surfaceparameterdirection

ComputMech(2016)58:747–767DOI10.
1007/s00466-016-1317-8ORIGINALPAPERSymmetryaspectsinstabilityinvestigationsforthinmembranesAndersEriksson1·ArneNordmark1Received:28February2016/Accepted:2July2016/Publishedonline:1August2016TheAuthor(s)2016.
ThisarticleispublishedwithopenaccessatSpringerlink.
comAbstractModellingofstructuralinstabilityproblemsisconsideredforthinsquaremembranessubjectedtohydro-staticpressure,withafocusontheeffectsfromsymmetryconditionsconsideredorneglectedinthemodel.
Ananaly-sisisperformedthroughgroup-theoreticalconceptsofthesymmetryaspectspresentinaatmembranewithone-sidedpressureloading.
Theresponseofthemembraneisdescribedbyitsinherentdifferentialeigensolutions,whichareshowntobeofvedifferenttypeswithrespecttosymmetry.
Adiscus-sionisgivenonhowboundaryconditionsmustbeintroducedinordertocatchalltypesofeigensolutionswhenmodellingonlyasubdomainofthewhole.
LackingsymmetryinaFEMmodelofthewholedomainisseenasaperturbationtotheproblem,andisshowntoaffectthecalculatedinstabilityresponse,hidingormodifyinginstabilitymodes.
Numericalsimulationsverifyandillustratetheanalyticalresults,andfurthershowtheconvergencewithmeshnenessofdiffer-entaspectsofinstabilityresults.
KeywordsBifurcation·Symmetry-breaking·Grouptheory·Meshing·Boundaryconditions1IntroductionSymmetryinstructuresisoftenaresultfromaestheticconsid-erationsorfromfunctionaloptimization.
Forinstance,manybiologicalstructurespossessahighdegreeofsymmetry.
Itiswellknownthattheoptimizedstructuralformsareverysen-sitivetoallkindsofimperfections.
ThispaperaimstodiscussBAndersErikssonanderi@kth.
se1KTHMechanics,RoyalInstituteofTechnology,OsquarsBacke18,10044Stockholm,Swedenhowconsideredorneglectedsymmetriesinthemodellingofaproblemsignicantlycanaffectthecomputationalresults.
Whenanalyzingastructure,itistemptingtoutilizeanexistingsymmetryforimprovedefciencyoraccuracy,andsymmetryhasbeenutilizedinmanycommonsolutionmeth-ods.
Perhapsmostprominentinclassicalanalysesforplatesandshells,theconsiderationofoddandevenfunctionsinseriessolutionswasanimportantaspect[42].
Itwasalsoearlieranimportantcompetenceforstructuralanalyststobeabletosplitgeneralloadingsintosymmetricandanti-symmetricsubcases,andthistopichasbeenanimportantpartofengineeringcurricula[13,andmanyothers].
Thesymmetryconsiderationshavealsobeenimpor-tantaspectsofthedevelopmentofniteelementmethods('FEM').
Inparticular,thiswasthecaseinearlierdays,whencomputerresourceswerecommonlylimitingtheanalyses,andsuchsimplicationswerenecessaryforobtainingresultswithmaximumaccuracy[33,45].
Eveniftheabovemethodsoccasionallyusedalsodiago-nalsymmetrysectionsinaCartesiansystem,forinstanceintheanalysisof18ofasquareplate,mosttraditionalsym-metryconsiderationshaveconsideredmirrorsymmetries(and,occasionally,anti-symmetries)withrespecttothethreeprincipalcoordinateplanes,byintroducingessentialbound-aryconditionsofzerodisplacements.
Similarreasoninghasguidedanalysesofcirculardomainsbyextractingarepetitivesectorofthefullgeometry.
Analyticaltreatmentsofthesymmetryaspectsofsquareandcircularpressurizedmembranesareverysimilar.
Thepresentworkfocussesonasquaredomain,basedontwoarguments.
First,itisbelievedthatthesquaredomainistheoneforwhichanengineerwouldbemostcondentinusingtheobviousmirrorsymmetriesintheaxisplanes,anditisthereforeimportanttopointtotheeffectsfromthisassump-tion.
Thesecondreasonisthatthesquaremembraneaffected123748ComputMech(2016)58:747–767byapressurehasasomewhatrichersetofinstabilities,duetothedifferencesbetweenthesideanddiagonalsymmetryplanes.
Forthepresentcontext,thecornersofthesquaredomaindonotleadtoanyparticularproblems.
Theabovesymmetryassumptionshavebeenbasedonanidenticationofasymmetricmodelgeometryaffectedbysymmetricloadingandotherboundaryconditions.
Inalinearstaticanalysis,thistypicallyimpliesthatthecalcu-latedresponsewillfulllthesamesymmetryconditions.
But,evenifthelinearstaticresponseofastructuremayshowtheconsideredsymmetryconditions,dynamicaswellasinsta-bilityresponsesdonotnecessarilyinheritthese.
Well-knownexamplesarevibrationandbucklingmodesofrectangularplates,butalsothefulldynamicresponseofasimplysup-portedbeamorframe,wheretheunsymmetriesinthesimplesupportscanaffectresultsthroughthemassdistribution.
Ageneralrecommendationhascommonlybeenoneofcautioninusingsymmetrysimplicationsinthesesituations.
Thepresentworkconsidersthesymmetryandregularityaspectsinthecontextofinstabilityanalysesofthinpres-surizedmembraneswithahighlysymmetricshape.
Thisclassofsimulationsisvalidforalargevarietyofthinthree-dimensionalinatablestructuresusedinseveralengineeringandmedicalcontexts[23,26,29].
