Computer Simulation Study of Collective Phenomena in Dense Suspensions of Red Blood Cells under Shear

168 Pages·2012·4.567 MB·English

by  Timm Krüger (auth.)

Computer Simulation Study of Collective
Phenomena in Dense Suspensions of Red
Blood Cells under Shear
Timm Krüger
Computer Simulation Study
of Collective Phenomena in
Dense Suspensions of Red
Blood Cells under Shear
Timm Krüger
Düsseldorf, Germany
ISBN 978-3-8348-2375-5 ISBN 978-3-8348-2376-2 (eBook)
DOI 10.1007/978-3-8348-2376-2
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Blut ist ein ganz besondrer Saft.
Johann Wolfgang von Goethe (Faust)
Abstract
Understandingtherheologyofbloodhasbeenascientifictaskfornearlyonecentury—forgood
reasons. Ontheonehand,bloodisvitalforthehumanbody,andsomediseaseslikemalariaor
sickle-cellanemiainterferewithitsproperfunctioning. Ontheotherhand,bloodisanexamplefor
adensesuspension. Uptonow,therheologicalanddynamicalpropertiesofsuchcomplexfluids
isnotcompletelyunderstood. Forthisreason,investigatingthepropertiesofdensesuspensions
ingeneralandthoseofdeformableparticlesuspensionsinparticularisofparamountimportance.
Duetothehighlycomplexboundaryconditionswhichcanbefoundinsuspensionsontheparticle
scale,analyticinvestigationsalonecannotclarifytheunansweredquestions. Especiallyinthe
firstdecadeofthiscentury,thecomputationalresourcesandavailablealgorithmshavebecome
mature enough to allow numerical studies of suspensions of deformable particles. This is the
primaryaimofthecurrentwork.
Inthepresentthesis,anumericaltoolisdevelopedwhichallowstosimulateparticulateblood
suspensions and to investigate their mechanical properties. Due to the separation of cellular
and molecular time and length scales, the basic idea is to follow a mesoscopic approach. The
red blood cells are resolved, and their deformation state is tracked explicitly. However, the
suspendingfluidisdescribedasacontinuummediumwithNewtonianproperties. Thelattice
Boltzmann method (chapter 5) is employed as Navier-Stokes solver for the suspending fluid,
whereasafiniteelementmethod(chapter7)isusedforthedescriptionoftheelasticredbloodcell
membranes. Afluid-structureinteractionalgorithmisrequiredtocoupletheparticlemotionand
thefluiddynamics. Theimmersedboundarymethod(chapter6)servesthispurposewell. The
modelisextendedinsuchawaythatdensesuspensions(upto65%volumefraction)ofO(1000)
deformable particles in arbitrary geometries can be simulated with reasonable computational
effort(chapter8).
Duringthecourseofthisthesis,anewboundaryconditionforthelatticeBoltzmannmethod
isintroduced(section5.4). Thisextensionofthebounce-backboundaryconditioncanbeused
to impose a well-defined shear stress to drive the fluid, even when its viscosity is not known.
Thisisparticularlyimportantwhenthestaticyieldstressofasuspensionistobeinvestigated.
Thestaticyieldstressisdefinedasthestressbelowwhichnoflowoccurs. Contrarily,shearing
asuspensionwithafiniteshearratedoesnotallowtofindthestaticyieldstress. Instead,the
dynamic yield stress (as the stress for vanishing small shear rates) may be obtained which is
usuallysmallerthanthestaticyieldstress.
InappendixB,itisarguedandshownviasimulationsthattheshearstressinthelatticeBoltzmann
methodisasecond-orderaccurateobservable. Ontheonehand,thefocusofsuchinvestigations
isusuallyonthevelocityasitisthemostrelevantobservableformanyhydrodynamicproblems.
Forsuspensionrheology,ontheotherhand,thestressplaysamoreimportantrole. Aconsistent
pictureofstressevaluationinimmersedboundarylatticeBoltzmannsimulationsisprovidedin
chapter9. Itisshownhowtheparticlecontributiontothesuspensionstresscanbecomputed
locally(inspaceandtime)andindependentlyfromthefluidstressormacroscopicassumptions.
