolationparameter
outwardnormalvector
physicalfluidpressure
Pecletnumber
elementalPecletnumber
pressurefluidtickinemarighthandside(data)oftimediscretisedflowequationsinconvec-
tiveformulation
righthandside(data)oftimediscretisedflowequationsindiver-
genceformulation
Reynoldsnumber
elementalReynoldsnumber
effectivereactioncoefficientinstabilisationoperator
Strouhalnumber
secondPiola-Kirchhoffstresstensor
timeintervalofinterest

xivtuGuvWSXxη,β,αδ,δθ,δBDF2
εκλiµµiνSνφFρSρσe0σFσSτeτeCτeMτχ

turelamencNo

timefluidvelocityfield
meshvelocityofALEmesh
controlvolumeinEuleriancoordinates
specificstrainenergyofthestructure
materialsystemofreference
spatialsystemofreference
parameterdeterminingtheversionofthestabilisation
parameterstemmingfromtimediscretisationoftheflowequations
strainratetensorofthefluid
diffusivityinadvection-diffusionproblem
amplificationcoefficientforithmode
yviscositfluidmicdynaitheigenvalueofaddedmassoperator
yviscositfluidtickinemaPoisson’sratioofthestructuralmaterial
scalartransportquantity
yitdensfluidydensitmassalstructurlowerboundoneffectiveelementalreactioncoefficientinconvective
ALEformulation
Cauchystressesofthefluid
structuralCauchystresstensor
generalstabilisationparameter
elementalstabilisationparameterforcontinuitystabilisation
elementalstabilisationparameterformomentumstabilisation
shearstressesofthefluid
ALEsystemofreference

1erChapt

Introduction

1.1Motivation

Problemandapplications
Fluid-structureinteraction(FSI)describesaverygeneralclassofphysicalproblems.A
widevarietyofFSIproblemsarisesinengineeringandtechnologybutalsoinunderstand-
ingandtreatmentofbiologicalphenomenaFSIplaysakeyrole.SampleFSIapplications
fromverydifferentbackgroundsaredepictedinfigure1.1.Thefirstsubfigureshowsa
Coriolismassflowmeterwhichservesasameasuringtoolfortheflowrateinapipe.The
amountofliquidpassingtheomega-shapedtubecanbededucedfromtubeoscillations
whichareinfluencedbythepassingflow.Massflowmetersatverydifferentscalesare
usedinamultitudeoftechnicalapplications.AnundesiredFSItakesplacewhenwide-
spanbridgesoscillateduetowindforces.Reliablepredictionsofthemutualinfluenceof
structuralmotionandsurroundingairflowmightwellreducetheneedforexpensivewind
channeltestsandeventuallyhelptoincreasethesafetyofbuildings.

Figure1.1:Coriolismassflowmeter,Cablestayedbridgeandhumanlung;source
andcopyrightofimagesfromlefttoright“RHEONIKMessger¨ateGmbH”[203]
MAGEBA[170]andHeinemann[124]

TheOndynatopofmicalallbtheehaviourengineeringwithinahapplicaumantionslungFSIasisomnipdepictedresenfigurtein1.1biocisloggoicalvernedphenobymena.the
interactionofairandsofttissue.
isticThesscaleseandinmatimenyamondrespaceapplicatiobutnsshareoftheFSIcvarhayllewngeidelyofbwitheingrespcoupledecttotheirproblems.character-Con-
tricsequenkyatlyndthetime-consuming.understandingHoandweveverentuatherellyistheansimimmenseulationinofterestsuchinproreliabblemsletendspredictionstobe
ofthebehaviourofFSIsystems.

1

2

Chapter1.Introduction

InafirstclassificationFSIphenomenaaredistinguishedinsurfacecoupledproblems
andcoincidevolumealongthecoupledcoprommonbleinms.terface,Whilei.e.inthethewetfirsbtclaoundassrytheofthefluidandstructurestructheturallatterdoclamainss
containsproblemslikeflowinporousmediawherethefluidandthestructurecommonly
occupyadomainofinterest.
Withinthisworksurface-coupledproblemsareconsidered.ExamplesofsuchFSIprob-
lemsintechnicalapplicationsincludebesidesthealreadymentionedonestanksloshing,
airbagdeployment,parachutedevelopment,wind-forceanalysisontallbuildingsoralso
earthquakeresponseanalysisofliquidstoragetankstonamebutaview.
Buttoolswhichareusedtosimulatethoseengineeringapplicationsarealsoapplica-
bletoenhancetheunderstandingofbiologicalprocesseswhichusuallyhappenonvery
differentscales.FSIoccursintheinteractionofbloodflowandvessels,airandlungor
bloodandheartvalve.Anincreasedunderstandinginparticularoftheselatterprocesses
givesrisetothehopethatmedicaltreatmentcanbeimproved,eventuallyincreasingthe
qualityoflifeforpatientsorevenhelpingtoreducethenumberofprematuredeathcases.
ItisthusnotsurprisingthattheincreaseincomputerpowerintermsofCPUand
memorybooststheeffortsmadetosimulateandpredictthephysicsofmultifieldphe-
nomenasuchasFSI.Thepresentworkhasbeenundertakenwithinthecollaborative
researchcentre“Sonderforschungsbereich404”onmultifieldproblemsattheUniversity
ofStuttgart.Anoverviewoverthenumericalapproachdevelopedwithinthisresearch
projectisgivenin[85].
Besidestheresearchinstitutesanimpressivenumberofscientificconferencesindicates
thehighinterestinthefieldofFSI.InthemeantimeFSIandothermultifieldmodules
havealsobeenincorporatedintocommercialsoftwarepackagesreflectingthepractical
importanceofthematter.

Modellingandapproaches

Modelsofvariouslevelsofcomplexityhavebeenderivedtopredictthebehaviourof
mechanicalsystemsoffluidsandstructuresinfluencingeachother.Aclassicalcivilengi-
AneeringsimpleproapprblemoachtoincludingsuchFSIaispplicatiosloshingnsindatesliqbauidckfilledtoHotankussnerunderine1a963rthquwhoakeprloaopding.osed
tomodeltheeffectofthesloshingfluidbyasystemofhorizontalsprings,massesand
dampcoupleing.acomplexSimilarapplicastructuraltionsmoaredelwithconsideredasimplebyaddeRammedrmasstsoapprorferacethaforl.[19the8]fluid.whoalso
andHostrweucvteruremasnhayllbeapplicasimulationsteddeserverealisticamorellyaccuraincludingtemolocaldelling.effectsIftheattinhetineractioterfacenofbofluidth
fieldshavetobemodelledatahighlevelofcomplexity.Thisisparticularlythecase
forproblemswherestructuralstressesaswellaspropertiesoftheflowareofinterestas
inparmanticularlyybiomecrequehanicastedlmaayrepplicatioquirens.acoFurmplextherflowproblemsdescriptiowhenreothefferingastructuralpredictionrespoofnsetheis
fluidforceswhichissufficientlyaccurate.
befoApproundacamohesngcommanybiningothersadvinancedmDettmerodelsforandthePerstric´[64uctural],aDoneandfluidetal.de[6scripti8],onEngelcan
andHeilGr[123],iebeHl¨ubn[72],erFetarhaal.t[130et],al.Le[77],TalFernlecandandezMoetural.o[16[80],6],FL¨¨orsteohnerretetal.al.[169[85],],

1.2.Scopeandobjective

3

Massjung[174],PipernoandFarhat[193,194],Tezduyaretal.[222]andWall
9].[22RammandWhiledoubtlessanimmenseamountofresearchefforthasalreadybeenputintoaccu-
ratesimulationofcoupledFSIproblemsthechallengehasbyfarnotbeensolvedtofull
pledsatisfactioformn.ulatManions.ySorecenundtmapublicathematicationslindicaformteulatiothepnsersisaretentrequirineterestdininotherdertotheoryguarofancou-tee
thatsimulationsofferreliableresults.
dernessAcoofmmonthecparharaticipatingcteristicofastructurlargee.Tclasshin-ofwallesurfacedcostructuresupledaFSIrepaproblemsrticularlyisthesensitivslen-e
tofluidforcesandtendtoexhibitlargedeformationwhichhighlyinfluencethedynamics
ofthetheflostructurw.e.ThTusheneffecumericaltofmonon-lineardelsneedelastoticmaincludeterialatbleaehastgeoviourmematyricalalsononbelineinatroritiesducedof
inlartogelytheavastructurailable.lAnsolvoveriferviewrofequired.modelsNumericalandfinitemethoedslemenfotrsthforiskindthin-wofalledstructurestructuressare
hasrecentlybeenpresentedbyBischoffetal.in[16].Withinthepresentworkthe
isfamilyemploofyednowhicnlinearhgothreseebackto-dimensionaworklsoffinitBe¨uchshellterelemetenal.tsa[36s,37]describintedhebyearly1Bischof990s.fF[14or]
two-dimensionalexamplesgeometricallynonlinearwallelementsareapplied.
whileInbtheeingpreatsentrest.conThtextusthefluidsterminclfluidudeliqdenotuidsesmasuchteriaaslswathatter,blocannotod,oilresistorglyshearcserotlresbutses
alsogasessuchasairasindicatedinfigure1.1.IntheregimeoflowMachnumber,
ini.e.thewhefluidneverthetheflocwharabehacteristicvesaflolmostwspeedisincompressibleconsiderably(seelowFerzerthaigerntheandspPeeedriofc´[81sound]).
Consequentlycompressibilityorincompressibilityisaproblemdependentpropertyrather
thanamaterialfeature.
ArestrictiontoincompressibleflowsofNewtonianfluidsputsthescopeofthefluid
fieldtotheincompressibleNavier-Stokesequations.Formosttechnicalapplications
thestressassisumptreasonaionoblyfaaccuratlinearre.Ielationannbumetbwereenoffluidothershearcasesstraasinforraexateandmpletheinblocorrodespfloowndingthe
assumptionofNewtonianfluidbehaviourallowstoobtainagoodfirstimpressionofthe
flowfieldandtoassesstheneedformoreappropriatematerialmodels.
Ontopofastructuralandflowsolvertheinteractionhastobemanaged,i.e.theideal
couplingalgorithmissought-after.Suchanidealalgorithmwouldbeefficientwhilenot
perfeaffectingctaalgorccurithmacyhasandnostatybilitetby.eenIteshatstoablished.beadmittedthatdespiteallimprovementsthe

1.2Scopeandobjective

WThisallwork[227]canwhoinsinotromeducesensedbtheeregtopicardedofaFSIfolloattw-upheofInstitutetheofcomprehensivStructuraeldisseMecrtatiohanicsnofat
theUniversityofStuttgart.WithinasubsequentworkofMok[182]thecouplingissue
d.cusefoaswWithinthisworktheflowproblemisdiscretisedbyfiniteelementsinspaceandfinite
Methodifferenced(FEmethoM)dstheinortimeiginofsimilawhicrhtodattheesstbacructuraktoltheequawtionorkss.ofAstHughabilisedesinthFiniteelate1Elemen970st

4

Chapter1.Introduction

(seee.g.[135])isusedtotreatconvectiondominatedcasesaswellastoovercometheinf-
supcondition.TheflowequationsareformulatedinanArbitraryLagrangeanEulerian
(ALE)schemewhichcombinesthepossibilityofconsiderabledeformationofthefluidfield
withacceptableaccuracyinparticularattheinterface.
Buildinguponanexistingalgorithmandcodethepresentworkisdedicatedtoarevisit
ofthemethodologicalfundamentalsandaimsatimprovementsinaccuracy,efficiencyand
stability.Thefocusissetontheflowsimulationonamovingflowfieldandthecoupling
ofthephysicalfields.InparticulartheALEflowsolverforadeformingdomaindeserves
specialattentiontoestablishaschemewhichisaccurateandreliable.Thereforetheeffect
ofdiscreteversionsofdifferentALEformulationsonvariousconservationlawsshallbe
considered.Inadditiontotheconservationoflinearmomentum,massandenergythe
matterofgeometricconservationhastobesettledinthiscontext.Closelyrelatedto
theseconservationlawsisthequestionofthenumericalstabilityoftherespectiveALE
schemes.Consideringthemodelproblemofadvection-diffusionthisstabilityissueshall
beclarified.Limitingtimestepssizeswithrespecttothemeshvelocitywillbederived
andinterpretedfordifferentALEformulationsoftheproblem.AdiscreteALEscheme
whichisstableirrespectiveofthemeshvelocityshallbeformulated.
Besidestheclarificationofthepreciseeffectofmeshvelocitythecommonoccurrence
ofatimedependentdomainandstabilisationofthefluidelementsshallbeconsideredby
revisitingthederivationofastabilisedfluidformulationfromavirtualbubblecontext.
Theambitionistoestablishanumericalschemeforflowondeformingdomainswhich
inheritsthedefinedstabilitywithrespecttomeshmotionthatisobtainedfortheunsta-
bilisedmodelproblem.Amodificationofthestabilisationtermsissearchedforwhich
guaranteesthatthestabilisedflowelementworksindependentlyofthemeshmotion.
AnALEschemedoesnotonlyintroduceameshvelocitybutgoesalongwithpoten-
tiallysignificantlydistortedelements.Theinfluenceofmeshdistortionontheaccuracyof
stabilisedfluidelementsshallbeassessedandelementformulationsshallbefoundwhich
arehighlyinsensitivetomeshdistortion.ArobustandreliableFSIalgorithmfurther
requiresthatallrelatedmodulesarenotjuststablebutofferaccurateresultsevenat
criticalparameterssuchasverysmalltimestepsorsteepgradientsinspaceandtime.
Inparticularverysmalltimestepsresultinadominatingzerothordertermwithinthe
semidiscreteequationwhichmightgiverisetolocaloscillationsinthevelocityandeven-
tuallyalsothepressurefield.Theeffectofthestabilisationofsuchoscillationsinthe
contextoftheunusualstabilisedFEMincludingazerothorderweightingfunctionwithin
thestabilisationtermsshallbeinvestigated.Altogetherthereliabilityofdifferentversions
oftheflowsolveratsuchcriticalparameterswillbeassessedandimproved.
AnotherimportantissueregardingthestabilityoftheentireFSIalgorithmrather
thanjusttheflowsolveristheso-calledartificialaddedmasseffect.Thiseffectisan
inherentinstabilityofweaklycoupledpartitionedFSIschemescouplingincompressible
flowandlight-weightstructures.ThistopictakenupfromthedissertationofMok[182]
wheretheeffectwasobservedandstudiednumerically.Adetailedstabilityanalysiswhich
wasmissedsinceshallbegivenhere.Thisanalysiswillshowwhymoreaccuratetime
discretisationontheparticipatingfieldsstrengthentheeffectoftheinstability.Further
thenatureofthestronginfluenceofthemassdensityratiobetweenfluidandstructure
clarified.ebwillThepresentworkfocusesprimarilyonmethodologicalaspectsandintendstoprovidea

erviewvO.1.3

5

profoundunderstandingoftheschemesunderconsideration.Numericalexamplesshallbe
reportedthroughouttheworkverifyingtheoreticalconsiderationsandprovidingfurther
insightintoparticularmatters.Somelargerapplicationswillbepresentedinafinal
chapter.InthiscontextmodellingaspectsandsimulationissuesofFSIproblemswill
particularlybeaddressed.

1.3Overview

Finishingthisintroductionanoutlineofthesinglechaptersshallbeprovidedofferinga
firstimpressionofthesubsequentmatters.
Inchapter2thegoverningequationsofthesinglefieldsaresummarised.Furtherthe
discretisationofthestructuralequationsinspaceandtimeisbrieflypresented.Theflow
equationsareintroducedinanEulerianframeworkwhereparticularemphasiseisputon
theconservationstructureoftherespectiveequations.Withinthechapteralsoanumber
ofmodelproblemsisintroducedwhichareusedsubsequentlytoanalyseparticulareffects
terest.inofChapter3isdevotedtotheALEschemeforflowondeformingdomains.Aconvective
andadivergenceALEformulationoftheflowequationsareintroduced.Acomparison
withrespecttotheconservationpropertiesofdiscreteversionsofthetwoalternative
formulationsisgivenandthestabilitywithrespecttothemeshmotionisconsidered.It
turnsoutthatanALEformulationcanbefoundwhichisstableirrespectiveofthemesh
.ycitelovWithinchapter4stabilisedfiniteelementmethodsareconsidered.Aninitialdiscus-
sionconcernsthereasonsofnumericaloscillationsandthustheneedforsomekindof
stabilisation.Residualbasedstabilisationmethodsforflowproblemsarereviewedand
afamilyofthestabilisationmethodswhichareapplicableforALEformulationsisin-
troducedalongwithallrequireddetails.Subsequentlytheperformanceofthestabilised
finiteelementmethodonmovingmeshesisconsidered.
Aninvestigationofstabilisedmethodsatcriticalparametersispresentedinchapter5
wherethecommonoccurrenceofstabilisationandverysmalltimestepsordistorted
meshesisconsidered.Bymeansofacoercivityanalysistheeffectofverysmalltimesteps
ondifferentversionsofthestabilisationisinvestigated.Anumericalexampleconfirms
thetheoreticalobservations.Inasubsequentpartofthischapternumericaltestsare
reportedtoevaluatethesensitivityofthemethodwithrespecttodifferentkindsofmesh
tion.distorThematterofchapter6isthecouplingoffluidandstructuralfield.Thecorrect
exchangeofinterfacedataisdiscussedinthefirstplacewhilesubsequentlyapopular
classofpartitionedFSIalgorithmsisdescribed.Ananalysisofthesequentiallystaggered
versionofthepartitionedalgorithmispresented.Thisanalysisprovidesanexplanationof
theinherentinstabilityofsequentiallystaggeredschemes,theartificialaddedmasseffect.
Withinchapter7somelargernumericalexamplesaregivenhighlightingthecapa-
bilitiesoftheapproach.Particularemphasiseisputonthecompletepresentationofall
modellinganddiscretisationdetailswhichhavebeencrucialtoobtainresults.
Itisthescopeofchapter8toofferanoverallsummaryoftheworkalongwithconclu-
sionsandindicationsoffuturework.WithintheappendixAsomefurtherinformation

6

vided.prois

Chapter

1.

troIniontduc

Inthefirstplaceaderivationofthebasickinematicformulaeispresentedand

someextramathematicalbackgroundisprovided.Subsequentlyadditionalresultdata

lumerican

test

cases

is

dendeapp

gcompletin

the

mplesexa

tedpresen

in

pterhac

5.

of

2erChapt

Governingequationsandmodel
msblepro

Thisrequircehadpterwithinreviewsthetpresehecontnwtinoruumk.Asmecthehanicafocuslbofasistohisfsowolidsrkaisndonstrtheucturesfluidtaondtheinexteratenct-
tionratherthanthestructuralpartofFSIproblemsthediscretisationofthestructural
equationsisalsobrieflycoveredwithinthischapter.
Subsequenttothegoverningequationsoftheflowproblemanumberofmodelproblems
ispresentedwhichareusedthroughoutthisworktohighlightparticulareffectsoftheflow
blem.pro

2.1Systemsofreference
Threedifferentsystemsofreferenceshallbeusedwithinthiswork.Structuraldeforma-
tionsaremostconvenientlydescribedintheso-calledLagrangeanormaterialformu-
lation.ThecorrespondingLagrangeancoordinatesystemdenotedbyXisassociated
withtheparticularmaterialpoints.
TheEulerianorspatialsystemofreferencedenotedbyxismostappropriatefor
purefluiddynamicsproblems.TheobserverinanEuleriansystemisfixedinspaceand
wfluidatchesdothemainfluidhastopabsseingco.nsiderWithined.aThusfluid-astrthirucdturesysteminoteractiofnreferencpreoblemχisainttimeroduced.dependenThist
ArbitraryLagrangean-Eulerianreferencesystemfollowsthemotionoftheflowatthe
respectiveboundarieswhiledeformingarbitraryinbetween.AnintroductiontotheALE
formulationcanbefoundin[69,227].WithinthepresentworktheALEformulationof
theflowequationsiscoveredinchapter3.

ructureSt22.Thissectionisdevotedtoabriefintroductiontothebasicstructuralcontinuumequa-
tions.Verboseexplanationscanbefoundinabroadvarietyoftextbooks.Inparticular
thefieldofstructuralcontinuummechanicsiscoveredbyanoverwhelmingamountof
literatureofwhichthebookbyMarsdenandHughes[173],theclassicaltextbookby
Malvern[171]andtheGermantextsbyStein[207]andalsotheonebyAltenbach
andAltenbach[2]shallbegivenheretonamebutaview.
Thebasicsofcontinuummechanicsarealsocoveredbymanybooksconcernedwith
numericalapproximationsofcontinuummechanicsequationsasforexampleinthesecond

7

8Chapter2.Governingequationsandmodelproblems
volumeofthefiniteelementseriesofZienkiewiczandTaylor[236]orthemonograph
byBonetandWood[26].

2.2.1Modelling
Thefluid-structureinteractionapplicationsconsideredhereincludelargestructuralde-
formationsandthusgeometricnonlinearitieshavetobeconsidered.Thereisabroad
rangeofproblemsdealingwiththin-walledstructureswherethestructuralstrainsremain
smallandthustheassumptionoflinearelasticmaterialbehaviourisjustified.However
theiterativealgorithmalsoallowstoincludenonlinearelasticmaterials.

2.2.2Kinematics
d0goTheesastructdisplacemeuralnmateriatldefinedinbitiallyytheofieldccupiesd(theX,t).domaTheinΩSstructu⊂Rralwithreferenced∈{2c,o3}nfigurandatiounder-nis
convenientlyidentifiedwiththeinitialconfigurationΩ0evenifotherchoicesareequally
wellpossible.ΩSdenotesthetimedependentdomainScurrentlyoccupiedbythestructure.
dTheXtodeforitsimagmatioeningtheradiencurtrenttensorFconfigurrelatatioesnavialineelementinthereferenceconfiguration
dx=FdX.(2.1)
ThedeformationgradienttensorFisgivenby
F=Gradx,(2.2)
whereGraddenotesthespatialgradientoperatorwithrespecttothereferenceconfigu-
raandtion,curreni.e.tX.conThefiguradefotion.rmationgradienttensorisatwo-fieldtensorconnectingmaterial
IncontrasttoFtherightCauchy-Greentensor
C=FT∙F(2.3)
ismighsytthmmetricusbeanduseddoesasanotstrconaintainmeasuretheroiftattheionalpamateriartloflathewaccodeformauntstionforathenyfactmore.thaItt
C=Iforrigidbodymotions.
norThemalisingGreenthe-LarighgtrangCaueanchy-GrstraineentensortensorEiswithdefinedrespectastoarigpropidberosdyttrainranslatmeasureions.by
E=1FT∙F−I.(2.4)
2AgradimensionaphicallinsettinterpretagittiorelatneosfthetheGredifferenceen-LaingtherangsquareanesstraofainsdeEfoshormedwsandthatinanundeformedone-
lineelementtothereferenceconfiguration,i.e.
E=1(dx)2−(dX)2.
2)X(d2

9ructureSt.2.22.2.3Constitutiveequation
ThesecondPiola-KirchhoffstresstensorSisobtainedfromthephysicalCauchy
stressesσSofthestructuralpointby
S=(detF)F−1∙σS∙F−T.(2.5)
Thisworkisrestrictedtohyperelasticmaterialspossessinganenergydensityfunction
withpotentialcharacter,i.e.satisfying
W˙S=∂WS..˙E.(2.6)
E∂AsthesecondPiola-KirchhoffstressesandtheGreen-Lagrangeanstrainsarea
workconjugatepairofstressesandstrainsthespecificstrainenergydensityrateisalso
givenbyW˙S=S..E˙.ThusthesecondPiola-Kirchhoffstressescanbederivedfrom
S=∂WS(d).(2.7)
E∂FwhicromhtheisbainheritedlanceobfyangtheularsecondmomenPiotumla-thKirechsymmetryhoffofstresstheCatensoru.chystressesisobtained
St.Venant-Kirchhoffmaterial
InthecaseoflinearSt.Venant-Kirchhoffmaterialthespecificstrainenergydensity
dependslinearlyuponthestrainsaccordingto
(4)S=C..E(2.8)
whereC(4)denotesthefourthordermaterialtensor.Theoversetnumberisusedtodistin-
guishthematerialtensorfromtherightCauchy-Greentensor.
DuetothesymmetryofSandE(4)andthepotentialcharacterofWSthenumberofin-
deptheefnodeurthntordermateriatelnsoparrofameterslinearinelaCsticitreducesyisgtivoen21.byAssumingfurtherisotropicbehaviour
Cijkl=λSgijgkl+µSgikgjl+gilgjk(2.9)
dependingontwoconstantsonly.Ifthemateriallawisexpressedwithrespecttoa
CartesiansystemofreferencethecontravijariantcoordinatesofthemeStricStensorgijcan
btoetheidenengtifiedinewitheringthemateriaKrloneckparaermetersdeltaYoδu.ngT’shemoLamduluse´EconstaandntsPoλissonand’sµarratioerνelSatedvia
SEνEλS=(1+νS)(1−2νS),µS=2(1+νS).(2.10)
Duetothelinearisationofthestress-strainrelationshipinherentintheSt.Venant-
Kirchhoffmaterialthismodelappliestosmallstrainelasticityonly.Iflargestrainsin
particularlargecompressivestrainsareexpectedmoreappropriatematerialmodelshave
hosen.cebto

10Chapter2.Governingequationsandmodelproblems
CompressibleNeo-Hookeanmaterial
Incontrasttothelinearelasticmateriallaw(2.8)aNeo-Hookeanmaterialiscapable
ofrepresentinganinfiniteinternalenergyatbothlimitsofinfinitedeformation,i.e.if
aportionofmaterialiseitherstretchedinfinitelylongorcompressedtojustonepoint.
CompressibleNeo-HookeanmaterialbehaviouryieldsthesecondPiola-Kirchhoff
sstresseSS=λ(detF)2−1C−1+µSI−C−1(2.11)
2depSendingSontheleftCauchy-GreentensorC(seee.g.Wriggers[234]).In(2.11)
λandµdenotetheLameconstants(2.10).

2.2.4Balanceoflinearmomentum
Newton’ssecondlawofmotionstatesthedynamicequilibriumatastructuralpointand
readsDρSDd−∙σS=ρSbS,inΩS×T,(2.12)
tDtDwhereD/DtdenotesthematerialtimederivativeandρSrepresentsthestructuralmass
densityinthedeformedconfiguration.Thebalanceequationappliesthepointswithin
thestructuraldomainΩSforalltimesofthetimeintervalT.Areformulationof(2.12)
withrespecttothereferenceconfigurationyields
ρ0d¨−Div(F∙S)=ρ0bSinΩS0×T(2.13)
whereDivdenotesthedivergenceintheLagrangeanreferencesystemandd¨represents
thesecondmaterialtimederivativeofthedisplacementfield,i.e.thematerialacceleration.
Incontrastto(2.12)whichreferstoΩSaweakformofequa0tion(2.13)hastobeintegrated
overthetimeindependentreferencestructuraldomainΩSwhichismuchmoreconvenient
asastartingpointofaFEMformulation.Correspondinglyρ0denotesthestructuralmass
densityreferringtoΩS.
Alongwithaconstitutiveequation(2.8)or(2.11)andthekinematicequation(2.3)
or(2.4)thebalanceoflinearmomentum(2.13)definesasystemofcoupledhyperbolic
partialdifferentialequationsgoverningthetemporalevolutionofthedisplacementfield
d,thestressesandstrains.

2.2.5Initialconditionsandboundaryconditions
Att=0theinitialstructuraldisplacementfieldaswellasitsfirstmaterialtimederivative,
thevelocity,isemployedtoserveasinitialconditions
d(t=0)=d0,d˙(t=0)=d˙0inΩS0.(2.14)
Theboundaryofthestructuraldomain∂ΩSisdecomposedintothedisjointportionsΓS,D
andΓS,Nwheredisplacementandtractionboundaryconditionsareprescribed,respec-
tively.Theboundaryportionssatisfy
∂ΩS=ΓS,D∪ΓS,NandΓS,D∩ΓS,N=∅.(2.15)

2.2.Structure11
Thesystemofdifferentialequations(2.13)(alongwiththekinematicandconstitutive
Direquatichlion)etispfoortrmionulaatedndΓwithtthehedisplaNeumanncementpartfieldofathesprimarystructuralbunknoownundaryth.usTΓheS,Disappliedthe
N,Sdisplacementboundaryconditionsaregivenby
d=d¯onΓS,D(2.16)
whileforceboundaryconditionsread
N∙S=TonΓS0,N,(2.17)
wherethevectorTisthepseudotractionvectorreferringtotheinitialconfigurationand
NdenotesthenormalvectoronΓS0,N.TheboundaryportionΓS0,Nrepresentstheimage
oftheNeumannboundaryΓS,Ninthereferenceconfiguration.

2.2.6Weakformulation
Inordertoderiveafiniteelementapproximationtothestructuralsystemofequations
anintegralformulationisrequired.Ifalloccurringenergiesexhibitpotentialcharacter
whicderivhedfromimpliesthatminimisingalsothethetotalmateriapoltenlawtialpossenergessesy.apotential,theweakformcanbe
Anotherapproachtotheprincipleofvirtualworkisobtainedfromtheformalappli-
cationofthemethodofweightedresidualswhichpavesthewaytoamuchwiderclassof
testfunctionsdifferingfromthevariationofthedisplacement.
Theweakformofthestructuralbalanceequation(2.13)istofindd∈VS,D×Tsuch
tthaρ0d¨,δdΩ0+(S,δE)ΩS0=ρ0bS,δdΩS0+(T,δd)ΓS0,Nforallδd∈VS,0(2.18)
SwherethespaceVS,D=d∈H1(ΩS0)|d=d¯onΓS0,Dcontainsallfunctionsthatsat-
isfytheDirichletboundaryconditionswhileallfunctionscomprisedinthespace
VS,0=d∈H1(ΩS0)|d=0onΓS0,DarezeroonΓS0,D.By(.,.)ΩS0and(.,.)ΓS0,Nthestandard
innerproductevaluatedoverthereferencedomainoritsNeumannboundaryportionis
denoted.FurtherδEdenotesthevariationoftheGreen-Lagrangeanstrainsduetoa
variationδdofthedisplacementfield.
Remark2.2.1Thesymbolδusedhererepresentsavariationandshouldnotbemixed
upequatwithions.theHoδwinevtreortducedheinameaningosubsequenfδistocbhaviopterusfroalomngthetheresptimeectivediscretisacontext.tionofthefluid
Remark2.2.2Theuseofthenotationδdfortheweightingfunctionofthestructural
balanceoflinearmomentumshallindicatethatavariationoftheunknowndisplacement
wfieldeakdforismulatutilisedion.toInserconvetasrasttestdifferenfuncttion.symbTolshiswillyieldsbeausedBubnotovrepres-Gaenttlerkinheunktypnoewnof
fieldanditscorrespondingweightingfunctiononthefluiddomain.Inthislattercase
oftheagtesenetralfunctioweignhistedunrederssiduatooldasmethoadgeneandralpointing(somewhattowarardsbitraPetry)rovfunc-Gationlerkininthetysepenseof
ds.metho

12

Chapter2.Governingequationsandmodelproblems

2.2.7Discretisationinspaceandtime
Thestructuralformulationemployedwithinthisworkisbasedonalargenumberof
previousworksinthefieldsofnonlinearstructuraldynamicsattheInstituteofStructural
Mechanics.Thusdetailedinformationcanbefoundamongothersinthedissertationsof
Bischoff[14],Kuhl[162]andGee[99]aswellasinKuhlandRamm[163].
Theintegralequation(2.18)whichrepresentstheunderlyingstructuralfieldequations
alongwiththecorrespondingboundaryconditionsisdiscretisedinspacebymeansoffinite
elements.Subsequentlyafinite-differencemethodisappliedtodiscretisetheresulting
differentialalgebraicsystemofequationsintime.Linearisationandaniterativesolution
methodisrequiredtoeventuallyobtainanumericalapproximationofthenonlinearpartial
differentialequationsgoverningthemotionofthestructure.

Discretisationinspace
Thefiniteelementmethodappliedhereisapowerfultoolinengineeringandparticularly
beeapplincaputbleinttoosttheructuralmethodmecsincehanicsitwasapplicafirstcasttions.inAntoaboalmostokbyinfiniteZienkiamoeunwicztofinw19ork67.hasA
broadoverviewcoveringthebasicsaswellasnumerousrecentdevelopmentscanbefound
inthesixtheditionofthefamousbookbyZienkiewiczandTaylor[236,238].One
shouldfurthermentionthemonographsbyBathe[9]andHughes[134].
AnearlybookcoveringthetheoryoffiniteelementsistheonebyOden[185].The
mathematicsoftheFEMaredealtwithbyBraess[28],BrennerandScott[30]oralso
bytheserieseditedbyCiarletandLions[44,45].Anothermathematicallyoriented
classicmethodsistfrohemamonogmaraphthematicabylCiapoinrlettof[43view].Acanrecenbefotlyundgivinenothevencerviewycloponediafiniteaelerticlemenbyt
literaBrennertureenandtitledCar‘sostmeenbsoenoks[29on].Cfinileateerlytlemenhists’acollectionndoisccupbyyingfarnomoretthacomplete.ntwoApaglistesofis
givenintheintroductionofthemonographbyGreshoandSani[107].
TheGalerkinweakform(2.18)isdiscretisedinspacebyreplacingthefunctionspaces
V0S,DandVS,0bydiscretespacesVSh,DandVSh,0,respectively.VSh,DandVSh,0contain
C-continuouspiecewisepolynomialfunctionsdefinedonthesingleelements.Within
thisworktheclassofLagrangeanpolynomialsaswellasSerendipityelementsare
considered.Thustheunknowndisplacementfielddisreplacedbytheapproximation
dh=Nd,δdh=Nδd,d¨h=Nd¨(2.19)
¨depwhereendeNntistnohedalmatrixdisplacemenofthetshapandeaccelefunctiornsationwhilevalues,dandrespdectivdenotelye.thevectorofalltime
thatIntroanarducingbitratryhevadiscriaretionteofapprothenoximadaltiondispla(2.19)cemeinntotsδthedisweapokssfoiblerm(2yields.18)aandsystemnotingof
nonlineardifferentialalgebraicequationswhichinmatrixnotationread
MSd¨+DSd˙+FSint(d)=FSext,(2.20)
SS˙pwhereositivethevisdefinitecousstdampingructuraltermsystemDdmahasssbmateenarixwdded.hileInFSin(2.2t0)Mrepresentsdenottehestinheternalsymmetricforce

2.2.Structure13
vectorthelinearisationofwhicheventuallyyieldsthetangentialstiffnessmatrix.FSis
theexternalforcevectorandcollectsthediscreterepresentationoftherighthandextside
8).(2.1inDiscretisationintime
Subsequenttothespatialdiscretisationfinitedifferenceschemesshallbeappliedtodiscre-
tisethesemi-discretesystem(2.20)intime.Ageneraloverviewovertimediscretisation
scHughhemesesfor[134].theFurtstructuraherlreferencessystemofareequatthevionsolumescanbbeyfoundHaireinrettheal.sound[111,t1extb12].ookTheby
timediscretisationofthenonlinearstructuralequationsshallyieldastableandaccu-
rateapproximationofthetemporalderivatives.Anonlinearversionofthegeneralised-α
timeintegrationschemeofChungandHulbert[42]hasshowntoofferthedesired
properties[162]andisusedhere.
Thegeneralised-αmethodisbasedontheNewmarkschemereplacingthenewest
displacementandvelocitybyexpressionsintermsofaccelerations.Denotingthetimestep
sizebyΔtthenodaldisplacementsandvelocitiesatthenewtimelevelareapproximated
ybdn+1=dn+Δtd˙n+Δt221−βd¨n+βd¨n+1(2.21)
d˙n+1=d˙n+Δt(1−γ)d¨n+γd¨n+1.(2.22)
Furtherthebalanceoflinearmomentumissatisfiedatanintermediatetimeleveltα
betweentnandtn+1
MSd¨α+DSd˙α+FSint(dα)=FSext,α,(2.23)
Swhereαspecialcarehastobetakeninthetreatmentofthenonlinearinternalforcevector
Fint(d)asshownin[60,163].
Displacements,velocitiesandaccelerationsofthestructureareinterpolatedbetween
thediscretetimelevelsby
dα=(1−αf)dn+1+αfdn,(2.24)
d˙α=(1−αf)d˙n+1+αfd˙n,(2.25)
d¨α=(1−αm)d¨n+1+αmd¨n.(2.26)
Theintegrationconstantsarechosensuchthattheoverallschemehasthedesiredspectral
radiusρ∞whilealsominimaldampingonlowfrequencymodesisensured.Evaluating
theparametersαm,αf,βandγaccordingto
2ρ∞−1ρ∞
αm=ρ∞+1,αf=ρ∞+1
β=41(1−αm+αf)2,γ=21−αm+αf
fyingissatand1αm≤αf≤2
yieldsasecondorderaccuratesystem.

Chapter2.Governingequationsandmodelproblems

14Chapter2.Governingequationsandmodelproblems
uidFl32.theWithinrequiredthisALEsectionfothermulabasictionsfluidofethequatiobalansncearofeinlineartroducemodmenintumEulerinaianfluidnotareationderivwhileed
theandbadiscsicequsseduatioincnshapterandno3.tatTheionsintenrequirtionedofwitthishinsectiothisnwisorktoraprothervidetahanbriefgivingpreasenprotationfoundof
intrpresenotaductiontionsinoftothethebrfundamenoadtotalpicofequacotionsntinuumusedhefluidrecamecnbhaenicfounds.SoinmeWalmorle[227]detailedand
3].[10vemeieraGrApresentationofthecontinuummechanicsoffluidflowalongwithhistoricalremarks
isuumgivenmecbyhanics.MalvernAmong[171]otherswhooonffeersmaaysoalsundocinotronsultductioFlntoechterboth,[82],fluidPirandonsonealiducon[19tin-6],
Warsi[230]andthebookbyFerzigerandPeri´c[81]coveringalsocomputational
techniquesbesidesthemechanicalbasis.

2.3.1Modelling
Theflowproblemshereareconsideredincompressible.Furtheritisfocusedatlowand
themoderavicinitteyofaReynoldstrucstnureumbtheersflowwheremaytheexhibitviscositavyiscocannousbtboundeneglaryelaycted.ertheInpadynamicsrticularofin
whichsignificantlyinfluencethephysicsoftheoverallcoupledproblem.Adiscussionofthe
modelrestrictionswhichalsoappliestothepresentworkhasbeengivenbyWall[227].

2.3.2Kinematics
Theprimarykinematicunknownofaflowproblemisthevelocityfieldu(x).Thesym-
metricgradientofu(x)iscalledstrainratetensor
ε(u)=1u+(u)T,(2.27)
2wherethegradientoperatordenotesspatialderivativeswithrespecttothefixedEu-
.xsystemianlerTheaccelerationofafluidparticleisgivenbythematerialtimederivativeofthere-
spectivevelocity.InanEulerianformulationhoweverthisvelocityisgivenasafunction
ofspatialratherthanmaterialcoordinates.Thusthematerialtimederivativeofthe
velocityfieldreads

XxDDtu=∂∂tu=∂∂tu+u∙u
containinganonlinearconvectiveterm.

.28)(2

2.3.3Constitutiveequation
AstheaCapourtiochynofstressefluidsatσFresattacannotfluidpcarroinytaninytoashearhydrforcesostatiticisreapressureso¯nablepandtoadecomptensoroseof

51Fluid.2.3shearstressesτ.DenotingtheidentitytensorbyIthisdecompositionreads
σF=−Ip¯+τ.(2.29)
DatingfromStokesistheassumptionthattheshearstresstensorτ=f(ε)isafunc-
tionofthestrainratetensorwhereStokesconsideredalinearrelationship.However,
flowswithalinearrelationbetweenshearstressesandstrainsarenowadayscalledNew-
ratonmetianers[171tot].woSy,λmmetryandµ,considerdeterminingationsthereducvisceotsitheynofumbtheeroffluid.indepThusendenthetcomaterianstlitutivpa-e
equationforaNewtonianflowreads
σF=−Ip¯+λtr(ε)I+2µε.(2.30)
Theassumptionthatmeanpressureofthefluidequalsthethermodynamicpressure¯p
holdsifandonlyifeithertheflowisincompressible,i.e.tr(ε)=0orthebulkviscosity
κ=λ+2/3µvanishes[171].
Withinthisworkincompressibilityisassumedyieldingtheconstitutiveequation
σF=−Ip¯+2µε.(2.31)
Thstressus×theotime.nlyTheremainingparametermateriaµliscoanslstoantetromefdthefluiddynamicistheviscosityviscosittoyµofdistingtheuishdimensitfrioomn
osityvisckinematictheµν=ρF(2.32)
whichisnormalisedbythefluidmassdensityρF.Similarlythekinematicpressurep=
¯p/ρFwillbeused.
2.3.4Conservationofmass
InanEuleriancontrolvolumevtherateofmasschangeofthemassmFenclosedinvis
directlyrelatedtotheinflowandoutflowtotherespectiveboundariesifnomasssources
orsinksarepresent
∂mF=∂ρFdv=−ρFu∙ndγ=−∙(ρFu)dv,(2.33)
∂txv∂tx∂vv
wherendenotestheoutwardnormalofthecontrolvolume.As(2.33)isvalidforevery
possiblecontrolvolumeoneobtainsthelocalformofthemassbalancereading
t∂x∂ρF+∙(ρFu)=0.(2.34)
Forflowswhichexhibitamaximalvelocitysignificantlysmallerthanthespeedofsound
intherespectivemedium,i.e.asmallMachnumbertheeffectofcompressibilitycanbe
neglected.Thusthemassbalancealsotermedcontinuityequationreducesto
∙u=0inΩF×T.(2.35)
ascalarbutpowerfulconditionomnipresentinthefluiddomainΩF.

16Chapter2.Governingequationsandmodelproblems
2.3.5Conservationoflinearmomentum
Convectiveform
Analogouslytothedynamicequilibriumatastructuralpoint(2.12)thebalanceoflinear
momentumatafluidparticleinsidethedomainΩFisgivenby
t∂xρF∂u+ρFu∙u−∙σF=ρFbFinΩF×T,(2.36)
wherebFdenotesspecificfluidbodyforces.In(2.36)usehasbeenmadeofthemate-
tiorialnacc(2.31)eleraandtionindividingEulerbyianthedforemnsitulatyionyields(2.2the8).EulerInsertingianformfurtherulationtheoftheconstitutivmomeenequtuma-
balanceoftheincompressibleNavier-Stokesequationsintheconvectiveform
t∂x∂u+u∙u−2ν∙ε(u)+p=bFinΩF×T(2.37)
ibilitwhicyhtishestraaccompainrnieatedbtensyortheεisconatintuityracelessequatensortion(2.3con5).tainingAsashearresultstraofinstheonlyinc.ompress-
Aninterpretationof(2.37)naturallyreferstoasinglefluidparticleatwhichforces
theduetoexternainerltia,bodyviscositforceys.andThetheformpressureulationgoftheradientmoaremenintumba(dynamlanceic)givenequilibriumin(2.37)withis
frequentlycallednon-conservativeformulation.
Divergenceform
Therespectiveconservativeordivergenceformisgivenby
u∂x∂t+∙(u⊗u−2νε(u)+Ip)=bFinΩF×T,(2.38)
equatwhereionthesugcongeststinuitanyinequaterpretation(2tion.35)refhaserrbingeentoeamploconyted.rolvThisolumevrersionatherofthantheabalancesingle
fluidparticle.Integrating(2.38)overanarbitrarycontrolvolumevwhichisfixedin
Euleriancoordinatesyields
vv∂xv∂∂tudv+(u⊗u−2νε(u)+Ip)∙ndγ=bFdv,(2.39)
whereGauss’theoremhasbeenapplied.Theintegralequation(2.39)revealswhy(2.38)
iscalledconservative:Atemporalchangeofthevelocityinacontrolvolumevisbalanced
bysourcesorsinksinvandfluxesovertheboundaryofthecontrolvolume.
Clearlybothformulationsareequalinthecontinuoussetting.Howevertheirdiscrete
solutionsdifferasinadiscretesettingthecontinuityequation(2.35)isapproximated
ratherthansatisfiedexactly.Thisopensthequestionofthebetterformulationorrather
thepropoernetieswhicofhvaisriousmoreadiscretepprovpriateersionsinofatheparflowticulareqcauatiose.nwTillhebeproblemdiscussedofincocnshaptervaterion3.

Fluid.2.3

17

2.3.6Conservationofenergy
Theenergyisanotherconservativequantityoftheflow.Fortheincompressibleflows
consideredhereenergyconservationisnotanindependentequation.Itisratherclosely
relatedtothebalanceoflinearmomentum(2.36).ThekineticenergyEkininsideacontrol
volumevcisgivenby
Ekin=1ρF|u|2dv=1ρFuv2,(2.40)
22vwhere|u|=√u∙udenotestheEuclidiannormofthevelocityand.vrepresentsthe
L2-norminthecontrolvolumev.Followingthelinesof[107]toderivetheenergybalance
equationtheequilibriumequation(2.36)ismultipliedbythevelocityuandintegrated
overvyielding
t∂xvρF∂u,u+ρFu∙u,uv−(∙σF,u)v=ρFbF,uv.(2.41)
Thefirsttermin(2.41)canberecognisedastemporalchangeofthekineticenergyin-
sidethevolumevwhiletheconvectivetermisreformulatedusingthecontinuityequa-
tion(2.35).Furthertheviscouscontributiontotheenergyrateisintegratedbyparts
yieldingaboundarytermandtheinternalenergydissipationrateσF..uwhichreduces
toτ..usincethepressuredoesnotcontributetotheinternalviscouswork.Usingthe
symmetryoftheshearstresstensorτthistermcanfurtherbereformulatedto
11τ..u=τ..2µτ=2µ|τ|2≥0.
Thusinsertingallthesetermsbackinto(2.41)andemployingGauss’divergencetheorem
conservationofthekineticenergyisobtainedaccordingto
∂tx22µv
∂Ekin+1ρF|u|2n,u∂v−(n∙σF,u)∂v+1||τ||v2=ρFbF∙udv.(2.42)
Equation(2.42)statesthatachangeofthekineticenergyinsidethecontrolvolume
isbalancedbytheinflowofkineticenergy,theworkdonebythetractionalongthe
boundaryofthecontrolvolume,theworkofbodyforcesandtheenergydissipatedby
internalfriction.From(2.42)itcanbeobservedthatineveryvolumewithhomogeneous
Dirichletboundaryconditionsandzerobodyforcesthekineticenergyhastodecay
.nouslytomonoAnalyticalsolutionsofeithertheconvectiveorthedivergenceformulationofthefluid
momentumbalancesatisfyenergyconservation.Howeverthisdoesnotnecessarilyapply
totherespectivespatiallydiscreteequationsasthediscretevelocityfieldwillnotbe
exactlydivergencefree.Tocarryconservationofenergyovertothediscreteversionthe
balanceoflinearmomentumhastobewrittenintheform(consultalso[107,197]inthis
)issue2t∂x∂u+u∙u+1∙uu−2ν∙ε(u)+p=bFinΩF×T.(2.43)
Theenergyconservationequation(2.42)canbederivedfrom(2.43)withoutrequiringan
exactlydivergencefreevelocityfield.Itisnoteworthythat(2.43)istheaverageofthe
convectiveformulationandthedivergenceformoftheincompressibleNavier-Stokes
equationsinafixedEuleriansystemofreference.

18Chapter2.Governingequationsandmodelproblems
2.3.7PropertiesoftheincompressibleNavier-Stokesequations
hTheyptrerbolicansien-partincaboolicpampressiblertialNadifferenvier-tialeStokquatioesnsequatfortionsheareaprimarynonlinearunknosywnstemfieldsofofmixedthe
velocityuandthekinematicpressurep.
Asaresultoftheincompressibilityalocalchangeinthepressureisinstantlycarried
totheentiredomain,i.e.thefluidexhibitsaninfinitespeedofsound.
oneTheisthinecosaddlempresspibleointNavierstructure-Stokofestheequatproblemionscconatauseindtbwyothemaincmixedhallenges.formulaThetionfirstin
velocityandpressure.Asecondproblemisthehyperbolicconvectivetermwhichalso
introducesanonlinearity.
Thepropertiesofaparticularflowcanbemeasuredbydimensionlessnumbersrepre-
sentingaratioofforceswhichgoverntheproblem.Themostfamousofthesemeasures
istheReynoldsnumbergivenby
Re=Lν|u|,(2.44)
whereLisacharacteristiclengthoftherespectiveflowproblemwhileuandνdenotea
representativescalarvelocityandthekinematicviscosity,respectively.TheReynolds
numReynoldbergivsenstumheberratiogetsofthestatlowionareryistheinertialinfluenceforcesoafndtheviscousellipticfoviscrces.ousThetermhig.herConsethe-
quentlyverythininternalandboundarylayermaydevelopathighReynoldsnumber.
AfurtherdimensionlessmeasurerequiredwithinthepresentworkistheStrouhal
numberdefinedby

LSt=tcu(2.45)
wherethecharacteristicperiodlengthtcenters.TheStrouhalnumberisadimensionless
frequencyrepresentingtheratiooftransientandstationaryinertialforces.
Anexhaustivediscussionofalternativeformulationsofthesingletermsaswellas
properexpressionstiesofinalterincompressiblenativevarviscoiablesusisflogivwenarbeyalsGroeshodiscusseanddbySaniRaninna[10ch7].erMain[20themat2].ical

2.3.8Initialconditionsandboundaryconditions
Theinitial0conditionofthemomentumbalanceisaninitialvelocityfieldintheinitial
ΩindomafluidF

u(t=0)=u0inΩF0(2.46)
whichhastobesolenoidal,i.e.tosatisfy∙u0=0toguaranteeawellposedproblem.
Itisimportanttonotethatthefactthattheinitialvelocityfieldhastobedivergence
freeexcludesimpulsivestartsofthevelocityperpendiculartotheboundary.
Thereisnoinitialconditionforthepressureinanincompressibleflow.Itratheracts
astheaveloLagcityrangfield.emForultiapliebrroadfordiscusthesioinconofmpresstheibilitinitiaylcoconnditionditiononeandshoinstanuldtlyconsultadjust[10s7].to

91Fluid.2.3Inaccordancetothestructuralboundaryconditionstheboundaryofthefluiddomain
∂ΩFisdecomposedinto
∂ΩF=ΓF,D∪ΓF,NwithΓF,D∩ΓF,N=∅(2.47)
wheretherespectiveDirichletandNeumannboundaryconditionsareprescribedas
u=u¯onΓF,Dandρ1Fn∙σF=honΓF,N.(2.48)
Herehdenotesthespecifictractionvector.
Theincompressibilitycondition(2.35)demandsthatonafixedcontrolvolumethe
inflowandoutflowbalance,i.e.

u∙ndγ=0.(2.49)
ΓFClearly(2.49)putsarestrictionifthefluidisfullyconstraintbyDirichlettypeof
ns.conditioryoundabItisofinteresttointerpretparticularboundaryconditionswithrespecttotheirphysical
.meaning•No-slipboundaryconditionscanbechosenastheNavier-Stokesequationsinclude
viscouseffects.Assumingthatthefluidparticlesattheboundarysticktothefixed
ormovingsurfaceDirichletboundaryconditionsforthevelocityareprescribed
inthenormalandtangentialcomponent.Aprescribedtangentialvelocitytypically
causesaboundarylayerwherethephysicsaredominatedbyinternalfriction.
•Slipbconsidered,oundaryi.e.cifnonditionsoviscosityconsequenoccurs.tlyIhanvethetobcaseeofusedsliifpbtheoundaryEulercoequatnditionsionstheare
firstequationof(2.48)isreplacedby

u∙n=u¯∙nonΓF,D.(2.50)
Farortificialviscousboundaflowrypoftroblemshedothemain.seboundaryconditionsmaybeusedtomodelan
•theSlipfluidwithfrpartictioniclesbareaoundaryllowedctosonditionsliptangareenactiallyhoicetobtheetwboeenundarytheablineovetbutwo.theThslipus
resultsinshearforcesduetoalinearfrictionlaw.Theseboundaryconditionsrequire
anuponadditiotheprnalopertymateriaoflthetypreespofectivpaesrameter.urfaceanThisdthefrictioviscnocosityofefficienthetfloheaw.vilySugdepgestedends
byGaldiandLayton[97]inthecontextoflargeeddysimulationssuchboundary
conditionshavebeenusedbyJohn,e.g.[148,151].
•Outflowboundaryconditionsareaparticularchallenge.Causedratherbytheneed
tolimitthecomputationaldomainthanbyanyphysicallyreasonableboundary
neitheressentialnornaturalboundarydataisavailable.Withinthisworkthe
pzeroopular‘doNeumannnobthing’oundabryoundaryconditiocon.nditionTheisqueusedstionwhicofhoutfloimplieswbtheoundaryassumptionconditionofas
isdiscussedbyHeywoodetal.[125].