Thesesituationsareoftenbothgeometricallynon-linearduetonitedeformationsandmateriallynon-linearthroughtheconstitutiverelationship.
Analyticalresultscanbeobtainedforseveralsimplegeome-tries[28,35],andnumericaltreatmentsarealsoavailable[3,4].
Severalaccompanyingaspectsareimportant,suchasloaddescriptions[16],contacts[1,21],anddynamics[15],butalsoinstabilitiesofmanyforms:limitandturningsolu-tions,bifurcations,andwrinkling.
Thethinmembranesareoftenmodelledashyper-elastic,withseveraldifferentformulations,eachwithanumberoffreeparameters.
Severalstudieshaveshownthattherela-tionsbetweentheconstitutiveparametersinamaterialmodelsignicantlyaffecttheresponseofsimulatedmembranes[11,30].
TheMooney–Rivlinmaterialmodel[27,34]isfre-quentlyshowntoproducereasonablygoodapproximationofstressesinamaterialatleastformoderatestrains,andiscomputationallyconvenient[5,18].
Thematerialscanbeaffectedbyinstabilityatcertainstressstates;suchinstabili-tiesarediscussedinliterature[17,20,24,andmanyothers].
Themostprominentinstabilityphenomenainthinmem-branesare,however,relatedtothegeometricnon-linearity,wheresignicantcongurationchangesresultfrompres-surization.
Forauid-loadedmembrane,differentstabilityconclusionsarereachedwhenauidlevelorauidvolumeiscontrolled[43].
Bifurcations,forinstancesymmetry-breakingdeformationmodescanalsooccurintheloadingprocess.
Wrinkling,correspondingtomoreorlesslocalizeddomainsofcompressivestressesalsooftenoccurfortheseproblems[36,39],andcanbeseenasrelatedtogeometriceffects.
Overallinstabilityofastructureduetoaconsidera-tionofwrinklingthrougharelaxedenergyformulation[32]orthetension-eldtheory[38],isshownin[31].
Generalaspectsof,primarily,geometricinstabilitiesarediscussedinclassicalworks[22,41],butalsoofteninthecontextofniteelementsimulations[2,andothers].
Thecurrentpaperrstgivesabriefreviewofthenumer-icalmodellingofthinmembranesusedfortheexperimentsbelow.
Then,adescriptionisgivenofthesymmetryaspectsrelevantfortheconsideredsquarestructureanditsFEMdiscretization.
Ananalyticaltreatmentshowshowasubdo-mainmodellingcanhideormodifyinstabilities,andhowthedomainsymmetryandaccompanyingsetsofboundaryconditionsforequilibriumsolutionandeigenmodeextrac-tionaffectinstabilityrepresentationandconclusions.
Asetofillustrativenumericalsimulationsaregiventodemonstratetherelevanceoftheanalyticalconclusions.
Afewconclud-ingremarksaregivenontheeffectsfromsymmetryinthemodellingofinstabilityaffectedstructures.
2MechanicalmodellingSophisticatedsimulationalgorithmsareneededfortheevalu-ationandinterpretationofinstabilityinageneralFEM-basedcontext.
Areliableniteelementformulationisneeded,whichisnotoverlysensitiveto,e.
g.
,thescalingoftheprob-lem.
Asolutionalgorithmmustalsobeused,whichcanisolateandidentifythecriticalsituations.
2.
1StructuralmodelAsshellmodelsarelessreliableformembraneanalyses,andarealsocomputationallydemanding,simulationsoftheloadingprocesshavebeenbasedpurelyonthemembranebehaviorinthepresentwork.
Themechanicalmodelwastherebyoneoflocalplanestressconditions,butina3Dset-ting.
Triangularelementswithlinearkinematicassumptionswereusedfordiscretizationofthemembrane[11].
Inbrief,theelementformulationisbasedonaTotalLagrangianform,assumingtheelementtokeepatinallcongurations.
Strainsandstressesareconstantwithintheelement.
Pre-stressingofthestructurewasintroducedbyprescribingdisplacementsofedgenodes,withanaccompanyingchangeinthickness.
AnisotropicincompressibleMooney–Rivlinmodelwasused,andbasedonastrainenergyformulationW=W(C)=c1I1(C)+c2I2(C),(1)whereI1andI2arethetworstinvariantsoftherightCauchy-GreendeformationtensorC.
Theconstitutivecon-stantswererelatedtoashearmodulusμanda'hardening123ComputMech(2016)58:747–767749parameter'k,accordingto:c1=12μ1+k,c2=kc1(2)Localplanestressconditionsandincompressibility,throughthethirdinvariantofC[18],werebuiltintotheformulation,givingexplicitconstitutiverelations.
Wrinklingisperhapsthemostcommoninstabilityinthinmembranes.
Relaxedstrainenergyforms[32],leadingtoatension-eldtheory[38],arecommonwaystotreattheselocaleffectsfromcompressivestresses,butthesecanleadtoglobalinstabilityinasimulationmodelwhenpressureactsnormaltoawrinkledmid-surface[31].
Aswrinklingeffectsarestronglydependentonauseddiscretizedmodel,wrin-klingconditionswerehereindicatedfromlocalstretchvaluesateachequilibriumsolutionfound,andcalculationsweredis-continued.
Contactswithhardorsoftsurroundingsurfacesandself-contactsbetweendifferentpartsofthemembranealsofrequentlyoccur[40],butwereveriednottoaffecttheresultsbelow.
