Introduction to Finite Element Analysis Formulation, Verification and Validation, First Edition

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P1:OTE/OTE/SPHP2:OTEJWST055-FMJWST055-SzaboFebruary16,20117:50PrinterName:YettoComeIntroductiontoFiniteElementAnalysisIntroductiontoFiniteElementAnalysis:Formulation,VerificationandValidation,FirstEdition.BarnaSzabóandIvoBabuška.©2011JohnWiley&Sons,Ltd.Published2011byJohnWiley&Sons,Ltd.ISBN:978-0-470-97728-6 P1:OTE/OTE/SPHP2:OTEJWST055-FMJWST055-SzaboFebruary16,20117:50PrinterName:YettoComeWILEYSERIESINCOMPUTATIONALMECHANICSSeriesAdvisors:RenédeBorstPerumalNithiarasuTayfunE.TezduyarGenkiYagawaTarekZohdi P1:OTE/OTE/SPHP2:OTEJWST055-FMJWST055-SzaboFebruary16,20117:50PrinterName:YettoComeIntroductiontoFiniteElementAnalysisFormulation,VerificationandValidationBarnaSzabo´WashingtonUniversityinSt.Louis,USAIvoBabuskaˇTheUniversityofTexasatAustin,USAAJohnWileyandSons,Ltd.,Publication P1:OTE/OTE/SPHP2:OTEJWST055-FMJWST055-SzaboFebruary16,20117:50PrinterName:YettoComeThiseditionfirstpublished2011c2011JohnWiley&Sons,LtdRegisteredofficeJohnWiley&SonsLtd,TheAtrium,SouthernGate,Chichester,WestSussex,PO198SQ,UnitedKingdomFordetailsofourglobaleditorialoffices,forcustomerservicesandforinformationabouthowtoapplyforpermissiontoreusethecopyrightmaterialinthisbookpleaseseeourwebsiteatwww.wiley.com.TherightoftheauthorstobeidentifiedastheauthorsofthisworkhasbeenassertedinaccordancewiththeCopyright,DesignsandPatentsAct1988.Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recordingorotherwise,exceptaspermittedbytheUKCopyright,DesignsandPatentsAct1988,withoutthepriorpermissionofthepublisher.Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprintmaynotbeavailableinelectronicbooks.Designationsusedbycompaniestodistinguishtheirproductsareoftenclaimedastrademarks.Allbrandnamesandproductnamesusedinthisbookaretradenames,servicemarks,trademarksorregisteredtrademarksoftheirrespectiveowners.Thepublisherisnotassociatedwithanyproductorvendormentionedinthisbook.Thispublicationisdesignedtoprovideaccurateandauthoritativeinformationinregardtothesubjectmattercovered.Itissoldontheunderstandingthatthepublisherisnotengagedinrenderingprofessionalservices.Ifprofessionaladviceorotherexpertassistanceisrequired,theservicesofacompetentprofessionalshouldbesought.RMATLABisatrademarkofTheMathWorks,Inc.andisusedwithpermission.TheMathWorksdoesnotwarrantRtheaccuracyofthetextorexercisesinthisbook.Thisbook’suseordiscussionofMATLABsoftwareorrelatedproductsdoesnotconstituteendorsementorsponsorshipbyTheMathWorksofaparticularpedagogicalapproachRorparticularuseoftheMATLABsoftware.LibraryofCongressCataloguing-in-PublicationDataSzabo,B.A.(BarnaAladar),1935-´Introductiontofiniteelementanalysis:formulation,verification,andvalidation/BarnaSzabo,IvoBabu´ska.ˇp.cm.Includesbibliographicalreferencesandindex.ISBN978-0-470-97728-6(hardback)1.Finiteelementmethod.I.Babuska,Ivo.ˇII.Title.TA347.F5S9792011620.00151825–dc222010051233AcataloguerecordforthisbookisavailablefromtheBritishLibrary.PrintISBN:9780470977286ePDFISBN:9781119993827oBookISBN:9781119993834ePubISBN:9781119993483MobiISBN:9781119993490Setin10/12TimesbyAptaraInc.,NewDelhi,India. P1:OTE/OTE/SPHP2:OTEJWST055-FMJWST055-SzaboFebruary16,20117:50PrinterName:YettoComeThisbookisdedicatedtoourteachersandstudents.Ifpeopledonotbelievethatmathematicsissimple,itisonlybecausetheydonotrealizehowcomplicatedlifeis.—JohnvonNeumann P1:OTE/OTE/SPHP2:OTEJWST055-FMJWST055-SzaboFebruary16,20117:50PrinterName:YettoComeContentsAbouttheAuthorsxiiiSeriesPrefacexvPrefacexvii1Introduction11.1Numericalsimulation21.1.1Conceptualization21.1.2Validation51.1.3Discretization71.1.4Verification81.1.5Decision-making91.2Whyisnumericalaccuracyimportant?111.2.1Applicationofdesignrules111.2.2Formulationofdesignrules121.3Chaptersummary142Anoutlineofthefiniteelementmethod172.1Mathematicalmodelsinonedimension172.1.1Theelasticbar172.1.2Conceptualization242.1.3Validation272.1.4Thescalarellipticboundaryvalueprobleminonedimension282.2Approximatesolution292.2.1Basisfunctions322.3Generalizedformulationinonedimension332.3.1Essentialboundaryconditions352.3.2Neumannboundaryconditions372.3.3Robinboundaryconditions372.4Finiteelementapproximations382.4.1Errormeasuresandnorms412.4.2Theerrorofapproximationintheenergynorm43 P1:OTE/OTE/SPHP2:OTEJWST055-FMJWST055-SzaboFebruary16,20117:50PrinterName:YettoComeviiiCONTENTS2.5FEMinonedimension442.5.1Thestandardelement442.5.2Thestandardpolynomialspace452.5.3Finiteelementspaces472.5.4Computationofthecoefficientmatrices492.5.5Computationoftherighthandsidevector522.5.6Assembly552.5.7Treatmentoftheessentialboundaryconditions582.5.8Solution612.5.9Post-solutionoperations622.6Propertiesofthegeneralizedformulation672.6.1Uniqueness672.6.2Potentialenergy682.6.3Errorintheenergynorm682.6.4Continuity692.6.5Convergenceintheenergynorm702.7Errorestimationbasedonextrapolation732.7.1Theroot-mean-squaremeasureofstress742.8Extractionmethods752.9Laboratoryexercises772.10Chaptersummary773Formulationofmathematicalmodels793.1Notation793.2Heatconduction813.2.1Thedifferentialequation833.2.2Boundaryandinitialconditions833.2.3Symmetry,antisymmetryandperiodicity853.2.4Dimensionalreduction863.3Thescalarellipticboundaryvalueproblem923.4Linearelasticity933.4.1TheNavierequations973.4.2Boundaryandinitialconditions973.4.3Symmetry,antisymmetryandperiodicity993.4.4Dimensionalreduction1003.5Incompressibleelasticmaterials1033.6Stokes’flow1053.7Thehierarchicviewofmathematicalmodels1063.8Chaptersummary1064Generalizedformulations1094.1Thescalarellipticproblem1094.1.1Continuity1114.1.2Existence1124.1.3Approximationbythefiniteelementmethod1124.2Theprincipleofvirtualwork1154.3Elastostaticproblems117 P1:OTE/OTE/SPHP2:OTEJWST055-FMJWST055-SzaboFebruary16,20117:50PrinterName:YettoComeCONTENTSix4.3.1Uniqueness1194.3.2Theprincipleofminimumpotentialenergy1274.4Elastodynamicmodels1334.4.1Undampedfreevibration1344.5Incompressiblematerials1404.5.1Thesaddlepointproblem1424.5.2Poisson’sratiolocking1424.5.3Solvability1434.6Chaptersummary1435Finiteelementspaces1455.1Standardelementsintwodimensions1455.2Standardpolynomialspaces1465.2.1Trunkspaces1465.2.2Productspaces1475.3Shapefunctions1475.3.1Lagrangeshapefunctions1485.3.2Hierarchicshapefunctions1505.4Mappingfunctionsintwodimensions1525.4.1Isoparametricmapping1525.4.2Mappingbytheblendingfunctionmethod1545.4.3Mappingofhigh-orderelements1565.4.4Rigidbodyrotations1565.5Elementsinthreedimensions1575.6Integrationanddifferentiation1585.6.1Volumeandareaintegrals1595.6.2Surfaceandcontourintegrals1605.6.3Differentiation1615.7Stiffnessmatricesandloadvectors1625.7.1Stiffnessmatrices1625.7.2Loadvectors1645.8Chaptersummary1646Regularityandratesofconvergence1676.1Regularity1676.2Classification1706.3Theneighborhoodofsingularpoints1736.3.1TheLaplaceequation1736.3.2TheNavierequations1756.3.3Materialinterfaces1836.3.4Forcingfunctionsactingonboundaries1856.3.5Strongandweaksingularpoints1926.4Ratesofconvergence1936.4.1Thechoiceoffiniteelementspaces1966.4.2Usesofaprioriinformation2016.4.3Aposteriorierrorestimationintheenergynorm2096.4.4Adaptiveandfeedbackmethods2116.5Chaptersummary212 P1:OTE/OTE/SPHP2:OTEJWST055-FMJWST055-SzaboFebruary16,20117:50PrinterName:YettoComexCONTENTS7Computationandverificationofdata2157.1Computationofthesolutionanditsfirstderivatives2157.2Nodalforces2177.2.1Nodalforcesintheh-version2177.2.2Nodalforcesinthep-version2207.2.3Nodalforcesandstressresultants2217.3Verificationofcomputeddata2227.4Fluxandstressintensityfactors2287.4.1TheLaplaceequation2287.4.2Planarelasticity2327.5Chaptersummary2358Whatshouldbecomputedandwhy?2378.1Basicassumptions2388.2Conceptualization:driversofdamageaccumulation2388.3Classicalmodelsofmetalfatigue2408.3.1Modelsofdamageaccumulation2438.3.2Notchsensitivity2468.3.3Thetheoryofcriticaldistances2488.4Linearelasticfracturemechanics2508.5Ontheexistenceofacriticaldistance2528.6Drivingforcesfordamageaccumulation2538.7Cyclecounting2548.8Validation2558.9Chaptersummary2579Beams,platesandshells2619.1Beams2619.1.1TheTimoshenkobeam2649.1.2TheBernoulli–Eulerbeam2699.2Plates2749.2.1TheReissner–Mindlinplate2769.2.2TheKirchhoffplate2809.2.3Enforcementofcontinuity:theHCTelement2829.3Shells2839.3.1Hierarchic“thin-solid”models2869.4TheOakRidgeexperiments2889.4.1Description2889.4.2Conceptualization2909.4.3Verification2919.4.4Validation:comparisonofpredictedandobserveddata2939.4.5Discussion2959.5Chaptersummary29610Nonlinearmodels29710.1Heatconduction29710.1.1Radiation29710.1.2Nonlinearmaterialproperties298 P1:OTE/OTE/SPHP2:OTEJWST055-FMJWST055-SzaboFebruary16,20117:50PrinterName:YettoComeCONTENTSxi10.2Solidmechanics29810.2.1Largestrainandrotation29910.2.2Structuralstabilityandstressstiffening30210.2.3Plasticity30610.2.4Mechanicalcontact31010.3Chaptersummary313ADefinitions315A.1Normsandseminorms315A.2Normedlinearspaces316A.3Linearfunctionals316A.4Bilinearforms316A.5Convergence317A.6Legendrepolynomials317A.7Analyticfunctions318A.7.1AnalyticfunctionsinR2318A.7.2AnalyticcurvesinR2318A.8TheSchwarzinequalityforintegrals319BNumericalquadrature321B.1Gaussianquadrature322B.2Gauss–Lobattoquadrature323CPropertiesofthestresstensor325C.1Thetractionvector325C.2Principalstresses326C.3Transformationofvectors327C.4Transformationofstresses328DComputationofstressintensityfactors331D.1Thecontourintegralmethod331D.2Theenergyreleaserate333D.2.1Symmetric(ModeI)loading333D.2.2Antisymmetric(ModeII)loading334D.2.3Combined(ModeIandModeII)loading335D.2.4Computationbythestiffnessderivativemethod335ESaint-Venant’sprinciple337E.1Green’sfunctionfortheLaplaceequation337E.2Modelproblem338FSolutionsforselectedexercises345Bibliography353Index359 P1:OTE/OTE/SPHP2:OTEJWST055-FMJWST055-SzaboFebruary16,20117:50PrinterName:YettoComeAbouttheAuthorsBarnaSzabo´isco-founderandpresidentofEngineeringSoftwareResearchandDevel-opment,Inc.(ESRD),thecompanythatproducestheprofessionalfiniteelementanalysissoftwareStressCheckR.PriortohisretirementfromtheSchoolofEngineeringandAp-pliedScienceofWashingtonUniversityin2006heservedastheAlbertP.andBlancheY.GreensfelderProfessorofMechanics.Hisprimaryresearchinterestisassuranceofqualityandreliabilityinthenumericalsimulationofstructuralandmechanicalsystemsbythefiniteelementmethod.Hehaspublishedover150papersinrefereedtechnicaljournals,severalofthemincollaborationwithProfessorIvoBabuska,withwhomhealsopublishedabookonˇfiniteelementanalysis(JohnWiley&Sons,Inc.,1991).HeisafoundingmemberandFellowoftheUSAssociationforComputationalMechanics.AmonghishonorsareelectiontotheHungarianAcademyofSciencesasExternalMemberandanhonorarydoctorate.IvoBabuskaˇ’sresearchhasbeenconcernedmainlywiththereliabilityofcomputationalanalysisofmathematicalproblemsandtheirapplications,especiallybythefiniteelementmethod.Hewasthefirsttoaddressaposteriorierrorestimationandadaptivityinfiniteelementanalysis.Hisresearchpapersonthesesubjectspublishedinthe1970shavebeenwidelycited.HisjointworkwithBarnaSzaboonthe´p-versionofthefiniteelementmethodestablishedthetheoreticalfoundationsandthealgorithmicstructureforthismethod.Hisrecentworkhasbeenconcernedwiththemathematicalformulationandtreatmentofuncertaintieswhicharepresentineverymathematicalmodel.Inrecognitionofhisnumerousimportantcontributions,ProfessorBabuskareceivedmayhonors,whichincludehonorarydoctorates,ˇmedalsandprizesandelectiontoprestigiousacademies. P1:OTE/OTE/SPHP2:OTEJWST055-FMJWST055-SzaboFebruary16,20117:50PrinterName:YettoComeSeriesPrefaceTheseriesonComputationalMechanicswillbeaconvenientlyidentifiablesetofbookscov-eringinterrelatedsubjectsthathavebeenreceivingmuchattentioninrecentyearsandneedtohaveaplaceinseniorundergraduateandgraduateschoolcurricula,andinengineeringpractice.Thesubjectswillcoverapplicationsandmethodscategories.Theywillrangefrombiomechanicstofluid-structureinteractionstomultiscalemechanicsandfromcomputationalgeometrytomeshfreetechniquestoparallelanditerativecomputingmethods.Applicationareaswillbeacrosstheboardinawiderangeofindustries,includingcivil,mechanical,aerospace,automotive,environmentalandbiomedicalengineering.Practicingengineers,re-searchersandsoftwaredevelopersatuniversities,industryandgovernmentlaboratories,andgraduatestudentswillfindthisbookseriestobeanindispensiblesourcefornewengineeringapproaches,interdisciplinaryresearch,andacomprehensivelearningexperienceincomputa-tionalmechanics.Thisbook,writtenbytwowell-recognized,leadingexpertsonfiniteelementanalysis,givesanintroductiontofiniteelementanalysiswithanemphasisonvalidation–theprocesstoascertainthatthemathematical/numericalmodelmeetsacceptancecriteria–andverification–theprocessforacceptabilityoftheapproximatesolutionandcomputeddata.Thesystematictreatmentofformulation,verificationandvalidationproceduresisadistinguishingfeatureofthisbookandsetsitapartfromothertextsonfiniteelements.Itencapsulatescontemporaryresearchonpropermodelselectionandcontrolofmodellingerrors.Anotheruniquefeatureofthebookisthatwithaminimumofmathematicalrequisitesitbridgesthegapbetweenengineeringandmathematically-orientedintroductorytextbooksintofiniteelements. P1:OTE/OTE/SPHP2:OTEJWST055-FMJWST055-SzaboFebruary16,20117:50PrinterName:YettoComePrefaceIncreasingly,engineeringdecisionsarebasedoncomputedinformationwiththeexpectationthatthecomputedinformationwillprovideareliablequantitativeestimateofsomeattributesofaphysicalsystemorprocess.Thequestionofhowmuchrelianceoncomputedinformationcanbejustifiedisbeingaskedwithincreasingfrequencyandurgency.Assuranceofthereliabilityofcomputedinformationhastwokeyaspects:(a)selectionofasuitablemathematicalmodeland(b)approximationofthesolutionofthecorrespondingmathematicalproblem.Theprocessbywhichitisascertainedthatamathematicalmodelmeetsnecessarycriteriaforacceptance(i.e.,itisnotunsuitableforpurposesofanalysis)iscalledvalidation.Theprocessbywhichitisascertainedthattheapproximatesolution,aswellasthedatacomputedfromtheapproximatesolution,meetnecessaryconditionsforacceptance,giventhegoalsofcomputation,iscalledverification.Thisbookaddressestheproblemsofverificationandvalidation.Obtainingapproximatesolutionsformathematicalmodelswithguaranteedaccuracyisoneoftheprincipalgoalsofresearchinfiniteelementanalysis.Animportantresultobtainedinthemid-1980swasthatexponentialratesofconvergencecanbeachievedthroughproperdesignofthefiniteelementmeshandproperassignmentofpolynomialdegreesforalargeandimportantclassofproblemsthatincludeselasticity,heatconductionandsimilarproblems.Thismadeitfeasibletoestimateandcontroltheerrorsofdiscretizationformanypracticalproblems.Atpresenttheproblemsofpropermodelselectionandcontrolofmodelingerrorsareattheforefrontofresearch.Theconceptsofhierarchicmodelsandmodelingstrategieshavebeendeveloped.Progressinthisareamakesmanyimportantpracticalapplicationspossible.Thedistinguishingfeatureofthisbookisthatitpresentsasystematictreatmentoffor-mulation,verificationandvalidationprocedures,illustratedbyexamples.Webelievethatusersoffiniteelementanalysis(FEA)softwareproductsmusthaveabasicunderstandingofhowmathematicalmodelsareconstructed;whataretheessentialassumptionsincorporatedinamathematicalmodel;whatisthealgorithmicstructureofthefiniteelementmethod;howthediscretizationparametersaffecttheaccuracyofthefiniteelementsolution;howtheaccuracyofthecomputeddatacanbeassessed;andhowtoavoidcommonpitfallsandmis-takes.Ourprimaryobjectiveinassemblingthematerialpresentedinthisbookistoprovideabasicworkingknowledgeofthefiniteelementmethod.AlinktothestudenteditionofaprofessionalFEAsoftwareproductcalledStressCheckRisprovidedinthecompanionweb-site(www.wiley.com/go/szabo)toenablereaderstoperformcomputationalexperiments.11StressCheckRisatrademarkofEngineeringSoftwareResearchandDevelopment,Inc.,St.Louis,Missouri,USA. P1:OTE/OTE/SPHP2:OTEJWST055-FMJWST055-SzaboFebruary16,20117:50PrinterName:YettoComexviiiPREFACEAnotherimportantobjectiveofthisbookistopreparereaderstofollowandunderstandnewdevelopmentsinthefieldofFEAthroughcontinuedself-study.EngineeringstudentstypicallytakeonlyonecourseinFEA,consistingofapproximately15weeksofinstruction(45lecturehours).Wehaveorganizedthematerialinthisbooksoastomakeefficientuseoftheavailabletime.Thebookiswritteninsuchawaythattheprerequisitesareminimal.Juniorstandinginengineeringwithsomebackgroundinpotentialflowandstrengthofmaterialsaresufficient.ForthisreasonthemathematicalcontentisfocusedontheintroductionoftheessentialconceptsandterminologynecessaryforunderstandingapplicationsofFEAinelasticityandheatconduction.Somekeytheoremsareproveninasimplesetting.WewouldliketothankDr.NormanF.Knight,Jr.andDr.SebastianNerviforreviewingandcommentingonthemanuscript.BarnaSzabo´WashingtonUniversityinSt.Louis,USAIvoBabuskaˇTheUniversityofTexasatAustin,USA P1:OSOJWST055-01JWST055-SzaboFebruary16,20117:52PrinterName:YettoCome1IntroductionEngineeringdecision-makingprocessesincreasinglyrelyoninformationcomputedfromapproximatesolutionsofmathematicalmodels.Engineeringdecisionshavelegalandethicalimplications.ThestandardappliedinlegalproceedingsincivilcasesintheUnitedStatesistohaveopinions,recommendationsanddecisionsbaseduponareasonabledegreeofengineeringcertainty.Codesofethicsofengineeringsocietiesimposehigherstandards.Forexample,theCodeofEthicsoftheInstituteofElectricalandElectronicsEngineers(IEEE)requiresmemberstoacceptresponsibilityinmakingengineeringdecisionsconsistentwiththesafety,health,andwelfareofthepublic,andtodisclosepromptlyfactorsthatmightendangerthepublicortheenvironmentandtobehonestandrealisticinstatingclaimsorestimatesbasedonavailabledata.Animportantchallengefacingthecomputationalengineeringcommunityistoestablishproceduresforcreatingevidencethatwillshow,withahighdegreeofcertainty,thatamathematicalmodelofsomephysicalreality,formulatedforaparticularpurpose,caninfactrepresentthephysicalrealityinquestionwithsufficientaccuracytomakepredictionsbasedonmathematicalmodelsusefulandjustifiableforthepurposesofengineeringdecision-makingandtheerrorsinthenumericalapproximationaresufficientlysmall.Thereisalargeandrapidlygrowingbodyofworkonthissubject.See,forexample,[38a],[68],[52],[51],[99].Theformulationandnumericaltreatmentofmathematicalmodelsforuseinsupportofengineeringdecision-makinginthefieldofsolidmechanicsisaddressedinadocumentissuedbytheAmericanSocietyofMechanicalEngineers(ASME)andadoptedbytheAmericanNationalStandardsInstitute(ANSI)[33].TheSimulationInteroperabilityStandardsOrganization(SISO)isanotherimportantsourceofinformation.Theconsiderationsunderlyingtheselectionofmathematicalmodelsandmethodsfortheestimationandcontrolofmodelingerrorsandtheerrorsofdiscretizationarethetwomaintopicsofthisbook.Inthischapterabriefoverviewispresentedandthebasicterminologyisintroduced.IntroductiontoFiniteElementAnalysis:Formulation,VerificationandValidation,FirstEdition.BarnaSzabóandIvoBabuška.©2011JohnWiley&Sons,Ltd.Published2011byJohnWiley&Sons,Ltd.ISBN:978-0-470-97728-6 P1:OSOJWST055-AppAJWST055-SzaboFebruary10,201113:36PrinterName:YettoComeAppendixADefinitionsThepropertiesofnorms,normedlinearspaces,linearfunctionalsandbilinearformsarelistedinthefollowing.Thesymbolsαandβdenoterealnumbers.InSectionA.6theLegendrepolynomialsaredefinedandinSectionA.8theSchwarzinequalityisderived.A.1NormsandseminormsThenormofafunctionorvectorisameasureofthesizeofthefunctionorvector.Normsarerealnon-negativenumbersdefinedonsomespaceX.Norms,denotedby·X,havethefollowingproperties:1.uX≥0.2.uX=0ifu=0.3.αuX=|α|uX.4.u+vX≤uX+vX.Thispropertyisknownasthetriangleinequality.AsnotedinSection2.4.1,afamiliarexampleofnormsisthedistanceinEuclideanspace.Inconnectionwithfiniteelementanalysis,thecommonlyusednormsarethemaximumnorm,definedinEquation(2.55);theL2norm,definedinEquation(2.56);andtheenergynorm,definedinEquation(2.53).Seminormssatisfyproperties1,3and4ofnormsbutdonotsatisfyproperty2.Insteadofproperty2seminormshavethepropertyuX=0u∈X¯⊂X,u=0.Forexample,uEdefinedbyEquation(2.53)isaseminormonthespacedefinedbyEqua-tion(2.38)whenc=0.InthiscaseX¯isthesetofconstantfunctionsontheinterval(0,).IntroductiontoFiniteElementAnalysis:Formulation,VerificationandValidation,FirstEdition.BarnaSzabóandIvoBabuška.©2011JohnWiley&Sons,Ltd.Published2011byJohnWiley&Sons,Ltd.ISBN:978-0-470-97728-6 P1:OSOJWST055-AppAJWST055-SzaboFebruary10,201113:36PrinterName:YettoCome316APPENDIXAA.2NormedlinearspacesAnormedlinearspaceXisafamilyofelementsu,v,...whichhavethefollowingproperties:1.Ifu∈Xandv∈Xthen(u+v)∈X.2.Ifu∈Xthenαu∈X.3.u+v=v+u.4.u+(v+w)=(u+v)+w.5.ThereisanuniqueelementinX,denotedby0,suchthatu+0=uforanyu∈X.6.Associatedwitheveryelementu∈Xthereisanuniqueelement−u∈Xsuchthatu+(−u)=0.7.α(u+v)=αu+αv.8.(α+β)u=αu+βu.9.α(βu)=(αβ)u.10.1·u=u.11.0·u=0.12.Associatedwitheveryu∈XthereisarealnumberuX,calledthenorm.ThenormhasthepropertieslistedinSectionA.1.A.3LinearfunctionalsLetXbeanormedlinearspaceandF(v)aprocesswhichassociateswitheveryv∈XarealnumberF(v).F(v)iscalledalinearfunctionalorlinearformonXifithasthefollowingproperties:1.F(v1+v2)=F(v1)+F(v2).2.F(αv)=αF(v).3.|F(v)|≤CvXwithCindependentofv.ThesmallestpossiblevalueofCiscalledthenormofF.A.4BilinearformsLetXandYbenormedlinearspacesandB(u,v)aprocessthatassociateswitheveryu∈Xandv∈YarealnumberB(u,v).B(u,v)isabilinearformonX×Yifithasthefollowingproperties:1.B(u1+u2,v)=B(u1,v)+B(u2,v).2.B(u,v1+v2)=B(u,v1)+B(u,v2).3.B(αu,v)=αB(u,v). P1:OSOJWST055-AppAJWST055-SzaboFebruary10,201113:36PrinterName:YettoComeAPPENDIXA3174.B(u,αv)=αB(u,v).5.|B(u,v)|≤CuXvYwithCindependentofuandv.ThesmallestpossiblevalueofCiscalledthenormofB.ThespaceXiscalledthetrialspaceandfunctionsu∈Xarecalledtrialfunctions.ThespaceYiscalledthetestspaceandfunctionsv∈Yarecalledtestfunctions.B(u,v)isnotnecessarilysymmetric.A.5ConvergenceAsequenceoffunctionsun∈X(n=1,2,...)convergesinthespaceXtothefunctionu∈Xifforevery>0thereisanumbernsuchthatforanyn>nthefollowingrelationshipholds:u−unX<.(A.1)A.6LegendrepolynomialsTheLegendrepolynomialsPn(x)aresolutionsoftheLegendredifferentialequationforn=0,1,2,...:(1−x2)y−2xy+n(n+1)y=0,−1≤x≤1.(A.2)ThefirsteightLegendrepolynomialsareP0(x)=1(A.3)P1(x)=x(A.4)1P(x)=(3x2−1)(A.5)221P(x)=(5x3−3x)(A.6)321P(x)=(35x4−30x2+3)(A.7)481P(x)=(63x5−70x3+15x)(A.8)581P(x)=(231x6−315x4+105x2−5)(A.9)6161P(x)=(429x7−693x5+315x3−35x).(A.10)716Legendrepolynomialscanbegeneratedfromtherecursionformula(n+1)Pn+1(x)=(2n+1)xPn(x)−nPn−1(x),n=1,2,...(A.11) P1:OSOJWST055-AppAJWST055-SzaboFebruary10,201113:36PrinterName:YettoCome318APPENDIXAandLegendrepolynomialssatisfythefollowingrelationship:(2n+1)P(x)=P(x)−P(x),n=1,2,...(A.12)nn+1n−1wheretheprimesrepresentdifferentiationwithrespecttox.Legendrepolynomialssatisfythefollowingorthogonalityproperty:⎧⎪⎪2+1⎨fori=j2i+1Pi(x)Pj(x)dx=(A.13)−1⎪⎪⎩0fori=j.AllrootsofLegendrepolynomialsarelocatedintheinterval−10and0000aκ(x0,y0)>0suchthat∞∞f=a(x−x)i(y−y)jand|a|(i+j)!ri+j≤κ(x,y)(A.14)ij00ij00i,j=0i,j=0wherer≤x2+y2.00Fromthisdefinitionitfollowsthatifasolutionisanalyticthenitsderivativesarebounded.Specifically,foranys,∂su≤s!rsκ(x,y),k=0,1,...,s.(A.15)∂xk∂ys−k00A.7.2AnalyticcurvesinR2Aplanecurveorarcisthesetofpointsdefinedasfollows:={(x,y)|x=x(t),y=y(t),t∈I¯=[−1,1]}.(A.16)Aplanecurveisanalyticifx(t)andy(t)areanalyticfunctionsoft∈I¯and22dxdy+>0forallt∈I¯.(A.17)dtdt P1:OSOJWST055-AppAJWST055-SzaboFebruary10,201113:36PrinterName:YettoComeAPPENDIXA319A.8TheSchwarzinequalityforintegralsDefinitionA.8.1Thefunctionf(x)definedontheintervala3.Exercise2.5.12⎡⎤2AEc2AEc++k0−+0⎢612⎥⎧⎫⎢⎢⎥⎥⎨k0d0⎬2AEc4AEc2AEc[C]=⎢⎢−++−+⎥⎥{r}=0.⎢12312⎥⎩0⎭⎣⎦2AEc2AEc0−++126Exercise2.5.14AssumethatforceFkisactingonnodekandnotractionload,thermalloadorspringloadisactingofelementsk−1andk.Letusconsiderthefollowingpossibilities:(k−1)(k−1)(k−1)(k)(k)(a)Fkisassignedtoelementk−1.Inthiscaseb2r2=b2Fkandb1r1=0wherethelowerindicesrefertothestandardelement-levelnumberingandtheupperindicesrefer(k−1)(k−1)totheglobalnumbering.(b)Fkisassignedtoelementk.Inthiscaseb2r2=0and(k)(k)(k)(k−1)(k)b1r1=b1Fk.Inordertoenforcecontinuityonthetestfunctionweassignb2=b1=bkandsumtheloadvectortermsmultipliedbybk.Therefore,followingtheassemblywewillhavebkFkineithercase. P1:OSOJWST055-AppFJWST055-SzaboFebruary10,201111:55PrinterName:YettoCome348APPENDIXFExercise2.5.16⎡√⎤⎧⎫⎧⎫29/50−421/35⎨a3⎬⎨33.068⎬⎢⎥⎣015/70⎦a4=5.376·√⎩⎭⎩⎭a50−421/35023/15Exercise2.5.18Denotinge=uEX−uFEandek=e(xk),selectingv=ϕk(x)asdefinedinFigure2.10,andapplyingtheGalerkinorthogonalityweget(AE)xk(AE)xk+1k−1kedx−edx=0fork=2,3,...,M()k−1xk−1kxkandhence(AE)k−1(AE)k−1(AE)k(AE)k−ek−1++ek−ek+1=0.k−1k−1kkFork=1andk=M(+1)wehavee1−e2=0and−eM()+eM()+1=0respectively.ThereforewehaveM()+1tridiagonalhomogeneousequations.Assumethatthees-sentialboundaryconditionisprescribedatx1.Inthatcasee1=0andhenceek=0fork=1,2,...,M()+1.Thesamewouldbetrueiftheessentialboundaryconditionwereprescribedatx=xM()+1.Exercise2.5.20LetT=T−T0=1Kandsolvetheproblemforu().Thegapclosesatu()=/2.Sinceu()isalinearfunctionofT−T0,wefindTc=T0+/(2u())where/(2u())isinK(kelvin)unitsbecauseitisthemultiplierof1K.Reactionforcebythedirectmethod:F(T)=AEu(0)−α(T−T)forTT0cc0ccwhereu(0)isu(0)evaluatedatT=T.cFEcReactionforcebytheindirectmethod:thedifferentialequationis−AE(u−αT)+cu=0.MultiplyingbyvandintegratingbypartswegetAE(u−αT)vdx+cuvdx=Fv()−Fv(0)00whereF=AE(u−αT),F=AE(u−αT).x=0x=0 P1:OSOJWST055-AppFJWST055-SzaboFebruary10,201111:55PrinterName:YettoComeAPPENDIXF349Selectingv=1−x/wehaveF=−(AEuv+cuv)dx−AEαT.00ThereforeF(T)=−(AEuv+cuv)dx−AEα(T−T)forTT0cc0c0whereucisuFEevaluatedatT=Tc.Exercise2.5.22FromthedefinitionofnodalforcesgivenbyEquation(2.91)wehavepk+1(k)(k)(k)(k)(k)f1+f2=k1,j+k2,jaj−(r¯1+r¯2).j=1ReferringtoEquation(2.71),+1(k)(k)2d(N1+N2)dNjk1j+k2j=κ(Qk(ξ))dξ=0.k−1dξdξThisisbecauseN1+N2=1.Bydefinition+1k−1r¯1+r¯2=f(Qk(ξ))(N1+N2)dξ+F0(N1(ξ0)+N2(ξ0))+r1+r2.2−1Exercise2.5.24Bydefinitionthenodalforcesforelement2are⎡⎤2AEc2AEcf(2)+−+a(2)1⎢⎢612⎥⎥1=f(2)⎣2AEc2AEc⎦a(2)2−++2126(2)(2)wherea1=a2iscomputedbythefiniteelementsolutionanda2=uistheprescribed(2)boundarycondition.Thenodalforcefistheestimatedreactionforceatx=.2Exercise2.5.26Theexactsolutionisu=x−xα.Thereforeu(0)=1.LetEXEXh=1/M().