InternationalConferenceontheApplicationofComputerScienceandMathematics - PDF document

1 InternationalConferenceontheApplicationo
InternationalConferenceontheApplicationofComputerScienceandMathematicsinArchitectureandCivilEngineeringK.Gurlebeck,T.LahmerandF.Werner(eds.)Weimar,Germany,04–06July2012T-SPLINEBASEDXIGAFORADAPTIVEMODELINGOFCRACKEDS.Sh.Ghorashi ,T.Rabczuk,J.J.RodenasandT.LahmerResearchTrainingGroup1462,Bauhaus-UniversitatWeimarBerkaerStr.9,99423Weimar,GermanyE-mail:shahram.ghorashi@uni-weimar.deKeywords:Crack,EXtendedIsoGeometricAnalysis(XIGA),T-splineBasisFunctions,LocalRenement,StressIntensityFactor.Safetyoperationofimportantcivilstructuressuchasbridgescanbeestimatedbyusingfractureanalysis.Sincetheanalyticalmethodsarenotcapableofsolvingmanycompli-catedengineeringproblems,numericalmethodshavebeenincreasinglyadopted.Inthispaper,apartofisotropicmaterialwhichcontainsacrackisconsideredasapartialmodelandtheproposedmodelqualityisevaluated.EXtendedIsoGeometricAnalysis(XIGA)isanewdevel-opednumericalapproach[1,2]whichbenetsfromadvantagesofitsorigins:eXtendedFiniteElementMethod(XFEM)andIsoGeometricAnalysis(IGA).Itiscapableofsimulatingcrackpropagationproblemswithnoremeshingnecessityandcapturingsingulareldatthecracktipbyusingthecracktipenrichmentfunctions.Also,exactrepresentationofgeometryispossi-bleusingonlyfewelements.XIGAhasalsobeensuccessfullyappliedforfractureanalysisofcrackedorthotropicbodies[3]andforsimulationofcurvedcracks[4].XIGAappliesNURBSfunctionsforbothgeometrydescriptionandsolutioneldapproximation.ThedrawbackofNURBSfunctionsisthatlocalrenementcannotbede

2 nedregardingthatitisbasedontensor-produc
nedregardingthatitisbasedontensor-productconstructsunlessmultiplepatchesareusedwhichhasalsosomelimitations.Inthiscontribution,theXIGAisfurtherdevelopedtomakethelocalrenementfeasiblebyusingT-splinebasisfunctions.Adoptingarecoverybasederrorestimatorintheproposedapproachforevaluationofthemodelqualityandperformingtheadaptiveprocessesisinprogress.Finally,somenumericalexampleswithavailableanalyticalsolutionsareinvestigatedbythedevelopedscheme. 1INTRODUCTIONIncreasinglydevelopmentofcomputersmadethepossibilitytoapplynumericalmethodsforsimulationofcivilstructuresasanalternativetoanalyticalmethodswhicharenotfeasibleinresolvingcomplexproblems.Thishasattractedmanyresearchers'interestsfordevelopingmoreaccurateandefcientcomputationalapproachesinthelastdecades.Fractureanalysisofstructuresisofgreatimportanceforestimationoftheirsafetyoperation.Areliableandefcientnumericalmethodisrequiredforanalysisofcrackedpartofastructure.Theremeshingnecessityandexistenceofasingulareldaroundacracktipinsimulationofcrackpropagationproblemsledtothedevelopmentofanewgenerationofcomputationalapproachessuchasmeshfreemethods[5-14]andtheextendedFEM(XFEM)[15-20]whichbelongstotheclassofPartitionofUnityMethods(PUM).Movingdiscontinuousproblemssuchascrackpropagationcanbeanalyzedbythesemethodswithouttherequrirementofremeshingorrearrangingofthenodalpoints.IntheXFEM,aprioriknowledgeofthesolutionislocallyaddedtotheapproximationspace.Thisenrichmentallowsforcapturingparticularfeaturessuchasdiscontinuiti