Thisapproachisespeciallyimportantwhenspatio-temporalfluctuationsoftheshearstressare
soughtafter. Thesefluctuationscarrysignificantinformationaboutthestatisticalpropertiesof
thesuspension.
The computational model is utilized to study the rheology of blood and its microdynamics
systematically(chapter10). Theinfluenceofthemostimportantcontrolparameters(theshear
viii
rate γ˙, the volume fraction Ht, and the red blood cell rigidity κS) is investigated. It is found
thatthemodelrecoverstheexperimentallyobtainedflowcurveforbloodatintermediatevolume
fractions. Theparticledeformabilitysignificantlyaffectsthemicrodynamicsinthesuspension:
Whenthesuspensionstressexceedsacertainthreshold,theredbloodcellsstarttotank-tread,
and an increased orientational ordering develops. The combined effect of tank-treading and
collectivealignmentisoneofthemechanismscontributingtotheshearthinningbehaviorofblood.
Thesimulationsprovideclearevidencefortheimportanceofacorrectmicroscopicdescription
oftheredbloodcellsinsimulations. Althoughsuspensionsofrigidparticlesalsoexhibitshear
thinningundersomecircumstances,themicroscopicbehaviorofthesuspensionsisfoundtobe
significantlydifferent. Aremarkableresultrelatedtothesimulationparameterspaceisfound.
Allrelevantresultscanbedescribedbytwo,ratherthanthreeparameters,thevolumefraction
Ht and the capillary number Ca∝γ˙/κS. Thus, the effect of varying particle rigidities can be
compensatedbytheshearrate. Aninterestingresultisthatsomeofthedata(e.g.,theparticle
tumbling frequency, the deformation, or the collective order parameter) collapse on a single
mastercurvewhenplottedasfunctionofthe‘corrected’capillarynumberCa∗∝σ/κS (σ isthe
totalsuspensionstress). Theseobservationscanbeexplainedbytheideathat,independentof
thesuspensionvolumefraction,tank-treadingredbloodcellsareawareoftheirneighborhood
onlyviathesuspensionstressσ.
Forthefirsttime,theshear-induceddiffusionofredbloodcellsinsimpleshearflowisinvestigated
as function of the above-mentioned control parameters (γ˙, Ht, κS). It is shown that the
deformability increases the shear-induced P´eclet number as compared to a system of rigid
particleswithcomparablevolumefraction(section10.7). Consequently,diffusivemotionsetsin
laterandislessefficientinmixingthesuspension. Itisalsofoundthatthefluctuationsofthe
shearrateandtheparticleshearstressarecorrelated(section10.8). Thehighertheshearrate,
thesmallerthestressandviceversa. Thisobservationmaypointatonepossiblemechanismfor
stressrelaxationwhichisnotrelevantforfrictionlesshardspheresystems: Whenparticlesare
lockedduringshearing,theycannotrotateanddeceleratetheambientfluid. Thestressincreases.
Atsomepoint,thestressissufficientlylargefortheparticlestobefreedagain. Theyincrease
theirangularvelocity,theshearrategrowsaswell,andthestressmayrelax.
Contents
Abstract vii
1 Motivation,aims,andoutlineofthethesis 1
1.1 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Aimsofthepresentwork. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Thesisoutline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
I Introduction 3
2 Introductiontocomplexfluidsandtheirrheology 5
2.1 Introductiontorheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Thecomplexityofcomplexfluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Rheologyofcomplexfluidsmadeofrigidparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Rheologyofcomplexfluidsmadeofdeformableobjects. . . . . . . . . . . . . . . . . . . . . . . . 9
3 Introductiontothephysicsofredbloodcellsandhemorheology 11
3.1 Introductiontothephysicsofredbloodcells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Introductiontohemorheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
II Numerical model for simulations of red blood cell suspensions 15
4 Physicalconsiderationsandingredientsforthenumericalmodel 17
4.1 Overviewofexistingnumericalapproaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Identificationoftherelevantphysicsforthepresenttask . . . . . . . . . . . . . . . . . . . . . . . 18
5 Fluidsolver: thelatticeBoltzmannmethod 21
5.1 IntroductiontothelatticeBoltzmannmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2 LBGKalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.3 InitialandboundaryconditionsinthelatticeBoltzmannmethod . . . . . . . . . . . . . . . . . . 26
5.3.1 InitialconditionsinthelatticeBoltzmannmethod . . . . . . . . . . . . . . . . . . . . . 26
5.3.2 BoundaryconditionsinthelatticeBoltzmannmethod . . . . . . . . . . . . . . . . . . . 27
5.4 Bounce-backboundarycondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.4.1 Velocitybounce-backboundarycondition . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.4.2 Shearstressbounce-backboundarycondition . . . . . . . . . . . . . . . . . . . . . . . . 31
5.5 EfficiencyandchoiceofsimulationparametersinthelatticeBoltzmannmethod . . . . . . . . . . 34
6 Fluid-structureinteraction: theimmersedboundarymethod 37
6.1 Overviewoftheimmersedboundarymethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.2 Governingequationsoftheimmersedboundarymethod . . . . . . . . . . . . . . . . . . . . . . . 38
6.3 Discretizationoftheimmersedboundarymethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.4 Connectionbetweentheimmersedboundarymethodandviscouscoupling . . . . . . . . . . . . . 41
7 Membranemodelandenergetics 43
7.1 Membranestrainandareadilationenergetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7.2 Membranebendingenergetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.3 Membranesurfacedilationenergetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.4 Membranevolumeenergetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
8 Advancedmodeldiscussions 49
8.1 Overviewofthecombinedsimulationalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
8.2 Conversionbetweenphysicalandlatticeunits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
x Contents
8.3 Membranemeshgeneration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
8.4 Benchmarktest: singlecapsuleinshearflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8.5 Initializationofdensesuspensions: distributionoftheparticlesinthesimulationbox . . . . . . . 60
8.6 Limitationsandrestrictionsofthenumericalmodel . . . . . . . . . . . . . . . . . . . . . . . . . . 62
8.7 Interactionsbetweennearbymembranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8.8 Wallslipandroughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
III Simulation results and interpretation 71
9 StressevaluationincombinedimmersedboundarylatticeBoltzmannsimulations 73
9.1 FluidstressevaluationinthelatticeBoltzmannmethod . . . . . . . . . . . . . . . . . . . . . . . 73
9.2 WallstressevaluationinthelatticeBoltzmannmethod. . . . . . . . . . . . . . . . . . . . . . . . 74
9.3 EvaluatingparticlestresseswithBatchelor’sapproach . . . . . . . . . . . . . . . . . . . . . . . . 74
9.4 Evaluatinglocalparticlestresseswiththemethodofplanes . . . . . . . . . . . . . . . . . . . . . 76
9.5 Benchmarktest: verificationofthestressevaluationmethods . . . . . . . . . . . . . . . . . . . . 79
9.5.1 Singlecapsuleinshearflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
9.5.2 Densesuspensioninshearflow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