(a)Thedirectmethod:uFE(h)−uFE(0)uEX(h)α−1uFE(0)===1−h.hh P1:OSOJWST055-AppFJWST055-SzaboFebruary10,201111:55PrinterName:YettoCome350APPENDIXFTheconditionthattherelativeerrormustnotexceed1%canbestatedasu(0)−u(0)EXFE=hα−1≤0.01.u(0)EXThereforeh≤10−2/(α−1)fromwhichwehaveh≤10−40orM()>1040.(b)Thenodalforcemethod:Bydefinition,thenodalforcesforelement1are(1)(1)F111−1uFE(x1)r1=−F(1)h−11u(x)r(1)2FE22whereh(1)α−2xα−1r=α(α−1)x1−dx=h.10hUsingu(x)=u(0)=0andu(x)=u(h)=u(h)=h−hα,wefindFE1FEFE2FEEX(1)(1)F=−1.Referringtothedefinitionofnodalforces,forthisproblemF=11−u(x)andwefindthatthenodalforcemethodyieldsu(0)=1,hencetheFE1FErelativeerroriszeroindependentofh.Exercise3.4.6Wecanselect,forexample,Tx=±1onx=±a,Tz=±1onz=±candTy=±yony=±bwiththeothertractioncomponentszero.Exercise3.4.10Lettheanglebetweenthepositivex-axisandtheunitnormalbeα.Thentheunitnormalisn=cosαex+sinαey≡nxex+nyey.Thetangentvectortisrotated90◦counterclockwiserelativetothenormal:t=cos(α+π/2)ex+sin(α+π/2)ey≡−sinαex+cosαey≡−nyex+nxey.BydefinitionT=Tnn+Ttt=(Tnnx−Ttny)ex+(Tnny+Ttnx)ey!!TxTywhichwastobeshown.Exercise6.3.61.Consideringthesymmetriccaseandα=3π/2,Equation(6.33)takestheformsinλα2−=0.λα3π P1:OSOJWST055-AppFJWST055-SzaboFebruary10,201111:55PrinterName:YettoComeAPPENDIXF351FigureF.1Exercise6.4.2:examplesofh-refinement,consistingof200and512elements.ReferringtoFigure6.5(pointD)weseethatthelowestrootliesbetweenπ/2andπ.UsingtherootfindingroutineinMathematica,orsomeotherprogram,wefindλα=2.56581916,henceλ=λ1=0.544483737.2.FromEquation(6.28)a3cos[(λ1−1)3π/4]≡Q1=−=0.543075579.a1cos[(λ1+1)3π/4]ThereforethesymmetrictermsinEquation(6.25)canbewrittenintheformofEqua-tion(6.40).TheequivalenceofEquation(6.40)andEquation(6.41)isdemonstratedbysubstitutingz=r(cosθ+isinθ)andz¯=r(cosθ−isinθ)intoEquation(6.41).Exercise6.4.2Partialsolution:havingcomputedfiniteelementsolutionscorrespondingtoasequenceofmeshes,similartothoseshowninFigureF.1,wehaveinformationsuchasthatshowninTableF.1.UsingTheorem2.6.3andEquation(2.102)wehavee2=π−π≈k2/N2β200E200200e2=π−π≈k2/N2β512E512512whereπ=−0.375722076321p2R2t/E.Wefindβ≈0.51.0TableF.1DataforExercise6.4.2.NELNπE/p2R2t02001079−0.3756277085122687−0.375685015 P1:OSOJWST055-AppFJWST055-SzaboFebruary10,201111:55PrinterName:YettoCome352APPENDIXFExercise8.3.61a+dσa+d1a23a4(d)∞σx=dσx(r,π/2)dr=d1+2r2+2r4draa1131+d/a+d2/(3a2)=σ∞1++21+d/a2(1+d/a)313d/a+2d2/(3a2)=σ∞1+2−1+d/a2(1+d/a)31=σ∞1+2+O(d/a).1+d/aLettingα=d,=ainEquation(8.14)andneglectingtermsoforderd/awegetσ(d)xKe=≈q¯(Kt−1)+1σ∞whereKt=3wasused.Therelativeerrorford/a=0.2is−4.46%.Exercise9.1.10ThesolutionisanalogoustothesolutionofExercise9.1.1.Lettingw−w(x)yv−v(x)yxxw(x,y)≡=,v(x,y)≡=wyw(x)vyv(x)andassumingthat=(x),wegetEIwvdx+cwvdx−ω2(x)(Iwv+Awv)dx=0.s000TheextractionfunctionforM0istheunitrotationatx=0,whichisv=−N2(ξ).Thesignisnegativebecauseapositivemomentwouldcausenegativerotation.Therefore3δEIδM0=B(w,v)=k12δ−k24=12·2Exercise9.2.4WehavetointerpretthepotentialP=:F(u)=FiuidV+TiuidS∂TinEquation(4.16)intermsofthedimensionallyreducedproblemoftheReissnerMindlinplate.LettingF1=F2=0,F3=q/dandu1=u2=0,u3=w,wehaved/2qFiuidV=wdzdxdy=qwdxdy.3D2D−d/2d2D 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P1:OTA/XYZP2:ABCJWST055-INDJWST055-SzaboFebruary11,20118:56PrinterName:YettoComeIndexaprioriestimate,72displacement,20,98adaptivemethods,211essential,29,35,58admissibledata,168,173flux,83admissiblefunctions,35,118,120,141force,21Airyequation,26homogeneous,20,35,93,134Airystressfunction,176kinematic,20,98analyticarc,168,194mixed,29analyticcurves,318naural,29analyticfunction,167,318Neumann,29,37,69,70,93,110,antisymmetry,21,29,175111assembly,55periodic,22,35,85,99asymptoticconsistency,277,285radiation,297augmentedLagrangianmethod,312Robin,29,37,93,110spring,21barforce,18symmetric,22,85basisfunction,29,32,39symmetry,99basisvectorstemperature,83curvilinear,283traction,98Bayesiananalysis,7Winklerspring,98beammodels,261boundarylayer,206,288,290BernoulliEulerbeam,269Timoshenkobeam,265CADtools,3bendingmoment,274calibration,4,24biharmonicequation,176,281characteristicfunction,135bilinearform,36,314characteristicvalue,135boundarycondition,83coefficientantisymmetric,22ofpermeability,92antisymmetry,85,99,126ofconvectiveheattransfer,84convection,83ofdynamicviscosity,105Dirichlet,29,35,93,110ofthermalconductivity,82,89,216IntroductiontoFiniteElementAnalysis:Formulation,VerificationandValidation,FirstEdition.BarnaSzabóandIvoBabuška.©2011JohnWiley&Sons,Ltd.Published2011byJohnWiley&Sons,Ltd.ISBN:978-0-470-97728-6 P1:OTA/XYZP2:ABCJWST055-INDJWST055-SzaboFebruary11,20118:56PrinterName:YettoCome360INDEXcoefficient(Cont.)discretization,2,7ofthermalexpansion,95displacementformulation,seeformulation,spring,98143coefficientmatrix,57displaygrid,216complexpotentials,176divergencetheorem,80,109,116concentratedforce,53,186domain,3,39conceptualization,2,24,237,241,driversofdamageaccumulation,238313conservationlaw,81effectivityindex,74,199,209consistencycondition,308eigenfunction,135constantModeI,178StefanBoltzmann,298ModeII,178constrainteigenpair,135rigidbody,120,125,139eigenvalue,135,178contact,312elementcontinuity,48,69constrainedlinearstraintriangle(CLST),C1,282291exactandminimal,48,282eight-nodequadrilateral,148contourintegral,161four-nodequadrilateral,148contourintegralmethod,231,233HsiehCloughTocher(HCT)triangle,contourplot,216282,291convergence,70,143,225,317nine-nodequadrilateral,149algebraic,74,193,198six-nodetriangle,149asymptotic,193,198three-nodetriangle,149exponential,193,198emissivity,298monotonic,128endurancelimit,241pre-asymptotic,198energynorm,seenorm,41radiusof,174energyreleaserate,233,332coordinatesenergyspace,seespace,34Eulerian,299equationsofmotion,97Lagrangian,299equationsofstaticequilibrium,97triangular,149equilibriumcrack,168dynamic,96cycleratio,seestressratio,240equivalentstress,306cyclicfrequency,135errorestimation,73,199,211errorindicator,212dAlembertsprinciple,96errorsdamageaccumulation,237conceptual,8Darcyslaw,92modeling,2,5,296dataofinterest,23,31ofapproximation,41degreesoffreedom,39,59ofdiscretization,2,7designrules,12ofidealization,2differentialvolume,159pollution,222,223,225differentiation,161programming,8dimensionalreduction,86exactsolution,43directorfunctions,261,284existence,seegeneralizedsolution,112Dirichlet,seeboundarycondition,93extensionoperator,212 P1:OTA/XYZP2:ABCJWST055-INDJWST055-SzaboFebruary11,20118:56PrinterName:YettoComeINDEX361extraction,2,8,212h-version,194extractionfunction,63,75,221,230HartfordCivicCenterArena,9hierarchicfiniteelementspaces,seefactorofsafety,11hierarchicspaces,193fatiguelimit,241hierarchicspaces,48,128,193,222fatiguestrength,241Hookeslaw,18,95,101feedbackinformation,222generalized,96feedbackmethods,211hp-extension,48,193fieldfunctions,261fillets,139,200incompressibilityfiniteelement,39conditionof,104hexahedral,157indexnotation,79pentahedral,157inequalitystandardquadrilateral,145Schwarz,34,70,319standardtriangular,145triangle,13,315tetrahedral,157initialcondition,83,84,99,134finiteelementmesh,39inputdata,168finiteelementmodel,9integration,159finiteelementsolution,43isoparametricmapping,seemapping,152uniquenessof,143finiteelementspace,seespace,196Jacobianflux,92determinant,159heat,81matrix,159,216fluxintensityfactor,213,228inverseof,161forcingfunction,185inadmissible,186Karm´anvortices,9´formulationKroneckerdelta,80displacement,143mixed,141,143L-shapeddomain,197,226Fourierslaw,81Lameconstants,95´fracturemechanics,232,250Laplaceequation,173freesurface,92Legendrepolynomial,45,152,317,322functionallinearfunctional,36linear,314linearindependence,32quadratic,68linearspace,70loadspectrum,243Galerkinorthogonality,43locking,143,287generalizedformulation,34shear,268generalizedsolution,111existenceof,112mappinguniquenessof,67,112,119,143byblendingfunctions,154Girkmannsproblem,227high-orderelements,156gradingfactor,48improper,160Greensfunction,76inverse,44,155,215isoparametric,152,153,156h-extension,48,193linearisoparametric,152h-method,194proper,159 P1:OTA/XYZP2:ABCJWST055-INDJWST055-SzaboFebruary11,20118:56PrinterName:YettoCome362INDEXmapping(Cont.)notchsensitivity,246quadraticisoparametric,152index,247subparametric,153numberingofbasisfunctions,55superparametric,153mappingfunction,44p-extension,48,193marginofsafety,12p-method,194materialstiffnessmatrix,119p-version,194Mathematica,130ParisKregion,250mathematicalmodel,2,14Parislaw,250hierarchicmodels,106,261path-independentintegral,229workingmodels,4periodicboundarycondition,seeboundarymatrixcondition,22Gram,50piecewiseanalyticfunction,168mass,51piezometrichead,92orthogonal,328planestrain,101stiffness,49generalized,101,252mechanicalcontact,310planestress,101,277membraneforce,274plasticity,306meshplateconstant,278geometric,48,189,197,198platesirregular,193Kirchhoffplate,280quasiuniform,48,194ReissnerMindlinplate,276radical,48,198Poissonsratio,95regular,193pollutionerrormetalfatigue,238seeerrors,222mixedformulation,seeformulation,141potentialenergy,32,68,111,127ModeIeigenfunctions,178signof,68modeofvibration,135potentialflow,81model,seemathematicalmodel,2potentialfunction,92modulusofelasticity,95prediction,27,28modulusofrigidity,seeshearmodulus,pressure,10596principaldirection,84,327Mohrcircle,276principalmoments,276MonteCarlomethod,3principalstress,327principleofminimumpotentialenergy,naturalfrequency,13568Navierequations,97applicationtobeams,263Neumann,seeboundarycondition,93applicationtoplates,278NewtonRaphsonmethod,25,155,215principleofvirtualwork,117nodalforce,64,192,217problemnodepoint,39,44inCategoryA,170,196irregular,193inCategoryB,172,196nominalstress,242inCategoryC,173,201norm,41,313processzone,238L2norm,42productspace,147,150,195energy,41anisotropic,287maximum,42,190projection,76 P1:OTA/XYZP2:ABCJWST055-INDJWST055-SzaboFebruary11,20118:56PrinterName:YettoComeINDEX363proportionallimit,23shellpull-backpolynomials,154hierarchicmodels,284hyperboloidal,287quadrature,160,321structural,283GaussLobatto,321,324shellmodelGaussian,130,321Naghdi,285quarter-pointelement,49NovozhilovKoiter,286singularpoint,168,225radiation,84geometric,168rateofconvergence,74,222neighborhoodof,172Rayleighquotient,136,138singularityRayleighRitzmethod,32,41degreeof,173,175,201regionofprimaryinterest,222,335strong,193,290regionofsecondaryinterest,222weak,193regularityoffunctions,167SleipnerAoffshoreplatform,10reliabilityofestimators,74smoothnessoffunctions,167representativevolumeelement,238solvability,143resolventset,304solverReynoldsnumber,105direct,61Richardsonextrapolation,211iterative,61rigidbodysourcefunction,92displacement,67,120spacerotation,120,156,301energy,34,118rigidbodyconstraint,seeconstraint,Euclidean,41,79139finiteelement,7,39,47,113,145,193,Ritzmethod,seeRayleighRitz196method,32span,32,33Robin,seeboundarycondition,93spectrum,135,304robustnesscontinuous,304oferrorestimators,209point,304SRQ,seesystemresponsequantity,2SNcurve,240standardelement,44,157saddlepoint,142stationaryproblems,83Saint-Venantsprinciple,187,337steadystateproblems,83secantmodulus,309Steklovmethod,183seminorm,67,315stiffnessmatrix,50separationofvariables,134globalunconstrained,57serendipityspaceStokesequations,105seetrunkspace,146strainshapefunctionvector,162Almansi,299shapefunctions,45engineeringshear,94three-dimensional,157equivalentelastic,307hierarchic,46Green,300Lagrange,45,147infinitesimal,93shearcorrectionfactor,264,266,277,279linear,301shearforce,274mechanical,18,95shearmodulus,96normal,93 P1:OTA/XYZP2:ABCJWST055-INDJWST055-SzaboFebruary11,20118:56PrinterName:YettoCome364INDEXstrain(Cont.)tractionload,seetractionforce,18shear,94tractionvector,325thermal,18,95trialfunction,29,34,35,39total,18,95trialspace,35,317volumetric,104trunkspace,146,150strainenergy,34anisotropic,287,290stresstwistingmoment,274residual,3,6,96resultant,265uncertainty,106RMSmeasureof,75aleatory,3,15,237transformationof,327epistemic,3,15,28,237stressintensityfactor,228,233,332quantification(UQ),295stressinvariants,327statistical,3stressratio,240uniqueness,seegeneralizedsolution,67stressstiffening,303,304unitsofphysicaldata,17stressvector,325StressCheck,77,130VaiontDam,10strongform,79validation,5,14,27,28,200,256superconvergence,76criteria,6surfaceintegral,161metric,6symmetry,21,29,137,175vectoraxial,89,102transformationof,328lineof,21verification,8,15,20,200,222mirrorimage,21virtualdisplacement,116systemresponsequantity,2virtualexperimentation,4,290,297,313virtualwork,116T-stress,180,233viscosity,105TacomaNarrowsBridge,9vonMisesstress,306Taylorseries,171temperatureWohlercurve,¨seeSNcurve,240absolute,84,297weakform,79testfunction,34,35,118,141Winklerspring,seeboundarycondition,testspace,35,31798thermalload,54thin-solidmodels,286ZienkiewiczZhuestimator,210traction,97ZZestimator,seeZienkiewiczZhutractionforce,18estimator,210 P1:OSOJWST055-01JWST055-SzaboFebruary16,20117:52PrinterName:YettoCome2INTRODUCTIONPhysicalMathematicalNumericalPredictionDecisionrealitymodelsolutionErrorsofErrorsofidealizationdiscretizationConceptualizationDiscretizationExtractionDecision-makingFigure1.1Themainelementsofnumericalsimulationandtheassociatederrors.1.1NumericalsimulationThegoalofnumericalsimulationistomakepredictionsconcerningtheresponseofphysi-calsystemstovariouskindsofexcitationand,basedonthosepredictions,makeinformeddecisions.Toachievethisgoal,mathematicalmodelsaredefinedandthecorrespondingnu-mericalsolutionsarecomputed.Mathematicalmodelsshouldbeunderstoodtobeidealizedrepresentationsofrealityandshouldneverbeconfusedwiththephysicalrealitythattheyaresupposedtorepresent.Thechoiceofamathematicalmodeldependsonitsintendeduse:Whataspectsofphysicalrealityareofinterest?Whatdatamustbepredicted?Whataccuracyisrequired?ThemainelementsofnumericalsimulationandtheassociatederrorsareindicatedschematicallyinFigure1.1.Someerrorsareassociatedwiththemathematicalmodelandsomeerrorsareassociatedwithitsnumericalsolution.Thesearecallederrorsofidealizationanderrorsofdiscretizationrespectively.Forthepredictionstobereliablebothkindsoferrorshavetobesufficientlysmall.Theerrorsofidealizationarealsocalledmodelingerrors.Conceptualizationisaprocessbywhichamathematicalmodelisformulated.Discretizationisaprocessbywhichtheexactsolutionofthemathematicalmodelisapproximated.Extractionisaprocessbywhichthedataofinterestarecomputedfromtheapproximatesolution.Someauthorsrefertothedataofinterestbythetermsystemresponsequantities(SRQs).1.1.1ConceptualizationMathematicalmodelsareoperatorsthattransformonesetofdata,theinput,intoanotherset,theoutput.Insolidmechanics,forexample,oneistypicallyinterestedinpredictingdisplacements,strainsandstresses,stressintensityfactors,limitloads,naturalfrequencies,etc.,givenadescriptionofthesolutiondomain,constitutiveequationsandboundaryconditions(loadingandconstraints).Commontoallmodelsaretheequationsthatrepresenttheconservationofmomentum(instaticproblemstheequationsofequilibrium),thestraindisplacementrelationsandconstitutivelaws.Theendproductofconceptualizationisamathematicalmodel.Thedefinitionofamath-ematicalmodelinvolvesspecificationofthefollowing:1.Theoreticalformulation.Theapplicablephysicallaws,togetherwithcertainsimplifica-tions,arestatedasamathematicalproblemintheformofordinaryorpartialdifferentialequations,orextremumprinciples.Forexample,theclassicaldifferentialequationforelasticbeamsisderivedfromtheassumptionsofthetheoryofelasticitysupplementedbytheassumptionthatthetransversevariationofthelongitudinalcomponentsofthe P1:OSOJWST055-01JWST055-SzaboFebruary16,20117:52PrinterName:YettoComeNUMERICALSIMULATION3displacementvectorcanbeapproximatedbyalinearfunctionwithoutsignificantlyaffectingthedataofinterest,whicharetypicallythedisplacements,bendingmoments,shearforces,naturalfrequencies,etc.2.Specificationoftheinputdata.Theinputdatacomprisethefollowing:(a)Datathatcharacterizethesolutiondomain.Inengineeringpracticesolutiondo-mainsareusuallyconstructedbymeansofcomputer-aideddesign(CAD)tools.CADtoolsproduceidealizedrepresentationsofrealobjects.Thedetailsofide-alizationdependonthechoiceoftheCADtoolandtheskillsandpreferencesofitsoperator.(b)Physicalproperties(elasticmoduli,yieldstress,coefficientsofthermalexpansion,thermalconductivities,etc.)(c)Boundaryconditions(loads,constraints,prescribedtemperatures,etc.)(d)Informationorassumptionsconcerningthereferencestateandtheinitialconditions(e)Uncertainties.Whensomeinformationneededintheformulationofamathemat-icalmodelisunknownthentheuncertaintyissaidtobecognitive(alsocalledepistemic).Forexample,themagnitudeanddistributionofresidualstressesisusuallyunknown,somephysicalpropertiesmaybeunknown,etc.Statisticalun-certainties(alsocalledaleatoryuncertainties)arealwayspresent.Evenwhentheaveragevaluesofneededphysicalproperties,loadingandotherdataareknown,therearestatisticalvariations,possiblyverysubstantialvariations,inthesedata.Considerationoftheseuncertaintiesisnecessaryforproperinterpretationofthecomputedinformation.Variousmethodsareavailableforaccountingforuncertainties.Thechoiceofmethoddependsonthequalityandreliabilityoftheavailableinformation.Onesuchmethod,knownastheMonteCarlomethod,istocharacterizeinputdataasrandomvariablesanduserepeatedrandomsamplingtocomputetheireffectsonthedataofinterest.Iftheprobabilitydensityfunctionsoftheinputdataaresufficientlyaccurateandsufficientlylargesamplesaretakenthenareasonableestimateoftheprobabilitydistributionofthedataofinterestcanbeobtained.3.Statementofobjectives.Definitionsofthedataofinterestandthecorrespondingpermissibleerrortolerances.Conceptualizationinvolvestheapplicationofexpertknowledge,virtualexperimentationandcalibration.ApplicationofexpertknowledgeDependingontheintendeduseofthemodelandtherequiredaccuracyofprediction,varioussimplifyingassumptionsareintroduced.Forexample,theassumptionsincorporatedinthelineartheoryofelasticity,alongwithsimplifyingassumptionsconcerningthedomainandtheboundaryconditions,arewidelyusedinmechanicalandstructuralengineeringapplications.Inmanyapplicationsfurthersimplificationsareintroduced,resultinginbeam,plateandshellmodels,planarmodelsandaxisymmetricmodels,eachofwhichimposesadditionalrestrictionsonwhatboundaryconditionscanbespecifiedandwhatdatacanbecomputedfromthesolution. P1:OSOJWST055-01JWST055-SzaboFebruary16,20117:52PrinterName:YettoCome4INTRODUCTIONIntheengineeringliteraturethecommonlyusedsimplifiedmodelsaregroupedintoseparatemodelclasses,calledtheories.Forexample,variousbeam,plateandshelltheorieshavebeendeveloped.Theformulationofthesetheoriestypicallyinvolvesastatementontheassumedmodeofdeformation(e.g.,planesectionsremainplaneandnormaltothemid-surfaceofadeformedbeam),therelationshipbetweenthefunctionsthatcharacterizethedeformationandthestraintensor(e.g.,thestrainisproportionaltothecurvatureandthedistancefromtheneutralaxis),applicationofHookeslaw,andstatementoftheequationsoftheequilibrium.Inundergraduateengineeringcurriculaeachmodelclassispresentedasathinginitselfandconsequentlythereisastrongpredispositionintheengineeringcommunitytovieweachmodelclassasaseparateentity.Itismuchmoreuseful,however,toviewanymathematicalmodelasaspecialcaseofamorecomprehensivemodel,ratherthanamemberofaconventionallydefinedmodelclass.Forexample,theusualbeam,plateandshellmodelsarespecialcasesofamodelbasedonthethree-dimensionallineartheoryofelasticity,whichinturnisaspecialcaseofalargefamilyofmodelsbasedontheequationsofcontinuummechanicsthataccountforavarietyofhyperelastic,elasto-plasticandothermateriallaws,largedeformation,contact,etc.Thisisthehierarchicviewofmathematicalmodels.Giventherichvarietyofchoices,modelselectionforparticularapplicationsisanon-trivialproblem.Thegoalofconceptualizationistoidentifythesimplestmathematicalmodelthatcanprovidepredictionsofthedataofinterestwithinaspecifiedrangeofaccuracy.Conceptualizationbeginswiththeformulationofatentativemathematicalmodelbasedonexpertknowledge.Wewillcallthisaworkingmodel.Thetermhasthesameconnotationandmeaningasthetermworkinghypothesis.Sincesubjectivejudgmentisinvolved,theformulationoftheinitialworkingmodelmaydifferfromexperttoexpert.Nevertheless,assumingthatsoftwaretoolsthatallowsystematicevaluationofmathematicalmodelswithrespecttoclearlydefinedobjectivesareavailable,itshouldbepossibleforexpertstoarriveatacloseagreementonthedefinitionofamathematicalmodel,givenitsintendeduse.VirtualexperimentationModelselectioninvolvessystematicevaluationoftheeffectsofvariousmodelingassumptionsonthedataofinterestandthesensitivityofthedataofinteresttouncertaintiesintheinputdata.Thisisdonethroughaprocesscalledvirtualexperimentation.Forexample,insolidmechanicsoneusuallybeginswithaworkingmodelbasedonthelineartheoryofelasticity.Theimpliedassumptionsarethatthestrainismuchsmallerthanunity,thestressisproportionaltothestrain,thedisplacementsaresosmallthatequilibriumequationswrittenwithrespecttotheundeformedconfigurationholdinthedeformedconfigu-rationalso,andtheboundaryconditionsareindependentofthedisplacementfunction.Onceaverifiedsolutionisavailable,itispossibletoexaminethestressfieldanddeterminewhetherthestressexceededtheproportionallimitofthematerialandwhetherthisaffectsthedataofinterestsignificantly.Similarly,theeffectsoflargedeformationonthedataofinterestcanbeevaluated.Furthermore,itispossibletotestthesensitivityofthedataofinteresttochangesinboundaryconditions.Virtualexperimentationprovidesvaluableinformationontheinfluenceofvariousmodelingassumptionsonthedataofinterest.CalibrationIntheprocessofconceptualizationtheremaybeindicationsthatthedataofinterestaresensitivefunctionstocertainparametersthatcharacterizematerialbehaviororboundarycon-ditions.Ifthoseparametersarenotavailablethencalibrationexperimentsmustbeperformed P1:OSOJWST055-01JWST055-SzaboFebruary16,20117:52PrinterName:YettoComeNUMERICALSIMULATION5forthepurposeofdeterminingtheneededparameters.Incalibrationthemathematicalmodelisassumedtobecorrectandtheparametersthatcharacterizethemodelareselectedsuchthatthemeasuredresponsematchesthepredictedresponse.Example1.1.1Ifthegoalofcomputationistopredictthenumberofloadcyclesthatcausefatiguefailureinametalpartthenoneormoreempiricalmodelsmustbechosenthatrequireasinputstressorstrainamplitudesandmaterialparameters.Oneofthewidelyusedmodelsforthepredictionoffatiguelifeinlow-cyclefatigueisthegeneralstrainlifemodel:σ¯fbca=(2N)+¯f(2N)(1.1)Ewhereaisthestrainamplitude,Nisthenumberofcyclestofailure,Eisthemodulusofelasticity,¯σfisthefatiguestrengthcoefficient,bisthefatiguestrengthexponent,¯fisthefatigueductilitycoefficientandcisthefatigueductilityexponent.TheparametersE,¯σf,b,¯fandcaredeterminedthroughcalibrationexperiments.See,forexample,[76].Severalvariantsofthismodelareinuse.Standardprocedureshavebeenestablishedforcalibrationexperimentsformetalfatigue.11.1.2ValidationValidationisaprocessbywhichthepredictivecapabilitiesofamathematicalmodelaretestedagainstexperimentaldata.Wewillbeconcernedprimarilywithproblemsinsolidmechanicsforwhichthepredictionscanbetestedthroughexperimentsespeciallydesignedforthatpurpose.Thisisaverylargeclassofproblemsthatincludesallmathematicalmodelsdesignedforthepredictionoftheperformanceofmass-produceditems.Thereareotherimportantproblems,suchastheeffectsofearthquakesandothernaturaldisasters,uniquedesignproblems,suchasdams,sitingofnuclearpowerplantsandthelike,forwhichthepredictionsbasedonmathematicalmodelscannotbetestedatfullscale.Insuchcasesthemodelsareanalyzedaposterioriandmodifiedinthelightofnewinformationcollectedfollowinganincident.Associatedwitheachmathematicalmodelisamodelingerror(illustratedschematicallyinFigure1.1).Thereforeitisnecessarytohaveaprocessfortestingthepredictivecapabilitiesofmathematicalmodels.Thisprocess,calledvalidation.isillustratedschematicallyinFigure1.2.PhysicalMathematicalNumericalsolutionPredictionrealitymodelandverificationConceptualizationComparepredictionwithexperimentNoCriteriaYesFailPassmet?Figure1.2Validation.1See,forexample,theInternationalOrganizationforStandardizationISO12106:2003andISO12107:2003. P1:OSOJWST055-01JWST055-SzaboFebruary16,20117:52PrinterName:YettoCome6INTRODUCTIONForavalidationexperimentoneormoremetricsandthecorrespondingcriteriaaredefined.Ifthepredictionsmeetthecriteriathenthemodelissaidtohavepassedthevalidationtest,otherwisethemodelisrejected.Inlargeprojects,suchasthedevelopmentofanaircraft,aseriesofvalidationexperimentsareperformedstartingwithcoupontestsforthedeterminationofphysicalpropertiesandfailurecriteria,thenprogressingtosub-components,components,parts,sub-assembliesandfinallytheentireassembly.Thecostofexperimentsincreaseswithcomplexityandhencethenumberofexperimentsdecreaseswithcomplexity.Thegoalistodevelopsufficientlyreliablepredictivecapabilitiessuchthattheoutcomeofexperimentsinvolvingsub-assembliesandassemblieswillconfirmthepredictions.Findingproblemslateintheproductioncycleisgenerallyverycostly.Inevaluatingtheresultsofvalidationexperimentsitisimportanttobearinmindthelimitationsanduncertaintiesassociatedwiththeavailableinformationconcerningthephysicalsystemsbeingmodeled:1.Thesolutiondomainisusuallyassumedtocorrespondtodesignspecifications(theblueprint).Inreality,parts,sub-assembliesandassembliesdeviatefromtheirspec-ificationsandthedegreeofdeviationmaynotbeknown,orwouldbedifficulttoincorporateintoamathematicalmodel.2.Formanymaterialstheconstitutivelawsareknownimperfectlyandonlyinsomeaveragesenseandwithinanarrowrangeofstrain,strainrate,temperatureandoverashorttimeintervalofloading.3.Theboundaryconditions,otherthanstress-freeboundaryconditions,arenotknownwithahighdegreeofprecision,evenundercarefullycontrolledexperimentalcondi-tions.Thereasonforthisisthattheloadingandconstraintstypicallyinvolvemechan-icalcontactwhichdependsonthecompliancesofthestructuresthatimposetheloadandconstraints(e.g.,testingmachine,millingmachine,assemblyrig,etc.)andthephysicalpropertiesofthecontactingsurfaces.Inotherwords,theboundaryconditionsrepresenttheinfluenceoftheenvironmentontheobjectbeingmodeled.Theneededinformationisrarelyavailable.Thereforesubjectivejudgmentoftheanalystintheformulationofboundaryconditionsisusuallyunavoidable.4.Duetothehistoryofthematerialpriortomanufacturingthepartsthatwillbeassem-bledintoamachineorstructure,suchascasting,quenching,extrusion,rolling,forging,heattreatment,coldforming,machiningandsurfacetreatmentresidualstressesexist,themagnitudeofwhichcanbeverysubstantial.Thedistributionofresidualstressmustsatisfytheequationsofequilibriumandthestress-freeboundaryconditionsbutotherwiseitisgenerallyunknown.See,forexample,[47],[48].5.Informationconcerningtheprobabilitydistributionofthedatathatcharacterizetheproblemandtheircovariancefunctionsisrarelyavailable.Ingeneral,uncertaintiesincreasewiththecomplexityofmodels.Remark1.1.1Morethanonemathematicalmodelmayhavebeenproposedwithidenticalobjectivesanditispossiblethatmorethanonemathematicalmodelwillmeetthevalidationcriteria.Inthatcasethesimplermodelis. P1:OSOJWST055-01JWST055-SzaboFebruary16,20117:52PrinterName:YettoComeNUMERICALSIMULATION7Remark1.1.2Duetostatisticalvariabilityinthedataanderrorsinexperimentalobserva-tions,comparisonsbetweenpredictionbasedonamathematicalmodelandtheoutcomeofphysicalexperimentsmustbeunderstoodinastatisticalsense.ThetheoreticalframeworkformodelselectionisbasedonBayesiananalysis.2Specifically,denotingamathematicalmodelbyM,thenewlyacquireddatabyDandthebackgroundinformationbyI,theprobabilitythatthemodelMisapredictorofthedataD,giventhebackgroundinformationI,canbewrittenintermsofconditionalprobabilities:Prob(M|D,I)≈Prob(D|M,I)×Prob(M|I).