3 esandsingularitieswhicharepresentintheso
esandsingularitieswhicharepresentinthesolutionexactly.Morerecently,anumericalapproachcalledextendedisogeometricanalysis(XIGA)[1,2]hasbeendevelopedforsimulationofstationaryandpropagatingcracksbyincorporatingtheconceptsoftheXFEMintotheisogeometricanalysis[21,22].Somesuperioritiesoftheisoge-ometricanalysisincomparisonwiththeconventionalFEMare:simpleandsystematicrene-mentstrategies,anexactrepresentationofcommonandcomplexengineeringshapes,robustnessandhigheraccuracy.XIGAhasalsobeensuccessfullyappliedforfractureanalysisofcrackedorthotropicbodies[3]andforsimulationofcurvedcracks[4].XIGAappliesNURBSfunctionsforbothgeometrydescriptionandsolutioneldapproxima-tion.ThedrawbackofNURBSfunctionsisthatlocalrenementcannotbedenedregardingthatitisbasedontensor-productconstructsunlessmultiplepatchesareusedwhichhasalsosomelimitations.Inthiscontribution,T-splinebasisfunctionsareappliedintheXIGAtomakelocalrenementfeasible.Finally,forqualityevaluationoftheproposedmodel,somenumericalsimulationswithavail-ableanalyticalsolutionsarestudied.2BASISFUNCTIONS2.1NURBSNon-uniformrationalB-splines(NURBS)areageneralizationofpiecewisepolynomialB-splinecurves.TheB-splinebasisfunctionsaredenedinaparametricspaceonaknotvector.Aknotvectorinonedimensionisanon-decreasingsequenceofrealnumbers:;᠀;:::;᠀istheistheknotindex,=1;:::;n+1istheorderoftheB-spline,andisthenumberofbasisfunctions.Thehalfopenintervalal᠀i;᠀iscalledtheknotspananditcanhavezerolengthsinceknotsmayberepeatedmorethan

4 once,andthe intervalal᠀1;᠀iscalledap
once,andthe intervalal᠀1;᠀iscalledapatch.Intheisogeometricanalysis,alwaysopenknotvectorsareemployed.Aknotvectoriscalledopenifitcontains+1repeatedknotsatthetwoends.Withacertainknotspan,theB-splinebasisfunctionsaredenedrecursivelyas,⤀=⤀= ⤀+ =1;:::=1;:::;nAB-splinecurveoforderisdenedby:⤀=istheB-splinebasisfunctionoforderarecontrolpoints,givenind-dimensionalspaceThenon-uniformrationalB-spline(NURBS)curveoforderisdenedas:⤀=⤀= aretheNURBSbasisfunctions,arethecontrolpointsandisthethatmustbenon-negative.Inthetwodimensionalparametricspacespace;1]2,NURBSsurfacesareconstructedbytensorproductthroughknotvectors;᠀;:::;᠀;᠀;:::;᠀.Ityieldsto:;᠀ k;lFormoredetailsonNURBS,referto[23].2.2T-splinesT-splinesisageneralizationofNURBSenablinglocalrenement[26,27].FordeningtheT-splinebasisfunctions,anindexspacecalledT-meshisdened.ItissimilartotheindexspacerepresentationofaNURBS,withthedifferencethatT-junctions,whichareverticesconnectingthreeedges,areallowed.AnexampleofT-meshisillustratedinFig.1.Itisnotedthateachline Figure1:AsampleofT-mesh.inthemeshcorrespondstoaknotvalue.Then,anchorsaredenedontheT-meshtoidentifythelocationofeachbasisfunction.Theyarelocatedattheintersectionsofknotlinesifthepolynomialorderisodd,otherwisetheirlocationareinthecenterofthecells.Regardlessofdegree,ananchorlocationisatthecenterofthesupportofafunctionintheindexspace.FordenitionofT-splines,localknotvectorsaredenedinsteadofusingtheglobalknotvectorssinceeachbas