Thekinematicassumptionsandthematerialmodelwereusedtoformulatethestructuralinternalforcescompletelyfromtheglobalstructuraldisplacementcomponentsdas:f=f(d)(3)BothdandfareN-dimensionalvectors,withNthenumberofconsidereddegreesoffreedominthediscretizedmodel.
2.
2PressureloadingTheloadingonapressurizedmembranecomesfromone-sidedgasoruidover-pressures,withsignicantdifferencesintheirformulations.
Anover-pressurefromgasψisuni-formoverallelements,givingthedisplacement-dependentstructuralexternalforcevectorp=p(d,ψ).
Ahydro-staticpressure,however,oftengivesaricherandmoreinterestinginstabilitybehavior,inparticularwhenthepressureisact-ingfrombelowonahorizontalmembrane[43].
AssuminggravityintheglobalnegativeZdirection,thepressureonanelementsurfaceisdescribedbyψ(z)=ρg(Zuidz)forz≤Zuid,withρtheuiddensity,gthegravitationalacceleration,andZuidtheuidsurfacelevel;zisthespatialpositionofapoint.
Consistentnodalloadsthengivesavectorofexternalforcesp=p(d,Zuid).
Ineithercaseofpressureload,theN-dimensionalvectorofglobalexternalforcesaredescribedbyoneprimaryloadparameterp=p(d,γ)(4)withγ≡ψorγ≡Zuid,respectively.
2.
3SimulationalgorithmsForaquasi-staticformulation,thediscretizedequilibriumbetweenexternalandinternalforcesdemandssolutionssat-isfyingthestructuralresidualequation:F(z)≡f(d)p(d,γ)=0(5)wherealsoFandpareofdimensionN,andzisan(N+1)-dimensionalvectorcollectingallvariables:z=(dT,γ)T(6)Thissystemdenesone-dimensionalcurvesegments,whicharerepresentedbyasequenceofequilibriumsolu-tionszi,(i=0,1,Oneparticularequilibriumsolutioniscontrolledbyanextraequation,whichcanbeinterpretedasthespecicationofonemechanicalvariable,orseenasanumericalbranchprogresscontrol.
Inthelattercase,thechoiceofbranchcontrolfunction[6,12],andstep-lengths[10],areimportantaspectsofequilibriumbranchfollowing.
ThevariationoftheresidualexpressioninEq.
(5)is:δF=KdKpδdδγp,γ(7)whereasubindexfollowingacommadenotesadifferential.
Theexpressiongivesthetangentstiffnessmatrix,K=KdKp(8)includingaload-dependentterm[37].
Importantpropertiesofanequilibriumsolutionaredescribedbythismatrix.
Differentformsofinstabilitiesalongthesebranchesarealsodetectedandclassied.
Inparticular,bifurcationsandlimitsolutionsarefundamentalforinterpretingstructuralresponseandstability.
2.
4StaticstabilityStabilityisanimportantaspectofloadedstructuralsystems,andcaninbroadtermsbedescribedintermsofthecapacityofthestructuretohandledisturbancesfromanestablishedequilibriumconguration,orasa'powertorecoverequi-librium'.
Inthissense,theengineeringconceptof'staticstability'referstotransitionsbetweenstaticanddynamicbehavior,andisapropertyofaparticularstaticequilibriumconguration.
Particularinterestistherebyalwaysdirectedtowardschangesofthestabilitywhenmodifyingtheloadparameter,withacriticalsolutionbetweenthesedomainsasthemostinterestingresult.
Thebasisforthisviewpointistheminimumofthetotalpotentialenergyforaparticularloadparameter.
123750ComputMech(2016)58:747–767Thisviewonstabilityisasimplication,andthestabilityofthestructureshouldratherbeseeninadynamicalcontext,whereanintroducedkinematicperturbationfromastableequilibriumcongurationofthestructurewillbelimitedovertime,andinparticularthatthedeviationfromthestaticequi-libriumcanbekeptbelowaspeciedmagnitudebylimitingtheperturbation.
Thestabilityofanequilibriumcongurationisintimatelyconnectedtotheeigensolutionsoftheproblem.
Forthedis-cretizedmodel,theeigenvaluesofthecurrenttangentialstiffnessmatrix,K,orofthecombinationofthismatrixwithacorrespondingmassmatrixMleadtothestabilityconclu-sions,withtheeigenvaluesandtheeigenvectors.
Itisnotedthat,aslongasthemassmatrixMispositivedenite,theeigenvaluesofthematrixKandthoseofthe(K,M)pairwillhavethesamesigndistributions,evenifthevaluesaredif-ferentanddependentonthemassdistributionconsidered.
Inthequasi-staticsetting,withonlytheKmatrixestab-lishedandanimplicitassumptionthatmassisequallydistributedonallnodes,theeigenvaluesaretherebyseenasthecurrentprincipalstiffnessvalues,relatedtoasetofdiscretizedorthonormaldisplacementvectors.
Withalleigenvaluespositive,thestructureisseenasstable,asthepotentialenergyisminimizedatthisloadparameter,andexternalworkisneededtoperturbtheequilibriuminanyspecieddirection.
Azeroeigenvalueindicatesthatasmallperturbationinaspecicdirectioncanoccuratcon-stantpotentialenergy.
Thepresentnumericalworkhasonlyconsideredthetangentstiffnessmatrixintheeigenvalueextraction,butithasbeenpreviouslynotedthatatleastasimpliedmassmatrixmustbeintroducedwhenconsider-ingconstrainedequilibriumformulationsresultingfrom,e.
g.
,two-parameterloaddescriptions[44].
2.
5ParameterdependenceFormembranes,akeyissueistheanalysisoftheparame-terdependenceinasimulatedresponse.
Forthis,speciallydesignedalgorithmscanintroduceauxiliaryvariables,rep-resentingaparameterizationofthestructureortheloading,andthensolvetheequilibriumproblem,anditscorrespond-ingstabilitypropertiesinthehigher-dimensionalspace.
Thegeneralizedbranch-followingalgorithmusedinthepresentworkisdiscussedin[8].
TheseformulationsuseNγpara-metersinthesolutionsetzT=[dT,T],andanextendedsetofequilibriumequations:G(z)=F(z)g(z)=0(9)whereg(z)isasetofNgequations,reectingadditionalcon-ditionsonthesoughtequilibriumsolutions.
AtypicalsettingwouldbetouseNg=Nγ1controlequationsforthespec-icationofthephysicalproblem,andonemoreequationforthenumericalplacementofeachsolutiononageneralizedequilibriumbranch.
ThesettinginEq.
(9)allowsforveryrobustandsystematicsolutionofthenon-linearequilibriumbranches,buthasalsobeenusedtoimmediatelyndcriticalequilibriumstates[7],andtofollowtheboundsforthefea-sibleregioninanoptimizationsetting[9],withouttheneedtosolveseveralcompleteload-responsebranches.
2.
6SymmetryaspectsofmodelandmeshSymmetryisseeminglywell-knowninstructuralmodelling,withageometricalbasisinthesensethatagureordomainisrepeated,oftenintheformofamirroredcopy.
Inthissense,ithasarelationtorepetition,whereseveralsubdomainsofawholeareidentical,andsimulationscanbesimpliedbyonlyconsideringonlyonesubdomainwithsuitableboundaryconditions.
Thiscangiveallequilibriumsolutionsfulllingtheseconditions.
Thesamesubdomainandboundarycon-ditionsalsogiveseigenmodeswhichfulllthesesymmetryconditions,showingthebifurcationsandthesecondarypathsemanatingatthem.
Withotherboundaryconditionsonthesamesubdomain,othersolutions,bifurcationsandsecondarypathswillbefound.
Inordertondallaspectsofinstabil-ityforagivenstructure,severalsetsofboundaryconditionsmaybeneededforthesolutions,andfortheeigenmodecal-culations.
Theeigenmodescantherebybesituatedinothersolutionspacesthantheequilibriumsolutionitself,some-whatlikethesituationappearingforabar,bucklingintoabendingresponseduetoapurelycompressiveforce.
Thesymmetryaspects,andtheireffectsonbothsolutionsandeigenmodes,willbefurtherdiscussedbelow.
Thepresentworkhasalsoconsideredanothersymme-tryaspectinthediscreteFEM-basedmeshingofthewholedomain.
Themeshingcankeepordestroyaninherentsym-metry,andcanbeseenasaperturbationtothecontinuousmodel.
Asetofmesheshavealloweddifferentclassesofsolutions.
Inparticular,someinstabilitymodesofastructurecanbehiddenorotherwiseaffectedbythesymmetryofthemeshused.
2.
7ConsidereddomainsandmeshesThepresentworkprimarilyfocussedonthehighlysymmetricdomainofasquareintheX–Y-plane.
Themaincharacteristicfeaturesofthestudiedproblemareitsshape,andtheuidpressureloading,whichisnotsymmetricinthedirectionofgravity,Fig.
1.
Thestudiesofdiscretizationsofaasquaredomaintherebyusedasetofsubdomains,butalsoarepresentativesetofmeshes,fulllingdifferentsymmetrypropertiespresentinthewholedomain,Fig.
2.
Thesubdomains,whicharegray-shadedinthegure,weredenotedaccordingto'gX',and123ComputMech(2016)58:747–767751XYZδ1δ2σ1σ2Fig.
1Geometryofstudiedsquaremembraneproblem,withsymme-tryaspectsmarked.
Materialcoordinates,withuidpressureactinginpositiveZdirection.
Notationforsymmetryplanesσ1,σ2,δ1,δ2isfur-therdiscussedbelowthemeshesaccordingto'mX(nnn)',where'mX'denotesthebasicmeshform,and'nnn'thenumberoftriangularele-mentsinthemesh.
Finermesheswerealwayscreatedbysuccessivelydividingeachtriangleintofourbyintroducingmid-pointsoneachelementedge.
Forpresentationpurposes,themid-pointofthesquarewasalwaysintroducedasonenodalpoint.
ThesubdomainsandmeshesinFig.
2havethefollowingsymmetryproperties,withfurtherdiscussionofthetermi-nologybelow.
Meshm1(andsubdomaing1):has4mirrorplanes,andrepeatswhenrotating2π/4.
Meshm2:2mirrorplanes(σ1,σ2);minimalrotation2π/2.
Meshm3:2mirrorplanes(δ1,δ2);rotation2π/2.
Meshm4:nomirrorplanes;rotation2π/4.
Meshm5:nomirrorplanes;rotation2π/2.
Meshm6:1mirrorplane(σ2);rotation2π/1.
Meshm7:1mirrorplane(δ1);rotation2π/1.
Meshm8:nomirrorplanes;rotation2π/1.
MeshmH(andgH):onehalfofthewhole,whichcanbecompletedeitherbymirroringinσ1orbyrotating2π/2.
MeshmD(andgD):onehalf:mirrorinδ1orrotation2π/2.
MeshmQ(andgQ):onequarter:successivemirrorsinσ1,σ2,orsuccessiverotations2π/4.
MeshmT(andgT):onequarter:successivemirrorsinδ1,δ2,orsuccessiverotations2π/4.