20

20Chapter2.Governingequationsandmodelproblems
•Apressureboundaryconditionisrequiredasthepressureisdeterminedbythe
incompressibleNavier-Stokesequationsuptoaconstant.Forpureflowproblems
thepressureconstantisusuallyfixedimplicitlybyaNeumannboundarycondition
ordirectlyataparticularpoint.Alternativelythemeanvalueofthepressurecanbe
fixed.TheinfluenceofthepressureboundaryconditioninacoupledFSIproblem
willbetoucheduponagaininsection2.4.
Furtherpossibleboundaryconditionssuchassymmetryboundaryconditions,periodic
boundaryconditionsorfreesurfaceswillnotbeusedwithinthiswork.Foranexhaustive
discussionofappropriateboundaryconditionsandtheireffectsonemayconsultGresho,
Saniandcoauthors,i.e.in[107,206].

2.4Couplingconditions
InordertoreachatthecoupledFSIproblemthestructuralandfluidfieldhavetobe
connectedalongtheinterfaceΓ.
Thesolutionoftheoverallproblemrequireskinematicanddynamiccontinuityacross
ΓforalltimesinT
x0S+d=x,d˙=u,d¨=u˙onΓ×T(2.51)
andn∙σS=n∙σFonΓ×T(2.52)
wherexdenotesthepositionofthefluidboundaryandx0Srepresentstheinitialposition
ofthestructuralpoint.Inmostcasesx0SwillbeidentifiedwiththeLagrangeancoor-
dinateX.In(2.51)no-slipboundaryconditionsattheinterfaceΓhavebeenassumed.
Ifslipshallbepossiblethenormalcomponentsofthekinematicvariablesarecoupled
.elyexclusivItisworthnotingthatthedynamiccouplingcondition(2.52)connectsthefluidpressure
levelwiththestructuralstresses.Thusforproblemswherethestructureisnotfully
immersedinthefluidthefreepressureconstantimplicitlydependsupontheexternal
structuralloads.Correspondinglytheinterfacedisplacementhastosatisfytheglobal
massconservationofthefluidfield.Particularlypartitionedsolutionapproachesmayfail
forsomeproblemsduetotheseimplicitcouplingconditions.Approachestosolvethis
incompressibilitydilemmaarediscussedbyK¨uttleretal.in[164].

2.5Modelproblems
Withinthissectionanumberofmodelproblemsisintroducedwhichwillusedsubse-
quentlytostudyparticularnumericaleffectsoftheflowfield.Thesedifferentialequations
canberegardedasspecialsimplificationsoftheincompressibleNavier-Stokesequations
whiletheyalsopossessaphysicalinterpretationontheirown.
TheStokesequationsmodellingcreepingflowstateanimportantsimplificationof
theincompressibleNavier-Stokesequations.TheStokesproblemisaclassicalmodel
problemtostudytheeffectoftheincompressibilitycondition.

2.5.Modelproblems21
Mixedconvective-diffusivetransportofaquantityismodelledbytheadvection-diffu-
sionequation.Ascalaradvection-diffusionproblemsufficestoinvestigatetheabilityofa
numericalmethodtoproperlyreproducetransportincludingsignificantconvection.

2.5.1Stokesproblem
Inthelimitofverysmallflowvelocitiesutheeffectoftheconvectivetermmightbe
neglected.SuchcreepingflowsaregovernedbytheincompressibleStokesequations
greadin

x∂∂tu−2ν∙ε(u)+p=bFinΩF×T(2.53)
∙u=0inΩF×T
alongwithappropriateinitial-andboundaryconditions.
WhilestillexhibitingtheproblemoftheincompressibilitytheStokesproblemis
linearsaddlepandointdoesstructurnotceoofntatheinainchoypermpressbolicibletermNavaniery-moStokre.esItisequatthusions.usedtomodelthe

2.5.2Transientadvection-diffusionequation
Theampleadvtheectioconcn-entratdiffusionionofequamilktioningoavcernsuptofhecotrffeeanspwhicorthofisastscairredlarup.quanThetityφmilkasfordiffusex-es
dependingonthegradientinmilkconcentrationwhileitismovedbytheunderlyingveloc-
inityafieldmediumattheswhicamehistimmoe.vingThewithfuncttheionvφelomacityyalsofieldareprese.ntthetemperaturedistribution
IntheconvectiveformulationonapotentiallytimedependentdomainΩttheadvection-
readsproblemdiffusion

t∂x∂φ+a∙φ−κΔφ=finΩt×T,(2.54)
φ(t=0)=φ0inΩ0,(2.55)
wherefdenotesasourceorsink,κthediffusivityandtheadvectivevelocityfieldais
free.genceerdivTheadvection-diffusionequationisaccompaniedbyappropriateDirichletandNeu-
mannboundaryconditionsgivenby
φ=φ¯onΓDn∙κφ=honΓN.(2.56)
Equation(2.54)isalinearmodelproblemof(2.37)combiningdiffusiveandconvective
traequatnspionort.inaItallolinearwscontotexstudyt.Astheapropmeasurertieseofoftheamratixedioofhyptheerbconolic-evectivllipteicanddifferendiffusivtiale
transportthePecletnumber

Pe=|a|L
κ

.57)(2

22

Chapter2.Governingequationsandmodelproblems

isused.ThisnumberistheanaloguetotheReynoldsnumberfortheNavier-Stokes
equationasgivenin(2.44).
Atemporallydiscretisedversionof(2.54)inconvectiveformulationtakestheform

φ+δa∙φ−δκΔφ=r,inΩt(2.58)
wherethesymbolrdenotestherighthandsidetermconsistingofbodyforcesandcon-
tributionsfromtemporaldiscretisationandδrepresentsacoefficientstemmingfromtime
discretisation.Equation(2.58)isalsotermedadvection-diffusion-reactionequation.The
thezerothunknoorderwntqeruanmistitycaφlleditself.reactivInetheascoinnctextertainoftreahiswctivoerkprothecessestermsoisurceusedstodepmoenddeluptheon
effectofthezerothordertermintroducedbytimediscretisationoftheflowequations.

Remark2.5.1Theparameterδdependsuponthetimestepsizeandthechoiceofthe
timediscretisationschemeasdefinedin(3.17).

2.5.3Singulardiffusion

Afurtherreductionoftheadvection-diffusion-reactionequation(2.58)yieldsthesingular
problemdiffusion

φ−δκΔφ=r,(2.59)
alongwithappropriateboundaryconditions.Differentialequationsofthiskindareob-
tainedfromtemporallydiscretiseddiffusionproblemsortransientheatconduction.The
singulardiffusionproblemisemployedtomodeltheeffectofthezerothordertermstem-
mingfromtimediscretisationanditsrelationshiptoellipticdiffusion.

3erChapt

Flowsolveronmovingmeshes

WithinthischaptertheALEformulationsoftheflowequationsarederivedanddiscussed.
SimilartotheconvectiveanddivergenceformulationinanEuleriansettingaconvective
anddivergenceformulationonamovingdomaincanbederived.Discretisingtheseformu-
lationsintimeandspaceyieldsschemeswithdifferentdiscretepropertiesandpotentially
differentdiscretesolutions.Inparticulartheconservationfeaturesofthesediscreteforms
considered.ear

3.1Introduction

Fluid-structureinteractionproblemshavetodealwithtemporallychangingdomains.
WhilethestructuralequationsarewritteninaLagrangeanformulationandthustrack
themovinginterfaceΓthebalanceoflinearmomentumintheflowfield(2.37)posesa
difficulty.TheuseofanALEformulationremovestheproblem.Itallowstodefinea
referencesystemfollowingthemovingboundarieswhilenotbeingattachedtothemotion
ofthefluidparticlesinsidethedeformingdomain.
ALEschemeshavegainedsignificantpopularityforsimulationswhichhavetotreat
FSItemporproallyblemscha[13nging1].Thedomainorigsainalnddevinpartelopmeniculartoffortheflowmethosimdulawastionsdonewithinfreetheconsurfacestextoofr
finitedifferencemethodsinthe1960s.Theformulationwaslateradoptedforfiniteelement
scpresenhemested.EbayrlyDoneaapplicaetationsl.[6wit7]hinmo197ving7andfiniteBelyelementscthkmesohesandhaveKennedamoyng[13]othersinb197een8.
In[1established3]thebfloywHirformtetulatioal.ninis1t97er4meindan‘quaasrticleti-Eulerian’.hatwasApparenreprinttedlyinthe199term7[1AL27,E12w8].as
Applicationsofthemethodinthefiniteelementcontextfromtheearly1980sandare
inreptorhetedbobokybyDoneaDoneainand[66,68Huer].tALEa[69me]thowheredsinalsothesomefinitemoreelementreferenceconstexttotarehecovhistoryered
ofALEmethodscanbefound.Anoverviewincludinginformationonmeshupdate
proceduresisalsogivenintheencyclopediaarticlebyDoneaetal.[70].
Itisobviouslydesirabletotransferthetemporalorderofaccuracyobtainedonafixed
referencesystemtothedeformingdomainproblem.AdditionallystabilityoftheALE
formulationisrequired.StabilityofdifferentALEformulationsisapersistentproblemof
interest[5,83,84,175].Thematterofaccurateandstablecomputationsondeforming
domaconservinsatioisnfurtherpropertcloyseonlymolinkvingedftortamesheofquestiorenference.ofgeoHowmetricever,consgeomerveatriction,coansnaervatdditioionnalof
anumericalschemeshouldnotbediscussedindependentlyofconservationoflinearmo-
mentumandenergyasnonumericalapproximationwillbeabletosatisfysimultaneously

23

24

Chapter3.ALEformulation

allconservationlawsexactly.
Withinthisworkthefocusissetontheflowsolverondeformingdomainsrather
thanthetreatmentofmeshmotionitself.Somecommentsonthemeshupdateshall
thusbeaddedhere.Variousmeshmotionschemeshavebeenproposedoverthelast
fewdecades.Forsimpleshapeddomainsheuristicorinterpolationmethodsastheone
describedin[156]areused.Suchapproachesmaybeformulatedratherefficient,i.e.
withouttheneedtosolvealargesystemofequations.Forcomplicatedgeometriesand
arbitrarymotionoftheboundarypseudo-structuralapproachesarefrequentlypreferred
fortheirgenerality.Pseudo-structuralmethodscanbefoundforexampleinChiandussi
etal.[39]andL¨ohnerandYang[168].Withinsuchschemeslocallyvaryingstiffnesses
areusedtomaintainanoptimalmeshqualityclosetothemovingboundary.Thesame
motivationisbehindtheideaofdiscontinuouspseudo-structuralschemesbasedonaspring
analogy.Insuchformulationsthemeshmotionisdeterminedfromthedisplacementof
asystemoftranslationalandpossiblytorsionalspringsattachedtotheedges,diagonals
andverticesoftheelements.Two-dimensionalapplicationsofthespringanalogycanbe
foundinBlom[18]andFarhatetal.[74].Thethree-dimensionalcaseisconsideredby
DegandandFarhatin[61]andveryrecentlybyMarkouetal.in[172].
Withinthepresentworkpseudo-structuralapproachesaswellasspring-typemesh
motionschemesareemployed.

3.2ALEformsoftheNavier-Stokesequations

lerAnaloiangouslysettintogttheherespconvectivectiveefoandrmuladivtionsergencecanfboermrecoofvtheeredonmomenatdefumobarminglancedinomain.anTheEu-
derivDoneaationandoftheHuertaALE[69].formAgulatioraphicanoflinthefloterpretawequationhastionsalsocanbbeenegivfoundeninbytheWalbolok[22b7].y

3.2.1TheALEsystemofreference
AsketchoftheALEsystemofreferenceisgiveninfigure3.1.Thespatialcoordinatesof
aparticularpointx∈ΩFaregivenbytheuniquemapping
x=ϕ(χ,t)forallt∈T.(3.1)
ThereferencesystemχtracksthemovingboundariesofΩFandisallowedtomove
arbitrarilyandindependentofthefluidflowinsidethedomain.Itcanbeidentifiedwith
aparticulartemporalconfigurationforexampletheinitialconfiguration.Withinthiswork
theelementalparameterspaceisconsideredasreferenceconfigurationinthediscretecase.

JacobiandeterminantoftheALEframeofreference
TheALEmapping(3.1)allowstointroducethestrictlypositiveJacobiandeterminant
x∂Jt=det∂χ(3.2)

3.2.ALEformsoftheNavier-Stokesequations

spacelysicaph

χ2ϕ(χ,t)

χ2χ1x2χ1spacereferencex1

ΩF

Figure3.1:SketchofALEmeasuringofthefluiddomain

52

relatingthedifferentialportionsofvolumeinthespatialandreferencedomainaccording
to

dΩF=JtdΩF0.(3.3)
ThereferencefluiddomainΩ0istimeindependent.Thetemporalchangeofthefluid
domainΩFiscausedbytheFnormalvelocityofitsboundaryaccordingto
∂∂Jt0G
∂tχΩFdΩF=ΩF0∂tχdΩF=∂ΩFu∙nd∂ΩF.(3.4)
Thelocalformof(3.4)yieldsthewellknownEulerexpansionformula[3]alsotermed
Euler’sformulabyWarsiin[230]
t∂∂Jt=Jt∙uG.(3.5)
χAInlesstheconillustratexttivofeaALEndfomorermulamtionsathemat(3.5)icalisderivcalledatiogenofometricthisceqonseruationvationisgivlawenin(GCL).ap-
A.1.endixp

MaterialtimederivativeintheALEframeofreference
Inordertoformulatetheflowequationsinthemovingframeofreferenceχthematerial
timederivativeofaquantityf(x(χ,t),t)shallbeexpressedinthedeformingreference
system.Applicationofthechainruleyields
Dt∂tχ∂χ∂tX∂tχ∂x∂χ∂tX
Df=∂f(χ,t)+∂f(χ,t)∂χ=∂f(χ,t)+∂f(x,t)∂x∂χ.(3.6)
Thematerialtimederivativeofthespatialcoordinatex,i.e.thetemporalchangeofthe
spatialpositionofamaterialpointisthevelocityu.Expressingthisderivativeaccording
to(3.6)inthemovingframeofreferencereads
u=Dx=∂x(χ,t)+∂x(χ,t)∂χ,(3.7)
Dt∂tχ∂χ∂tX

26

where

Chapter3.ALEformulation

wherethetemporalchangeofthespatialpositionofaGreferencepoint∂x(χ,t)/∂t|χcan
beidentifiedwiththevelocityofthereferencesystemu.Inserting(3.7)into(3.6)yields
theALEexpressionofthematerialtimederivativeofafunctionf
Df=∂f(χ,t)+u−uG∙f.(3.8)
Dt∂tχ
Equation(3.8)alsotermedfundamentalALEequationallowstoexpressthematerial
timederivativeofaquantityfasatemporalchangeoffinthereferencesystemanda
convectivetermaccountingfortherelativemotionofthissystem.Asaspecialcaseof(3.8)
thematerialtimederivativeinEuleriancoordinates(2.28)isrecoveredbysettingχ=x
whichimpliesuG=0.

3.2.2Convectiveformulation
BymeansofthefundamentalALEequation(3.8)anALEformulationofthemomentum
balanceequation(2.37)iseasilyobtainedreading
t∂χ∂u+u−uG∙u−2ν∙ε(u)+p=bFinΩF×T.(3.9)
InALE(3.9eq)theuationEuleremployianedmaonterialthevelotimecityderivfieldativu.ehasbeenreplacedbythefundamental
TheconvectiveformulationintheEulerianorALEformiscommonlyusedasabase
Itforisalsospatialthediscretisatiopreferredvnbersioynwmeansithinofthefinitepreselemenentwtsork(a.sforEquatioexamplen(3.9in)is[103,freq107ue,n10tly9,227termed]).
non-conservativeALEformulation.
Remark3.2.1Allspatialderivativesoccurringin(3.9)refertotheEuleriansystemx
whileonlythetimederivativehasbeentransferedtothemovingframeofreferenceχ.
domaConsequeninformtlyulathetioncontasinwuitell.yequation(2.35)remainsunchangedandappliestothemoving

3.2.3Divergenceformulation
canAnalobegooustbtained.otheToEulerserveianthisformpurpulatoseionthe(2.38)convaectivdiveergtermenceinor(co3.9)nseisrvativreformeformulatedof(3.9)
u−uG∙u=∙u−uG⊗u+u∙uG,(3.10)
wherethecontinuityequation∙u=0hasbeenused.Employing(3.10)and(3.5)a
divergenceformoftheALEformulationisrecovered.
∂(∂Jttu)+Jt∙u−uG⊗u−2νε(u)+Ip=JtbFinΩF0×T(3.11)
χOnadeformingdomaintheconvectiveandthedivergenceformofthemomentumbalance
conequatvectivione(ortermindeebutdofalsoevtheerytimeconservderivatiaontive.law)Thisdifferhasanotsigonlynificainnttheimpaactppeonatranceheofdiscretethe
versionsoftherespectiveequations.

3.3.Discretisation

72

Remark3.2.2AlternativelytousingthefundamentalALEequation(3.8)theconvective
anddivergenceALEformulationofaconservationlawcanalsobederivedfromintegral
formsasithasbeendoneforexamplein[86].

3.3Discretisation

Thetimefloarewperequaformedtions(atareleastdiscretisedtosomeinaextent)sequenindeptialmaendennnetlyr,ai.end.bydiscretisatiodifferentnmethoinspaceds.Thisand
allowstoemploythefiniteelementmethodforthespatialdiscretisationandtouseafinite
allodifferencewingtomethosimdultaneointime.uslyAdisncraetislternateallivetodimensiothisnsapprbyoacahprfiniteovideelemenspace-tctoimencept.elemenThists
22appr4]fooarchmorepursuedrecenatamongpplicatioothersns).byAsthepacegroup-timeofformTezdulatuyionarfor(seemoe.g.ving[213,domain218]orproblems[221,
hasmethoalsdobbaseenedproonpthoesedphbyysicalMasudentropandyvaHughriablesesisin[17used.6]TwherehemaiandiscondisatdvinanuoustageGaofspalerkince-
timeformulationsisthesignificantlyincreasedsizeoftheresultingequationsystems.
Inparticularwhenhigherorderaccuracyintimeisdesiredsequentialdiscretisationis
preferred.Withrespecttothestabilisedfiniteelementflowformulationintroducedinchapter4
thetimediscretisationisperformedpriortothediscretisationinspace.Incontrastto
theusualsequentialdiscretisationinspaceandsubsequentlyintime,i.e.themethodof
bylines,anathisrticleapproofRacohthisealsointe193r0med[204].horizontalmethodoflinesorRothemethodinspired

3.3.1Discretisationintime
Introduction
WithrespecttotimetheincompressibleNavier-Stokesequations(3.9)and(2.35)
poseacoupledproblemcomposedofapartialdifferentialequationandtheconstraint
oftheincompressibility.Anticipatingdiscretisationinspaceonenotesthatthediscrete
Navier-Stokesequationsyieldadifferential-algebraicproblem.Thustheincompress-
ibleNavier-Stokesequationsmightberegardedasinfinitelystiffpartialdifferential
equation.Aroughdefinitionofstiffnessofordinarydifferentialequations(ODE)intime
isthepresenceofaverylargerangeofcharacteristictimescales.Andindeedwithout
specificallydeterminingthelargesttimescaleonecaneasilyobservethattheincompress-
ibilityconditionintroducinganinfinitespeedofsoundalsointroducesaninfinitelysmall
le.castimeRelaxationoftheincompressibilityconditionwillbeintroducedbyusingastabilised
finiteelementmethodforspatialdiscretisation.Thuseventuallytheresultingdiscrete
equationswillbeoflargebutfinitestiffness.
Timeintegrationofstiffproblemsrequiresimplicitmethods.Forthepresentproblem
wherethestiffnessisrelatedtotheincompressibilityimplicitmethodshavetobeusedat
leasttointegratethepressurefield.Explicitmethodswoulddemandastepsizeableto
resolvethedynamicsofthesmallestoccurringtimescale.Thereforesuchmethodsshow

28Chapter3.ALEformulation
tremendouslyinefficientonstiffproblems.Howevernotonlyefficiencybutalsostability
requirementslimitthepossibletimediscretisationmethodsforstiffproblems[112].
timeThethereformethoedhaofsctohoicebetostifflydiscretisestableattheleast.(incompressibleSeeGearNav[98]ier-foraStokesdefinitionequatofionsstiffin
stability.)AsubsetofstifflystablemethodsareA-stable.ApplyinganA-stablemethod
EODlineartheto

y˙(t)=−λy(t)λ∈C,Re(λ)<0(3.12)

nequatyieldsionlim(3n.12→∞)ycan=be0regaindeprdedendenatmoodeflthofeanfixedadvtimeectivstepe-diffussizeiveeΔqt.uatioDnahwhelquistrethe’stestreal
trapartnspofoλrt.represenConsulttsGrtheeshodiffusivandeSaneffecit[10and7]theforanimaginaexplarynatioconnoftributiothisnanamologdelsy.advective
theorTheemtkimenownbydiscretisatiothenanmescsehemescondDemploahyedlquistwithinbarrthisierwowhicrkharestatesfurtherthatanlimitedA-stabyblea
ormdeultirsteA-sptmableethoscddohemeesisnottheexceewelldoknorderwntwo.traItpezfurtoidalhersarule.ystFhatortahedetamostileadccurdiscateussionsecondof
stiffnessandstablenumericalstepmethodsthereaderisreferredtothebookbyHairer
andWanner[112].
Thepresentworkthusaimsatasecondorderaccuratetimediscretisationofthein-
compressibleNavier-Stokesequationsonadeformingdomainwhichis(atleastcondi-
tiousetnally)oinstategblerate.AntheanalysisstabilisedcompaFEMringofthetemporNaalvierdisc-retStokisaestionscequathemesionsforcanthebepartfounicdulainr
DettmerandPeric´[62].

θp-One-ste

TheAppliedone-step-ontheθgenemethoraldfirstisoonerderoftODEheiny˙tegr=f(atioy,nt)scone-shemetsepcoθnsidetimereindtegrwitatiohinnythisieldswork.

n+1ny−y=θf(yn+1,tn+1)+(1−θ)f(yn,tn),(3.13)

whereΔtdenotesthetimestepsize.ThemethodisA-stableif1/2≤θ≤1and
containsthespecialcasesofbackwardEuler(BE)timediscretisationatθ=1andthe
trapezoidalrule(TR)forθ=1/2.TheTRisasecondorderaccurateschemewhile
adeviationfromθ=1/2yieldsamethodwhichisjustfirstorderaccurate.Another
advantageoftheTRisthatitisfreeofspuriousdamping.Itisthusthemethodofchoice
forlongtimesimulations.Howeverthelackofnumericaldampingfrequentlyresultsin
atendencytospuriousoscillationscausedbyunresolvedtransientsorinitialinaccuracy.
Asthenumericalstabilityof(3.13)increaseswithincreasingparameterθaslightshift
ofθtowardsahighervaluefrequentlycurestheoscillationsbutsacrificessecondorder
.cyaccura

29

3.3.Discretisation29
Secondorderbackwarddifferencing
Thesecondorderrepresentativeofthebackwarddifferentiationformulae(BDF2)applied
tothegeneralfirstorderODEy˙=f(y,t)reads
yn+1−yn=1yn−yn−1+2f(yn+1,tn+1).(3.14)
Δt3Δt3
Themethodcombinessecondorderaccuracyandsomenumericaldissipation.Itthus
guaranteesthaterrorsarealwaysdamped.BDF2isA-stable(ormorepreciselyeven
L-stablesee[112]).AdisadvantageofBDF2isitsneedforastartingalgorithm.
Themethodhasbeenpreferredwithinthisworkforitsrobustnessandaccuracy.

Timediscretisationoftheconvectiveform
Thetemporallydiscretisedequationiscastintoaunifiedconcept.Applying(3.13)and
(3.14)totheconvectiveformoftheALEmomentumbalance(3.9)andthecontinuity
yieldsionequatun+1+δun+1−uG,n+1∙un+1−2ν∙εun+1+pn+1=rn+1inΩFn+1,
.15)(3δ∙un+1=0inΩFn+1,(3.16)
whereδrepresentsascalardependingonthediscretisationscheme:
2δθ=θΔt,δBDF2=3Δt.(3.17)
Thevectorvaluedfunctionrcontainshistorytermsofthevelocityanddependsonthe
timeintegrationscheme.Possiblebodyforcesfarealsoincludedinraccordingto
rθn+1=δθbF,n+1+(1−θ)Δtu˙n+unrnBDF2+1=δBDF2bF,n+1+4un−1un−1.
33Equation(3.15)canbeinterpretedasadifferentialequationinspacewhichreferstoa
differentdomainineverytimestep.
Remark3.3.1Thetimeparameterδintroducedin(3.16)doesnotemergefromtem-
poraldiscretisation.Itisratherintroducedtoscalethecontinuityequationforbetter
conditioningoftheresultingeffectivefluidcoefficientmatrixandtopreservethelastbit
ofsymmetryofthemixedformulation.

Timediscretisationofthedivergenceform
TimeintegrationofthedivergenceALEform(3.11)requirestodiscretisethetemporal
changeoftheproductJtuyielding
Jn+1un+1+δJn+1∙un+1−uG,n+1⊗un+1−2νε(un+1)+Ipn+1=rdn+1inΩF0,
.18)(3δJn+1∙un+1=0inΩF0,

30Chapter3.ALEformulation
conwheretributiothenabsoigvenificandefinitiotlyndiffersoftfrheomparatheometerneδobtainedapplies.fortheHowconevverectivtheeforighrmtulathandion.sideIt
dsreawnordn,θ+1=δθJn+1bF,n+1+Jnun+Δt(1−θ)(J˙u)n(3.19)
and

nrdn,θ+1=δθJn+1bF,n+1+Jnun+Δt(1−θ)(J˙u)(3.19)

rdn,+1BDF2=δBDF2Jn+1bF,n+1+34Jnun−31Jn−1un−1(3.20)
forone-step-θandBDF2,respectively,includingtermsthatrefertopreviouspositionsof
thereferencesystem.ComparedtotheconvectiveALEequationthisresultsinaslightly
increasedeffortaspreviousnodepositionshavetobetracked.

Remark3.3.2Thesubscript‘d’isusedhereandsubsequentlytodenotesymbolsthat
refertothedivergenceformulationanddifferfromtherespectivetermoftheconvective
form.

3.3.2Discretisationinspace
theSpatialconservdiscratetionisaprotionpebrytiesofmeanstheofconfinitveectiveelemeandntsdiviserginetronceducedALEfoherrmeinulaotionrderontothediscfullyuss
discretisedequations.WiththisinmindtheGalerkinweakformisconsideredinthis
section.Additionalstabilisationtermswhichcompletethespatialdiscretisationwillbe
regatermsrdedarein(atclehaptasterto4.someSufficeextenitt)hereindetopassendenuretofthatthetheinsfundamentabilitiestalreconservquiringatiosntpropabilisatertiesion
oftheequationsandthepotentiallossofstabilityduetomeshmotion.
presenFinitetwoelerk.menAtpresendiscretisattationionofisthecovdetaeredilsandsubsequenprotlyblemstoethnecounextenteredtorequirnethedwforaytheof
GrusingeshofiniteandelemenSanits[10for7]flowhicwhprohasblemsalreacandybbeeenfoundmenintionedtheexseverahaustivltiemes.monogrTheaphboboky
btheythirdDoneavandolumeofHuertheta[69classic]balsooocoksvonerstheFEMsubbyjecZiteninckieludingwiczALEandfoTrmayulalortions.[237]Fsurhotheruld
bementionedalongtheengineeringbooksonthesubject.Amoremathematicalviewon
thealsotheFEMvoforlumebincompryGiesrasibleultfloandwsRacanvbiaret[10found2].inthebookbyGunzburger[109]or

Spatialdiscretisationofconvectiveform
Inordertodiscretisethesemi-discreteconvectiveALEequation(3.15)inspacethedo-
mainΩFisdividedintonon-overlappingpatches,theelements.Thespatialdiscretisation
maintains0itstopologywhilefollowingthedeformationofthedomain.Thereferencedo-
mainΩFisthenassociatedwiththeelementalparameterspaces,suchthatJtisidentified
withtheelementalJacobiandeterminant.
TodefinethediscreteGalerkinformthefiniteelementspacesVFh,0⊂H01(ΩF)and
VFh⊂H1(ΩF)areselected,whereVFhsatisfiesthetimedependentDirichletboundary

3.3.Discretisation31
conditionsoftheproblemwhileallfunctionsinVFh,0arezeroonΓF,D.Thepressureis
takenfromthespacePh⊂L2(ΩF)ofsquareintegrablefunctions.
Thediscretevariationalstatementthenreads:seekthesolutionatu∈VFh,n+1,p∈
Pnh+1atthenewtimeleveln+1suchthat
Bgal({u,p},{v,q})=rn+1,vΩn+1+δhn+1,vΓn+1forall(v,q)∈(VFh,0,n+1,Pnh+1),
FF,N(3.21)
wherethediscreteoperatorinconvectiveformBgal({u,p},{v,q})isgivenby
Bgal({u,p},{v,q})=(u,v)ΩFn+1+δu−uG,n+1∙u,vΩFn+1+(δ2νε(u),ε(v))ΩFn+1
−(δp,∙v)ΩFn+1−(δ∙u,βq)ΩFn+1.(3.22)
Itispointedoutthatthesuperscriptn+1isdroppedattheunknownvelocityandpressure
fieldatthenewtimelevelforclarityofthepresentation.Theparameterβ∈{−1,1}
carriesthesignofthepressuretestfunctionq.Thisparameterisintroducedherewith
respecttothegeneralstabilisedformulationthatshallbeconsideredinchapter4.
From(3.21)and(3.22)itcanbeobservedthatallinnerproductsrefertotheactualtime
instant,i.e.thetimediscretisationoftheconvectiveALEformulationdoesnotinterfere
withthetemporalmeshevolution.

).23(3.24)(3

Discreteflowequationsinmatrixnotation
Denotingthevectorsofunknownnodalvelocityvaluesbyuandthecorrespondingnodal
pressurevaluesbyp,i.e.boldfacenon-serifuprightletterswhicharegenerallyusedhere
fordiscretematricesandvectors,thefluidsystemofequationscanconvenientlybecast
intomatrixformreading
MFu+δN(u)+δKFu+δGp=fbF+fhF(3.23)
δGTu=0,(3.24)
wherefluidmassmatrixisobtainedfromthebilinearform
vTMFu=(u,v)ΩFn+1.
Thefluid‘stiffness’isderivedfromthediscretenonlinearconvectivetermreading
FvTN(u)=(u−uG,n+1)∙u,vΩn+1
andasymmetricpositivedefinitematrixstemmingformtheviscousterm
vTKFu=(2νε(u),ε(v))ΩnF+1.
Thediscretegradientoperatorisobtainedfrom
vTGp=−(p,∙v)ΩFn+1

Chapter3.ALEformulation

32Chapter3.ALEformulat
whilethediscretedivergenceoperatorisgivenby
qTGTu=−β(∙u,q)ΩFn+1.
ThevectorfbFcollectstherighthandsidetermsfromhistoryvaluesandbodyforces
vTfbF=rn+1,vΩFn+1
whilefhFrepresentstractions
vTfhF=hn+1,vΓFn,N+1.
Ondeformingdomainsallcoefficientmatricesaretimedependent.

Spatialdiscretisationofdivergenceform
Atimespatialdiscretisatiodiscretisnastiocnheme.oftheIndivtheergencecaseforofmone-ulationstep-(θ3.1it8)readepds:endsfindupothenthevelopartcityiculaandr
pressureu∈VFh,n+1,p∈Pnh+1atthenewtimeinstantn+1suchthat
nBgal,d({u,p},{v,q})=δbF,n+1,vΩFn+1+(un,v)ΩFn+Δt(1−θ)(J˙u),vΩ0
F+δhdn+1,vΓFn,N+1forall(v,q)∈VhF,0⊂H01(ΩF0),Ph⊂L2(ΩF0).
.25)(3VhAfter,timep∈PdischretsucisahttionhatbyBDF2theGalerkinweakformulationreads:seeku∈
+1n+1,nFBgal,d({u,p},{v,q})=δbF,n+1,vΩFn+1+34(un,v)ΩFn−31un−1,vΩFn−1
+δhdn+1,vΓFn,N+1forall(v,q)∈VFh,0⊂H01(ΩF0),Ph⊂L2(ΩF0).
.26)(3Inthebothreferenccasesethesystem,weighi.et.ingonthefunctionselemenfortalvpeloaracitymetanderspacepressuresandareintegradefinedtedowithverrespdoectmainsto
referringtodifferenttimelevels.Thediscreteoperatorinthedivergenceformisgivenby
Bgal,d({u,p},{v,q})=(u,v)ΩFn+1−(∙u,q)ΩFn+1
+δ−u−uG,n+1⊗u+2νε(u)−Ipn+1,vΩFn+1(3.27)
applyingtoeitheroftheabovecases.
Remrefersarkto3the.3t.3otalInthemomendivtumergencefluxorincludingconservtheativeconfovecrmtiv(e3.27one).theThusintegrtheatiocorrnesbpyondingparts
Neumannboundaryconditionis
hdn+1=n∙σ−un+1−uG,n+1⊗un+1=hn+1−n∙un+1−uG,n+1⊗un+1.(3.28)

3.4.Conservationlawsondeformingdomains33
OnboundarieswithalocalLagrangeanformulation,i.e.u=uGorwherethevelocity
vvectoectorrhis.taAltengenrnativtialtelyototheb(3.2o7)undaryitisptheossibletotaltousemomenatwumeakfluxformequaulatiolsnthofettrahedivctionergenceforce
itformisnotwherestrictlytheinnecestegratsaryiontobyintparegtsrateexclusivtheconelyvreectivfersetotermthebyvisparcousts.Ifandsucprheanssuretapproermacash
isfollowedtheNeumannboundaryconditionisnotaltered.

canRemalsoarkb3e.3use.4dThetowderiveighetedafinitresiduaevlofolumethedivmethoergdencbeycALEhofoosingrmvulattoionbe(3.25elemen)ort(3.2wise6)
constant.Insnuc+1hacasethesecondlinein(3.27)vanishesanditremainstobalancethe
totalfluxeshdovertheboundariesofthecontrolvolumes.

3.4Conservationlawsondeformingdomains
ItisworthwhiletoopenthediscussionofconservationwiththewordsofHughesand
coadesiruthorable,sp[137]:ossibly“Lethelpful,ustakeandctheerptainlyointofnotviewharmful.here”thatindicalocaltingcthaonsertdisvationcreteisatconservleasta-
tionisnotunconditionallymandatory.Whilethestatedsentencereferstotheconserved
quamomenntitytum,whicithisisaofctintuallyeresttodiscretised,simulti.e.aneouslyinthehpavreseenatlocaokseattothethevarioconservusatioothernofcolinensaer-r
vationlawswhicharemetonthewayofdiscretisingtheflowequationsonadeforming
in.domaThematterofconservationpropertiesofdiscreteequationsariseswhendiscreterep-
resentemptaoraltionsevofolutionconservofaaqtiouanntlaitywsfarecancobensidered.writtenaInsgeneralsuchalawgoverningthe
∂∂tf+∙F=s(3.29)

ormoreobviously

∂∂tvcfdvc+∂vcn∙Fd∂vc=vcsdvc(3.30)
statsourcingetterhatmstheandtemptheoralfluxeschangeFpofassingftheinsidecontrtheolfixedvolumecontrboloundarvolumey.vcisbalancedbya
Itiseasytoobservethatthedivergenceformulationofthemomentumbalancein
servEulerationianlafowsrmofulathetiontyp(2.3e8)(3.2as9).wellFasurtithesrALEconservversionation(3.1laws1)powhicsehsphavecialebcaeensesotoucfcohedn-
sofararetheenergyconservationwithinthefluid(2.42),thegeometricconservation
lawneither(3.5)conandtainingtheaconstempervatoralioncofhangmaessof(2.3the5)conseequatrvedion.qTualitheylattnoerrsopurceosingtearspmse.cialcase
Clearlythestructuralmomentumbalanceequation(2.12)isanotherconservationlaw
evenifrarelycalledso.Structuralengineersareusedtoprefertheideaofforceequilibrium
ratherthanconservationoflinearmomentumintheabovesense.

Chapter3.ALEformulation

34Chapter3.ALEformulation
3.4.1Conservationoflinearmomentum
Asthedivergenceformulationoftheflowbalanceappearsintheform(3.29)itisfrequently
termedconservativeformulationwhiletheconvectiveformiscallednon-conservative(see
e.g.FormaggiaandNobile[83,84]).Therigorousequalityofthetwoislostassoon
asthecontinuityequationisapproximatedratherthanstrictlysatisfied.
Usingaconservativeinsteadofanon-conservativeformulationoftherespectiveequa-
tionsfordiscretisationensuresthattheconservationpropertyofthequantityofinterest
iscarriedovertothediscreteversion.Forequation(3.11)thismeansthatthetemporal
changeoflinearmomentumwithinoneelementisbalancedbytheelementalbodyforces
andthetotalmomentumfluxeshdacrosstheelementboundaries.Thisconservation
propertycannotdirectlybeobservedanymorewhentheconvectiveformulation(3.9)is
discretisedwhichsuggeststheideaofforcebalanceatatimeinstantratherthanconser-
vationofaquantityovertime.However,itisworthnotingthatconservationdoesnot
necessarilyimplyaccuracy(evenifitevidentlydoesnotcontradictiteither).Quoting
Hughesagainonemayevenstatethat“advectiveformsareoftenpreferredongrounds
ofaccuracydespiteviolationofconservation”[144].
Therearetwomainreasonstoprefertheconservativeformoftheequations.Thefirst
oneistheintentiontouseafinitevolumemethodwhichreliesonadivergenceexpression.
Thesecondreasonisthedesiretosimulateproblemswithshockdiscontinuitieswhich
maybepresentforhighspeedcompressibleflow(seeZienkiewiczetal.[237]).
Asneitherofthereasonsisapparentforthepresentworktheconvectiveformulation
isusedpreferablyhere.NeverthelesssomepropertiesofdivergenceformulationsofALE
schemesshallalsobeconsideredinthesequel.

Remark3.4.1Usingthedivergenceformulation(3.11)impliesdiscreteconservationof
linearmomentuminanintegralsense.Ityields,however,neitherenergynorgeometric
conservationonthesamecontrolvolume.Bothconservationlawsarespoiledbydis-
cretisationerrors.Thusthemoreneutraltermsdivergenceformulationandconvective
formulationratherthanconservativeschemeandnon-conservativeschemearepreferred
rk.owthiswithin

Remark3.4.2ReturningtotheinitiallycitedpaperbyHughesetal.anotheraspect
hastobementionedwhenconservationisconsidered.ContinuousGalerkinmethodsare
frequentlyaccusedofnotbeinglocallyconservativeirrespectiveoftheformofthediscre-
tisedequations.Hughesandcoauthors[137]clarifythisissuedemonstratingthatlocal
conservationiswellachievedbycontinuousGalerkinapproximationsiftheconservative
fluxesarededucedfromtheconsistentnodalforces.

3.4.2Geometricconservation
GeometricconservationposesadifficultyparticularlywhenthedivergenceALEformula-
tionpressible(3.11)floiswbutapplied.arisesThealsomawtterhenofcompregeometricssiblefloconservwatioproblemsnisnoontrmoestrvingicteddotomainsincom-are
investigated.Consequentlytheneedforgeometricconservationwaspointedoutfirstby
ThomasandLombard[223]in1979whosolvedtheGCLnumericallyalongwiththe

3.4.Conservationlawsondeformingdomains

53

flowequationsusingdifferenceoperators.InthesequeltheGCLwasdiscussedinrelation
withtheaccuracyoffinitevolumeorfinitedifferencemethodsbyFarhatetal.[108,161].
Adesirablepropertyofanumericalschemeisthatitisabletopredictexactlya
constantflow.LesoinneandFarhatshowedthatthisconditionissatisfiedifthescheme
underconsiderationsatisfiesadiscreteversionoftheGCL[167].Discretegeometric
conservationslaws(DGCL)forfinitevolumeschemesweresuspectedtobenecessary
forthestabilityoftheoverallscheme[75,76,73]whileitwasalsoshownthatitis
neithernecessarynorsufficienttoguaranteethatthediscreteschemeonamovingdomain
preservestheorderofaccuracyintimeoftherespectivefixedgridversion[101].Boffi
andGastaldiprovedthattheGCLisneithernecessarynorsufficientforstability[25].
Neverthelessitremainsdesirabletosatisfygeometricconservationatleastforincom-
pressibleflowwhereitiscloselyrelatedtomassconservation.
TheGCLcanbesatisfiedindifferentways.UsingaconvectiveALEformulationofthe
flowequationsintrinsicallysatisfies(3.5)exactlyasshownforexamplein[86].Whena
divergenceformulationisemployedtheGCLcaneitherbestatedasaseparateequation
tobesolvede.g.asin[72,223]oradiscreteGCLcanbeusedtodeterminesomefree
timeintegrationparameterssubsequenttotemporaldiscretisationasdonein[75,101].
Subsequentlythegeometricconservationpropertiesofthediscreteschemes(3.22)
and(3.27)areconsideredbytestingtherequirementthataconstantsolutionhasto
bereproducedexactly.

Convectiveformulation
Toshowthatgeometricconservationisassuredinthediscreteconvectiveformaspatial
andtemporalconstantsolutionu¯isinsertedintothediscreteequation(3.21).Auniform
vtraelocitctions.yfieldThisusadepccompaniendingedonbytheatimepressurediscretisafieldtionthatsccahemerriesthethisboyieldsdyforcesandsurface
(u¯,v)ΩFn+1=(u¯,v)ΩFn+1and(u¯,v)ΩFn+1=(34u¯,v)ΩFn+1−(31u¯,v)ΩFn+1(3.31)
forone-stepθandBDF2,respectively.Consequentlygeometricconservationreducesto
thesamedemadomandinΩthat,ati.e.leatasttheallsamemassinstanliketterminstime.in(3This.22)cohavenditiontobiseinmettegrbyatedtheoverdiscretethe
Fformulation(3.21).
oftheThecoemplonvyectivedetimeALEdisfocrermtisulaationtionscisheme.inherenInttlyeregeostinglymetricallygeocmetriconservaconservtiveairrtionespisectivsat-e
isfiedindependentofthemeshvelocitywhichdoesnotoccurin(3.31).Nevertheless
thedeterminationofuGhasasignificantimpactontheorderofaccuracyoftheoverall
heme.sc

Determinationofthemeshvelocity
ThemeshmotionisgovernedbythefirstorderODE
t∂∂x=uG
χ

.32)(3

36

Chapter3.ALEformulation

lawhicw(3his.5).aItvehasctortoequabetionremarkedincludingherethethatininformathetionderivofatiothenoscfathelargGCLeometric3.5theconsdeervfinitationion
ofthemeshvelocity(3.32)hasalreadybeenutilised.WithinnumericalFSIschemesthe
newpositionofthemeshxn+1isdeterminedbyameshmotionalgorithmandthustreated
asknownhere.Hence(3.32)canbediscretisedintimeyielding
G,n+1xn+1−xn1−θG,nG,n+13xn+1−4xn+xn−1
u=θΔt−θuandu=2Δt(3.33)
forone-stepθandBDF2,respectively.Settingθ=1inthefirstequationof(3.33)recovers
thepopularstepwiseconstantmeshvelocity
G,n+1xn+1−xn
=utΔasbackwardEulertimediscretisationofthemeshmotionODE(3.32).Bothwaysof
discretising(3.32)intimesatisfythegeometricconservationcondition(3.5).
Howevercarehastobetakenifatemporallysecondorderdiscretisationofthefluid
velocityisdesired.Inthiscasethemeshmotionequationhastobeintegratedatleast
bsecepondossoiblerdertouseaccurattheelytra.pAnezoaidalnalysruleis(ofθt=he1lo/2)calfortruncathetionmeshmoerrorvemenindicatteswhilethatusingitshouldBDF2
forthesolutionoftheNavier-Stokesequationsandstillobtainsecondorderconvergence.
Thisisindeedthecase,however,thetrapezoidalruletendstointroduceoscillationsinto
thescheme.ThereforethesecondequationofG,n(+13.33)isusedhereforsecondorderaccurate
schemestoobtainthenewmeshvelocityu.
Subsequentlyanumericalinvestigationconfirmingtheaccuracyobservationsisre-
ed.ortpRemark3.4.3Itisnotnecessarytousethesametimeintegrationschemeforthemesh
vcanelocitalsoybthaetchoissenusedproforvidedtheisoffmomenersthetumdesiredbalance.accuraAnycyo.therimplicitmultistepmethod

NumericalaccuracycheckforconvectiveALEformulation
Inordertoshowthatsecondorderaccuracyintimecanbeobtainedfromthediscrete
wconellvknoectivwenformdrivulaenctionavit(3y.22flo)wanwhereumericalthepaexamprametleerissinusedvestigahereted.aregivTheenprinoblemfigureis3the.2.
onThethecavittopyoispaccupiesrabolictheinunitspacesquaanredandfollowsthesinhorizousonidaltalfflounctwinionxintime-directionaccorpresdingcribtedo
ux(x,t)=4(x−x2)sin(π/4t).Nine-nodedquadraticelementsinspaceareused.To
investigateamovingmesh,thehorizontalmeshmiddlelineasindicatedinfigure3.2is
moreactivedonvertforceicallyFfoatllothewingtopymo(ft)the=0ca.35vitsyin(atπ/2timet).t=The1.0conisvinergvenceestigaoftedtheforhorizodifferenntalt
xtemporaldiscretisationsofthemeshvelocityuG.
Forbothlinesinfigure3.3theoveralltimesteppingschemehasbeenBDF2.The
figureconfirmsthatusingconstantmeshvelocitieswithinthesingletimestepseffectively
thesacrificesnewgoridnevorelodercityofretempstoresortalheovaccuracyerall.tempUsingoraalseconcovndergoencrderetoaccuratesecondscorhedermeontomoobtvingain
mesheswithoutintroducingextraeffort.

3.4.Conservationlawsondeformingdomains37
ux(x,t)
materialdata:ν=0.01
Fxρ=1.0
u=0ym(t)mesh:20×20nine-nodedelements
0=uy0=uxFigure3.2:Drivencavityflowexample-problemdescription
010.01meshvelocitywithBE
1000.0rerro1e-05Fx2meshvelocitywithBDF2
1e-061e-0710Δ1t1001000
Figure3.3:Errorinhorizontalreactionforce
Divergenceformulation
ToALEfindequattheinionsheraentspatigeometrallyanicdcotensmpervoatrallyioncopropnstanerttsiesooflutiontheu¯diviserinsgenceertedfoinrmtoulattheionofdiscretethe
equation(3.25),i.e.theone-step-θcaseisconsidered.Asallspatialvelocitygradients
vanishoneobtains
(u¯,v)ΩFn+1−θΔtu¯−uG,n+1⊗u¯,vΩFn+1=(u¯,v)ΩFn+Δt(1−θ)J˙nu¯,vΩF0
N,F+Δthdn+1−hn+1,vΓn+1,(3.34)
thewhereconthevectivpresesurefluxfieldtermisbalainncestegrathetedbobydypartsforcesawhicndhtrgivesactions.Tosimplify(3.34)further
u¯−uG,n+1⊗u¯,vΩn+1=n∙u¯−uG,n+1⊗u¯,vΓn+1+∙uG,n+1u¯,vΩn+1,
FF,NF(3.35)

38Chapter3.ALEformulation
wherethefactthatu¯isconstantinspacehasbeenused.Inserting(3.35)backintothe
weakformulation(3.34)theboundarytermscanceldueto(3.28)andtheweakform
(u¯,v)ΩFn+1−θΔt∙uG,n+1u¯,vΩFn+1=(u¯,v)ΩFn+Δt(1−θ)J˙nu¯,vΩ0(3.36)
Fisobtained.Forarbitraryu¯theEulerequationof(3.36)isadiscreterepresentationof
thegeometricconservationlaw(3.5)
n+1nJ−J=θJn+1∙uG,n+1+(1−θ)J˙n.(3.37)
tΔEquation(3.37)hastobesatisfiedinordertoenablethediscretescheme(3.25)torepro-
duceanuniformvelocityfieldexactly,i.e.tomaketheschemegeometricallyconservative.
AsimilarexpressioncanbeobtainedforthediscreteschemeemployingBDF2time
discretisation(3.26).Insertingaconstantsolutionin(3.26)yieldsaftersomealgebra
Jn+1−Jn=1Jn−Jn−1+2Jn+1∙uG,n+1.(3.38)
Δt3Δt3
Thediscretegeometricconservationlaws(3.37)and(3.38)aretemporallydiscretised
versionsoftheGCL(3.5)bythesameschemethathasbeenusedforthegoverning
equationsthemselves.However,discretisingthemeshvelocitybythesameschemethat
hasbeenusedfortheALEformulationoftheNavier-Stokesequationsdoesgenerally
notalsosatisfy(3.37)or(3.38)astheJacobiandeterminantJt=det(∂x/∂χ)isa
nonlinearfunctionofthecurrentspatialpositionx.
Discretegeometricconservationlawsfortemporallysecondorderschemesandfinite
volumediscretisationsondeformingdomainshavebeendevelopedbyFarhatetal.e.g.
01].1,[75in

3.4.3Conservationofenergy
IncontrasttotheEuleriansettingconsideredinsection2.3.6theeffectofdomainmotion
upontheenergybalanceequationshallbeinvestigatedhere.Conservationofenergyin
adiscreteschemeiscloselylinkedtostability[107,197].Thustheenergyconservation
propertyofthediscreteschemesfortheincompressibleNavier-Stokesequationsisof
terest.inticularparToinvestigatetheenergyconservationpropertiesoftheALEversionsofthemomentum
balance(3.9)and(3.11)thecontinuousformulationsareconsidered.Theeffectofthe
spatialdiscretisationisaccountedforbykeepinginmindthatthecontinuitycondition
∙uwillnotbestrictlyvalidinadiscretescheme.
ThekineticenergyEkinenclosedinavolumeoffluidwhichisfixedinthereference
readsχsystemEkin=1ρF|u|2dΩF=1ρF|u|2JtdΩF0.(3.39)
2ΩF2ΩF0
FHencethetemporalchangeofkineticenergyinsideΩFisgivenby
∂Ekin=ρF∂u∙uJtdΩF0+ρ|u|2∙uGJtdΩF0,(3.40)
∂tχΩF0∂tχ2ΩF0
wherethegeometricconservationlaw(3.5)hasbeenused.

93

3.4.Conservationlawsondeformingdomains39
Convectiveformulation
Toderivetheenergyconservationequationonadeformingdomaintheconvectivemo-
mentumbalanceismultipliedbythevelocityuandintegratedoveradomainfixedinthe
ieldingysystemreference∂u
ρF∙udΩF+ρFu−uG∙u∙udΩF−∙σ∙udΩF
ΩF∂tχΩFΩF
=ρFbF∙udΩF.(3.41)
ΩFTheconvectivetermin(3.41)canbeintegratedbypartsandreformulatedusingGauss’s
remtheoenceergdivu−uG∙u∙udΩF=−1|u|2∙udΩF+1|u|2∙uGdΩF
ΩF2ΩF2ΩF
+1|u|2u−uG∙nd∂ΩF
2Ω∂FInsertingthisinto(3.41)andreformulatingtheviscousterminthesamewayasin(2.42)
yields

∂Ekin−ρF|u|2∙udΩF+ρF|u|2u−uG∙nd∂ΩF
1FF2∂tχ2ΩF2∂ΩF
−∂ΩFn∙σ∙ud∂ΩF+2µτΩF=ρΩFb∙udΩF(3.42)
wherethetemporalchangeofthekineticenergyaccordingto(3.40)hasbeenidentified.
Equallytotheenergybalanceonafixedreferencesystem(2.42)equation(3.42)states
thatthetemporalchanceofenergyinsideΩFisbalancedbyaninflowofenergy,thework
donebytractionsandbodyforcesandtheenergywhichisdissipatedbytheinternal
friction.Howeverthesecondterminthefirstlineof(3.42)whichvanishesforstrictly
divergencefreevelocityfieldsmightspoilthecorrectbalanceinaspatiallydiscretised
version.Thusconservationofenergycannotberecoveredexactlyinadiscretescheme
basingontheconvectiveALEformulation(3.9).

Divergenceformulation
Asimilarenergybalanceexpressionto(3.42)isobtainedbystartingfromthedivergence
ALEform(3.11),multiplyingitwiththevelocityfielduandintegratingoveradomain
whichisconstantinthereferencesystem.Inthiscasetheresultingenergybalancereads
∂Ekin+ρF|u|2∙udΩF+ρF|u|2u−uG∙nd∂ΩF
1∂tχ2ΩF2∂ΩF
FF2−∂ΩFn∙σ∙ud∂ΩF+2µτΩF=ρΩFb∙udΩF,(3.43)
i.e.thesignoftheenergyerrortermρF/2ΩF|u|2∙udΩFhaschanged.

40Chapter3.ALEformulation
Equation(3.43)highlightsthatdiscreteschemesderivedfromthedivergenceALE
pformoten(3tial.11)energywillalsoerrorbeisofunabletheofsameexactlyorderofconservingmagtnitudehefeneorrgythe.Itdivfurergencethershofowrmsandthatthethe
convectiveformulationprovidedthatthedeviationformexactfulfilmentofthecontinuity
equationiscomparable.