5 isfunctionhasthecompactsupportof+1)+1)kn
isfunctionhasthecompactsupportof+1)+1)knots.Asil-lustratedinFig.2,localknotvectorsineachdirectionaredenedbyhorizontallyorverticallymarchingfromtheanchorsbackwardandforward[27].Afterwards,eachbasisfunctioncanbedenedusingtheEqs.2,3and7anditscorrespondinglocalknotvectors.Inordertorenethemesh,knotinsertionprocessisperformed.Itconsistsofaddingnewknotstothepresentmesh/T-meshandcorrespondingly,modifyingandaddingsomecontrolpoints.FormoreinformationaboutT-splineandlocalrenement,readersarereferredto[26,3EXTENDEDISOGEOMETRICANALYSISExtendedisogeometricanalysis(XIGA)isanewlydevelopedcomputationalapproachwhichusesthesuperioritiesoftheextendedniteelementmethod(XFEM)withintheisogeometricanalysis.Itiscapableofcrackpropagationsimulationwithouttheremeshingnecessitysinceelementedgesaredenedindependentofthecracklocation.SolutionȀeldapproximationisextrinsicallyenrichedbytheHeavisideandbranchfunctionsforcrackfaceandsingulareld(aroundthecracktip)modeling,respectively.;᠀⤀=;᠀;᠀;᠀ThersttermintherighthandsideisstandardIGAapproximation.;᠀aretheT-Splinebasisfunctionsofordersdirections,respectively,atthepoint;᠀ (a) (b) Figure2:Schematicviewofdeninglocalknotvectorsfortheanchor:(a)quadraticpolynomialorder:;᠀;᠀;᠀;᠀;᠀;᠀;(b)cubicpolynomialorder:;᠀;᠀;᠀;᠀;᠀;᠀;᠀;᠀intheparametricspacespace;1]Ȁ[0;1].fajgarethevectorsofadditionaldegreesoffreedomwhicharerelatedtothemodelingofcrackfaces,arethevectorsofadditionaldegreesoffreedomfor

6 modelingthecracktip,isthenumberofnonzero
modelingthecracktip,isthenumberofnonzerobasisfunctionsforagivenknotspan,isthenumberofbasisfunctionswhichhavebeenselectedasbranchenrichedbasisfunctions.Theycanbeselectedusingthetopologicalenrichmentstrategyorgeometri-calenrichmentone.Intopologicalenrichmentscheme,thebasisfunctionswhichcontainthecracktipintheirinuencedomainsareselectedasthebranchenrichedbasisfunctionswhileingeometricalenrichmentmethod,branchenrichedbasisfunctionsconsistofthebasisfunctionschosenfromthepreviousstrategyandtheoneswhichareselectedaccordingtoconsideringaconstantdomainaroundthecracktip.Inthiscontribution,geometricalenrichmentmethodisadoptedandacirculardomainwithapredenedradiusatthecentercracktipisconsideredandbasisfunctionswhoseinuencedomainscontainthecracktipandwhoseanchorslocatedinthecircleareselectedasbranchenrichedbasisfunctions.isthenumberofbasisfunctionsthathavecrackfaceintheirsupportdomainsandhavenotbeenselectedasbranchenrichedbasisfunctions.isthegeneralizedHeavisidefunction[24],⤀=istheunitnormalvectorofcrackalignmentinpointonthecracksurfacewhichisthenearestpointto;᠀InEq.8,=1arethecracktipenrichmentfunctionswhoserolesarerepro-ducingthesingulareldaroundcracktips, rsin 2;p rcos 2;p rsinsin 2;p rsincos 2(10)5 Figure3:AmodeIcrackmodelinaninniteplate.r;ሀarethelocalcracktippolarcoordinateswithrespecttothetangenttothecracktipinthephysicalspace.Readersarereferredto[2]formoreinformationaboutXIGAformulationandimplementa-4NUMERICALEXAMPLESInthissection,twonumericalexampl

