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NURBS 等几何分析开山之作

Isogeometricanalysis:CAD,finiteelements,NURBS,exactgeometryandmeshrefinementT.J.R.Hughes*,J.A.Cottrell,Y.BazilevsInstituteforComputationalEngineeringandSciences,TheUniversityofTexasatAustin,201East24thStreet,1UniversityStationC0200,Austin,TX78712-0027,UnitedStatesReceived28September2004;accepted20October2004AbstractTheconceptofisogeometricanalysisisproposed.BasisfunctionsgeneratedfromNURBS(Non-UniformRationalB-Splines)areemployedtoconstructanexactgeometricmodel.Forpurposesofanalysis,thebasisisrefinedand/oritsorderelevatedwithoutchangingthegeometryoritsparameterization.Analoguesoffiniteelementh-andp-refinementschemesarepresentedandanew,moreefficient,higher-orderconcept,k-refinement,isintroduced.Refinementsareeas-ilyimplementedandexactgeometryismaintainedatalllevelswithoutthenecessityofsubsequentcommunicationwithaCAD(ComputerAidedDesign)description.Inthecontextofstructuralmechanics,itisestablishedthatthebasisfunctionsarecompletewithrespecttoaffinetransformations,meaningthatallrigidbodymotionsandconstantstrainstatesareexactlyrepresented.Standardpatchtestsarelikewisesatisfied.Numericalexamplesexhibitoptimalratesofconvergenceforlinearelasticityproblemsandconvergencetothinelasticshellsolutions.Ak-refinementstrategyisshowntoconvergetowardmonotonesolutionsforadvection–diffusionprocesseswithsharpinternalandboundarylay-ers,averysurprisingresult.Itisarguedthatisogeometricanalysisisaviablealternativetostandard,polynomial-based,finiteelementanalysisandpossessesseveraladvantages.2005ElsevierB.V.Allrightsreserved.Keywords:NURBS;Finiteelementanalysis;CAD;Structuralanalysis;Fluiddynamics;Meshrefinement;Convergence;Boundarylayers;Internallayers;Geometry;Shells;h-refinement;p-refinement;k-refinement1.IntroductionInthispaperweintroduceanewmethodfortheanalysisofproblemsgovernedbypartialdifferentialequationssuchas,forexample,solids,structuresandfluids.Themethodhasmanyfeaturesincommon0045-7825/$-seefrontmatter2005ElsevierB.V.Allrightsreserved.doi:10.1016/j.cma.2004.10.008*Correspondingauthor.Tel.:+1512232775;fax:+15122327508.E-mailaddress:hughes@ices.utexas.edu(T.J.R.Hughes).Comput.MethodsAppl.Mech.Engrg.194(2005)4135–4195www.elsevier.com/locate/cmawiththefiniteelementmethodandsomefeaturesincommonwithmeshlessmethods.However,itismoregeometricallybasedandtakesinspirationfromComputerAidedDesign(CAD).Aprimarygoalistobegeometricallyexactnomatterhowcoarsethediscretization.AnothergoalistosimplifymeshrefinementbyeliminatingtheneedforcommunicationwiththeCADgeometryoncetheinitialmeshisconstructed.YetanothergoalistomoretightlyweavethemeshgenerationprocesswithinCAD.Inthisworkweintro-duceideasinpursuitofthesegoals.Itisinterestingtonotethatfiniteelementanalysisinengineeringhaditsoriginsinthe1950sand1960s.Aerospaceengineeringwasthefocalpointofactivityduringthistime.Bythelate1960sthefirstcommercialcomputerprograms(ASKA,NASTRAN,Stardyne,etc.)appeared.Subsequently,thefiniteelementmethodspreadtootherengineeringandscientificdisciplines,andnowitsuseiswidespreadandmanycom-mercialprogramsareavailable.Despitethefactthatgeometryistheunderpinningofanalysis,CAD,asweknowittoday,haditsoriginslater,inthe1970sand1980s.Ahighly-recommendedintroductorybook,withhistoricalinsights,isRogers[1].ThisperhapsexplainswhythegeometricrepresentationsinfiniteelementanalysisandCADaresodifferent.Majorfiniteelementprogramsweretechnicallymaturelongbeforemod-ernCADwaswidelyadopted.Presently,CADisamuchbiggerindustrythananalysis.AnalysisisusuallyreferredtoasComputerAidedEngineering(CAE)inmarketresearch.ItisdifficulttopreciselyquantifythesizeoftheCAEandCADindustriesbutcurrentestimatesarethatCAEisinthe$1–$2billionrangeandCADisinthe$5–$10billionrange.Thetypicalsituationinengineeringpracticeisthatdesignsareencap-sulatedinCADsystemsandmeshesaregeneratedfromCADdata.Thisamountstoadoptingatotallydif-ferentgeometricdescriptionforanalysisandonethatisonlyapproximate.Insomeinstancesmeshgenerationcanbedoneautomaticallybutinmostcircumstancesitcanbedoneatbestsemi-automatically.TherearestillsituationsinmajorindustriesinwhichdrawingsaremadeofCADdesignsandmeshesarebuiltfromthem.Itisestimatedthatabout80%ofoverallanalysistimeisdevotedtomeshgenerationintheautomotive,aerospace,andshipbuildingindustries.Intheautomotiveindustry,ameshforanentirevehi-cletakesaboutfourmonthstocreate.Designchangesaremadeonadailybasis,limitingtheutilityofana-lysisindesignifnewmeshescannotbegeneratedwithinthattimeframe.Onceameshisconstructed,refinementrequirescommunicationwiththeCADsystemduringeachrefinementiteration.Thislinkisoftenunavailable,whichperhapsexplainswhyadaptiverefinementisstillprimarilyanacademicendeavorratherthananindustrialtechnology.Thegeometricapproximationinherentinthemeshcanleadtoaccuracyproblems.Oneexampleofthisisinthinshellanalysis,whichisnotoriouslysensitivetogeometricimperfections;seeFig.1.ThesensitivitytoimperfectionsisshowninFig.1binwhichthebucklingloadofageometricallyperfectcylindricalshelliscom-paredwithshellsinwhichgeometricimperfectionsareintroducedwithmagnitudesof1%,10%,and50%ofthethickness.Asmaybeseen,thereisaveryconsiderablereductioninbucklingloadwithincreasedimperfection.Sensitivitytogeometryhasalsobeennotedinfluidmechanics.Spuriousentropylayersaboutaerody-namicshapeswerethebaneofcompressibleEulersolversinthe1980sand1990s.Theproblemanditssolu-tionwereidentifiedinthethesisofBarth[2].Piecewiselinearapproximationsofgeometryweretherootcause.Smoothgeometrycompletelyeliminatedtheentropylayersevenwhentheflowfieldswereapprox-imatedbylinearelementsonthecurvedgeometry;seeFig.2.Thisresultexplainswhymethodswhichem-ploysmoothgeometricmappingsarewidelyusedinairfoilanalysis(see[3]).Itisalsowellknownincomputationalfluiddynamicsthatgoodqualityboundarylayermeshessignificantlyimprovetheaccuracyofcomputedwallquantities,suchaspressure,frictioncoefficient,andheatflux;seeFig.3.Theconstructionoffiniteelementgeometry(i.e.,themesh)iscostly,timeconsumingandcreatesinac-curacies.ItisclearfromthesmallersizeoftheCAEindustrycomparedwiththeCADindustrythatthemostfruitfuldirectionwouldbetoattempttochange,orreplace,finiteelementanalysiswithsomethingmoreCAD-like.ThisdirectionwastakeninthedevelopmentoftheRASNAprogramMechanica,inwhichexactgeometryinconjunctionwithap-adaptivefiniteelementprocedurewasutilized.However,thelackofsatisfactionoftheisoparametricconceptledtotheoreticalquestionswhichwereaddressedinlaterversions4136T.J.R.Hughesetal./Comput.MethodsAppl.Mech.Engrg.194(2005)4135–4195ofthecodebyabandoningtheexactgeometryinfavorofhigh-orderpolynomialapproximations[7].TheuseofafixedpolynomialapproximationtogeometryhasbeenshownbySzaboetal.