AcombinedALEformulation
Acomparisonof(3.42)and(3.43)suggeststhatanenergyconservingdiscreteformulation
couldbederivedfromanaverageoftheconvectiveanddivergenceformulation.Thisis
wellknownforEulerianflowformulationswhereityieldstheform(2.43).Consult
GreshoandSani[107]oralsoQuateroniandValli[197]foradetaileddiscussionin
theEuleriancase.AlsoondeformingdomainstheaverageoftheconvectiveALEform
andtherespectivedivergenceformulationyieldsadiscretelyenergyconservingscheme.
Howeverinthiscaseitisnotonlytheconvectivetermwhichisaffectedby‘averaging’
thetwoequationsbutalsothetimederivative,i.e.themassterm.Theenergyconserving
formulationreads
FF
2∂tχ∂tχ2
ρ∂(Jtu)+Jt∂u+JtρFu−uG∙u+Jtρ∙u−uGu
+Jt∙σ=JtρFbFonΩF0×T.(3.44)
Unfortunately(3.44)containstimederivativesofdifferentfieldsandisthusnotaccessible
tostraightforwardtemporaldiscretisation.Neverthelessitisoftheoreticalinterestasit
indicateshowtodesignanunconditionallystableschemeonadeformingdomain.

3.5StabilityofALEformulations

andNumericalstableinscohemesrdertoforbteheconvapproergenximat.tionAlsoofforbinitialoundarvalueyvpralueoblesproblemsrequiretobeconsistencyconsistenandt
willstabilitalwyaysarebhesaighlytisfieddesirbeyd.theanaConsistenclyticalyissolutobvioion.usforweightedresidualmethodswhich
ThestabilityoftemporallydiscretisedALEformulations,however,isapersistentmat-
terformofulatioconcern.nwasInpo[83ten,8tially4]Funstaormable.ggiaAccoandrdingNobitoletheconcludelatterpathaptertthehecondivveergctivenceeALEfor-
mstabilitulationycanbdiscretisedeshown.intimeHowbevyerBEitisisthealsoonlystatsedcthemehereforthatwhicthehauncouthorsnditio“wernaletempactualorlyal
theunablequestiotonfindofastabitestlitcyaseofALEwherethebasedschesimulationmeshavewouldbeenblow-upmade”.byBadReceniatconandtributioCodinnsa[5]to
5].[17MasudandStabilityofadiscreteproblemrequiresthatforafiniterighthandsidethesolution
doesnotexhibitunboundedgrowth.Thusitdemandsthatthediscreteoperatorcanbe
inbutvernotedtanwhicehcessaryisthecocasenditioniftheforoperstabilitatory.isIncotheercive.preseHnotwevconer,textcoofercivittheyincoisampressufficiensiblet
Navier-Stokesequationsacoercivityestimatecanalsobeinterpretedasalowerbound

3.5.StabilityofALEformulations41
ontpressurehefoenergrceyswhicwhichhaisreduegenerattoedavifelocinertiaitylfieldforces,uactconalvecongtivtheeforces,directionviscousofu.forcesand
Thesubsequentanalysisisconsideredwithstabilitywithinonetimestepwhichisa
inkind[5,of8l3,ocal84].stabilitCondityions(witharerespoectbtainedtotime)hereinthaconttrasguarttoantloeeng-termthatthestaopbiliteratyorcaconsiderednbe
invertedwithinaparticularstep.Thustheseconditionscouldalsobetermedsolvability
conditiontodistinguishitclearlyfromastabilityconditionobtainedfromalong-term
.lysisanaystabilitStabilitywithinonetimestepisnecessarybutnotsufficientforlong-termstability
whichishardtoshow.Thepresentconsiderationsshedlightonwhyandhowthemesh
motionaffectsstabilityofanumericalscheme.
Astabilityanalysisisperformedhereonthemodelproblemofthetransientadvection-
diffusionequation(2.55).Thisallowstoassesstheeffectofthedomainmotiononthe
Stokstabilitesyinequatdepions,endeni.et.ofthetheveloothercity-predifficultiesssurecoconuplingtainedandinthethenonlineaincomprritesy.sibleNavier-

3.5.1StabilityoftheconvectiveALEformulationoftheadvec-
tion-diffusionequation
Thediscretemodelprobleminconvectiveformulation
TheconvectiveALEformofthemodelproblemreads
t∂χ∂φ+a−uG∙φ−κΔφ=finΩt×T(3.45)
withaninitialfieldφ(t,χ)=φ0(χ)andhomogeneousDirichletboundaryconditions
φ=0on∂Ωtassumedforbrevityofthepresentation.Afterdiscretisationintimeby
one-step-θorBDF2thisyieldsineverytimestep
φn+1+δan+1−uG,n+1∙φn+1−κΔφn+1=rn+1inΩn+1(3.46)
wheretheparameterδisgivenby(3.17)andtherighthandsidetermrn+1dependson
theparticularchoiceofthetimediscretisationschemeaccordingto
rnθ+1=δθfn+1+(1−θ)Δtφ˙n+φn,rnBDF2+1=δBDF2fn+1+34φn−31φn−1.
ThustheGalerkinweakformisgivenby:findφ∈V0h,n+1⊂H1suchthat
(φ,ω)Ωn+1+δan+1−uG,n+1∙φ,ωΩn+1+δκ(φ,ω)Ωn+1=rn+1,ωΩn+1(3.47)
issatisfiedforallωinV0h,n+1⊂H1whichisthediscretespacethatsatisfiesthehomoge-
neousDirichletboundaryconditions.

Coercivityanalysis
ItisworthnotingthattheconvectiveALEformulationofthepresentmodelproblem(3.46)
isnotaffectedbythemotionofthemeshwithinonetimestep.Theproblemrather

42Chapter3.ALEformulation
takestheformofanadvection-diffusion-reactionequationwithanunderlyingvelocity
fielda−uGwhichisnotdivergencefree.Thestabilitylimitsofconvectiveproblems
withnon-solenoidalvelocityfieldsarewellknown,consultforexampleKnabnerand
Angermann[157],andtransfertothepresentcaseasshallbeshownsubsequently.
Aweakformulationoftherighthandsideof(3.47)definesthebilinearform
Bmod(φ,ω)=(φ,ω)Ωn+1+δan+1−uG,n+1∙φ,ωΩn+1+δκ(φ,ω)Ωn+1(3.48)
coercivityofwhichisrequiredforthestabilityofthediscreteproblem(3.47).Inserting
yieldsφ=ωBmod(φ,φ)=φΩ2n+1+δan+1−uG,n+1∙φφdΩt+δκφΩ2n+1.(3.49)
Ω+1nThesecondtermin(3.49)canbereformulatedbyintegrationbypartsandemploying
∙a=0.Usingfurtherthattheboundaryintegraltermvanishesduetotheboundary
conditionsoneobtains
an+1−uG,n+1∙φφdΩn+1=1∙uG,n+1|φ|2dΩn+1.(3.50)
Ωn+12Ωn+1
Inserting(3.50)into(3.49)allowstoobtainthecoercivityestimate
Bmod(φ,φ)≥1+δ1min∙uG,n+1φΩ2n+1+δκφΩ2n+1.(3.51)
2whichissharpif∙uGisconstantinspace,i.e.ifthedomainexpandsorcontracts
.mlyuniforThustheformulationisstablewithinonetimestepif
1+δ1min∙uG,n+1>0(3.52)
2issatisfied,i.e.ifthecoefficientoftheL2-normin(3.51)isstrictlyboundedawayfrom
zero.Thisconditiononthesolvabilityofconvectiondominatedproblemscanalsobe
].[157infoundConsequentlyadiscreteschemebasedontheconvectiveALEformulationisonlycon-
ditionallystable.AsimilarresulthasalsobeenobtainedbyBadiaandCodinawho
considerthelong-termstabilityoftheconvectiveALEformulationin[5].

Interpretationofstabilityresult
Howeverthesituationisnotasbadasitmightseem.Afirstsoothingobservationisthe
factthatapositivedivergenceofthemeshvelocitywillneveryieldinstabilitybutrather
stabilisethebilinearform.Thuselementcontractionhastobeconsidereddangerous.
Recallingthevalueofδforone-step-θandBDF2timediscretisationupperlimitsonthe
timestepsizeareobtained.Thetimestepsizehastoobey
32Δt<θmax|∙uG,n+1|andΔt<max|∙uG,n+1|(3.53)

3.5.StabilityofALEformulations43
forlimitone-saccortep-dingθtoandB(3.5D3)F2,isorespbserecvtedivelyfor,toBEtyieldimeainsttegableration.methoAssd.umingThelineastrictestrapprotimestxima-ep
tioninspaceandtimethedivergenceofthemeshvelocityofanelementinddimensions
thatshrinksuniformlycanbeexpressedby
n+1n∙uG,n+1=dhΔthn−+1h,(3.54)
whichisnegativeforacontractingelementwithasizeofhn+1=γhnandγ<1.The
whicderivhatioisanofligned(3.5in4)gloisbalillustrcooatedrdinaintefiguredirections.3.4whereHowaever,rectaastngularhedivergelemeencnteisoftheconsideredmesh
n2timet1timetn+1x-velocityofnodei:
hxnhxn+1uxiG,n+1=xin+1Δt−xin
2134x-derivative
yofhorizontalvelocity:
34uGx,x,n+1=uxG1,n+1hn−+1uGx2,n+1
xx+1nnFigurresultinge3.4:meshSkvetcelohocitfylinederivarativfiniteeaselemensumingtwithinlinearintheterptimeolationstepinfromtimetatondtspaceandthe
velocityactuallyisameasureofavolumeratioitisclearthat(3.54)appliesforall
elementswhichexpandandshrinkuniformly.
InsertingthisdivergenceexpressionintothetimesteplimitobtainedforBEtime
discretisation(3.53a)withθ=1yieldsthemaximalallowablecontractionwithinone
timestep.Dependingonthedimensionoftheproblemdthebilinearformremains
coerciveforauniformcontractionsatisfying
d/(2+d)<γ.(3.55)
Thusinthethree-dimensionalcasetheelementcancontracttonearly60%oftheprevious
lengthhnineverydirectionandtheschemeisstillstableonthedeformingdomain.
Schemeswithθ<1orBDF2areevenmorepermissive.Similarlymorerapidcontraction
ispinstabilossibleityofinalocwonervectivedimensionaALElcformases.ulationcoPresumablyuldnotthisbeistheconfirmedreasonnwhyumericallythe[8susp4].ected
Itismorecomplicatedtoevaluate(3.53)forhigherorderschemes,whenmorethan
tcanwobetimeconstrlevelsuctedareinforvolvhiged.heroArdermucschhemormeesbunfayvouraassumingblethadivterganenceeolemenfttheexpameshndsveinloconeity
timestepandcontractsrapidlyinthesubsequentone.Iftheapproximationofthemesh
velocitywithinoneelementislinearinspacewhichmeansthateitherlinearelementsare
andusedcenfortrethenomedessh,amodivtionergenceorqofuadratheticemeshlemenvtelosacitreyinvowithinlvedanwitelehpmenterfectlycanbplaecdeedducededge
b)..33(3fromacThaosampleracteristictheflengthollowofinghn+1caseattshallimebelevecolnnside+1red:andawthraselargere-dimensionalinthepreelevmeniouststhasep

44Chapter3.ALEformulation
hn=γhn+1withγ>1andofthesamesmallsizeatthetimeleveln−1,i.e.hn−1=hn+1.
Theexpansionandsubsequentcontractionshallhappenuniformlyinspacewhichisthe
worstcase.Thusonecoordinateofthemeshvelocityaccordingto(3.33b)evaluatesto
uG=2(1−γ)hn+1.
tΔInathree-dimensionalcasethisyieldsanegativedivergenceofthemeshvelocityof
∙uG=6(1−γ).
tΔItshallfurtherbeassumedthatBDF2hasbeenusedtodiscretisetheoverallequations
andthus(3.53b)hastohold.Themaximalexpansionfactorγofsuchascenariois
γ=1.5whichmeansthattheschemeisatitslimitifanelementuniformlyexpands
tohn=1.5hn−1andshrinksbacktohn+1=hn−1withinthenextstep.Howeversuch
ascenarioforthemeshmotionshouldirrespectiveofstabilityissuesnotbeconsidered
trustworthy.
TheseresultstransfertotherespectiveALEformulationoftheincompressibleNavier-
Stokesequationsifthedivergenceofthemeshvelocityisreplacedby−∙u−uG,n+1
toaccountforthedivergenceerrorofthediscretevelocityfield.Henceitisadvisableto
checkthestabilitylimits(3.53)duringacomputationandissuingawarningthatcauses
theusertorestarttheproblemwithasmallertimestepifnecessary.

3.5.2StabilityofthedivergenceALEformulationoftheadvec-
tion-diffustionequation
Thediscretemodelproblemindivergenceform
AnalogouslytotheNavier-StokesequationsadivergenceALEformofthemodelprob-
lemcanbefoundreading
t∂χ∂(Jtφ)+Jt∙a−uGφ−κφ=finΩ0×T(3.56)
accompaniedbyinitialconditionsandhomogeneousDirichletboundaryconditions.A
temporaldiscretisationofthisdivergenceexpressionyields
Jn+1φn+1+δJn+1∙an+1−uG,n+1φn+1−µφn+1=rdn+1inΩ0,(3.57)
whererdcontainsrighthandsidetermsanalogouslytotherespectivetermsfortheflow
momentumbalanceindivergenceform(3.20).
nrdn,θ+1=δθJn+1fn+1+Jnφn+(1−θ)ΔtJ˙φ,(3.58)
rdn,B+1DF2=δBDF2Jn+1fn+1+4Jnφn−1Jn−1φn−1.(3.59)
33TheGalerkinweakformulationofthediscretisedproblemreads:findφ∈V0h(Ω0)such
ttha000Jn+1φn+1,ωΩ+δJn+1∙an+1−uG,n+1φn+1,ωΩ+δκJn+1φn+1,ωΩ
=rdn+1,ωΩ0forallωinV0h(3.60)

3.5.StabilityofALEformulations45
issatisfiedwhereV0hdenotesthediscretespacethatsatisfiesthehomogeneousDirichlet
ns.conditioryoundab

Coercivityanalysis
Equation(3.60)definesthebilinearform
Bmod,d(φ,ω)=(φ,ω)Ωn+1+δ∙an+1−uG,n+1φ,ωΩn+1+δκ(φ,ω)Ωn+1
.61)(3coercivityofwhichisrequiredforstability.Insertingω=φyields
Bmod,d(φ,φ)=φΩ2n+1+δ∙an+1−uG,n+1φφdΩn+1+δκφΩ2n+1.
Ω+1n.62)(3Evaluatingthesecondtermin(3.62)gives
∙an+1−uG,n+1φφdΩn+1=−1∙uG,n+1|φ|2dΩn+1,
Ωn+12Ωn+1
where∙a=0and(3.50)havebeenused.Thisreformulationyieldsthestabilityresult
ofthedivergenceALEformulation
2Bmod,d(φ,φ)≥1−δ1max∙uG,n+1φΩ2n+1+δκφΩ2n+1.(3.63)
Theestimate(3.63)issharpifthedomaincontractsorexpandsuniformly.Thusthe
stabilitylimitobtainedfortheconservativeALEschemeisgivenby
21−δ1max∙uG>0(3.64)
showingthatadiscreteschemeobtainedfromthedivergenceALEformisalsocondi-
tionallystable.Thisresultisremarkablysimilartothestabilityconditionobtainedfor
theconvectiveALEformulation.HoweverthedivergenceALEformulationmightget
unstableatsuddenelementexpansionratherthancontraction.Transferring(3.64)intoa
specifictimesteplimityields
32Δt<θmax(∙uG,n+1)andΔt<max(∙uG,n+1)(3.65)
forone-step-θandBDF2,respectively,whichapplyforpositivedivergenceofthemesh
velocityonly.
Inordertoestimatethepracticalmeaningofthetimesteplimits(3.65)ad-dimensional
elementisconsideredagainsubjecttouniformexpansion.Theinitiallengthhnis
stretchedtohn+1=γhnwithγ>1.Assuminglinearinterpolationinspaceandtimethe
divergenceofthemeshvelocityisevaluatesto
∙uG,n+1=dhn+1−hn=dγ−1.(3.66)
hn+1ΔtγΔt

46

Chapter3.ALEformulation

Thelowesttimesteplimitin(3.65)obtainedforBEstatesthatthedivergenceofthe
meshvelocity(3.66)hastobesmallerthan2/Δt.Ifthedimensionislowerthanthree
thisissatisfiedforallfiniteγ.Inthethreedimensionalcasestabilityisconditionalupon
γ<3.Thusathree-dimensionalelementmaystablyundergoanexpansiontouptothree
timestheoriginallengthineachdirectionwithinonetimestep.Clearlythislimithasnot
beenmetinpracticalcomputations.
SimilartowhatisobservedfortheconvectiveALEformulationfurtherunstablecases
canbeconstructedforhigherdimensionalinterpolationsofthemeshvelocity.However
suchscenariosarelikewisenotofhighinterestinpracticalcomputations.
Remark3.5.1(oncomparison)Theobviousunsymmetrybetweenelementcontrac-
tionandexpansioniscausedbythefactthatthedivergenceoftheactualmeshvelocity
∙uG,n+1referstothemostrecentconfiguration.Thusanelementcontractingtozero
willalwaysyieldaninfinitelynegativedivergenceofthemeshvelocityandthuscertainly
violatethecondition(3.53)witheverytimestep.Howeverbothofthelimitingtime
stepsizeshardlydeterminethemaximaltimestepforagivenphysicalproblemasrapid
changesoftheelementlengthsarecausedbyrapidmotionofthefluid-structureinter-
faceΓwhichindicateshighlytransientdynamics.Suchsituationsanywaydeservesmall
timestepstoresolvethephysicalproblemadequately.

3.5.3Anunconditionallystableformulationondeformingdo-
nsmaiAcomparisonofthestabilityestimatesobtainedfortheconvectiveformulation(3.51)
andformtheulatiodivnergofeconncesfervormation(3.6la3)wsonindicatesdeforminghowtodomaconstructins.AnanenergyunconditcoionallynservingstaALbleEfoALErm
resultsfromaveragingtheconvectiveanddivergenceALEformulation,however,this
partialdifferentialequationcannotstraightforwardlybediscretisedintime.
Anunconditionallystablediscreteformulationisobtainedbyaveragingthediscrete
convectiveanddivergenceALEform(3.21)and(3.25)or(3.26)forone-step-θandBDF2,
respectively.Inthecaseofthepresentmodelproblemthismeansavheragingoftheweak
forms(3.47)and(3.60)yieldingthediscreteproblemoffindingφ∈V0(Ωn+1)suchthat
(φ,ω)Ωn+1+δan+1−uG,n+1∙φ,ωΩn+1+1δ∙an+1−uG,n+1φ,ωΩn+1
121
+δκ(φ,ω)Ωn+1=2rn+1,ωΩn+1+2rdn+1,ωΩ0(3.67)
issatisfiedforallω∈V0h.Indeedrepeatingthestabilityanalysisoneobtainscoercivityof
thebilinearformBmod,avdefinedbythelefthandsideof(3.67).Therespectiveanalysis
yieldsaftersomealgebra
Bmod,av(φ,φ)=φΩ2n+1+δκφΩ2n+1(3.68)
confirmingthattheproblem(3.67)canstablybeintegratedirrespectiveofthetimestep
sizeorthemeshmotion.
Howeverthereisnothingforfree.Whileguaranteeingstabilitywithrespecttothemesh
motiontheweakform(3.67)haslosttheexactgeometricconservationpropertyofthe
convectiveALEformaswellastheconservationpropertyofthedivergenceformulation.

rySumma.3.6

3.6Summary

47

aTherecontinareuousdifferensettintgwaallystotheseformformulautelationsconservareationequal,lawsondiscretemovevingrsionsdomains.thereofWhileexhibitin
conservdifferenceatiosnwithlawsrespinherenecttotlythesatisexactfiedbyfullfillmendifferenttofALcoEnserformvationulatiolansws.isAgivensummarintaybleof3the.1.

Table3.1:ShortsummeryofinherentconservationpropertiesofdifferentALEformula-
nstioALEconservationofconservationofconservationofconservationof
formulationenergymomentummassgeometry
divergenceformnoyesyesno
convectiveformnonoyesyes
averageformyesnoyesno

IthasbeenshownthattheconvectiveALEformulationinherentlysatisfiesgeometric
energyconserv.atioScnhemewhilesbaasingondiscretisatiothenconerrvecorteivnetersALEthecoformnseularvatiotionnshaofrelineathermotempmenoraltumoandrder
ofaccuracyoftheirfixedgridcounterpartassoonasthemeshvelocityisinterpolated
high.tlynsufficieservAatiotempnoforallymomendistum.cretisedHowscevheer,meofextratheeffodrtivhasergencetobfeormmauladetionhereguainraonrderteestoexaactpprocon-ach
thegeodivmetricergenceconservformatiodon.notSimisalarlytisfytoenergytheconvconservectivatioenformexactulatioly.nBoththeschediscretemesvaerresionscondi-of
uptioonnallythestadivbleergencewhereoanftheuppermeshbovundelooncittyheaswtimeellassteponsiztehecanparbeticularobtainedtimewhichdiscretisatdependsion
heme.scSimilartowhatiswellknownforEulerianmethodsanenergy-conservingALEformu-
lattimeiondiscancretbeisoedvbtainedersionsbyaofvetrheagingcontvheectivconevecandtivedivaergndedivncefoergencermulafortionmaulatiotempn.Avorallyeragdising-
creteschemeisobtainedwhichisunconditionallystableonmovingdomains.

4erChapt

Stabilisedfiniteelementmethodsfor
incompressibleflow

Withinthischapterthetemporallydiscretisedflowequationsarediscretisedinspace.
Alongwiththespatialdiscretisationstabilisationhastobeintroduced.Thisneedforsta-
arebilisatiodiscussnofed.theAfterdiscretebrieflyflowequatreviewingionsistheexplareasoninedforandtheanoumbsebrveredofnstaumericalbilisatioonscillamethotionsds
somemoreorlesspopularmethodstocircumventtheseoscillationsareconsidered.The
whicmainhpartaredisofthecussedcahapterndaisdevnalysedotedintosomethedetafamilyil.ofresidualbasedstabilisationmethods
aThedeformingfocusdoissetmainonsucinhttrohatducingnoaaddndfoitionarmlulastatingbilitaystarestrictiobilisednsfinitearise.elementmethodon

4.1Wigglesandtheneedforstabilisation

ItiswellknownthatfiniteelementmethodsbasedonweakGalerkinformssuch
as(3.21),(3.25)or(3.26)frequentlyfailassoonassignificantconvectionoccurs.An-
otherimportantchallengeinincompressibleflowistheincompressibilityconditionand
thustheroleofthepressure.Inparticularacloselookattheweakforms(3.22)and(3.27)
revealsthatthepressurevariableactsasaLagrangemultipliertoenforcetheincom-
pressibilitycondition(2.35).Thustherestrictionsimposedbytheinf-supconditionor
LBB(Ladyshenskaya-Babuˇska-Brezzi)conditionhavetobeconsidered.
Increasingtheflowvelocityonagivenmesh,i.e.increasingtheconvectivetransport,
aswellasviolatingtheinf-supconditionintroducesspuriousoscillationsalsotermed
wiggles.GreshoandSani[107]giveanexhaustivediscussiononsuchwigglesand
wiggle-generatingsituationswhiletheyinsistthatwigglesareaself-diagnosisproperty,
indicatinginappropriateresolutionandthedifficultyofthenumericalmethodtocome
upwithasuitableapproximationofthephysicalproblem.Theriskofover-stabilisation
bydampingoutnumericalproblemswhileatthesametimechangingthephysicsofthe
problemsolvedhastobekeptinmindwhenstabilisationmethodsshallbedeveloped.
Thissectionisdevotedtobrieflyhighlightingthecauseoftheoccurringunphysical
oscillations.Formoreexhaustiveexplanationsandexamplesonemightconsultthedis-
sertationsofGravemeierandWall[103,227]aswellasthebookbyDoneaand
Huerta[69].WithrespecttotheLBBconditionmathematicalliteratureonmixedvari-
ationalproblemsandfluiddynamicsasforexamplethemonographbyGiraultand
Raviart[102]ortheonebyGunzburger[109]shouldbeconsulted.Astandardrefer-
enceformixedmethodsproblemsisfurtherthebookbyBrezziandFortin[32].

49

Chapter4.Stabilisation

50Chapter4.Stabilisation
4.1.1Convectiondominatedproblems
Mathematicalanalysis
Numericaloscillationsinducedbydominatingconvectionareconvenientlyanalysedby
meansoftheadvection-diffusionequation(2.54).Thustheresultofacoercivityanal-
ysisofthediscretetransientadvection-diffusionproblemwithhomogeneousDirichlet
boundaryconditionsobtainedinsection3.5isrecalled.From(3.51),(3.63)and(3.68)it
canbeobservedthatirrespectiveoftheunderlyingALEformulation(andindeedalsoon
fixedgrids)thetransientadvection-diffusionproblemyieldsacoercivityestimateofthe
formBmod(φ,φ)≥CφΩ2n+1+δκφΩ2n+1(4.1)
whereCisapositiveconstant.Thusthecorrespondingnumericalmethodisstablein
thesensethatthecorrespondingdiscretecoefficientmatrixisnon-singular.Howeverthe
convectiveterma∙φdoesnotcontributetothecoercivity.Soiftheinfluenceofthe
convectionismuchlargerthantheeffectofthediffusiongovernedbyκthe1coercivity
estimate(4.1)mightbearbitrarilysmallindicatingthatthecontrolovertheHnormof
thesolutionisgraduallylostleavingthediscretesolutionwiththefreedomtooscillate.
Inthespecialcaseofastationaryadvection-diffusionproblemacoercivityconsideration
yields

Bmod,stationary(φ,φ)=κφΩ2n+1
furtherhighlightingthepotentiallossofstabilityatsmallκcomparedtotheconvection,
i.e.athighelementalPecletnumberPee=ah/2κ.Anerrorestimateforstationary
advective-diffusivetransportobtainedbyHughes[132]reads
e=O(1+Pee)hk
wherekdenotesthepolynomialorderoftheelement.Thusthegradientoftheerror
e=φh−φdependsonPeeindicatingthatoscillationsarewellpossibleatconvection
dominatedflowswhiletheboundbehavesoptimalfordiffusiondominatedproblems.

Propertiesandinterpretationofconvectivewiggles
Spuriousconvectionoscillationsemanatedatunresolvedgradientstendtospreadoverthe
entiredomain.Inparticularintimedependentcalculationsthesolutionispollutedby
suchwigglesgloballyafterjustafewsteps.
InordertoincreasetheunderstandingofthedeficiencyofGalerkinschemesin
convectiondominatedproblemsandindicatewaystoovercomeit,differentinterpretations
oftheproblemhavebeengivensomeofwhichshallbesummarisedhere.
•AnalysingthediscreteequationsinonedimensionDoneaandHuerta[69]show
thatcentredschemesareunder-diffusivecomparedtoschemeswhichwouldyield
nodallyexactsolutions.Thissuggestsstabilisationbyaproperamountofadditional
diffusion.lumerican

4.1.Wigglesandtheneedforstabilisation51
•Itcanalsobearguedthatthesolutionataparticularnodeattimeleveln+1is
muchmoreinfluencedbythebehaviourofthefunctionφupstreamthanbythe
trasituatnspoionrtdoshouldwnstreamthusatincludetimeanlevelupn.windingAnofumericalthewescighhemetingtosimfunctionulateraadvtherthectivane
tred.nceeingb•Anotcausedherbyviewonon-resolvntheedphproblemysicsr,evi.e.ealsbadlythattherepresewiggntedlynhigherumericamolades.pproxThusimatiointnro-is
ducingadditionalsubscaledegreesoffreedommightcurethedeficiencies.
•Atermfurtheroftheaforpproamchais∙φinspir2eisdbymissing;thecoerenriccivithingytheestimawteeigh(4.1ting)infwhicunctiohnanspaceadvectsucivhe
thatitincludesaconvectivetermwouldyieldthedesiredcontrolinthebilinear
form.

4.1.2SpuriouspressureoscillationsandtheLBBcondition
Inordertostudytheroleoftheinf-supconditiontheStokesequations(2.53)are
recalled.AdiscreteweakformoftheStokesproblemreadsonafixeddomain:findthe
velocityu∈VFhandpressurefieldp∈Phatthenewtimelevelsuchthat
(u,v)ΩF+δ(2νε(u),ε(v))ΩF−δ(p,∙v)ΩF−δ(q,∙u)ΩF
=rn+1,vΩF+δhn+1,vΓF,Nforall(v,q)∈VFh,0,Ph(4.2)

issatisfied.Inmatrixnotationthesystemmightbewrittenas
δGT0p=0.(4.3)
MF+δKFδGufF
Thediscretesystem(4.2)isstableifthefiniteelementspacesVFhandPhandthusthe
elementsarechosensuchthatthediscreteversionofthewell-knownLBBorinf-sup
conditionissatisfied.Thisconditionreads(seeforexampleGunzburger[109]):Given
anyp∈Ph,

−(p,∙v)
0=vsup∈VFh,0|v|1≥γp,(4.4)
wherethehpositiveconstantγisindependentofthemeshsizehandtheparticularchoice
oftestp∈Pfunction.TheviscoconnditiotaninedstaintesVFh,tha0twhicforheveryyieldsaadmiscontrsibleibutionnon-zeorfothepressurepressurefieldtoathveelotocittaly
internalenergy.
EmployingthematrixG,thediscreteLBBconditioncanbewritten
TGpv0=vsup∈V0h|v|1≥γp.(4.5)
raEquatnkwhicionh(4is.5)shoclearlywsathatnecinessarythecaconditseofionLBBfortstaheblesolvabilitdiscretisayoftionthethediscretematrixsysGteismof(4.3full)
aswellasdiscreteversionsoftheincompressibleNavier-Stokesequations.

52

Chapter4.Stabilisation

Unfortunatelytheinf-supconditionisfrequentlyviolatedfordesirablecombinations
ofvelocityandpressurespaces.Howevertherearetwodifferentgradesofviolation.A
numberofelementscanbefoundwhichmightbeused–atleastwithsomecareconcerning
btheefoboundundaryinthecolistnditioofpnsos–sibleeveneleifmentheytsdofornotthesatisfyincompr(4e.4).ssibleExamNavplesiero-fsucStokheselemeequatntsionscan
givenbyGreshoandSaniin[107].Suchelementsviolatetheinf-supconditioninthe
sensethattheydonotexhibitastabilityconstantγindependentofthemeshsizeh.Thus
theymightgetunstableinthelimith→0,acaserarelymetinpracticalcomputations.
Amuchmoresevereproblemoccursif(4.5)isviolatedsuchthatthediscreteoperatorG
noexhibitsuniqueasraonklutiondeficienccanbye.Inobtthisained.casetheConsequencoeffictlyientthedismatrixcretine(4.3)solutiownillofbethepsingularesrsureand–
whilepossiblyaccompaniedbyareasonablevelocityapproximation–exhibitsdevastating
spuriousoscillations.Examplesofsuchspuriouspressuremodeshavebeencomputede.g.
byWall[227].

4.2Stabilisation–omnipresentinflowsolvers

Wherevstabilisaertionfloiswconsproblemsidered.withOccasionaconsiderablellyitcmighonvtenotctivebetracallednsport‘stahavebilisatiotobn’ebutdealtratherwith
comeextraaefforlongthastermedbeen‘upwmadeindining’oorrder‘batoovlancingercomediffusiothen’.unstableHowevweriginglyallsolutiothosenocasesfasomepure
Bubnov-Galerkinscheme.
AnalogouslytherestrictionimposedbytheLBBconditionoccursassoonasanincom-
scpressiblehemeifproitcesalloswsistoconsidered.circumventAndthisitisgconditioenernallyoptakeningenastheanfaulldvranangtageeofofvaeloncityumericaland
pressurespacecombinations.
Beforeintroducingtheresidualbasedstabilisationmethodswhichhavebeenused
primarilywithinthisworkashortoverviewoversomealternativestabilisationmethods
floshallwbsolvegersivinen.Itdetail.isfarThebeyinondtenttionhesiscorapetheroftothepresehighlignhttwthaorktttoheinretroisnoducestaablewideflowrangsoelverof
withwithoutfiniteextreleaeffomenrt.ts.TheHowefovcuser,viseryputosimilarnstabilisamethodstionaremethouseddsawhenppliedfiniteinvconjuncolumestoionr
differencemethodsareconsideredwhereanalogousinstabilitiesarise.

4.2.1Convectionstabilisation

Inspiredbythefactthatinformationispropagatedbyanunderlyingvelocityfield,upwind
schemeshavebeenproposedforconvectiondominatedflow.Inthecontextofweighted
residualmethodsupwindingisintroducedbychoosingaweightingfunctionwhichputs
moreweighttotheupstreamthantothedownstreampartoftheflow,i.e.aPetrov-
Galerkinschemeisemployed.AdjustingtheamountofupwindingtothePeclet
numberallowstoobtainnodallyexactvaluesfortheone-dimensionaladvection-diffusion
problem[69,237].
Thesituationismoredifficultinthehigherdimensionalcase.Acertainamountof
upwindingcorrespondstoaparticularportionofartificialdiffusivityintroducedinthe

4.2.Stabilisation–omnipresentinflowsolvers

35

scwindhemediffusio[69,2n’,37].i.e.Butartinifictroialducingdiffusivitnyumericalperpviscoendicularsitygotoestahelongconvecwithtivethevelocitso-calledysev‘croerelyss-
harmingtheaccuracyoftheapproximation.
In(SUPG)ordermethotodovhaercosbmeeendevtheseelopeddifficultiesbyHughtheesandstreamlinecolleaguesupwindin[13Pe5,tr1o36v-,1Ga53].lerkinThe
themethofadmilyshaollfberesidualtouchedbaseduponstaagbilisainaintionSectiomethon4.3ds.asAthisnextisooneneisofthetheGaearlylerkinmemb/Lerseast-of
Squares(GLS)methodwhichhasalsobeenproposedbyHughesandcoworkers[140].

Finiteincrementcalculus
Stabilisationtermscanbederivedbyemployingtheso-calledfiniteincrementcalculus
suggestedbyO˜nate[188].Recentformulationsofstabilisationbasedonfinitecalculus
canbefoundin[190,191]wherethemethodhasalsobeenusedtotreatincompressible
flowsonmovingdomains.Applicationofthemethodtoshiphydrodynamicsispresented
9].[18inThebasicideaoftheformulationshallbeexplainedbrieflyhereonthescalaradvection-
nequatiodiffusiona∙φ−κΔφ−f=0inΩ.(4.6)
Inviewofthedesirednumericalsolutionof(4.6)bymeansofadiscretemethodwhich
considersfiniteportionsofthedomaintheresidual(4.6)isexpandedinspaceinaTaylor
yieldingseriesa∙φ−κΔφ−f+δ∙(a∙φ−κΔφ−f)=0inΩ,(4.7)
2whereδisaspatialdistancevectorpointinginthedirectionoftheexpansion.Itis
smalleroraslongasanelementallengthandscalesthestabilisationterms.Inorderto
ensurestreamlineorientedstabilisationtheexpansionisperformedinthedirectionofthe
underlyingvelocityfield,i.e.δ=ca/|a|wherecisaconstanttobechosenappropriately.
CharacteristicGalerkinprocedure
Anotherwayofobtainingstabilisationtermsistheso-calledcharacteristic-Galerkin
procedureextensivelyusedin[237],wherethepathofafluidparticleiscalleditscharac-
teristic.ThisapproachdepartsfromtheLagrangeanformulationofthetransientflow
problemwhichfortheadvection-diffusionequation(2.54)reads
t∂∂φ(x(t),t)−κΔφ=f(x),(4.8)
Xwheretheconvectivetermdisappearsandthematerialtimederivativeofthetransport
variableisinvolved.Atemporaldiscretisationof(4.8)bytheone-step-θmethodthen
yieldstermstobeevaluatedatdifferentspatialpositions
φxnn+1+1−φxnn=θΔt[κΔφ+f]xnn+1+1+(1−θ)Δt[κΔφ+f]xnn(4.9)

54

Chapter4.Stabilisation

wherethesubscriptsxn+1andxndenotetheactualandthepreviouspositionofthe
whicfluidhparishigticle.hlyundeDiscretisingsirable.(4.9Th)usthedirectlyoldinvaluespaceφnrnisequiresapproaLaximagtedrangbyeaanTaymeshlormotioseriesn
xexpansionbackwardinspacetruncatedafterthesecondterm.
φxnn≈φxnn+1−δ∙φxnn+1+21δ∙Hφxnn+1∙δ+OΔt3
n+1nwherewithinthethetimedistancestepvectoandrcanδ=bexobta−inedxif=ana¯aΔvterisagethecownavyectivofethevelocrespityeca¯tivewithinparticlethe
timestepisavailable.ThesymbolHdenotestheHesseanmatrix,i.e.secondspatial
.esativderivSimilarlythediffusiveandloadtermattheoldtimelevelandspatialpositionxnare
expandedinaTaylorseriesyielding
Δφxnn≈Δφxnn+1−δ∙Δφxnn+1+OΔt2,
fxnn≈fxnn+1−δ∙fxnn+1+OΔt2.
Introducingtheseexpansionsbackinto(4.9)yieldsaschemewithpossiblesecondorder
temporalaccuracyreading
φn+1−φn=θΔt[κΔφ+f]n+1+(1−θ)Δt[κΔφ+f]n−Δta¯∙φn
1+2Δt2a¯∙Hφn∙a¯−(1−θ)Δt2a¯∙(κΔφn+fn)(4.10)
wherealltermsrefertothepresentpositionsoftheparticlesxn+1.Inthefirstline
of(4.10)itcanbeobservedthatthetransformationoftheLagrangeanequation(4.8)
toanEulerianmeshviaTaylorseriesexpansionseffectivelyreintroducestheconvective
term.Additionallythesecondlineof(4.8)emergescontainingtermsthatenhancethe
stabilityoftheformulation.Inparticularthefirsttermonthesecondlineof(4.8)offers
thedesiredstabilisationdiffusioninthestreamlinedirection.
ItrestrictedisnotewtoaorthtempytorahatlathecccuracyhaupracteristictosecoprondceorderduredueinttohetheformulatruncattionionssketcinhedtheaTbaoyvloeisr
series.Themethodofcharacteristicsfurtherappliestotransientproblemsexclusively.

Flowconditionedbasedinterpolation
AnalternativewaytocopewithconvectiondominationhasbeensuggestedbyBathe.
Withintheschemeproposedin[10]anexponentialweightingfunctionforthevelocityis
usedwhichdependsupontheunderlyingflowfieldandthusintroducesanappropriate
wamoasunmotofdifieupdsucwindinghthatwitthehincoeacnvhectiveleemenvelot.citInyaitselfsubsequenisintterppuolatbliceadbtionyexp[11]onenthetialmethotriald
functionsdependinguponthevelocityfield.
Recentdevelopmentsofthemethodcanbefoundin[158,159,160].

4.2.2CircumventingtheLBBcondition-pressurestabilisation
Severalmethodshavebeenproposedtocircumventtherestrictionsimposedbytheinf-
supcondition.Thisisdesirableinparticularforloworderapproximationsofpressure

4.2.Stabilisation–omnipresentinflowsolvers

55

andvelocitymostofwhichdonotsatisfytheLBBcondition.Howeverloworderpairs
ofvelocityandpressureareofspecialinterestastheyyieldsmallbandwidthsandalow
numberofnonzerovaluesperlineofthecoefficientmatrixalsointhree-dimensional
.blemsproItisfurtherregardedageneraladvantageifthechoiceofthediscretevelocityand
pressurespacecanbemadefreely.Thissignificantlyenlargesthesetofpossibleelements
forincompressibleflowsimulations.Typicallytherelaxationoftheincompressibility
conditionyieldscontributionsonthepressuremaindiagonalblockoftheoverallcoefficient
matrixmakingthismatrixsignificantlyeasiertosolvebyiterativesolvers.Inotherwords
aslightrelaxationoftheincompressibilityconditionweakensthepressureshockwhich
resultsfromadeviationfromavelocityfieldwhichisdivergencefreeintheproperweak
sense.Theissueofstabilisedversusaprioristableelementsshalltoucheduponlater
againinthecontextofresidualbasedstabilisationmethods.
Penaltyformulationshavebeenusedtoformulatetheincompressibilityconstraintal-
lowingtouncouplethepressureandvelocityequation(seee.g.[135]).Incontrastto
thesemethodsstabilisationaimsatrelaxingtheincompressibilityconditionsuchthat
LBBincompatiblespacescansafelybeused.
Pressurestabilisationcanbedesignedbasingonavarietyofprojectionschemes.Oneof
thepossibilitiesistorelaxthecontinuityequationinthecontextofoperatorsplittingpro-
jectionmethodssuchasthecharacteristic-basedsplit(CBS)algorithmofZienkiewicz
etal.[237]whichisbrieflydescribedinthesequel.Thepossibilityofpressurestabilisation
inthecontextofprojectionschemesfortransientproblemshasalreadybeenshownby
KawaharaandOhmiyain[152].
AsanextensionoftheoperatorsplittingmethodsCodinaandBlascoproposeda
pressurestabilisationmethodfortheStokesequationswhichdoesnotrelyonthetime
discretisationofthetransientproblem[54].Thispressuregradientprojectionmethod
analysedbyCodinaandcoworkersin[17,55,56]isdesignedforelementswithdiscon-
tinuouspressurespaces.Theincompressibilityconstraintisthenrelaxedbyusingthe
differencebetweenthediscontinuouspressuregradientanditsprojectionontothecon-
tinuousvelocityspace.However,thisstabilisationmethodcomesalongwiththeneedfor
aninversionofaglobalmatrixinordertoperformtheprojection.
ResidualbasedstabilisationmethodsfortheStokesproblemhavebeenproposedby
Hughesandcoworkers[139].Comparisonsofstabilisedfiniteelementsandaprioristable
schemeshaveshownthatstabilisedformulationscompetewellwiththeirinherentlystable
counterpartsasshownbyNorburnandSilvesterin[184].
RecentworkonthePressurePoissonstabilisationhasbeenreportedbyBochevet
0].2[8,al.

Pressurestabilisationwithinprojectionbasedalgorithms
Apopularfamilyofflowsolversisbasedonprojectionandoperatorsplitting.Onewell-
knownrepresentationthereofistheso-calledcharacteristic-basedsplit(CBS)algorithm
describedinthethirdvolumeofthefiniteelementtrilogyofZienkiewiczetal.[237].A
StokcompaesrisonequatofionsCBScanandbeGfoLSundstin[59abilisatio]nwhereusedCodintostaaandbiliseZitehenkiinceowiczmpressshoiblewthatNavbotierh-
methodsintroducesimilarstabilisingterms.

56Chapter4.Stabilisation
ThebasicideaoftheCBSalgorithmandsimilarschemesistoemploytheHelmholtz
everydecompvectorositionfieldalsocanknownuniquelyasbethedecofundampomesendtalintoatheoremcurl-freeofvvectoectorrcafieldlculusandthestatinggradienthatt
ofascalarfield.Inthepresentcontextthisallowstodecomposetheunknownsolenoidal
velocityfielduaccordingto

u=u∗−ϕ(4.11)
wherethescalarfieldϕiscloselyrelatedtothekinematicpressurep.Employing(4.11)
thesolutionoftheincompressibleNavier-Stokesequationsmightbesplitintotwo
steps;firstsolvingfortheintermediatevelocityfieldu∗whileignoringtheincompressibil-
itycondition.Thisstepcanbedoneinanexplicitorsemi-implicitmanner.Subsequently
thefinalsolutionuisobtainedbyprojectingu∗intothespaceof(discretely)divergence
freefunctions.ThisprojectionmethodgoesbacktotheworksofChorininthecontext
offinitedifferencemethods[40,41].ItisalsodescribedbyTemamin[212].Theprojec-
tionmethodorcloselyrelatedschemesisalsotermedfractionalstepmethod,(operator)
splittingmethodorpressurecorrectionmethod.
TheversionoftheprojectionmethodwhichisusedwithintheCBSalgorithmdescribed
inthebookbyZienkiewiczetal.[237]canbeformulatedsuchthattheLBBcondition
iscircumvented.Thebasicideaoftheformulationshallbesketchedherebymeansofthe
Stokesproblematemporallydiscretisedversionofwhichreadsonafixedgrid
un+1−δ2ν∙ε(un)+δpn+1=rn+1inΩF(4.12)
∙un+1=0inΩF,(4.13)
wheretheviscoustermistreatedexplicitly.Inviewof(4.11)equation(4.12)isrewritten
rmfotheinun+1+δpn+1=δ2ν∙ε(un)+rn+1=u∗(4.14)
whichcanbesolvedexplicitlyfortheintermediatevelocityfieldu∗entirelyignoringthe
pressure.Takingthedivergenceof(4.14)allowstoobtainanellipticequationforthe
dingaerpressure

δΔpn+1=∙u∗(4.15)
∗uthaistevcanenbetuallysolvoedasbtainedsooninasauthirdisstknoepwn.fromThtheustheinitialdesiredsplit,divi.e.ergencefreevelocityfield
un+1=u∗−δpn+1.
Fromrestrictioansspatofiallytheinf-discretisedsupvconditioersionnofhavtheebabeenoveacircumlgovrithmented.itAcanbmateorixbservedrepresenthattattheion
dsrea12)(4.ofMFun+1+δKFun+δGpn+1=fF.(4.16)
Fromthematrixexpressionsof(4.14)and(4.15),i.e.from
MFun+1+δGpn+1=MFu∗andδHpn+1=GTu∗,

.16)(4

4.2.Stabilisation–omnipresentinflowsolvers

75

whereHdenotesthenegativediscreteLaplaceoperatoronthepressure,theintermediate
nodalvelocitiesu∗canbeeliminatedyieldingtheequation
GTun+1+δGTMF−1G−δHpn+1=0.(4.17)
Combining(4.16)and(4.17)showsthattheabovesplitalgorithmcorrespondstosolving
thediscretesystem
MFδGn+1FFn
GTδGTMF−1G−Hpun+1=f−0δ,Ku(4.18)
whichisnonsingularirrespectiveofthechoiceofthediscretevelocityandpressurespace
duetothestrictlypositivedefinitematrixH[237].Thisadditionalfreedomisgained
fromarelaxationoftheincompressibilityconditionbythedifferenceoftwodiscrete
thatrepresenthistarelaxtionsatioofnthedepLaendsplaoncetheanotpimeeratostepronsizetheandpresthussure.theItfoharmstoulabetiondonoted,eshonotwwevoerrk,
tocircumventtheLBBconditionwhenastationarysolutionissought.
Thealgorithmisdoubtlessaninterestingoneinparticularasitallowstosolvean
implicitequationforthepressurewhilethediffusive(andconvective)operatorareinte-
gratedexplicitly.Howevertheproblemsoftheprojectionmethodarehiddenwithinthe
details.Inparticulartheproperchoiceoftheboundaryconditionsfortheintermediate
velocityfieldu∗ischallenging.Theinterestedreadermayconsultalso[107,227]inthis
ext.tconWithintheCBSalgorithmtheprojectionideapresentedherehasbeencombinedwith
theconvectionstabilisationresultingfromthecharacteristicprocedurewhichissketched
insection4.2.1.Adetaileddescriptionofthisalgorithmcanbefoundinthethirdvolume
ofthefiniteelementtrilogyofZienkiewiczetal.[237].

Pressurestabilisationbasedonpolynomialpressureprojection
withPressurpareproticularjectionrespectmethotodsthehalovweestboeenrdervdesignedelocityb-ypressureDohrmanpairsnsucandhasBochlinearevve[19lo,c6ity5]
andpressurespress.urIteworapproks,hoximawevtioner,toaswstabilellaissealllinearequalveloordecitiesrpaaloirsngofvewithlocitpiecyandewisepressconstaure.nt
elemenThetsbaswithicidearespoecftthetothemetinfhod-supcancobenditioexplan(4.4ined).byWhileconsideringnotsatisthefyidngethisficiencycoofnditionssuch
thelinearequalorderinterpolationspacesobeyaweakerinequalityreading
−(p,∙v)
0=v∈supVFh,0|v|1≥γ1p−γ2hp,(4.19)
forTheanlastypter∈mPhon,whetherreighγ1tandhandγ2sideareoftw(4op.19)ositivquaencotifiesnstanthetsLBBindependendeficiencytofohft[19he,e2q09].ual
orderinterpolationspaces.Itcanbeshownthatthistermcanbeboundedfromabove
yb

γ3hp≤p−Πpforallp∈Ph

.20)(4

58

Chapter4.Stabilisation

wherestabiliseγ3isequalaporodesitivreincoterpnstolaanttions,indepΠendendenottesofthetheproelemenjectiontleongptherahtor[19for].aInproojecrdertionto
ontothediscontinuouspolynomialspacewhichisoneorderlowerthantheinterpolation
spaceforvelocityandpressureandwhichisdefinedonthesingleelementaldomainsΩe,
i.e.Πp∈[Pm]=q∈L2(Ω)|q|Ωe∈Pm(Ωe);forallΩe
wherePmdenotesthepolynomialspaceoforderm.
TheL2-projectionisthengivenby
(r,Πp−p)ΩF=0forallr∈[Pm].
Inordertoremovetheinf-supdeficiency(4.19)andhencestabilisethediscreteweak
formbilineartheblempro

1qTCp:=2ν(p−Πp,q−Πq)(4.21)
isaddedtotheweakformoftheincompressibleStokesequations(4.2).Duetothe
discontinuityoftheprojectionthestabilisedmethodfitsintoafiniteelementcodewith
justminormodificationsonelementlevel.Analogouslyto(4.3)thediscretestabilised
dsreablempro

MF+δKFδGufF
δGT−δCp=0,
whichshowsthatthepolynomialpressureprojectionallowstoretainthesymmetryof
theStokesproblem.Furtherthemethoddoesnotrequiretocalculatehigherorder
derivativesandcomesalongwithouttheneedforastabilisationparameter.Itmight,
however,beusefultoscaletheamountofstabilisationintroduced.Suchascalingallows
tousetheamountofstabilisationrequiredtoavoidzeropressuremodeswhileensuring
thatthecontinuityequationisnotrelaxedmorethannecessary.Howevertheproper
amountofstabilisationishardtodetermineanditmightberegardedadisadvantage
ofthemethodthattheadditionaltermsdonotvanishwhentheanalyticalsolutionis
insertedintotheweakform.Unlikeresidualbasedstabilisationmethodsitisthusnot
t.consisten

4.3Residualbasedstabilisationmethods

andResidualaccuracybasedforstatranspbilisatioortnprmethooblemsdshawithvebeenconsideradeveloblepedadvinoectiordern.toSuggcoemstedbinebyconsistencyHughes
wandascodevweloorkpeersd.firstSimilaintrheconmethotextdsofhavadveecbteenive-diffpropusioosednbequyatiothen[1same35]thegroupSUPGinordermethotod
twocircumideasventintheorderLBBtocsotnditioabilisentheintheincoStokmpressesibleproNablemvier-[139].StokAescomequatbinatioionsngoofesthebacsek
toUpwindTezdPeuytarrovet-aGal.[21lerkin9],where/PressuretheapproaStabilisedchwPeasttreorv-medGaSUPGlerkin/).PSPAG(reviewoStreamlinefthe

4.3.Residualbasedstabilisationmethods

95

historyofresidualbasedstabilisationmethodscanbefoundinBrezzietal.[33],thebook
byDoneaandHuerta[69]oralsointhecomprehensivedissertationofWall[227].
AlsotheencyclopediaarticlebyHughesetal.onmultiscaleandstabilisedmethods[143]
shouldbementionedinthiscontext.Anoverviewofstabilisedfiniteelementmethods
fortheadvection-diffusionproblemandthehistorythereofhasrecentlybeengivenby
Francaetal.[94].
AcloserelationshipbetweenresidualbasedstabilisationandastandardGalerkin
approachenrichedwithbubblefunctionshasbeenestablishedbyBrezzietal.[31]and
furtherelaboratedforexamplein[6,34].Basingonthisinsighttheso-calledunusual
stabilisedfiniteelementmethod(USFEM)hasbeenproposedbyFrancaandcoau-
thors[91,92,96].Acomparisonofstabilisationmethodsfortheadvection-diffusion
equationhasbeengivenbyCodinain[47]wherethecloserelationshipofdifferentsta-
bilisationmethodsishighlighted.Itisshownin[47]thatmostclassicalstabilisation
methodscanbewrittenasresidualbasedapproachesanddiffermainlyinthestabilisa-
tionoperatorandthestabilisationparameter.Codinaalsocontributedsignificantlyto
theanalysisofthesemethods,soin[17,48,51,53,55,57].
Residualbasedstabilisationmethodsarecloselyrelatedtoerrorestimationasthe
stabilisationisdesignedsuchthatoptimalconvergenceratescanbeachieved.Thiscon-
nectionoferroranalysisanddesignoffiniteelementmethodsyieldsnumericalschemes
thatrequiremeshdependentterms[116].Abasicintroductiontofiniteelementerror
analysiscanbefoundin[210]whereStewartandHughesalsogivespecialattention
totheadvection-diffusionequation.ArecentnoteofZienkiewiczshedslightonthe
backgroundoferrorestimationinproseratherthanbyformulas[235].Aposteriorierror
estimationinleast-squaresstabilisedfiniteelementschemeshasbeencoveredbyRan-
nacherin[201]andmorerecentlyHaukeetal.consideredaposteriorierrorestimation
inavariationalmultiscaleapproachin[122].
RecentdevelopmentsofresidualbasedstabilisationmethodsincludetheworkbyCo-
dinaandcoworkerswhoproposedtouseorthogonalsubscalesforstabilisation[50,52,53,
58].Whitingetal.usedahierarchicalbasistoformulatestabilisedmethods[232,233].
AcomparisonoftheclassicalresidualbasedSUPG/PSPGmethodandrecentdevelop-
mentsofsymmetricstabilisationtechniqueshasbeenpublishedbyBraacketal.[27].
InthispaperthegeneralisedOseenproblemisanalysedinafixedgridsetting.
Veryfewanalysiscanbefoundofnumericalschemesconsideringstabilisedfiniteele-
mentmethodsondeformingALEdomainsoneofwhichisaveryrecentpaperofBadia
andCodina[5].ApplicationsofastabilisedALEformulationsforfluid-structurein-
teractionorproblemswithmovingboundarieshavebeenreportedbyKhurramand
Masud[154]andTezduyar[217].
Withinthepresentworkresidualbasedstabilisationisemployedandanalysedinthe
contextofALEmethods.Additionalparametersareintroducedinordertoensurethatno
destabilisingeffectresultsfromthecoexistenceofstabilisationtermsanddomainmotion.