(1.2)Inotherwords,Bayestheoremrelatestheprobabilitythatamathematicalmodeliscorrect,giventhemeasureddataDandthebackgroundinformationI,totheprobabilitythatthemeasureddatawouldhavebeenobservedifthemodelwerefunctioningproperly.See,forexample,[74].ThetermProb(M|I)iscalledpriorprobability.ItrepresentsexpertopinionaboutthevalidityofMpriortocomingintopossessionofsomenewdataD.ThetermProb(D|M,I)iscalledthelikelihoodfunction.Inthisviewcompetingmathematicalmodelsareassignedprobabilitiesthatrepresentthedegreeofbeliefinthereliabilityofeachofthecompetingmodels,giventheinformationavailablepriortoacquiringadditionalinformation.Inlightofthenewinformation,obtainedbyexperiments,thepriorprobabilityisupdatedtoobtainthetermProb(M|D,I),calledtheposteriorprobability.AnimportantandhighlyrelevantaspectofBayestheoremisthatitprovidesaframeworkforimprovementoftheprobabilityestimateProb(M|D,I)basedonnewdata.1.1.3DiscretizationThefiniteelementmethod(FEM)isoneofthemostpowerfulandwidelyusednumericalmethodsforfindingapproximatesolutionstomathematicalproblemsformulatedsoastosimulatetheresponsesofphysicalsystemstovariousformsofexcitation.Itisusedinvariousbranchesofengineeringandscience,suchaselasticity,heattransfer,fluiddynamics,electromagnetism,acoustics,biomechanics,etc.Inthefiniteelementmethodthesolutiondomainissubdividedintoelementsofsim-plegeometricalshape,suchastriangles,squares,tetrahedra,hexahedra,andasetofbasisfunctionsareconstructedsuchthateachbasisfunctionisnon-zerooverasmallnumberofelementsonly.Thisiscalleddiscretization.Detailswillbegiveninthefollowingchapters.Thesetofallfunctionsthatcanbewrittenaslinearcombinationsofthebasisfunctionsiscalledthefiniteelementspace.Theaccuracyofthedataofinterestdependsonthefiniteelementspaceandthemethodusedforcomputingthedatafromthefiniteelementsolu-tion.Associatedwiththefiniteelementsolutionareerrorsofdiscretization,asindicatedinFigure1.1.Itisnecessarytocreatefiniteelementspacessuchthatthedataofinterestcomputedfromthefiniteelementsolutionarewithinacceptableerrorboundswithrespecttotheircounterpartscorrespondingtotheexactsolutionofthemathematicalmodel.Thedataofinterest,suchasthemaximumdisplacement,temperature,stress,etc.,arecomputedfromthefiniteelementsolutionuFE.Thedataofinterestwillbedenotedbyi(uFE),i=1,...,n,inthefollowing.Theobjectiveistocomputei(uFE)andtoensure2ThomasBayes(17021761). P1:OSOJWST055-01JWST055-SzaboFebruary16,20117:52PrinterName:YettoCome8INTRODUCTIONthattherelativeerrorsarewithinprescribedtolerances:|i(uEX)−i(uFE)|≤τi(1.3)|i(uEX)|whereuEXistheexactsolution.OfcourseuEXisnotknowningeneral,butitisknownthati(uEX)isindependentofthefiniteelementspace.Theerrorini(uFE)dependsonthefiniteelementspaceandthemethodusedforcomputingi(uFE).Theerrorsofdiscretizationarecontrolledthroughsuitableenlargementofthefiniteelementspaces,andbyvariousproceduresusedforcomputingi(uFE).1.1.4VerificationVerificationisconcernedwithverifyingthat(a)theinputdataarecorrect,(b)thecomputercodeisfunctioningproperlyand(c)theerrorsinthedataofinterestmeetnecessaryconditionstobewithinpermissibletolerances.Commonerrorsininputareincorrectlyentereddata,suchasmixedunitsanderrorsindataentry.Sucherrorsareeasilyfoundinacarefulreviewoftheinputdata.Theprimaryresponsibilityforensuringthatthecodeisfunctioningproperlyrestswiththecodedevelopers.However,computercodestendtohaveprogrammingerrors,especiallyintheirlessfrequentlytraversedbranches,andtheusersharesintheresponsibilityofverifyingthatthecodeisfunctioningproperly.Inverificationaccuracyisunderstoodtobewithrespecttotheexactsolutionofthemathematicalmodel,notwithrespecttophysicalreality.TheprocessofverificationofthenumericalsolutionisillustratedschematicallyinFigure1.3.Thetermextractionreferstomethodsusedforcomputingi(uFE).Detailsarepresentedinthefollowingchapters.Remark1.1.3Verificationandvalidationarepossibleonlywhenthemathematicalmodelisproperlyformulatedwithrespecttothegoalsofcomputation.Forexample,inlinearelasticitythesolutiondomainmustnothavesharpre-entrantcornersoredgesifthegoalofcomputationistodeterminethemaximumstress.Pointconstraintsandpointforcescanbeusedonlywhencertaincriteriaaremetetc.Detailsaregiveninthefollowingchapters.Unfortunately,usingmathematicalmodelswithoutregardtotheirlimitationsisacommonlyoccurringconcep-tualerror.MathematicalNumericalExtractionandmodelsolutionerrorestimationDiscretizationNoErrorsYesPredictionacceptable?Figure1.3Verificationofthenumericalsolution. P1:OSOJWST055-01JWST055-SzaboFebruary16,20117:52PrinterName:YettoComeNUMERICALSIMULATION9Remark1.1.4TheprocessillustratedschematicallyinFigure1.1isoftenreferredtoasfiniteelementmodeling.Thistermisunfortunatebecauseitmixestwoconceptuallydifferentaspectsofnumericalsimulation:thedefinitionofamathematicalmodelanditsnumericalsolutionbythefiniteelementmethod.1.1.5Decision-makingThegoalofnumericalsimulationistosupportvariousengineeringdecision-makingpro-cesses.Thereisanimpliedexpectationofreliability:onecannotreasonablybasedecisionsoncomputedinformationwithoutbelievingthattheinformationissufficientlyreliabletosupportthosedecisions.Demonstrationofthereliabilityofmathematicalmodelsusedinsupportofengineeringdecision-makingisanessentialpartofanymodelingeffort.Infact,theroleofphysicaltestingistocalibrateandvalidatemathematicalmodelssothatavarietyofloadcasesanddesignalternativescanbeevaluated.Inthefollowingweillustratetheimportanceofthereliabilityofnumericalsimulationprocessesthroughbriefdescriptionsoffourwell-documentedexamplesoftheconsequencesoflargeerrorsinpredictioneitherbecauseimpropermathematicalmodelswereusedorbecauselargeerrorsoccurredinthenumericalsolution.Additionalexamplescanbefoundin[61],[62].Undoubtedly,therearemanyundocumentedinstancesofsubstantiallossattributabletoerrorsinpredictionsbasedonmathematicalmodels.Example1.1.2TheTacomaNarrowsBridge,thefirstsuspensionbridgeacrossPugetSound(WashingtonState,USA),collapsedonNovember7,1940,fourmonthsafteritsopening.Windblowingat68km/hcausedsufficientlylargeoscillationsinthe853mmainspantocollapsethespan.Untilthattimebridgesweredesignedonthebasisofequivalentstaticforces.Thepos-sibilitythatrelativelysmallperiodicaerodynamicforces(theeffectsofKarm´anvortices)´3maybecomesignificantwasnotconsidered.TheKarm´anvorticeswerefirstanalyzedin1911´andtheresultswerepresentedintheGottingenAcademyinthesameyear.¨4ThedesignerswereeitherunawareofthoseresultsordidnotseetheirrelevancetotheTacomaNarrowsBridge,thefailureofwhichwascausedbyinsufficienttorsionalstiffnesstoresisttheperiodicexcitationinducedbyKarm´anvortices.´Example1.1.3TheroofoftheHartfordCivicCenterArenacollapsedonJanuary18,1978.Theroofstructure,measuring91.4by109.7m(300by360ft),wasaspaceframe,aninnovativedesignatthattime.Itwasanalyzedusingamathematicalmodelthataccountedforlinearresponseonly.Furthermore,theconnectiondetailsweregreatlysimplifiedinthemodel.Inlinearelastostaticanalysisitisassumedthatthedeformationofastructureisnegligiblysmallandhenceitissufficienttosatisfytheequationsofequilibriumintheundeformedconfiguration.Theroofframewasassembledontheground.Oncetheroofwasliftedintoitsfinalposition,itsdeflectionwasmeasuredtobetwiceofwhatwaspredictedbythemathematicalmodel:3TheodorevonKarm´an(18811963).´4VonKarm´an,Th.andEdson,L.,´TheWindandBeyond:TheodorevonKarm´an,PioneerinAviationand´PathfinderinSpace,Little,Brown,Boston,MA,1967,pp.211215. P1:OSOJWST055-01JWST055-SzaboFebruary16,20117:52PrinterName:YettoCome10INTRODUCTIONWhennotifiedofthiscondition,theengineersexpressednoconcern,explainingthatsuchdiscrepancieshadtobeexpectedinviewofthesimplifyingassumptionsofthetheoreticalcalculation.5Subsequentinvestigationidentifiedthatrelianceonanoversimplifiedmodelthatdidnotrepresenttheconnectiondetailsproperlyandfailedtoaccountforgeometricnonlinearitieswastheprimarycauseoffailure.Example1.1.4TheVaiontDam,oneofthehighestdamsintheworld(262m),wascom-pletedintheDolomiteRegionoftheItalianAlps,100kmnorthofVenice,in1961.OnOctober9,1963,afterheavyrains,amassivelandslideintothereservoircausedalargewavethatovertoppedthedambyupto245mandsweptintothevalleybelow,resultinginthelossofanestimated2000lives.6Thecourtsfoundthat,duetothepredictabilityofthelandslide,threeengineerswerecriminallyresponsibleforthedisaster.Thedamwithstoodtheoverloadcausedbythewave.Thisincidentservesasanexampleofafullscaletestofamajorstructurecausedbyanunexpectedevent.Example1.1.5TheconsequencesoflargeerrorsofdiscretizationareexemplifiedbytheSleipneraccident.Thegravitybasestructure(GBS)oftheSleipnerAoffshoreplatform,madeofreinforcedconcrete,sankduringballasttestoperationsinGandsfjorden,southofStavenger,Norway,onAugust23,1991.Theeconomiclosswasestimatedtobe700milliondollars.ThemainfunctionoftheGBSwastosupportaplatformweighing56000tons.TheGBSconsistedof24caissoncellswithabaseareaof16000m2.Fourcellswereelongatedtoformshaftsdesignedtosupporttheplatform.ThetotalconcretevolumeoftheGBSwas75000m3.TheaccidentoccurredastheGBSwasbeingloweredtoadepthofapproximately99m.Failurefirstoccurredintwotriangularcells,calledtri-cells,nexttooneoftheshafts.WhentheGBShittheseabed,seismiceventsmeasuring3ontheRichterscalewererecordedintheStavengerarea.7Thereisgeneralagreementamongtheinvestigatorsthattheaccidentwascausedbylargeerrorsinthefiniteelementanalysis,thegoalofwhichwastoestimatetherequirementsforreinforcementoftheconcretecellsbysteelbars:Theglobalfiniteelementanalysisgavea47%underestimationoftheshearforcesinthetri-cellwalls.Thiserrorwascausedbytheuseofacoarsefiniteelementmeshwithsomeskewedelementsusedforanalysisofthetri-cellwalls.85Levy,M.andSalvadori,M.,WhyBuildingsFallDown:HowStructuresFail,W.W.Norton,NewYork,2002.6See,forexample,Hendron,A.J.andPatten,F.D.,TheVaiontSlide.USCorpsofEngineersTechnicalReportGL-85-8(1985).7Jacobsen,B.,ThelossoftheSleipnerAPlatform,Proceedingsofthe2ndInternationalOffshoreandPolarEngineeringConference,InternationalSocietyofOffshoreandPolarEngineers,Vol.1,1992.8Rettedal,W.K.,Gudmestad,O.T.andAarum,T.,DesignofconcreteplatformsafterSleipnerA-1sinking,Proceedingsofthe12thInternationalConferenceonOffshoreMechanicsandArcticEngineering,Vol.1,OffshoreTechnology,pp.309310,ASME,1993. P1:OSOJWST055-01JWST055-SzaboFebruary16,20117:52PrinterName:YettoComeWHYISNUMERICALACCURACYIMPORTANT?11Acheckoftheglobalresponseanalysisrevealedseriousinaccuraciesintheinterpretationofresultsfromfiniteelementanalysesgivingashearforceinacriticalsectionofthecellwallthatwaslessthan60%ofthecorrectvalue.91.2Whyisnumericalaccuracyimportant?AnumberofdifficultiesassociatedwithaccuraterepresentationofarealphysicalsystembymathematicalmeanswerenotedinSection1.1.2.Giventhesedifficulties,itmayseemreasonabletoask:Ifwedonotknowtheinputdatawithsufficientaccuracy,thenwhyshouldwebeconcernedwiththeaccuracyofthenumericalsolution?Inansweringthisquestionweconsidertwoimportantareasofapplicationofmathematicalmodels:theapplicationofdesignrulesandtheformulationofdesignrules.Itisshowninthefollowingthatbothrequireestimationandcontrolofthenumericalaccuracy.1.2.1ApplicationofdesignrulesDesignanddesigncertificationinvolveapplicationofexistingdesignrules,establishedbyvariouscodes,regulationsandguidelines.Thedesignrulesaretypicallystatedintheformofrequiredminimumfactorsofsafety:limFS:=≥(FS)design(1.4)max(uEX)whereFSistherealizedfactorofsafety,lim>0isthelimiting(nottoexceed)valueofsomeentity(suchasmaximumbendingmoment,maximumstress,etc.),max(uEX)>0istheexactvalueofthesameentitycorrespondingtotheexactsolutionofthemathematicalmodeland(FS)designistheminimumvalueofthefactorofsafetyspecifiedbytheapplicabledesignrules.Itisthedesignersresponsibilitytoensurethattheapplicabledesignrulesarefollowed.Wewilldenotebymax(uFE)thevalueofmaxcomputedfromthefiniteelementsolution.Letussupposethat,duetonumericalerrors,itispossibletoguaranteeonlythattherelativeerrorisnotgreaterthanτ:|max(uEX)−max(uFE)|≤τ,0≤τ<1;(1.5)max(uEX)inotherwords,max(uFE)mayunderestimatemax(uEX)by100τ%.Thereforewehave1max(uEX)≤max(uFE).(1.6)(1−τ)OnsubstitutingthisexpressionintoEquation(1.4),weobtainlim(FS)design≥·(1.7)max(uFE)1−τ9Holand,I.,TheSleipneraccident,inFromFiniteElementstotheTrollPlatform-IvarHoland70thAnniversary,K.Bell,editor,TheNorwegianInstituteofTechnology,Trondheim,pp.157168,1994. P1:OSOJWST055-01JWST055-SzaboFebruary16,20117:52PrinterName:YettoCome12INTRODUCTIONOncomparingEquation(1.7)withEquation(1.4)itisseenthat,tocompensatefornumericalerrorsinthecomputationofmax(uFE),itisnecessarytoincreasetherequiredfactorofsafetyto(FS)design/(1−τ).Forexample,iftheaccuracyofmax(uFE)canbeguaranteedto20%(i.e.,τ=0.20)then(FS)designmustbeincreasedby25%.Since(FS)designwaschosenconservativelytoaccountfortheuncertainties,theeconomicpenaltiesassociatedwithusinganincreasedfactorofsafetygenerallyfaroutweighthecostsassociatedwithguaranteeingtheaccuracyofthedataofinteresttowithinasmallrelativeerror(say5%).Applicationofdesignrulesisataskofverification,thatis,verificationthatthecorrectdataareused,thecomputercodeisfunctioningproperlyandthetoleranceτinEquation(1.5)issufficientlysmall.Remark1.2.1Inaerospaceengineeringthedesignrequirementsarestatedintermsofminimumacceptablemarginsofsafety(MS).Bydefinition,MS=FS−1.Remark1.2.2TheFederalAviationRegulations(FAR),issuedbytheFederalAviationAdministrationoftheUSDepartmentofTransportation,stateinSec.25.303thatafactorofsafetyof1.5mustbeappliedtotheprescribedlimitload.Economicconsiderationsdictatethattherealizedfactorofsafetyshouldnotbemuchlargerthan(FS)design.Thisisespeciallytrueinaerospaceengineeringwhereavoidanceofweightpenaltiesmakesitnecessarytoensurethattherealizedfactorsofsafetyarereasonablyclosetothemandatedfactorsofsafety.Example1.2.1Theyieldstrengthinshearofhotrolled0.2%carbonsteelis165MPaandtheusualfactorofsafetyforstaticloadsis1.65(sothattheallowablemaximumshearstressis100MPa).10Ifthenumericalcomputationscouldunderestimatethemaximumshearstressbyasmuchas20%thenthefactorofsafetywouldhavetobeincreasedto2.06,thatis,theallowablemaximumvaluewouldbereducedto80MPa.1.2.2FormulationofdesignrulesTheresponsibilityforformulatingdesignrulesrestswithcommitteesofexpertsappointedbyprofessionalsocietiesandregulatorybodies.Someofthedesignruleshavebeenevolvingforalongtimewhileothersarestillintheirearlystagesofdevelopment.Forexample,theASMEBoilerandPressureVesselCodedatesbackto1914.Ithasbeenadoptedinpartorinitsentiretybyall50statesoftheUnitedStates.ThecodeisbeingupdatedbytheBoilerandPressureCommitteewhichmeetsregularlytoconsiderrequestsforinterpretations,revision,andtodevelopnewrules.Ontheotherhand,designrulesforstructuralcomponentsofaircraftmadeoflaminatedcompositesareintheirearlystagesofdevelopmentTheformulationandrevisionofdesignrulesareapplicationsoftheprocessofvalidationdiscussedinSection1.1.2andschematicallyillustratedinFigure1.2inthefollowingsense.Theobjectofdesignisthephysicalreality.Themathematicalmodelisthecollectionofphysicallawsthatexpertsconsiderrelevanttodecision-making.Theformulasandrulesare,respectively,themetricsandcriteriausedinthepredictionthattheobjectofdesignwill10See,forexample,Popov,E.P.,EngineeringMechanicsofSolids,2ndedition,PrenticeHall,UpperSaddleRiver,NJ,1998. P1:OSOJWST055-01JWST055-SzaboFebruary16,20117:52PrinterName:YettoComeWHYISNUMERICALACCURACYIMPORTANT?13functionasintendedwhenthestatedcriteriaaresatisfied.Inlightofnewexperience,themodel,metricsandcriteriaarerevised.Formulationofdesignrulesinvolvesdefinitionofcertainentitiesk(k=1,2,...),suchasthemaximumprincipalstress,somespecificcombinationsofstressandstraincomponents,etc.,thatcharacterizefailureandthecorrespondinglimitingvalues.Inthefollowingthesubscriptkwillbedroppedandthediscussionwillbeconcernedwithagenericdesignrule,thatis,thedeterminationoflimandevaluationoftheassociateduncertainties.Thefactorofsafetyisdeterminedonthebasisofassessmentofuncertaintiesandconsiderationoftheconsequencesoffailure.Supposethatamathematicalmodelpredictsthatfailurewilloccurwhenreachesitscriticalvaluelim.First,asetofcalibrationexperimentshavetobeperformedwiththeobjectivetodeterminelim.Second,anothersetofexperimentshavetobeconductedtotestwhetherfailurecanbepredictedonthebasisoflim.Thesearevalidationexperiments.Ingeneralcannotbeobserveddirectly,thereforeitmustbeinferredfromcorrelationsbetweencomputeddataandexperimentalobservations.(j)LetYijbetheithidealobservationofthejthexperimentandletφi(uEX)bethecorrespond-ingfunctional11computedfromtheexactsolutionu(j)sothatiftherewerenoexperimentalEXerrorsandthemathematicalmodelwerecorrectthenwewouldhave(j)Yij−φi(uEX)=0.(j)Duetoexperimentalerrorsweactuallyobserveyijandcompareitwithφi(uFE),thefinite(j)elementapproximationtoφi(uEX).LetuswriteexpYij=yij±eijand(j)(j)feaφi(uEX)=φi(uFE)±eijwhereeexp(resp.efea)istheexperimental(resp.approximation)error.Thenijij(j)exp(j)feayij−φi(uFE)=Yij∓eij−φi(uEX)±eij.Usingthetriangleinequality,wehave(j)(j)expfea|yij−φi(uFE)|≤|Yij−φi(uEX)|+|eij|+|eij|.(1.8)apparenterrortrueerrorThisresultshowsthatintestingamathematicalmodelitisessentialtohaveboththeexperi-mentalerrorsandtheerrorsofdiscretizationundercontrol,otherwiseitwillnotbepossibletoknowwhethertheapparenterrorisduetoanerrorinthehypothesis,errorsinthenu-mericalapproximation,orerrorsintheexperiment.Furthermore,meansfortheestimation11Afunctionalisarealnumberdefinedonaspaceoffunctions.Inthepresentcontextafunctionalisarealnumbercomputedfromtheexactsolutionorthefiniteelementsolution. P1:OSOJWST055-01JWST055-SzaboFebruary16,20117:52PrinterName:YettoCome14INTRODUCTIONandcontrolofdiscretizationerrors,intermsofthedataofinterest,mustbeprovidedbythecomputercode.Theaimofexperimentsneedstoincludethedevelopmentofreliablestatisticalinformationonthebasisofwhichthefactorofsafetyisestablished.1.3ChaptersummaryTheprincipalaimofthisbookistopresentthetheoreticalandpracticalconsiderationsrelevantto(a)thevalidationofmathematicalmodelsand(b)verificationofthedataofinterestcomputedfromfiniteelementsolutions.Somefundamentalconceptsandbasicterminologywereintroduced,asfollows.MathematicalmodelAmathematicalrepresentationofaphysicalsystemorprocessintendedforpredictingsomesetofresponsesiscalledamathematicalmodel.Ittransformsonesetofdata,theinput,intoanotherset,theoutput.ConceptualizationConceptualizationisaprocessbywhichamathematicalmodelisdefined,foraparticu-larapplication.Conceptualizationinvolves(a)applicationofexpertknowledge,(b)virtualexperimentationand(c)calibration.DiscretizationDiscretizationisaprocessbywhichamathematicalproblemisformulatedthatcanbesolvedondigitalcomputers.Thesolutionapproximatestheexactsolutionofagivenmathemati-calmodel.ValidationValidationisaprocessbywhichthepredictivecapabilitiesofmathematicalmodelsaretested.Ideally,experimentsareconductedespeciallytotestwhetheramathematicalmodelmeetsnecessaryconditionsforacceptancefromtheperspectiveofitsintendeduse.Ifthepredictionsbasedonthemodelarenotsufficientlyclosetotheoutcomeofphysicalexperimentsthenthemodelisrejected.Thequalityofpredictionsisevaluatedonthebasisofoneormoremetricsandthecorrespondingcriteria,formulatedpriortotheexecutionofvalidationexperiments.Ofcourse,thecomparisonsmusttakeintoaccountuncertaintiesinthemodelparameters,theaccuracyofnumericalsolutionsanderrorsintheexperiments.Itisgenerallyfeasibletoperformfullscalevalidationexperimentsformass-produceditems.Forone-of-a-kindobjects,suchasdams,bridges,buildings,etc.,itisnotfeasibletoperformfullscalevalidationexperiments.Insuchcasesthemodelisanalyzedaposterioriinlightofnewinformationcollectedfollowinganincident. P1:OSOJWST055-01JWST055-SzaboFebruary16,20117:52PrinterName:YettoComeCHAPTERSUMMARY15VerificationVerificationisaprocessbywhichitisascertainedthatthedataofinterestcomputedfromtheapproximatesolutionmeetthenecessaryconditionsforacceptance.Verificationisunderstoodinrelationtotheexactsolutionofamathematicalmodel,notinrelationtothephysicalrealitythatthemathematicalmodelissupposedtorepresent.UncertaintiesItisimportanttodistinguishbetweenstatistical(aleatory)andcognitive(epistemic)uncer-tainties.Intheprocessofcalibrationthemathematicalmodelisassumedtobecorrectandtheparametersthatcharacterizethemodelareselectedsuchthatthemeasuredresponsematchesthepredictedresponse.Consequentlyaleatoryandepistemicuncertaintiesareunavoidablymixed.Epistemicuncertaintiescanbereducedwhenthepredictivecapabilitiesofalternativemodelsaresystematicallytestedandrevisedasnewexperimentalinformationbecomesavail-able.Thisinvolvesreinterpretationoftheavailableexperimentaldatawithreferencetotherevisedmodel. P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome2AnoutlineofthefiniteelementmethodInthischapteranoutlineofthefiniteelementmethod(FEM)ispresentedinaone-dimensionalsetting.Itwillbegeneralizedtotwoandthreedimensionsinsubsequentchapters.ThroughoutthebooktheunitsofphysicaldatawillbeidentifiedintermsofthestandardSI1notation.Anyconsistentsetofunitsmaybeused,however.2.1MathematicalmodelsinonedimensionTheformulationofmathematicalmodelswillbediscussedinChapter3.HereasimplemathematicalmodelthatwillserveasthebasisforthediscussionoftheconceptualandalgorithmicaspectsoftheFEMisformulated.2.1.1TheelasticbarTheelastostaticresponseofanelasticbodyischaracterizedbythedisplacementvectoru=ux(r)ex+uy(r)ey+uz(r)ez(2.1)whereristhepositionvectorr=xex+yey+zez(2.2)whereex,ey,ezareorthonormalbasisvectors.Thedisplacementuandthepositionvec-torraremeasuredinunitsofmeters(m).Wewillbeconcernedwithahighlysimplified1SystemeInternationald’Unit`es(InternationalSystemofUnits).´IntroductiontoFiniteElementAnalysis:Formulation,VerificationandValidation,FirstEdition.BarnaSzabóandIvoBabuška.©2011JohnWiley&Sons,Ltd.Published2011byJohnWiley&Sons,Ltd.ISBN:978-0-470-97728-6 P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome18ANOUTLINEOFTHEFINITEELEMENTMETHODmodelofathree-dimensionalelasticbody,theelasticbar.Fortheelasticbarweassumethatu=ux(xex)ex,thatis,onlyonedisplacementcomponentisnon-zeroandxexistheonlyin-dependentvariable.Inordertosimplifynotationwewillwriteu(x)insteadofu=ux(xex)ex.Weassumethatthecentroidalaxisofthebariscoincidentwiththex-axis.Wedenotethelengthofthebarby(m).Themathematicalmodelofanelasticbarisbasedonequationsthatrepresentthestrain–displacementrelationship,thestress–strainrelationshipandequilibrium:1.Thestrain–displacementrelationship.Thetotalstrainisdu≡=≡u.(2.3)xdxThetotalstrainisthesumofthemechanicalstrainmandthethermalstraint=αTwhereα=α(x)≥0isthecoefficientofthermalexpansion(1/Kunits).Thereforethemechanicalstrainis=−=u−αT.