MeshmO(andgO):oneeighth:successivemirrorsinσ1,δ1.
3AnalyticaltreatmentThissectionshowshowsymmetrycanbetakenadvantageofwhenndingequilibriumsolutionsandtheirstabilityandbifurcations.
Thisappliesbothtoacontinuousmodelandtom2(6)m1(8)m3(12)mQ(2)m4(16)m5(14)mH(4)m6(10)m7(10)mT(2)m8(9)mD(4)mO(1)Fig.
2Symmetrycasesusedinexamples.
Basicmesheswithnumberofelementsinparenthesis.
Meshesaresuccessivelyrenedbydividingeachelementintofourbyplacingnewnodesonelementedges,givingmeshesofthesametype,butwithvaryingelementsizesandnumbers.
Grayareasinguresdenoteamodelledsubdomain,withfurtherdiscus-sionbelow.
Thesesubdomainsaredenotedaccordingto'gX',followingthemeshdenition'mX'intheguresadiscretizedsimulationmodel.
Theanalysisisperformedbyusinggrouptheoryconcepts[19,46],butdiffersinitsappli-cation.
Whiletherstreferencedevelopsthegeneraltheory,andappliesthistothesymmetrypropertiesofpolygonspacetrusses,thepresentworkfocussesonacontinuousdomain,whichisdiscretizedintoaniteelementmesh.
Inrelationtotheelementdiscretization,twomainsituationsareinvesti-gated.
Firstly,itisforsuchproblemsettingstemptingtousea123752ComputMech(2016)58:747–767subdomainofthefullproblemgeometrytogetherwithasym-metrygrouptoreconstructasolutionforthefullgeometry.
Whilecomputationallyefcient,however,thisidearestrictsthesetofsolutionsthatarepossibletond.
Secondly,dis-cretizingacontinuousmodelmightactasaperturbationtotheproblem,similartominorchangesoftheproblemgeometry,whichmaylowerthesymmetryoftheproblem.
Evenifthepresenttreatmentdiscussesaratherspecicproblem,namelythenon-linearresponseofathinmembranetohydro-staticpressureloading,theresultsarerelevantforallnon-linearsimulationsonplanesquaredomains.
Inthiscontext,thereaderisremindedthatasymmetrygroupisagroupwhereeachelementisanisometrictrans-formationof3Dspaceoreldsin3Dspace.
Foranitesizedobject,thesymmetrygroupisapointgroup,whereeachtransformationleavesapointxed.
Thecompositionruleforthegroupisthatabmeansdoingthebtransforma-tionrst,followedbytheatransformation,andtheresultisanothertransformationinthegroup.
3.
1ThesymmetrygroupC4vTheuidloadedsquaremembranehasthesymmetrygroupofasquarepyramid,C4v.
Thegroupconsistsofthefourrotationsr0,r1,r2,andr3aboutthepositivez-axisby0,1/4,1/2,and3/4ofafullturn,whichiscounter-clockwisewhenseeninacommonx-y-planeview.
Additionally,fourmirrorss0,s1,s2,ands3existinthey=0,x=y,x=0,andx=yplanes.
Theseplaneswillbedenotedσ1,δ1,σ2,andδ2,respectively,andareshowninFig.
1.
Theelementr0istheidentityelementofthegroup.
Asanabstractgroup,C4visthedihedralgroupwith8elements,sothecompositionrulesarerjrk=rj+kmod4,rjsk=sj+kmod4,sjrk=sjkmod4,andsjsk=rjkmod4.
Thesubgroups,theirelements,anddescrip-tionsoftheirsymmetrypropertiesareshowninTable1,togetherwithapointertothesubdomainsgXinFig.
2whichcanbeusedtogetherwiththesubgrouptorepresentthefullgeometry.
Asthesubgroupsconstituteahierarchy,pointerstothenextlowersubgroupsarealsogiven.
3.
2UsingsubdomainstondequilibriumsolutionsEquilibriumsolutionsdonotingeneralinheritthefullsym-metryC4voftheproblemitself,evenifthisisthecaseforsolutionsonthefundamentalbranch.
Solutionsoutsidethisbranchmaybecharacterizedbyanysubgroup,downtothecompletelyunsymmetricC1.
Thesolutionforasubdomain,togetherwithamatchingsymmetrysubgroup,Table1,canbeusedtoreconstructasolutiononthefulldomainifcorrectboundaryconditionsareimposedwhenobtainingit.
Thereconstructedsolutionwillnecessarilyshowatleastthesymmetryofthesubgroupusedinreconstruction,whichmeansthatequilibriumsolu-tionbranchesoflowersymmetryareunreachable.
Asanexample,thesubdomainofclass'gH',Fig.
2,canbeusedtogetherwiththeC1v(σ1)subgroup.
Thenon-identityelements0istherebyusedtoreconstructthefullsolution.
Toensurecontinuityofthemembrane,thexandzdeformedpositionsmustbesymmetricabouttheσ1plane.
OnlyequilibriumsolutionsthatobeytheC1v(σ1)orhigher(C2v(σ),C4v)symmetriescanbefoundthisway,whileallpotentialsolutionsoutsidethis,e.
g.
,anysolutionswiththemid-pointmovingintheydirectionwillbehidden.
If,insteadthesamesubdomainofclass'gH'iscombinedwiththeC2subgroup,continuityrequiresthat(x,y,z)atapointonσ1mustbeequalto(x,y,z)atthepointrotatedbytherotationr2,i.
e.
,halfaturn.
Thisisthusanon-localboundaryconditionexceptatthecenterpoint,wheretheconditionisx=y=0.