4.3.1Virtualbubbles
AderivationofthestabilisingtermsfrombubbleeliminationofaclassicalBubnov-
Galerkinschemeappearstobetheleastmiraculousone.Thisderivationrevealsthat
thestabilisingtermscanbeinterpretedasstaticallycondensedbubbles,i.e.theeffect

60Chapter4.Stabilisation
ofsmallerscales.ThetermvirtualbubbleswasfirstusedbyBaiocchiatal.[6]and
indicatesthatanenrichmentbybubblefunctionswhicharenotspecificlydefinedisused
toachievethedesiredstabilisationproperties.Baiocchietal.showthatundercertain
conditionsthereexistsabubblethestaticcondensationofwhichyieldsthestabilisation
terest.inoftermThederivationofastabilisationmethodfrombubbleenrichmentmakesuseofthe
linearityoftheunderlyingoperatorandisthusperformedontheadvection-diffusion
equation(2.54).Inaccordancewiththescopeofthisworkthisderivationisperformedon
theconvectiveALEformulationofthemodelproblem.Thusdomainmotionisconsidered
whichyieldsanadditionaltermcomparedtofixedgridstabilisationmethods.
Theweakformofthetemporallydiscretisedadvection-diffusiononadeformingdomain
isgivenbyequation(3.47)reading
a(φ,ω)=rn+1,ωΩn+1
wherethebilinearforma(φ,ω)containsaninertiaterm,aconvectivetermandtheviscous
contributionaccordingto
a(φ,ω):=(φ,ω)Ωn+1+δan+1−uG,n+1∙φ,ωΩn+1+δκ(φ,ω)Ωn+1.(4.22)
Thesuperscriptn+1hasbeendroppedontheunknowntransportvariableatthenew
timelevelforbrevity.
Ratherthanapproximatingtheunknownfunctionφandtheweightingfunctionωby
theusualpiecewisepolynomialfunctions(ormappingsthereofintothephysicalspace)
therespectivefunctionsarenowtakenfromthespace
VBh=ω∈H1(Ω0)ω|Ωe,0∈Pk(Ωe,0)⊕B(Ωe,0),(4.23)
definedoverthereferencedomain,i.e.theelementalparameterspaces.VBhisenrichedby
elementwisebubblefunctionsωB∈B(Ωe,0)satisfyinghomogeneousDirichletboundary
conditionsontheelementalboundariesωB=0on∂Ωe,0.Thelinearindependenceofthe
polynomialandthebubblespaceallowsanadditionaldecompositionofthediscretetrial
andweightingfunctionsaccordingto
φh=φk+φBandω=ωk+ωB.(4.24)
Usingthedecomposition(4.24)intheweakform(3.47)allowstosolve(atleastformally)
forthebubbledegreesoffreedom.DuetothebubblepropertyofωBthebubbleequations
decoupleandcanbesolvedindependentlyonthesingleelementswhere
a(φk,ωB)+a(φB,ωB)=rn+1,ωBΩe(4.25)
hastobesatisfiedforallωB∈B.Notingthatafterintegratingtheviscoustermbyparts
δκ(φk,ωB)Ωe=δκ(φk∙n,ωB)∂Ωe−δκ(Δφk,ωB)Ωe
theelementalboundarytermvanishesduetothezeroboundaryconditionofωBonthe
elementboundaries,(4.25)canbereformulatedto
a(φB,ωB)=rn+1−φk−δan+1−uG,n+1∙φk+δκΔφk,ωBΩe.(4.26)


4.3.Residualbasedstabilisationmethods

16

Equation(4.26)isavariationalprobleminthebubblespacewhichhasauniquesolution.
Therighthandsidedrivingthebubbleonasingleelementcaneasilyberecognisedas
thenegativeresidualoftheoriginaladvection-diffusion-reactionproblem
R(φk)=φk+δan+1+uG,n+1∙φk−δκΔφk−rn+1
inthepolynomialspace(3.46).Hencewheneverthepolynomialspacesufficestode-
scribethecorrectsolutionwithinoneelementthecontributionoftherespectivebubble
vanishes.Thispropertyeventuallytransferstotheconsistencyofthestabilisedfinite
elementmethodderived.
Thebubbleinoneparticularelementisthenfoundbysolving(4.26),i.e.
φB=−SBPB(R(φk)),(4.27)
wherePBdenotestheL2-projectionontothebubblespaceBandSBisthesolution
operatorinthebubblespace.Inotherwordsthemagnitudeofthebubblewithinone
elementisfoundbyperformingtwosteps.FirstthebubblewhichisclosesttoR(φk)on
theelementeinaleast-squaressenseissearched.Inasecondstepthenegativeofthis
bubbleistakenastherighthandsideofanadvection-diffusionprobleminthebubble
spaceontheelementandthisproblemissolvedasindicatedbythesolutionoperatorSB.
ThesofoundbubbleφBcannowbeusedinthesecondpartoftheweakformwhichis
weightedbythepolynomialtestfunctionωk.
a(φk,ωk)=rn+1,ωkΩn+1−a(φB,ωk)
Afterintegratingsometermsbypartsandinserting(4.27),thediscreteweakproblem
reads:findφkinPksuchthat
a(φk,ωk)=rn+1,ωkΩn+1+SBPB(R(φk)),Lstab(ωk)forallωk∈Pk(4.28)
issatisfied,wheretheoperatorLstabisgivenby
Lstabωk=1+δ∙uG,n+1ωk−δa−uG,n+1∙ωk−δκΔωk.(4.29)
Inequation(4.28)theweakformulationoftheoriginaladvection-diffusionequationis
augmentedbytheperturbationtermSBPB(R(φk)),Lstab(ωk)providingadditionalsta-
bility.Onafixedgrid,wherethemeshvelocityvanishestheoperatorLstabistheadjoint
oftheoriginaladvection-diffusionoperator.
IthasbeenshownbyBaiocchietal.[6]thatundercertainconditionsthereexistsa
bubblespaceBsuchthat
SBPB(R(φk)),Lstab(ωk)Ωe=τeR(φk),Lstab(ωk)Ωe,(4.30)
i.e.theprojectionofR(φk)ontothebubblespace,thesubsequentsolutionofthebub-
bleproblemontheelementandaprojectionbackintothespaceoftheoperatorLstab
reproducesalinearoperatorforaparticularbubblefunction.
Assumingthatsuchabubblehasbeenchosenandinserting(4.30)backinto(4.28)
yieldsthestabilisedweakproblemtofindφk∈Pksuchthat
a(φk,ωk)−τeR(φk),Lstab(ωk)Ωe=rn+1,ωkΩn+1(4.31)
eissatisfied.Theeffectofthebubbleisnowcondensedintheelementalstabilisation
parameterτewhichwillbediscussedinsection4.3.7.

62

Chapter4.Stabilisation

Remark4.3.1Asimilarapproachalsoworkstoderivestabilisationtermsforthein-
compressibleStokesproblem[7].

Remark4.3.2Thestabilisationoperator(4.29)derivedforthemodelproblemona
deformingdomaincontainsthedivergenceofthemeshvelocitywithinthecoefficientof
thezerothorderterm.Thisadditionaltermisrequiredtopreservethestabilityofthe
methodif∙uG,n+1<0whileitdestabilisesassoonasanelementexpands,i.eif
∙uG,n+1>0.ItwillthusbemodifiedwithinthestabilisationoftheNavier-Stokes
ions.equat

Remark4.3.3Thevirtualbubbleapproachisfrequentlyalsotermedconceptofresidual-
asfreeitsuggbubblesestsasfthatorexaanamplenalyintical[138,fine20s5].caleHoswoevlutioernthe-atlatterleasttermonisthesomewhatelementalmislevleael-dingis
availablewhichhardlyhappenstobethecaseinreal-worldproblems.

4.3.2Alookbeyond–thevariationalmultiscalemethod
Thevirtualbubblederivationofastabilisedfiniteelementmethodrevealsthatresidual
basedstabilisationcanberegardedaspecialapplicationofamuchwiderapproach,the
variationalmultiscalemethod.ThemethodwasproposedbyHughesin1995[133]and
isaverygeneraltooltotreatphysicalproblemsthesolutionofwhichexhibitsamultiscale
iour.vehabThebasicideaofthevariationalmultiscalemethodisasplitofthetrialandweighting
prospacesposedintoinordesubspacesrtoderefesigrnringformtoulatiodifferennstwhscicahlestreat[133,larg1e38].resolvInitiaedllyscalethesinmethothedusualwas
wsolvayedwhilescaleaddituponionalthelargtermseroarenescoofnsinidereterest.dHoaccounwevtinerg,forthethemethoeffedctisofbythefarnotsmall,limitedunre-
there.Itallowsasplitofthesolutionintomorethantwoscales.Forinstanceathree
levelmethodfortheincompressibleNavier-Stokesequationshasbeendevelopedby
Gravemeier[103,105,106].Thereadistinctionismadebetweenlargeresolvedscales,
smallresolvedscalesandunresolvedscales.
Akeyapplicationofthevariationalmultiscalemethodisturbulentflowwhereawide
ratiscalengeofmethoscalesdiswasinvproolvpeod.sedAblarygeeHughddyessimandulacotionworkdeverselo[1p41ed,142from]andthevalsoariattheionawlorkmoul-f
tioGrnalamvemeierultiscale[10m3,et1ho06d]fotrearlatsminarturbulenandtfloturbulenws.Atflostatweofcanbtheeartfoundreviniew[1o04n].theHowvarevia-er,
tothevarstructuriatioalnalprmoblemsultiscaleincludemethodtheisdevnotatelopmenalltoflimitedmethotodsfluidformecincohanics.mpressibleApplicamateri-tions
alsdifferen[178,t177scales]orasalsodonetheinmo[145].dellingAnofostrvucterviewuralofnostatenlinearitiesoftheartcausedconbytributiophenomenanstotheon
theoryofmultiscaleandstabilisedmethodscanbefoundinthespecialissueeditedby
].[90ancaFrinterpIntheretedlighasttofhethevinfluenceariationaoflmunresolvultiscaleedfinemethoscalesdtheuponstabilisattheionresolvedtermscalein(4.31)represencantedbe
hbimpyorthetanptolynomialsimplificatiofinitenshaelemenvebteenspacemadeV.inHoorwdeevretoritarrivhasetoatb(4ek.31)ept.inFirstlymindthethatsttaticwo
eliminationofthebubbletermsreliesontheassumptionthatthefinescalecontributionis

4.3.Residualbasedstabilisationmethods

36

local,i.e.restrictedtoeachelement.Asecondsimplificationislessobvious.Introducing
onestabilisationparameterperelementeffectivelyreducesthefine-scalespacewithin
eachelementtoaone-dimensionalspace.Consequentlythestabilisationtermisfarfrom
representingtheeffectoftheunresolvedscalesexactly.
Neverthelessthederivationofstabilisationtermsfromavariationalmultiscaleapproach
revealsthattheseadditionaltermsareanappropriatemodeloffinescaleeffectsrather
thanapurelynumericalorarbitrarystabilisationtool.

4.3.3Generalisationandrelatives
Thestabilisationmethodresultingfromthevirtualbubbleapproachcanbeextendedand
generalisedtoawiderclassofmethodswhichareusedwithinthepresentwork.Applying
thestabilisationobtainedintheprevioussectiontotheincompressibleflowproblemof
interestyieldsthevariationalproblemoffindingu∈VFh,n+1,p∈Pnh+1suchthat
Bgal({u,p},{v,q})−τMeRM(u,p),LMstab(u,{v,q})Ωe
e+(τCeRC(u),LC(v))Ωe(4.32)
e=rn+1,vΩFn+1+hn+1,vΓFn,N+1forall(v,q)∈(VFh,0,n+1,Pnh+1),
whereecountsallelementsofthetriangulation.TheGalerkinweakformBgalin(4.32)
istheonedefinedin(3.22).Thefirststabilisationtermintheaboveweakformulation
correspondstotheoneobtainedin(4.31)whilethelatteroneintroducedbyFrancaand
Hughes[95]providesadditionalstabilityathighReynoldsnumbers[12].
Theresidualofthemomentumequationinconvectiveformandtheresidualofthe
continuityequationatthenewtimelevelread
RM(u,p)=u+δu−uG,n+1∙u−2ν∙ε(u)+p−rn+1,(4.33)
RC(u)=∙u,(4.34)
whichcanberegardedpurelyspatialdifferentialequations.Ithastoberemarkedthere
thattheunknownvelocityandpressurefieldatthenewtimelevelisdenotedbyuandp
forclarity.Dataandrighthandsidetermsremainlabelledbyatimelevelsuperscript.
Aswellastheresidualsthestabilisationoperatorsarebaseduponthetemporally
discretisedequation.Thegeneralformofthemomentumstabilisationoperatorwhichis
consideredhereisgivenby
LMstab(u,{v,q})=rGηv+δ−u−uG,n+1∙v−α2ν∙ε(v)+βq(4.35)
whileLCstab(v)=LC(v)=∙v(4.36)
isusedonthecontinuityequation.Incontrasttotheconvectivetermin(4.36)abubble
condensationyieldsatermoftheform
−∙u−uG,n+1⊗v

64

Chapter4.Stabilisation

asitcanbeobservedfrom(4.29).Thistermisreformulatedto
−∙u−uG,n+1⊗v=−∙u−uG,n+1v−u−uG,n+1∙v(4.37)
andthefirstcontributionontherighthandsideof(4.37)isaddedtothezerothorder
termofLMstab,i.e.itisincludedintheeffectivereactioncoefficientrG.However,while
+1,nGtheu−divuG,n+1ergence>of0theitmacoynvspectivoilevstabilitelocityyifu−∙uu−uGhelps,n+1to<e0.nsureThstausbilittheyeffeforctive∙
reactioncoefficientrGin(4.35)isdefinedby
rG=min1−δ∙u−uG,n+1,1.(4.38)
TheinfluenceofrGonthestabilityofthestabilisedproblemwillbediscussedagainin
.4.4iontsecTheparametersα∈{−1,0,1}andη∈{1,0}withinthegeneraloperator(4.35)along
withthepreviouslyintroducedβfromequation(3.22)allowtodistinguishavarietyof
closelyrelatedbutdifferentstabilisationmethods.Anoverviewoverthepropertiesof
thedifferentschemesinthecaseofthestationaryStokesproblemhasbeengivenby
Bochevin[8].Allofthepossiblemethodsareconsistentinthesensethatsufficiently
smoothsolutionsofthestrongequations(3.15)and(3.16)satisfythestabiliseddiscrete
)..32(4formDuetothenonlinearityoftheNavier-Stokesequationsthestabilisationopera-
tor(4.35)dependsontheunknownvelocityfielduandhenceaddstothenonlinearityof
thestabilisedequation.Asaresultallcoefficientmatricesofthestabiliseddiscretefluid
equationsdependonthevelocity.
Twotermsofthemomentumstabilisationoperator(4.35)areofparticularimpor-
tance.Thefirstoneisthegradientoftheweightingfunctionforthepressurewhichpro-
videspressurestabilisation.AresidualbasedstabilisationoftheStokesproblemwhere
thestabilisationoperatorisgivenbyqhasbeentermedPSPG,PressureStabilised
Petrov-Galerkinmethod.Thesecondnecessarilyrequiredcontributionofthesta-
bilisationoperatoristheconvectivetermservingtostabiliseconvectioninducedwiggles.
UsingthistermexclusivelytostabiliseadvectiveproblemsyieldstheSUPGmethod[136].
Thezerothordertermgiveninthefirstlineofthestabilisationoperator(4.35)isofpar-
ticularimportanceinthecontextofverysmalltimestepsandwillbeconsideredin
section5.2.Alltheremainingtermsin(4.35)donotdirectlycontributetostabilisethe
problemandcanthusbeplayedaroundwithwhichyieldsanentirefamilyofstabilisa-
tionmethods.Thenotationemployedwithinthisworktodistinguishthesinglefamily
membersissummarisedinTable4.1.
•UsingUSFEMα=1,β=−1andη=1onafixedgridrecoverstheadjointL∗M(u,{v,q})=
L+(u,{v,q})oftheoriginaltemporallydiscretisedNavier-Stokesmomen-
tumoperator(3.15).ThisversionofthemethodwhichisalsotermedUnusualSta-
bilisedFiniteElementMethod(USFEM)wasintroducedbyFrancaandFarhat
in[91]andiscloselyrelatedtobubblefunctions[6,92].Extendingthisnotation
withinthepresentworkallmethodswithα=1withintheoperator(4.35)shallbe
denotedbythesuperscript‘USFEM’.
•Onafixedmeshthenegativeoftheoriginaloperatorcanberecoveredbysetting
η=−1,α=−1andβ=−1whichwouldthusyieldthecorrectGLSversionofthe

4.3.Residualbasedstabilisationmethods65
stabilisation.However,aswillbeshowninsection5.2η=−1furtherincreasesin-
stabilitiesduetodominatingzerothordertermsandthusyieldsanunstablemethod.
NeverthelessmodifiedGLSversionsareofinterestwithη∈{1,0}wherethesuper-
script‘GLS’subsequentlydenotesallstabilisationmethodsofthetype(4.32)with
1.=α−•Astherequiredstabilisationofconvectionandpressureinstabilitiesisindepen-
dentofthezerothorderweightingtermwithinthestabilisationoperatoritappears
interestingtoconsideralsostabilisationbyoperatorswhicharederivedfromthe
correspondingstationaryproblemi.e.η=0.Thesubscript‘+’or‘0’atthestabili-
sationoperatororthestabilisedformreferstothesignofηanddenotesthefullor
stationaryoperator,respectively.

Table4.1:Notationusedtodistinguishdifferentversionsofthestabilisationoperator
notationαηstabilisationoperator
L+USFEM11rGv+δ−u−uG,n+1∙v−2ν∙ε(v)−q
L+GLS−11rGv+δ−u−uG,n+1∙v+2ν∙ε(v)−q
L0USFEM10δ−u−uG,n+1∙v−2ν∙ε(v)−q
L0GLS−10δ−u−uG,n+1∙v+2ν∙ε(v)−q
Remark4.3.4IthasbeenshownbyCodinathatsubscalebasedmethods,whichin
inthethepresensensetconthattextthecorresstabilisatpondsiontoisinthevarianUSFEMtwithvrersioespnectoftotheachangestabilisaoftionvaareriablessupwhenerior
usedforlinearsystemsofconvection-diffusion-reactionequations[49].
Remark4.3.5Aninterestingmethodextendingtheconceptofvirtualbubblestabilisa-
tiomethonhadsforrecenthetlyincobeenmprespropsibleosedNabyvierCodin-Stokaandescoequatworkeionsrst[5he2,y57].notonlyDerivingpayaspstaecialbiliseatd-
tentiontothenonlineartermwhichopensadoortoturbulencebutalsosuggesttotrack
thefinescalesolution,i.e.thebubblepart,intime.Theapproachcanberegardedto
beturnedoverhalfwayfromastabilisedmethodtoafullmultiscaleformulation.On
thethisapricepproaofchadditoffersionalanintereleganalntvwaayriablestoabrepolishresenthetingdeptheendencysubscalesofaastatprtionaeviousrysotimelutionlevonels
thetimestepsizeotherwiseinherentinstabilisedmethodsandoffersthehopeformore
.accuracyalortempRemark4.3.6Analternativetoderivingastabilisedformulationfromeliminationof
virtsatioualnmethobubbleddegrwhereeestheofweighfreedomtingisftounctregionardforittasheaveloPetcitryoisv-v˜Ga=v−lerkinτMetLyMpstaeb(ofu,dis{v,qcreti-})
ratherthanvandthemodifiedpressuretestfunctionreadsq˜=q+τCeLCstab(v).
Remark4.3.7Acommentonequalorderinterpolationforvelocityandpressureinthe
contextofresidualbasedstabilisationappearsappropriatehere.Ithasalreadybeenmen-
tionedthatrelaxationoftheincompressibilityconditionyieldsadvantageswithrespect
totheanpresiterentativcehasopterlutionitisoffurthetherobresultingvioussysthatetmLBofBequatstableions.elemenFromtsthemighinittwialellstilldiscussiorequirenof

66

Chapter4.Stabilisation

stabilisationwheneveritcomestohighlyconvectiveflows.Itisthusregardedaparticu-
laradvantageofresidualbasedstabilisationmethodsthatbothpotentialinstabilitiesare
treatedwithinaunifiedframeworkwhichisfurtherconsistentatleastforhigherorder
ts.elemenReconsideringthederivationofresidualbasedstabilisationfromvirtualbubblefunc-
tionsfurthershowsthattheprincipaldifferenceofthepresentstabilisedfiniteelement
andLBBstableelementsisnotasfundamentalasitmightseematthefirstsight.A
popularwaytostableelementsistoenhanceanunstableonewithinternalvelocityde-
greesoffreedomsuchasthefirstorderMINIelementwhichisenrichedbyacubicbubble.
Staticcondensationofsuchinternaldegreesoffreedomyieldstermsthatmightbetermed
.stabilisation

4.3.4Acommentonconservationinstabilisedproblems

Itappearsinterestingtoreconsidertheissueofconservationoflinearmomentumfor
thestabiliseddiscreteproblem.Herestrictequivalenceofconvectiveanddivergence
formulationislostasthecontinuityequationisrelaxedbystabilisationtermsratherthan
weaklysatisfied.Thusconservationoflinearmomentuminthesensethattheacceleration
ofaportionoffluidisexactlybalancedbytheintegralforcesovertheboundariesis
approximatedratherthanstrictlysatisfied.
Anapproachtoreestablishconservationinthecontextofastabilisedmethodappears
firstinapaperbyTayloretal.[211],canbefoundinthedissertationofWhiting[231]
andiseventuallyexplainedinapaperbyHughesandWells[144].Thereitisshown
howglobalconservationcanapproximatelybepreservedthroughasmallresidual-based
modificationofconventionalstabilisedformulations.Thiscorrectionisbasedonmultiscale
considerationsandthefactthatanelementalfine-scalevelocityisapproximatedbythe
negativecoarse-scaleresidualwithinthiselementweightedbythestabilisationparameter.
Howevertheadditionaltermwhichisintroducedtoreestablishconservationisofadvective
typeandneedstobestabiliseditself.Consequentlyfurtherhigherorderstabilisation
termsentertheformulation.
WithinthepresentworkthemodificationofHughesandWellshasnotbeenused.
Conservationwhilebeingdesirableisnotacrucialmatterforviscousflowsatlowor
moderateReynoldsnumbers.Thusthepriceoffurtherstabilisationtermsincludingan
additionalstabilisationparameterappearstohighforthegainofacloserapproximation
n.atioconservofIthasalsotobementionedthattheseadditionaltermsincludesecondderivativesof
thevelocityanditsweightingfunction.Consistencyofthemethodincludingtheeffectof
largelyrestoredconservationcanthusbeexpectedonlyforhigherorderelements.

4.3.5Stabilisedequalorderelementsanddiscretelydivergence
onsfunctiefre

Priortoproceedingwithamatrixnotationofthestabiliseddiscreteflowproblema
somewhatun-pleasingissueshallbediscussed.Ithasbeenmentionedinchapter2that
aninitialvelocityfieldneedstobedivergencefreetodefineawellposedproblem.Inthe

4.3.Residualbasedstabilisationmethods

76

discretesettingthisdemandsthattheinitialvelocityfieldhastobediscretelydivergence
freeinthesenseimposedbythecorrespondingLagrangemultiplierspace,i.e.thespace
sure.prestheofStabilisationopensthewayforabroadvarietyofvelocityandpressurespacecombi-
nations.Howeveratthesametimethiscomplicatestheimportantissueofcorrectinitial
conditions.Thisquestionalsoariseswheneveravelocityfieldhastobemappedfromone
meshtoanother,i.e.whenremeshingtakesplace.Acloselyrelatedmatteristhedeter-
minationoftheunavoidableinterpolationerroronagivenmeshor-whichisessentially
thesameproblem-thenorminwhichacertaindiscretesolutionisoptimal.
WheneverLBBstableelementsareemployedanappropriatediscreteinitialvelocity
fieldu0,hisobtainedfromaL2-projectionofaknownvelocityfieldu0undertheconstraint
ofincompressibility.ThisconditionisimposedbyaLagrangemultiplierfieldspanned
bythesamefunctionsasthediscretepressure.Thusu0,h∈VEhandλ∈Pharedetermined
fromastationarypointoftheaugmentedfunctional
2Πu0,h,λ=1u0,h−u02−∙u0,h,λ,(4.39)
wherethediscretespaceVEhcontainsallthosefunctionsthatsatisfythenormalcomponent
ofthevelocityboundaryconditionn∙u0,h=n∙u0onΓF,Dinanappropriateweaksense.
Avariationofthefunctional(4.39)yieldsthesaddlepointproblem
u0,h+λ,v−(n∙v,λ)ΓF,N=u0,v(4.40)
−∙u0,h,q=0,
wherev∈V0handq∈PhdenotethevariationofthevelocityfieldandLagrange
multiplier,respectively.
EquivalentlytotheincompressibleNavier-Stokesequationsthemixedproblemde-
finedby(4.40)isunstableforcombinationsofthespacesVEhandPhthatdonotsatisfy
theLBBcondition.Consequently(4.40)cannotbeinvertedforequal-orderinterpolation
ofvelocityandpressure.Acorrectwaytoprojectaknowninitialvelocityfieldtoa
particularmeshthatworksforstabilisedfiniteelementswithequalorderinterpolation
couldnotbefoundintheliterature.Apparentlytheproblemisfrequentlycircumvented
ratherthansolved.Simulationsarestartedfromanincorrect(notdiscretelydivergence
free)initialconditionandthecorrespondingpeakinthepressurerequiredtoadjustthe
incompressibilitywithinthenexttimestepissimplyignored.
Howevertheinterpretationofstabilisedmethodsinthevirtualbubblecontextgives
somehintshowtheproblemshouldbetackled.Ifthevirtualbubblespacewasknown
anadmissibleprojectioncouldbefoundbyprojectingontoavelocityinthediscrete
polynomialspaceenrichedbybubblefunctionsVBh.Suchanapproachcouldbeformulated
asa‘stabilisedprojection’andwouldallowtocircumventtheLBBconditionandto
computeinitialconditionswhicharediscretelydivergencefreeinthesensethatsuitsthe
discretespacesofvelocityandpressurewhichareused.
Unfortunatelytheshapeofthevirtualbubbleremainsuntold.Consequentlythecer-
tainlyunsatisfactorystartfromanillposedproblemhastobechosenwheneveraninitial
velocityfieldisrequired.Mostsimulationshoweverstartfromazerovelocityfieldand
phase.tingstarademand

68Chapter4.Stabilisation
Neverthelesstheissueofcorrectprojectionontoavelocityfieldwhichisdivergence
freeinaproperweaksenseisofhigherinterestasitmightseematthefirstglance.An
L2-projectionlike(4.40)doesnotrequirethattheoriginalvelocityfieldu0issolenoidal.
InordertoabbreviatethestartphaseofafluidorFSIsimulationafunctionu0might
thusbeconstructedthatsatisfiestheDirichletboundaryconditionsandisasclose
aspossibletoanestimatedvelocityfunctionwithinthedomain.Aprojectionofsucha
functionontoadiscretelydivergencefreevectorfieldcouldpossiblyserveasagoodinitial
conditionandispotentiallymuchfasterthanacceleratingtheflowfromzero.

4.3.6Stabilisedformulationinmatrixnotation
Evaluatingtheweakform(4.32)withdiscretefiniteelementspacesVFh,n+1andPnh+1
spannedbynodalbasedpolynomialfunctionsallowstoobtainadiscretematrixrepresen-
tationcorrespondingto(3.23)and(3.24)intheunstabilisedcase.Thestabiliseddiscrete
matrixequationsread
MF(u)u+δKF(u)u+δN(u)+δG(u)p=fbF+fhF(4.41)
GTMu+δGKT(u)u−δCp=fC,(4.42)
whereantopbarindicatesthattherespectivematrixcontainscontributionsemerging
fromstabilisationterms.ThusallmatricesdependuponthestabilisationparametersτMe
andsuchonthefluidviscosity,theflowregimeandthetimestepsize.Additionallythe
stabilisationaddsdifferenttermsonthetwoG-matricesandintroducesanacceleration
dependenttermintothesecondlinealtogetherdestroyingtheformalsymmetryofthe
system.Thestabilisedmassmatrixreads
vTMF(u)u=(u,v)ΩFn+1−ητMe(u,rGv)Ωe+τMeu,δu−uG,n+1∙vΩe
ee+ατMe(u,δ2ν∙ε(v)Ωe.
eThemassmatrixcontributiontothecontinuityequationisgivenby
qTGTMu=−βτMe(u,δq)Ωe,
ewhilethestabilisedviscoustermiscomposedbythesum
vTKF(u)u=(2νε(u),ε(v))ΩFn+1+τCe(∙u,∙v)Ωe
e+ητMe(2ν∙ε(u),rGv)Ωe
e−τMe2ν∙ε(u),δu−uG,n+1∙vΩe
e−ατMe(2ν∙ε(u),δ2ν∙ε(v))Ωe,
e

4.3.Residualbasedstabilisationmethods69
wherethecontinuitystabilisationhasbeenadded.Furthertheconvectivepartisdeter-
omrfminedvTN(u)=(u−uG)∙u,vΩn+1−ητMeu−uG,n+1∙u,rGvΩ
eF
e+τMeu−uG,n+1∙u,δu−uG,n+1∙vΩ
e
e+ατMeu−uG,n+1∙u,δ2ν∙ε(v),
Ωeewhilethediscretegradientoperatorinthestabilisedcasereads
TqTGK(u)u=−β(∙u,q)Ωn+1−βτMeu−uG,n+1∙u,δqΩ
eFe+βτMe(2ν∙ε(u),δq)Ωe.
eThestabiliseddiscretedivergenceoperatorisgivenby
TvG(u)p=−(p,∙v)ΩFn+1−ητMe(p,rGv)Ωe
e+τMep,δ(u−u)∙vΩe+ατMe(p,δ2ν∙ε(v))Ωe,
G,n+1
eeandthestabilisingpressurematrixisobtainedfrom
TqCp=βτMe(p,δq)Ωe.
eSimilartothematricesalsotherighthandsidesin(4.41)and(4.42)containadditional
termsemergingfromstabilisation.Therighthandsidevectorofthemomentumequation
isasumoftwocontributionsthefirstofwhichstemsfrombodyforcesandhistoryterms
alongwiththerespectivestabilisationandreads
vTfb=rn+1,vn+1−ητMern+1,rGv+τMern+1,δ(u−uG,n+1)∙v
ΩFΩeΩe
F
ee+ατMern+1,δ2ν∙ε(v)Ωe,
ewhilethesecondaccountsforexternaltractions
vTfhF=δhn+1,vΓn+1.
N,FAlsothecontinuityequationgetsarighthandsidecontributionfromstabilisationterms
greadinqTfC=−βτMern+1,δq.
Ωee



70Chapter4.Stabilisation
Itremainstoremarkthatallthestabilisationtermshavetobeintegratedovertheactual
elementdomainattimeleveln+1.
codeWithinatthetheInstitutpresenetwoforktheStructuraabolveMectermshanicshaveccbareenatre-imincludingplemenatedinlinearisattotheionreseofarctheh
stabilisationtermsexceptthestabilisationparametersτMeandτCewhichhavenotbeen
prolinearisblemsed.withConsequenalmosttlyconstaquantdraticstabilisaconvtionergenceparameters.withintheflowsolverisobservedfor

4.3.7Stabilisationparameter
Thestabilisationparameterisacrucialingredientoftheentirestabilisedmethod.Conse-
HoquenwetvlyersthetablemetdehodfinitisfrionseqofuenτMtlyecranditicisτCeedfcanorbeforequiringundawhichproblemworkwdepellendenindeptparendenameter.tlyof
problem.thebeeFnorombtained.convergenceAccordingstudiestottheheresprequirecteivdeorlimitderocafsetheτMs,ethasabilisatiotosatnispafyrameterτM,ehas
hh2
τMe=O|u−euG|forRee1andτMe=OνeforRee1,(4.43)
whereReedenotestheelementalReynoldsnumber.Inthelimitofadominatingzeroth
orprodeprortiotermnal,i.eto.ifδ.veryThussmaforllttheimepresstepsentafreormulaconsideredtionwhicthehstaisbabilisatiosedonntermstimehadiscveretotisbede
operatorsτMehastobeoforderoneinδ.
manVyariocaseussexhibitdefinitionsjustinsligparthticulardifferenceforτs.MeRhaecveentbcoeennprotributposeionsdintothethelitepropraturereevwaluathichionin
ofstabilisationparametersinthecontextofUSFEMhavebeenmadebyFrancaand
Valentin[96],BarrenecheaandValentin[7],Codina[47,51,52,58]aswellas
tothegdeterroupmineofstaTezdbilisuyatioarn[1,pa21ram4,ete215rs,2directly16].Infrom[220]elemenTezdtaluyarmatriceandsandOsawveactoprors,poi.e.se
circumventingthedeterminationofelementlengths.
asτFcanurtherbefocommenundintstohenthedissertahistotionryoofftheWalstalbilis[22a7].tionInevpaeryrametcaesertaodausefulywstidelyabilisatknownion
parindicatameterethathasontomesobeyhestheuptolimitmobderaehavteiourmeshaccordistodingrtiotonthe(4.43).influenceNumericaoflthesimpartulaiculationsr
choiceofthestabilisationparameterisaminorissueprovidedthattheparameterexhibits
thecorrectoverallassymptoticsandexcessiveover-stabilisationisavoided.
Theparameterusedpreferablywithinthisworkisacombinationofthestabilisation
parameterderivedfrombubblecondensationbyBarrenecheaandValentinforthe
linearStokesproblemin[7]andbyFrancaandValentinforthereaction-advection-
workdiffusiontoyieldequastationbleinfo[96rm].ulaThetionsparainthemeterALEwhiccahseshasbreadseengeneralisedwithinthepresent

2heτMe=minh2ξ+4δνξ,σ0e
21eme

.44)(4

4.3.Residualbasedstabilisationmethods71
wherethenewelementalparameterσ0eisthelowerboundonthegeneralisedreaction
coefficientoftheconvectiveALEformulation

1for∙u−uG,n+1≤0
σ0e=1−δ/2max∙u−uG,n+1for∙u−uG,n+1>0.(4.45)
Forthefixedgridvariantofthemethodjust∙uhastobeconsideredwhichisclose
tozeroandsoσ0e≈1andisthushardlysignificantin(4.44).Fordeformingmeshes
thedivergenceisdominatedby∙uG,n+1andthusthefirstlinein(4.45)corresponds
tocontractingelementswhilethesecondlinecoversthecaseofexpandingelements.
u−u>0thestabilityofthemethodisaffectedbythemeshmotionandthe
InthisG,ncas+1eσ0ewillneverberelevantin(4.44).Oncontractingelements,where∙
influenceofσ0ehastobeconsidered.StabilityofconvectiveALEformulationswith
solenoidalconvectivevelocityisconditionalupon(3.52)asshowninsection3.5.1.For
thepresentcasethisconditiontransfersto
21−δ1max∙u−uG,n+1>0,(4.46)
i.e.thediscretisationerrorinthedivergenceofthefluidvelocityhastobeconsidered.
From(4.46)itcanbeobservedthatthecoefficientσ0eisstrictlypositive.
Thestabilisationparameter(4.44)obeystheassymptotics(4.43)andisstrictlysmaller
thanone.Thisiscausedbythefactthatdiscretisationintimewasperformedpriorto
discretisationinspaceandthusstabilisationisbaseduponthetimediscretisedresiduals
andstabilisationoperator.
Theparametersξ1andξ2dependontheeffectsdominatingtheflowinaparticular
otaccordingtelemenξ1=max(re,1)ξ2=max(Ree,1),(4.47)
whereredenotestheratiooftheviscous,secondordertermtothezerothorderterm
introducedbythetimeintegration.Theratiosareobtainedfrom
re=4δνandRee=me|u−uG|he.
mehe22ν
TheelementalReynoldsnumberReeisevaluatedattheelementcentre.TheEu-
clidiannormofthevelocity|u−uG|isameasureoftheconvectiveterm,wherethe
parametermecarriestheinfluenceoftheparticulardiscretisation.Itisdefinedby(see
alsoFrancaandValentin[96])

me=min13,Ce(4.48)
whereCeisthelargestconstantsatisfyingtheinverseestimate
Cehe2ΔvΩ2e≤vΩ2eforallv∈VFh,e,(4.49)
hhanddeterminaVF,etiondenotofesmetheandrestrthusictiontheofshatherpdiscreteconstantspCeaceisVFdiscussetothdeinseelemectiontne5..3.1The.exact

72

Chapter4.Stabilisation

AccordingtotheestimatespresentedbyHarariandHughesin[116],me=1/3
andme=1/12isusedforlinearandquadraticelements,respectively.HoweverUSFEM
simulationsrelyonthecorrectparametermeaccordingto(4.48)inordertoguaranteea
stablemethod.ThedeterminationofthecorrectconstantCein(4.49)willbediscussed
subsequentlyinthecontextofdistortedelementsinsection5.3.1.
ForthestabilisationparameterofthecontinuityequationτCethedefinitionpresented
byCodinain[52]isadoptedreading
2τCe=δ4ν2+c2|u−uG|he(4.50)
c1whichhasbeenderivedfromaFourieranalysis.Theconstantsc1andc2satisfyc22≤c1
andareexplainedfurtherin[52].Withinthesimulationspresentedinthisworkc1=2.0
andc2=1.0havebeenused.AnalogouslytoτMetheparameterτCeisevaluatedonceat
theelementcentreandthereforetreatedasanelementalconstant.
Asthestabilisedmethodisdefinedonthetemporallydiscretisedequation(3.15)and
residual(4.33)thetimediscretisationparameterδentersthestabilisationparameter.In
thecaseofastationaryoperator(i.e.η=0)theentirestabilisationoperatorismultiplied
bythefactorδstemmingfromtemporaldiscretisation.Absorbingthisfactorintothe
stabilisationoperatorτMeyieldsastabilisationparameterofthedimensionofatime,the
familiar‘intrinsictimescale’.Howeverasalsothefullstabilisationoperatorcontaininga
zerothordertermshallbeconsideredthedimensionlessstabilisationparameter(4.44)is
.erhepreferred

4.3.8Elementlengthdefinitions
Thecharacteristicelementlengthhehasapotentiallysignificantimpactontheactual
amountofstabilisationemployedasattheviscouslimitthestabilisationparameteris
proportionaltohe2whileitislinearintheelementlengthattheconvectivelimit.Especially
inthecontextofmeshdistortionorhighlyelongatedelementstheelementlengthdefinition
hastobechosencarefully[181].
Variousdefinitionshavebeensuggestedanddiscussedintheliterature.Anoverview
canbefoundin[227]whereitissuggestedtousedifferentelementlengthswithinthe
differentterms.Ageometric‘isotropic’elementlengthdefinition(forexamplethesquare
rootoftheelementalarea)issuggestedwhentheviscoustermsdominate,whileastream-
lengthisusedforconvectiondominatedflows.Inchapter5avarietyofdefinitionsfor
thecharacteristicelementlengthsarecomparedbynumericalinvestigations.Thesingle
earnsdefinitioi.squarerootofelementarea,i.e.he=√Ae
ii.elementlengthinflowdirectionaccordingto[227]evaluatedonceatelementcentre,
iii.approximateelementlengthinflowdirectionasdefinedbyCodinain[58]hk=
h0|u|/|u0|,wherethesubscript0referstothereferenceconfiguration,and
iv.elementlengthforanisotropicmeshesasdefinedbyCodinain[58],wherethe
smallesteigenvalueoftheoperatorBistakenascharacteristicelementlength.B

4.3.Residualbasedstabilisationmethods73
stemsfromthepolardecompositionoftheJacobianJoftheisoparametricmapping
totheelementdomain,i.e.J=BZ,whereBissymmetricandpositive-definite
whileZisorthogonal.
FurtheranimplicitdefinitionoftheelementlengthwhichhasbeensuggestedbyTaylor
etal.[211]shallalsobeconsidered.
v.Employingthecovariantcoordinatesofthemetrictensorgijofthemappingfrom
globalCartesiancoordinatestotheelementparametersthestabilisationparameter
isgivenby

τMe=4+δ2uigijuj+cν2gijgij−21,
wheretheconstantcissetto36andto60forlinearandquadraticelements,
respectively[231,232].Thecorrespondingstabilisationparameterforthecontinuum
equationisgivenby

τCe=(8τMetr(gij))−1.

parAdditiotsofthenallythecompuelementatiotnslengtrephorteddefinitioin[1ns81]invareerstigepateatededbyinsMittectioaln[185.3.1]areTheseconsidereddefinitionsas
earvi.minimalelementlengthgivenby
√2Ahe=he,min=max(hdiaeg)(4.51)

.51)(4

vii.maximalelementlengthdefinedastheedgelengthofasquarewithadiagonalof
max(hdiag)

he=he,max=max√(hdiag)(4.52)
2Whiletheelementlengthdefinitionsi.,iv.andv.tovii.arepurelygeometricalthestream-
length(ii.andiii.)dependsuponthevelocityandhenceaddstotheoverallnonlinearity.
Consequentlytheconvergencerateofthefluiditerationsdecreaseswhenthesestabilisation
parametersareemployed.Incomplexsituationsconvergencemayevenbelost.Inorder
tofixthisproblemlinearisationofthestabilisationparameterwithrespecttothevelocity
couldbeperformed.Numericalobservationsindicatehoweverthatstreamlengthcompu-
tationisnotessentialandgeometricaldefinitionsofthecharacteristicelementlengthmay
workequallywell.

4.3.9Residualbasedstabilisationandlinearelements
ThediscretefiniteelementspacesVFhandPhwithinthestabilisedform(4.32)canbe
ofarbitrarypolynomialorder.Neverthelessthecheapestandmostpopularversionisto
uselinearelements.Howeversuchelementsdonotallowtoproperlyrepresentsecond

74

Chapter4.Stabilisation

isderiveffectivativeselycon‘felt’tainedwitinhinthetheresidual.stabilisaThtionusthetermsresofidualinelaorfethelemenmotsmenreducestumeqtouationwhich
RMlin(u,p)=u+δu−uG,n+1∙u+p−rn+1.
alsoThewincoithinmpletelythestresabilisatolvedionsecoopnderadetorrivasastivoesonareasUSFEMincludedornotGLSonlytypinetheofresidualstabilisatbution
wiseighemplotingytederm(i.e.areαa=n0).additionaNumericallsourceobservofinaatioccnsuracyindicateifbilineathatrorthesetrilinearderivaeletivesmenintstheare
ed.yemploAsdiminishesthesoatintrospatiadulcoedrtempoconsistencyralerrorrefinemenscat.lesWhilewithnotthestabilisaaffectingtionconvpaergencerameterraτtesMeitit
ispresentforparticulardiscretisationswithlinearelements.Thusinsuchsimulations
aportionoftheerrorisobservedwhichscalesdirectlywiththestabilisationparameter.
Consequentlyitisofparticularimportancetoemployastabilisationparameterassmall
aspossiblewhenlinearelementsareusedinconjunctionwithresidualbasedstabilisation
methods.Howeveralowerboundonthestabilisationparameterisrequiredtoprevent
artificialpressuremodesresultingfromtheviolationoftheinf-supcondition.
Improvementswhichreintroducesomeinfluenceoftheviscouspartoftheresidual
hacategveboryeenofs‘vugariatgestedionalbycrimes’.JansenOneetwaal.yinto[1cop47e]witwhilehlinearsuchmoelemendificattsandionsfaresiduallinltobasedthe
thestabilisamaximationlarespmountonsiblyofcouldstabilisabettionocontwithinroltheanelemenstabilisattisionlimitparaedtometer,atoi.e.tlerableoensuramoeunthat.t
Hoyieldweverfrequenthistwmeouldshnotorefinemennlytrequirasseotoondeasfinesteeptheertogralerabledientsamoevounlvteofinstathebilisasolutiotionn.butItathlsous
appearsworthtopaythepriceandgoforhigherorderelementsinstead.Thesuperiority
ofbiquadraticelementsforresidualbasedstabilisedproblemshasalreadybeenmentioned
inanearlypaperonstabilisationbyFrancaetal.[93]wheretheadvection-diffusion
modelproblemaswellastheStokesproblemareconsidered.
150].HigherStaboilisrderedfinitfiniteeeelemenlementtswmethoithadshierforarcflowhicalprbasisoblemshahavevebbeeneendevusedelopbyedJobyhnWhit-[149,
ing[231]andappearasaninterestingchoice.

4.4Stabilityofastabilisedmethodonamovingmesh

ItremainstoshowthatthestabilityoftheconvectiveALEformulationobtainedforan
unstabilisedmodelprobleminsection3.5.1alsoappliesforconsistentlystabilisedNavier-
Stokesequationsonadeformingdomain.StabilityoftheconvectiveALEformulationof
themodelproblemisguaranteedif(3.52)issatisfiedandthusthereactioncoefficientσ0e
isstrictlypositive.
Anupperboundonthetimestepsizetointegratethemodelproblemonadeforming
domainhasbeenobtained.Repeatingtheanalysisofsection3.5.1fortheconvective
ALEformulationoftheincompressibleNavier-Stokesequationsshowsthatwithslight
restrictionscondition(3.52)alsotransferstothepresentcase,wherethecorresponding
condition(4.46)hastobesatisfied.Theanalysispresentedhereisrestrictedtothecon-
vectiveformulationhowevertheresultsobtainedinsection3.5cansimilarlybetransfered
intheothercases.

4.4.Stabilityofastabilisedmethodonamovingmesh75
Withrespecttotheformofthestabilisationoperator(4.35)twodifferentcasesshall
btoeηco=0nsideranded.α=Theβfir=st−v1.ariantAdditioisanallyclasasicalfullGLSUSFEMwherevtheersionparaofmetheterinstabilisat(4.35)ionarsehasetll
beofinteresttheparametersofwhichareη=α=1,andβ=−1.Inparticularthe
analysisofthislattercaseisbasedonthenewlyintroducedparametersrandσwhich
dependuponmeshmotion.InthislattercasetheanalysisfurtherreliesonGthea0edditional
conditclearlyionnotthatsartisfiedGandinσthe0earegeneralconstacase.ntwHoitwhinevertheasthesinglerespeelemenctivets.aThisnalysiscisononditiotvneryis
Nevsharperthitelesscanabeprosuspveforectedthethatgeneralstabilitcaseycisonouldtlonotstbeevenfoundwhenwithinthiscothiswnditionork.isrelaxed.
elemenAsthetalstconstaabilisatntaionndpcaaranthmetuserbeistaevkenaluaouttedofotncheepreerspectivelemenetinittegcanrals.beThetreaanatedlysisasanis
basedonsomebasicinequalitieswhicharealsogiveninappendixA.2.3.

4.4.1CoercivityofGLSstabilisedALEformulation
SGLCoandαercivit=yβof=the−1formrequiresB0that({u,ap}set,{ovf,qw})eighdetingfinedbfuncyt(4ions.32)vwithandqthecanparbeametersfoundηsuc=h0
thattheformisboundedfrombelow.Insertingv=u∈VFh,0,n+1andq=p∈Pnh+1yields
B0GLS({u,p},{u,p})=uΩ2Fn+1+δu−uG,n+1∙u,uΩFn+1+δ2νuΩ2Fn+1
+τMeδu−uG,n+1∙u−δ2ν∙ε(u)+δpΩ2e
e−τMeu,−δu−uG,n+1∙u+δ2ν∙ε(u)−δpΩe
e+τCe∙uΩ2e.(4.53)
eTherearetwotermsin(4.53)whichmayhaveadestabilisingeffectthefirstofwhichis
btheeintconvegraectivtedebtyerpamrtsintheyieldingfirstline.Similarlytowhatwasdonein(3.50)thistermcan
δu−uG,n+1∙u,uΩn+1=−δ1∙u−uG,n+1u,uΩn+1
F2F+δ21n∙u−uG,n+1u,uΓFn+1.(4.54)
Theboundarytermin(4.54)vanishesatDirichletboundarieswheretheweightingfunc-
tiowherenisn∙zero.uIt=dnis∙auppGe,na+1rsfurtherapplies.atAalltoFSIutfloinwbterfacesoundaandrieslocwhealreLang∙urang>enan∙buG,noundar+1iethes
boundarytermaddsstability.Freeinflowboundarieswheretheboundarytermin(4.54)
wsequenouldstlyubttheractbostaundarbilityyteryieldmcaill-npobseedboovunderallfroprombblemselowabyndarzeroe,thi.e.usitnotcanbeapplicable.omitted.Con-
oftheThemefirstshtverelomcitiny(4is.54)negativsubtrae,cti.e.ssifttheabilitryesonpaectivpaerticularelementiselemeconntteifractingthe.divThusergencethe
elementalcontributionoftheconvectivetermcanbeestimatedby
2δu−uG,n+1∙u,uΩe≤δ1∙u−uG,n+1L∞(Ωe)uΩ2e.(4.55)

76Chapter4.Stabilisation
Theremainingtermin(4.53)whichisnotnecessarilypositivecaneasilybeboundedfrom
abovebyemployingtheCauchy-Schwarzinequality(A.15)andtheε-inequality(A.16).
Oneveryelemente

u,−δu−uG,n+1∙u+δ2ν∙ε(u)−δpΩe
≤uΩeδu−uG,n+1∙u−δ2ν∙ε(u)+δpΩe
eε4≤εuΩ2e+1δu−uG,n+1∙u−δ2ν∙ε(u)+δpΩ2(4.56)
holdsforeverystrictlypositiveε.Settingε=1/2andusing(4.55)and(4.56)in(4.53)
yields

1B0GLS({u,p},{u,p})≥σ0e−τMeuΩ2e+δ2νuΩ2n+1
2Fe12
2+τMeδu−uG,n+1∙u−δ2ν∙ε(u)+δpΩe
e2+τCe∙uΩe(4.57)
e

where(4.45)hasbeenemployed.ThuswithτMe≤σ0ethedesiredlowerboundontheGLS
stabilisedformisobtained.Theestimate(4.57)explainstherestrictionofthemomentum
stabilisationparameterwithrespecttoσ0ethathasbeenintroducedin(4.44).However
whilebeingformallynecessarytoobtaintheabovestabilityestimatethisrestrictionis
hardlysignificantinpracticalcomputations.
Asaconsequenceof(4.57)thetimesteprestrictionswithrespecttothemeshvelocity
∙uisreplacedby−∙u−u.
obtGained,n+1insection3.5.1alsoapplyforGt,nhe+1GLSstabilisedNavier-Stokesequationsif

4.4.2CoercivityofUSFEMstabilisedALEformulation

USFEMoperatorandinitialdiscussion

Itturnsouttobemoredifficulttoobtaincoercivityoftheunusualstabilisedform
B+USFEM({u,p},{v,q}).Inthiscasetheinfluenceofanegativedivergenceofthemesh
velocityenteringthestabilisedproblemviatheparameterrGhastobeconsidered.
Usingtheparametersη=α=1andβ=−1andinsertingv=u∈VFh,0,n+1and

4.4.Stabilityofastabilisedmethodonamovingmesh77
q=p∈Pnh+1yields
B+USFEM({u,p},{u,p})=uΩ2Fn+1+δu−uG,n+1∙u,uΩFn+1+δ2νuΩ2Fn+1
e+τMeδu−uG,n+1∙u+δpΩ2
e−τMeδu−uG,n+1∙u+δp,(rG−1)uΩe(4.58)
e−τMerGu2−(1+rG)δ2ν(u,∙ε(u))
e+δ24ν2∙ε(u)2Ωe
+τCe∙uΩ2e,
ewheretheeffectivereactioncoefficientrGaccordingto(4.38)differsfromoneonlywhen
lonegacaltivcoe.ntraHowctionevertaduekestoplace.thegeInnerathelstacasebilitofyraconditiopidlnoss(3of.52e)tlemenhetparavolumemeterrrGGcansatisfiesbe
|rG|≤1.(4.59)
In(4.58)ithasbeenassumedthatrGisconstantoveranelementwhichmeansthat
intheterpelemenretatiotnshrpurpinksosore.conThistractsassumptionuniformlyisifneconeessaryignoresforthethedivanaerlysisgencewhileerrortheofuresultfor
isnotsharpandnumericalexperienceindicatesthatthemethodmaywellbegenerally
stableevenifthiscouldnotbeshownhere.
Consideringtheobviouszeroordercontributionsin(4.58)theswitchintheeffective
reactioncoefficientrGcanbeexplained.FromtheGalerkintermsoneobtainsthe
elementalcontribution
euΩ2−δ∙u−uG,n+1u,u(4.60)
2whereasthestabilisationtermsyieldanelementalcontributionof
−τMeuΩ2e−δ∙u−uG,n+1u,u(4.61)
iftermrGinconttheainsfotheurthderivlineatofive(4of.58t)heandmethshusvelothecitpy.otentialExpression(4destabilising.61)folloeffectwsforfomthethemixedfirst
termshasnotyetbeencondsidered.AsthestabilisationparameterτMemayapproach
∙u−u<0.ConsequentlyrG=1isusedinthiscase.
oneforδ>G,n0+1instabilitywouldbeobtainedinthecaseofexpandingelements,i.e.if
Boundingsingleterms
Inordertoshowthat(4.58)isstrictlypositiveandthustheproblemisstableallthe
mixedsubtraprocted.ductsThuswhicthehspingoletentiaterllymsaresubtractconsideredstabilitinyahakveindtoofbewborost-caundedsescfromenaario.boveand
potenThetiallyproductdestaonbilistheingtinhirdconlinetraofcting(4a.58)reas.vanisOnhesevforeryalleleelemenmenttitstcanhatbeexpabondundedandfritomis

78Chapter4.Stabilisation
abovebymeansoftheCauchy-Schwarzinequality(A.15)andtheε-inequality(A.16)
withε=1/2accordingto
δu−uG,n+1∙u+δp,(rG−1)uΩe
Ωe≤δu−uG,n+1∙u+δp|rG−1|uΩe
22≤1|rG−1|δu−uG,n+1∙u+δpΩ2e+1|rG−1|uΩ2e(4.62)
Themixedproductwithinthefourthlineof(4.58)istreatedinasimilarway
(1+rG)δ2ν(u,∙ε(u))Ωe≤(1+rG)uΩeδ2ν∙ε(u)Ωe
≤ε(1+rG)uΩ2e+(1+rG)δ2ν2∙ε(u)Ω2e,
εforEmploallεying>0.Ffurtherromthe(4.5in9)vitersecanbinequaeolitbseyrv(4ed.49)thaatllothewscotoereforfficienmt1ulate+rGisnevernegative.
22(1+rG)δ2ν(u,∙ε(u))Ωe≤ε(1+rG)u2Ωe+(1+rG)εδCνh2uΩ2e.(4.63)
eeTotermco(4mplete.55)isthetrecalledreatmenandtotheftinheversesingleinequatermslitytheiseusedstimateagaofintotheobtaGainlerkinconvective
22δ24ν2∙ε(u)Ω2e≤δCe4hν2uΩ2e.(4.64)
eEquippedwithboundsforallthetermsthecoefficientsofthesinglenormscanbeevalu-
ated.Heretwocaseshavetobedistinguishedthefirstofwhichconsiderselementswhich
contractwithinthepresentstep,i.e.whichexhibitrG<1.