(2.4)mtNotethatstrainisdimensionless.2.Thestress–strainrelationship.InonedimensionHooke’slawstatesthatthestressisproportionaltothemechanicalstrain:σ≡σ=E=E(u−αT)(2.5)xmwhereE=E(x)>0isthemodulusofelasticity(MPaunits).Sincestrainisdimen-sionless,stressisalsoinMPaunits.Positivestressiscalledtensilestress,negativestressiscalledcompressivestress.3.Equilibrium.Itisassumedthatthestressisconstantoverthecross-sectionalarea.TheequilibriumequationsarewrittenintermsofthebarforceFbdefinedbyF:=σdydz=σA=AE(u−αT)(2.6)bAwhereA=A(x)>0(m2)istheareaofthecross-section.Thebarmaybesubjectedtodistributedforcesand/orvolumeforcesTbinN/munitsandtractionsexertedbyelasticsprings:Ts:=c(d−u)(2.7)whered=d(x)(m)isdisplacementimposedonthedistributedspringandc=c(x)≥0isthespringrateinN/m2units.WhentheforceTaccountsforvolumeforcesthenbitisunderstoodthatthevolumeforceshavebeenintegratedoverthecross-sectionalareaandthereforeTbisinN/munits.Distributedforcesarealsocalledtractionforcesortractionloads. P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeMATHEMATICALMODELSINONEDIMENSION19Figure2.1Barelement.ReferringtoFigure2.1,andconsideringtheequilibriumofanisolatedpartofthebaroflengthx,wewriteFb+Tbx+Tsx=0.NotethatFb,TbandTsarevectors.Fbispositiveinthedirectionofthepositive(outward)normaltothecross-section.Thisisconsistentwiththeaction–reactionprinciplewhichstatesthatifaforceactsuponabody,thenanequalandoppositeforcemustactuponanotherbody.ThebarelementinFigure2.1was“cut”fromthebarandthereforeequalandoppositeforcesactonthematingcross-sections.Thebarelementissaidtobeintension(resp.compression)whenFbispositive(resp.negative).AssumingthatFbisacontinuousanddifferentiablefunction,dFbFb=x+O(x).dxLettingx→0,wehavetheequilibriumequation:dFb+Tb+Ts=0.(2.8)dxOncombiningEquations(2.6),(2.7)and(2.8),theordinarydifferentialequationthatmodelsthemechanicalresponseofelasticbarstoappliedtractionforcesisobtained:ddud−AE+cu=Tb+cd−(AEαT)onx∈I(2.9)dxdxdxwhereIrepresentsthesetofpointsxthatlieintheinterval00(resp.k>0)isthespringconstant(inN/munits)atx=0(resp.x=)andd0(resp.d)isadisplacementimposedonthespringatx=0(resp.x=).Ofcourse,thedisplacement,forceandspringboundaryconditionsmayoccurinanycombination.Symmetry,antisymmetryandperiodicityUnderspecialconditionsthemathematicalproblemcanbeformulatedonasmallerdomainandextendedbysymmetry,antisymmetryorperiodicitytothefulldomain.Thelineofsymmetryisalinethatpassesthroughthemid-pointoftheintervalandisperpendiculartothex-axis.Bysymmetrywewillunderstandmirrorimagesymmetry.AscalarfunctiondefinedonIissaidtobesymmetricifinsymmetricallylocatedpointswithrespecttothelineofsymmetrythefunctionhasthesameabsolutevalueandthesamesign.Ascalarfunctionissaidtobeantisymmetricifinsymmetricallylocatedpointswithrespecttothelineofsymmetrythefunctionhasthesameabsolutevaluebutoppositesign.Vectorfunctionsaresymmetricifinsymmetricallylocatedpointstheabsolutevaluesareequalandthebasisvectorcomponentsnormaltothelineofsymmetryhaveoppositesense;thebasisvectorcomponentsparalleltothelineofsymmetryhavethesamesense.Forexample,assumingthatthey-axisiscoincidentwiththelineofsymmetry,inFigure2.3(a)visthesymmetricimageofu.Figure2.3Symmetryandantisymmetryofavectorfunctionintwodimensions.PointsAandAareequidistantfromthey-axis. P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome22ANOUTLINEOFTHEFINITEELEMENTMETHODFigure2.4Elasticbar:symmetric,antisymmetricandperiodicloadingandboundaryconditions.Vectorfunctionsareantisymmetricifinsymmetricallylocatedpointstheabsolutevaluesareequalandthebasisvectorcomponentsnormaltothelineofsymmetryexhavethesamesense;thebasisvectorcomponentsparalleltothelineofsymmetryhaveoppositesense.Forexample,assumingthatthey-axisiscoincidentwiththelineofsymmetry,inFigure2.3(b)vistheantisymmetricimageofu.InmanyinstancesthecoefficientsAE(x)andc(x)aresymmetricfunctionswithrespecttothelineofsymmetry.IfinsuchcasesTb(x),d(x),αT(x)andtheboundaryconditionsarealsosymmetric(resp.antisymmetric)thenthesolutionisasymmetric(resp.antisymmetric)functionwithrespecttothelineofsymmetry.Whenthesolutionisasymmetricorantisym-metricfunctionthentheproblemcanbesolvedonhalfoftheintervalandextendedtotheentireintervalbysymmetryorantisymmetry.Thedisplacement,forceandtractionarevectorfunctions.Inthecaseoftheelasticbarthesevectorfunctionshaveonlyonenon-zerocomponent,whichisperpendiculartothelineofsymmetry.ExamplesofsymmetricandantisymmetricloadingandconstraintsareshowninFigures2.4(a)and2.4(b).Inthesymmetriccasetheboundaryconditionisu(/2)=0.IntheantisymmetriccasetheboundaryconditionisFb(/2)=0.WhenAE(x),c(x),Tb(x),d(x)andαT(x)areperiodicfunctions,thelengthoftheperiodbeing,thatis,(AE)x=0=(AE)x=,c(0)=c(),Tb(0)=Tb(),d(0)=d(),αT(0)=αT(),u(0)=u()andF(0)=F(),thenthesolutionisaperiodicfunctionandtheboundaryconditionsaresaidtobeperiodic.Thesolutionobtainedfortheinterval(0,)isextendedto−∞0areconstantsandTb=0,d=0,T=0.Letu(0)=uˆ0,Fb()=k(d−u()).Writedownthesolutionforthisproblem. P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome24ANOUTLINEOFTHEFINITEELEMENTMETHOD2.1.2ConceptualizationWehaveformulatedmathematicalmodelssuitableforpredictingstaticresponsesofelasticbars.Wetacitlyassumedthatthephysicalpropertiesandboundaryconditionsweregiven.Inmanypracticalapplicationsnotalloftheneededinformationisavailable.Thereforeitisnecessarytoperformandinterpretcalibrationexperiments.Theprocedureisillustratedbythefollowingexample.Example2.1.3Oneofthemethodsusedforensuringthatthefoundationsofalargebuildingaresufficientlystifftoresistthedeadandliveloadswithoutundergoingexcessivesettlementistodrivelargeelasticbars,calledpiles,intothesoil.Supposethattwoexpertswereconsultedonthequestionofhowtoestimatethestiffnessofapileandbothexpertsagreedthatthemathematicalmodelshouldbebasedonthefollowingdifferentialequation:−AEu+cu=0,AEu(0)=F,u()=0(2.16)0wherecrepresentstheactionofthesoilonthepile.Thegoalistopredictthedisplacementu0atthetopofthepileasafunctionoftheappliedaxialforce.ThenotationisshowninFigure2.5.BothexpertsrecommendedusingthenominalvalueforthemodulusofelasticityofsteelE=200GPa;however,oneexpertrecommendedthatcshouldbetreatedasaconstantandtheotherexpertrecommendedthatc=kx,wherekisaconstant,shouldbeassumed.Inotherwords,differentmathematicalmodelswereproposedforthesameproblem.InthefollowingwerefertotheseasModelAandModelB,respectively.Inordertodeterminec,Figure2.5Example2.1.3:notation. P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeMATHEMATICALMODELSINONEDIMENSION25anHP305×110testpile2wasdrivenintothesoiltothedepthof12.0m.Thecross-sectionalareais1.402×10−2m2.Thepulltestyieldedthefollowingresults.WhentheappliedforceF0is200kNthenthemeasuredupwarddisplacementu0is9.0mm;atF0=300kN,u0=13.5mm;atF0=400kN,u0=18.0mm.Inotherwords,theexperimentalmeasurementsyieldedF0/u0=22.22kN/mm.TheobservationthattheratioF0/u0isconstantisconsistentwiththeassumption,madebybothexperts,thatthelineardifferentialequation(2.16)isareasonablemathematicalmodelofthepiledrivenintothesoil.IftheratioF0/u0changedsubstantiallywiththeappliedloadthenthismodelwouldhavetoberejectedonthebasisoftheoutcomeofcalibrationexperiments.CalibrationofModelAEquation(2.16)canberewrittenasc−u+λ2u=0,u(0)=F/AE,u()=0whereλ:=(2.17)0AEthesolutionofwhichisF0coshλu(x)=sinhλx−coshλx.λAEsinhλNotethatwiththesignconventionadoptedinSection3.4.4andillustratedinFigure2.2,theupwarddisplacementisnegative,thatis,u(0)=−u0.Thereforetheforce–displacementrelationshipisλsinhλF=AEu(0)=AEu(2.18)00coshλwhichcanbewrittenasF0λsinhλG(λ):=−AE=0.(2.19)u0coshλForthethreedatapairsF0/u0=22.22kN/mmwasmeasured.WeneedtofindλsuchthatG(λ)=0.Variousrootfindingmethodsareavailable.OneofthemostcommonlyusedmethodsistheNewton–Raphsonmethod.3Inthismethodweselectatrialvalueforλ,denotedbyλ,andcomputethecorrespondingG(λ)andG(λ):=(dG/dλ).Thechoiceofλ111λ=λ11mustbesuchthat(dG/dλ)λ=λ1=0.Wethencomputeλk+1fromG(λk)λk+1=λk−fork=2,3,...G(λk)2Thisdesignationindicatesthatthecross-sectionisH-shaped,thenominaldepthofthecross-sectionis305mmandthemassisapproximately110kg/m.3SirIsaacNewton(1642–1727),JosephRaphson(1642–1727). P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome26ANOUTLINEOFTHEFINITEELEMENTMETHODTable2.1Thederivativesofu(x).nDnu(x)Dnu(0)0u−u01DuF0/AE2λ2xu03λ2u+λ2xDu−λ2u042λ2Du+λ2xD2u2λ2F/AE053λ2D2u+λ2xD3u064λ2D3u+λ2xD4u−4λ4u075λ2D4u+λ2xD5u10λ4F/AE0andcontinuetheprocessuntilλk+1−λkissufficientlysmall.Bythismethodwefindλ=2.6112×10−2m−1andfromthedefinitionofλgiveninEquation(2.17)wehavec=1912kN/m2.ThiscompletesthecalibrationstepforModelA.CalibrationofModelBCalibrationofModelBinvolvessolutionoftheproblem−AEu+kxu=0,AEu(0)=F,u()=0(2.20)0whichwillbewrittenask−u+λ2xu=0,u(0)=F/AE,u()=0whereλ:=.(2.21)0AEThisisknownastheAiryequation,4seeforexample[64].WewilluseaTaylorseries5expansiontofindanapproximatesolution.WedenotethenthderivativeofubyDnu.Thederivativesforn=0,1,...,7areshowninTable2.1.Weseethatfork≥3Dku=(k−2)λ2Dk−3u+λ2xDk−2u.TheTaylorseriesexpansionofu(x)isFλ22λ2F4λ410λ4Fu(x)=−u+0x−ux3+0x4−ux6+0x7−···.000AE3!4!AE6!7!AELettingu()=0wegetFλ2λ2Fλ4λ4F0=0−u2+03−u5+06−···.(2.22)00AE23AE3072AE4SirGeorgeBiddellAiry(1801–1892).5SirBrookTaylor(1685–1731). P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeMATHEMATICALMODELSINONEDIMENSION27ThereforeweneedtofindλsuchthatF1λ23λ46λ22λ450G(λ)≈++−−=0.(2.23)u0AE3AE72AE230UsingtheexperimentalresultF/u=22.22kN/mm,wefindλ≈1.0767×10−2m−3/2and00hencek≈325.1kN/m3.Inthisexampletheconceptualdevelopmentofamathematicalmodelwasillustratedinasimplesetting.ModelAandModelBdifferbythedefinitionoftheconstantc.Thecharacterizingparameterscandkweredeterminedbycalibration.Calibrationispartoftheconceptualizationprocessbecausedefinitionofthemathematicalmodeldependsoninformationobtainedbycalibrationexperiments.Exercise2.1.3Determinewhetherusing4significantfiguresintheestimatek≈325.1kN/m3inExample2.1.3isjustified.2.1.3ValidationMakingapredictionbasedonamathematicalmodelconcerningtheoutcomeofaphysicalexperiment,thentestingtoseewhetherthepredictioniscorrect,iscalledvalidation.Validationinvolvesoneormoremetricsandthecorrespondingcriteria.Themetricsandcriteriadependontheintendeduseofthemodel.FortestingthemodeldescribedinExample2.1.3wedefinethemetrictobetheratioF0/u0andthecriterionisthecorrespondingtolerance.Validationisillustratedbythefollowingexample.Example2.1.4OnexaminingthepulltestdatainExample2.1.3,weseethateach100kNincrementintheappliedforceresultedina4.5mmincrementindisplacement.Thereforetheassumptionthatthepileissupportedbyadistributedlinearspringisconsistentwiththeavailableobservations.However,itisnotpossibletodeterminefromtheseobservationshowthespringcoefficientcvarieswithx.Letusassumethatasecondpile,drivento8.5mdepth(i.e.,=8.5inFigure2.5),istobetested.BasedonModelAandModelBwepredictthetestresultsshowninthesecondandthirdcolumnsofTable2.2andwestateourcriterionasfollows:amodelwillberejectedifthedifferencebetweenthepredictedandobservedvaluesofF0/u0exceedsthetoleranceof5%.Table2.2Predictedandobserveddata.F0Appliedu0ModelAu0ModelBu0Experiment(kN)(mm)(mm)(mm)20012.517.416.530018.826.025.140025.034.733.7 P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome28ANOUTLINEOFTHEFINITEELEMENTMETHODLetussupposethatweobservethesetofdisplacementsshowninthefourthcolumnofTable2.2.SincetheratiopredictedbyModelA,(F/u)A=16.0kN/mm,andtheobserved00predratio,(F0/u0)obs=11.9kN/mm,differbymorethan5%,ModelAisrejected.Ontheotherhand,theratiopredictedbyModelB,(F/u)B=11.5kN/mm,andtheobservedratiodiffer00predbylessthan5%.ThereforeModelBpassesthevalidationtest.Remark2.1.4Example2.1.4illustratessomeofthedifficultiesassociatedwithvalidationofmathematicalmodels.Typicallyonlyaverylimitednumberofexperimentalobservationsareavailable.Theinformationbeingsought,inthiscasec(x),isnotobservabledirectlybutmustbeinferredfromsomeobservableinformation.Ifforce–displacementdatawereavailableforonedepthonlythenitwouldnotbepossibletodecidewhethercisconstantornot.Basedontwopiletestsofdifferinglengths,itwaspossibletorejectthehypothesisthatcisaconstantandestablishthattheavailableinformationisconsistentwithlinearvariationoftheformc=kx,butitwasnotpossibletoestablishwithcertaintythatcvarieslinearly.Theprobabilitythatamodeladequatelyrepresentsphysicalrealityincreaseswiththenumberofsuccessfulpredictionsoftheoutcomesofindependentexperiments,buttheinherentepistemicuncertaintycannotberemovedcompletelybyanynumberofexperiments[38a],[53].Infact,itispossibletoconstructseveralmodelsthatmatchagivensetofobservations.Inengineeringandscientificapplicationsthesimplestmodelispreferred.Remark2.1.5InordertofocusonthemainpointsofcalibrationinExample2.1.3andvalidationinExample2.1.4,theinputdataandphysicalobservationsweretreatedwithoutconsiderationoftheirstatisticalaspects.Sincethereareuncertaintiesinmodelparameters,comparingpredictionswiththeoutcomeofexperimentsshouldbeunderstoodinastati-sticalsense.Letusassumethat,havingconsideredstatisticaluncertaintiesintheinputdata,wepredictalog-normalprobabilitydensityfunction(pdf)forthematerialconstantkinExample2.1.4.Letusassumefurtherthatthecriterionforrejectionwassetatthe95%confidenceinterval.Wemakeanexperimentalobservationandcomputekexp.Letusassumethatkexpfallswithinthe95%confidenceinterval.Thisshowsthattheoutcomeoftheexperimentisconsistentwiththepredictionbasedonthemodelatthe95%confidencelevel.Thisshouldnotbeinterpretedtomeanthatweare95%confidentthatthemodelisvalid.Whatthismeansisthatthechancethatavalidmodelwouldberejectedis5%.Thechanceofrejectingavalidmodelwouldbereducedbysettingtheconfidenceintervalat(say)99%;however,thechanceofnotrejectinganinvalidmodelwouldbeincreased.Exercise2.1.4UsingthecalibrationresultsforModelBinExample2.1.3,predicttheF0/u0ratioforapiledriventoadepthof17.5m.2.1.4ThescalarellipticboundaryvalueprobleminonedimensionEquation(2.9)isasecond-orderellipticordinarydifferentialequation(ODE).InChapter3itwillbeshownthatthemathematicalmodelofsteadystateheatconductioninabarwillresultinasecond-orderellipticODEalso.Althoughthephysicalmeaningsoftheunknownfunctionsandthecoefficientsdiffer,themathematicalproblemisessentiallythesame.For P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeAPPROXIMATESOLUTION29thisreasonwewillfocusonthemathematicalproblem−κu+cu=f(x)on0≤x≤(2.24)whereκ(x)≥κ0>0,c(x)≥0andf(x)areboundedfunctionssubjecttotherestrictionthattheindicatedoperationsaredefined.TheboundaryconditionsareanalogoustothosedescribedinSection2.1.1,butinthemathematicalliteraturetheyareknownbydifferentnames.Thedisplacementboundarycon-ditioniscalledtheessentialorDirichletboundarycondition.6TheforceboundaryconditioniscalledtheNeumannboundarycondition.7ThespringboundaryconditioniscalledthemixedorRobinboundarycondition.8TheNeumannandRobinboundaryconditionsarealsocallednaturalboundaryconditions.AlthoughEquation(2.24)maybeunderstoodtorepresentanelasticbar,whereuisthedisplacementvector,orheatconductioninabar,whereuisthetemperature,ascalarfunction,symmetryandantisymmetryaretreateddifferently:whenuisascalarfunctionthenthesymmetryboundaryconditionisu(/2)=0andtheantisymmetryconditionisu(/2)=0.SeeRemark2.1.2.2.2ApproximatesolutionAbriefintroductiontoapproximationbasedonminimizingtheerrorofanintegralexpressionispresentedinthefollowing.ConsidertheproblemgivenbyEquation(2.24)withtheboundaryconditionsu(0)=u()=0andletusseektoapproximateubyun,definedasfollows:nu:=aϕ(x)ϕ(x):=xj(−x)(2.25)njjjj=1suchthattheintegral1I:=κ(u−u)2+c(u−u)2dx(2.26)nn20isminimum.Itwillbeshowninthefollowingthatminimizationoftheerrorinthesenseofthisintegralwillallowustofindanapproximationtotheexactsolutionuwithoutknowingu.Thefunctionuniscalledatrialfunction.Thefunctionsϕj(x)arecalledbasisfunctions.Selectionofthetypeandnumberofbasisfunctionswill,ofcourse,influencetheerrorofapproximationu−un.Discussionofthispointispostponedinordertokeepourfocusonthebasicalgorithmicstructureofthemethod.6JohannPeterGustavLejeuneDirichlet(1805–1859).7FranzErnstNeumann(1798–1895).8VictorGustaveRobin(1855–1897). P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome30ANOUTLINEOFTHEFINITEELEMENTMETHODNotethatϕj(0)=ϕj()=0,henceunsatisfiestheprescribedboundaryconditionsforanychoiceofthecoefficientsai.Fromtheminimumconditionwehave∂I=0:κ(u−u)ϕ+c(u−u)ϕdx=0,i=1,2,...,n.(2.27)nini∂ai0Recallingtheproductrule,wewriteκuϕi=κuϕi−(κu)ϕiandsubstitutethisexpressionintoEquation(2.27)toobtainκuϕ−κuϕ+−(κu)+cuϕdx−κuϕ+cuϕdx=0ix=ix=0inini000f(x)wherethefirsttwotermsarezeroonaccountoftheboundaryconditions.Thisequationcanbewrittenasκuϕ+cuϕdx=f(x)ϕdx,i=1,2,...,n.(2.28)ninii00ObservethatEquation(2.28)representsnalgebraicequationsinthenunknownsai.Thereforeweareabletocomputeanapproximationtou(x)withoutknowingu(x),sinceonlythegivenfunctionf(x)isneeded.Specifically,Equation(2.28)isequivalentto[K]{a}={r}(2.29)where{a}:={aa...a}Tandtheelementsof[K]and{r}are,respectively,12nk:=κ(x)ϕ(x)ϕ(x)+c(x)ϕ(x)ϕ(x)dx(2.30)ijijij0ri:=f(x)ϕi(x)dx.(2.31)0Example2.2.1ConsidertheproblemonI=(0,),−u+u=xu(0)=u()=0,andassumethatthegoalistodetermineu(0).Let=1.Theexactsolutionofthisproblemis2e2eu=−sinhx+xandthereforeu(0)=1−≈0.14908e2−1e2−1 P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeAPPROXIMATESOLUTION31whereeisthebaseofthenaturallogarithm.Wewillseektoapproximateuusingthebasisfunctionsϕj(x)giveninEquation(2.25)withn=2.Therefore1111k=[(ϕ)2+ϕ2]dx=(1−2x)2+x2(1−x)2dx=111100301111k=k=[ϕϕ+ϕϕ]dx=(1−2x)(2x−3x2)+x3(1−x)2dx=122112120060111k=[(ϕ)2+ϕ2]dx=(2x−3x2)2+x4(1−x)2dx=2222007and111r=xϕdx=x2(1−x)dx=110012111r=xϕdx=x3(1−x)dx=.220020Theproblemisthentosolvethesystemoflinearequations:11/3011/60a11/12=.11/601/7a21/20Onsolvingweobtaina1=0.14588,anda2=0.16279,thereforetheapproximatesolutionisu=u=0.14588x(1−x)+0.16279x2(1−x)n2andhenceu(0)=0.14588andtherelativeerrorisn|u(0)−u(0)|n=0.021(2.1%).|u(0)|Inthisexampletheexactsolutionwasknownandhencetherelativeerrorinthedataofinterestcouldbecomputed.Ingeneraltheexactsolutionisnotknown,thereforetherelativeerrorinthedataofinteresthastobeestimated.Methodsoferrorestimationwillbediscussedinsubsequentchapters.Exercise2.2.1Determinetherelativeerrorof(u)fortheproblemsolvedinExamplenx=2.2.1.Remark2.2.1Inengineeringcomputationsthegoalistodeterminesomedataofinterest.Thedataofinterestaretypicallynumbersorfunctionsthatdependonthesolutionu(x)and/oritsfirstderivative.Forexample,ifEquation(2.24)isunderstoodtorepresentanelasticbarthenwemaybeinterestedincomputingthereactionforceatx=0,definedbyF=(κu).If,0x=0 P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome32ANOUTLINEOFTHEFINITEELEMENTMETHODontheotherhand,Equation(2.24)isunderstoodtorepresentheatconductionthenwemaybeinterestedintherateofheatflowexitingthebaratx=0,whichisdefinedbyq=−(κu).0x=0Wewillbeinterestedinfindinganapproximatesolution,computingthedataofinterestfromtheapproximatesolution,asillustratedbyExample2.2.1,andestimatingtherelativeerrorinthedataofinterest.Remark2.2.2ObservethatEquation(2.29)canbeobtainedbyminimizingthequadraticexpression1π(u):=κ(u)2+c(u)2dx−fudxnnnn2001={a}T[K]{a}−{a}T{r}(2.32)2withrespecttoai.Thereforethefunctionthatminimizesπ(un)isalsoclosesttotheexactsolutionuinthesensethattheerrordefinedbytheintegralexpressionofEquation(2.26)isminimized.ThismethodisknownastheRayleigh–Ritzmethod9orsimplyastheRitzmethod.Thefunctionalπ(u)iscalledpotentialenergy.Exercise2.2.2Computethecoefficientsa1anda2ofExample2.2.1byminimizingπ(un)withrespecttoa1anda2.2.2.1BasisfunctionsWedefinedthepolynomialbasisfunctionsϕ(x):=xj(−x),j=1,2,...,n,inEqua-jtion(2.25)andsoughttominimizeEquation(2.26)withrespecttothecoefficientsajofthesebasisfunctions.Thisledtothedefinitionofnalgebraicequationsinnunknowns,repre-sentedbyEquation(2.29).ThesolutionofEquation(2.29)isunique,providedthat[K]isanon-singularmatrix.Inordertoensurethat[K]isnon-singular,thebasisfunctionsmustbelinearlyindependent.Bydefinition,asetoffunctionsϕj(x)(j=1,2,...,n)islinearlyindependentifnajϕj(x)=0j=1impliesthataj=0forj=1,2,...,n.Itislefttothereadertoshowthatϕj(x)(j=1,2,...,n)arelinearlyindependent.Givenasetoflinearlyindependentfunctionsϕj(x)(j=1,2,...,n),thesetoffunctionsSdefinedby⎧⎫⎨n⎬S:=un|un=ajϕj(x)⎩⎭j=1iscalledthespanandϕj(x)arebasisfunctionsofS.9LordRayleigh(JohnWilliamStrutt;1842–1919),WalterRitz(1878–1909). P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeGENERALIZEDFORMULATIONINONEDIMENSION33Wecouldhavedefinedotherpolynomialbasisfunctions,forexample,nu:=cψ(x)ψ(x):=x(−x)i.(2.33)niiii=1Whenonesetofbasisfunctions{ϕ}:={ϕϕ...ϕ}Tcanbewrittenintermsofanother12nset{ψ}:={ψψ...ψ}Tintheform12n{ψ}=[B]{ϕ}(2.34)where[B]isaninvertiblematrixofconstantcoefficients,thenbothsetsofbasisfunctionsaresaidtohavethesamespan.Thefollowingexercisedemonstratesthattheapproximatesolutiondependsonthespanofthebasisfunctions,notonthebasisfunctions.Exercise2.2.3SolvetheproblemofExample2.2.1usingthebasisfunctionsψ1(x)=x(1−x),ψ2(x)=x(1−x)(1−2x)andshowthatthesolutionu2=b1ψ1(x)+b2ψ2(x)isthesameasthesolutioninExample2.2.1.Inthisexercisethespanisthesetofpolynomialsofdegree3,subjecttotherestrictionthattheyvanishattheboundarypoints.Exercise2.2.4Letϕ(x)=xi(−x),ψ(x)=x(−x)iandii33un=aiϕi(x)=ciψi(x).i=1i=1Determinethematrix[B]asdefinedinEquation(2.34)and,assumingthatthevaluesofaiaregiven,findanexpressionforciintermsofai(i=1,2,3)and[B].2.3GeneralizedformulationinonedimensionWehaveseeninSection2.2thatitwaspossibletoobtainanapproximatesolutiontoadifferentialequationwithoutknowingtheexactsolution.ThisdependedonaseeminglyfortuitouschoiceoftheintegralexpressionIandzeroboundaryconditions,allowingustoreplacetheunknownexactsolutionwiththeknownfunctionffollowingintegrationbyparts.InthissectionthereasonsforthechoiceofIareexplainedinageneralsetting,withoutrestrictionontheboundaryconditions.OnceagainourstartingpointisEquation(2.24)−κu+cu=f(x) P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome34ANOUTLINEOFTHEFINITEELEMENTMETHODsubjecttoboundaryconditionstobediscussedlater.Letusmultiplythisequationbyanarbitraryfunctionv(x)definedonI=(0,)andintegrate:−κu+cuvdx=fvdx.(2.35)00Clearly,ifuisthesolutionofEquation(2.24)thenthisequationwillbesatisfiedforallvforwhichtheindicatedoperationsaredefined.Integratingthefirsttermbyparts,−κuvdx=−κuv−κuvdx00=−κuv+κuv+κuvdxx=x=00wehaveκuv+cuvdx=fvdx+κuv−κuv.(2.36)x=x=000Notethattheintegrand(κu)vbecameκuvplustwoboundaryterms.Thisequationwillbethestartingpointforourdiscussionofthegeneralizedformulation.Thespecificstatementofthegeneralizedformulationforaparticularproblemdependsontheboundaryconditions.Someusefuldefinitionsandnotationareintroducedinthefollowing.Definition2.3.1Thefunctionu(resp.v)inEquation(2.36)iscalledthetrial(resp.test)function.Definition2.3.2Inonedimensionthestrainenergyisdefinedby1U(u)=κ(u)2+cu2dx(2.37)20whereκ≥κ0>0andc≥0.ThereforeU(u)≥0.Whenc=0thenU(u)=0onlywhenu=0intheintervalI=(0,).Whenc=0thenU(u)=0whenu(x)=CwhereCisanarbitraryconstant.Definition2.3.3Theenergyspace,denotedbyE(I),isthesetoffunctionsdefinedontheintervalIthatsatisfytheinequalityE(I):=uU(u)≤C<∞(2.38)whereCissomepositivenumber.Foranyu∈E(I)andv∈E(I)theintegralexpressionsinEquation(2.36)aredefined.ThisfollowsfromtheSchwarzinequality,10seeAppendixA.10HermannAmandusSchwarz(1843–1921). P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeGENERALIZEDFORMULATIONINONEDIMENSION35Definition2.3.4Whenu(0)=uˆ0and/oru()=uˆarespecifiedontheboundariesthentheboundaryconditioniscalledanessentialorDirichletboundarycondition.ThefunctionsinE(I)thatsatisfytheessentialboundaryconditionsarecalledadmissiblefunctions.ThesetofalladmissiblefunctionsiscalledthetrialspaceandisdenotedbyE˜(I).Thisnotationshouldbeunderstoodasfollows:(a)Ifessentialboundaryconditionsarespecifiedatx=0andx=thenE˜(I):={u|u∈E(I),u(0)=uˆ0,u()=uˆ}.(2.39)CorrespondingtoE˜(I)isthetestspaceE0(I)definedasfollows:E0(I):={u|u∈E(I),u(0)=0,u()=0}.(2.40)(b)Ifanessentialboundaryconditionisspecifiedonlyatx=0thenE˜(I):={u|u∈E(I),u(0)=uˆ0}(2.41)E0(I):={u|u∈E(I),u(0)=0}.(2.42)(c)Ifanessentialboundaryconditionisspecifiedonlyatx=thenE˜(I):={u|u∈E(I),u()=uˆ}(2.43)E0(I):={u|u∈E(I),u()=0}.(2.44)(d)Iftheessentialboundaryconditionsarehomogeneous,thatis,uˆ0=0,uˆ=0,thenE˜(I)=E0(I).(e)IfessentialboundaryconditionsarenotprescribedoneitherboundarythenE˜(I)=E0(I)=E(I).(f)IfperiodicboundaryconditionsareprescribedthenboththetrialandtestspacesareEˆ(I)={u|u∈E(I),u(0)=u()}.(2.45)Remark2.3.1NotethatE˜(I)isnotalinearspace.RefertoAppendixA,SectionA.2.ItisseenthatE˜(I)doesnotsatisfycondition1whereasE0(I)andEˆ(I)satisfyalloftheconditionsofSectionA.2.2.3.1EssentialboundaryconditionsEssentialboundaryconditionsareenforcedbyrestriction.ThiswasdoneinthespecialcasediscussedinSection2.2wherethehomogeneousessentialboundaryconditionsuˆ0=uˆ=0wereusedandthebasisfunctionsweredefinedinEquation(2.25)sothattheboundaryconditionsprescribedonuweresatisfiedforanarbitrarychoiceofthecoefficientsai.Theknownboundaryconditionsareimposedonthetrialfunctionsuandthetestfunctionvissettozeroontheboundarypointswhereessentialboundaryconditionswereprescribed.Inthiswaytheboundaryterms(thetermsinthesquarebracketsinEquation(2.36))vanish P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome36ANOUTLINEOFTHEFINITEELEMENTMETHODandthegeneralizedformulationisstatedasfollows:“Findu∈E˜(I)suchthatκuv+cuvdx=fvdxforallv∈E0(I).”(2.46)00B(u,v)F(v)WewillusetheshorthandnotationB(u,v)forthelefthandsideandF(v)fortherighthandside,asindicatedinEquation(2.46).B(u,v)isasymmetricbilinearform,thatis,itislinearwithrespecttoeachofitsargumentsandB(u,v)=B(v,u),andF(v)isalinearfunctional.ThepropertiesofbilinearformsandlinearfunctionalsaregiveninAppendixA.AlternativelywecanselectanarbitraryfunctionufromE˜(I)andwriteu=u¯+u(2.47)whereu¯∈E0(I).Clearly,theprescribedboundaryconditionsaresatisfiedforanychoiceu¯∈E0(I).SubstitutingEquation(2.47)intoEquation(2.36),thegeneralizedformulationcanbestatedasfollows:“Findu¯∈E0(I)suchthatκu¯v+cuv¯dx=fvdx−κ(u)v+cuvdx(2.48)000B(u¯,v)F(v)forallv∈E0(I).”Example2.3.1Letusstatethegeneralizedformulationforthefollowingproblem:−u=(2+x)exu(0)=1,u(2)=−1.InthiscaseE˜(I)={u|u∈E(I),u(0)=1,u(2)=−1}.Letusselectu=1−xandsubstituteu=u¯+uintoEquation(2.36).Thestatementofthegeneralizedformulationisnow:“Findu¯∈E0(I)suchthatB(u¯,v)=F(v)forallv∈E0(I)”where222B(u¯,v):=u¯vdx,F(v):=(2+x)exvdx−(−1)vdx.000Example2.3.2InthisexampleitisshownthatEquation(2.36)leadstothesameequationsasthoseobtainedinSection2.2.ToobtainanapproximationtothesolutionofEquation(2.24),wesubstituteunfromEquation(2.25)foruandsimilarlysubstitutevnforv:nv:=bϕ(x),ϕ(x):=xi(−x)niiii=1wherebi,i=1,2,...,n,areasetofarbitrarynumbers.Sincevn(0)=vn()=0,thetermsinthesquarebracketsinEquation(2.36)vanishandwehave{b}T[K]{a}={b}T{r} P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeGENERALIZEDFORMULATIONINONEDIMENSION37where{b}:={bb...b}Tandthedefinitionsof[K]and{r}arethesameasinEquations12n(2.30)and(2.31).Equivalently,{b}T([K]{a}−{r})=0.Sincethisrelationshipmustholdforanychoiceof{b},wemusthave[K]{a}={r},whichisexactlythesameastheresultobtainedinSection2.2withkij(resp.ri)definedbyEqua-tion(2.30)(resp.Equation(2.31)).Exercise2.3.1Showthat(a)B(u1+u2,v)=B(u1,v)+B(u2,v)(b)B(u+v,u+v)=B(u,u)+2B(u,v)+B(v,v).2.3.2NeumannboundaryconditionsWhenuormorecommonlyF=κuisprescribedonaboundarythentheboundaryconditioniscalledaNeumannboundarycondition.ThetreatmentofNeumannboundaryconditionsisstraightforward.LetF=(κu)andF=(κu)begiven.SubstitutingFandFinto0x=0x=0Equation(2.36),thegeneralizedformulationisstatedasfollows:“Findu∈E(I)suchthatB(u,v)=F(v)forallv∈E(I)”whereB(u,v)=κuv+cuvdx,F(v)=fvdx+Fv()−Fv(0).