7 esareinvestigatedbytheproposedapproach.T
esareinvestigatedbytheproposedapproach.TherstonecontainsamodeIcrackwhiletheotherincludesamixedmodecrack.Newknotsaddedforrenementsatisfytheconditionsofanalysis-suitableT-splines[28].Basisfunctionsofcubicorderareconsidered.Gaussquadraturerulewith4x4Gausspointsfornormalelementsisutilized.Forintegrationoversplitandtipelements,sub-trianglesandalmostpolartechniqueswith13and7x7Gausspointsforeachsub-trianglesareadopted.4.1ModeIcrackmodelintheinniteplateAninniteplateincludingastraightcrackunderpurefracturemodeIisconsidered,asde-pictedinFigure3.Theplateisinplanestrainstate.Then,alocalnitesquaredomainABCDwhichincludesthecracktipinthecenterisdened.ThedomainABCD,whichincludesthe=5mmpartofthecrack,issmallerthanthecracklength=200mmintheinniteplate.ThesizeofthisanalyticaldomainABCDismm.Otherparametersare:Young'smod-=10,Poisson'sratio=0andprescribeduniaxialstress=10Theanalyticalsolutionforthedisplacementandstresseldsintermsoflocalpolarcoordi-natesinareferenceframer;ሀcenteredatthecracktipare:r;ሀ⤀= p 2KI Ep rcos 222cos2 r;ሀ⤀= p 2KI Ep rsin 222cos2 2(11)6 Table1:Errornorms(inpercent)ofthethreemodelsbeforeandafterlocalrenementusingNURBSandT-splines. model localrened basisfunctions controlpoints elements DOFs errornorm(%) L2 energy I no NURBS 64 25 272 0.1341 2.4945 yes NURBS 140 77 500 0.0698 2.1441 yes T-spline 112 77 444 0.0706 2.1595 II no NURBS 324 225 1044 0.0516 1.6823 yes NURBS 680 527 2488 0.0101 0.9486 yes T-spline 442 391 1820 0

8 .0101 0.9252 III no NURBS 784 625 2592 0
.0101 0.9252 III no NURBS 784 625 2592 0.0230 1.2012 yes NURBS 1620 1377 5864 0.0039 0.6358 yes T-spline 972 891 3848 0.0040 0.6238 r;ሀ⤀= p ᤀr 21sin 2sin3 yyr;ሀ⤀= p ᤀr 1+sin 2sin3 r;ሀ⤀= p ᤀr 2cos 2cos3 2(12)whereKI=op ᤀaisthemodeIstressintensityfactor.Analyticaldisplacementeld(Eq.11)isprescribedontheboundariesexceptforthecrackboundary.Unlikehomogeneousessen-tialboundaryconditions,inhomogeneousboundariescannotbeimposedinastraightforwardapproachinisogeometricanalysis;becausethenon-interpolatingnaturesofNURBSandT-splinesdonotallowforsatisfactionofthekroneckerdeltaproperty.Forimpositionofessentialboundaryconditions,theleast-squaresminimizationmethod[1]isapplied.Threemodelswithuniformlydistributedelementsareconsidered:modelIwithments,modelIIwithelements,andmodelIIIwithelements.Forthispurpose,theh-renement(knotinsertion)processisutilized.Inordertolocallyrenethemesharoundthecrack,theelementsintersectedwiththecrackarechosenforuniformrenementinamesh.BothNURBSandT-splinebasisfunctionsareappliedforeachmodel.Meshandele-mentsforthemodelIIIbeforeandafterlocalrenementaredisplayedinFig.4.Theexact(ofdisplacement)andenergyerrornorms(inpercent)ofallmodelsaregiveninTable1.ItisobservedthatinsomecasesthemodelswhicharelocallyrenedbyusingT-splinesresultinevenmoreaccurateresultsthanthoseobtainedbyusingNURBS,althoughmuchlessnumberofcontrolpointsanddegreesoffreedomareapplied.4.2InclinedcentercrackinasquareplateunderuniaxialtensionMixedmodestressinten