[8]tobelimiting.Assolutionpolynomialorderisincreased,theerrorplateausatsomelevelandcannotbefurtherreduced(seeFig.4).Theseriousnessofthisresultiscompoundedbythefactthatcomputedquantitiesdefinedonboundariesareusuallythemostimportantonesinengineeringapplications,andthisiswheregeometricerrorsaremostharmful.Furthermore,mostfiniteelementanalysesarestillperformedwithlow-orderelementsforwhichgeometricerrorsarelargest.ThesuccessofRASNA,whichwaslateracquiredbyParametricTechnologyCorporation(PTC),aCADcompany,wasduetoitstightlinkagewithCADFig.1.Thinshellstructuresexhibitsignificantimperfectionsensitivity:(a)facetedgeometryoftypicalfiniteelementmeshesintroducesgeometricimperfections[4]and(b)bucklingofcylindricalshellwithrandomgeometricimperfections[5].(a)(b)Fig.2.IsodensitycontoursofGLSdiscretizationofRinglebflow.(a)IsoparametriclinearLagrangeelementapproximation:bothsolutionspaceandgeometryspacearerepresentedbypiecewiselinearfunctions.(b)Super-isoparametricelementapproximation:solutionspaceispiecewiselinear,whilegeometryispiecewisequadratic.Smoothgeometryavoidsspuriousentropylayersassociatedwithpiecewise-lineargeometricapproximations(from[6]).T.J.R.Hughesetal./Comput.MethodsAppl.Mech.Engrg.194(2005)4135–41954137geometryand,perhapsmoreimportantly,itsconsequentabilitytoprovideadaptivep-refinementandthusmorereliableresults.Thepresentmethodologyissimilarlyinspired,butattemptstomorefaithfullyadheretoCADgeometryandeliminatethefiniteelementpolynomialdescriptionentirely.(Thep-methodisde-scribedinSzaboandBabusˇka[9]andSzaboetal.[8].)TheapproachwehavedevelopedisbasedonNURBS(Non-UniformRationalB-Splines),astandardtechnologyemployedinCADsystems.WeproposetomatchtheexactCADgeometrybyNURBSsur-faces,thenconstructacoarsemeshof‘‘NURBSelements’’.Thesewouldbesolidelementsinthree-dimen-sionsthatexactlyrepresentthegeometry.1Thisisobviouslynotatrivialtaskandonethatdeservesmuchstudybutwhenitcanbeaccomplisheditopensadoortopowerfulapplications.SubsequentrefinementdoesnotrequireanyfurthercommunicationwiththeCADsystemandissosimplethatitmayfacilitatemorewidespreadadoptionofthistechnologyinindustry.Thereareanaloguesofh-,p-,andhp-refinement0.111010100100100010000pg=3,p=1,...,8pg=4,p=1,...,8pg=8,p=1,...,8Relativeerrorinenergynorm[%]DegreesoffreedomFig.4.ConvergencestudyoftheScordelis–Loroofproblem:pgrepresentsthepolynomialdegreeofgeometryrepresentation,pcorrespondstothepolynomialdegreeoftheapproximationspace(from[10]).1Weassumethatunnecessaryfeatures,suchastherivetsonanairplanewing,areremovedfromthegeometrypriortomeshgenerationforanalysis.Featureremovalisanecessarybutcomplexprocess.Fig.3.Qualityboundarylayermeshessignificantlyimproveaccuracy(from[3]).4138T.J.R.Hughesetal./Comput.MethodsAppl.Mech.Engrg.194(2005)4135–4195strategies,andanew,higher-ordermethodologyemerges,k-refinement,whichseemstohaveadvantagesofefficiencyandrobustnessovertraditionalp-refinement.Allsubsequentmeshesretainexactgeometry.Throughout,theisoparametricphilosophyisinvoked,thatis,thesolutionspacefordependentvariablesisrepresentedintermsofthesamefunctionswhichrepresentthegeometry.Forthisreason,wehavedubbedthemethodologyisogeometricanalysis.NURBSarenotarequisiteingredientinisogeometricanalysis.Wemightenvisiondevelopingisogeo-metricproceduresbasedon‘‘A-patches’’(see[11–16])or‘‘subdivisionsurfaces’’(see[17–19]).However,NURBSseemtobethemostthoroughlydevelopedCADtechnologyandtheoneinmostwidespreaduse.ThebodyofthispaperbeginswithatutorialonB-splines(B-splinesaretheprogenitorsofNURBS),fol-lowedbyoneonNURBS.WethendescribeananalysisframeworkbasedonNURBS.Thisisfollowedbysampleapplicationsinlinearsolidandstructuralmechanicsandsomeintroductorycalculationsinfluids,namely,onesinvolvingclassicaltestcasesfortheadvection–diffusionequation.Variousrefinementstrategiesarestudiedand,incasesforwhichexactelasticitysolutionsareavailable,optimalratesofconvergenceareattained.Thestructuralproblemsincludesomeapplicationstothinshellsmodeledassolids.Theapproachisseentohandlethesesituationsremarkablywell.Inthefluidcalculations,weemploytheSUPGformulationandconsiderdifficulttestcasesinvolvinginternalandboundarylayers.Weobservethat,byemployinghigh-order,k-refinementstrategies,convergencetowardmonotonesolutionsisobtained.Thissurprisingresultseemstocontradictnumericalanalysisintuitionsandsuggeststhepossibilityoflineardifferencemethodsthataresimultaneouslyrobustandhighlyaccurate.Weclosewithconclusionsandsuggestionsforfuturework.2.B-splinesandNURBS2.1.KnotvectorsNURBSarebuiltfromB-splines.TheB-splineparametricspaceislocalto‘‘patches’’ratherthanele-ments.Patchesplaytheroleofsubdomainswithinwhichelementtypesandmaterialmodelsareassumedtobeuniform.Aknotvectorinonedimensionisasetofcoordinatesintheparametricspace,writtenN={n1,n2,...,nn+p+1},whereni2Ristheithitknot,iistheknotindex,i=1,2,...,n+p+1,pisthepoly-nomialorder,andnisthenumberofbasisfunctionswhichcomprisetheB-spline.Remark.Theconventionwewilladoptisthattheorderp=0,1,2,3,etc.,referstoconstant,linear,quadratic,cubic,etc.,piecewisepolynomials,respectively.Thisistheusualterminologyinthefiniteelementliterature.Whatwerefertoas‘‘order’’isusuallyreferredtoas‘‘degree’’inthecomputationalgeometryliterature.Ifknotsareequally-spacedintheparametricspace,theyaresaidtobeuniform.Iftheyareunequallyspaced,theyarenon-uniform.Morethanoneknotcanbelocatedatthesamecoordinateintheparametricspace.Thesearereferredtoasrepeatedknots.Aknotvectorissaidtobeopenifitsfirstandlastknotsappearp+1times.OpenknotvectorsarestandardintheCADliterature.Inonedimension,basisfunctionsformedfromopenknotvectorsareinterpolatoryattheendsoftheparametricspaceinterval,[n1,nn+p+1],andatthecornersofpatchesinmultipledimensionsbuttheyarenot,ingeneral,interpolatoryatinteriorknots.Thisisadistinguishingfeaturebetweenknotsand‘‘nodes’’infiniteelementanalysis.2.2.BasisfunctionsB-splinebasisfunctionsaredefinedrecursivelystartingwithpiecewiseconstants(p=0)Ni;0ðnÞ1ifni6n<niþ1;0otherwise:ð1ÞT.J.R.Hughesetal./Comput.MethodsAppl.Mech.Engrg.194(2005)4135–41954139Forp=1,2,3,...,theyaredefinedbyNi;pðnÞnniniþpniNi;p1ðnÞþniþpþ1nniþpþ1niþ1Niþ1;p1ðnÞ:ð2ÞDerivativeswithrespecttospatialcoordinatesmaybecomputedbywayofstandardtechniquesde-scribedinHughes[20,Chapter3].Aninitialexampleoftheresultsofapplying(1)and(2)toauniformknotvectorispresentedinFig.5.Notethat,forp=0and1,thebasisfunctionsarethesameasforstan-dardpiecewiseconstantandlinearfiniteelementfunctions,respectively.However,forpP2,theyaredif-ferent.QuadraticB-splinebasisfunctions(andNURBSbasisfunctions,aswillbeshownlater)areidenticalbutshifted.Thisisincontrastwithquadraticfiniteelementfunctionswhicharedifferentforinternalandendnodes.This‘‘homogeneous’’patterncontinuesaswegotohigher-orderB-splinesandmayresultinsignificantadvantagesinequationsolvingoverfiniteelementfunctions,whicharequite‘‘heterogeneous’