(2.49)000Notethattherearenorestrictionsonuorvattheboundarypoints.Remark2.3.2Whenc=0andNeumannboundaryconditionsareprescribedthen,sinceEquation(2.49)mustholdforallchoicesofv∈E(I),itmustholdforv=CwhereCisaconstant.Thereforewemusthavefdx+F−F0=0.(2.50)0Thismeansthatf,F0andFcannotbeassignedarbitrarily.Thetractionsactingonthebarandthebarforcesactingontheboundarypointsmustbeinequilibrium.2.3.3RobinboundaryconditionsAlinearcombinationofuanduisgivenattheboundary:(κu)=β(u(0)−U)x=000(κu)=β(U−u())x=whereβ0>0,β>0,U0,Uareinputdataanalogoustothespringratesk0,kandspringdisplacementsd0,ddefinedinSection2.1.1.SubstitutingtheseexpressionsintoEqua-tion(2.36),thegeneralizedformulationisonceagainstatedasfollows:“Findu∈E(I)such P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome38ANOUTLINEOFTHEFINITEELEMENTMETHODthatB(u,v)=F(v)forallv∈E(I)”whereB(u,v):=κuv+cuvdx+βu(0)v(0)+βu()v()00F(v):=fvdx+β0U0v(0)+βUv().0Therearenorestrictionsonuorvattheboundarypoints.ThespringboundaryconditiondescribedinSection2.1.1isaRobinboundarycondition.Example2.3.3AnycombinationofDirichlet,NeumannandRobinboundaryconditionsmaybeprescribed.Forexample,letusconsidertheproblem−κu+cu=f(x)u(0)=uˆ0,(κu)x==β(U−u()).InthiscaseE˜(I)isdefinedbyEquation(2.41),E0(I)isdefinedbyEquation(2.42)andB(u¯,v):=κu¯v+cuv¯dx+βu()v()0F(v):=fvdx+βUv()−κ(u)v+cuvdx00whereuisanarbitraryfixedfunctionfromE˜(I).Forexample,wemayselectu=uˆ(1−0x/)orsimplyu=uˆ.0Thegeneralizedformulationofthisproblemisstatedasfollows:“Findu¯∈E0(I)suchthatB(u¯,v)=F(v)forallv∈E0(I).”Theexactsolutionisthenu=u¯+u.Exercise2.3.2Statethegeneralizedformulationforthefollowingproblem:−κu+cu=f(x)(κu)x=0=−qˆ0,u()=0.Exercise2.3.3Statethegeneralizedformulationforthefollowingproblem:−κu+cu=f(x)(κu)x=0=β0(u(0)−U0),u()=uˆ.Exercise2.3.4Statethegeneralizedformulationforthefollowingproblem:−κu+cu=f(x)(κu)x=0=−qˆ0,(κu)x==β(U−u()).2.4FiniteelementapproximationsWehaverecastadifferentialequationintheformofageneralizedformulationwhichreads:“Findu∈E˜(I)suchthatB(u,v)=F(v)forallv∈E0(I).”Itmayappearthatnothinghasbeengained.Thisproblemismoredifficulttosolvethanthedifferentialequationwas,since P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeFINITEELEMENTAPPROXIMATIONS39thereareaninfinitenumberoftrialfunctionsforuthatmustbetestedagainstaninfinitenumberoftestfunctionsv.Oneofthemainadvantagesofthegeneralizedformulationisthatitservesasaframeworkforobtainingapproximatesolutions.Toobtainanapproximatesolutionweconstructafinite-dimensionalsubspaceofE(I),aswehavedoneinSection2.2,whereweselectednun=ajϕj(x)j=1withn=2.ThefamilyoffunctionsthatcanbewritteninthiswaywillbedenotedbyS(I).Thefunctionsϕj(x),calledbasisfunctions,willbedefinedsuchthatS(I)⊂E(I).ThenumbernisthedimensionofS(I).WewillusethenotationS˜(I):=S(I)∩E˜(I);S0(I)=S(I)∩E0(I).ThedimensionofspaceS0(I),thatis,thenumberoflinearlyindependentfunctionsinS0(I),iscalledthenumberofdegreesoffreedomanddenotedbyN.IntheFEMthespaceSisconstructedbypartitioningthesolutiondomainintoelementsanddefiningpolynomialbasisfunctionsontheelements.Approximationspacessoconstructedarecalledfiniteelementspaces.Aparticularpartitioniscalledafiniteelementmeshandwillbedenotedby.ThenumberofelementswillbedenotedbyM().Asimpleillustrationisgiveninthefollowingexample.Example2.4.1TypicalfiniteelementbasisfunctionsinonedimensionareillustratedinFigure2.6wherethedomainI=(0,)ispartitionedintothreeintervals(i.e.,M()=3),calledfiniteelements,andthepolynomialdegreesp1=2,p2=1andp3=3areassigned.Thelengthoftheelementsisdenotedbyk,k=1,2,3.Therearefournodepoints,labeledxi,i=1,2,3,4.Therearesevenbasisfunctions,labeledϕ1(x),...,ϕ7(x).Thenumberingofthebasisfunctionsisarbitrary;however,itisgoodpracticetonumberthembypolynomialdegree.Thefirstfourbasisfunctionsarepiecewiselinear.Forexample,⎧x−x1⎪⎪⎪⎪ifx1≤x≤x2⎪⎪x2−x1⎨ϕ2(x)=x3−xifx0EByEquation(2.57),B(e,v)=0,thereforeforanyv=0wehavee+v2>e2EEEwhichwastobeproven.ThistheoremshowsthattheselectionS(I)isofcrucialimportance,sincetheerrorofapproximationisdeterminedbyS(I).ThistheoremalsoshowsthatifuEXhappenstolieinS(I)thenuFE=uEX.Furthermore,thetheoremshowsthatifweconstructasequenceoffiniteelementspacesS1⊂S2⊂···⊂Smandcomputethecorrespondingfiniteelement(1)(2)(m)solutionsu,u,...,u,thentheerrormeasuredintheenergynormwilldecreaseFEFEFEmonotonicallywithrespecttoincreasingm.2.5FEMinonedimensionInthissectionthekeyalgorithmicprocedurescommontoallfiniteelementcomputerprogramsareoutlinedinthesimplestsetting.Althoughthediscussioncoverstheone-dimensionalcaseonly,analogousproceduresapplytotwo-andthree-dimensionalproblems.2.5.1ThestandardelementInordermakecomputationofthecoefficientmatricesandloadvectorssuitableforim-plementationinacomputerprogram,thecomputationsareperformedelementbyelement.Inonedimensionthekthelementischaracterizedbythenodepointsxkandxk+1ofthemesh.ThemeshisthesetofelementsIk:={x|xk1,k=1,2,...,M()+1.M()ItwillbedemonstratedinChapter6thatforalargeandimportantclassofproblems,theidealmeshesaregeometricmesheswhenthemeshisfixedandthepolynomialdegreeofelementsisincreased,andtheidealmeshesareradicalmesheswhenthepolynomialdegreeisfixedandthenumberofelementsisincreased[70].Exercise2.5.4Considerthefamilyofmappingfunctions11x=Q(α,ξ):=ξ(ξ−1)x+(1−ξ2)(αx+(1−α)x)+ξ(1+ξ)xkkkk+1k+122 P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeFEMINONEDIMENSION49where1/4<α<3/4.Showthatbylettingα=1/2theshapefunctionN2ismappedinto(x−xk)/kandallmappedshapefunctionsarepolynomials.Showalsothatbyletting√α=3/4theshapefunctionN2ismappedinto(x−xk)/k.Inonedimensionthismappingisnotadmissiblebecausethemappedshapefunctiondoesnotlieintheenergyspace.Similarmappings,however,areadmissibleintwoandthreedimensionswhereelementsmappedbyanalogousfunctionsarecalledquarter-pointelements.Thisexerciseillustratesthatwhenthemappingislinear(α=1/2)themappedshapefunctionsarepolynomials.2.5.4ComputationofthecoefficientmatricesIntheFEMthecoefficientmatricesarecomputedelementbyelement.Thesecomputationsproduceelement-levelmatricesthatare“assembled”inaseparatestep.Theprocedureisoutlinedinthefollowing.ComputationofthestiffnessmatrixThefirsttermofthebilinearformB(un,vn)iscomputedasasumofintegralsovertheelementsM()xk+1κ(x)uvdx=κ(x)uvdxnnnn0k=1xkwherenisthenumberofbasisfunctionsthatspanthefiniteelementspace,seeSection2.4.Wewillbeconcernedwiththeevaluationoftheintegral⎛⎞)*xk+1xk+1pk+1dNpk+1dNκ(x)uvdx=κ(x)⎝aj⎠bidx.nnjixkxkj=1dxi=1dxTheshapefunctionsNiaredefinedonthestandarddomainIst.Thereforetheindicateddifferentialoperationscannotbeperformeddirectly.Usingdddξ2d2dxk+1−xkk==≡anddx=dξ≡dξ,dxdξdxxk+1−xkdξkdξ22wherek:=xk+1−xkisthelengthofthekthelement,wehave⎛⎞)*xk+12+1pk+1dNpk+1dNκ(x)uvdx=κ(Q(ξ))⎝aj⎠bidξ.nnkjixkk−1j=1dξi=1dξDefining+1(k)2dNidNjkij=κ(Qk(ξ))dξ,(2.71)k−1dξdξ P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome50ANOUTLINEOFTHEFINITEELEMENTMETHODthefollowingexpressionisobtained:xk+1pk+1pk+1κ(x)uvdx=k(k)ab≡{b}T[K(k)]{a}.(2.72)nnijjixki=1j=1(k)Thetermsofthecoefficientmatrixkarecomputablefromthemapping,thedefinitionijoftheshapefunctionsandthefunctionκ(x).Thematrix[K(k)]iscalledtheelementstiffnessmatrix.Observethatk(k)=k(k),hence[K(k)]issymmetric.ThisfollowsdirectlyfromtheijjisymmetryofB(u,v)andthefactthatun∈S,vn∈Sandthesamebasisfunctionsareusedforunandvn.IntheFEMtheintegralsareevaluatedbynumericalmethods.NumericalintegrationisdiscussedinAppendixB.Thenumberofintegrationpointsmustbesufficientlylargesoastoensurethattheerrorofapproximationisnotinfluencedsignificantlybytheerrorsinnumericalintegration.Intheimportantspecialcasewhereκ(x)=κkisconstantonIk,itispossibletocompute[K(k)]onceandforall.Thisisillustratedbythefollowingexample.Example2.5.1Whenκ(x)=κkisconstantonIkandthehierarchicshapefunctionsdefinedinSection2.5.2areused,then,exceptforthefirsttworowsandcolumns,theelementalstiffnessmatrixisperfectlydiagonal:⎡⎤1/2−1/200···0⎢1/2000⎥⎢⎥2κk⎢⎢100⎥⎥[K(k)]=.(2.73)⎢10⎥k⎢⎥⎢....⎥⎣(sym.)..⎦1Exercise2.5.5Assumethatκ(x)=κkisconstantonIk.UsingtheLagrangepolynomials(k)(k)definedinSection2.5.2forp=2,computek11andk13intermsofκkandk.ComputationoftheGrammatrixThesecondtermofthebilinearformisalsocomputedasasumofintegralsovertheelements:M()xk+1c(x)unvndx=c(x)unvndx.0k=1xkWewillbeconcernedwithevaluationoftheintegral⎛⎞)*xk+1xk+1pk+1pk+1c(x)unvndx=c(x)⎝ajNj⎠biNidxxkxkj=1i=1⎛⎞)*+1pk+1pk+1=kc(Q(ξ))⎝aN⎠bNdξ.kjjii2−1j=1i=1 P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeFEMINONEDIMENSION51Defining+1(k)kmij:=c(Qk(ξ))NiNjdξ,(2.74)2−1thefollowingexpressionisobtained:xk+1pk+1pk+1c(x)uvdx=m(k)ab={b}T[M(k)]{a}(2.75)nnijjixki=1j=1where{a}:={aa...a}T,{b}T:={bb...b}and12pk+112pk+1⎡⎤(k)(k)m11m12···m1,pk+1⎢⎥⎢(k)(k)⎥⎢m21m22···m2,pk+1⎥[M(k)]:=⎢⎥.⎢......⎥⎢...⎥⎣⎦(k)(k)mpk+1,1mpk+1,2···mpk+1,pk+1(k)Thetermsofthecoefficientmatrixmarecomputablefromthemapping,thedefinitionofijtheshapefunctionsandthefunctionc(x).Thematrix[M(k)]iscalledtheelementalGrammatrix15ortheelementalmassmatrix.Observethat[M(k)]issymmetric.Intheimportantspecialcasewherec(x)=cisconstantonI,itispossibletocompute[M(k)]onceandforkkall.Thisisillustratedbythefollowingexample.Example2.5.2Whenc(x)=ckisconstantonIkandthehierarchicshapefunctionsdefinedinSection2.5.2areusedthentheelementalGrammatrixisstronglydiagonal.Forexample,forpk=5theelementalGrammatrixis⎡√√⎤2/31/3−1/61/31000⎢√√⎥⎢2/3−1/6−1/31000⎥⎢√⎥⎢⎥⎢2/50−1/5210⎥(k)ckk⎢⎢√⎥⎥M=⎢2/210−1/745⎥.(2.76)2⎢⎥⎢(sym.)⎥⎢⎥⎢⎥⎣2/450⎦2/7715JorgenPedersenGram(1752–1833).¨ P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome52ANOUTLINEOFTHEFINITEELEMENTMETHODRemark2.5.2Forpk≥2asimpleclosedformexpressioncanbeobtainedforthediagonaltermsandtheoff-diagonalterms.UsingEquation(2.68)itcanbeshownthat+1(k)ckk12mii=(Pi−1(ξ)−Pi−3(ξ))dξ22(2i−3)−1ckk2=,i≥3(2.77)2(2i−1)(2i−5)andalloff-diagonaltermsarezerofori≥3,withtheexceptions(k)(k)ckk1m=m=−√,i≥3.(2.78)i,i+2i+2,i2(2i−1)(2i−3)(2i+1)Exercise2.5.6Assumethatc(x)=ckisconstantonIk.UsingtheLagrangeshapefunctions(k)(k)definedinSection2.5.2forp=2,computem11andm13intermsofckandk.2.5.5ComputationoftherighthandsidevectorComputationoftherighthandsidevectorinvolvesnumericalevaluationofthefunctionalF(v),giventhatv∈S0.Inparticular,wewriteM()xk+1F(vn)=f(x)vndx=f(x)vndx.0k=1xkTheelement-levelintegraliscomputedfromthedefinitionofvnonIk:x+1)pk+1*pk+1k+1k(k)(k)(k)f(x)vndx=f(Qk(ξ))biNidξ=biri(2.79)xk2−1i=1i=1where+1(k)kri:=f(Qk(ξ))Ni(ξ)dξ(2.80)2−1whichcanbecomputedfromthemapping,thegivenfunctionf(x)andthedefinitionoftheshapefunctions.Example2.5.3Letusassumethatf(x)isalinearfunctiononIk.Inthiscasef(x)canbewrittenas1−ξ1+ξf(x)=f(xk)+f(xk+1)=f(xk)N1(ξ)+f(xk+1)N2(ξ)22 P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeFEMINONEDIMENSION53and,assumingthatthehierarchicshapefunctionsdefinedinSection2.5.2areused,+1+1(k)k2kkr1=f(xk)N1dξ+f(xk+1)N1N2dξ=(2f(xk)+f(xk+1))2−12−16+1+1(k)kk2kr2=f(xk)N1N2dξ+f(xk+1)N2dξ=(f(xk)+2f(xk+1))2−12−16+1+1(k)kkr3=f(xk)N1N3dξ+f(xk+1)N2N3dξ2−12−1k3=−(f(xk)+f(xk+1)).62Exercise2.5.7Assumethatf(x)isalinearfunctiononIkandthehierarchicshapefunctions(k)(k)definedinSection2.5.2areused.Computerandshowthatr=0fori>4.Hint:Make4iuseofEquation(2.68).Exercise2.5.8Letx−xkf(x)=fksinπ,x∈Ikk(k)wherefkisaconstant.Computer5numericallyintermsoffkandkusingthree,fourandfiveGausspoints.SeeAppendixB.UsethehierarchicbasisfunctionsdefinedinSection2.5.2.Exercise2.5.9Assumingthatf(x)isalinearfunctiononIandtheLagrangeshapefunctions(k)definedinSection2.5.2forp=2areused,computer.1LoadingbyaconcentratedforceAconcentratedforceF0actingonanelasticbaratx=x0isunderstoodasasurfaceloadingT(x)definedby⎧⎪⎨0if0T0.(a)ExplainhowyouwoulddeterminethetemperatureTcatwhichthegapcloses.(b)AssumingthatT≤Tc,explainhowyouwouldcomputethereactionforceatAbythedirectandindirectmethods.(c)AssumingthatT>Tc,explainhowyouwouldcomputethereactionforceatAbythedirectandindirectmethods.Exercise2.5.21WriteanadhoccomputerprogramforsolvingtheprobleminExample2.5.6foranarbitrarynumberofelements.Plottherelativeerrorsforu(2)computedby(a)FEthedirectmethodand(b)theindirectmethodversusthenumberofelements.Comparethenumberofelementsneededforreducingtherelativeerrortounder1%.NodalforcesThevectorofnodalforcesassociatedwithelementk,denotedby{f(k)},isdefinedasfollows:{f(k)}=[K(k)]{a(k)}−{r¯(k)}k=1,2,...,M()(2.91)where[K(k)]isthestiffnessmatrix,{a(k)}isthesolutionvectorand{r¯(k)}istheloadvectorcorrespondingtotractionforces,concentratedforcesandthermalloadsactingonelementk.Thesignconventionfornodalforcesisdifferentfromthesignconventionforthebarforce:whereasthebarforceispositivewhentensile,anodalforceispositivewhenactinginthedirectionofthepositivecoordinateaxis.Figure2.15Exercise2.5.20:notation. P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeFEMINONEDIMENSION65Figure2.16Exercise2.5.22:notation.Exercise2.5.22AssumethathierarchicbasisfunctionsbasedonLegendrepolynomialsareused.ShowthatwhenAEisconstantandc=0onIkthen(k)(k)(k)(k)f+f=r+r1212independentlyofthepolynomialdegreepk.Considerboththermalandtractionloads.Thisexercisedemonstratesthatnodalforcesareinequilibriumindependentlyofthefiniteelementsolution.Thereforeequilibriumofnodalforcesisnotanindicatorofthequalityoffiniteelementsolutions.Exercise2.5.23Showthatthesumofnodalforcescomputedfornodekfromelementk−1andfromelementkiszerounlessaconcentratedforceFkisactingonnodek,inwhichcasethesumisequaltoFk.Illustratethisbyasimpleexampleanddemonstratethatthesumofforcesactingonanodepointiszero.Hint:Letv=ϕk(x)showninFigure2.10.Exercise2.5.24RefertoExercise2.5.12.Assumethatthefiniteelementsolutionisavailable.Writedownanexpressionforthecomputationofnodalforcesforelement2.Remark2.5.4WehaveseeninTheorem2.4.2thattheerrorintheenergynormdependsonthechoiceofthefiniteelementspaceS,whichdependsonthechoiceofdiscretizationcharacterizedbythemesh,thepolynomialdegreesassignedtotheelementspandthemappingfunctionsQ.Ofcourse,thisistrueforalldatacomputedfromthefiniteelementsolution.ItwasnotedinSection1.1.4thatitisnecessarytoverifythatthedataofinterestaresubstantiallyindependentofthediscretization.Itisofgreatpracticalimportancetoensurethattherelativeerrorsinthedataofinterestarewithinacceptablebounds.ProceduresforerrorestimationarediscussedinSection2.7.Remark2.5.5LetusconsidertheproblemofEquation(2.24)withc=0andκ(x)=κkconstantonIk.Letusassumethatthemappingislinearandu(0)=uˆ0isoneoftheboundaryconditions.Inthiscasethefiniteelementsolutionatthenodepointsistheexactsolution.Toshowthiswedefineϕ(x)∈S0asfollows:kxϕ1(x)=1−for0≤x1,onI=(0,1)withtheboundaryconditionsu(0)=u(1)=0.Thegoalistoestimateu(0)usingtheFEMwithinanerrortoleranceof1%.Letα=1.05.Usingauniformfiniteelementmeshandp=1,whatisthenumberofelementsM()neededwhen(a)u(0)iscomputedbythedirectmethodand(b)u(0)iscomputedbythenodalforcemethod?Hint:MakeuseofthefactthatuFE(xk)=uEX(xk)fork=1,2,...,M()+1,seeRemark2.5.5. P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComePROPERTIESOFTHEGENERALIZEDFORMULATION672.6PropertiesofthegeneralizedformulationSomeofthekeypropertiesofthegeneralizedformulationandthefiniteelementsolutionarepresentedinthefollowing.Althoughthesepropertiesarepresentedintheone-dimensionalsettingonly,theyareapplicabletotwoandthreedimensionsaswell,unlessnotedotherwise.2.6.1UniquenessThemodelproblemgivenbyEquation(2.24)hasbeenreplacedbythecorrespondinggen-eralizedformulation.Thefollowingtheoremestablishesthatthesolutionofthegeneralizedformulationisunique.Theorem2.6.1Thefunctionu∈E˜(I)thatsatisfiesB(u,v)=F(v)forallv∈E0(I)isuniqueinthespaceE(I).Proof:Thetheoremisprovenbycontradiction.Assumethattherearetwofunctionsu1,u2inE˜(I),u1=u2,thatsatisfyB(u1,v)=F(v)B(u2,v)=F(v)forallv∈E0(I).SubtractingthesecondequationfromthefirstwegetB(u−u,v)=0forallv∈E0(I).12Since(u−u)∈E0(I),wemayselectv=u−u,inwhichcaseB(u−u,u−u)=121212120.InviewofEquation(2.53)thisisequivalenttou1−u2E=0,whichcontradictstheassumptionthatu1=u2inE˜(I).Remark2.6.1NotethatuniquenessisunderstoodinthespaceE(I).Supposethatc=0inEquation(2.24)andNeumannboundaryconditionsarespecified,subjecttoEquation(2.50).Thenifu1isasolutionthenu2=u1+Cisalsoasolution,whereCisanarbitraryconstant.Inthiscasetheenergynormcannotdistinguishbetweentwofunctionsthatdifferbyanarbitraryconstant.Thisisseendirectlyfromthedefinitionoftheenergynorm,givenbyEquation(2.53):1u−u2=(u−u)2dx=0.12E1220Thereforethesolutioncanbedeterminedonlyuptoanarbitraryconstant.16Inmechanicsthishasasimplephysicalinterpretation:theconstantCrepresentsrigidbodydisplacement.16Suchanormiscalleda“seminorm.” P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome68ANOUTLINEOFTHEFINITEELEMENTMETHOD2.6.2PotentialenergyAnimportantpropertyofthegeneralizedformulationisthatthesolutionminimizesaquadraticfunctional,calledthepotentialenergy.Thisisprovenbythefollowingtheorem.Theorem2.6.2Thefunctionu∈E˜(I)thatsatisfiesB(u,v)=F(v)forallv∈E0(I)min-imizesthequadraticfunctionalπ(u),calledthepotentialenergy:1π(u):=B(u,u)−F(u)(2.92)2onthespaceE˜(I).Proof:Foranyv∈E0(I),v=0,wehaveE1π(u+v)=B(u+v,u+v)−F(u+v)211=B(u,u)+B(u,v)+B(v,v)−F(u)−F(v)221=π(u)+B(u,v)−F(v)+B(v,v).20>0Thereforeanyadmissiblenon-zeroperturbationofuwillincreaseπ(u).Thistheoremisknownastheprincipleofminimumpotentialenergy.Remark2.6.2Whereasthestrainenergyisalwayspositive,thepotentialenergymaybepositive,negativeorzero.2.6.3ErrorintheenergynormTherelationshipbetweentheerrorintheenergynormandtheerrorinpotentialenergyisestablishedbythefollowingtheorem.Oneofthemethodsusedforestimatingtheerrore=uEX−uFEintheenergynormisbasedonthistheorem.Theorem2.6.3u−u2=π(u)−π(u).(2.93)EXFEEFEEX P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComePROPERTIESOFTHEGENERALIZEDFORMULATION69Proof:Writinge=u−uandnotingthate∈E0(I),wehaveEXFE1π(uFE)=π(uEX−e)=B(uEX−e,uEX−e)−F(uEX−e)211=B(uEX,uEX)−F(uEX)−B(uEX,e)+F(e)+B(e,e)220=π(u)+e2EXEwhichisthesameasEquation(2.93).2.6.4ContinuityBydefinition,u(x)iscontinuousonI¯:={0≤x≤}ifforany>0wecanfindaδ()suchthat|u(x2)−u(x1)|≤if|x2−x1|≤δ(),x1,x2∈I¯.(2.94)Firstweshowbyanexamplethatadiscontinuousfunctioncannotlieintheenergyspace.Specifically,letusconsiderthecontinuousfunctionu(x)showninFigure2.17andcomputetheintegralx0+xc−c2(c−c)222121(u)dx=dx=.0x0xxLettingx→0,thefunctionbecomesdiscontinuousandthevalueoftheintegralisinfinity.Withonequalification,anyfunctionu(x)∈E(I)iscontinuousandboundedonI¯bytheenergynormuE.Thequalificationisthatwhenc(x)=0andNeumannboundaryconditionsarespecified,subjecttoEquation(2.50),thenu=Cisasolution(whereCisanarbitraryconstant),henceu(x)isnotboundedbytheenergynorm.SeeRemark2.6.1.Thestatementthatu(x)∈E(I)isboundedonI¯byuEisunderstoodtomeanthatthereexistsaconstantC,independentofu(x),suchthatforanyx∈I¯thefollowinginequalityholds:|u|≤CuE.Figure2.17Notation. P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome70ANOUTLINEOFTHEFINITEELEMENTMETHODWenowprovecontinuityforthecaseκ=1,c=0.Letusassumethatx2≥x1.Thenx2u(x)−u(x)=u(x)dx(2.95)21x1wheretheprime()representsdifferentiationwithrespecttox.ApplyingtheSchwarzin-equality(seeAppendixA,SectionA.8)wegetx21/2x21/2√21/2|u(x2)−u(x1)|≤dxu(x)dx=|x2−x1|2uE.(2.96)x1x1Thereforeifweselectδ()<2/(2u2)thentheconditionofcontinuityissatisfied.ThatEu(x)isboundedonI¯byuEfollowsdirectlyfrom(2.96)forthecaseu(x1)=0.Thistheoremholdsinonedimensiononly.Remark2.6.3Whereasallfunctionsu∈E(I)arecontinuousandbounded,du/dxhastobeneithercontinuousnorbounded.ForthisreasonNeumannboundaryconditionscannotbeenforcedbyrestriction.Exercise2.6.1Considerfunctionsoftheformu(x)=xαontheinterval01/2.2.6.5ConvergenceintheenergynormIntheexamplediscussedinSection2.2thebasisfunctionswereselectedtobepolynomialsthatsatisfiedthehomogeneousessentialboundaryconditions.Thetacitassumptionwasthatinsomesenseun→uasn→∞.ConvergenceinanormedlinearspaceXisunderstoodtomeanthatforany>0thereisann,dependenton,suchthatu−unX<.Inthefollowingweassumethatu(x)iscontinuousandu(x)≤C<∞onI¯:=max[0,].LetuspartitionI¯intonsegmentsanddenotethekthnodepointbyxk.Letusinterpolateu(x)byapiecewiselinearfunctionu¯n(x)suchthatu¯n(xk)=u(xk).Anexamplewheren=4isshowninFigure2.18(a).Onthekthsub-intervalIk:=(xk,xk+1)wehaveFigure2.18Linearinterpolation:notation. P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComePROPERTIESOFTHEGENERALIZEDFORMULATION71x−xkx−xk+1u¯n=u(xk+1)−u(xk)x∈Ik,k=1,2,...,nhkhkwherehk:=xk+1−xkisthelengthofthekthelement.Themaximumvalueofhkwillbedenotedbyh.Wefirstshowthath2u−u¯≤u.(2.97)nmaxmax8Toprovethisinequality,weconsidertheerroronthekthsub-interval,showninFig-ure2.18(b).Sincee¯k(x):=u(x)−u¯n(x)forx∈Ik,e¯k(x)vanishesattheendpointsofthesub-intervalI.Furthermore,byassumptionuiscontinuousonI,thereforethereisapointkkx¯where|e¯|ismaximumande¯(x)=0.Weexpande¯aboutthispointintoaTaylorseries.kkkkkLetusassumethatxk+1−x¯k≤hk/2andwrite1e¯(x)=0=e¯(x¯)+e¯(x¯)(x−x¯)+e¯(ξ)(x−x¯)2ξ∈Ikk+1kkkkk+1kkkk+1kk20wherethelasttermistheerrortermoftheTaylorexpansion.Fromthisrelationshipwehave1max|e¯(x)|=|e¯(x¯)|=|e¯(ξ)|(x−x¯)2ξ∈I.kkkkkk+1kkkx∈Ik2Sinceu¯=0wehavee¯=u.Also,sincex−x¯≤h/2wehavenkk+1kkh2max|e¯|≤kuk=1,2,...,n.kmaxx∈Ik8Ifxk+1−x¯k>hk/2thenweexpresse¯k(xk)andobtainthesameresult.Onreplacinghkwithh,Equation(2.97)followsdirectlyfromthisresult.Theestimatefore¯isobtainedfromtheTaylorexpansionofe¯(x)atx=x¯:kL2(Ik)kke¯(x)=e¯(x¯)+e¯(ξ)(x−x¯)=u(ξ)(x−x¯).kkkkkkkk0Therefore|e¯|≤u|x−x|kmaxkandxk+1xk+1h3(e¯)2dx≤u2(x−x)2dx=u2k.kmaxkmaxxkxk3 P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome72ANOUTLINEOFTHEFINITEELEMENTMETHODOntheentiredomain,M()h31M()(e¯)2dx≤u2k≤h2u2h.kmaxmaxk033k=1k=14M()Sincek=1hk=,wehave(e¯)2dx≡u−u¯2≤h2u2.(2.98)knL2(I)3max0Wearenowinapositiontoobtainanestimatefortheerroroftheinterpolantintheenergynorm.Bydefinition,1u−u¯2=κ(u−u¯)2+c(u−u¯)2dx.nE(I)nn20UsingEquations(2.97)and(2.98),forsufficientlysmallhwehaveu−u¯≤ChunE(I)maxwhereCisaconstantthatdependsonκ,c,butisindependentofhandu.Theorem2.4.2(seepage43)statesthatthefiniteelementsolutionminimizestheerrorintheenergynormonthespaceS˜(I).Thereforeu−unE(I)≤u−u¯nE(I)andu−u≤Chu.(2.99)nE(I)maxErrorestimatesofthistypearecalledaprioriestimates.17ThisestimateshowsthatthefiniteelementsolutionconvergestotheexactsolutionintheenergynormgiventheassumptionthatuiscontinuousandboundedonI¯.Italsoshowshowfasttheerrorintheenergynormdecreasesasthemeshisrefinedsothatthesizeofthelargestelementhapproacheszero.Thisestimateholdsforallh,andforsufficientlysmallhtheinequality(2.99)becomesanapproximateequality.Convergencecanbeprovenforanyu∈E(I),see,forexample,[70].Abriefoverviewofaprioriestimatesfortwo-andthree-dimensionalproblemsispresentedinSection6.4.Remark2.6.4Itispossibletoproveconvergenceofthefiniteelementsolutioninothernorms,suchasu−uandu−u,butthisisbeyondthescopeofthisbook.nmaxnL2(I)Exercise2.6.2Theestimate(2.99)wasderivedforlinearshapefunctions.Obtainananal-ogousestimateforquadraticshapefunctionsundertheassumptionthat|u(x)|≤C<∞.Exercise2.6.3RepeatExercise2.6.2assumingthat|u(x)|≤C<∞but|u(x)|canbearbitrarilylarge.17Aprioriestimatesareobtainedthroughdeductivereasoning,basedoncertaincharacteristicsofaproblemclass.Inthisinstancetheproblemclassischaracterizedbyu(x)beingcontinuousandboundedonI¯. P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeERRORESTIMATIONBASEDONEXTRAPOLATION732.7ErrorestimationbasedonextrapolationComputedvaluesofthepotentialenergycorrespondingtoahierarchicsequenceoffiniteelementspacescanbeusedforestimatingtheerrorintheenergynormbyextrapolation.ByTheorem2.6.3wehaveu−u2=π(u)−π(u).(2.100)EXFEEFEEXForalargeandimportantclassofproblemstheerrorintheenergynormisproportionaltoN−βwhenNissufficientlylarge:kuEX−uFEE≈(2.101)Nβwherekissomepositiveconstant.Anerrorestimatecanbebasedonthisrelationship.UsingEquation(2.100)weobtaink2π(uFE)−π(uEX)≈.(2.102)N2βTherearethreeunknowns:π(uEX),kandβ.AssumethatwehaveasequenceofsolutionscorrespondingtoSp−2⊂Sp−1⊂Sp.Letusdenotethecorrespondingpotentialenergyvaluesbyπp−2,πp−1,πpandthedegreesoffreedombyNp−2,Np−1,Np.Wewilldenoteπ:=π(uEX).Withthisnotationwehavek2πp−π≈(2.103)2βNpk2πp−1−π≈2β.(2.104)Np−1OndividingEquation(2.103)by(2.104),weget2βπp−πNp−1πp−πNp−1≈orlog≈2βlog.πp−1−πNpπp−1−πNpRepeatingforp−1andp−2,itispossibletoeliminate2βtoobtainQ−1πp−ππp−1−πNp−1Np−2≈whereQ:=loglog.(2.105)πp−1−ππp−2−πNpNp−1Thisequationcanbesolvedforπtoobtainanestimatefortheexactvalueofthepotentialenergy. P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome74ANOUTLINEOFTHEFINITEELEMENTMETHODWiththeestimatedvalueofπitispossibletoestimatetherelativeerrorintheenergynormusingEquation(2.100).Bydefinition,therelativeerrorintheenergynormisuEX−uFEE(er)E=(2.106)uEXEwhereuEXEisestimatedfromuFEcorrespondingtothefiniteelementspaceofthelargestnumberofdegreesoffreedom:1uEXE≈B(uFE,uFE).2Therelativeerrorisusuallyreportedaspercenterror.Thisestimatorhasbeentestedagainstanumberofproblemsforwhichtheexactsolutionsareknown.Itwasfoundthattheestimatorworkswellforawiderangeofproblems,includingmostproblemsofpracticalinterest,butitcannotbeguaranteedtoworkforallproblems.Thequalityofanestimatorismeasuredbytheeffectivityindexθ,definedastheestimatederrordividedbythetrueerror:(uEX−uFEE)est.