Now,reconstructedsolutionsarerestrictedtohaveC2,C2v(σ),C2v(δ),C4,orC4vsymmetry,whichisadifferentrestrictiontothefullsolutionspace.
BothTable1SubgroupsofC4v,theirelementsandadescriptionofthesymmetrypropertiesSubgroupElementsDescriptionSubdomainChildrenC4vr0,r1,r2,r31/4rotationandallmirrorsgOC2v(σ),C4,C2v(δ)s0,s1,s2,s3C4r0,r1,r2,r31/4rotationandnomirrorsgQ,gTC2C2v(σ)r0,r2,s0,s2MirrorsinσplanesgQC1v(σ1),C1v(σ2),C2C2v(δ)r0,r2,s1,s3MirrorsinδplanesgTC1v(δ1),C1v(δ2),C2C2r0,r21/2rotationandnomirrorsgH,gDC1C1v(σ1)r0,s0Mirrorintheσ1planegHC1C1v(δ1)r0,s1Mirrorintheδ1planegDC1C1v(σ2)r0,s2Mirrorintheσ2planeC1C1v(δ2)r0,s3Mirrorintheδ2planeC1C1r0Trivial(no)symmetryg1—Thefourthcolumnstateswhichsubdomainscanbecompletedbytheelement.
'Children'denotethenextlowergroupsofsymmetryinthehierarchy.
Underlinedelementsaregeneratorsofthesubgroup123ComputMech(2016)58:747–767753approachesareabletoreconstructsolutionswithC2v(σ)orC4vsymmetry,buteachapproachcontainssomesolutionsnotfoundintheother,andneitherapproachcanndsolu-tionswithC1,C1v(δ1),C1v(σ2),orC1v(δ2)symmetry.
Thecombinationofsubdomainandsubgroup,anditsassociatedboundaryconditionstherebygivethelimitsforreachablesolutions.
Thenumericalsimulationsbelowfurtherdemon-stratetheconsequencesofthis.
3.
3RepresentationsofC4vandeigenmodesThestabilityofanequilibriumsolutioncanmostoftenbedeterminedbylinearstabilityanalysis.
Onlywhenalinearstabilityanalysisndseigenvaluesequaltozero,anon-linearanalysisisneededtodeterminestability.
Thespaceoflin-earizeddeviationsfromanequilibriumsolutionisequaltothespacespannedbythecompletesetofeigenmodesofthesolution.
Thenon-zeroeigenvaluesandtheireigenmodeshapesdependonwhatinertiapropertiesareassumedfortheproblem,butlinearstabilityorinstability,i.
e.
,thenumberofnegativeeigenvaluesisindependentofthewaymassisintroduced,aslongasnon-zeromassisgiventoallpartsofthestructure.
Thus,auniformmassdistributionwasassumedhereforthecontinuousmodel,whereasequalnodemasseswereusedforthediscretizedmodel.
Thesymmetryoftheequilibriumsolutionwillcarryovertotheproblemofndingeigenmodes.
Sincetheeigenmodeproblemisalinearproblem,theelementsofthesymmetrygroupwillactaslinearoperatorsonthespacespannedbytheeigenmodes,accordingtogrouprepresentationtheory,previouslydevelopedandappliedin[19].
Aresultisthattheeigenspaceforaproblemwithnitesymmetrygroupcanbewrittenasadirectsumofnite-dimensionalspaces,eachcorrespondingtoacertaintypeofrepresentation.
Findingtheserepresentationsisanon-trivialtask,buttheresultisknownfortheC4vgroupanditssubgroups.
Inthefollowing,anequilibriumsolutionwithC4vsymmetryisassumed.
Thecorrespondingresultsforequilibriumsolutionswithlowersymmetryaresimpler,butwillnotbepresentedinthispaper.
GivenanequilibriumsolutionwithC4vsymmetry,grouprepresentationtheorystatesthattheeigenspacecanbewrittenasadirectsumoftwo-orone-dimensionalsubspacesofthefollowingvetypes:IAtwo-dimensionalsubspace.
Wehavechosenabasisconsistingofoneeigenmodeφ1symmetricundertheC1v(σ1)subgroup,andoneeigenmodethatistherstrotatedby1/4turn:φ2=r1φ1.
Ingeneral,eigenmodesinthisspaceuφ1+vφ2haveonlyC1symmetry,unlessu=0,v=0or|u|=|v|,inwhichcaseaC1vsymmetryispresent.
IIAone-dimensionalsubspacewithonebasisvectorφsymmetricundertheC2v(σ)subgroup.
IIIAone-dimensionalsubspacewithonebasisvectorφsymmetricundertheC2v(δ)subgroup.
IVAone-dimensionalsubspacewithonebasisvectorφsymmetricundertheC4subgroup.
VAone-dimensionalsubspacewithonebasisvectorφsymmetricundertheC4vsubgroup.
Examplesofeigenmodesofthedifferenttypes,shownonadeformedmembranecongurationcanbeseeninFig.
3.
Foreachoftheinnitenumberofeigenvalues,thecor-respondingeigenspacewillbeoneoftheabovetypes,andthuseacheigensolutionmaybelabelledasoneofthevetypes,witheigenvaluesoftypeIdouble,andtheotheronessingle.
Thesepropertiesoftheeigenvaluesareindependentofwhetherthemodelisdiscretizedornot,aslongasthedis-cretizationpreservestheC4vsymmetryoftheproblem,andtherebyhasequilibriumsolutionswithC4vsymmetry.
Theeigenvaluesofanequilibriumsolutionwilldenethestiffnesspropertiesofeacheigenmodedirection,orthesec-ondordervariationofpotentialenergywhenfollowingthismodedirectionfromtheequilibriumconguration.