’.Anexampleofquadraticbasisfunctionsforanopen,non-uniformknotvectorispresentedinFig.6.Notethatthebasisfunctionsareinterpolatoryattheendsoftheintervalandalsoatn=4,thelocationofarepeatedknot,whereonlyC0-continuityisattained.Elsewhere,thefunctionsareC1-continuous.Ingeneral,basisfunctionsoforderphavep1continuousderivatives.Ifaknotisrepeatedktimes,thenthenumberofcontinuousderivativesdecreasesbyk.Whenthemultiplicityofaknotisexactlyp,thebasisfunctionisinterpolatory.01234501N1,0ξ01234501N2,0ξ01234501N3,0ξ01234501N1,1ξ01234501N2,1ξ01234501N3,1ξ01234501N1,2ξ01234501N2,2ξ01234501N3,2ξFig.5.Basisfunctionsoforder0,1,2foruniformknotvectorN={0,1,2,3,4,...}.01234501N1,2N2,2N3,2N5,2N6,2N8,2N7,2N4,2ξFig.6.Quadraticbasisfunctionsforopen,non-uniformknotvectorN={0,0,0,1,2,3,4,4,5,5,5}.4140T.J.R.Hughesetal./Comput.MethodsAppl.Mech.Engrg.194(2005)4135–4195ImportantpropertiesofB-splinebasisfunctionsare:(1)Theyconstituteapartitionofunity,thatis,"nXni1Ni;pðnÞ1:ð3Þ(2)ThesupportofeachNi,piscompactandcontainedintheinterval[ni,ni+p+1].(3)Eachbasisfunctionisnon-negative,thatis,Ni,p(n)P0,"n.Consequently,allofthecoefficientsofamassmatrixcomputedfromaB-splinebasisaregreaterthan,orequalto,zero.2.3.B-splinecurvesB-splinecurvesinRdareconstructedbytakingalinearcombinationofB-splinebasisfunctions.Thecoefficientsofthebasisfunctionsarereferredtoascontrolpoints.Thesearesomewhatanalogoustonodalcoordinatesinfiniteelementanalysis.Piecewiselinearinterpolationofthecontrolpointsgivestheso-calledcontrolpolygon.Ingeneral,controlpointsarenotinterpolatedbyB-splinecurves.Givennbasisfunctions,Ni,p,i=1,2,...,n,andcorrespondingcontrolpointsBi2Rd;i1;2;...;n,apiecewise-polynomialB-splinecurveisgivenbyCðnÞXni1Ni;pðnÞBi:ð4ÞAnexampleisshowninFig.7forthequadraticbasisfunctionsconsideredpreviously.Notethatthecurveisinterpolatoryatthefirstandlastcontrolpoints,duetothefactthattheknotvectorisopen,andalsoatthesixthcontrolpoint,duetothefactthatthemultiplicityoftheknotn=4isequaltothepolynomialorder.Notealsothatthecurveistangenttothecontrolpolygonatthefirst,last,andsixthcon-trolpoints.ThecurveisCp1=C1-continuouseverywhereexceptatthelocationoftherepeatedknot,n=4,whereitisCp2=C0-continuous.Fig.7.B-spline,piecewisequadraticcurveinR2.Controlpointlocationsaredenotedby•.BasisfunctionsandknotvectorasinFig.6.T.J.R.Hughesetal./Comput.MethodsAppl.Mech.Engrg.194(2005)4135–41954141ImportantpropertiesofB-splinecurvesare:(1)Theyhavecontinuousderivativesoforderp1intheabsenceofrepeatedknotsorcontrolpoints.(2)Repeatingaknotorcontrolpointktimesdecreasesthenumberofcontinuousderivativesbyk.(3)AnaffinetransformationofaB-splinecurveisobtainedbyapplyingthetransformationtothecontrolpoints.2Werefertothispropertyasaffinecovariance.2.4.h-refinement:knotinsertionTheanalogueofh-refinementisknotinsertion.Knotsmaybeinsertedwithoutchangingacurvegeomet-ricallyorparametrically.GivenaknotvectorN={n1,n2,...,nn+p+1},letn2nk;nkþ1beadesirednewknot.Thenewn+1basisfunctionsareformedrecursively,using(1)and(2),withthenewknotvectorNfn1;n2;...;nk;n;nkþ1;...;nnþpþ1g.Thenewn+1controlpoints,fB1;B2;...;Bnþ1g,areformedfromtheoriginalcontrolpoints,{B1,B2,...,Bn},byBiaiBiþð1aiÞBi1;ð5Þwhereai1;16i6kp;nniniþpni;kpþ16i6k;0;kþ16i6nþpþ2:8>>><>>>:ð6ÞKnotvaluesalreadypresentintheknotvectormayberepeatedasabovebutasdescribedinSection2.2,thecontinuityoft

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