Coefficientsforcontractingelements
Firstthecoefficientofu−uG,n+1∙u+δpΩ2eshallbelookedat.Using(4.58)
and(4.62)oneobtains
1τMe1−2rG−1L∞(Ωe)=τMeσ0e.(4.65)
SecondthecoefficientoftheL2-normofthevelocityuΩ2eisconsidered.Summarising
therespectivetermsfrom(4.58),(4.55),(4.62)and(4.63)theelementalcoefficient
1−1δ∙u−uG,n+1L∞(Ωe)
2−τMe1−δ∙u−uG,n+1−τMeδ∙u−uG,n+1L∞(Ωe)
21
−τMeε2+δ∙u−uG,n+1(4.66)
isdeterminedwherethefirstlinecontainsGalerkintermswhilethecoefficientsstem-
mingfromstabilisationaregiveninthesubsequentlines.Recallingthathereapositive

4.4.Stabilityofastabilisedmethodonamovingmesh79
divergenceofu−uG,n+1isconsideredandusing(4.45)thecoefficientofuΩ2ecanbe
yieldingulatedreformσ0e−τMeσ0e−2τMeεσ0e.(4.67)
Expression(4.67)isminimisedifthestabilisationparametertakesonitsmaximalvalue.
With

.68)(4

2heτMe≤h2+4δν(4.68)
emeandε=δν/(mehe2)thecoefficientofuΩ2ecanbeboundedby
νδ2τMeσ0e2.(4.69)
hmeeFinallythecoefficientoftheviscoustermhastobeevaluated.From(4.58),(4.63)and
(4.64)coefficientofuΩ2ecanbedeterminedto
222δ2ν−τMe2−δ∙u−uG,n+1δν−τMeδ4ν(4.70)
εCehe2Cehe2
usingfurther(4.45)andε=δν/(mehe2)yieldstheelementalcoefficient
δ22ν
δ2ν1−τMeσ0e−τMeCh2
eewhichisagainminimalforthemaximalpossiblestabilisationparameter(4.68).Thusa
lowerboundfortheviscouscoefficientreads
22ν4δτMemehe2(4.71)

whereσ0e≤1hasbeenused.
Coercivityestimate
Accordingtothedefinitionofthestabilisationoperator(4.35)andthecoefficientrG(4.38)
thedivergenceofthemeshvelocitywithinthezerothordertermofthestabilisation
operatorisomittedforexpandingelements.WithrG=σ0e=1itcanbeobservedthat
thepreviouslyobtainedestimatesalsoholdforexpandingorrigidelements.
Summarisingtheseresultsforallelementsyieldsthelowerboundontheunusualsta-
bilisedoperatoronamovingmeshreading
B+({u,p},{u,p})≥τMeσ0emehe2uΩe+τMemehe2uΩe
USFEM2δν2δ24ν22
eee+τMeσ0eδu−uG,n+1∙u+δpΩ2(4.72)
e+τCe∙uΩ2e.
e

80

Chapter4.Stabilisation

Theestimate(4.72)showsthattheunusualstabilisedversionofthestabilisationmethod
canbegeneralisedinanALEframeworkwhentheadditionalassumptionofuniform
expansionorcontractionismade.Thepresentworkintroducesadistinctionbetween
expandingandcontractingelementsviatheparameterrGin(4.38)whichalsoappliesto
thecloselyrelatedelementalparameterσ0e(4.45).Thisdistinctioneventuallyallowsthat
acoercivityestimateoftheform(4.72)canbefound.
Thepresentanalysisrevealsthattheconditionalstabilityasobtainedinsection3.5.1
whichistypicalforconvectiveALEformulationsalsoappliestotheresidualbasedsta-
bilisedflowformulation.Thusnofurtherrestrictionsduetothemeshmotionhavetobe
dealtwithifthedivergenceerrorofthevelocityisconsideredalongwiththedivergence
ofthemeshvelocity.Thediscussionaboutmaximaltimestepsizesgiveninsection3.5.1
appliestothestabilisedincompressibleNavier-Stokesproblem.

4.5Summary

Stabilisationisanomnipresentmatterinflowsimulationswhicharebasedonthein-
compressibleNavier-Stokesequations.Therearetwoclassicalneedsforstabilisation
thefirstofwhichisdominatingconvectivetransportwhilethelatteroccursiftheLBB
conditionshallbecircumvented.Variousmethodscanbefoundtostabiliseconvection
basedwiggles.Allthoseschemeshaveincommonthattheycanbeinterpretedasakind
ofupwindingorartificialviscosityintroducedbasingonameshdependentparameter.
Pressurestabilisationcanbeachievedbysomekindofrelaxationoftheincompressibility
condition.TheneedforpressurestabilisationcanbeavoidedwhenLBBstableelements
ed.yemploearResidualbasedstabilisationmethodsareameanstodealsimultaneouslywithbothef-
fects.Thestabilisationisconsistentanddoesnotaffectconvergenceratesevenforhigher
orderelements.Furtheritisrobustifthecorrectstabilisationparameterisemployed.
Forversionsofthestabilisationwhichdonotincludeazerothordertermwithinthesta-
bilisationoperatorstabilityalsoondeformingdomainsisstraightforward.Ifsuchaterm
isusedprovablestabilityonmovingdomainsrequiresadistinctionbetweenexpanding
andcontractingelementsaswellastheassumptionofuniformexpansionorcontraction
oftheelements.However,astheestimateisnotsharpthelatterassumptionmaynot
berequiredinpracticalapplications.Inparticularstabilisedformulationswithoutzeroth
ordertermswithinthestabilisationoperatorcanbeusedsafelyondeformingdomains
astheydonotdegenerateaccuracyorstabilitypropertiesoftheoriginalunstabilised
formulationsindependentlyofthemeshmotion.

5erChapt

Stabilisedfiniteelementmethodsat
criticalparameters

Withinthischapterthebehaviourofthestabilisedflowsolveratverysmalltimesteps
isconsideredanddifferentversionsofthestabilisingschemearecompared.Theeffi-
ciencyofanALEflowsolversignificantlyreliesontheperformanceofthemeshmotion
scheme.However,successfulmeshmotionnecessarilyintroducesasignificantamountof
meshdistortion.Asubsequentnumericalinvestigationthusregardstheaccuracyofflow
simulationsobtainedondistortedmeshes.

5.1Introduction

WithrespecttoFSIapplicationstheflowsolverdoesnotonlyneedtobestableona
deformingALEdomainbutalsoneedstoofferreliableresultsatcriticalparameters.
Unfortunatelystabilityinthesensethatthesystemcanbesolveddoesnotguarantee
thatasmoothapproximationisobtained.Whileaflowsimulationmaycopewitha
wigglysolutionwhichisdampedoutafterafewstepsanerroronceintroducedintoan
FSIcomputationhasapotentiallyverysignificantimpactonthecoupleddynamics.In
particularverythinstructuresarehighlysensitivetoslightlydifferingfluidforces.
Twoveryimportantcriticalsituationsshallbeconsideredhere.Thefirstoneregards
veryfinetimestepsandinparticularahightemporalresolutiononameshwithunaltered
spatialmeshsize.Temporalrefinementonagivenmeshmaywellbeofinterestespecially
inthree-dimensionalFSIproblemswheretheoverallproblemsizeisunfavourablylimited
bycomputationalresources.Neverthelessahighlytransientbehaviourofthestructureas
itisobservedinlimitsituationssuchassnap-throughorbucklingmaywelldeservevery
steps.imetfineAnothercrucialissueofALEmethodsistheinherentneedtosolvetheflowequations
onapotentiallyheavilydistortedmesh.Anumberofquestionsariseinthiscontextthat
arehardtoaccessanalytically.Afirstmatteristhegeneralproblemofhowfastasolution
deterioratesatsuccessivemeshdegeneration.Inthepresentcontextofastabilisedflow
formulationitmightfurtherbeofinterestifdifferentversionsofthestabilisationexhibit
poorerorbetterbehaviouronadistortedmesh.Aspecialcaseofmeshdistortionare
highlystretchedelementsasusedtoresolveboundarylayers.Aninvestigationofthe
influenceofthechoiceoftheelementlengthhewithinthestabilisationparameterhasbeen
reportedbyMittalin[181].Someofthetestcasesreportedthereshallberepeatedhere
alongwithanumberofadditionaltestsconcerningthebehaviourofquadraticelements
inthesamesituation.

81

82Chapter5.Criticalparameters
5.2Residualbasedstabilisationatsmalltimesteps
Stabilisedflowformulationshaveshowntoworkwellforawidevarietyofapplications.
Howeversituationsareencounteredwhereunphysicaloscillationsareobtainedwithina
fullystabilisedformulation.Inparticularwigglesemergingwhenthetimestepisreduced
onagivenspatialmeshhavebeenreported.
Suchobservationscontradicttheintuitiveexpectationthattemporalrefinementim-
provconsistenesortatmetholeastd.doMoesrenotimpoharmrtanatlyntheseumericaloscillaapprotionsximahatioventowhicbehisunderstoobtaodinedpropwitherlya
inthecontextoffluid-structureinteractionproblemswhereincriticalsituationsthestruc-
turesmallmatimeyexspteerpsienceandatendshighlytotberansientincrediblybucklingseornsitivesnap-throwithrespughectphasetothewhicflohwdictabehatesvviour.ery

Motivatingexample
AdrivtenypicacalvitysituaprotionblemawheretearlyoscillatiotimeasnsodepictedccurinwhenFigsmaurell5t.1.imeThestepsfigureareshouswsedisthetcahevitlidy
problemdiscretisedby20×20linearelementsinspaceandBDF2intimeafter20time
stepsofΔt=0.003andakinematicviscosityofν=0.001.Thehorizontaltopvelocity
haswhenbteenheinctimereasestepdorlinearlythevisincostimeityareuptoureducedx=0.furt02.her.TheAreguladepictedromescshillaaligtionnedsgetwithworsethe
flowdirectionandperpendiculartothefloweasestheoccurrenceofsuchinstabilities.

10.8t0.6heigh0.40.20-0.01-0.0050velo0.cit005yux0.010.0150.02
Figure5.1:Velocityoscillationsondrivencavityproblem

refinemenSimilarttowiggwlesardsarethefrebqouenundarytlyodobseresvednotinthesufficetovicinitreysoolvfetheno-slipwabllgoundaradienriest.ifthemesh

Potentialinstabilitiesatsmalltimestepsduetotransientstabilisationterms
Bochevetal.investigatepotentiallydestabilisingeffectsatsmalltimestepscausedbythe
thesestabilisaauthortionsartermsguethatthemselves.residualAnalysingbasedsttheabilisatiotransienntmethoStokdseshavproebeblemenindev[21,elop2ed2,f2o4]r
stationaryoperatorswhichdonotcontainaninertiaterm.Applyingthesamekindof
stabilisationtotransientproblemsrequirestoincludethemasstermintotheresidual
termswithinonthethestmassabilisatmaiontrixtermareinnecessordearrytotheyretainmaycodensisstatenbilisecy.theWhilediscretetheseequastabilisattions.ionIn

5.2.Residualbasedstabilisationatsmalltimesteps

38

particularithasbeenshownthattheuseofapurelyspatialmomentumstabilisation
parameterwhichforthepresentformulationtakestheform
τMe=c1he2(5.1)
δwithapositivestabilityconstantc1yieldsastabilityconditionlinkingthespatialand
temporaldiscretisation.Thestabilityconditioninducedbythetransienttermsthenreads
δ2he<c1(5.2)
limitingthemeshsizedependingonthetimestep.Violating(5.2)whileastationary
stabilisationparameteroftheform(5.1)isusedyieldspressureinstabilities.However
employingastabilisationparameterwhichatthelimitofsmalltimestepsisoftheorder
ofonecircumventsaconditionlike(5.2).Theonlyproblemremainingisthefactthat
atthelimitofδ→0forfixedelementsizehethestabilisationtermsdiminishwithδ
eventuallyrecoveringtheunstabilisedGalerkinformulation.
Remark5.2.1Thespatialparameterin(5.1)hastobenormalisedbythetimeconstant
δasinthestapresebntformulationoftheresidualbasedstabilisationmethodthestabilisation
operatorLMisdefinedoneorderhigherinδthaninthecitedreferences.
IthasfurtherbeenshownbyBochevetal.in[23]thatSUPGmethodsusedforthe
transientadvection-diffusionproblemdonotintroduceaninstabilitycausedbythemass
termsemergingfromstabilisation.
Withinthepresentworkatransientdefinitionstabilisationparameterisemployedwhile
are-enteringofunstablepressuremodeshasnotbeenobservedinpracticalcomputations.
Thedestabilisingeffectofthetransientstabilisationtermsisthereforeregardedaminor
.issueHoweverthereisafurtheraspectthatshouldbementionedhere.Thechoiceofa
transientstabilisationparameterhastheconsequencethatstationarysolutionsdepend
uponδandthusuponthetimestepsizeassoonasδdropsbelowacertainlimit.Conse-
quentlycareshouldbetakentoemploytimestepsizeslargeenoughwhensteadystates
areapproached.Thisgoesalongwithefficiencyconsiderationswhichalsostronglysuggest
largertimestepsizeswhensteadystatesolutionsshallbeapproximated.
Amoreelegantandmoreexpensivewaytoavoidthisdependencyistoutilisetime
dependentsubscales,i.e.takeintoaccountthetemporalchangeofthebubblepartofthe
solution.ThisapproachwhichwassuggestedbyCodinain[52]andanalysedbyCodina
etal.in[57]requiresthattheactualfinescalesolutionofthevelocityiscomputedfroma
nonlinearequationandstoredattheelementalGausspoints.Thismethodwhichishalf
wayturnedoverfromastabilisedformulationtoapropermulti-scaleanalysisallowsto
recoverstabilityirrespectiveofthetimestepsizeandstationarysolutionsindependentof
step.timetheHoweverinthecontextoffluid-structureinteractionsteadystatesolutionsarehardly
everconsideredanyway.

Instabilitiesatsmalltimestepsduetodominatingzerothorderterms

vAelomocitreywseviterehinprotheblemtempisordueallytothediscretiseddominamoncemenofthetumzeroequatiothonrder(3.1term5)orofathelsoinunk(no3.18wn)

84

paCritical5.Chapterrameters

whichoccursatsmalltimesteps.Atverysmalltimesteps(3.15)canberegardeda
singularperturbedproblemwherethespatialderivativesofuhavelosttheirinfluence
comparedtotheundifferentiatedzerothorderterm.Solutionsofsingularperturbed
problemsfrequentlyexhibitcharacteristicsteepboundarylayersthepropernumerical
approximationofwhichposesachallenge.Indeedthevelocityoscillationsobservedin
figure5.1areinducedbyadominatingzerothorderterm.
Inordertore-establishcontroloverthederivativesinthecontextofdominatingzeroth
ordertermsHarariandHughes[117]proposedtouseaGLSapproachinconjunction
withagradientleastsquares(GGLS)stabilisation.Whileachievingthedesiredstabilisa-
tioneffectsthismethodintroducesnotonlyafurtherstabilisationparameterbutalsoa
leastsquareformofthegradientofthedifferentialequationofconcernandthusturnsout
rathercumbersomeandexpensive.Itfurtherrequiresatleastthirdorderfiniteelement
spacesifallthestabilisationshallbeconsistent.
Amoreappealingwaytodealwithdominatingzerothordertermsarisesinthecontext
ofconsistentlystabilisedmethods.Aderivationofstabilisationtermsfrombubblecon-
densationasperformedinsection4.3.1yieldsazerothordertermwithinthestabilisation
operatorassoonasanundifferentiatedtermiscontainedwithintheoriginaldifferential
equation.TheapproachwhichsuggeststosubtractGalerkinliketermshasbeenpro-
posedasamethodapplicablefordominatingzerothordertermproblemsfromitsearly
dayson(seee.g.Francaetal.in[91,92]).
Similarmethodscontainingazerothordertermwithinthestabilisationoperator,i.e.
η=1havebeenusedforproblemswithzerothordertermsasinBarrenecheaand
Valentin[7],Codina[51]andFrancaandValentin[96].In[113,114]Harari
suggeststoemploystabilisationoftheunusualtypetodampoutoscillationsinadiscrete
versionofamodifiedHelmholtzproblemwhichemergesfromtimediscretisationofa
transientdiffusionequation.InaveryrecentcontributiontotheissuebyHarariand
Hauke[115]theadvection-diffusion-reactionproblemisconsideredandstabilisationof
thetime-discretisedproblemissuggested.
ThemethodhasbeenextendedbyHauke[118]todealwithnegativereactionterms.
HaukeandDoweidarareconcernedwiththeadvection-diffusion-reactionequationalso
in[119,120,121]whereanexplicittreatmentofthediffusionandadvectionisproposed
alongwithanimplicittreatmentofthelumpedmassandzerothorderterms.
However,carehastobetakenifthiskindofstabilisationisappliedalongwithlinear
elements.Inthiscaseanimmensestabilisationeffectisobservedwhichoriginatesfrom
theinabilityoflinearelementstoproperlyapproximatesecondderivativesandpotentially
significantlychangestheproblemtobesolved.

5.2.1Dominatingzerothorderterms
Modelproblemsingulardiffusion
Thesingulaefferdctiffofusioanzerotprohblemorder(2.59term)whicdohminatcanbingetintheerdiffepretedrenastialthetermstempisoramoldelleddiscretisatbyionthe
ofthetransientdiffusionproblem
∂∂tφ−κΔφ=0,(5.3)

.3)(5

5.2.Residualbasedstabilisationatsmalltimesteps85
δiswhereproptheorrightionatlhandtothesidetimein(2.59step.)resultsfromtemporaldiscretisationandtheparameter
ampleInousedrderbtoyhigHarahlighrittheandnareptureortedofinthe[11ins3]tisabilityemplotheyed.one-Ondimeadonsiomainnalnoftheumericallengex-th
xL=the0isunknoassumedwnfieldwhileφiszerointeterpretmperaedaturestisheprestempcriberaedtureatxof=aL.rod.ThePinitiaerfectltempinsulatioeraturnate
0iswithaconstantimetasttφep.sizeFoollofΔwingt=1/Hara100riκ.[11The3]stohelutionTRoisftheusedfullyfordistimecretiseddiscretisatioproblenmaaloftengr
onetimestepobtainedfromaspatialdiscretisationwithfivelinearelementsisdepicted
inandfigtheure5.2solutioanlongofthewithtODEhe(2ana.59)lyticalosobtainedlutionafterofthediscretisatiopartialnindifferentime.tialequation(5.3)
1.2discretesolutionwith5linearelements
1

0.80.60φφ/0.40.2

discretesolutionwith5linearelements

analyticalsolutionofPDE

spatiasemidislscroetelutionequatofion

00x/L0.20.40.60.81
Figure5.2:Modelproblem,solutionsatdifferentlevelsofdiscretisation
Thedifferencebetweenthetwosmoothcurvesinthediagram5.2canbeidentifiedas
thetemporaldiscretisationerrorwithinthefirsttimestep.Asthetrapezoidalruleisa
convergentscheme,thiserrordecreasesifthetemporaldiscretisationisrefined.Forthe
giventimestepsize,however,thespatialdiscretisationof(2.59)hastoconvergetothe
solutionanalyticinspaceanddiscreteintime,givenbythedottedline.Inthiscontext
theovershotapparentinthefiniteelementsolutioncanbeinterpretedasanecessary
resultofthemethod’sattempttoreproduceboth,thefunctionvaluesanditsspatial
derivativesonacoarsemesh.
Interestinglytheslopeoftheanalyticalsolutionatx=Lisalmostmatchedbythe
fullydiscreteapproximation,i.e.inthepresentmodelproblemthespatialandtemporal
discretisationerroralmostcancelwithrespecttothisderivative.Howeverthiseffectdoes
notappeartobeareliablebaseforanumericalmethod.
Iftheovershotinthediscretesolutionshallberemovedbyastabilisationthereremains
thequestionfortheidealstabilisedsolutiononthegivenmesh.Anevidentgoalof
stabilisationistoreproducenodalexactvalues.Thismeans,however,correctingthe
functionvalueswhileincreasingtheerrorsinthederivatives.Forthepresentexamplethis
hastheconsequenceofanunderestimatedthermalflowatx=L.Fortheincompressible

86

rameterspaCritical5.Chapter

Navier-Stokesproblemtherespectivestabilisationyieldsanon-oscillatorysolutionthat
underestimatestheviscoustractions.

Remark5.2.2Theabovediscussionrevealsthatitismisleadingtoblamesmalltime
stepsProblemsfortwhehicohdoccurringnotexhibitoscillatiostensepgrawhichdienartse(forinitiatedexamplebytheunresolvKim-edMoinspatiaprolgrblem)adiencants.
besolvedproperlybyalmostarbitrarilysmallΔt.

Remark5.2.3Acrudewaytogetridofthiskindofoscillationsismassmatrixlumping.
Howeverthismeansincreasingsmoothnessnotaccuracyandsacrificestheconsistencyof
thescheme.Furthermoreamassmatrixemergingfromastabilisedformulationofthe
incompressibleNavier-Stokesequationscontainstermsintroducedbythestabilisation
whicharenotsymmetric.Thereisnostraightforwardlumpingtechniqueapplicablein
es.scahsuc

Detectionofoscillationsinducedbyzerothordertermsbyreactionforces

Theandawigreglesinducedinducbyedbyunresdoolvminatedgraingdienzertsothinoparderrticulartermsgtradienypicallytspoccerpurenclodicseulatorbtootheundariesflow
direction.Theflowgradientperpendiculartoano-slipboundaryisintimatelyrelatedto
therespectivereactionshearforce.Thusshearboundaryforcesareusedasameasureof
theaccuracyofaschemeinthesmalltimestepregime.
firstThereoneisaretotwopdetermineossibilitiesandintotegracomputetethesucresphsecheativrereactioshearnfostressesrcesinaloangFEtheMbosettingundary.A.
Howeverthesestressesdependuponvelocityderivativesandhavetobeexpectedone
orderlowerinspatialaccuracythanthevelocityapproximation.
AmuchmoreelegantwaytocomputereactionforcesinaFEMcontextistodeter-
mineconsistentnodalreactionforcesbysummarisingtheelementalnodeforcesalongthe
bandoundafitryin.toCoansnodeistentbasnoeddaldatfoarcesstructure.sharetheTheoserderfoofrcesaccaurreacyalsooftusedheforprimaFSIryvcoauplingriables
purposeasdescribedinchapter6.
Obviouslybothpossibilitiesofdeterminingtheshearforceshavetoconvergetothe
samevalueatmeshrefinement.Withinthepresentworkthereactionforcehasbeen
withincomputaedbonoundabothrywlaayyserforandsowmehicprhareoblems.dueThetoresdominaultsobttingzainederothoindicaterdertethatrmsosgocillaalotionsng
withthetwotypesofreactionforcesconvergingfrombothsides.Insmoothproblems
convergencefromthesamesidewasobserved.Unfortunatelythisobservationcouldnot
yetbeconfirmedbyanytheoreticalanalysis.Clarifyingthisissuecouldhelptoestablish
anindicatorforlocalmeshrefinementatunresolvedgradients.Whilethedifference
betindicatweenorinittegrappatedearstostressebespaandrticularconslyistentappronodepriatfoercestoscaignnalgenerallyinsufficienbetregalocalrdedanresoluterroionr
inspacecomparedtothetimestep.Incontrasttogeneralrelationshipsofspatialand
temporalresolutionsuchas(5.2)anindicatorbasedonintegratedstressesandconsistent
nomeshedesfoinrcesthewvoiculdinityofconsidersteepthegrphadienysicalts.Tcohiswnfiguraouldbtion,ei.e.usefulasindicatesmallttheimeneedstepsforcanfinerbe
employedsafelyonsmoothproblems.

5.2.Residualbasedstabilisationatsmalltimesteps87
5.2.2Acloserlookatstabilisationfordominatingzerothorder
srmteStabilisationparametersatsmalltimesteps
Inthelimitofδ→0andhe=const,i.e.whenthetimestepisrefinedbutthespatial
meshsizeisretained,theratiooftheviscousforcestotheinertiaforcesrepresentedbythe
parameterredecreaseswithδ.Hencethefirstswitchparameterin(4.47)isξ1=1.Thus
dependingontheelementalReynoldsnumberReetherearetwocasestodistinguishfor
thestabilisationparameterτMegiveninequation(4.44)atthesmalltimesteplimit.If
significantconvectionisencounteredthestabilisationparameterisgivenby
heδ→lim0τMe=he+2δ|u−uG|=1(5.4)
whileintheothercasethelimit
2heδ→lim0τMe=he2+4mδeν=1(5.5)
isobtained.Thecontinuitystabilisationparameter(4.50)vanishesatthesmalltimestep
limitlimδ→0τCe=0.

Effectofzerothorderterminthestabilisationoperator
Thefollowingconsiderationdoesnotclaimtobearigorousmathematicalanalysisbut
ratheraroughcalculationgivinganimpressionofthebasiceffectoftheinclusionofa
zerothordertermwithinthestabilisationoperator.
Inordertounderstandtheeffectofazerothodertermwithinthestabilisationoperator,
i.e.η=1in(4.35),thestabilisedformulation(4.32)iswritteninitspossiblyshortest
way.Thecontinuitystabilisationisignoredforbrevityofthepresentdiscussionandthe
fulladjointstabilisationoperatorL+USFEMisappliedsuchthatthestabilisedformofthe
momentumbalancereads
(RM(u,p),v)ΩF−RM(u,p),τMeL+USFEM(v)Ωe=0(5.6)
ewherethefirstinnerproduct(afterintegrationbyparts)yieldstheGalerkinterms.
Thestabilisationmightnowbesplitupintotheadditionalreactionstabilisationterm
andonetermwhichusesthestationarystabilisationoperatoraccordingto
(RM(u,p),v)ΩF−RM(u,p),τMe1−δ∙u−uG,n+1vΩe

e−RM(u,p),τMeL0USFEM(v)Ωe=0(5.7)
eThestabilisationtermsduetothezerothorderweightingfunctionwithinthestabilisation
operatorareGalerkinliketermsscaledbythestabilisationparameterτMeandaterm
dependingonthemeshmotion.Itshallbeassumedforthemomentthatthesetermsare
constantandtheequalforallelements.Iffurtherhigherorderelements(k≥2)areused

88

paCritical5.Chapterrameters

thesuchGathatalllerkintermslikeoftermsthefrresidualomtheRMfirst(u,plin)ecaofnb(5er.7)eprcanesenbteedcollewithinctedtheandaftstabilisederdivfoisiorm,n
bytheterm1−τMe1+δ∙uG,n+1thestabilisedproblem(5.7)isequivalentto
(RM(u,p),v)Ωe−RM(u,p),τMmoedL0USFEM(v)Ωe=0(5.8)
eewiththemodifiedelementalstabilisationparameter
τeMdmoτMe=1−τMe(1−δ∙(u−uG,n+1)).(5.9)
Equation(5.8)showsthattheadditionalreactionstabilisationwhichsubtractsGalerkin
anlikenewtermscanstabilisingbereeducedffect,toitaratmoherdifiedaffecstatsthebilisatiorationparabetwemeter.enGaThuslerkinitdoesandnotistantrobilisingduce
terms.stabilisaIntionpaparticularameterritcaincreanbesesforobserveddecreasingfrom(5.9)timeandsteps.(5.4)Inorthe(5.5)limitingthatcatheseofmoδ→dified0
Gathewealerkinkformterms(5.8b)ecoismesdoneglminatedigible.bytheConsequenstabilisatiotlythentermssolutioniswhilegovtheernedbinfluencyneofumericalthe
dampingratherthanthephysicsifsmalltimestepsalongwithreactionstabilisationare
ed.yemploisaFctivromated(5.9)itwithincanafnurtherelemenbetisobservexcedlusivthaelyttgohevearneddditionabyltahemountimetofstepstasizebilisaratiotherntwhichanh
loofcalunresosolvlutionedprgraopdienerties.tsbutThauslsaonwherincreaseetheofvelocitstabilisatyfieionldbdoehaesvnotessonlymooothlyccur.inThreusgionsthe
methodexhibitsastrongtendencytoheavilychangethephysicsoftheproblemathand.
theThesecondsituatioderivnaistiveveesnwincorseludedwheninthelinearresidualelementsRaM(reu,pused).whicWhilhearetheunafirstbletotermofrepresenthet
stabilisedequation(5.7),theGalerkinterm,hasbeenintegratedbypartstheadditional
‘reactionstabilisation’termhasnot.Thusintheformulation(5.7)allGalerkinterms
arereducedexcepttheviscousonewhichcontainssecondderivativeswithintheoriginal
operator.SothestabilisationincreasestheviscositywithintheGalerkintermson
enantirelyelemenduetaltotbaseheandinconthsusistencyyieldsofmuclinearhmorelemenesmotsoandthdosolutioesnotns.givHoewaeverreliablethiseffebasectfoisr
astabilisationscheme!

5.2.3Coercivityanalysisofadvection-diffusion-reactionmodel
mbleproTofullyunderstandthestabilisingeffectofUSFEMandtheGLSmethodinthecon-
textoflargezerothordertermstheadvection-diffusion-reactionequationinafixedgrid
formulationisemployed.Theresultsobtainedonthescalaradvection-diffusion-reaction
equationalsoholdforthevectorvaluedcasewhichiscoveredin[88].Numericalobserva-
tionsalsoshowthatthetheoreticalpredictionsarewelltransferabletotheincompressible
Navier-Stokesequations.
Anaccuratecoercivityanalysisofanumericalmethodallowstogetanideaofthenorm
inwhichtheapproximationtothesolutioniscontrolled.Thusunphysicaloscillationsare
indicatedbyimproperbalancesofdifferenttermswithinacoercivityestimate.

5.2.Residualbasedstabilisationatsmalltimesteps89
TheconvectiveALEformulationofthescalaradvection-diffusion-reactionproblem
(2.58)isconsideredreading
Lφ=rinΩwithφ=0on∂Ω,(5.10)
wherehomogeneousboundaryconditionsareassumedforclarity.ThelinearoperatorL
ybdefinedisLφ=φ+δa∙φ−δκΔφ(5.11)
whileanalogoustotheNavier-Stokescasethestabilisationoperator
Lstabω=ηω−δa∙ω−αδκΔω(5.12)
used.isFromthecoercivityanalysisofaGalerkinweakformoftheproblem(5.10)whichwas
givenin(4.1)notonlyapotentialinstabilityatdominatingconvectioncanbeobserved
but2alsothattheweakunstabilisedformofthe1modelproblemBmodisdominatedbyan
L-normatsmallδwhilethecontrolintheHsenseisgraduallylost(4.1).Asaresult
oscillationsinducedbyhighgradientsmayspoilthesolution.
ThestabilisationparameterτeistheoneproposedbyFrancaandValentin[96].
2hτe=he2ξ(re)+e2δκξ(Pee)withξ(x)=max(x,1)and(5.13)
mdmoeremod=2δκ,Pee=me|a|he.
mehe22κ
eIfssmallmaallndtimeτecanstepstaarkeetcowonsidediffererdentin(5.1forms3)tdepheendingratioofonvtishecoausamounndtofinertiaadvectforcesion.rmodWithis
onlyminoradvectionthestabilisationparameteraccountsforthezerothordertermonly
andwillbetermedbyτe0whileτaaincludesadvection.Theparametersread
τe0=1+2δκ2andτea=1+δ|a|.(5.14)
−1−1
mehehe
Theelementalparametermeistheonedefinedin(4.48).
USFEM–thefulloperator
Theunusuallystabilisedbilinearformoftheadvection-diffusion-reactionproblemisgiven
ybBmodUSFEM,+(φ,ω)=Bmod(φ,ω)−Lφ,τeLmodUSFEM,+ωΩe,(5.15)
ewithη=1andα=1
LmodUSFEM,+ω=ω−δa∙ω−δκΔω(5.16)
accordingto(5.12)whichistheadjointoperatorofL.

90Chapter5.Criticalparameters
Insertingtheweightingfunctionω=φinto(5.15)yields
BmodUSFEM,+(φ,φ)=φ2+δκφ2(5.17)
−τeφ2−δ2a∙φ2+δ2κ2Δφ2−2(δκΔφ,φ)Ωe.
eInordertoshowthat(5.17)isstrictlypositiveandtoassesstheratioofthedifferent
termsthesinglecontributionstotheoperatorhavetobeevaluated.
From(5.17)itcanbeobservedthatanumberoftermsintroducedwithinthestabilisa-
tionactuallysubtractstability.Employingtheinverseinequality(4.49)thedestabilising
Laplaceantermcanbeboundedby
122ΔφΩe≤Cehe2φΩe.(5.18)
Forpurelylinearelementsthedestabilisingsecondderivativesvanish.Inallothercases
theestimateissharpinthesensethatwithineveryelementafunctionφ∈Vehalongwith
aconstantCecanbefoundsuchthattheinverseinequality(4.49)isanequality.
Apotentiallydestabilisingeffectisalsoduetothelasttermin(5.17).Bymeansof
theCauchy-Schwarzinequality(A.15)theabsolutevalueofthemixedtermcanbe
boundedfromaboveby
(δκΔφ,φ)Ωe≤CcsδκΔφΩeφΩe
≤√CcsδκφΩeφΩe,(5.19)
hCeewheretheinverseinequality(4.49)hasbeenusedagain.TheconstantCcs<1isintro-
ducedinto(5.19)inordertoincludethesharpcase.TheCauchy-Schwarzinequality
issharpifthetermsconsideredarelinearlydependent.Thediscretefiniteelementspace
ononeelementVehisspannedbypolynomialfunctionsanddoesnotcontaintheeigen-
functionoftheLaplaceoperator.ThustheconstantCcsisstrictlysmallerthanone.
Applyingfurthertheε-inequality(A.16)yields
222Cehe4ε
(δκΔφ,φ)Ωe≤εCcsδ2κφΩ2e+1φΩ2eforallε>0,(5.20)
wheretheparameterεallowstoshiftweightbetweentheL2normandthegradientnorm.
Withtheaimofobtainingacorrectestimateoftheratioofbothtermsεcannotbechosen
arbitrarily.Theε-inequalityhasasharpcasedependingonεwhichforthepresentcase
requirestheconstanttotakeonthevalue
√CeheφΩe
εs=2CcsδκφΩe.(5.21)
Employingfurtherasecondinverseestimatereading
C0ehe2φΩ2e≤φΩ2eforallφ∈Veh(5.22)
allowstoboundthesharpvalueforεfrombelowby
√2εs≥CeC0ehe.(5.23)
κδC2cs

5.2.Residualbasedstabilisationatsmalltimesteps
3)(5.2ofInstead

19

2ε=Cehe(5.24)
2Ccs2δκ
shallbeusedwithinthepresentderivation.Inserting(5.18)and(5.20)backintothe
coercivitycondition(5.17)yields
τBmodUSFEM,+(φ,φ)≥1−τe−2eεφ2
e+δκ−τeδ2κ2−τe2εCcs2δ2κ2φ2+τδ2a∙φ2
eCehe2Cehe2Ωe
Insertingalsotheparameterεaccordingto(5.24)andthestabilisationparameteraccord-
ingto(5.14)allowstoobtainthesinglecoefficients.Inthecaseofminoradvectionthis
yields2εhe2Cehe2meCe
1−τe0−τe01=τe0τe0−1−1−Ccs2δκ=τe0δκ2−Ccs2
forthecoefficientofφ2and
22222
CeheCeheCehehemeCe
δκ−τe0δκ2−τe02εCcsδ2κ=δκτe0τe0−1−δκ2−1=δ2κτe0κ22−1
forthecoefficientofφ2.
Intheothercasewhentheadvectiondominatesthediffusivetermsimilarresultsare
ngreadidecomput

21−τea−τea1=δτea|a|−C2csκ
222222εheheCe
δκ−τeaδκ2−τea2εCcsδ2κ=δ2κτea|a|−2κ.
CeheCeheheheCe
Duetothedefinitionoftheelementalparameterme(4.49)bothexpressionsarepositive.
RecallingCcs<1theorem1canbeconcludedimmediately.
Theorem1Ifthestabilisedbilinearform(5.15)employstheadjointstabilisationoper-
atorLmodUSFEM,+alongwiththestabilisationparameter(5.13)andthemethodisformulated
onafixedgridthanthereexistsapositiveelementbasedconstantαesuchthat
BmodUSFEM,+(φ,φ)≥τeαeδφ2+δ2κφ2+δ2a∙φ2e,(5.25)
ewheretheconstantisgivenby
αe=.(5.26)
hκe2m2e−C1eifPee≤1
|hae|−he2κCeifPee>1

92

rameterspaCritical5.Chapter

Acomparisonof(5.25)withthecoercivityoftheGalerkinweakform(4.1)showsthe
wellknowneffectofadvectionstabilisation.Itcanalsobeobservedthattheorderinδ
ofthezerothordertermhassuccessfullybeenreduced.Howevertheestimate(5.25)also
indicatesthattheorderofthederivativetermshasbeendecreasedaswell.Thustheratio
ofthesetermsremainsunchanged.Neverthelesstheorderreductionhasanimportant
impactasitensuresthattheadvectionstabilisationtermremainscomparabletotheother
termswhichguaranteesstablecomputationsintheadvectiondominatedregimeevenat
smalltimestepsizes.
Howeverthewarningshouldberepeated;themethod’ssuccessisbasedonanamount
ofstabilisationincreasingasthetimestepdecreases.Thustheriskofover-stabilisation
isratherpresentwheneverη=1isused.
Remark5.2.4Theeffectoflinearelementsontheweakformandstabilitymightbeof
interest.Equation(5.17)revealsthatthestabilisationalsointroducespotentialsources
ofinstabilities.Theonlytermwithinthecurlybracketsin(5.17)whichdefinitelyadds
stabilityistheadvectiveterm.Approximating(5.15)withlinearelementsincapableof
reproducingsecondderivativeseffectivelyyields
BmodUSFEM,+,lin(φ,φ)=φ2+δκφ2−τeφ2−a∙φ2e
ewithoutthetermscontainingLaplaceoperators.Thustwopotentialsourcesofinsta-
bilitiesareexcludedwhilefullconsistencyhasbeensacrificed.Clearlythereducedform
BmodUSFEM,+,liniscoerciveforallτe<1whichisalwayssatisfied.Thustheinconsistencyof
linearelementssignificantlyaddsstabilityandyieldsthesmoothingeffectof‘reaction
tion’.stabilisaThepricewhichispayedintermsofaccuracyhoweverispotentiallyimmense.From
(5.13)itcan−1beobservedthatforsmallδthestabilisationparametercanbeexpressedas
τe=(1+)withasmallparameter.Assuminglinearelementswhicharezeroforall
secondderivativesthefullyUSFEMstabilisedbilinearformyields
BmodUSFEM.+,lin(φ,ω)=τe{(φ,ω)+δ(a∙φ,ω)+(1+)δκ(φ,ω)
e+δ2(a∙φ,a∙ω)+δ(φ,a∙ω)e
wherethefirstlinecontainstheGalerkinliketermsandthesecondlinetheremaining
advectionstabilisation.Thustheeffectiveresidualisalinearcombinationofthesums
φ+δa∙φ−(1+)δκΔφ−randφ+δa∙φ−r
wherethecontributionofeithersumdependsontheamountofadvectioninvolved.For
smallδandthussmallthepotentiallossofaccuracyofthisapproachisobvious.

USFEM–thereducedoperator
ThestabilityresultofthefullystabilisedUSFEMgivenintheorem1motivatestheuse
ofthesimplerstabilisationoperator
LmodUSFEM,0ω=−δa∙ωφ−δκΔω(5.27)

.27)(5

5.2.Residualbasedstabilisationatsmalltimesteps93
whichisthestabilisationoperatorfortherespectivestationaryproblem.Bymeans
of(5.27)thestabilisedbilinearform
BmodUSFEM,0(φ,ω)=Bmod(φ,ω)−Lφ,τeLmodUSFEM,0ωe(5.28)
eisdefined.Performingananalysisverysimilartothepreviouscaseyieldsthefollowing
coercivityresult.

Theorem2Ifthereducedstabilisationoperator(5.27)alongwiththeprUSFEMeviouslydefined
stabilisationparameter(5.13)areemployedthestabilisedbilinearformBmod,0(5.28)is
defineddescribingafixedmeshproblemwhichsatisfiesthecoercivitycondition
BmodUSFEM,0(φ,φ)≥τeαeφ2+δκφ2+2a∙φ2e,(5.29)
δ2
ewherethepositiveelementalconstantisgivenby
21+m2eδhκe2−Ceδhκe2ifPee≤1
=α.30)(5.e21+δh|ae|−Ceδhκe2ifPee>1
Acomparisonoftheestimate(5.29)andthecoercivityresultobtainedfortheoperator
thatincludesazerothorde2rterm(5.25)clearlyshowsthattheeffectofthistermis
areductionofboth,theLtermandthederivative.Whenthereducedstabilisation
opexperaectedtorinisethempslomayelld,timei.e.thesteplimit.estimate(5.29)holds,advectionoscillationshavetobe

GLSmethod–thefulloperator
ItisfurtherofinteresttoinvestigatealsothestabilityoftheoriginalGLSmethodapplied
totheadvection-diffusion-reactionequationintheregimeofdominatingreactioncaused
byverysmalltimestepsizes.IncontrasttoUSFEM,thestabilityoftheGLSmethod
doesnotdependuponthecorrectlydeterminedparametermeandthelattermethodis
thusbetterapplicabletomovingmeshschemes.TheGLSstabilisedformreads
BmoGLdS,−(φ,ω)=Bmod(φ,ω)+(Lφ,τeLω)e.(5.31)
ei.e.theoriginaloperatorisemployedforstabilisationpurpose.Formalcoercivityofthis
least-squareformiseasilyshownas
BmoGLdS,−(φ,φ)=δκφ2+φ2+τeLφe2.(5.32)
eHoweveracloseanalysisrevealsthatthisprocedureyieldsaratherunfavourablestability
normwhichgivesrisetoevenhigheroscillationwhenthetimestepisveryfine.Thusthe
methodisnotofpracticalinterest.

5.ChapterrameterspaCritical

94Chapter5.Criticalparameters
GLS–themodifiedoperator
ToretaintheadvectionstabilityintheGLScontext
LmoGLdS,+ω=ω−δa∙ω+δκΔω(5.33)
isusedtoserveasstabilisationoperator.Theoperatoriscalledmodifiedasincomparison
tothethezerotneghativordereofthetermorisiginarevlersedoperatohere.rwhicThishopyieldserataorcacompnnotletebeGLSderivemethodfrdomtheasigmnultiof
scaleanalysisnordoesityieldaproperleast-squaresform.Ithasratherbeencombined
byusingthestationaryleast-squaresoperatorandsubtractingazerothordertermin
ordertobalancetheequationsinthedominatingreactionscheme.Bymeansof(5.33)
thestabilisedform
BGLmodS,+(φ,ω)=Bmod(φ,ω)−Lφ,τeLmoGLdS,+ωe(5.34)
eisdefined.Inthiscaseacoercivityanalysisyieldsthefollowingestimate.
Theorem3ThestabilisedbilinearformBmoGLdS,+(φ,ω)definedin(5.34)satisfiestheco-
onditionccivityerBmoGLdS,+(φ,φ)≥τeαeδφ2+δ2κφ2+δ2δκC+ehCeh2a∙φ2,(5.35)
2
eeeewherethepositiveelementalconstantisgivenby
m2eκh2eifPee≤1
αe=|hae|ifPee>1.(5.36)

.36)(5

Thestabilisedmethodgivenby(5.34)sharesthestabilitypropertiesofthefullUSFEM
formulation(5.15)whilebeinginsensitivewithrespecttotheelementalparameterme.
Theroleofmewillbeaddressedagaininthesubsection5.3.1.
ByRemmeaarkns5of.2.5theAinversegeneralcoestimatemment(5.2c2)oitncerningisinevtheeryabcaosveepestimossibleatestohatrasnsftoerbestabilitadded.y
fromtheL2normtothederivative.Thisreflectsthefactthatonafiniteelement,
i.e.inConsequenafixedtlyndiscreteumericalosspacecillathetionswillfunctionnecesandsaitsrilyaffderivectatbivoesthartheecloselyfunctiontieitdselftoasgether.well
asitsderivatives.
Withintheaboveestimatescarehasbeentakentoproperlysorttheinfluenceofthe
particulartermsinordertoobtainanindicationofthenorminwhichthesolutionactually
iscontrolled.Andindeedtheresultscanbeconfirmednumericallyasshownbythe
subsequentexample.

5.2.4Exampleatsmalltimesteps
isThecolidnsidereddrivenascavitydepictedisinusedfigurtoen5.3.umericallyThexverify-directiontheaofnalyttheictoalpveloresults.cityAis2-Dprescribcavitedy

5.2.Residualbasedstabilisationatsmalltimesteps

59

ux(x,t)=(sin(4πt−π/2)+1)−2x2+2xcm/s
fort<0.25slater:topflowconstantintime
u=0timestepsize:1.0×10−3s
0=uyfluid:mercurymeshes:
νF=1.14×10−3cm2/s20×20linearelements
xu=0ρF=13.57904g/cm310×10quadraticelements
domain:1.0cm×1.0cmreference:50×50quadraticelements
Figure5.3:Problemsettingofdrivencavityexample

parabolicinspaceandincreaseswithasinfunctionintime.Theparabolicprofileinspace
isservgivesentoinavfigoidureco5.3.rnersingularitiesinthepressureresult.Materialanddiscretisationdata
Differentmeshesandstabilisationmethodsareappliedtocomparetheresults.The
bmenehats.viouroNine-nofdedbilinearquadrelemenatictsiselemeninvtsestigareatedaoppliednaonarelativdiselycretisacoarsetionomeshf10of×2100×eleme20nele-ts
whichyieldsthesamenumberofunknownscomparedtothelinearversion.Toobtain
accurateresultsforreferencepurposethedomainisdiscretisedby50×50nine-noded
quacompadraticrison.elemeThisnts.forceTheistanghoenrizontiatlaltoreathectionboforceundaryaatndthethtouspofunaffectedthecabvityytheispruseedssure.for
Itqualitonlyyofdeptheendsapprouponximatthevioneloofcitythederivevloacittivyesgraanddienistth∂uusx/∂any.appTheropriatotaltehorizmeasureontalforfotherce
isobtainedfromthesumoftherespectiveconsistentnodalforces.
Theproblemischosensuchthataninitiallyverysteepgradientoccursatthetopof
thecavitywhichcannotberesolvedbythecoarsemeshesused.Afterthecirculatingflow
insidesignificathentlycavitdecyrdeveaseselopandedacoandrsethemeshestopveloshouldcitybdoeesablenottooffeacceleratraereasofurthernablethesolutiogradienn.t
isThedepictedreinsultingfigurereactio5.4.nFfororcemothebtaineddiagrabmyinthefigdiffeureren5.4ta)meswhicheshandhasbestabilisaencotionmputedmethowithds
linearelementsitcanbeobservedthatinthiscasefullUSFEMstabilisationincludingthe
zerothorderweightingtermwithinthestabilisationoperatoryieldsdevastatinglywrong
largeresults.inaThemplitudeheavilyevenoveforretstimaimestedlatertviscosithany0.125yieldssawhenreaactsionmootherforceprowhicfilehisismdevucheloptoedo
inUSFEMtheostariginalbilisproationblem.canAnalsobeimmenseobservaccuraedincyfigprureoblem5.5,ofwhiclinearhshowelementhattsirusedrespwectivithefullof
thelowviscosityofmercuryaStokesflowtypeofbehaviourisobtained.
IncontrastlinearelementsstabilisedbymeansofastationaryUSFEMoperatorbehave
inmuchfigurebetter5.4.withWhileresptheyecttaolsothefailhortoizonproptalerlytopaforpprocexasitimatecanbtheeevobservolutioednofrofmthethebdiagroundaamry
beforcereasoduringnablythewellinitresiallyolvved.eryNevtransienerthelesstperioadvdecttheyiontcatcypehupwigglasessocaonnnotasbtheerphemoysicsvedcanon
thecoarsemeshasfigure5.6shows.
Thediagraminfigure5.4b)displaystheresultsobtainedonameshofbiquadratic
wellelemenonts.theIncoathisrsecasmesebhocothnfirmingstabilisedthavtersionstheextrappraoximastabilisatetionthereaeffectctionoffothercezerothreasonaorderbly

96Chapter5.Criticalparameters
termelementswithintoathepprostaximatebilisatiosecondnoperderivatorativisesto.alarImpressionsgeextenofttheduevetolotcitheyandinabiliprestyofsurelinefieldar
obtainedonthequadraticmesharegiveninfigures5.7and5.8.
b)a)00-0.1xF-0.5referencesolution-0.2fullUSFEMstabilisation
-1USFEMstabilisationwith-0.3
-1.5reducedoperator-0.4-0.5USFwithEMreducestabildisopaterionator
cfortoptalizonhore-0.9
-0.6-2-2.5fullUSFEMstabilisation-0.7-0.8referencesolution
-300.10.20.30.4time0.5ins0.60.70.80.91-100.10.20.30.40.50.60.70.80.91
sintimeFigure5.4:Temporalevolutionofthehorizontaltopforcein10−5Nobtaineda)fromthe
linearandb)thequadraticelementscomparedtothereferencesolution
compInothenentatfigurestime5.5tb)=to0.515.8b)s.tThehecvoelolourcityscaalerrofowsratherepholottedrizontaonlvtheelocithoyrizonissettaltove−loc0.it05y
to+0.65cm/sinallimages.Thereferencesolutionobtainedonthefinequadraticmesh
isgiveninfigure5.9b).Thecorrespondingfigures5.5a)to5.9a)depictthehorizontal
velocityprofileatt=0.15salongaverticalcutinthecentreofthecorrespondingvelocity
e.figurAllresultsobtainedonthecoarsemeshesdisplayvelocityandpressureoscillations
exceptthefullyUSFEMstabilisedlinearelementsinfigure5.5whichgivesaperfectly
smoothbutinaccurateresultasacomparisonoffigure5.9reveals.Thisconfirmsthat
Athemacompajorrisonpartofoffigurreactioe5n.7b)staandbilisatio5.8b)nishowdueevertorevtheealsthatinconsistencywhileothfelineafullyrelemenstabilisedts.
version5.7exhibitsoscillationsonlyinthevicinityofthetoplayer,thestationarystabil-
isationoperatoryieldsadditionalconvectiontypeinstabilitieswhichspreadalloverthe
in.domaThisinterpretationoftheresultsisalsosupportedbythepressurefieldsdepictedin
figure5.5c)to5.8c)and5.9c)wherethesolutioninthealmoststationaryregimeat
t=1.0sisshown.ThefullyUSFEMstabilisedlinearelementsyieldasmoothbutwrong
tharesultnthewhichreferencegoesalosolungtion.withAllaproteshersurepresfieldsurefieldsexhibitingusefivtheetosametencolotimesurscahigherlebetvwalueseen
−0.1and+0.24Pa.
latInionsallinfigtheurespreitssurecanfurfieldtherwhicbehoinbsetrhevedcasethaotfvfullyelocitystabilisedoscillatiobiqunsgadraoaticlongelewithmentsoscareil-
restrictedtothevicinityofthesteeptopgradient.Meshrefinementinthisareayields
correctresultsatlowestpossiblecostwhenfullyUSFEMstabilisedquadraticelements
ed.yemploearThecoarsegridresultsdepictedinfigure5.5c)to5.8c)highlightalsoadifficultyto
accuratelyrepresentthecontinuityequation(2.35)whichcanbeobservedfromnonclosing
yieldsstreamlaines.significanThistisarelasidexateionffectofoftheacolargntineuitstayequabilisatiotion.nparaThusmetteherτuseMkrcloseshouldtobeonewwhicarnedh

c)

79

5.2.Residualbasedstabilisationatsmalltimesteps97
c)b)a)100.20.40.6
s/cm0.80.60.40.20Figure5.5:SolutionobtainedwithlinearelementsandfullUSFEMoperator:a)profile
ofonhohorizonrizontaltalvveloelocitcityyinatatv=ert0.ic15alsccut)tstreahroughmlinestheoncentrepressureatta=tt0.=151s.0b)svelocityarrows
c)b)a)1-0.200.20.40.6
s/cm0.80.60.40.20Figure5.6:SolutionobtainedwithlinearelementsandreducedUSFEMoperator:a)
arprorofilewsofonhohorizorizonntatallvveeloloccitityyinatatv=ert0.ic15alsccut)thrstreaoughmlinestheoncentrepressureatta=tt0.=151s.0b)svelocity

andremindedtobetterbalancetemporalandspatialdiscretisationwhenthestabilisation
parameterapproachesonetooclosely.
Fromthestreamlinesinfigure5.7c)itcanfurtherbeobservedthatthefullUSFEM
stabilisationalsoinconjunctionwithquadraticelementsdoesindeedintroducemore
numericaldissipationasthevortexcentreismovedtowardsthecentreofthecavity
comparedtothereferencesolution5.9c).
WhenGLStyperatherthanUSFEMstabilisation(i.e.α=−1in(4.35))isemployed
theresultsareidenticaltotheUSFEMonesexceptthatthecomputationislesssensitive
totheparameterme.Onthefinemeshusedforreferenceallstabilisationmethodsyield
almostidenticalresults.