Ingen-eral,theselinearizeddeviationsaredescribedbyuφ1+vφ2fortypeIanduφfortheothercases.
Representationtheoryshowshowthesedeviationsareaffectedbytheactionofeachgroupelement,Table2.
Forexample,therotationr3actingonaneigenmodeoftypeItransformsageneraldirection[u,v]Tintothedirectiondescribedby[v,u]T.
Asafur-therexample,allmirrorelementswillchangethecoefcientuofaneigenmodeoftypeIVtou,i.
e.
,theuφisthenanti-symmetricwithrespecttothemirrors.
3.
4EigenvalueboundaryconditionsforsubdomainsTable2canbeusedtoidentifyhowboundaryconditionscanbeintroducedoneigenmodestoincludesomeofthetypesandexcludeothers.
Forexample,usingthesubdomain'gQ'andenforcingsymmetryforaneigenmodeontheσ1planeandanti-symmetryontheσ2plane,meansthataneigenmodemustbeunchangedbytheactionofthes0groupelementandmustchangesignundertheactionofthes2element.
ConsideringeigenmodesoftypesII,III,IV,orV,lookingatthes0columnofTable2,therstconditionforcesu=u,i.
e.
,u=0fortypesIIIandIV,whereasthesecondconditionbythes2columnforcesu=0fortypesIIandV.
Thus,nonon-zeroeigenmodeoftypesII–Vcanfulllboththeseboundaryconditions.
ConsideringeigenmodesoftypeI,bothcolumnss0ands2requirev=0withnorestrictionsonu.
AnyeigenmodeevaluatedusingtheseboundaryconditionsmustthereforebeoftypeIandhavetheformuφ1.
Usingforsubdomain'gQ'insteadsymmetryonbothplanes,meaningthattheeigenmodemustbeunchangedbytheactionofboths0ands2,thenewconditionsbecome:{u=u,v=v},{u=u,v=v}fortypeI,{u=u},123754ComputMech(2016)58:747–767I:II:III:IV:V:Fig.
3ExampleeigenmodesoftypesI–V,indicatedbyarrows,Addi-tionally,thezcomponentisshownbycoloring.
FortypeI,eigenmodeφ1isshowninthegure.
Allmodesareplottedonadeectedshapecorrespondingtothethirdbifurcation,cf.
thenumericalsolutionbelow{u=u}fortypesIIandV,and{u=u},{u=u}fortypesIIIandIV.
EigenmodesuφoftypesIIorV,butnoothers,arethuscompatiblewiththeseboundaryconditions.
Asystematicusageofboundaryconditionsintheeigen-modeevaluationscantherebyoftengivethecompletesetofeigenmodes.
Forthesubdomain'gQ'andaC2vsymmetry,anintroductionofsymmetryconditionsonedgesσ1andσ2willadmittypesIIandV,whereasanti-symmetricconditionswillgivetypesIIIandIV.
Symmetryononeoftheedgesandanti-symmetryontheotherwilladmitthetwotypesI.
Foursimulations,withtheseconditionssuccessivelyintroduced,willtherebyfacilitatetheevaluationofthefundamentalsolu-tionbranch,andadmitthesolutionoftheeigenmodesforthesamesetsofboundaryconditions,asalltypesofmodesarereachablefromatleastoneofthesimulations.
Similarreasoningconcerningsymmetryconditionsandtheeigenspacetypesforsubdomain'gO',Figs.
1,2,leadstoTable3.
Itisnotedthattheseboundaryconditionsonlyapplyfortheeigenmodes,whereasthefundamentalsolutioncanbeobtainedbyimposingsymmetryonbothedgesσ1andδ1,whichonlywillgivetypeVlimitsolutionsimmediately.
Forthissubdomainchoice,alleigenmodescannotbecalculatedwiththesameboundaryconditionsasarevalidfortheequi-libriumsolutions.
BotheigenmodesmustthusbesolvedforsimultaneouslyfortypeI,sincethemodescouplethroughtheboundaryconditionatδ1.
Acompletesetofeigensolu-tionscanthusbeobtainedbysolvingtheeigenvalueproblemonthesmallerdomainthroughfoursolutionswithdifferentsimpleboundaryconditionsonthemodeledges,andthenonesolutionforbothtypescoupled.
Inthediscretizedver-sionofthiscase,thelattercaseneedsasolutionvectorofdoublesize.
3.
5BifurcationoftypeIforaconservativesystemInrelationtothestudiedproblem,therstbifurcation,whichcorrespondstoadoublevanishingeigenvalue,typeI,isofparticularinterest,andthiswillbeamainfocusforthenumer-icalexperimentsbelow.
Thisequilibriumsituationwillbeanalyzedhere,presentingaheuristicviewontheworkbyIkedaetal.
[19].
ForacriticalequilibriumsolutionwithC4vsymmetry,andadoublezeroeigenvalueoftypeI,atwo-dimensionalinvari-antcentermanifoldtangenttothezeroeigenspaceexists.
Theoriginalequilibriumsolutionisrepresentedbyu=v=0.
Thetwocoordinatestransformaseigenmodeamplitudes,Table2.
Sincethesystemisconservative,apotentialfunctionV(u,v)givestheresponse.
ThefunctionmustbeinvariantunderalltransformationsofC4v,andtheonlysecondordertermisu2+v2,asallothertermswillnotbeinvariantunders0(uu,vv)ands1(uv,vu),whichtogethergenerateC4v,accordingtoTable2.
Similarconsiderationofthefourthorderterms,leadsto,e.
g.