10 Rheologyandmicroscopicbehaviorofredbloodcellsuspensions 85
10.1 Simulationsetupanddataevaluationremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.2 Characterizationofparticledeformation,orientation,androtation . . . . . . . . . . . . . . . . . 88
10.3 Suspensionviscosityandshearthinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
10.4 Particlerotation: tumblingandtank-treading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
10.5 Particledeformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
10.6 Particlealignmentandorientationalordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
10.7 Particledisplacements: ‘ballistic’anddiffusivemotion . . . . . . . . . . . . . . . . . . . . . . . . 107
10.8 Shearstressfluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
11 Conclusionsandoutlook 121
11.1 Summaryofowncontributionsandconclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
11.2 Outlookandsuggestionsforfutureresearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
IV Appendix 127
A Conventions,abbreviations,andsymbols 129
A.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.2 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.3 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
B Chapman-EnskoganalysisandadvancedlatticeBoltzmanncalculations 133
B.1 Chapman-Enskoganalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
B.1.1 Chapman-Enskoganalysisinthepresenceofforces . . . . . . . . . . . . . . . . . . . . . 133
B.1.2 DiffusivescalinganditsrelevancefortheconvergenceoftheLBGKalgorithm . . . . . . 137
B.1.3 ErrortermsoftheLBGKequationanditsconvergencetotheNavier-Stokesequations . 138
B.1.4 Benchmarktest: ConvergencefortheTaylor-Greenvortexflow . . . . . . . . . . . . . . 140
B.2 Recoveryandinitializationofnon-equilibriumpopulations . . . . . . . . . . . . . . . . . . . . . . 144
C Derivationofthemembraneforces 145
C.1 Derivationofthestrainforce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
C.1.1 Displacementgradienttensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
C.1.2 Strainforce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
C.2 Derivationofthebendingforce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
C.3 Derivationofthesurfaceforce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
C.4 Derivationofthevolumeforce. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Bibliography 153
Acknowledgments 165
Curriculumvitaeandpublications 167
List of Figures
3 Introductiontothephysicsofredbloodcellsandhemorheology 11
3.1 Schematicshapeofaredbloodcell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5 Fluidsolver: thelatticeBoltzmannmethod 21
5.1 D3Q19latticeforthelatticeBoltzmannmethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2 UnknownpopulationsatboundariesinthelatticeBoltzmannmethod . . . . . . . . . . . . . . . 28
5.3 Comparisonofhalf-wayandwetboundaryconditionsinthelatticeBoltzmannmethod . . . . . . 29
5.4 StaircaseshapeofaninclinedwallinthelatticeBoltzmannmethod. . . . . . . . . . . . . . . . . 30
5.5 Bounce-backboundaryconditioninthelatticeBoltzmannmethod . . . . . . . . . . . . . . . . . 31
5.6 MomentumexchangeatasolidwallinthelatticeBoltzmannmethod. . . . . . . . . . . . . . . . 33
6 Fluid-structureinteraction: theimmersedboundarymethod 37
6.1 EulerianandLagrangianmeshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2 Velocityinterpolationandforcespreadingintheimmersedboundarymethod . . . . . . . . . . . 40
6.3 Discretedeltafunctionsfortheimmersedboundarymethod . . . . . . . . . . . . . . . . . . . . . 42
7 Membranemodelandenergetics 43
7.1 Deformationofamembranefaceelement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
8 Advancedmodeldiscussions 49
8.1 Schemeofthecombinedsimulationalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8.2 Membranemeshgenerationbysubdivision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
8.3 Examplesofsphereandredbloodcellmeshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
8.4 Tank-treadingcapsule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.5 Deformationparameterandinclinationangleforatank-treadingcapsule,part1. . . . . . . . . . 56
8.6 Deformationparameterandinclinationangleforatank-treadingcapsule,part2. . . . . . . . . . 58
8.7 Convergenceofthedeformationparameterandtheinclinationangle . . . . . . . . . . . . . . . . 59
8.8 Columninitializationofaredbloodcellsuspensionsimulation. . . . . . . . . . . . . . . . . . . . 60