9 sityfactorsforasquareplatewithacenterinc
sityfactorsforasquareplatewithacenterinclinedcrackunderremoteuniaxialtensilestress(Fig.5)areinvestigated.Theplateisinplanestressstate,with=10=0.Sincetheplatedimensionsarelargeincomparisontothecracklength,thenumericalresultscanbereasonablycomparedwiththeanalyticalsolutionofinniteplate.Forthepredenedloading,theexactmixedmodestressintensityfactorsare: ᤀaḻKII ᤀaisthecrackinclinationanglewithrespecttothehorizontalline. (a) (b) (c) (c) Figure4:MeshandelementsofthemodelIIIbeforeandafterlocalrenement:(a)mesh/elementsbeforelocalre-nement;(b)mesh/elementsafterlocalrenementusingNURBS;(c)meshafterlocalrenementusingT-splines;(c)elements(forintegration)afterlocalrenementusingT-splines. Figure5:Geometryandloadingofasquareplatewithacenterinclinedcrack.Table2:Errors(%)ofcomputedmixed-modeSIFsfordifferentcrackinclinationangles,(degree). KI II NURBS T-spline NURBS T-spline 0 0.4339 0.4339 - - 15 0.3786 0.3786 0.2912 0.2913 30 0.477 0.477 1.0711 1.07 45 0.4759 0.4759 1.0388 1.0387 60 0.6498 0.6526 1.3027 1.3039 75 0.4746 0.4746 1.1389 1.1389 SinceDirichletboundaryconditionishomogeneousinthisexample,nospecictechniqueisutilizedforimpositionofessentialboundaryconditions.Fordiscretizingthemodel,rstlyuniformlydistributedelementsareconstructedusingtheh-renement,thentheelementslocatedinin;6]Ȁ[4;6]areselectedforuniformrenementinamesh.BothNURBSandT-splinebasisfunctionsareappliedforanalysis.ModeldiscretizationsformodelswhichuseNURBSandT-splinebas

10 isfunctionsareillustratedinFigs.6and7,re
isfunctionsareillustratedinFigs.6and7,respectively.Differentinclinationangleshavebeenmodeledusingthetwoaforementioneddiscretizations.ItisinterestingtonotethatthebothresultedinverysimilarmixedmodeSIFswhile2304controlpointsand2025elementsaremodeledfortherstdiscretizationand1504controlpointsand1465elementsaremodeledforthesecondone.Errors(%)ofthecomputedSIFsaregiveninTable2andtheexactandcomputednormalizedmixedmodeSIFsareillustratedinFig.8.ThecomputedSIFsareclosetotheexactSIFs.5CONCLUSIONInthiscontribution,theXIGAmethodhasbeenfurtherdevelopedbyusingtheT-splinebasisfunctions.Thismethodiscapableoflocalrenementwhichisnecessaryforadaptiveprocedure. (a) (b) Figure6:DiscretizationofasquareplatewithacenterinclinedcrackusingNURBS:(a)mesh;(b)controlpoints. (b) (c) Figure7:DiscretizationofasquareplatewithacenterinclinedcrackusingT-splines:(a)mesh;(b)controlpoints;(c)elementsforintegration. Figure8:AnalyticalandcomputednormalizedmixedmodeSIFsforseveralcrackinclinationanglesAdoptingarecoverybasederrorestimatorintheproposedapproach(whichisinprogressbytheauthors),canmakeXIGAarobustandpracticalmethodforfractureanalysisofstructures.ACKNOWLEDGEMENTThisresearchissupportedbytheGermanResearchInstitute(DFG)viaResearchTrain-ingGroup鑅瘀aluationofCoupledNumericalPartialModelsinStructuralEngineering(GRK1462)”,whichisgratefullyacknowledgedbytheauthors.[1]E.DeLuycker,D.J.Benson,T.Belytschko,Y.BazilevsandM.C.Hsu,X-FEMinIso-geometricAnalysisforLinearFractureMechanics.InternationalJour

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