θ:=.(2.107)(uEX−uEXE)trueOfcourse,theeffectivityindexcanbecomputedonlyforthoseproblemsforwhichtheexactsolutionisknown.Evaluationofanestimatorinvolvesthesolutionofavarietyofsuchproblems.Anestimatorisgenerallyconsideredtobereliableif0.8<θ<1.2formostproblems.Remark2.7.1LetusdivideEquation(2.101)byuEXEtoobtainuEX−uFEEk¯=(er)E≈uENβEXwherek¯:=k/uEXE.Thereforelog(er)E≈logk¯−βlogN.Ifweplot(er)Evs.Nonalog–logscalethen,iftheassumption(2.101)iscorrect,wewillseeastraightlineofslope−β.Theconvergence(2.101)iscalledalgebraicconvergenceandβiscalledtherateofconvergence.2.7.1Theroot-mean-squaremeasureofstressWhenc=0andtheboundaryconditionsdonotincludespringboundaryconditionsthentheerrorintheenergynormiscloselyrelatedtotheroot-mean-square(RMS)errorinstress.We P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeEXTRACTIONMETHODS75definetheRMSmeasureofstressS=S(σ)inonedimensionasfollows:#1S(σ):=σ2AdxwhereV:=Adx.(2.108)V00Fromthisdefinitionandthedefinitionofstrainenergyforthebarwehave112S2(σ)=σ2Adx=A(Eu)2dx≤maxE(x)u2E(I)V0V0Vx∈I2≥minE(x)u2.(2.109)E(I)Vx∈ITherefore22minE(x)uE(I)≤S(σ)≤maxE(x)uE(I).(2.110)Vx∈IVx∈IInthegeneralcasewhenc=0and/ork0=0,k=0thenthereisapositiveconstantCthatdependsonE(x),c(x),k0,kandVsuchthatS(σ)≤CuE(I).(2.111)Exercise2.7.1DenotetherelativeerrorintheRMSstressby(er)S,themaximum(resp.minimum)ofE(x),x∈I,byEmax(resp.Emin)andassumethatc=0andtheboundaryconditionsdonotincludespringboundaryconditions.Showthat""Emin/Emax(er)E≤(er)S≤Emax/Emin(er)E.(2.112)ThisexercisedemonstratesthatwhenEisconstant,c=0andeitherforceordisplacementboundaryconditionsareprescribed,thentherelativeerrorintheRMSstressequalstherelativeerrorintheenergynorm.2.8ExtractionmethodsWehaveseeninExample2.5.8thattheindirectmethodusedforcomputingu(0)yieldedaFEmuchsmallererrorthanthedirectmethod.Thereasonsforthisarediscussedinthefollowing.LetussayweareinterestedincomputingsomefunctionalQandletusassumethatwecanfindafunctionwsuchthattheexactvalueofQ,denotedbyQEX,isQEX=B(uEX,w)−F(w).(2.113)Thefunctionwiscalledtheextractionfunction.Itdoesnothavetolieintheenergyspace;however,theindicatedoperationsmustbedefined.ThefiniteelementapproximationofQEXisdenotedbyQFE:QFE=B(uFE,w)−F(w).(2.114) P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome76ANOUTLINEOFTHEFINITEELEMENTMETHODThereforetheerrorinQisQEX−QFE=B(eu,w)(2.115)wheree:=u−u.NotethateliesinE0(I).Defineg∈E0(I)suchthatuEXFEuEXB(g,v)=B(w,v)forallv∈E0(I).(2.116)EXThefunctiongistheprojectionofwontothespaceE0(I).Byselectingv=eandusingEXuthesymmetryofthebilinearform,Equation(2.115)canbewrittenasQEX−QFE=B(eu,gEX).(2.117)LetgbetheprojectionofgontothetestspaceS0(I):FEEXB(g,v)=B(g,v)forallv∈S0(I).(2.118)FEEXSincev=g∈S0(I),bytheGalerkinorthogonality(seeEquation(2.57))wehaveFEB(eu,gFE)=0.ThereforewecanrewriteEquation(2.117)asQEX−QFE=B(eu,gEX)−B(eu,gFE).(2.119)0Denotingeg:=gEX−gFEweobtainQEX−QFE=B(eu,eg).(2.120)UsingtheSchwarzinequality(seeSectionA.8)weget|QEX−QFE|=|B(eu,eg)|≤2euE(I)egE(I).(2.121)ThisequationshowsthattheerrorinQdependsontheerrorinthefiniteelementsolutioneuandtheerroreg.Thereforethefiniteelementspacehastobedesignedsuchthatbotherrorsaresmall.NotethatgEXdoesnothavetobeknown,itisoftheoreticalimportanceonly.IfegE(I)convergestozeroatapproximatelythesamerateaseuE(I)then|QEX−QFE|convergestozeroataboutthesamerateastheerrorinstrainenergy,whichistwicetherateofconvergenceoftheerrorintheenergynorm.Amethodofcomputationforsomefunctionalissaidtobesuperconvergentwhenthedataofinterestconvergetotheirlimitvalueatapproximatelythesamerateasthestrainenergy.Remark2.8.1SuperconvergentmethodsutilizeextractionfunctionsconstructedsoastoapproximatetheappropriateGreen’sfunction.18Fordetailswereferto[5].18GeorgeGreen(1793–1841). P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoComeCHAPTERSUMMARY772.9LaboratoryexercisesThefollowingchaptersdealwiththeFEMintwoandthreedimensions.Toperformbasicex-ercises,thereaderwillneedtousethestudenteditionofFEAsoftwareproductStressCheck,19whichisprovidedwiththisbook.AtthispointthereadershouldbecomeacquaintedwiththekeyfeaturesofStressCheck.ThebestwaytostartistoreadtheGettingStartedGuide,whichcanbefoundunderthemainmenuheading“Help.”Chapter2ofthisguideprovidesinformationaboutthemostimportantfeaturesoftheuserinterface.Chapter4isatutorialthatprovidesinformationaboutthepreparationofinputdataforproblemsintwo-andthree-dimensionalelasticity,executionofthesolution,andpost-processingprocedures.Chapter3oftheGettingStartedGuideprovidesanoverviewofStressCheck’sHandbookLibrary.TheHandbookLibrarycontainsanumberofproblemsdefinedintermsofparametersinmuchthesamewayasinconventionalengineeringhandbooks.TheprincipaldifferenceisthatherethesolutionsareobtainedbytheFEM.Thisallowsproblemsoffargreatercomplexityandvarietytobeformulatedintermsofparameters.Thefiniteelementmesheschangeautomaticallywiththegeometricparameters,thereforehandbookusersdonotneedtobeconcernedwithmeshgeneration.Thereaderisencouragedtoexplorethehandbooklibraryanduseitforguidancewhenformulatingandsolvingexerciseproblemsinthefollowingchapters.HavinggainedsomefamiliaritywithStressCheck,thereaderwillfinddetailedinformationintheMasterGuide.20TheMasterGuideiscomposedoffourparts.Part1,theUsers’Guide,providesdetailedinformationabouttheuserinterface,post-processingandthehandbookframework.Part2,theModelingGuide,explainsproceduresforthecreationofgeometricentitiesintwoandthreedimensionsandforautomaticgenerationoffiniteelementmeshes.Part3,theAnalysisGuide,providesinstructionsonthepreparationofdataforthevarioustypesofanalysissupportedbyStressCheck.Part4,theAdvancedGuide,providesinformationaboutfracturemechanicsapplications,nonlinearanalysisprocedures,thesolvers,andothertopicsthatareofinteresttoadvancedusers.SpecifictopicscanbelocatedbymeansoftheIndexwhichcanbeaccessedthroughtheBookmarkssectionoftheMasterGuide.2.10ChaptersummaryFundamentalconcepts,proceduresanddefinitions,essentialforunderstandingthefiniteele-mentmethod,werepresentedinasimplesetting:1.Thegeneralizedformulation,itsdependenceontheboundaryconditions,treatmentofnaturalandessentialconditions,definitionsoftheenergyspace,variousnormsandthepotentialenergyarefundamentaltothefiniteelementmethod.2.Theapproximatesolutionandhencetheerrorofapproximationisdeterminedbythefiniteelementspacecharacterizedbythefiniteelementmesh,thepolynomialdegreesoftheelementsandthemappingfunctions.19StressCheckisatrademarkofEngineeringSoftwareResearchandDevelopment,Inc.,St.Louis,Missouri.20TheMasterGuidecanbefoundunderthemainmenuheading“Help.” P1:OSOJWST055-02JWST055-SzaboFebruary18,20117:2PrinterName:YettoCome78ANOUTLINEOFTHEFINITEELEMENTMETHOD3.Thefiniteelementsolutionisuniqueandminimizestheerrorinenergynorm,seeTheorem2.4.2onpage43.4.Allinformationgeneratedbythefiniteelementmethodresidesinthestandardbasisfunctions,calledshapefunctions,theircoefficientsandthemappingfunctions.5.Theerrorsinthedataofinterestdependonhowthedataarecomputedfromthefiniteelementsolution.Incomputingthefirstderivativetheindirectmethodwassubstantiallymoreaccuratethanthedirectmethod. P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoCome3FormulationofmathematicalmodelsTheformulationofmathematicalmodelsisintroducedonthebasisoflinearheatconduction,linearelastostaticandviscousflowproblems.Linearmodelsareusedveryfrequentlyinengi-neeringpractice.Neverthelesstheyshouldbeviewedasspecialcasesofnonlinearmodelsthataccountformaterialnonlinearities,geometricnonlinearities,mechanicalcontact,radiation,etc.NonlinearmodelsarediscussedinChapter10.Forthepurposesofthefollowingdiscussionamathematicalmodelisunderstoodtobeastatementofamathematicalproblemintheformofoneormoreordinaryorpartialdifferentialequationsandspecificationofthesolutiondomainandboundaryconditions.Thisiscalledthestrongformofthemodel.BeginningwithChapter4,generalizedformulations,alsoknownasweakforms,willbediscussed.3.1NotationTheEuclideanspaceinndimensionswillbedenotedbyRn.TheCartesian1coordinateaxeswillbelabeledx,y,z(incylindricalsystemsr,θ,z)andavectoruinRnwillbedenotedeitherbyuoru.Forexample,u≡u≡{uuu}TrepresentsavectorinR3.xyzWewillemploytheindexnotationalsowheretheCartesiancoordinateaxesarelabeledx=x1,y=x2,z=x3.Theindexnotationwillbeintroducedgraduallyinparallelwiththeconventionalvectornotationsothatreaderscanbecomefamiliarwithit.Thebasicrulesofindexnotationaresummarizedinthefollowing:1.Afreeindexisunderstoodtorangefromonetothree(intwospatialdimensionsfromonetotwo).1ReneDescartes(inLatin:RenatusCartesius;1596–1650).´IntroductiontoFiniteElementAnalysis:Formulation,VerificationandValidation,FirstEdition.BarnaSzabóandIvoBabuška.©2011JohnWiley&Sons,Ltd.Published2011byJohnWiley&Sons,Ltd.ISBN:978-0-470-97728-6 P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoCome80FORMULATIONOFMATHEMATICALMODELS2.Thepositionvectorofapointisxwhereasinconventionalnotationitis{xyz}T.Aigeneralvectora≡{aaa}Tiswrittensimplyasa.xyzi3.Twofreeindicesrepresentamatrix.Thesizeofthematrixdependsontherangeofindices.Thus,inthreedimensions,⎡⎤⎡⎤a11a12a13axxaxyaxzaij≡⎣a21a22a23⎦≡⎣ayxayyayz⎦.a31a32a33azxazyazzTheidentitymatrixisrepresentedbytheKronecker2deltaδ,definedasfollows:ij1ifi=jδij=0ifi=j.4.Indicesfollowingacommarepresentdifferentiationwithrespecttothevariablesiden-tifiedbytheindices.Forexample,ifu(xi)isascalarfunctionthen∂u∂2uu,2≡,u,23≡·∂x2∂x2∂x3Thegradientofuissimplyu,i.5.Repeatedindicesrepresentsummation.Forexample,thescalarproductoftwovectorsaiandbjisaibi≡a1b1+a2b2+a3b3.Ifui=ui(xk)isavectorfunctioninthree-dimensionalspacethen∂u1∂u2∂u3ui,i≡++∂x1∂x2∂x3isthedivergenceofui.Repeatedindicesarealsocalled“dummyindices,”sincesummationisperformedandthereforetheindexdesignationisimmaterial.Forexample,aibi=akbk.Theproductoftwomatricesaijandbijiscij=aikbkj.6.ThetransformationrulesforCartesianvectorsandtensorsarepresentedinAppendixC.Example3.1.1Thedivergencetheoreminindexnotationisui,idV=uinidS(3.1)∂whereuiandui,iarecontinuousonthedomainanditsboundary∂,niistheoutwardunitnormalvectortotheboundary,dVisthedifferentialvolumeanddSisthedifferentialsurface.Wewillusethedivergencetheoreminthederivationofgeneralizedformulations.Exercise3.1.1Outlineaderivationofthedivergencetheoremintwodimensions.Hint:ReviewthederivationofGreen’stheoremandcastitintheformofEquation(3.1).2LeopoldKronecker(1823–1891). P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoComeHEATCONDUCTION813.2HeatconductionTheproblemofheatconductionisrepresentativeofanimportantclassofproblemsinengi-neeringandphysics,calledpotentialflowproblems,thatinclude,forexample,theflowofviscousfluidsinporousmediaandelectrostaticphenomena.Furthermore,themathematicalformulationofpotentialflowproblemshasanalogieswithsomephysicalphenomenathatarenotrelatedtopotentialflowproblems,suchasthesmalldeflectionofmembranesandtorsionofelasticbars.Mathematicalmodelsofheatconductionarebasedontwofundamentalrelationships:theconservationlawandFourier’slaw.Thesearedescribedinthefollowing:1.Theconservationlawstatesthatthequantityofheatenteringanyvolumeelementoftheconductingmediumequalsthequantityofheatexitingthevolumeelementplusthequantityofheatretainedinthevolumeelement.Theheatretainedcausesachangeintemperatureinthevolumeelementwhichisproportionaltothespecificheatoftheconductingmediumc(inJ/(kgK)units)multipliedbythedensityρ(inkg/m3units).Thetemperaturewillbedenotedbyu(x,y,z,t)wheretistime.Theheatflowrateacrossaunitareaisrepresentedbyavectorquantitycalledheatflux.TheheatfluxismeasuredinW/m2unitsandwillbedenotedbyq=q(x,y,z,t):={q(x,y,z,t)q(x,y,z,t)q(x,y,z,t)}T.Inadditiontoheatfluxxyzenteringandleavingthevolumeelement,heatmaybegeneratedwithinthevolumeelement,forexample,fromchemicalreactions.TheheatgeneratedperunitvolumeandunittimewillbedenotedbyQ(inW/m3units).ApplyingtheconservationlawtothevolumeelementshowninFigure3.1,weobtaint[qxyz−(qx+qx)yz+qyxz−(qy+qy)xz+qzxy−(qz+qz)xy+Qxyz]=cρuxyz.(3.2)Figure3.1Controlvolumeandnotationforheatconduction. P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoCome82FORMULATIONOFMATHEMATICALMODELSAssumingthatuandqarecontinuousanddifferentiable,andneglectingtermsthatgotozerofasterthanx,y,z,t,weget∂qx∂qy∂qz∂uqx=x,qy=y,qz=z,u=t.∂x∂y∂z∂tOnfactoringxyzttheconservationlawisobtained:∂qx∂qy∂qz∂u−−−+Q=cρ·(3.3)∂x∂y∂z∂tInindexnotation∂u−qi,i+Q=cρ·(3.4)∂t2.Fourierslawstatesthattheheatfluxvectorisrelatedtothetemperaturegradientinthefollowingway:∂u∂u∂uqx=−kxx+kxy+kxz(3.5)∂x∂y∂z∂u∂u∂uqy=−kyx+kyy+kyz(3.6)∂x∂y∂z∂u∂u∂uqz=−kzx+kzy+kzz(3.7)∂x∂y∂zwherethecoefficientskxx,kxy,...,kzzarecalledcoefficientsofthermalconductivity(measuredinW/(mK)units).Itiscustomarytowritekx:=kxx,ky:=kyy,kz:=kzz.Thecoefficientsofthermalconductivitywillbeassumedtobeindependentofthetemperatureu,unlessotherwisestated.Denotingthematrixofcoefficientsby[K],Fourier’slawofheatconductioncanbewrittenasq=−[K]gradu.(3.8)Thematrixofcoefficients[K]issymmetricandpositivedefinite.Thenegativesignindicatesthatthedirectionofheatflowisoppositetothedirectionofthetemperaturegradient;thatis,thedirectionofheatflowisfromhightolowtemperature.InindexnotationEquation(3.8)iswrittenasqi=−kiju,j.(3.9)Forisotropicmaterialskij=kδij. P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoComeHEATCONDUCTION833.2.1ThedifferentialequationCombiningEquations(3.3)through(3.7),weget∂∂u∂u∂u∂∂u∂u∂ukx+kxy+kxz+kyx+ky+kyz∂x∂x∂y∂z∂y∂x∂y∂z∂∂u∂u∂u∂u+kzx+kzy+kz+Q=cρ(3.10)∂z∂x∂y∂z∂twhichcanbewritteninthefollowingcompactform:∂udiv([K]gradu)+Q=cρ·(3.11)∂tInindexnotation∂u(kiju,j),i+Q=cρ·(3.12)∂tInmanypracticalproblemsuisindependentoftime,thatis,therighthandsideofEquation(3.11)andhenceEquation(3.12)iszero.Suchproblemsarecalledstationaryorsteadystateproblems.Thesolutionofastationaryproblemcanbeviewedasthesolutionofsometime-dependentproblemwithtime-independentboundaryconditionsatt=∞.InformulatingEquation(3.11)weassumedthatkijaredifferentiablefunctions.Inmanypracticalproblemsthesolutiondomainiscomposedofsubdomainsithathavedifferentmaterialproperties.InsuchcasesEquation(3.11)isvalidoneachsubdomainandontheboundariesofadjoiningsubdomainscontinuoustemperatureandfluxareprescribed.Foracompletedefinitionofamathematicalmodelinitialandboundaryconditionshavetobespecified.Thisisdiscussedinthefollowingsection.3.2.2BoundaryandinitialconditionsThesolutiondomainwillbedenotedbyanditsboundaryby∂.Wewillconsiderthreekindsofboundaryconditions:1.Prescribedtemperature.Thetemperatureu=uˆisprescribedonboundaryregion∂u.2.Prescribedflux.Thefluxvectorcomponentnormaltotheboundary,denotedbyqn,isprescribedontheboundaryregion∂q.Bydefinition,qn:=q·n≡−([K]gradu)·n≡−kiju,jni(3.13)wheren≡niisthe(outward)unitnormaltotheboundary.Theprescribedfluxon∂qwillbedenotedbyqˆn.3.Convectiveheattransfer.Onboundaryregion∂cthefluxvectorcomponentqnisproportionaltothedifferencebetweenthetemperatureoftheboundaryandthe P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoCome84FORMULATIONOFMATHEMATICALMODELStemperatureofaconvectivemedium:qn=hc(u−uc)(x,y,z)∈∂c(3.14)wherehisthecoefficientofconvectiveheattransferinW/(m2K)unitsanduisthecc(known)temperatureoftheconvectivemedium.Thesets∂u,∂qand∂carenon-overlappingandcollectivelycovertheentireboundary.Anyofthesetsmaybeempty.Theboundaryconditionsaregenerallytimedependent.Fortimedependentproblemsaninitialconditionhastobeprescribedon:u(x,y,z,0)=U(x,y,z).ItispossibletoshowthatEquation(3.10),subjecttotheenumeratedboundaryconditions,hasauniquesolution.Stationaryproblemsalsohaveuniquesolutionswiththeexceptionthatwhenfluxisprescribedovertheentireboundaryofthenthefollowingconditionmustbesatisfied:QdV=qndS.(3.15)∂Thisiseasilyseenbyintegrating(kiju,j),i+Q=0(3.16)onandusingthedivergencetheorem,Equation(3.1)andthedefinition(3.13).NotethatifuiisasolutionofEquation(3.16)thenui+Cisalsoasolution,whereCisanarbitraryconstant.Thereforethesolutionisuniqueuptoanarbitraryconstant.Inadditiontothethreetypesofboundaryconditionsdiscussedinthissection,radiationmayhavetobeconsidered.Whentwobodiesexchangeheatbyradiationthenthefluxisproportionaltothedifferenceofthefourthpoweroftheirabsolutetemperatures,thereforeradiationisanonlinearboundarycondition.Theboundaryregionsubjecttoradiation,denotedby∂r,mayoverlap∂c.RadiationisdiscussedinSection10.1.1.Inthefollowingitwillbeassumedthatthecoefficientsofthermalconductivity,thefluxprescribedonq,andhcanducprescribedoncareindependentofthetemperatureu.Ingeneral,thisassumptionisjustifiedinanarrowrangeoftemperatureonly.Exercise3.2.1DiscussthephysicalmeaningofEquation(3.15).Exercise3.2.2Showthatincylindricalcoordinatesr,θ,ztheconservationlawisoftheform1∂(rqr)1∂qθ∂qz∂u−−−+Q=cρ·(3.17)r∂rr∂θ∂z∂tUsetwomethods:(a)applytheconservationlawtoadifferentialvolumeelementincylindricalcoordinatesand(b)transformEquation(3.3)tocylindricalcoordinates.Exercise3.2.3Showthattherearethreemutuallyperpendiculardirections(calledprin-cipaldirections)suchthattheheatfluxisproportionaltothe(negative)gradientvector. P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoComeHEATCONDUCTION85Hint:Considersteadystateheatconductionandlet[K]gradu=λgraduandthenshowthattheprincipaldirectionsaredefinedbythenormalizedcharacteristicvectors.Remark3.2.1TheresultofExercise3.2.3impliesthatthegeneralformofmatrix[K]canbeobtainedbyrotationfromorthotropicmaterialaxes.Exercise3.2.4ListallofthephysicalassumptionsincorporatedintothemathematicalmodelrepresentedbyEquation(3.10)andtheboundaryconditionsdescribedinthissection.3.2.3Symmetry,antisymmetryandperiodicityAscalarfunctionissaidtobesymmetricwithrespecttoaplaneifinsymmetricallylocatedpointswithrespecttotheplanethefunctionhasequalvalues.Onaplaneofsymmetryqn=0.Afunctionissaidtobeantisymmetricwithrespecttoaplaneifinsymmetricallylocatedpointswithrespecttotheplanethefunctionhasequalabsolutevaluesbutofoppositesign.Onaplaneofantisymmetryu=0.Inmanyinstancesthedomainandthecoefficientshaveoneormoreplanesofsymmetryandthesourcefunctionandboundaryconditionsareeithersymmetricorantisymmetricwithrespecttotheplanesofsymmetry.Insuchcasesitisoftenadvantageoustoexploitsymmetryandantisymmetryintheformulationoftheproblem.When,[K],Qandtheboundaryconditionsareperiodicthenaperiodicsectorofhasatleastoneperiodicboundarysegmentpairdenotedby∂+and∂−.Oncorrespondingpointsppofaperiodicboundarysegmentpair,P+∈∂+andP−∈∂−,theboundaryconditionsareppu(P+)=u(P−)andq+=−q−.Periodic,symmetricandantisymmetricboundaryconditionsnnareillustratedinthefollowingexample.Example3.2.1Figure3.2representsaplate-likebodyofconstantthickness.Itcanbedividedintofivesectorsasillustrated.Letusassumethatonthecylindricalboundaryrepresentedbytheinnercircle(∂(i))constantfluxisprescribedandontheboundaryrepresentedbytheoutercircle(∂(o))constanttemperatureisprescribed.Theplanarsurfacesparalleltothexyplaneareperfectlyinsulated.OnthesurfacesrepresentedbytheellipticalboundariesletCsinϕfor−π/2≤ϕ≤π/2qn=0forπ/2<ϕ<3π/2whereCisconstantandtheformulaisunderstoodtobegiveninthelocalcoordinatesystemofeachellipse.Inthiscasethesolutionisperiodicandthesolutionobtainedforonesectorcanbeextendedtotheothersectors.Periodicboundaryconditionsareprescribedon∂+and∂−.ppIfqn=Csinϕisprescribedontheellipticalboundariesinthelocalcoordinatesystemofeachellipsethenthesolutionissymmetricwithrespecttothey-axis,seeFigure3.2(b).Ifqn=Ccosϕisprescribedontheellipticalboundariesinthelocalcoordinatesystemofeachellipsethenthesolutionisantisymmetricwithrespecttothey-axis. P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoCome86FORMULATIONOFMATHEMATICALMODELSFigure3.2NotationforExample3.2.1.Exercise3.2.5ConsiderthedomainshowninFigure3.2(a).Assumethatu=0ontheboundaryrepresentedbytheinnercircle(∂(i))andq=0ontheboundariesrepresentednbyellipses.Prescribesinusoidalfluxesontheouterboundary(∂(o))thatwillresultin(a)periodic,(b)symmetricand(c)antisymmetricsolutions.3.2.4DimensionalreductionInmanyimportantpracticalapplicationsreductionofthenumberofdimensionsispossiblewithoutsignificantlyaffectingthedataofinterest.Inotherwords,amathematicalmodelinoneortwodimensionsmaybeanacceptablereplacementforthefullythree-dimensionalmodel.PlanarproblemsConsidertheplate-likebodyshowninFigure3.3.Thethicknesstzwillbeassumedconstantunlessotherwisestated.Themid-surfaceisthesolutiondomainwhichwillbedenotedby.Figure3.3Notationfortwo-dimensionalmodels. P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoComeHEATCONDUCTION87Thesolutiondomainliesinthexyplane.Theboundarypointsof(shownasadashedline)willbedenotedby.Theunitoutwardnormaltotheboundaryisdenotedbyn.Theheatconductionproblemintwodimensionsis∂∂u∂u∂∂u∂u∂ukx+kxy+kyx+ky+Q¯=cρ(3.18)∂x∂x∂y∂y∂x∂y∂twherethemeaningofQ¯dependsontheboundaryconditionsprescribedonthetopandbottomsurfaces(z=±tz/2)asdescribedinthefollowing.Equation(3.18)representsoneoftwocases:1.Thethicknessislargeincomparisonwiththeotherdimensionsandboththematerialpropertiesandboundaryconditionsareindependentofz,thatis,u(x,y,z)=u(x,y).Thisisequivalenttothecaseoffinitethicknesswiththetopandbottomsurfaces(z=±tz/2)perfectlyinsulatedandQ¯=Q.2.Thethicknessissmallinrelationtotheotherdimensions.Inthiscasethetwo-dimensionalsolutionisanapproximationofthethree-dimensionalsolutionthatcanbeinterpretedasthefirsttermintheexpansionofthethree-dimensionalsolutionwithrespecttothezcoordinate.ThedefinitionofQ¯dependsontheboundaryconditionsonthetopandbottomsurfacesasexplainedbelow:(a)Prescribedflux.Letusdenotetheheatfluxprescribedonthetop(resp.bottom)surfacebyqˆ+(resp.qˆ−).Notethatpositiveqˆisheatfluxexitingthebody.ThennnamountofheatexitingthebodyoverasmallareaAperunittimeis(qˆ++qˆ−)A.nnDividingbyAtz,theheatgeneratedperunitvolumebecomes1Q¯=Q−(qˆ++qˆ−)·(3.19)nntz(b)Convectiveheattransfer.Letusdenotethecoefficientofconvectiveheattransferatz=t/2(resp.z=−t/2)byh+(resp.h−)andthecorrespondingtemperatureofzzcctheconvectivemediumbyu+(resp.u−);thentheamountofheatexitingthebodyccoverasmallareaAperunittimeis[h+(u−u+)+h−(u−u−)]A.Thereforecccctheheatgeneratedperunitvolumeischangedto1Q¯=Q−[h+(u−u+)+h−(u−u−)]·(3.20)cccctzOfcourse,combinationsoftheseboundaryconditionsarepossible.Forexample,fluxmaybeprescribedonthetopsurfaceandconvectiveboundaryconditionsmaybeprescribedonthebottomsurface.Inthetwo-dimensionalformulationuisassumedtobeconstantthroughthethickness.Therefore,prescribingatemperaturehasmeaningonlyifthetemperatureisthesameonthetopandbottomsurfaces,aswellasonthesidesurface,inwhichcasethesolutionistheprescribedtemperature.Exercise3.2.6UsingthecontrolvolumeshowninFigure3.4,deriveEquation(3.18)fromfirstprinciples. P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoCome88FORMULATIONOFMATHEMATICALMODELSFigure3.4Controlvolumeandnotationforheatconductionintwodimensions.Exercise3.2.7Assumethattz=tz(x,y)>0isacontinuousanddifferentiablefunctionandthemaximumvalueoftzissmallincomparisonwiththeotherdimensions.Assumefurtherthatconvectiveheattransferoccursonthesurfacesz=±t/2withh+=h−=hzcccandu+=u−=u.UsingacontrolvolumesimilartothatshowninFigure3.4,butaccountingcccforvariablethickness,showthatinthiscasetheconservationlawforheatconductionintwodimensionsis∂∂∂u−(tzqx)−(tzqy)−2hc(u−uc)+Qtz=cρtz·(3.21)∂x∂y∂tRemark3.2.2InExercise3.2.7thethicknesstzwasassumedtobecontinuousanddif-ferentiableon.Iftziscontinuousanddifferentiableovertwoormoresubdomainsof,butdiscontinuousontheboundariesofthesubdomains,thenEquation(3.21)isapplicableoneachsubdomainsubjecttotherequirementthatqntziscontinuousontheboundariesofthesubdomains.Example3.2.2TheplanviewofaconductingmediumisshowninFigure3.5.Thethicknesstzisconstant.Thetopandbottomsurfaces(z=±tz/2)areperfectlyinsulated.Onthesidesurfaces(x=±b,y=±c)thetemperatureisconstant(u=uˆ0).Letkx=ky=k,kxy=0andQ=Q0wherekandQ0areconstants.Thegoalistodeterminethestationarytemperaturedistribution.Themathematicalproblemistosolve∂2u∂2uk++Q0=0∂x2∂y2ontherectangulardomainshowninFigure3.5withu=uˆ0ontheboundary.Figure3.5SolutiondomainforExample3.2.2. P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoComeHEATCONDUCTION89Thesolutionofthisproblemcanbedeterminedbyclassicalmethods:3Qb2∞(−1)(n−1)/2coshnπy/2b0u=uˆ0+161−cosnπx/2b.(3.22)kπ3n3coshnπc/2bn=1,3,5,...Thisinfiniteseriesconvergesabsolutely.Itisseenthattheclassicalsolutionofthisseeminglysimpleproblemisrathercomplicatedandinfacttheexactsolutioncanbecomputedonlyapproximately,althoughthetruncationerrorcanbemadearbitrarilysmallbycomputingasufficientlylargenumberoftermsoftheinfiniteseries.Exercise3.2.8Assumethatthecoefficientsofthermalconductivitykx,kxy,kyaregiven.ShowthatinaCartesiancoordinatesystemxy,rotatedcounterclockwisebytheangleαrelativetothexysystem,thecoefficientswillbe⎡⎤⎡⎤⎡⎤⎡⎤kxkxycosαsinαkxkxycosα−sinα⎣⎦=⎣⎦⎣⎦⎣⎦·kyxky−sinαcosαkyxkysinαcosαHint:Ascalar{a}T[K]{a},where{a}isanarbitraryvector,isinvariantundercoordinatetransformationbyrotation.AxisymmetricproblemsInmanyimportantpracticalproblemsthetopologicaldescription,thematerialpropertiesandtheboundaryconditionsareaxiallysymmetric.Asimpleexampleisastraightpipeofconstantwallthickness.Insuchcasestheproblemisusuallyformulatedincylindricalcoordinatesand,sincethesolutionisindependentofthecircumferentialvariable,thenumberofdimensionsisreducedtotwo.Inthefollowingthez-axiswillbetheaxisofsymmetryandtheradial(resp.circumferential)coordinateswillbedenotedbyr(resp.