98

b)

rameterspaCritical5.Chapterc)

c)b)a)1-0.200.20.40.6
s/cm0.80.60.40.20Figure5.7:SolutionobtainedwithquadraticelementsandfullUSFEMoperator:a)
profileofhorizontalvelocityinaverticalcutthroughthecentreatt=0.15sb)velocity
arrowsonhorizontalvelocityatt=0.15sc)streamlinesonpressureatt=1.0s
00.2a)0.40.6b)c)
1s/cm0.80.60.40.20Figure5.8:SolutionobtainedwithquadraticelementsandreducedUSFEMoperator:a)
profileofhorizontalvelocityinaverticalcutthroughthecentreatt=0.15sb)velocity
arrowsonhorizontalvelocityatt=0.15sc)streamlinesonpressureatt=1.0s
00.2a)0.40.6b)c)
1s/cm0.80.60.40.20Figure5.9:Solutionobtainedonfinereferencemesh:a)profileofhorizontalvelocity
inaverticalcutthroughthecentreatt=0.15sb)horizontalvelocityatt=0.15sc)
streamlinesonpressureatt=1.0s

5.3.Residualbasedstabilisationondistortedmeshes99
5.3Residualbasedstabilisationondistortedmeshes
parALEedtomethofixeddsgridinevitablyflowcomputacomealongtionsthiswithintmeshroducesmotiontwoaandtdditiohusnalmeshchallenges.distortion.TheCom-first
oneproisblemisrelatedcaustoedbtheythemeshneedmototionsolveitselftheandflowhasequatiobeennscoonvperedoteinnctiallyhapterhea3.vilyAdistosecondrted
s.mesheThesensitivityofstabilisedmethodswithrespecttodistortedandunfavourablyshaped
testselemenwhictsisharehardalsotorepaccessortedanainlytica[89]allyre.desThigusnednumetoricacompaltesretthecasespaerforeusedrmanceofhere.differenTheset
elemenstabilisedtsonfotrmypicaulaltiondistorviattionhemoedes.lementTalhepashaperameterofthemee.lemenIntspartenicuterslartheforrethesidualUSFEMbased
versionofthestabilisationmethodtheexactdeterminationofthisparameteriscrucial
forthestabilityofthemethod.

5.3.1SensitivityofUSFEMstabilisationvariant
Influenceoftheparameterme
NumericalinvestigationsrevealthattheUSFEMimplementationwithα=1andafixed
isparcausedameterbmyeasgetstunstaabilisatioblenattermaacertainndcalevneloeasilyfmeshbeshodistortwnation.theThemoobsedelrvproedbleminstabilitofany
unusualstabiliseddiffusionproblemgivenby:findφ∈Vhsuchthat
(κφ,ω)−(κΔφ,τeκΔω)Ωe=(f,ω)−(f,τeκΔω)Ωeforallω∈V0h.
ee.37)(5Theproblem(5.37)representsanunusualstabilisedformulationoftheLaplaceequation
−theκΔproφp=erf.cIthoicedefinesoftheasymmestabilisatrictionbilineparaarmetformerBτe(.φ,ωThe)colowestercivityeigenofvawhicluehofdepthisendsbilineauponr
formisboundedfrombelowby
B(φ,φ)=κφ2−τeκ2ΔφΩ2e.
eEmployingtheinverseinequality(4.49)oneobtains
2B(φ,φ)≥κ−Cτeeκh2φΩ2e
eenconditiotheyieldingτeκ<Cehe2.(5.38)
Intheviscouslimitofthestationarycasethestabilisationparametergivenbydefini-
tion(5.13)reducesto
τe=he2me.
κ4

010

Critical5.Chapterrameterspa

Togetherwith(5.38)thisyieldsthestabilitycondition
me<4Ce.(5.39)
Thecondition(5.39)issatisfiedformeaccordingto(4.48).Thusforaproperuseof
theUSFEMondistortedmeshesitprovesessentialtoemploythecorrectdefinitionof
theparametermeratherthanworkingwithafixedconstantasitiscommonlydonefor
convenienceandefficiency.
Theparametermeentersthestabilisationparameterinordertoaccountfortheratio
ofthefirstandsecondderivativesinthefiniteelementspace.Elementdistortionhighly
influencesthisratioandthusaccuratedeterminationofmeisrequiredifUSFEMshallbe
usedonALEmeshes.
ThispotentialinstabilitydisappearsassoonasanSUPG(α=0)orGLS(α=−1)
methodisusedratherthantheunusualstabilisedversion.Inthesecasesthedestabilising
termiseithernotpresent(SUPG)orisaddedratherthansubtractedandhenceadds
tothestability(GLS).Theunconditionalstabilityofthestabilisedmethodobtained
bysettingα=1,denotedGLSwithinthepresentwork,hasalreadybeennoticedby
DouglasandWangin1989[71]andisofparticularinterestinthecontextofdeforming
s.meshe

Determinationoftheparameterme
ThedeterminationoftheexactconstantCesatisfyingtheinverseestimate(4.49)fora
particularelementeyieldsaneigenvalueproblem.Estimatesoftheconstantforavariety
ofdifferentelementshavebeenprovidedbyHarariandHughesin[116].Thenorms
ofthegradientandtheLaplaceandefinesymmetricpositivedefinite(orsemi-definite)
elementaloperatormatricesby
vΩ2e=veTKeveandΔvΩ2e=veTLeve,
wherevedenotesthenodaldegreesoffreedomwithrespecttothenodalbaseofVFh,e,
thetrialspaceoftheelementewhichhastobefreeofrigidbodymodes.Themaximal
eigenvalueλe,1ofthegeneralisedeigenvalueproblem
det(Le−λe,iKe)=0(5.40)
yieldsthesharpconstantCe=(λe,1he2)−1.
AsthecorrectvalueoftheparameterCecanbecrucialinUSFEMthesolutionofone
eigenvalueproblemperelementisrequired.InthecaseofamovingALEmeshastable
USFEMimplementationisobtainedonlyifCeisdeterminedforeveryelementaftereach
meshmotionstepwhichmakesthealgorithmratherexpensiveandshouldbeavoided.
AlternativelyGLSorSUPGversionsofthestabilisationmightbeusedwhicharestable
irrespectiveoftheparameterme.

5.3.2Kim-Moinflow
ANatvwier-Stoko-dimensionaeslequatflowionsproisblemknoforwnwhicdateshanfroamnalyKimticalandsolutionMoinofinthe1985inco[155mpres].sibleThe

5.3.Residualbasedstabilisationondistortedmeshes

110

problemisstatedbyprescribingtheexactvelocityaccordingto(5.41)and(5.42)along
theboundaryofthefluiddomain.
TheKim-MoinmodelproblemissolvedontheunitsquareΩF=[0,1]×[0,1]and
comparedtoitsanalyticalsolutionwhichisgivenby
ux(x,y,t)=−cos(aπx)sin(aπy)e−2a2π2tν(5.41)
uy(x,y,t)=sin(aπx)cos(aπy)e−2a2π2tν(5.42)
p(x,y,t)=−41(cos(2aπx)+cos(2aπy))e−4a2π2tν.(5.43)
Animpressionoftheflowfieldisgiveninfigure5.10wherevelocityvectorsonthecor-
respondingpressurefieldaredepicted.Accordingto(5.41)-(5.43)velocityandpressure
fieldremaininspaceanddecreasemonolithiclyintime.

Figure5.10:Kim-Moinflow-velocityvectorsonpressurefield

Thesolution(5.41)-(5.43)isaproductofaspatialandatemporalfunctionwhich
toallothewsdecaeasyyofscalingtheosofthelution.errorTheinerrororderstreporemoortedvesubthesequentemptorlyalareerroabrsodecaluteyspatwhicialhiserrdueors
ybdefined

erru:=uh−u0e2a2π2tν
errp:=ph−p0e4a2π2tν.(5.44)
Characteristicmaximalsolutionvaluesofthespatialvelocityandpressurefieldsare
uL∞(ΩF)=1.0andpL∞(ΩF)=0.5suchthat(5.44)mightalsobereadasrelative
rs.erroThecalculationshavebeenperformedwiththeparametera=2.0andakinematic
viscosityofν=0.01.BDF2hasbeenusedfortemporaldiscretisationwithatimestep
sizeofΔt=0.01.Theerrorsafter100timestepsarecomparedindifferentcases.

Remark5.3.1ThedeterminationofthespatialerroroftheKim-Moinproblemopens
theveryinterestingquestionoftheunavoidablediscretisationerroronagivenmesh.
AnapproximationofthiserrorcouldbeobtainedifanadmissibleL2-projectiontothe

102Chapter5.Criticalparameters
discretelydivergencefreespacewouldhavebeenfound.Thisissueturnsouttobedifficult
inthecontextofstabilisedmethodsasdiscussedinsection4.3.5.Howevertheproblem
goesevenfurther.Thenumericalsolutiononagivenmeshwillnotbeclosesttothe
analyticalsolutioninanL2-sense.Sodoesanormexistinwhichtheerrorisminimised
andwhatdoesitlooklike?Thedeveloperofnumericalschemesmightalsoturnaround
thisquestionandaskforthenorminwhichthediscretesolutionshouldbeoptimal.
Unfortunatelynosatisfyinganswertothesequestionshasbeenfound.Theproblem
en.opnsremai

Sensitivitytomeshdistortion–linearandquadraticelements
Inafirsttestcasedifferentdiscretisationswith32×32linearelementsand16×16
quadraticnine-nodedelementsarecompared.Bothmeshesyieldthesamenumberof
wns.unknoWithinthisexamplenosteepgradientsoccur.Thusthestabilisationmethodswhich
arecomparedaretheonesdefinedbyη=0,α=−1,β=−1(GLS)andη=α=0,
βterm,=−1i.e.η(SUPG)=0,bαot=h1withandfixedβ=me−.1FisurtherusedanwhereUSFEmMisvagrrianaduatllywithoutreducedzerotdohownrderto
e1/200inordertoobtainaconvergentsolution.Inallcasesthestabilisationparameteris
calculatedonceperelementattheelementcentre.
Twodifferentdistortionmodesareinvestigatedasdepictedinfigure5.11b)andc).
c)b)a)

Figure5.11:Meshesusedforerrorevaluationwithzoomareaa)undistortedmeshfor
reference,b)distortionmode1andc)distortionmode2

Thefirstmodedegeneratesthequadraticalelementstotrapezoidalswhilethesecond
introducesveryslenderrhombuseswhichturnedouttoappeareasilywhenthemesh
moves.Thedistortedquadraticelementsworkonthesamenodaldistributions,i.e.the
quadraticmeshexhibitsperfectlyplacededgeandcentrenodes.
Animpressionoftheerrordistributionintwoexamplesisgiveninfigure5.12wherethe
absolutevalueofthedifferenceofnumericalandanalyticalsolution,i.e.|uh−u|e2a2π2tν
isplotted.ThefigureshowsresultsobtainedwithlinearGLSstabilisedelementswith
stabilisationparameteraccordingtodefinitioni.Theerrordistributionontheundistorted
meshandthemeshindistortionmode2ispresented.Inthelattercasethemaximal
deviationfromtheanalyticalsolutionismorethan250timeslargerthanontheregular
mesh.Howevertheerrordistributionitselfappearsremarkablysimilarandcloselyrelated

5.3.Residualbasedstabilisationondistortedmeshes103
tothepatternofthesolution.Thisindicatesthatinbothcasestheprimaryerrorsource
isnumericaldissipationyieldingafasterdecayoftheflow.
b)a)

1700.0

06245.

0.00.0
Figure5.12:Distributionofabsolutevalueofnormalisedvelocityerrorobtainedwith
linearelementsafter100timestepsa)ondistortionmode0andb)ondistortionmode2

Velocityandpressureerrorsaccordingto(5.44)havebeencomputedonthethree
meshesforthedifferentstabilisationparameterslistedinsection4.3.8.Acompletetable
oftheseerrorvaluesisgivenintheappendixA.3.Asthechoiceofthestabilisation
parameterisofminorinfluenceagraphicalcomparisonofthevelocityandpressureerror
obtainedwiththestabilisationparameter(4.44)andtheelementlengthi)isgivenin
diagrams5.13and5.14.

0.20.008mode0mode1mode2
0.0070.15S0.006GLurer0.0050.004SUPGMEUSF
0.10.003SGLSUPGMEUSF
0.0020.0010.050linquadlinquadlinquad
0linearquadraticlinearquadraticlinearquadratic
mode0mode1mode2
quaFigurdrate5ic.13:elemenVelotscitaynderroGLS,robtaSUPGinedandonUSFEMdifferentcolynfiguradistotiortednofmesthehesstwithabilisatlineaionrand

gradienThetsuperrorserioritareyofconsquaiderdrated.icInelemenaccordatsnceobservwithedin(5the.44)seadnoiagramsrmalisediscoH1nfirmedseminormifalsoof

410

0.2

rameterspaCritical5.Chapter

mo2de

2demo0.150.0250.02prer0.10.015mode0mode1
0.010.050.005SGLSUPGMEUSFGLSSUPGMEUSF
0linquadlinquadlinquad
0linearquadraticlinearquadraticlinearquadratic
mode0mode1mode2
quaFigurdrateic5.14:elemenPretsassurenderroGLS,robtSUPGainedandonUSFEMdifferentcolynfiguradistortiotednofmetheshesstwithabilisatlinearionand

theerrorcanbedefinedwhichisindependentofthetemporaldecay
graderru:=uh−u0e2a2π2tν
graderrp:=ph−p0e4a2π2tν.(5.45)
TheH1seminormoftheerrorcalculatedwiththeelementlengthdefinitioniandfor
thethreedifferentmodesofmeshdistortionisgiveninthetables5.1and5.2.These
errorsconfirmthepreviousobservationsandshowaparticularsensitivityofthepressure
gradientwithrespecttomeshdistortion.Interestinglythissensitivityevenincreaseswhen
theSUPGversionofthestabilisationisemployed.
Table5.1:VelocityandpressureerrorinnormalisedH1seminormondifferentlydistorted
meshesoflinearelements
graderrugraderrpgraderrugraderrp
modeGLSGLSSUPGSUPG
00.3570030.3563850.3566710.356563
10.4553230.4487260.4327840.431555
22.8948647.0052692.04958515.375787

Fromtheseresultsanumberofobservationscanbemade.
•Withthesamenumberofunknownsquadraticelementsperformsubstantiallybetter
thanlinearones.
•Onaperfectlysquaredmeshtheperformanceoflinearelementsdependsuponthe
valueαwithinthestabilisationoperator.Thequadraticelementisindependent
thereof.Thishighlightstheimpactoftheinconsistencyoflinearelements.

5.3.Residualbasedstabilisationondistortedmeshes

510

Table5.2:VelocityandpressureerrorinnormalisedH1seminormondifferentlydistorted
meshesofquadraticelements
graderrugraderrpgraderrugraderrp
modeGLSGLSSUPGSUPG
00.0428450.0828130.0430540.083754
210.40.005595993310.10.909121392990.00.258818108790.12.81093860530

•Meshdistortionsignificantlyincreasestheerrorsobtainedwithlinearelements.The
meshdistortionofmode2hasadevastatingeffectonlinearelements.
•Quadraticelementsareinsensitivetothedistortionmode1andonlyslightlyaffected
bythedistortionmode2.
Ontheheavilydistortedmeshinmode2USFEMcalculations,i.e.α=1arepossibleonly
forsomestabilisationparameterandinthosecasesrequireverysmallparametersme.A
sideeffectthereofisinsufficientpressurestabilisationandthusanincreasedpressureerror
forUSFEMcalculationsonthedistortionmode2.
Theaboveallowstodrawtheconclusionthatsubstantialmeshdistortioncanbecoped
withwhenhigherorderelementsareemployedwheretheedgeandcentrenodesare
perfectlyplaced.FurtherGLSoralsoSUPGversionsofthestabilisation,i.e.α=−1or
α=0,respectively,shouldbepreferred.

Misplacededgeandcentrenodes
InasecondtestcasetheKim-Moinproblemdiscretisedby16×16nine-nodedelements
isusedtoaccessthesensitivityoftheelementwithrespecttomisplacededgenodes.
Misplacementofedgeandcentrenodescanhaveahugeimpactontheaccuracyreached
withhigherorderelements.Howeverusuallythereisnoneedtoallowarbitraryplacement
ofthenon-cornernodesinelementsinsidethedomain.Onlyatcurvedboundariesedge
nodesofhigherorderelementshavetobeadjusted.
Neverthelesstheeffectofmisplacededgenodesintheinteriorofthedomainshallbe
consideredinordertoinvestigatetheerrorsensitivityduetosuchdisplacements.The
interestisfocusedhereonedgenodesdisplacedperpendiculartotheundistortedelement
edgewhilestillresidingonthecentreofthenowcurvededge.Theelementmidnodesare
placedatthecentresofthedistortedelements.
Thedistortedmeshesunderconsiderationaredepictedinfigure5.15wheretheoffset
oftheedgenodeperpendiculartotheundistortedelementedgedisvariedbetween0≤
d≤1.5hwherehdenotesthedistanceoftwoadjacentnodesintheoriginalmesh,i.e.
halftheoriginalelementlength.
Thesquarerootoftheelementalareaistakenaselementlengthandtheerrorsare
computedaccordingto(5.44).Thediagramsinfigure5.16showtheevolutionofthetime
normalisedvelocityandpressureerrordependingonthedistortiondandobtainedwith
GLSorSUPGvariantsofthestabilisation.
Fromthediagramsinfigure5.16itcanbeobservedthatsmallormoderatedisplace-

610

rameterspaCritical5.Chapter

Figure5.15:Distortedquadraticelementmesheswithoffsetedgenodesfromlefttoright:
distortionmode3anddistortionmode4

mentsoftheedgenodeperpendiculartothestraightedgecanbedonesafelywithout
introducingvelocityorpressureerrorsduetomeshdistortion.Hugedeflectionsofthe
edgenodeyieldcleardeteriorationoftheresults.ItisfurtherobservedthattheSUPG
typeofthestabilisationmethod,i.e.α=0yieldssignificantlysmallererrorsinvelocity
andpressurethanGLSwhenhighedgenodeoffsetareconsidered.Badlyapproximated
secondderivativesareabletoaffectquadraticelementsalsoifhighdistortionhastobe
considered.Inmostpracticalcases,however,goodgeometryapproximationshouldbe
possiblebyplacingtheedgenodewith−0.5h≤d≤0.5h,i.e.withanoffsetofuptoa
quarteroftheelementlengthandusingSUPGorGLS.
b)a)

rerurerpGLS

0.04erruerru
errperrp
0.03GLSGLSerroredisalmnor0.01SUPGSUPG
0.02000.2edge0.4n0.6odee0.8lon1gation1.2d1.41.600.2edge0.4n0.6ode0.e8lon1gation1.2d1.41.6
Figure5.16:Evolutionofthenormalisederrordependingupontheoffsetdoftheedge
nodesindistortiona)mode3andb)mode4

AdifferentbehaviourisobservedifanUSFEMversionofthestabilisationisemployed,
i.e.ifα=1(wherestillη=0isused).Inthiscaseanelementalconstantmedetermined
inaccorfigureding5.15to(a5.40re)displahasytoedbeinemfigploureye5.d.17Thealongerrworsithobttheainedevoluontiontheoftwtoheparmeshesameterdepictedme.
modeThe3ardiagemramuchinlafigurrgeretha5.1n7tshohosewsofthatallothethererrorscases.obtAdditainedforionallyUSFEMtheandpressuretheerrordistorthereion
significantlyexceedsthecorrespondingvelocityerror.Thiseffectisduetothenecessary
dropdecreasesintheandpasodorameterestmhee.influenceAlongowithfthearsetducabilisatioedmenttheerms.stabilisaConstionequenpatlyarameterlimiτtMis,e

5.3.Residualbasedstabilisationondistortedmeshes

3demo

710

0.25reru0.2rerp0.15parametermode3
0.1me4demo0.05000.20.40.60.811.21.41.6
Figure5.17:EvolutionofthenormalisederrorobtainedwithUSFEMstabilisationfor
distortionmodes3and4alongwiththeparameterme

4demo

reachedatwhichinsufficientpressurestabilisationoccursandzeropressuremodesbegin
tospoilthesolution.However,suchpressuremodesarehighlymeshdependentandthus
onlyobservedforthedistortionmode3whereallelementsexhibitthesamedistortion.
Inpracticalsituationsthiswillrarelybethecase.Neverthelesstheseresultsconfirm
thatSUPGorGLSsimulationsaresomewhatpreferablecomparedtoUSFEMwhichmay
requireadeterminationofmewithineverytimestepwhenthemeshismoving.

5.3.3Flowpastcylinder
IncompressibleflowpastarigidcylinderataReynoldsnumberofRe=100isaclassical
testcase.Theproblemhasbeenusede.g.byBrooksandHughes[35]forthevalidation
oftheSUPGmethodandithasbeeninvestigatedwithrespecttolinearelementsofhigh
aspectratiobyMittal[181].Thepresentinvestigationparticularlyreferstothatlatter
paper.Partsofthecomputationsreportedin[181]arerepeatedhereandsomefurther
testcasesareconsidered.Thuscomparisonsofthebehaviouroflinearandquadratic
elementscanbepresented.
Thegeometryandthemeshdatausedisdepictedinfigure5.18.Threedifferent
meshesoflinearfour-nodedelementsareemployeddifferingonlyintheregioncloseto
thecylinder.Thenumberofelementsalongthediagonallineais36,48and90yielding
atotalnumberofelementsof4424,5192and7880forthelinearmeshesA4,B4andC4,
respectively.Theelementsalongaareconcentratedtowardsthecylindersuchthata
maximalaspectratiooftheorderofmagnitudeof101,103and105resultsforthethree
respectivemeshesA4,B4andC4.
AsecondsetofthreemeshesA9,B9andC9ofquadraticnine-nodedelementsisdefined
inthesamewaybytakinghalfthenumberofelementscomparedtothelinearcasealong
everyedgeyieldingintotalaquarteroftherespectivelinearelements.
Thefluidhasaviscosityofν=0.005andadensityofρF=1.0.Ahorizontalinflow
velocityofu∞=0.5isprescribed.Atthetopandbottomboundarytheflowisallowedto
slipfrictionlessalongthewall.FollowingMittalinallcasesatimestepofΔt=0.125
hasbeenused.GLSstabilisation,i.e.η=0.0,α=−1.0andβ=−1.0isemployed.
Forverystretchedelementstheelementdefinitionsgiveninsection4.3.8significantly
differ.Inordertoaccessthesedifferencestheelementlengthdefinitionsaccordingtoi,vi

810

inflowvelocityu∞=0.5
d8d8

a4016108d22.5d

rameterspaCritical5.Chapter

cylinderdiameterd=1
10dieleffmenerentstlyfialonnegdmesiagonhesalinathegreyregion
16memeshshB4:A4:4836
90C4:shme10freeoutflowboundary

Figure5.18:Flowpastcylinder,geometryandmeshdata

andvii,i.e.basedonthesquarerootoftheelementalareaandtheminimalandmaximal
elementlengths,respectively,arecompared.Intable5.3theStrouhalnumbersforthe
differentcasesarepresented.TheStrouhalnumberisdefinedby
dSt=tcu∞(5.46)
andrepresentsadimensionlessfrequency.Thedurationofaperiodintheoscillationsof
theliftforceisthecharacteristictimetcemployedtoevaluate(5.46).Themaximaland
minimalobtainedStrouhalnumberdifferslessthan3%whichisjusthalfthevariation
obtainedbyMittalontheQ1Q1meshin[181].

Table5.3:Strouhalnumberfordifferentelementlengthmeasuresandondifferent
smeshenumbernumberaspectelementlengthelementlengthelementlength
meshofofratioviviii
elementsnodesStrouhalStrouhalStrouhal
numbernumbernumber
A444244558100.169490.168780.16913
B4519253261030.168070.167010.16736
C4788080141050.168780.168070.16842
A911064558100.171310.171310.17167
B9129853261030.170940.170580.17058
C9197080141050.171310.171310.17131

Itisfurtherobservedthatquadraticelementsinallcasesyieldaslightlyhigher
Strouhalnumberindicatingthatevenfortheveryfinemeshesemployedaroundthe
ofcylinderthedraquagcodraticefficienelemetonntsthearecyablelindertooofferbtainedmorewaitchcuraquatedratresicults.elemenAlsotstishehigmeanhervtaluehan
whenlinearelementsareused.
Thetemporalevolutionofthecoefficientsofliftanddragaccordingto
Cl=ρF2uF∞ldandCd=ρF2uF∞dd

5.3.Residualbasedstabilisationondistortedmeshes109
whereFlandFddenotetheliftanddragforce,respectively,obtainedwithdifferent
elementlengthdefinitionsarecomparedinthediagrams5.19to5.22forthemeshesA4
andC9.TherespectiveresultsobtainedonthemeshesB4,C4,A9andB9aregivenin
theappendixA.4.Infigure5.22thepressureprofileobtainedonthemeshC9andwith
minimalelementlengthviisgiventohighlighttheslightoscillationobtainedinthiscase.
Thiswastheonlysignofan‘instability’thatcouldbefoundinthisinvestigation.Using
theothertwodefinitionsfortheelementlengthhkonthemeshC9yieldsverysimilar
results.Thelegendinfigure5.19alsoappliestoallthefollowingdiagrams.
0.40.320.240.160.08l0C-0.08-0.16-0.24-0.32-0.4

05101520253035404550
timeelementlengthvi,he,min
elementlengthvii,he,ma√x
elementlengthi,he=Ae
Figure5.19:LiftcoefficientobtainedonmeshA4withthreedifferentelementlength
definitionswithinthestabilisationparameter

1.4241.4161.408dC1.41.3921.38405101520253035404550
timeFigure5.20:DragcoefficientobtainedonmeshA4withthreedifferentelementlength
definitionswithinthestabilisationparameter
Additionallythenormalisedpressuredistributionalongthecylinderisgivenforevery
case.Thenormalisedpressureiscomputedfrom
Cp=2(p−2p0)+1
ρu∞

rameterspaCritical5.Chapter

110Chapter5.Criticalparameters
10.50p-0.5C-1-1.5-2-1-0.500.51
xFigure5.21:PressureprofilealongthecylinderobtainedonmeshA4withthreedifferent
elementlengthdefinitionswithinthestabilisationparameter
wherep0denotesthepressureattheleadingpointofthecylinder.FollowingMittal
in[181]thepressuredistributionisprojectedtothehorizontal,i.e.x-direction.The
pressuredistributionisevaluatedatatimeinstantwhentheliftcoefficientreachesitsa
maximumvalue.
110.80.50.60.400.2pC0-1-0.9-0.8
-1-1.5-2-1-0.500.51
xFigure5.22:Slightlyoscillatingpressureprofilewithclose-upviewobtainedonMeshC9
withminimalelementlengthvi
Theresultsgiverisetoanumberofobservations.
•Theinfluenceoftheelementlengthwithinthestabilisationparameterisonlyminor
eveninthecaseofhighlystretchedelements.
•Whenlinearelementsareemployedadeviationintheliftcoefficientfortheelement
lengthhe,maxisobserved.Thisindicatesslightover-stabilisation.
•Incontrasttotheresultsreportedin[181]noinstabilitiesareobserved.Eventhe
pressureprofileremainscorrectapartfromatinydispositiontooscillationsbetween
cornerandedgenodepressurevalueswhentheminimalelementlengthisusedon

10.80.60.40.20-1

0.9-

0.8-

5.3.Residualbasedstabilisationondistortedmeshes
0.40.320.240.160.08l0C-0.08-0.16-0.24-0.32-0.40510152025
time1.4241.4161.408dC1.41.3921.3840510152025

51015202530
time

35

35

4540

4540

50

111

50

05101520253035404550
timeelementlengthvi,he,min
elementlengthvii,he,ma√x
elementlengthi,he=Ae
Figure5.23:LiftanddragcoefficientsobtainedonmeshC9withthreedifferentelement
lengthdefinitionswithinthestabilisationparameter

mesh.icdratquaa•Quadraticelementsyieldevenbetterresults.Theyaresurprisinglyinsensitiveto
thechoiceoftheelementlengthaslongassufficientpressurestabilisationisensured.
Whilestillofferingverygoodresultshighlystretchedelementsyieldbadlyconditioned
matricesandthusthesolutionmayrequireparticularconsideration.Theiterativesolver
packageAZTEChasbeenusedforthemeshesA4,B4,C4andA9whileadirectsolver
hadtobeemployedfortheremainingcasesB9andC9.Howeverastheconditionproblem
isscalingrelatedoftothesinglrespeecnotivdeseorlinesdegrofeestheofoverafreedomllofsystemhigohlyfstequaretctionshedcouldelemenbtseaintropproprducediate
removingtheproblemifsuchelementshavetobeused.
bestIn.It[181is]thusMittsuppalosdraedwsherethethattconclusionheinaccuracythattheatusehighofaaspectminimaraltioelemenobservtedblengtyhMittworksal
ismainlyduetotheinconsistencyoflinearelements.Thiseffectincreasestheamountof
numericaldissipationalongwiththeincreaseofthestabilisationitself.

211

5.4Summary

rameterspaCritical5.Chapter

StableFSIalgorithmsrequireaflowsolverwhichisreliablealsoatcriticalparameters.
Atverysmalltimestepsizesandsteepvelocitygradientsperpendiculartotheflow
numericaloscillationsareobservedinsimulationswithstabilisedfinitefluidelements.
Theseoscillationsareduetoadominanceoftheinertiatermcomparedwithviscous
effectsandindicateasingularperturbedproblem.Comparedtothedominatingmass
termtheconvectionstabilisationbecomesinsufficientallowingconvectiontypeofwiggles
toenterthesimulation.
Usingastabilisationvariantwhichcontainsazerothordertermwithinthestabilisation
operatorreducesthedominatingtermwhichhasarevitalisingeffectontheconvection
stabilisation.However,thisstabilisationvariantmustnotbeusedinconjunctionwith
linearelementswhereitmayhaveadevastatingeffectontheaccuracy.
Asasecondcriticalparameterdistortedmeshesareconsidered.Consistentlystabilised
finiteelementsarepotentiallyveryinsensitivetomeshdistortionprovidedthatquadratic
andhigherorderelementswithperfectlyplacededgenodesandmidnodesareused.
Themethodfurthershowsinsensitivetotheactualchoiceoftheelementlengthwithin
thestabilisationparameter.Evenwhenhighlystretchedelementsareusedverylittle
influencesareobservedinparticularwhenhigherorderelementsareemployed.Forlinear
elementsanerrorwhichscaleswiththeamountofstabilisationhastobeexpecteddue
totheinherentinconsistencyofsuchelements.

6erChapt

coFSIngupli

theThiscstructurhapteralisdevfield,otedthetofluidaninfievldestigaandtiontheofintheterasolvctioneroforftthehetwcouploedwhicprohappblemearsaincludingsan
additionalchallenge.BuildingupontheworkofMok[182]andWall[227](also[183,
22par8])tofsomethiscimphaptroveremeisnintsteofndedthetoFSIclacorifytuplinghealgartificialorithmaddearedmasspresentedeffect.here.ThisTheeffectmainis
responsibleforaninherentinstabilityofsequentiallycoupledFSIalgorithmsappliedto
systemsoflight-weightstructuresandincompressibleflows.

6.1Introduction

AvarietyofschemeshasbeenproposedandusedtosolveFSIandothersurfacecoupled
problems.Anoverviewoverrecentdevelopmentscanbefoundinthespecialissueon
FSI[187]wherevariousapproachesarediscussed.AsummaryofdifferentFSIproblems
treatedbyreducedmodelscanbefoundintheencyclopediaarticlebyOhayon[186]
wherethestructural-acousticproblembutalsosloshinginelasticstructuresistreated.
Monolithicschemeswhichrequirethattheentirecoupledproblemiseventuallycast
intoonesystemofequationshavebeenreportedforexamplebyH¨ubneretal.[130,226]
andHeil[123].Howeversuchschemessufferfromthedisadvantagethatpotentiallyrather
badlyconditionedoverallmatricesmaybeobtained.Apossiblyevenmoreimportant
drawbackliesinthefactthatsuchschemesdonotallowtocoupleexistingfluidand
structuralsolversbutratherrequireaspecifiedoverallframework.
Ahigherpopularityhasbeengainedbypartitionedschemeswhichallowtousespecif-
icallydesignedcodesonthedifferentfields.Smallerandbetterconditionedsubsystems
aresolvedinsteadofoneoverallproblem.AmongothersthegroupofTezduyar[208,
221,224],DettmerandPeric´[63,64],Matthiesetal.[179,180],Piperno[193,194]
andRanketal.[199,200]employpartitionedapproachestocomputesurfacecoupled
systems.Asignificantamountofworkinthefieldofpartitionedcouplingschemesisdue
toFarhatandhisgroup[77,195].
Sequentiallystaggeredschemesareaparticularlyappealingsubclassofpartitioned
algorithms.Howevertheseschemesexhibitaninherentinstabilitywhenusedonfluid-
structureinteractionproblemswhereincompressibleflowsareconsidered.Surprisingly
theinstabilitydependsuponthedensitiesoffluidandstructureandalsoonthegeometry
offlavtouherindomatointhe[38co,18mputa2,183tion,e2v28].enifbotClearlyhparsequentitionstialtcohemsuplingelvesinaretrosolducvesedanimplicitlyexplicit.
Thusrestrictionsonthetimestephavetobeexpected.Observationshowevershow
thatdecreasingthetimestepsizecausesanincreaseoftheinstability.Theinstability

311

411

uplingcoFSI6.Chapter

ismajorinherenparttsinofthetheschefluidmeactitsaelfsaanndextrhasabmaeenssonnamedthe‘astrucrtificialturaladeddedgreesmassofeffect’freedomsinceat
thecouplinginterface.Insequentiallystaggeredschemesthefluidforcesdependupon
apopredictedrtionofstructuraincorlrectinterfacouplingceforces.displacemenIttsisrathisther‘athanrtificial’theconcorrecttriboutionnesatondththeuscocontuplingain
whichyieldstheinstability.
FSI[79Already].Thein1977maximaFellipptimeaetstepal.sizerepobtortedainedanuppthereerdeplimiteondsnonthettihemeratiostepofsizeforstructuralacousticand
fluidmassdensitybutitisindependentofthetemporaldiscretisationscheme.Applying
theresultsofFelippaetal.tothelimitcaseofincompressibleflowwhichgoesalong
withaninfinitespeedofsound,predictsimmediateinstabilityirrespectiveofthetime
size.stepItshallbeshowninthesubsequentanalysisthattheproblemposedbytheartificial
addedmasseffectisnotquiteasbadasthat.

6.2PartitionedFSIalgorithmanditsdetails

ThepresentworkisexclusivelyconcernedwithpartitionedFSIapproaches.Withinsuch
schemesitisthetaskofthecouplingalgorithmtoensurethatthekinematiccoupling
condition(2.51)aswellasthedynamiccontinuity(2.52)aresatisfied.
Partitionedstrongcouplingschemesconvergetothesolutionofthemonolithicscheme
andarethusabletosatisfydiscreteversionsofthekinematicanddynamiccoupling
conditionsexactly.Howevertheseschemesrequiresub-iterationsandthusintermsof
efficiencyweakorloosecouplingpartitionedschemescouldbepreferred.Thelatter
approachmanageswithjustonesolutionofeitherfieldpertimestepbutitconsequently
lacksaccuratefulfilmentofthecouplingconditions.
Subsequentlyanunifiedapproachforageneralpartitionedalgorithmisdescribedwhich
canbespecifiedtoalooselyorstronglycoupledversion.Thestronglycoupledoriterative
schemeofthealgorithmicframeworkforFSIhasbeenproposedbyLeTallecetal.[166].

6.2.1Ageneralpartitionedalgorithm
Thealgorithmofconsiderationisrestrictedtosynchronoustimediscretisationswithequal
timestepsizeΔtonbothfields.Additionallytothetimestepsuperscriptnthesubscripti
denotingtheiterationnumberinthestronglycoupledversionoftheschemeisintroduced.
tureTheandalgofluidrithmgivisenbabysed(2on.23)theandtemp(4.4ora1),lly(4.42and)respspatiaectivllyelydisc.retTheisedequatsubscriptionsΓofstrdenotesuc-
therestrictionofavectorofnodalvaluestothoselinesreferringtointerfacedegreesof
freedom.nhaveStatobrtingeftakreomninknoordwnertosolutioconsmputeuptothethenewtimecoupledleveltsothelutionfolloattnwing+1.aSplgorithmicecificationsstepsof
thesinglestepsaregiveninsubsequentsections.
1.Startstep.Computn+1estructuralpredictordΓn,P+1=dΓn,0+1inΩSfortheinterface
displacementsatt,seti=0.

6.2.PartitionedFSIalgorithmanditsdetails115
2a.Meshstep.SolvefluidmeshonmeshdomainΩGfornewpositionsrin+1+1dΓn,i+1.
2b.ComputenewgridvelocityuiG+1,n+1dΓn,i+1accordingto(3.33).
3a.Fluidstep.DeriveanewfluidvelocityalongwetsurfaceuΓn,i+1+1dΓn,i+1tobeused
asDirichletboundaryconditiononthefluiddomainaccordingtosection6.2.3.
3b.IntegratefluidvectorsandmatricesonΩn+1andsolvenonlineardiscreteflowequa-
tionsiterativelyfornewfluidvelocityuiFn+1+1dΓn,i+1andpressurefieldpin+1+1dΓn,i+1.
Considerexternalfluidforcesifpresent.
3c.DeterminefluidforcesontheinterfacefΓn,i+1+1dnΓ,i+1.
4.Structuralstep.Solvestructuralequations(2.23)iterativelyfornewstructural
displacementd˜in+1+1wherethefluidforcesfΓn,i+1+1dΓn,i+1andexternalloadsareexerted
structure.theon5.Foriterativeschemesonly:Determinerelaxationparameterωi∈R+accordingto
themethodsreportedin[183,228].
6.Foriterativeschemesonly:Computerelaxedupdateofpredictedinterfaceposition.
dΓn,i+1+1=ωid˜Γn,i+1+1+(1−ωi)dΓn,i+1(6.1)
7.Foriterativeschemesonly:Checkconvergence.Ifnotconverged,thenseti→i+1
andcontinuewith2.
8.Proceedtonexttimestepbysettingn→n+1
IniterativelystaggeredschemesthepredictedinterfacedisplacementdΓn,P+1canberegarded
thezerothiterationandisthusalsotermeddΓn,0+1instep1oftheabovealgorithm.With
regardtotheanalysisoftheartificialaddedmasseffectinlooselycoupledschemesthe
formernotationispreferredinthesequel.

6.2.2Structuralpredictor
Avarietyofpredictorscanbeemployedtofindastartingguessfortheinterfacedisplace-
mentinthenewtimestep.Theeasiestversionisgivenbythesimplechoice
dΓn,P+1=dΓn,(6.2)
whichiszerothorderaccurateintimeandhasbeenproposedforpartitionedFSIpro-
ceduresbyFelippaetal.in[78,79].Insequentiallystaggeredschemestheaccuracyis
determinedbythelowestorderingredientandthusapredictorofhigherorderthanthe
aboveishighlydesirable.
AfirstorderaccuratepredictionwhichhasbeensuggestedbyPipernoin[192]is
omrfainedobt

dΓn,P+1=dΓn+Δtd˙Γn.
Asecondorderaccuratepredictorreading
dΓn,P+1=dΓn+Δt3d˙Γn−1
22

˙dn−Γ1

.3)(6

.4)(6

FSI6.Chapteruplingco

116Chapter6.FSIcoupling
canalsobefoundin[192].
Whiletheactualchoiceofthepredictordoesnotinfluencethesolutionwhensub-
ititeradoestionsnotaorenlyperformeddetermineithathesaordersignificanofttempimpactoralaonccuracysequenbuttialstalsoaggeredhighlyschemes.influencesTherethe
actualonsetoftheinstabilityduetotheartificialaddedmasseffect.Adetailedanalysis
ofthiseffectispresentedinsection6.3.

6.2.3Fluidvelocityboundarycondition
Thestructuralinterfacedisplacementobtainedinstep1oftheabovealgorithmneedsto
betransformedintoaninterfacevelocityinordertoprovidetheDirichletboundary
conditionforthefluidfield.Withaninterpolationwhichisfirstorderaccurateintime
thisvelocityisobtainedfrom

n+1nuΓn+1=dΓ,P−dΓ,P.(6.5)
tΔEquation(6.5)isabackwardEulerdiscretisationofthefluidvelocityatthecoupling
FainvterfaelocitceyΓ.whichAccuraiscteonsistconserventatwithionofthethetimesizediscofrtetheisatiofluidnscdomahemeinΩoftherequirfluidestofield.emploThey
timediscretisationsschemesone-step-θmethod(3.13)andBDF2(3.14)canbecastinto
formgeneralthe

a¯=un+1−un,(6.6)
tΔwherea¯isanaverageaccelerationwithinthetimestepfromtimelevelnton+1which
isassumedconstantinbothdiscretisationschemeswhilebeingapproximateddifferently
ineithercase.Thustheinterfacedisplacementofthefluidfieldcanbeobtainedfrom
integratinga¯accordingto
tn+1t
a¯dτ+undt=rn+1−rn(6.7)
nnttwherekinematiccontinuityrequiresthatthefluidandstructuralmotioncoincidewithin
thetimestep,i.e.rn+1−rn=dΓn,P+1−dΓn,P.Evaluating(6.7)andusing(6.6)yieldsa
secondorderdiscretisationofthefluidboundaryvelocityreading
n+1nuΓn+1=2dΓ,P−dΓ,P−uΓn.(6.8)
tΔEquation(6.8)canbeinterpretedasatrapezoidalrulefortheinterfacevelocityandit
correctlypreservesthesizeofthefluiddomain,i.e.itextendsgeometricconservation
totheboundaryofthedomain.Consequently(6.8)posesthepreferablefluidboundary
conditionwhichshouldbeusedwheneverpossible.Despitethedesirablepropertyof
geometricalcorrectness(6.8)exhibitsthetendencyofthetrapezoidalruletooscillations
andhasthustobereplacedbythedissipatingcondition(6.5)insomesensitivecases.

711

6.2.PartitionedFSIalgorithmanditsdetails117
6.2.4Structuralforceboundarycondition
Carehasalsotobetakentoobtainthefluidforceswhicharetransferedtothestructure.
Forflowswithverylowviscositythecouplingisoccasionallyperformedbyjustexerting
thefluidpressuresonthestructureaccordingto

Fp=−p¯ndγ.(6.9)
ΓForlowandmoderateReynoldsnumberflowswhichareconsideredheretheflowfriction
alongthestructureshouldnotbeneglectedandthus

Fσ=Γ(2µε(u)−p¯I)∙ndγ(6.10)
shouldbeevaluatedratherthan(6.9).0However(6.10)suffersfromseveraldrawbacks.
AsthevelocitiesareapproximatedbyCcontinuousfiniteelementshapefunctionsthe
viscoustractionsexhibitjumpsattheelementbordersandconsequentlytheevaluation
ofDue(6to.10)thedoesneednottofitvcaerlcyuwlateellintospatiatheldenorivdaaltivbasedesofdatathesvteloructurecityuofathefiniteforceselemenFσtcofurtherde.
exhibitanorderofaccuracyinspacewhichisoneorderlowerthantheapproximationof
thevelocity.
nodalMucfohrcemorsewhichaccuracyarecalsoanbstroenglysqueezedoutrecommendedofthebyfluidGresolutioshonandbyevSanaluiinating[107].consistenNodalt
forcesaredefinedinthediscretesetting.Inthefullystabilisedcasegivenin(4.41)these
forcescanbeobtainedfrom
fΓF=δ1MΓF(u)u+KΓF(u)u+NΓ(u)+GΓ(u)p−δ1fbF,Γ,(6.11)
wherethesubscriptΓdenotestherestrictionoftherespectivematrixorvectortothe
linescorrespondingFtotheinFterfacedegreesoffreedom.Itisworthnotingthatthenodal
offluidthefodisrcecretevectoflorwfequatandionsalsofnorΓasmaliseddefinebdytinhesectionfluid4.3densit.6yexρF.pressestConsequenherightlythitandhastsidehe
unitforceperdensityandneedstobemultipliedbythefluiddensityinordertoobtain
thephysicalcouplingforceswhichwillbedenotedbyfΓ.
Forcesaccordingto(6.11)haveanumberofdesirableproperties.
•Consistentnodalforcessharetheorderofaccuracyofthepressureandvelocity
n.imatioxoappr•Forcesaccordingto(6.11)fitintoanelementbaseddatastructureandareeasyto
t.implemen•Consistentnodalforcesaccountfortheinfluenceofthestabilisationandthusthe
effectofthevirtualbubbleenrichingtheapproximationspace.
Whennostabilisationisappliedanequivalentexpressionto(6.11)canbefoundbycon-
).3.23(sidering

8116.3Artificialaddedmasseffect

coFSI6.Chapterupling

6.3.1Introduction
Sequenstructurtiaeslstaexhibitggeredanscinsthemesabilitaywppliedhichtohasbeenincompressiblenamedfloawsrtificialinteraactddeingdmawithssligheffect.t-weigThisht
effectalreadymentionedbyWalletal.in[228]andLeTallecandMouroin[166]
hasbeeninvestigatedbymeansofareducedmodelproblemin[38]whereitisshownthat
theonsetoftheinstabilitycanbepredictedwellwithinthesimplifiedproblem.In[38]
aCausisimplifiednetavl.rersioefonrmofulathetethePressure-continPouousissonfluidequatproblemionwhicandhperforeducesrmtotheir−Δpana=0lysiswithon
appropriateboundaryconditions.Theyeventuallygiveadiscreteaddedmassoperator
intermsofthediscretisedPressure-Poissonequation.
Theanalysispresentedsubsequentlyfollows[87]andshowshowtoobtainanexpression
ofthediscreteaddedmassoperatorintermsofmatriceswhichareusuallycomputed
withinfluidcodessolvingtheincompressibleNavier-Stokesequationsfortheprimary
unknoearlierwnsonsetuofandthep.Itinstabilitfuryther.Alsoshowsthewheffecytmoreofaresiduaccurlatebasedtimestadisbilisacretisationtionforyieldselemenants
notsatisfyingtheinf-supconditionisconsidered.
Accordingtothepropertiesoftheinstabilitythefollowingobservationshaverepeatedly
beenreported.
•WithdecreasingΔttheinstabilityoccursearlier.
•ofThethemastassggraterioedbetwesystem.enfluidThealandrgstreructheturemasshasaratioρsignificanF/ρSttheinfluenceworseotnhetheinstabilitstabilityy
gets.•ityNumericalwhileincreobservasedationsstructurindicalastetiffnethatsscincreaausessedafluidlightviscositdecreasyingincreffect.easestheinstabil-
•Theactualonsetofunconditionalinstabilitydependsupontheparticularcombina-
tionoftemporaldiscretisationitemswherethemostinaccuratecombinationyields
beststabilityproperties.

6.3.2AddedmassoperatorforLBBstablefluidelements
Basicdiscreteequations
Inodiscretisedrdertoflopwerformequataionsgeneralstillconstabtinilityuousanainlysistimeofaretherecoquired.upledproTheblemtinfluenceheospatiafthelly
particulartimediscretisationshallbeconsideredsubsequently.Forinf-supstableelements
wherenostabilisationisappliedthetimecontinuousmatrixequationsread
MFu˙+N(u)+KFu+Gp=fΓF(6.12)
GTu=0(6.13)
wherethematricesandvectorsaredefinedinsection3.3.2.Inviewoftheintended
analysisnobodyforcesandtractionsareconsideredin(6.12).Thustherighthandside
exclusivelyconsistsofFSI-couplingforcesalongΓ.Equivalentlythestructuralsystemof

911

6.3.Artificialaddedmasseffect119
readsionsequatMSd¨+FSint(d)=−fΓ(6.14)
loawheredonthethedampingstructure.termAshathesbsteenructuralneglected.equatTheionsstwereructuraldiscretiseinterfdacedirectlyforcespitsosreighthetohandnly
sideconsistsofreal,physicalforces.

Simplificationsrequiredfortheanalysis
Somefurthersimplificationsaremadeinordertoeasetheanalysis.
•Thediscretisationoffluidandstructuralfieldisassumedconformingalongthe
Γ.ceterfain•Bothfieldequationsareregardedaslinear.
•Theinfluenceofmeshmotionandthusthechangeofallcoefficientmatricesis
ible.neglig•ThestructuraldensityρSremainsconstant.
•Nophysicalstructuraldampingisassumed.
Thefirstoneoftheaboveassumptionssimplyeasesthepresentationwhiletheothersare
reasonableastheinstabilityofconsiderationisobservedveryearlyinthecomputation
whennosignificantnonlinearityhasbeenbuiltup.Inordertoperformaneigenvalue
analysisoftheamplificationoperatorofthecouplingproceduretheassumptionoflinearity
isfurthercrucialasnonlineareigenvaluesfornonlinearoperatorsarenotdefined.However
theinsightobtainedfromtheanalysisalsotransferstothenonlinearproblem.
Tosimplifyfurtherthefluidstiffnessandconvectivetermin(6.12)areomitted.Thisis
reasonableatverysmalltimestepswherethebehaviouroftheflowisgovernedbyinertia
andincompressibility.Thusthissimplificationallowstohighlightclearlythereasonof
theinstability.

