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首页 Isogeometric analysis CAD, finite elements, NUR…

Isogeometric analysis CAD, finite elements, NURBS, exact geometry and mesh refinement.pdf

Isogeometric analysis CAD, fini…

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简介:本文档为《Isogeometric analysis CAD, finite elements, NURBS, exact geometry and mesh refinementpdf》,可适用于高等教育领域,主题内容包含Isogeometricanalysis:CAD,finiteelements,NURBS,exactgeometryandmeshrefinement符等。

Isogeometricanalysis:CAD,finiteelements,NURBS,exactgeometryandmeshrefinementTJRHughes*,JACottrell,YBazilevsInstituteforComputationalEngineeringandSciences,TheUniversityofTexasatAustin,EastthStreet,UniversityStationC,Austin,TX,UnitedStatesReceivedSeptemberacceptedOctoberAbstractTheconceptofisogeometricanalysisisproposedBasisfunctionsgeneratedfromNURBS(NonUniformRationalBSplines)areemployedtoconstructanexactgeometricmodelForpurposesofanalysis,thebasisisrefinedandoritsorderelevatedwithoutchangingthegeometryoritsparameterizationAnaloguesoffiniteelementhandprefinementschemesarepresentedandanew,moreefficient,higherorderconcept,krefinement,isintroducedRefinementsareeasilyimplementedandexactgeometryismaintainedatalllevelswithoutthenecessityofsubsequentcommunicationwithaCAD(ComputerAidedDesign)descriptionInthecontextofstructuralmechanics,itisestablishedthatthebasisfunctionsarecompletewithrespecttoaffinetransformations,meaningthatallrigidbodymotionsandconstantstrainstatesareexactlyrepresentedStandardpatchtestsarelikewisesatisfiedNumericalexamplesexhibitoptimalratesofconvergenceforlinearelasticityproblemsandconvergencetothinelasticshellsolutionsAkrefinementstrategyisshowntoconvergetowardmonotonesolutionsforadvection–diffusionprocesseswithsharpinternalandboundarylayers,averysurprisingresultItisarguedthatisogeometricanalysisisaviablealternativetostandard,polynomialbased,finiteelementanalysisandpossessesseveraladvantagesElsevierBVAllrightsreservedKeywords:NURBSFiniteelementanalysisCADStructuralanalysisFluiddynamicsMeshrefinementConvergenceBoundarylayersInternallayersGeometryShellshrefinementprefinementkrefinementIntroductionInthispaperweintroduceanewmethodfortheanalysisofproblemsgovernedbypartialdifferentialequationssuchas,forexample,solids,structuresandfluidsThemethodhasmanyfeaturesincommon$seefrontmatterElsevierBVAllrightsreserveddoi:jcma*CorrespondingauthorTel:fax:Emailaddress:hughesicesutexasedu(TJRHughes)ComputMethodsApplMechEngrg()–wwwelseviercomlocatecmawiththefiniteelementmethodandsomefeaturesincommonwithmeshlessmethodsHowever,itismoregeometricallybasedandtakesinspirationfromComputerAidedDesign(CAD)AprimarygoalistobegeometricallyexactnomatterhowcoarsethediscretizationAnothergoalistosimplifymeshrefinementbyeliminatingtheneedforcommunicationwiththeCADgeometryoncetheinitialmeshisconstructedYetanothergoalistomoretightlyweavethemeshgenerationprocesswithinCADInthisworkweintroduceideasinpursuitofthesegoalsItisinterestingtonotethatfiniteelementanalysisinengineeringhaditsoriginsinthesandsAerospaceengineeringwasthefocalpointofactivityduringthistimeBythelatesthefirstcommercialcomputerprograms(ASKA,NASTRAN,Stardyne,etc)appearedSubsequently,thefiniteelementmethodspreadtootherengineeringandscientificdisciplines,andnowitsuseiswidespreadandmanycommercialprogramsareavailableDespitethefactthatgeometryistheunderpinningofanalysis,CAD,asweknowittoday,haditsoriginslater,inthesandsAhighlyrecommendedintroductorybook,withhistoricalinsights,isRogersThisperhapsexplainswhythegeometricrepresentationsinfiniteelementanalysisandCADaresodifferentMajorfiniteelementprogramsweretechnicallymaturelongbeforemodernCADwaswidelyadoptedPresently,CADisamuchbiggerindustrythananalysisAnalysisisusuallyreferredtoasComputerAidedEngineering(CAE)inmarketresearchItisdifficulttopreciselyquantifythesizeoftheCAEandCADindustriesbutcurrentestimatesarethatCAEisinthe$–$billionrangeandCADisinthe$–$billionrangeThetypicalsituationinengineeringpracticeisthatdesignsareencapsulatedinCADsystemsandmeshesaregeneratedfromCADdataThisamountstoadoptingatotallydifferentgeometricdescriptionforanalysisandonethatisonlyapproximateInsomeinstancesmeshgenerationcanbedoneautomaticallybutinmostcircumstancesitcanbedoneatbestsemiautomaticallyTherearestillsituationsinmajorindustriesinwhichdrawingsaremadeofCADdesignsandmeshesarebuiltfromthemItisestimatedthataboutofoverallanalysistimeisdevotedtomeshgenerationintheautomotive,aerospace,andshipbuildingindustriesIntheautomotiveindustry,ameshforanentirevehicletakesaboutfourmonthstocreateDesignchangesaremadeonadailybasis,limitingtheutilityofanalysisindesignifnewmeshescannotbegeneratedwithinthattimeframeOnceameshisconstructed,refinementrequirescommunicationwiththeCADsystemduringeachrefinementiterationThislinkisoftenunavailable,whichperhapsexplainswhyadaptiverefinementisstillprimarilyanacademicendeavorratherthananindustrialtechnologyThegeometricapproximationinherentinthemeshcanleadtoaccuracyproblemsOneexampleofthisisinthinshellanalysis,whichisnotoriouslysensitivetogeometricimperfectionsseeFigThesensitivitytoimperfectionsisshowninFigbinwhichthebucklingloadofageometricallyperfectcylindricalshelliscomparedwithshellsinwhichgeometricimperfectionsareintroducedwithmagnitudesof,,andofthethicknessAsmaybeseen,thereisaveryconsiderablereductioninbucklingloadwithincreasedimperfectionSensitivitytogeometryhasalsobeennotedinfluidmechanicsSpuriousentropylayersaboutaerodynamicshapeswerethebaneofcompressibleEulersolversinthesandsTheproblemanditssolutionwereidentifiedinthethesisofBarthPiecewiselinearapproximationsofgeometryweretherootcauseSmoothgeometrycompletelyeliminatedtheentropylayersevenwhentheflowfieldswereapproximatedbylinearelementsonthecurvedgeometryseeFigThisresultexplainswhymethodswhichemploysmoothgeometricmappingsarewidelyusedinairfoilanalysis(see)Itisalsowellknownincomputationalfluiddynamicsthatgoodqualityboundarylayermeshessignificantlyimprovetheaccuracyofcomputedwallquantities,suchaspressure,frictioncoefficient,andheatfluxseeFigTheconstructionoffiniteelementgeometry(ie,themesh)iscostly,timeconsumingandcreatesinaccuraciesItisclearfromthesmallersizeoftheCAEindustrycomparedwiththeCADindustrythatthemostfruitfuldirectionwouldbetoattempttochange,orreplace,finiteelementanalysiswithsomethingmoreCADlikeThisdirectionwastakeninthedevelopmentoftheRASNAprogramMechanica,inwhichexactgeometryinconjunctionwithapadaptivefiniteelementprocedurewasutilizedHowever,thelackofsatisfactionoftheisoparametricconceptledtotheoreticalquestionswhichwereaddressedinlaterversionsTJRHughesetalComputMethodsApplMechEngrg()–ofthecodebyabandoningtheexactgeometryinfavorofhighorderpolynomialapproximationsTheuseofafixedpolynomialapproximationtogeometryhasbeenshownbySzaboetaltobelimitingAssolutionpolynomialorderisincreased,theerrorplateausatsomelevelandcannotbefurtherreduced(seeFig)Theseriousnessofthisresultiscompoundedbythefactthatcomputedquantitiesdefinedonboundariesareusuallythemostimportantonesinengineeringapplications,andthisiswheregeometricerrorsaremostharmfulFurthermore,mostfiniteelementanalysesarestillperformedwithloworderelementsforwhichgeometricerrorsarelargestThesuccessofRASNA,whichwaslateracquiredbyParametricTechnologyCorporation(PTC),aCADcompany,wasduetoitstightlinkagewithCADFigThinshellstructuresexhibitsignificantimperfectionsensitivity:(a)facetedgeometryoftypicalfiniteelementmeshesintroducesgeometricimperfectionsand(b)bucklingofcylindricalshellwithrandomgeometricimperfections(a)(b)FigIsodensitycontoursofGLSdiscretizationofRinglebflow(a)IsoparametriclinearLagrangeelementapproximation:bothsolutionspaceandgeometryspacearerepresentedbypiecewiselinearfunctions(b)Superisoparametricelementapproximation:solutionspaceispiecewiselinear,whilegeometryispiecewisequadraticSmoothgeometryavoidsspuriousentropylayersassociatedwithpiecewiselineargeometricapproximations(from)TJRHughesetalComputMethodsApplMechEngrg()–geometryand,perhapsmoreimportantly,itsconsequentabilitytoprovideadaptiveprefinementandthusmorereliableresultsThepresentmethodologyissimilarlyinspired,butattemptstomorefaithfullyadheretoCADgeometryandeliminatethefiniteelementpolynomialdescriptionentirely(ThepmethodisdescribedinSzaboandBabusˇkaandSzaboetal)TheapproachwehavedevelopedisbasedonNURBS(NonUniformRationalBSplines),astandardtechnologyemployedinCADsystemsWeproposetomatchtheexactCADgeometrybyNURBSsurfaces,thenconstructacoarsemeshof‘‘NURBSelements’’ThesewouldbesolidelementsinthreedimensionsthatexactlyrepresentthegeometryThisisobviouslynotatrivialtaskandonethatdeservesmuchstudybutwhenitcanbeaccomplisheditopensadoortopowerfulapplicationsSubsequentrefinementdoesnotrequireanyfurthercommunicationwiththeCADsystemandissosimplethatitmayfacilitatemorewidespreadadoptionofthistechnologyinindustryThereareanaloguesofh,p,andhprefinementpg=,p=,,pg=,p=,,pg=,p=,,RelativeerrorinenergynormDegreesoffreedomFigConvergencestudyoftheScordelis–Loroofproblem:pgrepresentsthepolynomialdegreeofgeometryrepresentation,pcorrespondstothepolynomialdegreeoftheapproximationspace(from)Weassumethatunnecessaryfeatures,suchastherivetsonanairplanewing,areremovedfromthegeometrypriortomeshgenerationforanalysisFeatureremovalisanecessarybutcomplexprocessFigQualityboundarylayermeshessignificantlyimproveaccuracy(from)TJRHughesetalComputMethodsApplMechEngrg()–strategies,andanew,higherordermethodologyemerges,krefinement,whichseemstohaveadvantagesofefficiencyandrobustnessovertraditionalprefinementAllsubsequentmeshesretainexactgeometryThroughout,theisoparametricphilosophyisinvoked,thatis,thesolutionspacefordependentvariablesisrepresentedintermsofthesamefunctionswhichrepresentthegeometryForthisreason,wehavedubbedthemethodologyisogeometricanalysisNURBSarenotarequisiteingredientinisogeometricanalysisWemightenvisiondevelopingisogeometricproceduresbasedon‘‘Apatches’’(see–)or‘‘subdivisionsurfaces’’(see–)However,NURBSseemtobethemostthoroughlydevelopedCADtechnologyandtheoneinmostwidespreaduseThebodyofthispaperbeginswithatutorialonBsplines(BsplinesaretheprogenitorsofNURBS),followedbyoneonNURBSWethendescribeananalysisframeworkbasedonNURBSThisisfollowedbysampleapplicationsinlinearsolidandstructuralmechanicsandsomeintroductorycalculationsinfluids,namely,onesinvolvingclassicaltestcasesfortheadvection–diffusionequationVariousrefinementstrategiesarestudiedand,incasesforwhichexactelasticitysolutionsareavailable,optimalratesofconvergenceareattainedThestructuralproblemsincludesomeapplicationstothinshellsmodeledassolidsTheapproachisseentohandlethesesituationsremarkablywellInthefluidcalculations,weemploytheSUPGformulationandconsiderdifficulttestcasesinvolvinginternalandboundarylayersWeobservethat,byemployinghighorder,krefinementstrategies,convergencetowardmonotonesolutionsisobtainedThissurprisingresultseemstocontradictnumericalanalysisintuitionsandsuggeststhepossibilityoflineardifferencemethodsthataresimultaneouslyrobustandhighlyaccurateWeclosewithconclusionsandsuggestionsforfutureworkBsplinesandNURBSKnotvectorsNURBSarebuiltfromBsplinesTheBsplineparametricspaceislocalto‘‘patches’’ratherthanelementsPatchesplaytheroleofsubdomainswithinwhichelementtypesandmaterialmodelsareassumedtobeuniformAknotvectorinonedimensionisasetofcoordinatesintheparametricspace,writtenN={n,n,,nnp},whereniRistheithitknot,iistheknotindex,i=,,,np,pisthepolynomialorder,andnisthenumberofbasisfunctionswhichcomprisetheBsplineRemarkTheconventionwewilladoptisthattheorderp=,,,,etc,referstoconstant,linear,quadratic,cubic,etc,piecewisepolynomials,respectivelyThisistheusualterminologyinthefiniteelementliteratureWhatwerefertoas‘‘order’’isusuallyreferredtoas‘‘degree’’inthecomputationalgeometryliteratureIfknotsareequallyspacedintheparametricspace,theyaresaidtobeuniformIftheyareunequallyspaced,theyarenonuniformMorethanoneknotcanbelocatedatthesamecoordinateintheparametricspaceThesearereferredtoasrepeatedknotsAknotvectorissaidtobeopenifitsfirstandlastknotsappearptimesOpenknotvectorsarestandardintheCADliteratureInonedimension,basisfunctionsformedfromopenknotvectorsareinterpolatoryattheendsoftheparametricspaceinterval,n,nnp,andatthecornersofpatchesinmultipledimensionsbuttheyarenot,ingeneral,interpolatoryatinteriorknotsThisisadistinguishingfeaturebetweenknotsand‘‘nodes’’infiniteelementanalysisBasisfunctionsBsplinebasisfunctionsaredefinedrecursivelystartingwithpiecewiseconstants(p=)NiðnÞifnin<niþotherwise:ðÞTJRHughesetalComputMethodsApplMechEngrg()–Forp=,,,,theyaredefinedbyNipðnÞnniniþpniNipðnÞþniþpþnniþpþniþNiþpðnÞ:ðÞDerivativeswithrespecttospatialcoordinatesmaybecomputedbywayofstandardtechniquesdescribedinHughes,ChapterAninitialexampleoftheresultsofapplying()and()toauniformknotvectorispresentedinFigNotethat,forp=and,thebasisfunctionsarethesameasforstandardpiecewiseconstantandlinearfiniteelementfunctions,respectivelyHowever,forpP,theyaredifferentQuadraticBsplinebasisfunctions(andNURBSbasisfunctions,aswillbeshownlater)areidenticalbutshiftedThisisincontrastwithquadraticfiniteelementfunctionswhicharedifferentforinternalandendnodesThis‘‘homogeneous’’patterncontinuesaswegotohigherorderBsplinesandmayresultinsignificantadvantagesinequationsolvingoverfiniteelementfunctions,whicharequite‘‘heterogeneous’’Anexampleofquadraticbasisfunctionsforanopen,nonuniformknotvectorispresentedinFigNotethatthebasisfunctionsareinterpolatoryattheendsoftheintervalandalsoatn=,thelocationofarepeatedknot,whereonlyCcontinuityisattainedElsewhere,thefunctionsareCcontinuousIngeneral,basisfunctionsoforderphavepcontinuousderivativesIfaknotisrepeatedktimes,thenthenumberofcontinuousderivativesdecreasesbykWhenthemultiplicityofaknotisexactlyp,thebasisfunctionisinterpolatoryN,ξN,ξN,ξN,ξN,ξN,ξN,ξN,ξN,ξFigBasisfunctionsoforder,,foruniformknotvectorN={,,,,,}N,N,N,N,N,N,N,N,ξFigQuadraticbasisfunctionsforopen,nonuniformknotvectorN={,,,,,,,,,,}TJRHughesetalComputMethodsApplMechEngrg()–ImportantpropertiesofBsplinebasisfunctionsare:()Theyconstituteapartitionofunity,thatis,"nXniNipðnÞ:ðÞ()ThesupportofeachNi,piscompactandcontainedintheintervalni,nip()Eachbasisfunctionisnonnegative,thatis,Ni,p(n)P,"nConsequently,allofthecoefficientsofamassmatrixcomputedfromaBsplinebasisaregreaterthan,orequalto,zeroBsplinecurvesBsplinecurvesinRdareconstructedbytakingalinearcombinationofBsplinebasisfunctionsThecoefficientsofthebasisfunctionsarereferredtoascontrolpointsThesearesomewhatanalogoustonodalcoordinatesinfiniteelementanalysisPiecewiselinearinterpolationofthecontrolpointsgivesthesocalledcontrolpolygonIngeneral,controlpointsarenotinterpolatedbyBsplinecurvesGivennbasisfunctions,Ni,p,i=,,,n,andcorrespondingcontrolpointsBiRdin,apiecewisepolynomialBsplinecurveisgivenbyCðnÞXniNipðnÞBi:ðÞAnexampleisshowninFigforthequadraticbasisfunctionsconsideredpreviouslyNotethatthecurveisinterpolatoryatthefirstandlastcontrolpoints,duetothefactthattheknotvectorisopen,andalsoatthesixthcontrolpoint,duetothefactthatthemultiplicityoftheknotn=isequaltothepolynomialorderNotealsothatthecurveistangenttothecontrolpolygonatthefirst,last,andsixthcontrolpointsThecurveisCp=Ccontinuouseverywhereexceptatthelocationoftherepeatedknot,n=,whereitisCp=CcontinuousFigBspline,piecewisequadraticcurveinRControlpointlocationsaredenotedby•BasisfunctionsandknotvectorasinFigTJRHughesetalComputMethodsApplMechEngrg()–ImportantpropertiesofBsplinecurvesare:()Theyhavecontinuousderivativesoforderpintheabsenceofrepeatedknotsorcontrolpoints()Repeatingaknotorcontrolpointktimesdecreasesthenumberofcontinuousderivativesbyk()AnaffinetransformationofaBsplinecurveisobtainedbyapplyingthetransformationtothecontrolpointsWerefertothispropertyasaffinecovariancehrefinement:knotinsertionTheanalogueofhrefinementisknotinsertionKnotsmaybeinsertedwithoutchangingacurvegeometricallyorparametricallyGivenaknotvectorN={n,n,,nnp},letnnknkþbeadesirednewknotThenewnbasisfunctionsareformedrecursively,using()and(),withthenewknotvectorNfnnnknnkþnnþpþgThenewncontrolpoints,fBBBnþg,areformedfromtheoriginalcontrolpoints,{B,B,,Bn},byBiaiBiþðaiÞBiðÞwhereaiikpnniniþpnikpþikkþinþpþ:>>><>>>:ðÞKnotvaluesalreadypresentintheknotvectormayberepeatedasabovebutasdescribedinSection,thecontinuityoft

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