θ).ReferringtotheresultofExercise3.2.2,andlettingqθ=0,theconservationlawis1∂(rqr)∂qz∂u−−+Q=cρ·r∂r∂z∂tSubstitutingtheaxisymmetricformofFourier’slaw,∂u∂uqr=−kr,qz=−kz∂r∂zwegettheformulationoftheaxisymmetricheatconductionproblemincylindricalcoordi-nates:1∂∂u∂∂u∂urkr+kz+Q=cρ·(3.23)r∂r∂r∂z∂z∂t3See,forexample,Timoshenko,S.andGoodier,J.N.,TheoryofElasticity,2ndedition,McGraw-Hill,NewYork,1951,pp.275–276. P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoCome90FORMULATIONOFMATHEMATICALMODELSOneormoresegmentsoftheboundarymaylieonthez-axis.Itisimpliedintheformulationthattheboundaryconditionisthezero-fluxconditiononthosesegments.Thereforeitwouldnotbemeaningfultoprescribeessentialboundaryconditionsonthoseboundarysegments.Toshowthis,consideranaxisymmetricproblemofheatconduction,thesolutionofwhichisindependentofz.Forsimplicityweassumethatkr=1.Inthiscasetheproblemisessentiallyone-dimensional:1ddur=0ri0du→0asri→0drr=independentlyofuˆianduˆo.Exercise3.2.10DeriveEquation(3.23)byconsideringacontrolvolumeincylindricalco-ordinatesandusingtheassumptionthatthetemperatureisindependentofthecircumferentialvariable.Exercise3.2.11Considerwaterflowinginastainlesssteelpipe.Thetemperatureofthewateris80◦C.Theoutersurfaceofthepipeiscooledbyairflow.Thetemperatureoftheairis20◦C.Theouterdiameterofthepipeis0.20manditswallthicknessis0.01m.(a)Assumingthatconvectiveheattransferoccursonboththeinnerandoutersurfacesofthepipeanduisafunctionofronly,formulatethemathematicalmodelforstationaryheattransfer.(b)Assumethatthecoefficientofthermalconductivityofstainlesssteelis20W/mK.Usingh(w)=750W/m2Kforthewaterandh(a)=10W/m2Kfortheair,determinethetemperatureccoftheexternalsurfaceofthepipeandtherateofheatlossperunitlength. P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoComeHEATCONDUCTION91HeatconductioninabarOne-dimensionalmodelsofheatconductionarediscussedinthissection.Thesolutiondomainisabar,oneendofwhichislocatedatx=0,theotherendatx=,andthecross-sectionalareaA=A(x)>0isacontinuousanddifferentiablefunction.One-dimensionalformulationsarejustifiedwhenthesolutionofthethree-dimensionalproblemrepresentedbytheone-dimensionalmodelisafunctionofxonlyand/orthecross-sectionAissmall.Ifthesolutionisknowntobeafunctionofxonlythenthebarisperfectlyinsulatedalongitslength.Ifthecross-sectionissmallthentheboundaryconditionsonthesurfaceofthebararetransferredtothedifferentialequation.Wewillassumeinthefollowingthatthecross-sectionalareaissmallandconvectiveheattransfermayoccuralongthebar,asdescribedinSection3.2.4.Inthiscasetheconservationlawis∂(Aq)∂u−−cb(u−ua)+QA=cρA∂x∂twherecb=cb(x)isthecoefficientofconvectiveheattransferofthebar(inW/(mK)units)obtainedfromhcbyintegration:cb=hcdswiththecontourintegraltakenalongtheperimeterofthecross-section.Thereforethediffer-entialequationofheatconductioninabaris∂∂u∂uAk−cb(u−ua)+QA=cρA·(3.25)∂x∂x∂tOneoftheboundaryconditionsdescribedinSection3.2isprescribedatx=0andx=.Example3.2.3Considerstationaryheatflowinapartiallyinsulatedbaroflengthandconstantcross-sectionA.ThecoefficientskandcbareconstantandQ=0.ThereforeEqua-tion(3.25)canbecastinthefollowingform:22cbu−λ(u−ua)=0,λ:=·AkIfthetemperatureuaisalinearfunctionofx,thatis,ua(x)=a+bx,andtheboundaryconditionsareu(0)=uˆ0,q()=qˆthenthesolutionofthisproblemisu=C1coshλx+C2sinhλx+a+bxwhere1qˆC1=uˆ0−a,C2=−+(uˆ0−a)λsinhλ+b.λcoshλk P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoCome92FORMULATIONOFMATHEMATICALMODELSExercise3.2.12SolvetheproblemdescribedinExample3.2.3usingthefollowingboundaryconditions:q(0)=qˆ0,q()=h(u()−Ua)whereqˆ0,h,Uaaregivendata.Exercise3.2.13Aperfectlyinsulatedbarofconstantcross-section,length,thermalcon-ductivityk,densityρandspecificheatcissubjecttotheinitialconditionu(x,0)=U0(constant)andtheboundaryconditionsu(0,t)=u(,t)=0.AssumingthatQ=0,verifythatthesolutionofthisproblemis∞1−cos(nπ)n2π2kxu=2U0exp−tsinnπ·nπ2cρn=1Itissufficienttoshowthatthedifferentialequation,theboundaryconditionsandtheinitialconditionaresatisfied.Remark3.2.3AsnotedinSection2.1.4,morethanonephysicalphenomenoncanbemodeledbythesamedifferentialequation.Forexample,flowofviscousfluidsinporousmediaisbasedontheconservationlawwherethefluxrepresentstheflowrateofanincompressibleviscousfluidperunitarea(m/sunits)andQrepresentsadistributedsourceorsinkinjectingorextractingfluid(1/sunits).ThepotentialfunctionurepresentsthepiezometricheadandtherelationshipbetweenthepiezometricgradientandthefluxisformallyidenticaltoFourier’slawgivenbyEquation(3.8),exceptthatitiscalledDarcy’slaw,4andtheelementsofmatrix[K]arecalledcoefficientsofpermeability(m/sunits).TheboundaryconditionsareanalogoustothelinearboundaryconditionsdescribedinSection3.2;thepiezometrichead,theflux,oralinearcombinationofthefluxandpiezometricheadmaybeprescribed.Inviscousflowproblemsthereisnophysicalanalogytoradiation.Ontheotherhand,oneoftheboundariesmaybeafreesurface.Onafreesurfacethepiezometricheadequalstheelevationhead,thatis,uequalstheelevationwithrespecttoadatumplane.Furthermore,understationaryconditionsthefluxvectoristangentialtothefreesurface.Thepositionofthefreesurfaceisunknownaprioriandmustbedeterminedbyaniterativeprocess.3.3ThescalarellipticboundaryvalueproblemInviewofRemark3.2.3,agenerictreatmentofdiversephysicalphenomenaispossiblewhentheircommonmathematicalbasisisexploited.Wewillbeconcernedwiththefollowingmodelproblem:−div([κ]gradu)+cu=f(x,y,z)(x,y,z)∈(3.26)where⎡⎤κxκxyκxz[κ]:=⎣κyxκyκyz⎦(3.27)κzxκzyκz4HenryPhilibertGaspardDarcy(1803–1858). P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoComeLINEARELASTICITY93isapositive-definitematrixandc=c(x,y,z)≥0.InindexnotationEquation(3.26)reads−(κiju,j),i+cu=f.(3.28)Wewillbeconcernedwiththefollowingboundaryconditions:1.EssentialorDirichletboundarycondition:u=uˆisprescribedonboundaryre-gion∂u.Whenuˆ=0on∂uthentheDirichletboundaryconditionissaidtobehomogeneous.2.Neumannboundarycondition:−[κ]gradu·n=qˆnisprescribedonboundaryregion∂q.Inthisexpressionnistheunitoutwardnormaltotheboundary,showninFigure3.3.Whenqˆn=0on∂qthentheNeumannboundaryconditionissaidtobehomogeneous.3.Robinboundarycondition:−[κ]gradu·n=hR(u−uR)isgivenonboundaryseg-ment∂R.InthisexpressionhR>0anduRaregivenfunctions.WhenuR=0on∂RthentheRobinboundaryconditionissaidtobehomogeneous.Theboundarysegments∂u,∂q,∂Rand∂parenon-overlappingandcollectivelycovertheentireboundary∂.Anyoftheboundarysegmentsmaybeempty.WewillconsiderrestrictionsofEquation(3.26)tooneandtwodimensionsaswell.Inonedimensionwewilluseddu−κ+cu≡−(κu)+cu=f(x)(3.29)dxdxonthedomainI=(0,)withboundaryconditionsprescribedontheendpoints.3.4LinearelasticityMathematicalmodelsoflinearelastostaticandelastodynamicproblemsarebasedonthreefundamentalrelationships:thestrain–displacementequations,thestress–strainrelationshipsandtheequilibriumequations.Theunknownsarethecomponentsofthedisplacementvectoru=ux(x,y,z)ex+uy(x,y,z)ey+uz(x,y,z)ez≡{u(x,y,z)u(x,y,z)u(x,y,z)}Txyz≡ui(xj).(3.30)1.Straindisplacementrelationships.Wewillintroducetheinfinitesimalstrain–displacementrelationshipshere.Adetailedderivationoftheserelationshipsispre-sentedinSection10.2.1.Bydefinition,theinfinitesimalnormalstraincompo-nentsare∂ux∂uy∂uzx≡xx:=,y≡yy:=,z≡zz:=(3.31)∂x∂y∂z P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoCome94FORMULATIONOFMATHEMATICALMODELSandtheshearstraincomponentsareγxy1∂ux∂uyxy=yx≡:=+22∂y∂xγyz1∂uy∂uzyz=zy≡:=+(3.32)22∂z∂yγzx1∂uz∂uxzx=xz≡:=+22∂x∂zwhereγxy,γyz,γzxarecalledtheengineeringshearstraincomponents.Inindexnotation,thestateof(infinitesimal)strainatapointischaracterizedbythestraintensor1ij:=ui,j+uj,i.(3.33)22.Stressstrainrelationships.Mechanicalstressisdefinedasforceperunitarea(N/m2≡Pa).Since1pascal(Pa)isaverysmallstress,theusualunitofmechan-icalstressisthemegapascal(MPa)whichcanbeinterpretedeitheras106N/m2oras1N/mm2.Theusualengineeringnotationforstresscomponentsisillustratedonaninfinites-imalvolumeelementshowninFigure3.6.Theindexingrulesareasfollows:facestowhichthepositivex-,y-,z-axesarenormalarecalledpositivefaces,theoppositefacesarecallednegativefaces.Thenormalstresscomponentsaredenotedbyσ,theshearstressescomponentsbyτ.Thenormalstresscomponentsareassignedonesubscriptonly,sincetheorientationofthefaceandthedirectionofthestresscomponentarethesame.Forexample,σxisthestresscomponentactingonthefacestowhichthex-axisisnormalandthestresscomponentisactinginthepositive(resp.negative)coordinatedirectiononthepositive(resp.negative)face.Fortheshearstresses,thefirstindexreferstothecoordinatedirectionofthenormaltothefaceonwhichtheshearstressFigure3.6Notationforstresscomponents. P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoComeLINEARELASTICITY95isacting.Thesecondindexreferstothecoordinatedirectioninwhichtheshearstresscomponentisacting.Onapositive(resp.negative)facethepositivestresscomponentsareorientedinthepositive(resp.negative)coordinatedirections.ThereasonforthisisthatifwesubdivideasolidbodyintoinfinitesimalhexahedralvolumeelementssimilartotheelementshowninFigure3.6,thenanegativefacewillbecoincidentwitheachpositiveface.Bytheaction–reactionprinciple,theforcesactingoncoincidentfacesmustbeequalinmagnitudeandoppositeinsign.Analternativenotationisσxx≡σx,σxy≡τxy,etc.,whichcorrespondsdirectlytotheindexnotationσij.ThemechanicalpropertiesofisotropicelasticmaterialsarecharacterizedbythemodulusofelasticityE>0,Poisson’sratio5ν,andthecoefficientofthermalexpansionα>0.Thestress–strainrelationships,knownasHooke’slaw,6are1x=σx−νσy−νσz+αT(3.34)E1y=−νσx+σy−νσz+αT(3.35)E1z=−νσx−νσy+σz+αT(3.36)E2(1+ν)γxy≡2xy=τxy(3.37)E2(1+ν)γyz≡2yz=τyz(3.38)E2(1+ν)γzx≡2zx=τzx(3.39)EwhereT=T(x,y,z)isthetemperaturechangewithrespecttoareferencetem-peratureatwhichthestrainiszero.Thestraincomponentsrepresentthetotalstrain;αTisthethermalstrain.Mechanicalstrainisdefinedasthetotalstrainminusthethermalstrain.InindexnotationHooke’slawcanbewrittenas1+ννij=σij−σkkδij+αTδij.(3.40)EETheinverseisσij=λkkδij+2Gij−(3λ+2G)αTδij(3.41)whereλandG,calledtheLameconstants,´7aredefinedbyEνEλ:=,G:=·(3.42)(1+ν)(1−2ν)2(1+ν)5SimeonDenisPoisson(1781–1840).6RobertHooke(1635–1703).7GabrielLame(1795–1870).´ P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoCome96FORMULATIONOFMATHEMATICALMODELSGisalsocalledtheshearmodulusandmodulusofrigidity.SinceλandGarepositive,therangeofPoisson’sratiois−1<ν<1/2.Typically0≤ν<1/2.ThegeneralizedHooke’slawstatesthatthecomponentsofthestresstensorarelinearlyrelatedtothemechanicalstraintensor:σij=Cijkl(kl−αklT)(3.43)whereCijklandαijareCartesiantensors.BysymmetryconsiderationsthemaximumnumberofindependentelasticconstantsthatcharacterizeCijklis21.Thesymmetrictensorαijischaracterizedbysixindependentcoefficientsofthermalexpansion.Thisisthemostgeneralformofanisotropyinlinearelasticity.AccordingtoEquation(3.43)theelasticbodyisstress-freewhenthemechanical(0)straintensoriszero.Theremaybe,however,aninitialstressfieldσpresentintheijreferenceconfiguration.Theinitialstressfield,calledresidualstress,mustsatisfytheequilibriumequationsandstress-freeboundaryconditions[47].Residualstressesmaybeintroducedbymanufacturingprocesses,8forging,machining,andvarioustypesofcold-workingoperations[48].Incompositematerialsthereisalargedifferenceinthecoefficientsofthermalexpansionofthefiberandmatrix.Residualstressesareintroducedwhenapartcoolsfollowingcuringoperationsatelevatedtemperatures.Whenresidualstressesarepresentwehave(0)σij=σij+Cijkl(kl−αklT).(3.44)Accuratedeterminationofresidualstressesisgenerallydifficult.3.Equilibrium.Consideringthedynamicequilibriumofavolumeelement,similartothatshowninFigure3.6,exceptthattheedgesareoflengthx,y,z,sixequationsofequilibriumcanbewritten:theresultantsoftheforcesandmomentsmustvanish.Assumingthatthematerialisnotloadedbydistributedmoments(bodymoments),considerationofmomentequilibriumleadstotheconclusionthatτxy=τyx,τyz=τzy,τzx=τxz;thatis,thestresstensorissymmetric.Assumingfurtherthatthecomponentsofthestresstensorarecontinuousanddifferentiable,applicationofd’Alembert’sprinci-ple9andconsiderationofforceequilibriumleadstothreepartialdifferentialequations:∂σ∂τ∂τ∂2uxxyxzx+++Fx−=0(3.45)∂x∂y∂z∂t2∂τ∂σ∂τ∂2uxyyyzy+++Fy−=0(3.46)∂x∂y∂z∂t2∂τ∂τ∂σ∂2uxzyzzz+++Fz−=0(3.47)∂x∂y∂z∂t2whereF,F,Farethecomponentsofthebodyforcevector(inN/m3units),isthexyzspecificdensity(inkg/m3≡Ns2/m4units).Theseequationsarecalledtheequations8Forexample,7050-T7451aluminumplatesarehot-rolled,quenched,overagedandstretchedbytheimpositionof1.5to3.0%strainintherollingdirection.9JeanLeRondd’Alembert(1717–1783). P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoComeLINEARELASTICITY97ofmotion.Inindexnotation∂2uiσij,j+Fi=·(3.48)∂t2Forelastostaticproblemsthetimederivativeiszeroandtheboundaryconditionsareindependentoftime.Thisyieldstheequationsofstaticequilibrium:σij,j+Fi=0.(3.49)3.4.1TheNavierequationsTheequationsofmotion,calledtheNavierequations,10areobtainedbysubstitutingEqua-tion(3.41)intoEquation(3.48).Inelastodynamicstheeffectsoftemperatureareusuallynegligible,hencewewillassumeT=0:∂2uiGui,jj+(λ+G)uj,ji+Fi=2·(3.50)∂tInelastostaticproblemswehaveGui,jj+(λ+G)uj,ji+Fi=(3λ+2G)α(T),i.(3.51)Exercise3.4.1DeriveEquation(3.51)bysubstitutingEquation(3.41)intoEquation(3.49).IndicatetherulesunderwhichtheindicesarechangedtoobtainEquation(3.51).Hint:(kkδij),j=(uk,kδij),j=uk,ki=uj,ji.Exercise3.4.2Derivetheequilibriumequationsfromfirstprinciples.Exercise3.4.3InderivingEquations(3.50)and(3.51)itwasassumedthatλandGareconstants.FormulatetheanalogousequationsassumingthatλandGaresmoothfunctionsofxi∈.Exercise3.4.4Assumethatistheunionoftwoormoresubdomainsandthematerialpropertiesareconstantsoneachsubdomainbutvaryfromsubdomaintosubdomain.Formulatetheelastostaticproblemforthiscase.3.4.2BoundaryandinitialconditionsAsinthecaseofheatconduction,wewillconsiderthreekindsofboundaryconditions:prescribeddisplacements,prescribedtractionsandspringboundaryconditions.Tractionsareforcesperunitareaactingontheboundary.Prescribeddisplacementsandtractionsareoftenspecifiedinanormal–tangentreferenceframe:10ClaudeLouisMarieHenriNavier(1785–1836). P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoCome98FORMULATIONOFMATHEMATICALMODELSFigure3.7Springboundarycondition:schematicrepresentation.1.Prescribeddisplacement.Oneormorecomponentsofthedisplacementvectorisprescribedonallorpartoftheboundary.Thisiscalledakinematicboundarycondition.2.Prescribedtraction.Oneormorecomponentsofthetractionvectorisprescribedonallorpartoftheboundary.ThedefinitionoftractionvectorisgiveninAppendixC.1.3.Linearspring.Alinearrelationshipisprescribedbetweenthetractionanddisplace-mentvectorcomponents.ThegeneralformofthisrelationshipisTi=cij(dj−uj)(3.52)whereTiisthetractionvector,cijisapositive-definitematrixthatrepresentsthedistributedspringcoefficients,djisaprescribedfunctionthatrepresentsdisplacementimposedonthespringandujisthe(unknown)displacementvectorfunctionontheboundary.Thespringcoefficientsc(inN/m3units)maybefunctionsofthepositionijxbutareindependentofthedisplacementu.Thisiscalleda“Winkler11spring.”kiAschematicrepresentationofthisboundaryconditiononaninfinitesimalboundarysurfaceelementisshowninFigure3.7undertheassumptionthatcijisadiagonalmatrixandthereforethreespringcoefficientsc1:=c11,c2:=c22andc3:=c33characterizetheelasticpropertiesoftheboundarycondition.Figure3.7shouldbeinterpretedtomeanthattheimposeddisplacementdiwillcauseadifferentialforceFitoactonthecentroidofthesurfaceelement.Suspendingthesummationrule,themagnitudeofFiisFi=ciA(di−ui),i=1,2,311EmilWinkler(1835–1888). P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoComeLINEARELASTICITY99whereuiisthedisplacementofthesurfaceelement.ThecorrespondingtractionvectorisFiTi=lim=ci(di−ui),i=1,2,3.A→0ATheenumeratedboundaryconditionsmayoccurinanycombination.Forexample,thedisplacementvectorcomponentu1,thetractionvectorcomponentT2andalinearcombinationofT3andu3maybeprescribedonaboundarysegment.Inengineeringpracticeboundaryconditionsaremostconvenientlyprescribedinthenormal–tangentreferenceframe.Thenormalisuniquelydefinedonsmoothsurfacesbutthetangentialcoordinatedirectionsarenot.Itisnecessarytospecifythetangentialcoordinatedirectionswithrespecttothereferenceframeused.TherequiredcoordinatetransformationsarediscussedinAppendixC.Theboundaryconditionsaregenerallytimedependent.Fortime-dependentproblemstheinitialconditions,thatis,theinitialdisplacementandvelocityfields,denotedrespectivelybyU(x,y,z)andV(x,y,z),havetobeprescribed:∂uu(x,y,z,0)=U(x,y,z)and=V(x,y,z).∂t(x,y,z,0)Exercise3.4.5Assumethatthefollowingboundaryconditionsaregiveninthenormal–tangentreferenceframex,xbeingcoincidentwiththenormal:T=c(d−u);T=i111112T=0.Usingthetransformationx=gx,determinetheboundaryconditionsinthex3iijjicoordinatesystem.(SeeAppendixC,SectionC.3.)3.4.3Symmetry,antisymmetryandperiodicitySymmetryandantisymmetryofavectorfunctionwithrespecttoalineareillustratedinFigure2.3.Thedefinitionsofsymmetryandantisymmetryofavectorfunctioninthreedimensionsareanalogous.Thecorrespondingvectorcomponentsparalleltoaplaneofsym-metry(resp.antisymmetry)havethesameabsolutevalueandthesame(resp.opposite)sign.Thecorrespondingvectorcomponentsnormaltoaplaneofsymmetry(resp.antisymmetry)havethesameabsolutevalueandopposite(resp.same)sign.Inaplaneofsymmetrythenormaldisplacementandtheshearingtractioncomponentsarezero.Inaplaneofantisymmetrythenormaltractioniszeroandthein-planecomponentsofthedisplacementvectorarezero.Whenthesolutionisperiodiconthenaperiodicsectorofhasatleastoneperiodicboundarysegmentpairdenotedby∂+and∂−.Oncorrespondingpointsofaperiodicppboundarysegmentpair,P+∈∂+andP−∈∂−,thenormalcomponentofthedisplacementppvectorandtheperiodicin-planecomponentsofthedisplacementvectorhavethesamevalue.Thenormalcomponentofthetractionvectorandtheperiodicin-planecomponentsofthetractionvectorhavethesameabsolutevaluebutoppositesign.Exercise3.4.6AhomogeneousisotropicelasticbodywithPoisson’sratiozerooccupiesthedomain={x,y,z||x|0}(3.53)whereω∈R2isaboundeddomain.Thelateralboundaryofthebodyisdenotedby={(x,y,z)|(x,y)∈∂ω,−/20}(3.54)andthefacesaredenotedbyγ±={(x,y,z)|(x,y)∈ω,z=±/2}.(3.55)ThenotationisshowninFigure3.8.Thediameterofωwillbedenotedbydω.Figure3.8Notation. P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoComeLINEARELASTICITY101Thematerialproperties,volumeforcesandtemperaturechangeactingonandtractionsactingonwillbeassumedtobeindependentofz.Thereforethexyplaneisaplaneofsymmetry.Itisassumedthattractionsarespecifiedontheentireboundary.Thisisusuallycalledthefirstfundamentalboundaryvalueproblemofelasticity.Whenthetractionsactingonγ±arezeroand/dω1thensuchproblemsareusuallyformulatedonωasplanestressproblems.Whenthenormaldisplacementsandshearingtractionsactingonγ±arezerothensuchproblemsareformulatedonωasplanestrainproblems,independentlyofthe/dωratio.When1/dωandthetractionsactingonγ±arezerothenthethree-dimensionalthermoelasticityproblemoniscalledthegeneralizedplanestrainproblem.Inageneralizedplanestrainproblemthestressresultantscorrespondingtoσzmustbezero:σzdzdy=0σzxdzdy=0σzydzdy=0.∂ω∂ω∂ωFordetailswereferto[14].PlanarelastostaticmodelsarespecializationsoftheNavierequations.Themid-planeisunderstoodtobeaplaneofsymmetry,thatis,uz(x,y,0)=0.Theformulationiswritteninunabridgednotationinthefollowing:1.Thelinearstraindisplacementrelationships.Theseare∂uxx:=(3.56)∂x∂uyy:=(3.57)∂y∂ux∂uyγxy:=+·(3.58)∂y∂xIntwo-dimensionalproblemstheshearstraincomponentsγyzandγzxandthecorre-spondingshearstresscomponentsτyz,τzxarezero.2.Stressstrainrelationships.Wewillbeconcernedwithtwospecialcases,thecaseofplanestress(σz=τyz=τzx=0)andthecaseofplanestrain(z=γyz=γzx=0).Applyingtheappropriaterestrictionstothethree-dimensionalHooke’slaw,fromEquation(3.34)andEquation(3.35),weobtainforplanestress⎧⎫⎡⎤⎧⎫⎧⎫1ν0⎨σx⎬⎨x⎬⎨1⎬E⎢ν10⎥EαTσy=⎣⎦y−1(3.59)⎩⎭1−ν21−ν⎩⎭1−ν⎩⎭τxy00γxy02and,lettingz=0inEquation(3.36),weobtainforin-planestrain⎧⎫⎡⎤⎧⎫⎧⎫⎨σx⎬λ+2Gλ0⎨x⎬⎨1⎬σy=⎣λλ+2G0⎦y−(3λ+2G)αT1·(3.60)⎩⎭⎩⎭⎩⎭τxy00Gγxy0 P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoCome102FORMULATIONOFMATHEMATICALMODELS3.Thestaticequationsofequilibrium.Theseare∂σx∂τxy++Fx=0(3.61)∂x∂y∂τyx∂σy++Fy=0(3.62)∂x∂ywhereF,FarethecomponentsofbodyforcevectorF(x,y)(inN/m3units).xyTheNavierequationsareobtainedbysubstitutingthestress–strainandstrain–displacementrelationshipsintotheequilibriumequations.Forplanestrain∂∂u∂u∂2u∂2uEα∂Txyxx(λ+G)++G+=−Fx(3.63)∂x∂x∂y∂x2∂y21−2ν∂x∂∂u∂u∂2u∂2uEα∂Txyyy(λ+G)++G+=−Fy.(3.64)∂y∂x∂y∂x2∂y21−2ν∂yTheboundaryconditionsaremostconvenientlywritteninthenormal–tangent(nt)referenceframeillustratedinFigure3.3.Therelationshipbetweenthexyandntcomponentsofdisplacementandtractionisestablishedbytherulesofvectortransformation,describedinAppendixC,SectionC.3.ThelinearboundarydescribedinSection3.4.2isdirectlyapplicableintwodimensions.Exercise3.4.8DeriveEquation(3.59)andEquation(3.60)fromHooke’slaw.Exercise3.4.9ShowthatforplanestresstheNavierequationsareE∂∂u∂u∂2u∂2uEα∂Txyxx++G+=−Fx(3.65)2(1−ν)∂x∂x∂y∂x2∂y21−ν∂xE∂∂u∂u∂2u∂2uEα∂Txyyy++G+=−Fy.(3.66)2(1−ν)∂y∂x∂y∂x2∂y21−ν∂yExercise3.4.10Denotethecomponentsoftheunitnormalvectortotheboundarybynxandny.ShowthatTx=Tnnx−TtnyTy=Tnny+TtnxwhereTn(resp.Tt)isthenormal(resp.tangential)componentofthetractionvector.AxisymmetricelastostaticmodelsAxialsymmetryexistswhenthesolutiondomaincanbegeneratedbysweepingaplanefigurearoundanaxis,knownastheaxisofsymmetry,andthematerialproperties,loading P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoComeINCOMPRESSIBLEELASTICMATERIALS103andconstraintsareaxiallysymmetric.Forexample,pipes,cylindricalandsphericalpressurevesselsareoftenidealizedinthisway.Theradial,circumferentialandaxialcoordinatesaredenotedbyr,θandzrespectivelyandthedisplacement,stress,strainandtractioncomponentsarelabeledwithcorrespondingsubscripts.Theproblemisformulatedintermsofthedisplacementvectorcomponentsur(r,z)anduz(r,z).1.Thelinearstraindisplacementrelationships.Theseare[87]∂urr:=(3.67)∂rurθ:=(3.68)r∂uzz:=(3.69)∂zγrz1∂ur∂uzrz=:=+·(3.70)22∂z∂r2.Stressstrainrelationships.Forisotropicmaterialsthestress–strainrelationshipisobtainedfromEquation(3.41):⎧⎫⎡⎤⎧⎫⎧⎫⎪⎪σr⎪⎪λ+2Gλλ0⎪⎪r⎪⎪⎪⎪1⎪⎪⎨σθ⎬⎢λλ+2Gλ0⎥⎨θ⎬EαT⎨1⎬=⎢⎥−.(3.71)⎪⎪σz⎪⎪⎣λλλ+2G0⎦⎪⎪z⎪⎪1−2ν⎪⎪1⎪⎪⎩⎭⎩⎭⎩⎭τrz000Gγrz03.Equilibrium.Theelastostaticequationsofequilibriumare[87]1∂(rσr)∂τrzσθ+−+Fr=0(3.72)r∂r∂zr1∂(rτrz)∂σz++Fz=0.(3.73)r∂r∂zExercise3.4.11WritedowntheNavierequationsfortheaxisymmetricmodel.3.5IncompressibleelasticmaterialsWhenν→1/2thenλ→∞,thereforetherelationshiprepresentedbyEquation(3.41)breaksdown.ReferringtoEquation(3.40),thesumofnormalstraincomponentsisrelatedtothesumofnormalstresscomponentsby1−2νkk=σkk+3αT.E P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoCome104FORMULATIONOFMATHEMATICALMODELSThesumofnormalstraincomponentsiscalledthevolumetricstrainandwillbedenotedbyvol:=kk.Defining1σ0:=σkk,3weget3(1−2ν)vol≡kk=σ0+3αT(3.74)Ewherethefirsttermontherightisthemechanicalstrainandthesecondtermisthethermalstrain.Forincompressiblematerials,thatis,whenν=1/2,vol=3αTisindependentofσ0.Thereforeσ0cannotbecomputedusingHooke’slaw.SubstitutingEquation(3.74)intoEquation(3.41)andlettingν=1/2weget2Eσij=σ0δij+(ij−αTδij).(3.75)3SubstitutingintoEquation(3.49),andassumingthatEandαareconstant,2E(σ0),i+(ij,j−α(T),i)+Fi=0.3Writing11ij,j=(ui,j+uj,i),j=(ui,jj+uj,ij)22andinterchangingtheorderofdifferentiationinthesecondtermwegetuj,ij=uj,ji=(uj,j),i=(jj),i=3α(T),i.Therefore,forincompressiblematerials,13ij,j=ui,jj+α(T),i.22TheproblemistodetermineuisuchthatE(σ0),i+ui,jj+α(T),i+Fi=0(3.76)3subjecttotheconditionofincompressibility,thatis,theconditionthatvolumetricstraincanbecausedbytemperaturechangebutnotbymechanicalstress,ui,i=3αT(3.77) P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoComeSTOKES’FLOW105andtheappropriateboundaryconditions.Ifdisplacementboundaryconditionsareprescribedovertheentireboundary(∂u=∂)thentheproblemissolvableonlyiftheprescribeddisplacementsareconsistentwiththeincompressibilitycondition:uinidS=3αTdV.∂ThisfollowsdirectlyfromintegratingEquation(3.77)andapplyingthedivergencetheorem.Exercise3.5.1Anincompressiblebarofconstantcross-sectionandlengthissubjectedtouniformtemperaturechange(i.e.,Tisconstant).Thecentroidalaxisofthebariscoincidentwiththex1-axis.Theboundaryconditionsareu1(0)=u1()=0.Thebodyforcevectoriszero.ExplainhowEquations(3.75)and(3.77)areappliedinthiscasetofindthatσ11=−EαT.Exercise3.5.2Writetheequationsofincompressibleelasticityinunabridgednotation.3.6StokesflowTheflowofviscousfluidsatverylowReynoldsnumbers12(Re<1)ismodeledbytheStokes13equations.ThereisaverycloseanalogybetweentheStokesequationsandtheequationsofincompressibleelasticitydiscussedinSection3.5.Influidmechanicstheaveragenormalstressisthepressure,p:=−σ0.ThevectoruirepresentsthecomponentsofthevelocityvectorandtheshearmodulusoftheincompressibleelasticsolidE/3isreplacedbythecoefficientofdynamicviscosityμ(measuredinNs/m2units):μui,jj=p,i−Fi(3.78)ui,i=0.