,theindependentfunc-123ComputMech(2016)58:747–767755Table2ThevetypesofeigenomderepresentationsofC4vTyper0r1r2r3s0s1s2s3I[u,v][u,v][v,u][u,v][v,u][u,v][v,u][u,v][v,u]IIuuuuuuuuuIIIuuuuuuuuuIVuuuuuuuuuVuuuuuuuuuEffectsoftheelementsinthegroupontheeigenmodeamplitudesTable3Boundaryconditionsforsolvingalltheveeigenmodetypesonthesubdomain'gO'betweeny=0andy=x,Figs.
1,2TypeBCatσ1BCatδ1Iφ1:S,φ2:AS(φ1+φ2):S,(φ1φ2):ASIIφ:Sφ:ASIIIφ:ASφ:SIVφ:ASφ:ASVφ:Sφ:SThenotation'S'denotessymmetricconditions,'AS'anti-symmetric.
ThetypeIsolvesthetwovectorssimultaneously,withbothindividualandcombinedboundaryconditionstionalterms(u2+v2)2andu46u2v2+v4asonechoiceofadditions.
Inpolarcoordinates,theconsideredtermsarer2,r4andr4cos(4θ).
Sincebotheigenvaluesofthesym-metricsecondderivativematrixareassumedtovanishatthecriticalsolution,thesecondordertermmustvanish,andthepotentialcanbewrittenasV(r,θ)=r4(A+Bcos(4θ))/24uptofourthorder,withA,Bassituationspecicconstants.
AssumingthereisoneparameterinthesystemwhichchangestheeigenvaluesoftheC4vsymmetricequilibriumsolution,aloadparametertermcanbeaddedtothepotential,as:V(r,θ,μ)=μr2/2+r4(A+Bcos(4θ))/24,(10)Withtheparameterchosensuchthatanincreasingloadfactorμchangestheequilibriumsolutionfromstabletounstable,thepotentialshowspositiveeigenvaluesforμ0atu=v=0.
WithB=0,and|A|=|B|,vanishingoftherstdifferen-tialofthepotentialinEq.
(10)denesveintersectingcurvesinthe(u,v,μ)space.
Eigenvaluesλareobtainedfromtheseconddifferentialsofthepotentialattheequilibriumsolu-tion.
Theresultsshowthefollowingforthebranches:–onebranchwithu=v=0anddoubleeigenvaluesλ1,2=μ.
ThisfundamentalbranchmaintainsC4vsymmetry.
–twosecondarybrancheswithμ=r2(A+B)/6andcos(4θ)=1.
Theeigenvaluecorrespondingtoavectoralongthebranchisvaryinglikeλ1=r2(A+B)/3,whiletheeigenvalueperpendicularlyvariesasλ2=2r2B/3.
ThetwobrancheshaveC1v(σ1)andC1v(σ2)symmetry,respectively,butareequivalentinotherrespects.
–twosecondarybrancheswithμ=r2(AB)/6andcos(4θ)=1.
Similarly,theeigenvaluesvaryasλ1=r2(AB)/3andλ2=2r2B/3.
ThebrancheshaveC1v(δ1)andC1v(δ2)symmetry,respectively,butareequivalentinotherrespects.
AlocalviewonthebranchespassingthroughthetypeIbifurcationsolutionisshownbyanexampleinFig.
4a,wherethesymmetrypropertiesofthesolutionsareindicated.
Thesignsoftherelevanteigenvaluesarealsoshownonthebranches.
Dependentonthebifurcationconsidered,i.
e.
,thevaluesofAandB,thecurvesmightalsobeturnedupwards.
Asthescalesofrandμarenotknown,onlythesignofAandthevalueofthequotientA/Bareimportantforthedescriptionofthebifurcationstudied,andtheseconstantscanbedeterminedexperimentally.
Forexample,ifboththesecondarybranchesexistfornegativeμvalues,thenA0:v2=u2+3b/B,μ=a+(AB)u2/3+(AB)b/(2B).
Ifb/B0:v=0,u2=3b/B,μ=a+(AB)b/(2B).
PitchforkbifurcationfromC2v(σ)toC1v(σ2):u=v=0,μ=ab.
PitchforkbifurcationfromC1v(σ2)toC1:Ifb/B0andu0(twodisconnectedbranches)orsgn(a/B)u3≤(3/2)|a/B|(asinglebranchconnectedtoaC1v(σ1)branch).
Secondarybifurcationscanalsobelocated,whereasec-ondaryC1branchconnectswithatertiaryC1v(σ1)branch.
Byseekingsolutionswhereμ(u)hasalocalextremum,alimitsolutioncanbelocatedontheotherC1v(σ1)branch,andpossiblylimitsolutionsontheC1branches.
Theseare:Pitchforkbifurcation:u3=3a/(2B),v=0,μ=(A3B)u2/6,A/B=3,A/B=3/5.
LimitsolutiononaC1v(σ1)branch:u3=3a/(A+B),v=0,μ=(A+B)u2/2,A/B=3.
LimitsolutionsonC1branches:u3=3(A+B)a/(8(AB)B),v2=u2+3a/(2Bu),μ=(AB)u2.
TheseexistsonlyforA/B>1orA/B<1(onelimitsolu-tiononeachdisconnectedbranch)or3/5AsA/B→3,thebifurcationandlimitsolutionontheC1v(σ1)branchjoins,andasA/B→3/5thebifurcationandthelimitsolutionsontheconnectedC1branchjoins.
Allbifurcationandlimitsolutionsscaleasu|a|1/3,μ|a|2/3.
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