Theaddedmassoperator
TheassumptionthatthecoefficientmatricesdonotchangeintimeallowstousetheALE
timederivativeofthedivergenceequation(6.13)yielding
Tt∂∂Gu=GT˙u=GTu˙=0.(6.15)
χThesimplifiedfluidsystemofequationsisnowsplitupintodegreesoffreedombelong-
ingtotheinteriorofthefluiddomainandothersattheinterface.Thesplitmatrices
andvectorsarelabelledbythesubscriptsIandΓindicatingtheinteriorandinterface,
respectively.Replacing(6.13)by(6.15)andusingthesimplificationsthefluidsystemof
readsionsequat

MIIMIΓGIu˙I0
FF
MFΓIMFΓΓGΓu˙Γ=fΓF.
GITGΓT0p0

.16)(6

012

uplingcoFSI6.Chapter

In(6.16)theinterfaceaccelerationisprescribedaccordingtothestructuralpredictor.
Thustheproblemcanbesolvedfortheremainingaccelerationsandpressureswhich
dependonu˙Γandaregivenby
p=GITMIFIGI−GITMIFIMFIΓ+GΓTu˙Γ,(6.17)
−1−1−1
u˙I=−MIFIMFIΓu˙Γ+MIFIGIGITMIFIGIGITMIFIMFIΓ−GΓTu˙Γ.
−1−1−1−1−1
.18)(6Employingthesecondlineofthesystem(6.16)thenormalisedinterfacialcouplingforce
vectorfΓFcanbeexpressedintermsoftheinterfaceaccelerationreading
fΓF=MFΓIMIFI−1GI−GΓGITMIFI−1GIGITMIFI−1MFIΓ−GΓT
−1
1−−MFΓIMIFIMFIΓ+MFΓΓu˙Γ.(6.19)
Duetothenormalisationoftheflowequationsthediscreteoperatorincurlybracketshas
thedimensionofavolume.Itcanbenormalisedbyacharacteristicvolumeofthesupport
ofafluidnodevFtorecoveradiscreterepresentationoftheaddedmassoperator.
MA:=FMΓIMIIGI−GΓGIMIIGIGIMIIMIΓ−GΓ
1FF−1TF−1−1TF−1FT
v−FMFΓIMIFIMFIΓ+FMFΓΓ(6.20)
v1−11
vEmployingtheaddedmassoperator(6.20)yieldsthephysicalfluidforcesatthecoupling
ryoundab

fΓ=ρFvFMAu˙Γ=mFMAu˙Γ(6.21)
wheremFdenotesacharacteristicmassofafluidnode,i.e.mF=ρFvF.Theadded
massoperatorMAcontainsthecondensedfluidequationsandmapsadimensionless
interfaceaccelerationontoanalsodimensionlessforcevectorattheinterfaceΓ.Thusthe
operatorispurelygeometrical.Itcanbeobservedfrom(6.20)thatthediscreteadded
massoperatorissymmetricandpositive.
Afurtherinterestingsimplificationistousealumpedfluidmassmatrix.Thentheoff-
diagonalblocksofthemassmatrixvanishandassumingaregularmeshthemainblocks
arereplacedby
MIFI=2vFIIIandMFΓΓ=vFIΓΓ(6.22)
expressionsimpletheyieldingMAl=IFΓΓ+2GΓGITGI−1GΓT(6.23)
forthelumpedaddedmassoperator.Expression(6.23)indicatesthattheeigenvaluesof
theaddedmassoperatorexceedone.Itcanfurtherbeobservedthatthediscretegradient
operatormatrixGIneedstohavefullrankiftheaddedmassoperatorexists,i.e.ifthe
flowequationshaveauniquesolution.

.22)(6

6.3.Artificialaddedmasseffect121
6.3.3Addedmassoperatorforstabilisedfiniteelements
Stabiliseddiscreteequations
Amatrixrepresentationofthestabilisedflowproblemstillcontinuousintimereplac-
ing(6.12)and(6.13)reads
MF(u)˙u+NF(u)+G(u)p=fΓF(6.24)
GTMu˙+GKT(u)u−Cp=0,(6.25)
wherethesinglematricesaredefinedinsection4.3.6.Asintheunstabilisedcasethefluid
momentumbalanceequation(6.24)hasbeennormalisedbythefluiddensityρF.
Somesimplifications
Inmade.orderThtousptheerformsystaenma(6.2nalysis4)sandim(6.2plific5)aistionsconsideredsimilartointthehelimitunstaofbiliseverdycasesmallhavtimeetostepbe
itsizeca.nRbeecaobsllingervtheedlimitfromof(6.2the5)orstabilisaalsotion(4.42)parathatmeterinthelimΔt→regime0τMeof=1smallfromtimesectionsteps5the.2.2
TTsmallleadingtimetermsstepsofthGeMandstiffneGssKtearremsofinthe(6.24same)areoorderfominofrmaginfluencenitudeincoΔt.mparedWhtoileftheorvmaesrsy
termsthisisnotthecasein(6.25).Inordertostillreducethesystem(6.24),(6.25)to
onerelatingpressuresandaccelerationsthecontinuityequation
∙u=0isreplacedbyδ∙u˙=0,(6.26)
whichcanbedoneprovidedthatadivergencefreeinitialvelocityfieldisused.The
substitution(6.26)doesnotchangethesystembutrathersomewhatanticipatestheeffect
oftemporaldiscretisation.
Alongwiththerestrictiontostabilisationvariantswithη=0,i.e.noadditionalze-
roandth(o6.25rder)interthemsmwithinalltimethestastepbilislimaittioinnoptermseraoftor,notdalhisacallowscelerattoionswriteu˙andthenodalsystem(pressure6.24)
valuespandtoobtainthesimplifiedsystem
MFu˙+Gp=fΓF(6.27)
GTu˙−Cp=0(6.28)
wherethebarhasbeendroppedonthematricesofthefirstlineindicatingthatthe
stabilisationTtermshereareofhigherorderinΔtandthusofminorinfluence.In(6.28)
thematrixGrepresentsthesum
qTGTu˙=qGTM+GT˙u=−βτMe(u˙,δq)Ωe−βδ(∙u˙,q)ΩFn+1.(6.29)
eThematricesGTMandGThavebeendefinedinsection4.3.6andsection3.3.2,respec-
tivequately.ionAs(6.2the8)βpara=1metermighβtbescalesasstheumedpressureherewittesthoutfunctionlossofagndenerthalitusy.theentirematrix

uplingcoFSI6.Chapter

122Chapter6.FSIcoupling
Addedmassoperatorinthestabilisedcase
Analogousto(6.16)thesystem
IMIFIMFIΓGIu˙0
TTMFΓIMFΓΓGΓu˙Γ=fΓF(6.30)
GIGΓ−Cp0
isobtainedwhichyieldsthefluidaccelerationandpressuredependingonaprescribed
interfaceaccelerationu˙Γaccordingto
p=GIMIFI−1GI+CGΓ−GIMIFI−1MFIΓu˙Γ(6.31)
T−1TT
1−u˙I=−MIFI−1MFIΓ+MIFI−1GIGITMIFI−1GI+C
GΓ−GIMIFI−1MFIΓu˙Γ.(6.32)
TT
Inserting(6.31)andF(6.32)intothesecondlineofthesplitsystem(6.30)andmultiplying
bythefluiddensityρyieldsthephysicalfluidforceatthecouplinginterfaceΓ
fΓ=ρFvFMAstabu˙Γ=mFMAstabu˙Γ(6.33)
withtheaddedmassoperatorinthestabilisedcasegivenby
MAstab:=vFMFΓIMIFIGI−GΓGIMIFIGI+C
1−1T−1−1
GITMIFI−1MFIΓ−GΓT(6.34)
−1FMFΓIMIFI−1MFIΓ+1FMFΓΓ.
vvWithalumpedmassmatrixtheoperatorreducesto
1−MAl,stab=IFΓΓ+2GΓGITGI+vFCGΓT.(6.35)
Theexpression(6.35)suggeststhatsomeoftheeigenvaluesoftheaddedmassoperator
mightexceedone.Whileastrictmathematicalproveofthepositivityofthesecond
termin(6.35)appearsdifficultorevenimpossibletheoperator(6.35)physicallydescribes
thesameas(6.23)andmapsaninterfaceaccelerationu˙Γontoaninterfaceforcevector.
Correspondingtothecompositionoftheaddedmassoperatorthisinterfaceforcevector
consistsoftwoparts.Thefirstonesareforcesduetonodalinertiawhichareobtainedfrom
thefirsttermin(6.35).Thesecondpartoftheseforcesareforces−due1toincompressibility
resultingfromthesecondtermof(6.35).If2GΓGITGI+vFCGΓTwasstrictlynon-
positive,i.eifonewouldobtain
T−1T
u˙T2GΓGIGI+vFCGΓu˙≤0forallpossibleu˙(6.36)

6.3.Artificialaddedmasseffect123
thiswouldbeequivalenttoexclusivelynegativeenergiesgeneratedbytheincompressibility
attheinterface.Inotherwords(6.36)wouldimplythatallpossibleinterfaceaccelerations
resultineithernointerfacialforcesorsuchoftheoppositedirectiontotheinterface
acceleration.Thisisphysicallystronglyunreasonable.Itisthusnotfar-fetchedtoassume
thatthesecondtermin(6.35)exhibitsatleastsomepositiveeigenvaluesandtheoverall
eigenvalueoftheaddedmassoperatorinthestabilisedcaseexceedsone.
Asthelargesteigenvalueoftheaddedmassoperatorgovernsthestabilitylimitthe
addedmassoperatorinthestabilisedcasedeservesacloserlook.

Discussionoftheinfluenceofstabilisation
caseWhile(6.2s3)eemingthelyopvereryatorsimilar(6.35)totheexhibitassoddedmemasssignificopaerntatordifferencesobtained.Ainfirstthedifferenceunstabilisedis
rathenk.presThusencetheofptheositivestabilisatmatrixionCmatrensuresixCthatwhicthheistermrequiredinbraasctkheetsmatcanrixbeGinvisernotted.ofIffullC
ischosentoosmallthisexpressionexhibitsverysmalleigenvalueswhichmightincrease
themaximaleigenvalueoftheaddedmassoperator.Thustheproperlychosenamount
ofstabilisationiscrucial.Analternativeinterpretationisthatduetostabilisationthe
matincomprrixeC.ssibilitInythiscosensenditionfluidisrestalaxedbilisaantioneffecttosawhicmehisextentcausedalsobytstabiliseshepresencetheaofrtificialthe
oradded(6.35)maasssalleffect.theovHower-linedevertmatherericesisnodependdirectlinearlyinfluenceonδoftwhilehethetimestepremainingsizeinmat(6.34rice)s
areindependentofthetimestepsize.Consequentlytheonsetofadramaticinstabilityat
reducedtimestepsizemaywellhavefurtherreasons.
Asecondimportantdifferencetotheunstabilisedcaseisthenowobtainedunsymmetry
oftheaddedmassoperatorcausedbyG=G.Acloserlookat(6.29)reveals

TqGu˙=−τMe(u˙∙n,δq)∂Ωe+δ(1−τMe)(∙u˙,q)Ωe
e=qG˜Tu˙+δ(1−τMe)qGeTu˙,(6.37)
ewhereGeTdenotesthecontributionofelementetoGT.Thusforδt→0andτMe→1
TthestabilisedversionofthecontinuityequationyieldsamatrixGwhichcontainsa
contributionoftheoriginaloperatormatrixGTthatvanishesforverysmalltimesteps
whereτMe→1.ItfurthercontainsaboundaryintegraltermG˜Temergingfromintegration
byparts.From(6.37)itcanbeobservedthatthecontinuityequationwithinthedomain
iscancelledatthesmalltimesteplimitwhiletheremainingboundarytermdemands
globalmassconservationbybalancingtheamountofinflowandoutflowwiththechanges
ofthedomainduetoadisplacementoftheinterfaceΓ.
Stabilityofthestabilisedflowproblem(4.32)hasbeenshowninseTction5.2.3bymeans
ofamodelproblem.ThisstabilitynowimpliesthattheexpressionGIGI+vFCin(6.35)
canbeinverted.Howeverduetotheunsymmetryofthismatrixwhichincreaseswith
decreasingtimestepsizetheminimaleigenvaluemightberathersmallcomparedtothe
unstabilisedcase.Thisyieldsapossiblylargemaximaleigenvalueoftheoveralladded

412

uplingcoFSI6.Chapter

massoperator.Anincreasedeigenvalueoftheaddedmassoperatorhoweverresultsinan
earlieronsetoftheinstabilityasitwillbeshowninsection6.3.4.
Numericalinvestigationsconfirmtheaboveinterpretation.AsthematrixGTMisre-
conquiredtributefortothecothensstaistencybilisingoftheeffectsstaofbilistahetionstabilisatmethodioninttermhestranitscanientbecaseomittedanddoewithosnoutt
sacrificingstabilitTyofthefluidequations.Inparticularithasbeenobservedthatremov-
ingthematrixGMandthusintroducingaconsistencyerrorrestoresthestabilityforsmall
timestepsizesatleastforsometimediscretisationschemes.Howeverthisomissionisnot
meanttobeusedforpracticalcomputationsbutratherunderstoodasameanstoclarify
theeffectofstabilisationontheartificialaddedmasseffect.
Aderivationofadiscreterepresentationoftheaddedmassoperatorrevealsthatwhile
inanunstabilisedcasethisoperatorispurelygeometricalinthestabilisedcaseitdepends
uponthetimestepsize.

6.3.4Influenceofthediscretisationintime
Introducingthephysicalcouplingforce(6.21)or(6.33)intothediscretelinearisedstruc-
turalequation(6.14)whichhasbeensplitintointerfacedegreesoffreedomatΓand
remainingonesyields

MISIMSIΓd¨IKISIKSIΓdI0
MSΓIMSΓΓd¨Γ+KSΓIKSΓΓdΓ=−mFMAu˙Γ,(6.38)
wherewithinthestaggeredschemethefluidinterfaceaccelerationu˙ΓSisobtainedfroma
structuralpredictionofthenewinterfacedisplacement.ThematrixKdenotesthestruc-
turaltangentstiffnessobtainedfromalinearisationoftheinternalstructuralforcesNS(d).
Equation(6.38)revealswhyMAisnamed‘addedmassoperator’.¨Identifyingthe
profluidduinctmterfaceFMacceleworrksatioansau˙ΓnawithdditionathelstrmauctssuralonintheterfainceterfaceacceleratdegreesionodfΓshofreedom.wsthatthe
AInthecontextofthecoupledFSIproblemtheoveralltimediscretisationschemeis
composedofthesingleschemesappliedonthefluidandstructuralfield,thestructural
predictorandthewayofdeterminingthefluidDirichletboundarycondition.Forthe
subsequentanalysisthetimediscretisationschemesemployedtosolveatemporallydis-
cretisedversionof(6.38)aredistinguishedintotwomainclasses.Oneclassofsuchoverall
timediscretisationmethodsyieldsfluidandstructuralinterfaceaccelerationsdepending
uponalimitednumberofpreviousinterfacepositions.Theseschemesshallbetermed
schemeswithlimitedrecursion.Ontheotherhandthereareschemeswhichyieldexpres-
sionsfortheaccelerationsofthetwofieldswhichdependuponallpreviouslycomputed
interfacepositions.Thoseformulationswillbecalledrecursiveoralsofullyrecursive.
Withinthesubsequentanalysisdiscretisationitemsofdifferentorderofaccuracyare
combined.Clearlythelowestordercontributiongovernstheoverallorderofaccuracy
makingsomeoftheconsideredschemesratherunattractive.Howeverthecombinations
servetoenhanceunderstandingratherthanbeingapplicableschemes.Itcanfurtherbe
observedthatacombinationofhigherorderdiscretisationitemsworsensthestability
.blemspro

125

6.3.Artificialaddedmasseffect125
Schemeswithlimitedrecursion
Thestabilityorinstabilityofthescheme(6.38)solvedinasequentiallystaggeredmanner
depstructurendsalupongenerathelisepard-αticulartimetimediscretisatidiscretisatioonscnhemeemploisyoed.btaineThedwmohenststmaableximalversionnofumericalthe
dissipationisinvolved,i.e.whenthespectralradiusoftheschemeissettoρ∞=0.0.This
rameterpahetyields

3αm=−1,αf=0,β=1,γ=2.
Usingtheseparametersin(2.21),(2.22)and(2.26)to(2.24)allowstoobtainanexpression
forthestructuralaccelerationintermsofdisplacementsreading
tΔd¨α=122dn+1−5dn+4dn−1−dn−2.(6.39)
Thefluidaccelerationu˙Γisalsoexpressedintermsofstructuraldisplacements.Using
backwardEulertimeintegration,thezerothorderinterfacepredictor(6.2)andthefirst
orderinterpolationoftheboundaryconditionattheinterfaceyields
u˙Γn+1=12dΓn−2dΓn−1+dΓn−2.(6.40)
tΔInserting(6.39)intothediscretelinearisedstructuralsystemofequationsgives
tΔMS122dn+1−5dn+4dn−1−dn−2+KSdn+1=fn+1,(6.41)
wherefn+1representsforcesonthestructureatthenewtimeleveln+1.Forvery
smalltimesteps(6.41)isdominatedbythestructuralmasstermwhilethestiffnesslooses
influence.Omittingthestiffnessandlumpingthemassterminatemporallydiscretised
versionof(6.38)allowstoreducethesystemtotheinterfacialdegreesoffreedomaccording
tomS2dΓn+1−5dΓn+4dΓn−1−dΓn−2+mFMAdΓn−2dΓn−1+dΓn−2=0.(6.42)
expandedintermsoftheeigenvectorsviofMA,i.e.dΓn=idinvi.Thescalarcoefficients
AstheaddedmassoperatorMAisarealpositivematrixallvectorsin(6.42)canbe
dihavetosatisfy
F2din+1−5din+4din−1−din−2+µimmSdin−2din−1+din−2=0,(6.43)
whereµirepresentstheitheigenvalueofMA.Insertingtheamplificationfactorλiwith
din+1=λidininto(6.43)yieldsthecharacteristicpolynomialof(6.43)associatedwithµi.
Fm2λi3−5λi2+4λi−1+µimSλi2−2λi+1=p(λi)=0(6.44)
Allsolutionsλiof(6.44)havetosatisfy|λi|≤1ifthescheme(6.42)isstable.Thediagram
infigure6.1showsanarrayofcurvesp(λi)forFanSincreasingmaximaleigenvalueofthe
addedmassoperatorandfixedmassratioofm/m=1.Thecurveexhibitsadouble

612

10

4

uplingcoFSI6.Chapter

54)iλ0(p3-5increasingµi
2µimF/mS=1
-10-2-1.5-1-0.500.511.52
λiFigure6.1:Polynomialp(λi)accordingto(6.44)fordifferentvaluesofµi

rootatλi=1whichisatypicalbehaviourofpolynomialsobtainedfromextrapolations.
Itcanbeobservedthatincreasingµi(orequivalentlyincreasingthemassratio)shiftsthe
lowestrootfurtherleftincreasingitsabsolutevalue.
Usingthedoublerootλi;1,2=1athirdrootisfoundbysolvingtheremaininglinear
ionequat

Fm2λi;3−1+µimS=0,
yieldingλi;3≤1/2.Thesystemisunstableifλi;3<−1whichgivestheinstability
ionconditFmmSimaxµi>3.(6.45)
Whilethethirdorderpolynomial(6.44)allowstobesolvedexactlythisisnolonger
possiblewhenmorecomplicatedtimediscretisationschemesareemployed.Observing
thatthepotentialinstabilityisfoundforλi<−1ratherthanλFi>S1theapproachin[38]
ispursuednotingthatp(−∞)=−∞andp(−1)=−12+4m/mµi.Achangeinsign
betweenp(−∞)andp(−1)indicatesasolutionp(λ∗)=0withλ∗<−1whichagain
yieldstheinstabilitycondition(6.45).
highAsnallumertheicalingrediendampingtsonofbtheothabothevescstructurhemealareandveryfluidgood-parnatisturedinvinolved,the(6.4sense5)isthata
verypermissiveresult.RepeatingtheanalysisbyusingBDF2(3.14)ratherthanBEto
discretisethefluidpartintimeyields
u˙Γn+1=123dΓn−7dΓn−1+5dΓn−2−dΓn−3.(6.46)
tΔ2Inthiscasethecharacteristicpolynomialisgivenby
F4λi4−10λi3+8λi2−2λi+µimmS3λi3−7λi2+5λi−1=p(λi)=0

6.3.Artificialaddedmasseffect
andyieldstheinstabilitycondition

712

Fm3mSimaxµi>2.(6.47)
ChangingfromfirstorderaccurateBEtoBDF2onthefluidpartoftheproblemresultsin
aninstabilityconditionwhichistwiceasrestrictive.Intable6.1theinstabilityconstants
Cinstoftheinstabilitycondition
FmmSimaxµi>Cinst(6.48)
obtainedwithgeneralised-αtimeintegration(withρ∞=0)ofthestructuraldomain,first
orderinterpolationattheinterfaceΓ(6.5)andthedifferentstructuralpredictors(6.2)-
(6.4)aresummarised.
Table6.1:InstabilityconstantCinstincondition(6.48)obtainedforsequentiallystaggered
fluid-structureinteractionschemesdependinguponthestructuralpredictors(6.2)-(6.4)
andthefluidtimediscretisationscheme
BDF2BErpredicto0thorder323
1storder53103
2ndorder3161

AnotItheshorwsremathatrkableincreasedresultaisccuthatracyswitcresulthingsinafromBsignificaEtonBtlyDF2eaorlierntheonsetfluidofthedomaininstabilitresultsy.
inaninstabilityconditiontwiceasrestrictive.
Remark6.3.1Moreaccurateextrapolationintimeyieldsacloserrestrictionofthesta-
orbilitderyconstapredictornt.sevenSimilartuallyobservcausedatiodivnshaergencevebofeenanmadeiterabtivyeaHunpproadcinh[145used]forwheremhiultiscalegher
structurremindsalofathenalywseis.llknoThiswnseffehrctinkingappearstabstoilitybedomainhereninftorinphigherolynoordermialbacextrakwpardolatdiffeionreandn-
tiationformulae[112].
ofRemchaarkracte6r.3ist.2icpInothelynopresmialsentconresultingtextforneommigtimhetbdiseincretisaterestedtioninscthehemesgenerawithlblimitedehavrioure-
iscursion.relatedtoAppatherenfatlyctsthaucthapocolynonstanmialstdoadisplacemenlwaystexcorhibitrespaondsdoubletozerorootvateloλcity=1andawhiclsoh
n.acceleratiozero

Schemeswithfullyrecursivecharacteristics
CouplingschemeswhichemploytheTRforfluidtimeintegrationorusethegeometrically
correctversionfortheDirichletboundaryconditiononthefluidstructureinterface(6.8)
arealittlemoredifficulttotreatthanthepreviousones.Suchalgorithmsdonotexhibit
anexpressionforthepredictedfluidinterfaceaccelerationintermsofalimitednumber

128Chapter6.FSIcoupling
inofterfapreviocepusinositioternsfacetobecodisplacemennsidered,tscoi.e.mparcycleabledotwonto(6.4t0)hebutinitiaralcothernditions.requireallprevious
Tosamplethefollowingschemeisconsidered:
•structure:generalised-αwithρ∞=0
fluid:•–timediscretisation:TR
–predictor:zerothorder(6.2)
–Dirichletboundarycondition:firstorder(6.5)
Heretheparticularinfluenceofthetrapezoidalruleemployedonthefluiddomainshallbe
orinvdeerstigaccatured.ateThepredictorresultsaobtndbainedoundforaryafullyconditionsecondareorevdeenrscmorehemerestwithrictivTRe.andasecond
Whilethestructuralaccelerationisgivenbytheexpression(6.39),thefluidacceleration
isobtainedfrom
u˙Γn+1=122dΓn−4dΓn+2dΓn−1−u˙Γn,
tΔthewhicschhemecannotthbusedepexpreendssseduponexclusivtheelyactuainlintimeterfacstep.eAssdisplaumceingmentthas.tThetheainitialmplificacondittionionof
includeszeroaccelerationoftheinterface,i.e.u˙Γ0=0thefirststep(n=0)yields
Fm2dΓ1−5dΓ0+MAmS2dΓ0=0
andthecharacteristicpolynomial
2λ−5+2µimSF=p(λ)=0.
mSFWithconditpion(λf=or−∞the)fir=st−∞stepwandhicph(λis=uns−t1)able=if−7+2µim/moneobtainstheinstability
FimaxµimmS>27(6.49)

issatisfied.Withn=1
Fm2dΓ2−5dΓ1+4dΓ0+MAmS2dΓ1−6dΓ0=0
isobtainedandsois
Fm2λ2−5λ+4+µimS(2λ−6)=p(λ)=0.
Asp(λ=−∞)=∞andp(λ=−1)=11−8µimF/mSthecorrespondinginstability
readsionconditF11mimaxµimS>8.(6.50)

6.3.Artificialaddedmasseffect

912

Analogouslythenext(n=2)stepgives
2dΓ3−5dΓ2+4dΓ1−dΓ0+MAmSF2dΓ2−6dΓ1+8dΓ0=0,
mwhichbymeansofthecharacteristicpolynomial
F2λ3−5λ2+4λ−1+µimS2λ2−6λ+8=p(λ)=0
myieldstheinstabilitylimitofthethirdstepreading
F3mimaxµimS>4.(6.51)
Theinstabilityconditionsofallfurtherstepsnisobtainedaccordingly.Itisgivenby
F12mimaxµimS>8nforn>1,(6.52)
whichshowsthattheschemewithfixedgeometryandmassratiobecomesunstableafter
alimitednumberofstepsirrespectiveofthemassratioortheaddedmassoperator
themselves.Similarresultscanbeobtainedwhenthecouplingcondition(6.8)isemployed.
Additionally,thecombinationofhigherorderingredients(timediscretisationscheme,
predictor,couplingcondition)andarecursiveschemedestabilisesevenfaster.
Thusafluidtimediscretisationschemewhichemploysthetrapezoidalrule(oralso
theone-step-θschemewithθ=1)orthegeometricallycorrectcouplingcondition(6.8)
cannotbeemployedinasequentiallystaggeredschemeifstablelong-timesimulationsare
tobeguaranteed.

6.3.5Consequencesoftheartificialaddedmasseffect
Summarisingthepreviousresultsitcanbestatedthatsequentiallystaggeredschemes
yieldaninstabilityconditionoftheform(6.48)wherethemaximaleigenvalueofthe
addedmassoperatormaxiµiislargerthanone.TheconstantCinstontheotherhandis
smallerthanoneforallschemeswithanaccuracywhichisofinterest.Withrespectto
themassratiodifferentcasescanbeconsidered.
•Fluidmassdensitylargeristhanstructuralmassdensity.Thisisthecaseforinter-
actionsofwaterwithlightstructuresabletoswimsuchasfoilsfromsynthetics.A
widevarietyoftechnicalfluid-structureinteractionprocessesfallsintothiscategory.
Suchproblemssimulatedbyasimplesequentiallystaggeredschemewilldefinitely
ble.unstaeb•Fluidandmassdensityareofcomparablemagnitude.Typicalapplicationsin-
cludebiologicalorphysiologicalflowslikebloodflow-vesselinteraction.Sequentially
staggeredsimulationsofsuchproblemsaretypicallyunstable.Infactinstabilities
observedinsequentialbloodflowsimulationswereonedrivingforcefortheinvesti-
gationoftheartificialaddedmasseffect[38,80].
•Thestructuralmassdensityissignificantlylargerthanthefluidone.Thisisthecase
formostinteractionsofstructureswithairflow.Hereconditionallystablesequential
computationsmightbepossible.Additionallytheseapplicationsfrequentlyrequire
compressibleflowwheretheartificialaddedmasseffecthasnotbeenobserved.

013

uplingcoFSI6.Chapter

Altogetherithastobeconcludedthattheartificialaddedmasseffectexcludessequential
partitionedschemesforawiderangeofinterestingapplications.

6.3.6Generalinstability
Itremainsthequestionifbyintelligentlytuningthediscretisationitemsaschemecanbe
definedwhichisstableirrespectiveofthemassratio.In[193,194]modificationsatthe
load-andmotiontransferinthecontextofcoupledaeroelasticproblemsaresuggested
toimprovetheaccuracyandstabilityoftheoverallschemewhichshowbeneficialin
particularforcompressibleflow.
Howeveritturnsoutthatwhenincompressibleflowisconsideredthesemodifications
mightpostponetheonsetoftheinstabilitywhilebeingunabletoactuallypreventit.
Irrespectiveoftheparticulartimediscretisationschemesthesequentialstaggeredcoupling
schemeitselfcarriesaninherentinstabilityasstatedbythefollowingtheorem.
Theorem4Foreverysequentiallystaggeredschemeconstructedasdescribedinsection
6.2.1,amassratiomF/mSexistsatwhichtheoverallalgorithmbecomesunstable.
Proof.Foreverysequentiallystaggeredschemethestructuralpredictorforthedis-
placementoftheinterfaceΓattimeleveln+1containspreviousstructuralinformation
uptotimelevelnonly.Thusthegeneralappearanceoftheschemeis
ajdΓj+MAmmSbkdΓk=0,(6.53)
n+1Fn
=0k=0jwhereaj,bk∈Rarethecoefficientsdefiningtheparticulartimediscretisationscheme.
Equation(6.53)yieldsthecharacteristicpolynomial
FpS(λ)+µimSpF(λ)=p(λ)=0,(6.54)
mwherethepolynomialdefinedbythetimediscretisationofthestructureisdenotedby
pS(λ)=jn=0+1ajλjwhilethepolynomialpF(λ)=kn=0bkλkcontainsthetemporaldis-
cretisationofthefluidpartition,thetypeofthestructuralpredictorandDirichletcoupling
velocity.ThepolynomialpS(λ)isonedegreehigherinλduetothesequentialstructure
oftheproblem.Thusoneobtains
pS(λ=−∞)
pF(λ=−∞)<0,
i.e.bothpolynomialsareofdifferentsignintheinfinitenegative.Asthepolynomialsare
continuousthisimpliesthatapointλ∗<−1canbefoundwhichsatisfies
pS(λ∗)
pF(λ∗)<0.
Consequentlythereexistsapositivecoefficientasuchthat
pS(λ∗)+apF(λ∗)=0.
Hencefora=maxiµimF/mSthecharacteristicpolynomial(6.54)exhibitsasolutionwith
|λ∗|>1andtheunderlyingschemeisunstable.

6.3.Artificialaddedmasseffect

113

6.3.7Furtherinfluencesontheartificialaddedmasseffect
Forasimplifiedcoupledmodelproblemanaddedmassoperatorcanbedefinedtheeigen-
valuesofwhichpreciselypredicttheonsetofinstabilitiesasshownbyCausinetal.
in[38].Realfluidstructureinteractionproblemshoweverintroduceasignificantnumber
ofadditionalinfluences.Therearephysicaleffectswhichbringmorecomplexity.

•Firstofallfluidviscosityandstructuralstiffnessthelatterpossiblyincludingcontri-
butionsfromgeometricnonlinearitiescannotbeignoredwithinsuchapplications.

•butionsfromgeometricnonlinearitiescannotbeignoredwithinsuchapplications.
Usingimplicittimediscretisationonbothfieldsresultsinanstabilisingeffectof
structuralstiffnessontheaddedmassinstability.Viscousfluidforceshoweverin-
creasetheinfluenceoftheinstability.Clearlybotheffectsdependuponthetime
stepΔtasareducedtimestepsizedecreasestheinfluenceofthestiffnessterms
comparedtothemassmatrices.
•Furthertherearenonlinearitiesduetotheconvectivefluidtermandalsomaterialor
geometricalnonlinearitiesofthestructuralbehaviour.Typicallysimulationsstart
fromareferenceconfigurationandzerovelocities.Nonlinearitiesnotdominating
withinthefirstfewstepsbuildupandprecludetheexistenceofalinearaddedmass
operatortheeigenvaluesofwhichcouldpreciselypredictthestabilityorinstability
ofacalculationfortheentiresimulationtime.
•Additionalnonlinearitiesalsoemergefromthegeometricalchangesduetothedis-
placementoftheinterfaceΓ.Changinggeometrymeanschangingintegrationdo-
mainandthusachangeofallcoefficientmatricesencountered.
•Frequentlycompressiblestructuresareemployedwhereachangeoftheeffective
structuraldensityρShastobeexpectedwhichmayalsoinfluencepotentialinsta-
bilities.

Thusthephysicalfluid-structureinteractionproblemishighlynonlinear.Howeverthe
insightgainedbyconsideringalinearisedversionandthestabilitythereofsignificantly
helpstounderstandtheinstabilityoftheproblemathand.Whiletheanalysisperformed
doesnotnecessarilyallowtopredicttheonsetoftheinstabilitypreciselyitexplainswhy
stableschemesfortheinteractionofincompressibleflowandlight-weightstructureshave
und.foeenbnot

6.3.8Numericalinvestigation
Theclassicaldrivencavityproblemequippedwithathinflexiblebottomisusedtonu-
mericallyinvestigatetheaddedmassinstabilitywithinafullfluidstructureinteraction
environment.Theexamplewhichistakenfrom[182,183,228]isdepictedinfigure6.2.
Thefluiddomainisdiscretisedby32×32stabilisedbilinearelements.Thestructureis
modelledby32×1geometricallynonlinearwallelements.
Theinfluenceofdifferentparametersanddiscretisationschemesontheonsetofthe
instabilitywithinthetimeintervalt∈[0s;100s]shallbeexamined.Todiagnosestability
orinstabilitythehistoryoftheverticalcomponentoftheFSIcouplingforceatnodeAis
monitored.Oscillationsinthecouplingforceindicateinstability.

213

FSI6.Chapteruplingco

yux(t)=1−cos25πtm/sfluidviscosity:νF=0.01m2/s
fluiddensity:ρF=1.0kg/m3
inflowfreeoutflowstructuredensity:ρS=500kg/m3
Young’smodulus:E=250N/m2
Poisson’sratio:νS=0.0
fluiddomain:ΩF=1m×1m
Aflexiblebottomxstructurethickness:t=0.002m
Figure6.2:Geometryandmaterialdataofdrivencavityexamplewithflexiblebottom

Thedefaultalgorithmisthemoststableschemetobefound.Generalised-αtime
integrationofthestructurewithzerospectralradiusandBEonthefluiddomainis
used.Furtherthesimplepredictor(6.2)andDirichletboundarycondition(6.5)are
employed.AtatimestepofΔt=0.1stheproblemcanstablybeintegratedintimeup
toatleast1000timesteps.

TheinfluenceofthestructuraldensityρSiscomparedforBEandBDF2time
discretisationonthefluiddomain.Fromthepredictionsummarisedintable6.1itis
expectedthatroughlyhalfthestructuraldensityrequiredtostablyintegratewithBDF2
sufficesifBEisusedonthefluiddomain.Startingfromthedefaultparametersettingand
decreasingthestructuraldensityρSthesimulationbecomesunstabletowardstheendof
theinvestigatedtimeintervalatρS=321kg/m3.Theonsetofthisinstabilityisdepicted
inthediagraminfigure6.3.

0.0030.00250.0020.0030.0015AFy0.0010.00280.0026
0.00240.00050.00220.00209092949698100
-0.0005020406080100
defaultρSS=500kg/m33timeins
reducedρ=321kg/m
Figure6.3:EvolutionoftheverticalcouplingforceatpointAforthedefaultconfiguration
ofparametersandaproblemwithlowerstructuralmassdensitybothevaluatedwithBE
timediscretisationofthefluid
AsimilarprocedureisrepeatedwithBDF2timediscretisationonthefluiddomain.In

6.3.Artificialaddedmasseffect133
thiscasetheproblemisunstableatρS=500kg/m3andthestructuraldensityisincreased
inuntilthethediagrsimamulaintionfigrureemains6.4.stableIncreasingwithinthethesttimeructurainltervmaalssofindensityterest.fromResρSul=tsa50re0kgsho/mwn3
toρS=550Skg/m3delays3theonsetoftheinstabilityandalsoslightlydampstheinstability
itself.Atρ=590kg/mnoinstabilityisobservedwithinthetimeintervalofinterest.

0.0030.00250.0020.0015AyF0.0010.0050.0040.0030.00050.0020.0010-0.0010-0.002-0.003-0.000551015202530
020406080100
sintimedefault,BEρS=500kg/m3BDF2,ρS=550kg/m3
BDF2,ρS=500kg/m3BDF2,ρS=590kg/m3
Figure6.4:EvolutionoftheverticalcouplingforceatpointAobtainedwiththedefault
parametersettingandproblemswithdifferentstructuralmassdensitiesobtainedwith
BDF2timediscretisationonthefluidfield

Theinfluenceofthestructuralstiffnessshallbeconsiderednext.Departingagain
fromthedefaultconfigurationwithBE,Δt=0.1sandρS=500kg/m3thestructural
stiffnessisreducedaimingataschemewhichbecomesunstablewithinthetimeinterval
ofinterest.However,thisisnottobereached,indicatingthatthedefaultparameter
configurationisactuallystableevenwithoutthehelpofthestructuralstiffness.

Theinfluenceofthestructuralpredictorisalsoworthtobeinvestigated.Employ-
ingthefirstorderaccuratepredictor(6.3)ratherthan(6.2)yieldsanimmediatelyunstable
schemeatρS=500kg/m3evenifBEisusedtointegratethefluidequationsintime.Us-
ingthesecondorderaccuratepredictor(6.4)thebehaviourgetsevenworseasshownin
thediagram6.5.Fromtable6.1anestimateofthestructuraldensitynecessarytostabilise
thesimulationcanbeobtainedbyusingtheratioofthestabilityconstantscalculatedwith
differentpredictors.ThezerothorderpredictorisstabledowntoρS=322kg/m3,thusthe
firstorderpredictorshouldbestableforstructuraldensitiesexceedingρS=1610kg/m3.
Thispredictionfitsverywellasthefirststablesimulationwiththefirstorderpredictoris
obtainedatρS=1635kg/m3.Similarobservationscanbemadewhenthesecondorder
accuratepredictorisusedwhichyieldsaninstabilityconditionninetimesasrigorousas
thezerothorderone.Thusthesmalleststructuraldensitywhichshouldallowastable

134Chapter6.FSIcoupling
computationisρS=2898kg/m3.ActuallyatρS=2300kg/m3nooscillationsareob-
servedwithinthetimeintervalofinterestwhileoscillationsoccurwithinthenextfew
.epststime

0.0030.00250.0020.0015AyF0.0010.00050-0.00050

0.010.00150.008AyF0.0060.0010.0040.0020.00050-0.0020-0.00400.511.52
-0.0005020406080100
sintimedefaultalgorithm:zerothorderpredictor
firstorderpredictor
secondorderpredictor
Figure6.5:EvolutionoftheverticalcouplingforceatpointAevaluatedwithdefaultpa-
rametersettinganddifferentpredictors:withinthefirstfewtimestepsviolentinstabilities
areobservedifhigherorderpredictorsareemployed

TheinfluenceofthetimestepsizeisinvestigatedbystartingagainfromBEtime
integrationonthefluiddomainandΔt=0.1swithρS=500kg/m3.Stablecomputations
uptot=100canbeobtainedforallΔt≥0.005swhererapidlygrowinginstabilities
areobservedforsmallertimesteps.Thetemporalevolutionoftheverticalcomponent
ofthecouplingforceatpointAisdepictedindiagram6.6.Duetotheprevioustest
regardingthestructuralstiffnessitissuspectedthatthesuddenonsetoftheinstabilityis
notcausedbythedecreasinginfluenceofthestructuralstiffnessbutratherbyanincrease
ofthemaxTimaleigenvalueofMAstabcausedbyadominatinginfluenceofthestabilisation
operatorGM.

TheeffectofstabiliTsationatsmalltimestepsisconsideredbyremovingthesta-
scalesbilisatiowithnopetheratorinfluencGM.eofThisthecatrusesansienatconsisterm.tencyRemoerrovingrthiswithintermthestaapparenbilisatiotlynrestoreswhich
thatemptorthealinststabilitabilityyforatvsceryhemessmallwhictimehhavstepsebeisencastausebledbyforthelargerinfluencetimeofstepsthethusstabilisatstressingion
ateterms.lyforAtsheimcorrulatioectnlywithstaabilisetimedformstepulatiosizenofcaΔntb=ep0.00erfo01srmedwhicwhenhgetsBE,theunstablezerothoimmedi-rder
predictor(6.2)andthefirstorderboundarycondition(6.5)areemployed.

6.3.Artificialaddedmasseffect

0.003

0.002

AyF0.001

0

135

0.004AFyinstabilityatΔt=0.003s0.003
0.0020.0040.0010.0030.0010.002-0.0010.00101919.219.419.6
0-0.002-0.003instabilityatΔt=0.004s
-0.00400.10.20.30.4
020406080100
defaultalgorithmΔt=0.1stimeins
ΔΔtt==00..004003ss
Figure6.6:EvolutionoftheverticalcouplingforceatpointAevaluatedwithdifferenttime
stepsizesandzoomontheregionswhereinstabilitiesoccur;temporallystableintegrations
areobtainedforallΔt≥0.005s
Someresultsaredepictedinthediagraminfigure6.7wherethesolidlinerepresentsa
computationwithdefaultparametersettingTbutatimestepofΔt=0.003sandwithout
theinfluenceofthestabilisationmatrixGM.Thesimulationhasbeenstoppedafter
10000stableTstepswithoutanysignofaninstability.Interestinglythesamecomputation
includingGMgetsunstablebeforetheabsolutetimet=0.4sisreachedasitcanbe
observedindiagram6.6.Asignificantlydifferentbehaviourisobservedifthefluidis
integratedbythetrapezoidalrule.Inthiscaseaschemewithinstationarycharacteristics
isobtainedwhichgetsunstableirrespectiveofthemaximaleigenvalueoftheaddedmass
operator.RemovingthestabilisationmatrixGTMslightlydelaystheonsetoftheinstability
butcannotpreventitsonset.

613

0.003

0.002

AyF0.001

0

uplingcoFSI6.Chapter

0.002AyF0.0010.0010-0.0010-0.00200.511.52
051015202530
Δt=0.003BE,GTMremotimevedins
Δt=0.1TR,GTMremoved
Δt=0.1TR,fullstabilisation
Figure6.7:EvolutionoftheverticalcouplingforceatpointAforconsistentlystabilised
TfluidandremovedstabilisationmatrixGM

6.4Stablepartitionedschemes
Fortheapplicationsconsideredheretheartificialaddedmasseffectexcludestheuseof
simplesequentiallystaggeredschemes.Inordertoderiveaconvergentmethodwhich
ishighlyparameterindependentfurtherefforthastobemade.Withinthisworkan
acceleratediterativelycoupledschemeisemployedwhichhasshowntoberobustand
.lereliab

6.4.1Iterativelystaggeredschemesandtheneedforrelaxation
theTheusesolutioofnosub-ftheiteraretiospnsectivalloewsmonotoconlithicvsergectheme.heovTherallus,solutiodynamicnandwithinevkinemaerytictimecontinstepuittoy
ofarethesatisiterfiedativeupsctoahemeisnotdiscretisatioalwanysergror.uaranHoteedwev[18er,2,it1ha83s].bIneenpaobservrticularedCathatusinconetvergal.e[3nce8]
haalsoveshoinfluwenncesthatthethecoarnvergtificialencaeddedpropermasstiesoeffecftthecorpresenrestpinondingsequentiteraiallytivstaelyggstageredscgeredhemesap-
isproacrequirh.edTheinanathellysisimitinof[3Δ8]tre→v0ealstothaobtatrinelaconxatvionergencewithoaftherelaxaiterationtionparaovermetertheωfie<lds1.
AndindeedevenfortransonicaeroelasticanalysisconvergenceproblemsatsmallΔtare
reportedasforexampleinarecentpaperbyMassjung[174].
gencemaConsequenkingtlytheresclaxathemeionsefeasible.rvestoBydochotwoosingimpaonrtaoptntimaljobs.ornearFirstlylyitoptimalenablesrelacoxatnvioner-
parmethoameterd.Thisitisasecondlynimpaccortaneleratfetesacoturenvasergiteraencetionsincreaosveringthethefieldsefficofiencyfluidoftandhenstructureumerical

713

6.4.Stablepartitionedschemes137
tendtoberathertimeconsuming.
Aheuristicanduser-inputdependentwaytogetconvergencehasbeenreportedby
Tezduyaretal.in[222].Withinthe‘block-iterativecoupling’approachdescribed
in[222]theiterationbetweenthefluidandstructuralfieldisperformedinthecoreloop,
i.e.thecouplingconditionsaresatisfiedforiteratesofthefluidandstructuralsolution.
Thusconvergenceoverthefieldscanbeachievedbyanartificialincreaseofthestructural
lefthandsidemassmatrix.Thestructuralresidualvectorisleftunchangedensuringthat
aconvergedsolutioniscorrect.Thusconvergencedifficultiesofthecoupledproblemare
shiftedontothestructuralequationswhiletheamountofshiftrequiredremainsa-priori
wn.unknoMethodstoobtainanappropriaterelaxationparameterautomaticallyhavebeende-
velopedbyMoketal.in[182,183,228].WithinthepresentworktheAitkenmethod
isusedpreferably.ThischeapandeasyapproachtodetermineωisbaseduponAitken’s
accelerationschemeforvectorsequencesaccordingtoIronsandTuck[146].Whileno
rigorousanalysisoftheconvergencepropertiesoftheAitkenmethodisavailableithas
showntoworkverywellinvariousnumericalapplications.
TheAitkenmethodprovidesapossiblespecificationofstep5.insection6.2.1where
thefollowingsub-stepshavetobeperformed.
5a.Determinethedifferenceofthepresentandpreviousinterfacepositionaccordingto
ΔdΓn,i+1+1=dΓn,i+1−d˜Γn,i+1+1.

5b.ComputeAitkenfactor
n+1n+1Tn+1
µin+1=µin−+11+µin−+11−1ΔdΓ,i−ΔdΓ,i+1Δ2dΓ,i+1fori>0,
ΔdΓn,i+1−ΔdΓn,i+1+1
wheretheinitialAitkenfactorofanewtimestepisgivenbyµ0n+1=µnmaxandthe
veryfirstfactorofasimulationisµ01=0.
5c.Obtainrelaxationparameterfrom
ωi=1−µin+1.
TheabovestepsinvolveglobalvectoroperationsonlyandarethuscheapintermsofCPU
andmemory.ItcanfurtherbeobservedthattheAitkenmethodiseasytoimplement.
AnalternativetotheAitkenrelaxationparameteristhemethodofsteepestdescent
orgradientmethodwhichcanbefoundin[183].Thisapproachinvolvesthesolution
ofanauxiliarycoupledproblemtodeterminetherelaxationparameterωandisthus
considerablymoreexpensive.Astrongadvantageofthegradientmethodistheavailability
ofaconvergenceanalysis.

6.4.2Projectionbasedsemi-implicitscheme
Whilesub-iterationswithproperrelaxationallowtointegratetheFSIproblemstablyin
timeaniterativelystaggeredschemerequiresasignificantnumberoffluidandstructure
fieldsolutionspertimestep.Moreefficientformulationsarethusdesirable.

813

uplingcoFSI6.Chapter

Theartificialaddedmassinstabilityisintrinsicallyrelatedtotheincompressibility
condition.Thefluidpressureishighlysensitivewithrespecttothecorrectlydetermined
fluid-structureinterfacepositionandyieldsheavilywrongcouplingforcesforslightly
incorrectpredictionsofthestructuralinterfacemotion.Thusimplicitcouplingofthe
pressureisunavoidable.
MotivatedbythisfactFern´andezetal.proposeasemi-implicitFSIcouplingscheme
in[80].SolvingtheflowequationsusingaChorintypeprojectionschemedecouples
thepressuresolutionfromtheinversionoftheremainingoperator.Thustheviscousand
convectivetermsaresolvedoncepertimestepwhiletheprojectionstepwhichyields
thepressurehastoberepeatediterativelyuntilconvergenceofthefluidandstructural
solutionisobtained.ConditionalstabilityoftheschemeisprovenbyFernandezetal.
].[80in

6.5Summary

ApartitionedFSIalgorithmisusedwhichcanbeformulatedinanefficientsequentially
partitionedversionandastableiterativelystaggeredvariant.Accurateexchangeofcou-
plingdatarequiresthatafluidinterfaceconditionisusedwhichisconsistentwiththe
fluidtimediscretisation.Nodalfluidforcesareexertedonthestructureratherthaninte-
gratedstresses.Theseforcesexhibittheorderofaccuracyoftheprimaryvariablesand
fitverywellintoanodalbaseddatastructure.
Theartificialaddedmasseffectwhichisaninstabilityofthesequentialversionof
thepartitionedalgorithmiscloselyrelatedtotheincompressibilityoftheflow.Froman
analysisofthecoupledsysteminstabilityconditionsarededuced.Theseconditionsshow
thattheinstabilitydependsuponthemaximaleigenvalueoftheaddedmassoperatorand
thetimediscretisationschemesusedonthefluidandstructure.Evenmoreimportant
theinstabilityishighlyinfluencedbytheratioofthefluidandstructuralmassdensity.
Wheneverfluidandstructuraldensitiesarecomparabletheinstabilityisalmostunavoid-
able.Onlyincaseswherethefluidismuchlighterthanthestructureconditionallystable
schemesmightbepossible.
Theanalysisfurtherrevealsthathighertemporalaccuracynecessarilyyieldsearlier
instability.Itisalsoshownthattherearediscretisationschemesforthefluidfieldasfor
examplethetrapezoidalrulewhichyieldaninstabilityconditionthatgetsincreasingly
restrictivewitheveryfurthertimestep.Suchschemesnecessarilyfailtowork.
Extendingtheideawhichyieldstheinstabilityconditionitcanalsobeshownthatno
unconditionallystablestaggeredalgorithmcanbedesigned.Consequentlystableparti-
tionedschemeshavetobeiterativelystaggeredwhereatleastthecouplingforcesdueto
fluidpressurehavetobetreatedimplicitly.

7erChapt

esExamplicalNumer

Inadditiontotheacademicexamplesusedinthepreviouschapterstohighlightpar-
ticularnumericaleffectsanumberofapplicationsoftheentireFSIinteractionscheme
arepresentedwithinthischapter.Theseexamplesarenotonlydesignedtodemonstrate
thecapabilityofthealgorithmbutalsotopresentsomecharacteristicsofFSIproblems
includingmodellingandsimulationissues.

7.1Introduction

7.1.1Generalalgorithmicinformationandmodelling
locitThrouyandghoutpressureresidualhavebasedbeenstabusedilisoednthefinitefluidelemendomatsin.withTheequalostructurrderalinfieldterpisoladistionofcretisedve-
bygeometricnonlinearwallelementsornonlinear,three-dimensionalshellelementsfor
two-dimensionalandthree-dimensionalproblems,respectively.Inallcasestheinteraction
ofthefluidandstructuralfieldisachievedbysubiterationswheretheAitkenrelaxation
strategyhasbeenemployed.Typicallyfouruptomaximalteniterationsoverthefields
arerequiredtoconvergethecoupledproblem.
Theexamplesdescribedherearegivenwiththefulldatarequiredtoreruntheprob-
lems.Thisnotonlyincludesgeometryandfullmaterialdataalongwithdiscretisation
informationbutalsodetailsofthemodellingprocess.These‘tricks’usuallynotpublished
inaretohothewevercomplexitcloselyyrelatedthereof.toItthealsoacoupledppearsnaturelikelyoftFSIhatprtheoblemsmodeandllingofferinformafurthertionginsigivhent
heremighthelpandinspirethesolutionofsimilardifficulties.

7.1.2Afewcommentsoncomputationaltools
AllapplicationshavebeensimulatedusingtheresearchcodeoftheInstituteofStructural
Mechanicscalledccarat.Contributionstothiscodeandimprovementsthereofhavebeen
madeintheprogressofthiswork.Theresearchfiniteelementprogramoftheinstitute
hadbeenrelaunchedintheprogramminglanguageCin2002asamuchwiderframework
thantheparentFORTRANcode.ThemajorworkhasbeendonebyGeewhoprovided
anexcellentcomputationalenvironmentwhichisgratefullyacknowledged.Almostevery
FSIexamplewhichwassimulatedduringthecourseofthisworkalsohighlyreliedonthe
restartfacilitywhichisbasedonabinaryinputandoutputmodule.Thisrestartoffers
animmenseflexibilitybyallowingnotonlyamodificationofmaterialparametersortime
stepsizesbutalsoachangeinNeumannandalsoDirichletboundaryconditionsorthe

139

014

ExamplesNumerical.7Chapter

numberofprocessorsusedtoruntheprobleminparallel.Thiscoremodulewasadded
toccaratbyK¨uttlerwhoalsoprovidedahighlyvaluablesupport.
ThedesignofthenewccaratallowstocompileasequentialoraparallelMPIbased
version.Inparticularthethree-dimensionalexampleshavebeenruninparallelontwoto
essors.cprofourIterativesolversandpreferablythesolverpackageAZTEC(see[225]forproductde-
tails)havebeenemployedtosolvetheresultingfluidsystemofequations.Inparticular
astabilisedbiconjugategradient(BiCGSTAB)methodalongwithasymmetricGauss-
SeidelpreconditioneroranincompleteLUpreconditionerhavebeenusedwithinmost
examples.AlternativelyaGMRESmethodalsoavailablewithintheAZTECpackagehas
beenemployed.
Themeshandstructuralsystemofequationstendstobesmallerandmuchlessde-
manding.Hereadirectsolverorforlargerproblemsaconjugategradientmethodavailable
withintheAZTECpackagehavedoneagoodjob.
TheprogramGIDofCIMNEinBarcelona[46]hasbeenusedforpreprocessingand
meshgenerationandtosomeextentalsointhepostprocessingstage,inparticularforthe
three-dimensionalproblems.Fortwo-dimensionalproblemsthevisualisationtoolvisual2
hasbeenemployed[110].