8.9 Self-intersectionofaredbloodcellduringinitialization. . . . . . . . . . . . . . . . . . . . . . . . 61
8.10 Randominitializationofadenseredbloodcellsuspensionsimulation . . . . . . . . . . . . . . . . 62
8.11 Problemsrelatedtotheimmersedboundarymethod . . . . . . . . . . . . . . . . . . . . . . . . . 64
8.12 Immersedboundarymethodandsmallnodedistances . . . . . . . . . . . . . . . . . . . . . . . . 67
8.13 Implementationofwallroughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
9 StressevaluationincombinedimmersedboundarylatticeBoltzmannsimulations 73
9.1 Timeevolutionofparticlestressandwallvelocityforasinglecapsule. . . . . . . . . . . . . . . . 80
9.2 Localstressesandviscosityforasinglecapsule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
9.3 Suspensionstressesforroughandsmoothwalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
9.4 Suspensionvelocitiesforroughandsmoothwalls . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9.5 Suspensionstress,viscosity,andvolumefractionprofiles . . . . . . . . . . . . . . . . . . . . . . . 84
10 Rheologyandmicroscopicbehaviorofredbloodcellsuspensions 85
10.1 Velocityanddensityprofilesforaredbloodcellsuspension . . . . . . . . . . . . . . . . . . . . . 87
10.2 Snapshotsofshearedredbloodcellsuspensionsatvariousshearrates . . . . . . . . . . . . . . . 89
10.3 Redbloodcellanditsinertiaellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
10.4 Tank-treadingandtumblingellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
xii ListofFigures
10.5 Schematicsofthedirectorandthenematicorderparameter . . . . . . . . . . . . . . . . . . . . . 92
10.6 Viscosityofredbloodcellsuspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.7 Rescaledviscosityofredbloodcellsuspensions,part1 . . . . . . . . . . . . . . . . . . . . . . . . 94
10.8 Effectivebulkvolumefractionsanddensityprofilesofredbloodcellsuspensions . . . . . . . . . 96
10.9 Comparisonofbloodviscositiesobtainedfromsimulationsandexperiments . . . . . . . . . . . . 97
10.10 Averagetumblingfrequenciesofsuspendedredbloodcells.. . . . . . . . . . . . . . . . . . . . . . 99
10.11 Tumblingandtank-treadingofasingleredbloodcellinshearflow . . . . . . . . . . . . . . . . . 100
10.12 Rescaledviscosityofredbloodcellsuspensions,part2 . . . . . . . . . . . . . . . . . . . . . . . . 101
10.13 Deformationprobabilitydistributionsforsuspendedredbloodcells . . . . . . . . . . . . . . . . . 102
10.14 Deformationmaximumofsuspendedredbloodcells . . . . . . . . . . . . . . . . . . . . . . . . . 103
10.15 Orderparameteranddirectorinclinationangleforsuspendedredbloodcells. . . . . . . . . . . . 104
10.16 Inclinationangleprobabilitydistributionsforsuspendedredbloodcells. . . . . . . . . . . . . . . 105
10.17 Meansquaredisplacementsofsuspendedredbloodcells . . . . . . . . . . . . . . . . . . . . . . . 109
10.18 Reduceddiffusivitiesofsuspendedredbloodcells . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
10.19 Relativeimportanceofredbloodcelldiffusionalongthey-andthez-axes . . . . . . . . . . . . . 111
10.20 Reducedshorttimevelocitiesofsuspendedredbloodcells . . . . . . . . . . . . . . . . . . . . . . 112
10.21 Relativeimportanceofredbloodcellshorttimevelocitiesalongthey-andthez-axes . . . . . . 113
10.22 Non-Gaussianparameterfordiffusionofsuspendedredbloodcells . . . . . . . . . . . . . . . . . 114
10.23 Displacementprobabilitydistributionsofsuspendedredbloodcells . . . . . . . . . . . . . . . . . 115
10.24 Timeevolutionoftheparticlestressfluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
10.25 Relativemagnitudeofparticlestressfluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
10.26 Distributionsofparticlestressfluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
10.27 Correlationdiagramsforthefluidandtheparticlestresses . . . . . . . . . . . . . . . . . . . . . . 119
10.28 Pearsonproduct-momentcorrelationcoefficientfortheshearrateandtheparticleshearstress. . 120
B Chapman-EnskoganalysisandadvancedlatticeBoltzmanncalculations 133
B.1 ConvergenceofthestressintheTaylor-Greenvortexflow . . . . . . . . . . . . . . . . . . . . . . 143
C Derivationofthemembraneforces 145
C.1 Deformationofamembranefaceelement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
C.2 Membranefaceandnodeconventionforthederivationofthebendingforce . . . . . . . . . . . . 150
C.3 Volumeandsurfacecontributionsofasinglemembraneface . . . . . . . . . . . . . . . . . . . . . 151