7.2Bridgecrosssectioninlaminarflow

Thisexampleisdesignedtosimulatethefundamentalbehaviouroftheflowaroundaplate
girderbridgecrosssection.TheproblemgeometryisinspiredbytheTacomaNarrows
bridgewhichisfamousforitscollapsein1940.
Itwasafindingofthestudiesinitiatedbythiscollapsethatvortex-sheddingexcitation
andflutterinstabilityofasuspensionbridgecanberepresentedbyaspring-supported
modelofthecrosssection.Comparedtoacoupledanalysisoftheentirebridgeandthe
surroundingthree-dimensionalflowthisofferssubstantialsavingswhilestillexhibitingthe
advantagesofanumericalsimulation.Incontrasttowindtunneltestsnumericalanalysis
maybeperformedmuchfasterandatlowerexpenses.Inparticularpreliminarystudies
ofgeneralshapesmightefficientlybeperformednumerically.

7.2.1Geometryandmaterialdata
Thegeometryandboundaryconditionsofthetwo-dimensionalproblemwhichareadopted
fromtheworkofH¨ubner[129]aredepictedinfigure7.1.Asectionof1mofthebridge
profileisfixedatitscentreinx-directionandsupportedbyaspringwithalinearstiffness
ofky=2.467kN/miny-direction.Additionallyatorsionalspringofkα=126.33kNm
isattached.Asfluid-rigidbodyinteractionisnotdirectlysupportedwithinthepresent
codethecrosssectionismodelledbyaverystiffstructurewithYoung’smodulusof
Eart=5×109kN/m2andsupportedbytwoverticallinearspringsofastiffnessofky/2
andahorizontaldistanceofd=14.31m.TwoadditionaltrussmembersofEartandunit
areaintroducedoneithersideofthebridgeprofileallowtoreachthisdistance.This
correspondstoalinearisationofthetorsionalstiffnessasdescSribedinthe2diplomathesis
ofHilchenbach[126].Withastructuralmassdensityofρ=823kg/mthetotalmass

7.2.Bridgecrosssectioninlaminarflow

slipboundarycondition
40.0m90.0m

m0.35eefrwfloout

m3.0outm4.2constant12.0m
wfloinycitelovux=10.0m/s
yxslipboundarycondition
Figure7.1:Geometryofbridgecrosssectioninflowfield

m0.35

114

ofthecrosssectionpermeterlengthofthebridgeis4000kg/mandatotalmomentof
inertiaofabout80000kgm2/mcanbeobtained.
Thecorrespondingeigenfrequencyoftheverticalcrosssectionalmodeisfy=0.125Hz
whileafreerotationwouldtakeplaceataneigenfrequencyoffα=0.20Hz.These
frequenciesareclosetothoseoftheTacomaNarrowsbridgereportedin[165].Thefluid
isratherviscouswithakinematicviscosityofν=0.08m2/sandthemassdensityofair
ρF=1.25kg/m3.
Altogetherthisexampleisverygood-naturedandanimmediateconvergenceofthe
iterationoverthefieldsisobtained.

7.2.2Modellinganddiscretisation
UsingthewidthofthebridgeL=12masacharacteristiclengthoftheflowaReynolds
numberofRe=1500resultsfortheproblem.Clearlywindpassingabridgecorrespondsto
muchhigherReynoldsnumbersandhastobetreatedashighlyturbulentflow.However,
duetothesharpedgesofthepresentcrosssectiontheflowpatternisgovernedbythe
bridgedeckgeometryandtheinfluenceoftheactualReynoldsnumberisminor.Itis
thusreasonabletoexpectthatthefundamentalcoupledbehaviourandtheself-excitation
ofthesystemcanbemodelledwhileanaccuratepredictionofawindflowpatternwillnot
beachieved.Thisexpectationisalsosupportedbystudiesreportedin[126,129]where
vortex-sheddingatdifferentcrosssectionsiscompared.Thedimensionlesscharacteristics
suchastheStrouhalnumberaswellasliftanddragcoefficientsoftheflowacrossthe
presentprofilefitverywelltovaluesobtainedformuchhigherReynoldsnumbersas
reportedbyLarsenin[165].In[165]alsocomparisonstowindtunnelsectionmodel
testsarepresented.
Atotalof16028bilinearfour-nodedelementsisusedtodiscretisetheflowfieldand208
structuralelementsareemployed.Thediscretisationofthemovingmeshmanageswith
15127bilinearelementsasapartofthefluiddomainismodelledinEulerianformulation.
Particularcarehasbeentakentoassurearegularmeshinthedirectvicinityofthebridge

ChapterExamplesNumerical.7

142Chapter7.NumericalExamples
file.proAtimestepsizeofΔt=0.02sisusedwhichallowsahighresolutionofthecoupled
dynamics.Startingthesimulationtheinflowvelocityisacceleratedaccordingto
ux=u∞1sinπt−π+1,
2s22isemployedwherethesurroundingflowspeedisgivenbyu∞=10m/s.Thisstart-up
generatesasmoothflowaccelerationwithinthefirsttwoseconds.

7.2.3Results
Theinflowvelocityof10m/ssufficestocauseanincreasingexcitationofthebridgedeck
whicheventuallyyieldsfailureifnoadditionaldampingoccurs.Thetemporalevolution
ofthetotalverticalfluidforceupontheprofileisshownindiagram7.2.Accordingly
theoverallfluidmomentisgivenindiagram7.3.Bothdiagramsshowanapparently
stablephasefromabout50sto100swheretheliftforceaswellasthefluidmoment
ofmomentumexertedonthebridgedeckisverysimilartothatoftherespectivefixed
1.51.00.5fluidicalertvkNinrcefo
0.0-0.5-1.0-1.550100150200timeins
Figure7.2:Temporalevolutionofthetotalverticalfluidforceexertedonthebridgecross
iontseccrosssection.Butassoonasthevortexsheddinghasfullydevelopedasecondfrequency
correspondingtothetorsionaloscillationofthebridgecrosssectionsetsin.Whilethe
initialstructuralmotionisaslightverticaltranslation,therotationalmodedominates
soonafter.Afterabout150sthiseffecteventuallytakesoverandanincreasingtorsional
oscillationisobserved.Therapidgrowthofthepeakfluidforceandangularmomentum
priortotheendofthesimulationiscausedbythelinearisationofthetorsionalspring.
Thetransitioninthedynamicscausedbythefluid-structureinteractioncanalsobe
expressedintheStrouhalnumbers.Usingtheheightofthecrosssectionascharacter-
isticlengthscale,i.e.L=2.4mtheinitialStrouhalnumberSt1=0.112equalsthat
ofthefixedcrosssection(see[126,129]).Afterthecoupledsystemhasdevelopedthe
StrouhalnumberisSt2=0.048.
Imagesoftheflowfieldalongwiththedisplacedstructurearegiveninfigure7.4aswell
asinthecolourchartinfigure7.5.Infigure7.4streamlinesareplottedonthepressure
fieldandthepressurescaleoffigure7.5appliesalsohere.Thetimeinstantsdepicted

143

7.2.Bridgecrosssectioninlaminarflow143
.015.0105.00.0mkNintumnmemoularang-15.0
-5.0-10.050100150200timeins
Figure7.3:Temporalevolutionofthetotalangularfluidmomentumexertedonthebridge
nectioscrosscorrespondtopeakvaluesintheliftforceandthetorsionalmomentasitcanbeobserved
fromthediagrams7.2and7.3.
t=75.0st=125.0s

t=175.0s

t=200.0s

Figure7.4:Streamlinesonpressurefield
Thehorizontalvelocityatdifferenttimeinstantsispresentedonthelefthandsideof
figure7.5.Itnicelyhighlightsthesignificantinfluenceofthestructuralmotiononthe
flowfield.Thehorizontalvelocityincreasesinthecourseofthesimulation.Thiseffect
canalsobeobservedforthepressurewhereinparticularzonesofnegativepressuregrow
instrengthasaresultoftherotationofthecrosssection.Themutualexcitationofcross
sectionandintensityofthevorticesisshowninfigure7.5.Thecloseupviewsonthe
righthandsideoffigure7.5furthergiveanimpressionofthemeshanditsmotion.These
resultsagreewellwiththoseobtainedbyH¨ubnerreportedin[129].

144

s50=t

ts010=

=ts015

20=ts0

ExamplesNumerical.7Chapter

horizontalvelocityuxinm/s
0215−s75=t

s512=t

s517=t

aPinpressure07015−Figure7.5:Snapshotsofbridgedeckatdifferentinstantsintime:Horizontalvelocity
withisolines(left)close-upviewofcrosssectionareawithvelocityarrowsonpressure
t)(righfield

7.3.Channelwithbackwardfacingstepandmembrane145
7.3Channelwithbackwardfacingstepandmem-
braneThepresentnumericalexamplehasbeendesignedforseveralpurposes.Itdemonstrates
thecapabilityofthefluid-structureinteractionschemetosimulatethehighlytransient
dynamicsofamembranestructureinteractingwithairflow.Itisalsousedtocompare
asimulationcomputedwithlinearelementstooneperformedwithquadraticelementsof
theSerendipitytype.

7.3.1Geometryandmaterialdata
ThegeometryoftheinitialproblemisdepictedinFigure7.6.Inordertorelaxthemem-
braneandthusallowingpurebendingdeformationwithoutinducingmembranestresses
thestructureisrelaxedinapreliminarystepbymovingthepointA0.1cmtotheleft.

0cm.inflowwithparabolicprofiletopandbottom:no-slipboundaryfreeoutflow
1membranet=0.0025cm
Atoinp5cm.02cm0.5cm5cm16.5cm
Figure7.6:Initialgeometryofbackwardfacingstepproblem

Thefluidhasthematerialdataofdryairatabout25oC,i.e.akinematicviscosityof
ν=0.146cm/sandadensityofρF=0.0012g/cm3.Thedeformationoftheslackmem-
branepassedbytheflowisgovernedbybendingandthussmallstrainswhichjustifiesthe
assumptionofalinearSt.Venant-Kirchhoffmateriallaw.ThestructuralYoung’s
modulusisE=1.0×108g/s2cmrepresentingasoftrubberlikematerialwhilePoisson’s
ratiohasbeensettoνS=0.2.ThestructuralmassdensityisρS=0.5g/cm3.
Theinflowvelocityisincreasedlinearlyintimewithinthefirst0.048softhecompu-
tation.Eventuallytheairentersthedomainwithamaximalvelocityof120cm/satthe
centreoftheinflowboundaryline.No-slipboundaryconditionsareappliedatthetop
andbottomofthechannel.

7.3.2Discretisationandinitialmembranerelaxation
Discretisationinspaceandtime
Inordertoavoidaninitiallycurvedstructurethebottommembraneiscreatedasa
plane.Fiveortwoelementsareusedinthicknessdirectionofthemembranewithlinear
orquadraticelements,respectively.Thislargelyreducesshearlocking.Thequadratic
structuralelementsoftheSerendipitytypeareunder-integratedby2×2Gausspoints.
Overall21200linearor5280quadraticSerendipityelementsareemployedonthreefields
yieldingatotalof21908or16547nodes,respectively.Priortotheactualcomputation
themembraneisrelaxedbymovingthepointAtotheleft.

ExamplesNumerical.7Chapter

146Chapter7.NumericalExamples
set-upialInitTheconditmoionsvingandactmaionofterialpopaintArameters.consistsTheseoftwostepsarepreliminarnotyusedstepstowmoithdelandifferingyrealbophundaryysical
actionbutrathersupplytheinitialstateforthesimulation.Theinflowboundarycondi-
tionisfixedtozeroduringthesesteps.WithinafirstpreliminarysteppointAmovesto
theleftatconstantvelocity.Atthesametimeaconstantexternalpressureattheout-
sidesomeoffluidthetomemleavbretanehedoensuresmain.thatAftertheafewstructurelargebuctimeklesstepstowaastardsticthesituainnertionpaisrtreacforcinghed.
Theproblemisrestartedfromthisstateforasecondpreliminarystepwheretheexternal
pressureisremoved.Thiscausesthefluiddomaintoenlargeagainandthusinflowatthe
kfreeeptcrooutfloswsedbtoaoundarsteadyyconditiostaten.andDeprospitevidesbteingheillinitiaplosedshathepeofstepthecanmebmerunbrane.withfingers
Theinitialstepsaremadegood-naturedbyevokingartificialdamping.Tothisend
BEcausesisusedsignificaonnthtenfluidumericalanddaavermpingyloinwbspothectralfields.radiusFonurthertheanincrstructuraeaselddfluidomainviscowhicsithy
of5.146cm/sisinitiallyemployed.
Afteradjustingtheviscositytothevalueofairandselectinglessdissipativetime
branediscretisatiointeractsnscwithhemesthetheflowsimulatioexhibitingnofalthemostphexcysicallusivelyproblembendingisstarted.deformaThetion.slacTkhusmethem-
structureisextremelysoftandsensitivetothefluid−4forces.Thehighlytransientnature
oftheproblemrequiresatimestepofΔt=1.0×10s.
Thedynamicbehaviourofthepresentmembraneproblemexhibitsaninitialsnap-
throughwhichisdampedbytheactionoftheflowbutneverthelesshighlytransient.
Consequentlyitisnecessarytoincreasetheinflowvelocityfastenoughtoavoidnegative
atthepressuresfreeodueutflotowthebsoundauddenryenlaandthrgemenusantofill-tpheosedfluidprodoblemain.m.Thiswouldcauseaninflow

7.3.3Resultsontwomeshes
Dynamicsofaphysicallyverysensitivesystem
Animpressionofthehighlytransientdynamicsofthecoupledproblemisgiveninthe
figures7.7and7.8wheretheabsolutevalueofthevelocityfieldisdepicted.Theseresults
areobtainedonthelinearandquadraticSerendipitymeshes,respectively.Itcanbe
observedthatthedynamicbehaviourcomputedonthetwodifferentdiscretisationsisini-
tiallyverysimilarwhilethetwoevolutionsseparatesoonaftertheinitialphaseof0.048s.
Thisseparationisalsoobviousfromtheverticaldisplacementofthemembranecentre
pointasdepictedindiagram7.9.Atthelatestafter0.125sthetwosimulationspredict
adifferentbehaviour.Inparticularacomparisonoftheclose-upviewofthemembrane
areaaspresentedinfigure7.10revealsasmootherflowbehaviourobtainedwithlinear
elements.Asfarastheflowfieldisconcernedthisadditionalsmoothnessiscausedby
numericaldampinginducedbytheinconsistencyoflinearelements.Thediscretisationby
quadraticSerendipityelementsadmitsthedevelopmentofmorevorticeswhichgovern
theoveralldynamics.Howeverdifferentstructuralstiffnessesseemtoplayasignificant
role,too.Theresultssuggestthatfivelinearelementsusedinmembranethicknessdirec-
tionmightnotsufficetoremoveshearlockingeffectsinparticularasthoseelementsare

7.3.Channelwithbackwardfacingstepandmembrane

714

verystretchedduetorestrictionsimposedbytheresolutionofthefluidfieldalongthe
.ceterfainInasensethesedistinctlydisjointresultsareunfavourableandindicatethatatleast
oneofthesolutionsisnotconverged.Theresultssuggestthatabetterapproximation
isobtainedbyquadraticelementswhileevenhereafinermeshwouldbebeneficial.A
convergencestudyverifyingthecoupleddynamicshasnotyetbeenperformeddueto
limitationsintheavailablecomputationalresources.Howeverthesuggestionshallbe
madeherethatevenwithfinerresolutionsinspaceandtimeauniquesolutionwillbe
hardtoobtain.Itisthecharacteristicofthephysicsgoverningthedynamicsofthepresent
problemthatitishighlynonlinear.Itisthusalsoduetothephysicalsensitivityofthe
coupledsystemthatsmallperturbationsimmenselyeffectthetemporalevolutionofthe
system.

Computationaleffort

nTheumberofcomputaunknotionalwnsreseffortultingrequiredfromonquadrbothaticdiscrSerendetisatioipitynsiscelemenompartsdoable.esnotTheparyeoff.ducedIt
ismatratherrix.baThuslancedthebytimeanincreasedconsumednbumybterheofsolvnonerzeornoevaitheluesrmewithinshistheocompaverallrable.coefficienOnta
poneersonatimelstepcomputtoeokrabwithoutIn5tel30s.coreCo2duonsequenprotlycessotherovT72era00llwithcomput2.0ingGHztimetheforsimoneulaoftiontheof
discretisationssimulatedformorethan4500timestepsonthesamemachineamounts
aboutamonth.

Performanceofthepartitionedsolver

Duedifficulttoytheforverythethinpartanditionedflexiblesolver.strucEvtenureifthetheprefluidsentmassmemdebransitneyisproblemmuchpsosesmaallersptecialhan
thestructuralonethisexamplecanbynomeansbetreatedbyasequentiallystaggered
scheme.Aconsiderableartificialaddedmasseffectisobservedwhichalsoharmsthe
iterativelystaggeredprocedure,i.e.alargenumberofiterationsisneededineverytime
step.Upto15oreven20iterationsoverthefieldsarerequiredtoobtainaconverged
coupledsolutionwhileanonmonotoneconvergencebehaviourisobservedinthisiteration.

814

t=0.005s

t=0.045s

t=0.085s

t=0.125s

t=0.165s

t=0.205s

t=0.245s

t=0.285s

t=0.325s

t=0.365s

t=0.405s

t=0.445s

.7Chapter

ExamplesNumerical

0130cm/s
Figure7.7:Evolutionofabsolutevalueofthefluidvelocity|u|obtainedonlinearelements

7.3.Channelwithbackwardfacingstepandmembrane

t=0.005s

t=0.045s

t=0.085s

t=0.125s

t=0.165s

t=0.205s

t=0.245s

t=0.285s

t=0.325s

t=0.365s

t=0.405s

t=0.445s

914

0130cm/s
Figure7.8:Evolutionofabsolutevalueofthefluidvelocity|u|obtainedonquadratic
tselemenipitySerend

015

1a

2a

3a

4a

5a

ExamplesNumerical.7Chapter0.6cm2blinearelements5a6a
0.44a3a2a0.26b5b01a4b-0.21bquadraticelements
3b-0.4-0.600.050.10.150.20.250.30.350.4s0.45
Figure7.9:Temporalevolutionofverticaldisplacementofmembranemidpoint
t=0.045s

1bt=0.125s

2bt=0.205s

3bt=0.285s

4bt=0.365s

5bt=0.445s

6b6a0130cm/s
Figure7.10:Close-upviewofmembranearea;streamlinesonabsolutevalueofvelocity
obtainedonthelinear(left)andquadratic(right)mesh

115

7.4.VibratingU-pipe151
7.4VibratingU-pipe
Coriolisflowmetersareanelegantwaytomeasurethemassflowrateinapipe.The
measuringunitinsidesuchaflowmeterisaflexibletubewhichinitsclassicalformis
U-shaped,clampedatbothendsandpassedbytheflow.Thepipeissubjecttoaforced
vibrationinthecantilevermodeatangularfrequencyωf.Thisoscillationinducesopposite
andtimedependentCoriolisforceswithinthefluidintheinflowandoutflowpartof
thetube.Thustheresultingvibrationisnotjusttheenforcedbendingbutaccompanied
byanamountoftorsiondependinguponthefrequencyratioft/ffwhereftdenotesthe
eigenfrequencyofthetorsionalmode.

7.4.1Geometryandmaterialdata
tubeGeometryisfullyandclamampedterialattparahein-metaersndofoutflothewbsampleoutndaries.ubearegiveninfigure7.11wherethe

geometryinflow

cm0.10

woutflo

shellthickness:t=0.1cm

ialermat:estructurE=1.0×108g/(s2cm)
diameterPoisson’sratio:0.4
cm0.1.0cmρS=1.5g/cm3
10fluid:ρF=0.998g/cm3
AyBcm5ν=0.01012cm2/s
cm.05.1xz1.5cm2.0cm1.5cm
Figure7.11:Geometryandmaterialofflowmetertube

Incontrasttotechnicalflowmeterswhicharemadeofmetalsherethetubematerialis
arubberlikecompressibleneo-Hookeantypeofmaterialwhichyieldslargedeflections
oftheoverallsystemandthusallowstohighlightthephysicaleffect.Gravitypointsin
negativez-direction.Graviatationalforcesofthewaterinsidethetubeareconsidered
whileeffectofgravitationontheshellitselfisneglected.Theinflowvelocityofthewater
insidethetubeisprescribedtouy=15cm/s.

ExamplesNumerical.7Chapter

152Chapter7.NumericalExamples
7.4.2Modellinganddiscretisation
Themeshofthetubeitselfconsistsof5120linearfour-nodedshellelementsenriched
binymelineaarnsofelementhets[1enhan5].cFedaurtherssumedthestraAssumedinmethoNatd(uralEAS)Straitonsremo(ANS)velomectkinghodisinherenusedt
toconditavoidioningparasasitictrdescribansvederseins[99he,ar10s0]twrains,ithai.e.scaremolingfavingctorshearofCloc=1king.0.0isScaemledplodiryeedctotor
improvetheconditioningoftheresultingstructuralsystemofequations.
partTheofthefluidfluiddomaindomainismesishedbaccompay11520niedstabyabiliseddeformatrilineblearhemeshxahedrafieldlconsiselementingts.ofA10ma24jo0r
thepseudoprostblemcructuraonsisltsoelemenf268ts.80eThuslementhetsovyieldingerallameshtotalwhileof949still24bdegeingreesoquiteffrceoedoarsemfoonr
threefields.Toresolvethehigherfrequenciesofinterestadequatelyasmalltimestepof
Δt=0.005sisused.
toArsionahalmormonicdeffotisrceaofppliedaonfrequencythetipofofffthe=7c.la685mpHzedctulobseeptoointhetinginzeigenfrequency-direction.ofThethe
forceisdistributedovertheareaofthepartofthepipewhichisparalleltothex-axis.
Inthisexamplethechoiceofsuitableboundaryconditionsforthefluidfieldposesan
inclassicalterestingDircichlhallengete.typCleareoflyboattheundaryinflocowbndition.oundaArytttheheofluidutflovwelo,cithoywisever,prescribthevedeloincitay
posecannoatbproeblempresctribhatedanasunmothiswdifiedouldpaimplrtitioicitnedlyprocedeterminedurethecannovthaolumendle.ofAntheefluidscapefandor
thisNeumannso-calledboundaryincompressibilitconditionyatdilemmatheoutflohasbweenwouldgivenbeinan[164easy].Spwecorkarifyingound.anaHodewqevuateer,
duetogravitypointinginz-directionthepressuredistributionattheendcrosssectionof
thestructureisunknown.Apriorideterminationofthestressdistributionattheoutflow
btheseoundarydifficultiessurfaceathelsofluidfailsduedomatointheisextranstendedientasnaturedepictedoftheinprofigurblem.e7.12.CircumAnvaendditiotingnalall
bofencotpanstanrttofgrathevitypippeotwenhictialhiswhicrighidiscallowshosentotoendz=the2ccm.omputaThusationalNeumanndomainbataoundalevryel
conditioncanbeprescribedhereanddeterminesthepressurelevelinsidethetube.

freewfloout

yvitagr?winfloFigure7.12:Isometricviewonfluid(left)andstructuraldomain(right);extensionofthe
fluiddomainallowstostatecorrectboundaryconditions

315

7.4.VibratingU-pipe153
7.4.3Results
Theresultingtubeoscillationisnotjusttheexpectedbendingbutalsoanincreasing
toduringrsionalthereplaupyandwhicdohwnisduecycles.toInthefigureCoriolis7.13theforcesevolutioinducnedofinthethevtertwoicaalrmsofdisplacementhepiptse
atthereferencecornerpointsAandBisdepictedalongwiththedisplacementdifference.

0.60.40.20-0.2-0.4emac-displzcminten-1.2-1.4
-0.6-0.8-1-1.65678timein9s101112
vverticalerticaldispldisplacacememenenttofofppoinointtAB
differenceofverticaldisplacementofpointAandpointB
Figure7.13:VerticaldisplacementatthetwopointsAandBalongwithdisplacement
differencebetweenthosepoints

11

12

Inordertoensurethewell-posednessoftheproblemtheflowpassingthetubeaswell
asthegravitationalforceshavetobebuiltupoveraperiodintime.Thisstartupwas
finishedat4.0swhenthesystemoscillatedinthefirstbendingmodeanditscorresponding
eigenfrequencyasitcanbeseeninthefirstpartofthediagraminfigure7.13.At5.5s
theperiodicverticaltipforceisswitchedon.Thisinitiatesaforcedvibrationonthefirst,
thebendingeigenmodeofthestructure.Theevolutionoftheincreasingcontributionof
thetorsionalmodecanbeobservedfromtheincreasingdisplacementdifferenceofthetwo
referencepointsAandBoftheshell.Ineverycycleaportionofthebendingenergyis
transferedtothetorsionalmoderesultinginanincreasingtorsionaloscillation.
Asaconsequenceofphysicalaswellasnumericaldampingthetorsionaloscillation
eventuallyreachesaconstantamplitudeandtheoverallsystemremainsstable.
Azoomintothediagram7.13isgiveninthefirstpartofcolourchart7.14wherefive
timeinstantsaremarkedindicatingtheconfigurationsdepictedsubsequently.Theview
onthedeformedshellclearlyshowsthetorsionaldisplacementcausedbytheCoriolis
forces.AU-pipefilledwithwateratrestdoesnotexcitetheunsymmetricpartofthe
structuralresponse.Correspondinglytheverticalfluidvelocityisdepictedoncutting
surfacesinthesecondcolumnoffigure7.14.Highvelocityinz-directionatthepointB
canbeobservedforthoseconfigurationswhichshowalmostnotorsionaldisplacement.
Inthethirdcolumnoffigure7.14theabsolutevalueofthevelocityisdepictedoncuts
ofthedeformedconfiguration.Itcanbeobservedthatthevelocityduetothetorsional
oscillationwelldominatesthelongitudinalflowvelocityinsidethetube.

415

0.40.20-0.21-0.4tenemacdisplerticalv-1.2
-0.6Atoinp-0.8-1-1.410.910.81t=10.95s

2t=11.05s

3t=11.15s

4t=11.25s

5t=11.35s

11

.7ChapterExamplesNumericaldifference

2534Btoinp11.311.2111.z-velocityincm/s
3030−

s11.511.4

|u|incm/s
370

Figure7.14:Detailofdiagram7.13anddeformedstructureviewedinz-direction,vertical
fluidvelocityonundeformedconfigurationandabsolutevalueoffluidvelocityondeformed
configuration(fromlefttoright)

8erChapt

SummaryandConclusions

8.1Summary

Thisworkisdevotedtoimprovementsofapartitionedfluid-structureinteractionalgo-
rithmwhichisbasedonanALEflowformulationwithstabilisedfiniteelementsanda
nonlinstanceeaardetastructuriledalsolvtheoreticaer.lInarevisitsensofethetheobfundamenjectivethaalssofbeenthetflowwofold.solverInandthetofirstsomein-
extenaccuratcyalsoandthestacobilituplingyproalpgorerties.ithmAshasabeenresultomadefthineseordermethotoedolostagicalblishandconsideraconfirmtionsitsa
numberofchangeswithinthecodeitselfaroseenhancingconvergenceandstabilityofthe
n.ulatioformmericalThemalgostorithmimportaandntdeatmailsbitionthereoofft.hisInwpaorkisrticulartotheincreasemattertheofungedeorstmetricandingconsofervtheatnionu-
kandeyitsquescotionnnecfotriontostablestaabilitndyreliaofbleflowalgosimularithms.tionsonWithindefothisrmingwodorkamainsstabilitappyearcritederasiona
intermsofamaximalallowabletimestepsizedependinguponthemeshvelocitycould
beconfirmedandinterpreted.Inparticulartheinterpretationrevealswhythepotential
applicainstabilitytions.ofcItonvhasectivefurtherALEbeescnhemeshoswnhashobweaenflosuspwsecotlvederbutonanotdeforbeenmingfounddomaininprahascticalto
beconstructedsuchthatitisstableindependentlyofthemeshvelocity.
theAnotalgoherrithmkeyistheproblemso-calledregaardingrtificialtheaddunderstaedmassndingofeffect.theThisistheoreticaanlinherenfundatmentalsinstabil-of
ityofsequentiallycoupledpartitionedFSIalgorithmsusedwithincompressibleflows.
Especiallythecombinationoflight-weightstructuresandincompressibleflowsyieldsan
bealmostestablishedimmediateexplablow-iningupothefweainstakbilitcouplingyandscidenhemes.tifyingWithinthepathiswrameterorksanitaisnalysisinfluecouldnced
bMoy.kThis[182].analyItsshoiswswconfirmshymorethenaccuratumericaletimeobsdiscervrationsetisatiodenascribccedeleraintettheheinstadissertatbilityionandof
ithighlightstheroleofthefluidandstructuralmassdensityratio.Consequencesofthe
atuseveorfyasmallstabilisedtimeFEMstepsfoharvtehebeenfluidfieldconsidered.intheInstacontebilitxtyoftheconditioartinsficiafolravarddediousmacossmebina-ffect
ptionsossibilitofytodiscretisatioconstructnscahesequenmestiacouldllybstaeggeestablredissched.hemeItwhhicashbewenouldprobveesdtablethatirretherespisectivnoe
ofthemassdensityratiooffluidandstructure.
Additionallythestabilisedfiniteelementformulationforfluidelementsondeforming
domainswasrevisitedwithinthiswork.Heresomemodificationscouldbeintroduced
iswhicnothguainfluencranteeedbthaytthethestastbilisabilitaytiownithterms.respecttoConsequenthemotlytiontheofstathebilityreferencepropertiessystemof

515

615

Chapter8.SummaryandConclusions

themodelproblemcanbetransferredtothestabilisedNavier-Stokesequationson
mains.dodeformingFurtherconsiderationsregardthebehaviourofthestabilisedfiniteelementmethod
forincompressibleflowsincriticalsituationssuchashighlydistortedmeshesandsteep
gradients.Numericaltestcaseswereemployedtogainthedesiredexperiencerequiredto
answerquestionswhicharehardtobeaccessedanalytically.
Themethodologicalconsiderationsbroughtanumberofchangeswithinthecode.
Alongwiththemodificationspartsofthecodehavebeenrewritteninordertoderive
andimplementafullylinearisedversionofthestabilisedfluidelement.Thecomputa-
tionoftheelementalrighthandsidevectorhasbeenreformulated.Boththesechanges
resultedinasignificantlyfasterconvergenceofthenonlineariterationswithinthefluid
field.Extraaccuracyforthecoupledproblemhasbeengainedbyintroducingconsistent
nodalforcesandacorrectlyobtainedflowvelocityboundaryconditionascouplingdata.
Additionallytheintroductionofsecondorderaccuratetimediscretisationschemesforthe
flowequationsopenedthedoorforafullysecondorderFSIalgorithm.
AltogethertheiterativelystaggeredFSIalgorithmbasedonanALEflowformulation
oftheincompressibleNavier-Stokesequationswithstabilisedfiniteelementsanda
nonlinearstructuralfiniteelementsolverisregardedassettledandtheoreticallysound.
Inparticularthestabilityissueoftheflowsolverondeformingdomainscouldbeclarified
andananalysisoftheartificialaddedmasseffecthasbeenpresented.

8.2Prospectus

Thefocusofthepresentworkistoimproveanexistingapproachandtodeepenthe
understandingofthemethodologicalfundamentals.NowaverygeneralpartitionedFSI
solverforincompressibleflowandhighlyflexiblestructuresisavailablewhichissecond
orderaccurate,stableandoffersreliableresultsforabroadvarietyofproblems.
Howeveraworklikethepresentprojectwillneverbefinishedinthesensethatall
questionswouldhavebeenansweredandallproblemssolved.Itisratherinterrupted
bythefactthattimehaspassed.Soavarietyofinterestingissuesremains.Continuing
thepresentinvestigationsonemightwishtoclarifytheexactinteractionoftheartificial
addedmasseffectwiththestabilisationofthefluidelements.Oronemightwellwonder
ifthereisanintelligentwaytohigherorderfluidelements.Andwhatwouldsuchan
elementlooklike?Isthereanythinglike‘optimalstabilisation’?
Andanevenhigherneedforfurtherprogressisfoundawayfromthefundamentals.It
isstillmosturgenttospeeduptheentirealgorithmwhichwouldallowlargerexamples
closertorealisticapplicationstobesimulatedatanacceptableaccuracy.Inparticular
thesimulationoflargescalethree-dimensionalproblemsarecurrentlyquiteatthelimit.
Modificationsofthecouplingapproacharepresentlyinvestigatedandshallmakethe
simulationmoreefficient.Inthiscontextitappearsinterestingtoworkoutinwhich
rangeofparameters,i.e.forwhichapplicationssemi-explicitcouplingschemesmightwork
well.Additionalspeed-upcouldalsobegainedbyoptimisingthesolveroftheresulting
nonlinearsystemofequations.Amatterwhichhasnotbeentouchedatallwithinthis
k.orwAstheoverallFSIalgorithmrunsstablyaveryinterestingchallengewouldbethe

spPro.8.2ectus

715

Asaimplementaprerequisitetionofaadastableptivitoyverallwithralgesporitecthmtowtheouldtemphaoveraltobutbealsotheestablishedspatialwhichresolutalloion.ws
differenttimestepsizeswithinthesinglefields.Analogouslynon-matchinggridsatthe
interfacewouldhavetobetreated.
Anotherveryimportantdirectionoffutureworkistheextensionoftheclassofproblems
thatcanbedealtwith.Itappearsverypromisingtocustomisethealgorithmtospecial
flowapplicas.Suctionshasucsphasecialisatiobuildingntoawindclasinsteorfsactioub-pron,pablemsrticulaarpppipearseflotowbetwproblemsofold.orOnbtheiologoneical
handthederivationandimplementationofadditionalmodelsornonlinearmateriallaws
isrequiredwhichwouldincreasethecomplexityofthemodel.Ontheotherhandthe
priorestrictioriknontowledgeapaaboutrticularthetprypeooblem.faThispplicatioconuldwopuldossiblyverylikreducelyeathellowtocomputbuiltatioinnalsomeefforta-
andatthesametimeincreasetheaccuracyofthesimulation.
Inanycasethepresentapproachappearstobepromisingandworthtobuilton.

AxendiApp

Somefurtherinformation

A.1Thekinematicformulaeinadeformingframeof
rencefereTwokinematicformulaethefirstofwhichisthegeometricconservationlaw(3.5)andthe
HosecwoendveristthehebasicReynoldcoosrdinattraensptraortnsfotheormatremionsareausndedrequirwithinedthisALEworkequawittionshoutheaderivvilyatiorelyn.
upontheseformulaeandthusaderivationshallbegivenhere.

A.1.1Geometricconservationlaw
ThestartsderivfromataionnexprforesthesiognofeometricthemeshconservdeterminaationlantworreadingEulerformulafollows[230]and
x∂iJtδij=∂χαajα,(A.1)
whereajαisthecofactorof∂xi/∂χαandasummationoverα∈1,2,3isapplied.Ina
three-dimensionalcaseoneobtainsthisfactorto
ajα=∂xk∂xl−∂xk∂xl,
∂χβ∂χγ∂χγ∂χβ
where{j,k,l}={1,2,3}and{α,β,γ}={1,2,3}andcyclicpermutationsthereof.From
(A.1)oneobtainsthepossibleexpressionsfortheJacobiandeterminantwithi=j
∂x1∂x2∂x3
Jt=∂χαa1α=∂χαa2α=∂χαa3α
ThusatimederivationofJtatafixedpointofreferencecanbeexpressedby
∂tχ∂χα
∂Jt=∂ukGakα,
where∂x/∂t|χ=uGhasbeenused.Applyingfurtherthechainruleyields
∂Jt∂ukG∂xj
∂tχ=∂xj∂χαakα(A.2)
Gu∂=∂xkjJtδjk(A.3)
=Jt∙uG,(A.4)
i.e.thedesiredgeometricconservationlaw.

915

)(A.2)(A.3)(A.4

endixAppA.endixApp

160AppendixA.Appendix
A.1.2Reynoldstransporttheorem
TheReynoldstransporttheoremallowstoexpressthetimederivativeofthespatial
integralofatimedependentfunctionwithrespecttoaspatiallymovingsystemofreference
∂∂tχVtf(x,t)dVt,
whereVtisavolumefixedinthereferencesystemχ.
Inordertoderivethetheoremthedifferentialvolumeelementistransferedbacktothe
referencesystemandtherelationx=x(χ)isemployedyielding
∂∂
∂tχVtf(x,t)dVt=V0∂tχ(f(x(χ),t)Jt)dV0
∂∂Jt
=V0∂tχf(x(χ),t)JtdV0+V0f(x,t)∂tχdV0(A.5)
Thetimederivativeinthefirsttermof(A.5)canbeevaluatedbythechainrule
∂f(x(χ),t)=∂f(x,t)∂x+∂f(x,t)(A.6)
∂tχ∂x∂tχ∂tx
Inserting(A.6)into(A.5)yields

∂tχVtV0∂tx
∂f(x,t)dVt=∂f(x,t)+uG∙f(x,t)+f(x,t)∙uGJtdV0,
where(A.4)andthedefinitionofthemeshvelocityhasbeenused.Thisequationcanbe
reformulatedasanintegraloverthetimedependentdomainVt

∂tχVtVt∂tx
∂f(x,t)dVt=∂f(x,t)+∙uGf(x,t)dVt(A.7)
andyieldstheReynoldstransporttheoreminanALEframeofreferencereading
∂tχVtVt∂txΓt
∂f(x,t)dVt=∂f(x,t)dVt+f(x,t)uG∙ndΓt.(A.8)
InthespecialcaseofanEulerianframeofreferencethetheoremallowstoexpressthe
materialtimederivativeoftheintegraloverafunctionf(x,t)andthusthearbitrary
systemχisidentifiedwiththematerialcoordinatesXyieldingthewellknownexpression
∂∂f(x,t)
∂tXVtf(x,t)dVt=Vt∂txdVt+Γtf(x,t)u∙ndΓt(A.9)
whichistheReynoldstransporttheoreminanEulerianframeofreference.

A.2.Somemathematicalbackground
A.2Somemathematicalbackground

A.2.1Thescalarproduct
Thenotation(a,b)Ωisusedshortfor
(a,b)Ω=ΩabdΩ,
whereaandbmayrepresenttensorsoffirstorsecondorder.

161

(A.10)

A.2.2Lax-Milgramlemma
ThelemmanamedaftertheHungarianmathematicianPeterDavidLaxandArturNorton
Milgramguaranteesthesolvabilityofavariationalproblemoftheform
a(u,v)=(f,v)forallv∈V.(A.11)
In(A.11)a(u,v)denotesabilinearformandvcanberegardedatestorweightingfunction
inaweakformulation.Theproblem(A.11)hasauniquesolutionifthebilinearformais
i.e.ounded,b|a(u,v)|≤Cuv(A.12)
andcoercivewhichmeansthat
a(u,u)≥cu2(A.13)
with0<c<C<∞isguaranteed.Forsymmetricpositivedefiniteoperatorsathe
constantcisanlowerboundonthelowesteigenvalueofathusensuringthattheleft
handsideof(A.11)isboundedawayfrombeingsingular.Furthertheinequality
)v,f(cu≤0=vsup∈Vv(A.14)
holds.Inmoregeneralcasescoercivityrequiresthatforeveryadmissiblesolutionu∈Vat
leastonetestfunctionvcanbefound,suchthata(u,v)>0.Forfurtherreferenceone
mightconsultforexample[4].

A.2.3Someinequalities
Someinequalitiesarerequiredincoercivityanalysesandshallthusbesummarisedhere.

Cauchy-Schwarzinequality
Fortwotermsaandb
(a,b)Ω≤aΩbΩ(A.15)
issatisfied.ThisCauchy-Schwarzinequalityfollowsfromgeometricalconsiderations.
Itissharpinthecasethatacanbeexpressedasa=λbwhereλisapositiveconstant.

162

endixAppA.endixApp

Theε-inequality
Theε-inequalitystatesthatforanyaandbandforanystrictlypositiveε
εa2+41εb2≥ab(A.16)
isequalsaittisfieyind.tIthecancaseofeasilya>be0derivandbed>0fromwhenathebinomiaparlformameterula.εThesatisfiesεinequalit=b/y(2ba).ecomesan

Inverseinequalities
Theinverseinequalitiesrequiredwithinthepresentworkarerelatedtoaparticularspatial
discretisationorrather,validonasingleelement.Inadiscretespaceitispossibletobound
higherderivativesbylowerones,i.e.
C0ehe2vΩ2e≤vΩ2eforallv∈Vhe(A.17)
withelemenapt.ositLikiveewiseelemenantalinequalitconstayntoneC0eowhicrderhhigdepherendscanupbeonstattheeodrderwhicahndisgeousedmetrytoofadjustthe
secthetionstabilisat4.3.7ionreadsparametertotheelementorderandgeometry.Theinequalitystatedin
Cehe2ΔvΩ2e≤vΩ2eforallv∈Veh.(A.18)
Theconstantsin(A.17)and(A.18)satisfyC>Casshownin[116].Thesharpconstant
withinaparticularelementcanbeobtained0efromethesolutionofanelementaleigenvalue
byproblem.BrennerForandmoreScottinforma[30tio].noninverseinequalitiesconsultforinstancethetextbook

A.3ErrorsinKim-Moinflow

TheL2errorsobtainedonthreedifferentlydistortedmesheswithavarietyofstabilisa-
tionparametersandvariantsofthestabilisationoperatoraregivenhere.Thesevalues
completetheKim-Moinproblemaspresentedinsection5.3.2.
Ontheheavilydistortedmeshinmode2noconvergentUSFEMsolutioncouldbe
obtainedformostofthestabilisationparameters.

A.3.ErrorsinKim-Moinflow

316

TableA.1:L2errorinvelocityandpressureonundistortedmeshesoflinearelementsfor
differentchoicesofthestabilisationparameter
τerruerrperruerrperruerrp
GLSGLSSUPGSUPGUSFEMUSFEM
i0.0051300.0028770.0046140.0024600.0042120.002287
ii0.0052670.0029190.0047290.0025050.0043190.002360
iii0.0051300.0028770.0046140.0024600.0042120.002287
iv0.0051300.0028770.0046140.0024600.0042120.002287
v0.0048150.0028850.0045240.0026760.0042740.002552
vi0.0051300.0028770.0046140.0024600.0042120.002287
vii0.0051300.0028770.0046140.0024600.0042120.002287

TableA.2:L2errorinvelocityandpressureonundistortedmeshesofquadraticelements
fordifferentchoicesofthestabilisationparameter
τerruerrperruerrperruerrp
GLSGLSSUPGSUPGUSFEMUSFEM
i0.0026240.0025600.0026170.0025440.0026310.002597
ii0.0026230.0025620.0026160.0025460.0026270.002543
iii0.0026240.0025600.0026170.0025440.0026310.002597
iv0.0026240.0025600.0026170.0025440.0026310.002597
v0.0026310.0025610.0026240.0025460.0026120.002526
vi0.0026240.0025600.0026170.0025440.0026310.002597
vii0.0026240.0025600.0026170.0025440.0026310.002597

TableA.3:L2errorinvelocityandpressureonmode1distortedmeshesoflinearelements
fordifferentchoicesofthestabilisationparameter
τerruerrperruerrperruerrp
GLSGLSSUPGSUPGUSFEMUSFEM
i0.0076130.0050440.0065140.0042660.0056500.003835
ii0.0076950.0049690.0065930.0041990.0057520.003802
iii0.0075130.0048390.0064600.0040830.0056200.003638
iv0.0073760.0048210.0064440.0042070.0056670.003829
v0.0069940.0048250.0063530.0043760.0057770.004045
vi0.0074220.0049130.0064560.0042510.0056540.003840
vii0.0077850.0051770.0065750.0043210.0056790.003908

416

AppA.endixAppendix

TableA.4:L2errorinvelocityandpressureonmode1distortedmeshesofquadratic
elementsfordifferentchoicesofthestabilisationparameter
τerruerrperruerrperruerrp
GLSGLSSUPGSUPGUSFEMUSFEM
i0.0026380.0025810.0026370.0025870.0026550.002662
ii0.0026360.0025810.0026350.0025900.0026910.002886
iii0.0026380.0025800.0026370.0025870.0026560.002662
iv0.0026420.0025830.0026410.0025890.0026630.002670
v0.0026480.0025840.0026440.002588––
vi0.0026420.0025830.0026410.0025890.0026620.002670
vii0.0026350.0025790.0026340.0025860.0026470.002656

TableA.5:L2errorinvelocityandpressureonmode2distortedmeshesoflinearelements
fordifferentchoicesofthestabilisationparameter
τerruerrperruerrp
SUPGSUPGGLSGLSi0.2323570.1853400.0435550.084645
ii0.1340090.1207650.0555460.058709
iii0.1089690.1022560.0576600.061274
iv0.0873370.0850320.0580180.057737
v0.0875310.0870890.0580850.059084
vi0.0873560.0850380.0580270.057756
vii0.4198510.2729360.0435160.155843

TableA.6:L2errorinvelocityandpressureonmode2distortedmeshesofquadratic
elementsfordifferentchoicesofthestabilisationparameter
τerruerrperruerrp
SUPGSUPGGLSGLSi0.0065580.0094510.0041980.010419
ii0.0061350.0082650.0046010.010309
iii0.0059100.0080460.0046240.010148
iv0.0049530.0077940.0042120.006309
v0.0056210.0087790.0040480.004463
vi0.0049520.0077930.0042120.006316
vii0.0069160.0103070.0045100.018528

51015202530
time

403545

50

A.4.Flowaroundrigidcylinder165
A.4Flowaroundrigidcylinder
ThesubsequentdiagramscompletetheresultsobtainedfromRe=100flowarounda
rigidcylinderreportedinsection5.3.3.Theliftcoefficientsanddragcoefficientsforthe
meshesB4,C4,A9andB9aregivenhere.
0.40.320.240.160.08l0C-0.08-0.16-0.24-0.32-0.405101520253035404550
time1.4241.4161.408dC1.41.3921.38405101520253035404550
time10.50p-0.5C-1-1.5-20.50-0.51-1xelementlengthvi,he,min
elementlengthvii,he,ma√x
elementlengthi,he=Ae
FigureA.1:LiftanddragcoefficientsandpressureprofileobtainedonmeshB4withthree
differentelementlengthdefinitionswithinthestabilisationparameter

51015202530
time

0x

354540

0.5

50

1

5

10

15

302520time

endixAppA.endixApp

403545

50

166AppendixA.Appendix
0.40.320.240.160.08l0C-0.08-0.16-0.24-0.32-0.405101520253035404550
time1.4241.4161.408dC1.41.3921.38405101520253035404550
time10.50p-0.5C-1-1.5-2-1-0.5x00.51
elementlengthvi,he,min
elementlengthvii,he,max
elementlengthi,he=√Ae
FigureA.2:LiftanddragcoefficientsandpressureprofileobtainedonmeshC4withthree
differentelementlengthdefinitionswithinthestabilisationparameter

5

10

15

302520time

0x

454035

0.5

50

1

A.4.Flowaroundrigidcylinder

51510

302520time

35

40

45

50

716

0.40.320.240.160.08l0C-0.08-0.16-0.24-0.32-0.405101520253035404550
time1.4241.4161.408dC1.41.3921.38405101520253035404550
time10.50p-0.5C-1-1.5-2-1-0.500.51
xelementlengthvi,he,min
elementlengthvii,he,ma√x
elementlengthi,he=Ae
FigureA.3:LiftanddragcoefficientsandpressureprofileobtainedonmeshA9withthree
differentelementlengthdefinitionswithinthestabilisationparameter

15105

302520time

35

0.5

40

45

50

1

816

5

10

15

202530time

35

AppA.endixAppendix

40

45

50

0.40.320.240.160.08l0C-0.08-0.16-0.24-0.32-0.405101520253035404550
time1.4241.4161.408dC1.41.3921.38405101520253035404550
time10.50p-0.5C-1-1.5-2-1-0.500.51
xelementlengthvi,he,min
elementlengthvii,he,ma√x
elementlengthi,he=Ae
FigureA.4:LiftanddragcoefficientsandpressureprofileobtainedonmeshB9withthree
differentelementlengthdefinitionswithinthestabilisationparameter

5

10

15

302520time

35

50.

40

45

50

1

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Index

ε-inequality,162ofstructures,8
ectionvconA-stability,28dominatedproblem,50
accuracy,36inducedoscillations,50
addedmassoperator,119,122instability,seeinducedoscillations
advection-diffusionequation,21,50,53,60stabilisation,52
advection-diffusion-reactionequation,22convective
ALEformulation,16,26
formulation,7,23–27term,14,54
42system,referencecouplingartificialaddedmasseffect,113,118conditions,20
117forces,Bubnov-Galerkinscheme,52crosswinddiffusion,52
backwardEulermethod,28
balancedeformingdomain,24,60
ofangularmomentum,9differential-algebraicproblem,27
oflinearmomentum,10
BBcondition,seeinf-supconditiondiscretegeometricconservationlaws,38
272,1n,discretisatioBDF2,29,32,35,36,42,82,101,116,126inspace,12,30
boundaryconditions,10,18intime,13,27
bubblecondensation,59divergenceformulation,16,26
functions,59Eulerianformulation,7,14
Cauchystresses,9elementlength,72
Cauchy-Schwarzinequality,161equalorderinterpolation,55,57,66
changingdomain,23finite
characteristicGalerkinprocedure,53elements,12,27,30,52
coercivity,42,45,50,75–80,88–161incrementcalculus,53
conservationvolumes,52
laws,33flow
ofenergy,17,38conditionedbasedinterpolation,54
oflinearmomentum,16,34equations,14
ofmass,15flowmeter,151
consistennodaltforces,117fundamentalALEequation,26
stabilisation,seeresidualbasedstabili-Galerkinweakform,seeweakformulation
sationGalerkin/Least-Squaresmethod,seeGLS
consistentnodalforces,86generalised-αmethod,13,125
continuummechanicsgeometricconservation,25,34,116
offluids,14GLS,53,65,75

218

318IndexHhigherelmholordertzelemendecompts,ositio66,7n,4,55102,146relaxreferenceation,1config36uration,8
horizontalmethodoflines,27residual
incompressibility,49,55,56basedstabilisation,52,53,55,58,73
incompressibleNavier-Stokesequations,freebubbles,62
seeNavier-StokesequationsStokesproblem,21,51,56,58
inf-supcondition,49–54,57Strouhalnumber,18
initialconditions,10,18second
instability,52Dahlquistbarrier,28
inverseinequality,162orderbackwarddifferencing,seeBDF2
41forces,shearLagrangeanformulation,7singulardiffusionequation,22
LBBlinearconditelemenion,ts,7se3,e84inf-,8sup8conditionsmalltimesteps,82
stabilisation,49,52
materialtimederivative,9,25parameter,70,87,89,107
ystabilitmeshdistortion,72,99–107ofALEformulations,40,74
velocity,35unconditional,46
methodoflines,27stiffpartialdifferentialequation,27
modelproblems,20SUPG,structur5al3material,9
Navier-Stokesequations,16systemofreference,7
Newtonianfluid,15time
nnonlinumericaealrity,64dependentsubscales,65
accuracycheck,36stepsizelimit,42,45
diffusion,50trapezoidalrule,28
example,36,94,100,107,131,153unresolvedgradients,50
one-step-53θ,116method,28,30,32,35,36,42,upUSFEM,winding5,9,51,64,5276,106
operatorsplitting,55
orthogonalsubscales,59variationalmultiscalemethod,62
Pecletnumber,21,50,52weakformulation,11
Petrov-Galerkinscheme,52weightedresidualmethod,52
partitionedalgorithm,114wiggles,49
151r,predictopressure51ions,oscillat57jection,proprostabilisajectiontion,metho54d,55
principleofvirtualwork,11
proPSPG,jection,6466,102
Reynoldsnumber,14,18,70

Name::Geburtsdatum:GeburtsortEltern:at:lit¨ionaNatamilienstand:F

1984-1986:
1986-1990:
1990-1992:
1992-1996:
96:19Juni1:200-9619November1998:
:0120August:0220uarJan2001-2002:
2:200erbSeptem0320uarJan2002-2006:

aufnsleLeb

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7719August5.aukwicZMatthiasK¨uttler,MariaK¨utter,geb.Dietrich
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