(3.79)Exercise3.6.1WritetheStokesequationsinunabridgednotation.Exercise3.6.2AssumethatvelocitiesareprescribedovertheentireboundaryforaStokesproblem(i.e.,∂u=∂).Whatconditionmustbesatisfiedbytheprescribedvelocities?Remark3.6.1Inthischapterwetreatedthephysicalpropertiessuchascoefficientsofheatconduction,thesurfacecoefficient,themodulusofelasticityandPoisson’sratioasgivenconstants.Readersshouldbemindfulofthefactthatphysicalpropertiesareempiricaldatainferredfromexperimentalobservations.Duetovariationsinexperimentalconditionsandotherfactors,thesedataarenotknownpreciselyandarealwayssubjecttorestrictions.Forexample,stressisproportionaltostrainuptotheproportionallimitonly;thefluxisproportionaltothetemperaturegradientonlywithinanarrowrangeoftemperatures.Infact,thecoefficientofthermalconductivity(k)istypicallytemperaturedependent.Forexample,inthecaseofAISI304stainlesssteelkchangesfromabout15toabout20W/m◦Cinthe12OsborneReynolds(1842–1912).13GeorgeGabrielStokes(1819–1903). P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoCome106FORMULATIONOFMATHEMATICALMODELStemperaturerangeof0to400◦C[11].Foranarrowrangeoftemperature(say100to200◦C)thesizeoftheuncertaintyinkisaboutthesameasthechangeinthemeanvalue.Therefore,usingaconstantvalueforkthisrangemaybe“goodenough.”Ignoringtemperaturedependenceforamuchwiderrangeoftemperaturescanleadtosubstantialerrors.Takingintoaccountthetemperaturedependenceofthecoefficientofthermalconductivityleadstotheformulationofnonlinearproblems,thesolutionsofwhicharefoundbyiterativelysolvingsequencesoflinearproblems.ThisisdiscussedinSection10.1.2.Analystsusuallyrelyondatapublishedinvarioushandbooks.Thepublisheddatacanvarywidely,however.Forexample,BabuskaandSilva,havingconsultedsevenreferences,foundˇthatinthesereferencesthecoefficientofthermalconductivityofpureironvariesbetween71.8and80.4W/(mK)[11].Publisheddataaretypicallyaccompaniedbylegalandtechnicaldisclaimers.3.7ThehierarchicviewofmathematicalmodelsUptothispointwehaveconsideredtheformulationofmathematicalmodelsintheformoflinearpartialdifferentialequationsandlinearboundaryconditions.Linearmodelsofheatcon-ductionandelasticityshouldbeviewedasspecialcasesofmodelswithlessrestrictiveassump-tions,suchasmodelsthataccountformaterialnonlinearities,whichinturnshouldbeviewedasspecialcasesofmodelsthatalsoaccountforradiation,mechanicalcontact,largestrainsanddisplacements,etc.Inthissense,anymathematicalmodelshouldbeviewedasaspecialcaseofamorecomprehensivemodel.Thisisthehierarchicviewofmathematicalmodels.Manypossiblemathematicalmodelscanbeconstructedfortherepresentationofmateriallaws.Forexample,manymodelshavebeenproposedtorepresentelastic–plastic,viscoelasticandviscoplasticbehavior.Eachcanbeviewedasspecialcaseofamoreelaboratemodelandthereforeamemberofahierarchicsequence.Amodelbasedontheassumptionsoflinearelasticityisaspecialcaseofanymodelthataccountsforplasticdeformation;however,therearemanymodelsofplasticdeformationthatarenotmembersofthesamehierarchicsequence.Thereforealternativesequencesofhierarchicmodelscanbeconstructedforthesimulationofaphysicalreality.Theeffectsofnonlinearitiesanduncertaintiesinmaterialpropertiesandboundarycon-ditionsonthedataofinterestareevaluatedintheprocessofconceptualizationdiscussedinSection1.1.1.Thisrequiresthatappropriatesoftwaretoolsbeavailablefortheconsidera-tionofnonlineareffects.AnintroductorytreatmentofnonlinearformulationsispresentedinChapter10.Ideallythechoiceofamathematicalmodelforaparticularpurposeinvolvesinformedjudgmentbasedonsystematicconsiderationofalternativechoicesandthequalityandreliabil-ityofavailableinformation.Inpracticethechoiceofmathematicalmodelsisofteninfluencedbythesubjectivepreferencesoftheanalystandthelimitationsoftheavailablesoftwaretoolsandothercomputationalresources.3.8ChaptersummaryTheformulationofmathematicalmodelsforlinearproblemsinheatconductionandelasticitywasdescribed.Amathematicalmodelwasunderstoodtobeoneormorepartialdifferential P1:OSOJWST055-03JWST055-SzaboFebruary9,201111:42PrinterName:YettoComeCHAPTERSUMMARY107equationsandtheprescribedinitialandboundaryconditions.AlternativeformulationswillbepresentedinChapter4.Themodelofheatconductionisbasedontheconservationlaw,afundamentallawofphysics,andsomeempiricalrelationshipsbetweenthederivativesofthetemperatureuandthefluxvectorwhicharesubjecttocertainlimitations.Forexample,thecoefficientsofthermalconductivitygenerallydependonthevalueofuaswellasthegradientofu.Onlywithinacertainrangeofuanditsgradientcanthesecoefficientsbetreatedasconstants.Similarly,themodelforelasticbodiesisbasedontheconservationofmomentum(instatics,theequationsofequilibrium)andanempiricallinearrelationshipbetweenthestressandstraintensors.Thelinearrelationshipholdsforsmallstrainsonly.Itisimportanttobearthelimitationsofaparticularmathematicalmodelinmindwhenconsideringitinthecontextofanengineeringdecision-makingprocess.Amathematicalmodelmustneverbeconfusedwiththephysicalrealitythatitwasconceivedtosimulate.Dimensionalreductionshouldbeunderstoodasaspecialcaseofthefullythree-dimensionalformulation.Itwasnotedthatmorethanonephysicalphenomenoncanberepresentedbyapar-tialdifferentialequation.Thisallowsgenericmathematicaltreatmentofvariousphysicalphenomena. P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoCome4GeneralizedformulationsTheideaofageneralizedformulationwasintroducedintheone-dimensionalsetting,anditsusefulnessforobtainingapproximatesolutionsbythefiniteelementmethodwasdemon-stratedinChapter2.Inthischaptergeneralizedformulationsintwoandthreedimensionsaredescribed.Theexamplesandexercisesinthischapterareintroductionstotheuseofthefiniteelementmethodforsolvinglinearproblemsinheattransferandelasticity.Itisassumedthatthereaderisfamiliarwithatleastonefiniteelementsoftwareproduct.ThesolutionspresentedherewereobtainedwithStressCheck.4.1ThescalarellipticproblemMultiplyingEquation(3.28)byatestfunctionvandintegrating,weget−(κiju,j),ivdV+cuvdV=fvdV.(4.1)Clearly,thisequationmustholdforarbitraryv,providedthattheindicatedoperationsaredefined.Thefirstintegralcanbewrittenas(κiju,j),ivdV=(κiju,jv),idV−κiju,jv,idV.Applyingthedivergencetheorem(Equation(3.1))weget(κiju,jv),idV=κiju,jnivdS∂IntroductiontoFiniteElementAnalysis:Formulation,VerificationandValidation,FirstEdition.BarnaSzabóandIvoBabuška.©2011JohnWiley&Sons,Ltd.Published2011byJohnWiley&Sons,Ltd.ISBN:978-0-470-97728-6 P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoCome110GENERALIZEDFORMULATIONSwhereniistheunitnormalvectortotheboundarysurface.ThereforeEquation(4.1)canbewritteninthefollowingform:−κiju,jnivdS+κiju,jv,idV+cuvdV=fvdV.(4.2)∂Itiscustomarytowriteqi:=−κiju,jandqn:=qini.Withthisnotationwehaveκiju,jv,idV+cuvdV=fvdV−qnvdS.(4.3)∂ThisisthegeneralizationofEquation(2.36)totwoandthreedimensions.AswehaveseeninSection2.3,thespecificstatementofageneralizedformulationdependsonthebound-aryconditions.Inthegeneralcaseu=uˆisprescribedon∂u(Dirichletboundarycondi-tion),qn=qˆnisprescribedon∂q(Neumannboundarycondition)andqn=hR(u−uR)isprescribedonR(Robinboundarycondition),seeSection3.3.WenowdefinethebilinearformasB(u,v):=κiju,jv,idV+cuvdV+hRuvdS(4.4)∂RandthelinearfunctionalasF(v):=fvdV−qnvdS+hRuRvdS.(4.5)∂q∂ROfcourse,if∂RisemptythenthelasttermsinEquations(4.4)and(4.5)arenotpresent.F(v)(resp.B(u,v))satisfiesallofthepropertieslistedinSectionA.3(resp.SectionA.4).ThespaceE()isdefinedbyE():={u|B(u,u)≤C<∞}andthenorm1uE:=B(u,u)2isassociatedwithE().ThespaceofadmissiblefunctionsisdefinedbyE˜():={u|u∈E(),u=uˆon∂u}.Hereweassumethat,correspondingtoanyu=uˆspecifiedon∂,thereisau∈E()usuchthatu=uˆon∂.ThisimposescertainrestrictionsonuˆandensuresthatE˜()isnotu P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoComeTHESCALARELLIPTICPROBLEM111empty.ThespaceoftestfunctionsisdefinedbyE0():={u|u∈E(),u=0on∂}.uThegeneralizedformulationisnowstatedasfollows:“Findu∈E˜()suchthatB(u,v)=F(v)forallv∈E0().”Afunctionuthatsatisfiesthisconditioniscalledageneralizedsolution.ThegeneralizedformulationisoftenbasedonTheorem2.6.2.TheexactsolutionofthegeneralizedformulationminimizesthepotentialenergydefinedbyEquation(2.92)onthespaceE˜(I).Alternatively,thepotentialenergyisdefinedby111(u):=κuudV+cu2dV+h(u−u)2dSij,i,iRR222∂R−fudV+qnudS.(4.6)∂qNotethatπ(u)differsfrom(u)onlybyaconstant.Thereforetheminimumπ(u)isthesameastheminimumof(u).Exercise4.1.1Considerthegeneralizedformulationofsteadystateheatconductionincylindricalcoordinatesinthespecialcasewhenthesolutiondependsonlyontheradialvariabler:rodudvduduk(r)rdr=rkv−rkv·ridrdrdrr=rodrr=ri(a)DerivethisformulationfromEquation(3.23)and(b)applythisformulationtoalongpipeofinternalradiusri,externalradiusrousingtheboundaryconditionsu(ri)=uˆiandduqn:=−k=hc(u−uc)atr=ro.drExercise4.1.2FollowingtheproofofTheorem2.6.2,showthatthegeneralizedformulationminimizes(u)givenbyEquation(4.6).4.1.1ContinuityIntwoandthreedimensionsu∈E()isnotnecessarilycontinuousorbounded.Forexample,thefunctionu:=log|logr|wherer:=(x2+y2)1/2isdiscontinuousandunboundedatthepointr=0yetitliesinE().Thishasimportantimplications:recallthatindiscussingconcentratedforcesinSection2.5.5continuityofthetestfunctionvwasinvoked.Thesameargumentcannotbeextendedtotwoandthreedimensions,henceconcentratedforcesandconcentratedfluxesareinadmissibledata.Similarly,pointconstraintsareinadmissibleexceptfortheenforcementoftheuniquenessofthesolutionwhenNeumannconditionsarespecifiedontheentireboundary. P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoCome112GENERALIZEDFORMULATIONSExercise4.1.3Letr:=(x2+y2)1/2and:={r|r≤ρ<1}.Showthatu:=logrdoes01notlieinE()butu2:=log|logr|does.Hint:u2=log(−logr)on.4.1.2ExistenceToguaranteetheexistenceofageneralizedsolution,itissufficienttoshowthat:(a)Thereexistpositiverealnumbersα,βsuchthatforanyxi∈αx2≤κxx≤βx2ijijwhereκisthematrixinEquation(3.27),xisavectorofrealnumbersandx2:=ijixixi.(b)Further,0≤c(x)≤γ,0≤h≤whereγ,∈R1andiRf2dV<∞,q2dS<∞,u2dS<∞.nRqR(c)When∂q=∂(i.e.,∂uand∂Rareempty)andc=0,thenthespecifiedfunc-tionsfandqnhavetosatisfythefollowingcondition:fdV−qndS=0.(4.7)∂ThisisbecauseE0()=E()andhencev=C(whereCisanarbitraryconstant)liesinthetestspace.ThereforeinEquation(4.4),B(u,C)=0andconsequentlyF(C)hastobezeroalso:F(C)=CfdV−qndS=0.∂Ifthisconditionissatisfiedthen,asshowninTheorem2.6.1inSection2.6,thesolutionisuniqueuptoanarbitraryconstant.(d)When∂u=∅thenuˆmustbesuchthatE˜isnotempty.Inonedimensionthisconditionwassatisfiedbecauseanyu∈E(I)iscontinuousonI¯.Intwoandthreedimensionsnoteveryu∈E()iscontinuouson¯(see,forexample,Exercise4.1.3).Itisnecessarytoprescribeuˆonlyonboundarysegmentsthathavenon-zerolengthintwodimensionsandnon-zeroareainthreedimensions.Inotherwords,uˆcannotbeprescribedatapointintwoandthreedimensions,oralongalineinthreedimensions.Exercise4.1.4ExplainthephysicalmeaningofEquation(4.7).4.1.3ApproximationbythefiniteelementmethodWhenseekinganapproximatesolutionbythefiniteelementmethod,denotedbyuFE,weconstructafinite-dimensionalsubspaceofE(),calledthefiniteelementspace P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoComeTHESCALARELLIPTICPROBLEM113Figure4.1SolutiondomainandfiniteelementmeshforExample4.1.1.S=S(, ,p,Q),whichisanalogoustothefiniteelementspaceinonedimension,de-scribedinSection2.5.3.FiniteelementspacesintwoandthreedimensionsaredescribedinChapter5.ThesubspaceS˜(resp.S0)isthecorrespondingfinite-dimensionalspaceofadmis-siblefunctions(resp.testfunctions).Theapproximationproblemisformulatedasfollows:“Findu∈S˜suchthatB(u,v)=F(v)forallv∈S0.”FEFEExample4.1.1Astainlesssteelpipeisinsulatedbypolyurethane(PU)foamwhichisencasedinapolyvinylchloride(PVC)pipe.Theoutsidediameterofthestainlesssteel(ss)pipeis250.0mm,thewallthicknessis22mm.TheoutsidediameterofthePVCcasingis450mm,thewallthicknessis20mm.Thestainlesssteelpipeiscarryingahotliquidat400K.ThePVCcasingiscooledbyconvection.Thephysicalpropertiesare:kPVC=0.14W/mK,hPVC=6.5W/(m2K),k=17W/mK,k=0.025W/mK.ThetemperatureoftheconvectivemediumssPUis300K.Determinethetemperaturedistributionandtherateofheatlossperunitlengthofpipe.Duetocircularsymmetry,thisisessentiallyaone-dimensionalproblem;however,wewillformulateitintwodimensions.Utilizingsymmetry,wedefinethesolutiondomainasa30◦sectorABCD,showninFigure4.1.Thematerialsareassumedtobeisotropic,thereforeκij=kδijwherek=k(x,y)isthethermalconductivity,apiecewiseconstantfunction.Inthiscasec=0andnoheatisgeneratedinthematerials,henceEquation(4.3)issimplifiedtoku,iv,idx1dx2=−qnvds.(4.8)TheboundaryconditionsarelistedinTable4.1.NotethatsegmentsABandCDlieonlinesofsymmetry.Thereforeqn=0andthelineintegralontherighthandsideiszeroontheseboundarysegments.OnsegmentDAtheboundaryconditionu=400Kisenforcedbyrestrictiononthespaceofadmissiblefunctionsandthetestfunctionsv∈E0()arezero,thereforethelineintegralontherighthandsideiszeroonthisboundarysegment.OnsegmentBCqn=6.5(u−300).ThereforeEquation(4.8)canbewrittenasku,iv,idx1dx2+6.5uvds=6.5×300vds.(4.9)BCBCB(u,v)F(v) P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoCome114GENERALIZEDFORMULATIONSTable4.1Example4.1.1:boundaryconditions.SegmentsDescriptionConditionAB,CDSymmetryqn=0BCConvectionqn=6.5(u−300)DATemperatureu=400ThedefinitionsforB(u,v)andF(v)areindicatedinEquation(4.9).Tocompletethegeneralizedformulationoftheproblem,wefirstdefinetheenergyspaceE():E():={u|B(u,u)≤C<∞}whereCissomepositiveconstant.ThenormassociatedwithE(),calledtheenergynorm,isdefinedby1uE:=B(u,u).2ThespaceofadmissiblefunctionsE˜()isdefinedbyE˜():={u|u∈E(),u=400onDA}andthespaceoftestfunctionsE0()isdefinedbyE0():={u|u∈E(),u=0on}.DAThegeneralizedformulationisstatedasfollows:“Findu∈E˜()suchthatB(u,v)=F(v)forallv∈E0().”InthefiniteelementapproximationthesolutiondomainABCDwaspartitionedintothreeelementswithcurvedboundariescorrespondingtothethreematerials(seeFigure4.1).AsequenceofsevensolutionswasobtainedusingasequenceoffiniteelementspacesS1⊂S⊂···⊂Scharacterizedbyuniformp-distributionsrangingfrom1to7(trunkspace)1on27thethree-elementmesh.Therelativeerrorintheenergynormwasestimatedbyextrapolation.Theestimatedrelativeerrorintheenergynormvs.thenumberofdegreesoffreedomNisshowninFigure4.2(a).Theestimatederrorintheenergynormisseentodecreaseveryrapidly:atp=3itisalreadywellunder1%.Theheatlossrateper30◦sectorperunitlengthiscomputedfromHL30◦=qndsBC1Theterm“trunkspace”isdefinedinSection5.2.1. P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoComeTHEPRINCIPLEOFVIRTUALWORK1151002.65p=1p=11022.60130.142.550.0150.001Estimatedlimit62.476W/m2.500.00017Heatlossper30deg.sector(W/m)3456p=7Estimatedrel.errorinenergynorm(%)25101002.4520406080Numberofdegreesoffreedom(N)Numberofdegreesoffreedom(N)(a)(b)Figure4.2(a)Estimatedrelativeerrorinenergynorm(%)vs.Nand(b)computedheatlossrateper30◦sector(W/m)inExample4.1.1.bynumericalquadrature.TheconvergenceoftheheatlossHL30◦(W/m)isshowninFigure4.2(b).TheestimatedlimitvaluewithrespecttoN→∞is2.476W/m.Ofcourse,thisnumbermustbemultipliedby12toobtaintheheatlossrateperunitlengthofpipe(29.7W/m).Exercise4.1.5Assumethatthevalueofthecoefficientofthermalconductivityofthepolyurethanefoam(kPU)inExample4.1.1isnotknownprecisely.Itisknownonlytobeintherange0.02to0.03W/mK.Computethecorrespondingrangeoftheheatlossrateperunitlengthofpipe.Exercise4.1.6FormulateandsolvetheproblemofExample4.1.1asanaxisymmetricproblem.4.2TheprincipleofvirtualworkConsidertheequationsofequilibrium(3.48)withoutinertiaforces:σij,j+Fi=0.(4.10)MultiplyEquation(4.10)byatestfunctionviandintegrateon:σij,jvidV+FividV=0.(4.11) P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoCome116GENERALIZEDFORMULATIONSObservethatiftheequilibriumequation(4.10)issatisfiedthenEquation(4.11)holdsforarbitraryvi,subjecttotheconditionthattheindicatedoperationsaredefined.Wewriteσij,jvidV=(σijvi),jdV−σijvi,jdVandusethedivergencetheoremtoobtainσij,jvidV=σijnjvidS−σijvi,jdV.∂Notingthatσijnj=Ti(seeEquation(C.3)inAppendixC),Equation(4.11)canbewrittenasσijvi,jdV=FividV+TividS.(4.12)∂Thisequationhasaphysicalinterpretation:thetestfunctionvicanbeinterpretedassomearbitrarydisplacementfield,independentoftheappliedbodyforceFiandtractionTi.Forthisreasonviiscalleda“virtualdisplacement.”Thetermsontherighthandsiderepresenttheworkdonebythebodyforceandthetractionforcesactingonthebody,collectivelycalled“externalforces.”Thelefthandsiderepresentstheworkdonebytheinternalstresses.Toseethis,refertoFigure4.3andassumethatvertexAoftheinfinitesimalhexahedralelement,thecoordinatesofwhicharexi,issubjectedtoavirtualdisplacementvi.Then,sinceviiscontinuousanddifferentiable,thefacelocatedatx1+dx1willbedisplaced,relativetopointA,byvi,1dx1andthevirtualworkdonebyσ11isdWσ11=(σ11dx2dx3)(v1,1dx1)=σ11v1,1dV.forcedisplacement(a)(b)Figure4.3Virtualdisplacementscorrespondingto(a)σ11and(b)σ13. P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoComeELASTOSTATICPROBLEMS117Similarly,thevirtualworkdonebyσ13isdWσ13=(σ13dx2dx3)(v3,1dx1)=σ13v3,1dVforcedisplacementetc.Theprincipleofvirtualworkstatesthat“thevirtualworkofexternalforcesisequaltothevirtualworkofinternalstresses.”Notethatsincethisresultisbasedontheequilib-riumEquation(4.10),itisindependentofthematerialpropertiesandthereforeholdsforanycontinuum.Equation(4.12)isthegenericformoftheprincipleofvirtualwork.Particularstatementsoftheprincipleofvirtualworkdependonthematerialpropertiesandtheboundaryconditions.Oftenanalternativeformof(4.12)isused:observethatthesumσijvi,jequalsthesumσ11v1,1+σ22v2,2+σ33v3,3plusthesumofpairslikeσ12v1,2+σ21v2,1.Sinceσij=σji,thiscanbewrittenas11(v)(v)σ12v1,2+σ21v2,1=σ12(v1,2+v2,1)+σ21(v2,1+v1,2)=σ1212+σ212122wherethesuperscript(v)indicatesthatthesearetheinfinitesimalstraintermscorrespondingtothetestfunctionvi,thatis,(v)1ij:=(vi,j+vj,i).2Therefore(v)σijvi,j=σijijandEquation(4.12)canbewrittenas(v)σijijdV=FividV+TividS.(4.13)∂Thisequationhasgreatimportanceintheformulationofmathematicalmodelsincontinuummechanics.Inthischapterwewillconsideritsapplicationtoproblemsinlinearelasticity.Exercise4.2.1StartingfromEquation(3.3),derivethecounterpartofEquation(4.13)forthestationaryheatconductionproblem.4.3ElastostaticproblemsOnsubstitutingEquation(3.43)intoEquation(4.13)weobtain(v)(v)CijklijkldV=FividV+TividS+CijklijαklTdV.(4.14)∂Thisisanapplicationoftheprincipleofvirtualworktoelastostaticproblems.Specificstatementsoftheprincipleofvirtualworkdependontheboundaryconditions.Let∂u P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoCome118GENERALIZEDFORMULATIONSdenotetheboundaryregionwhereui=uˆiisprescribed;let∂TdenotetheboundaryregionwhereTi=Tˆiisprescribed;andlet∂sdenotetheboundaryregionwhereTi=kij(dj−uj)(i.e.,springboundarycondition)isprescribed.Letusdefine(v)B(u,v):=CijklijkldV+kijujvidS(4.15)∂sandF(v):=FividV+TˆividS+kijdjvidS∂T∂s(v)+CijklijαklTdV(4.16)whereu≡uiandv≡vi.IntheinterestofsimplicitywewillassumethatthematerialconstantsE,νandαarepiecewiseconstant.ThespaceE(),calledtheenergyspace,isdefinedbyE():={u|B(u,u)≤C<∞}(4.17)andthenorm1uE:=B(u,u)(4.18)2isassociatedwithE().ThespaceofadmissiblefunctionsisdefinedbyE˜():={ui|ui∈E(),ui=uˆion∂u}.Notethatthisdefinitionimposesarestrictionontheprescribeddisplacementconditions:forthereasonsdiscussedinSection4.1.2,therehastobeaui∈E()sothatui=uˆion∂u.ThespaceoftestfunctionsisdefinedbyE0():={u|u∈E(),u=0on∂}.iiiuThegeneralizedformulationbasedontheprincipleofvirtualworkisstatedasfollows:“Findu∈E˜()suchthatB(u,v)=F(v)forallv∈E0().”Exercise4.3.1WhenthematerialisisotropicwecansubstituteEquation(3.41)intoEqua-tion(4.12)toobtain(v)(v)λkkii+2GijijdV=FividV+TividS∂E(v)+αTiidV.(4.19)1−2ν P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoComeELASTOSTATICPROBLEMS119Wedefinethedifferentialoperatormatrix[D]andthematerialstiffnessmatrix[E]asfollows:⎡⎤∂00⎢∂x⎥⎢∂⎥⎢⎥⎡⎤⎢⎢0∂y0⎥⎥λ+2Gλλ000⎢⎢∂⎥⎥⎢⎢λλ+2Gλ000⎥⎥⎢00⎥⎢λλλ+2G000⎥[D]:=⎢⎢∂∂∂z⎥⎥[E]:=⎢⎢⎥⎥.⎢0⎥⎢000G00⎥⎢⎢∂y∂x⎥⎥⎣0000G0⎦⎢0∂∂⎥00000G⎢⎥⎢∂z∂y⎥⎣∂∂⎦0∂z∂xFurthermore,wedenoteu≡{u}:={uuu}Tandv≡{v}:={vvv}T.ShowthatxyzxyzEquation(4.19)canbewritteninthefollowingform:([D]{v})T[E][D]{u}dV={v}T{F}dV+{v}T{T}dS∂⎧⎫⎨1⎬∂vx∂vy∂vzEαT+1dV(4.20)∂x∂y∂z⎩⎭1−2ν1where{F}:={FFF}Tisthebodyforcevectorand{T}:={TTT}Tisthetractionxyzxyzvector.Remark4.3.1Inthegeneralanisotropiccase,representedbyEquation(4.14),wehave([D]{v})T[E][D]{u}dV={v}T{F}dV+{v}T{T}dS∂+([D]{v})T[E]{α}TdV(4.21)wherethematerialstiffnessmatrix[E]isasymmetricpositive-definitematrixwithatmost21independentcoefficientsand{α}:={ααα2α2α2α}T.1122331223314.3.1UniquenessThegeneralizedformulationbasedontheprincipleofvirtualworkisuniqueintheenergyspaceE().TheproofofuniquenessgivenbyTheorem2.6.1inSection2.6isapplicabletotheelasticityprobleminthreedimensions.Uniquenessintheenergyspacedoesnotnecessarilymeanuniquenessofthedisplacementfieldu.When∂uand∂sarebothemptythentherearesixlinearlyindependenttestfunc-tionsinE0()=E()forwhich(v)=0andhenceB(u,v)=0.Threeofthesefunctionsij P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoCome120GENERALIZEDFORMULATIONScorrespondtorigidbodydisplacements:(v)(1)T11=0:vi=c1{100}(v)(2)T22=0:vi=c2{010}(v)(3)T33=0:vi=c3{001}andthreecorrespondtoinfinitesimalrigidbodyrotations:(v)(4)T12=0:vi=c4{−x2x10}rotationaboutx3(v)(5)T23=0:vi=c5{0−x3x2}rotationaboutx1(v)(6)T31=0:vi=c6{x30−x1}rotationaboutx2wherec1,c2,...,c6arearbitraryconstants.Consequentlythebodyforcevectorandthesurfacetractionsmustsatisfythefollowingsixconditions:(k)(k)F(v)=0:FividV+TividS=0k=1,2,...,6.(4.22)∂Thephysicalinterpretationoftheseconditionsisthatthebodymustbeinequilibrium,thatis,thesumofforcesandthesumofmomentsmustbezero.Thesolutionisuniqueuptorigidbodydisplacements.Inordertoensureuniquenessofthesolution,“rigidbodyconstraints”areimposed;thatis,therigidbodydisplacementfunctionsareeliminatedfromthespaceoftestfunctions:0(k)E()={vi|vi∈E(),vi=0,k=1,2,...,6}.(4.23)Thevaluesofrigidbodydisplacementsarearbitrary,thereforethespaceofadmissiblefunc-tionsisE˜()={u|u∈E(),u(k)=uˆ(k),k=1,2,...,6}(4.24)iiii(k)whereuˆarearbitraryrigidbodydisplacements,usuallychosentobezero.iRigidbodyconstraintsareenforcedbysettingsixdisplacementcomponentsinatleastthreepointstoarbitraryvalues.Theusualprocedureisasfollows.Threenon-collinearpoints,labeledA,B,CinFigure4.4,areselectedarbitrarily,subjectonlytotherestrictionthatthedisplacementfunctionmustbecontinuousatthosepoints.ACartesiancoordinatesystemisassociatedwiththesepointssuchthatpointsAandBlieonaxisX1,axisX3isperpendiculartotheplanedefinedbythepointsABCandaxisX2isperpendiculartoaxesX1,X3.The(A)(A)(A)(B)(B)(C)displacementcomponentsU,U,U,U,UandU,showninFigure4.4,are123233assignedarbitraryvalues,usuallyzero.Thiswillensurethatthecoefficientsoftherigidbodydisplacementfunctionsareuniquelydefined. P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoComeELASTOSTATICPROBLEMS121Figure4.4Rigidbodyconstraints:notation.Forexample,letting(A)(A)(A)U1=C1−C4X2+C6X3=0(A)(A)(A)U2=C2+C4X1−C5X3=0(A)(A)(A)U3=C3+C5X2−C6X1=0preventsrigidbodytranslation.Letting(B)(B)(B)U2=C2+C4X1−C5X3=0(B)(B)(B)U3=C3+C5X2−C6X1=0preventsrigidbodyrotationaboutaxesX3andX2respectively,andletting(C)(C)(C)U3=C3+C5X2−C6X1=0preventsrigidbodyrotationaboutaxisX1.Thisisequivalenttowriting⎡⎤⎧⎫100000⎪⎪C1⎪⎪⎢010−a00⎥⎪⎪C⎪⎪⎢⎥⎪⎪⎨2⎪⎪⎬⎢⎢00100a⎥⎥C3⎢010b00⎥C=0(4.25)⎢⎥⎪⎪⎪⎪4⎪⎪⎪⎪⎣00100−b⎦⎪⎪C5⎪⎪⎩⎭0010c0C6wherethedeterminantofthecoefficientmatrixisc(a+b)2whichisnonzeroforanychoiceofthenon-collinearpointsA,B,Candthereforeallrigidbodydisplacementmodesvanish.Ofcourse,thereareotherwaysofpreventingorspecifyingrigidbodydisplacements.Itisnec-essaryonlythatthesolutionatthepointswhererigidbodyconstraintsareprescribedmustbecontinuousandthedeterminantofthecoefficientmatrix,analogoustothatinEquation(4.25),mustbenon-zero. P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoCome122GENERALIZEDFORMULATIONS3Figure4.5IllustrationforExample4.3.1andExercise4.3.2.Inmanycases∂uand/or∂sarenotemptybuttheprescribedconditionsdonotprovideasufficientnumberofconstraintstopreventallrigidbodydisplacementsand/orrotations.Inthosecasesitisnecessarytoprovideasufficientnumberofrigidbodyconstraintstopreventrigidbodymotion.SuchacaseisillustratedinExample4.3.2.Example4.3.1Athinelasticplate-likebodywithacircularhole,showninFigure4.5,isloadedbyconstantnormaltractionsT0.Theequilibriumconditions(4.22)areobviouslysatisfied.InthiscasetherearesixrigidbodymodesandthespaceofadmissiblefunctionsisgivenbyEquation(4.24)andthespaceoftestfunctionsbyEquation(4.23).Letting(A)(A)(A)(B)(B)(C)u=u=u=u=u=u=0123233therigidbodyconstraintsareenforced.Example4.3.2Ifthecenterofthecircularholeinthethinelasticplate-likebodyofExample4.3.1islocatedatx1=/2thenthesolutionissymmetricwithrespecttotheplanex1=/2andtheproblemmaybeformulatedonthehalfdomainshowninFigure4.6(a).Ona(b)(a)Figure4.6(a)Domain,Example4.3.2;(b)planarmodel,Example4.3.3. P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoComeELASTOSTATICPROBLEMS123planeofsymmetrythenormaldisplacementcomponentunandtheshearingstresscomponentsarezero.Settingun=u1=0ontheplaneofsymmetrypreventsdisplacementinthex1directionandrotationsaboutthex2-andx3-axes,butdoesnotpreventdisplacementsinthex2andx3coordinatedirections,norrotationaboutthex1-axis.Onemayset(B)(B)(C)u=u=u=0233topreventrigidbodydisplacementsnotpreventedbytheplaneofsymmetry.Remark4.3.2Inplanarproblemsthemid-planeistreatedasaplaneofsymmetry,hencethereareatmostthreerigidbodydisplacementmodes:thein-planedisplacementcomponentsandrotationaboutanarbitrarynormaltothemid-plane.TheprobleminExample4.3.2canbeformulatedasaplanarproblem.Inthatcasetheimpliedzeronormaldisplacementofthemid-planepreventsdisplacementinthex3directionaswellasrotationaboutthex1-andx2-axes.Therefore,onlyrigidbodyconstrainthastobeimposed,forexample,byletting(A)u2=0.NotethatinFigure4.6(b)thecoordinateaxesx1,x2areunderstoodtolieinthemid-plane.Exercise4.3.2RefertoFigure4.5.RelocatepointAto{000}andpointCto{w0}.Assumethatthefollowingconstraintsarespecified:(A)(A)(A)(B)(B)(C)u=u=u=u=u=u=0.123233WritedownthesystemofequationsanalogoustoEquation(4.25)anddeterminewhetherthecoefficientmatrixisoffullrank.Exercise4.3.3RefertothenotationinFigure4.7.Thestressresultantsactingonthecircularboundaryofacylindricalbodyatz=0aretheforcevectorcomponentsdenotedbyFx,Fy,FzandthemomentvectorcomponentsdenotedMx,My,Mz.Figure4.7NotationforExercise4.3.3. P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoCome124GENERALIZEDFORMULATIONSUsingformulasavailableinintroductorytextsonthestrengthofmaterials,thetractionvectorcomponentscorrespondingtoFy,Fz,MxandMzactingonthecircularboundaryatz=0are2MzyTx(x,y)=−(4.26)πr404Fy22MzxTy(x,y)=2(1−(y/r0))+4(4.27)3πrπr00Fz4MxyTz(x,y)=+·(4.28)πr2πr4001.AssumethatonlythebendingmomentMxandtheshearingforceFyarenon-zero.Determinethetractionvectorcomponentsactingonthecross-sectionatz=−and,usingsymmetryandantisymmetryasappropriate,sketchthesmallestdomainonwhichthisproblemcanbesolved.Specifythenecessaryrigidbodyconstraints.2.AssumethatonlythebendingmomentMx,theshearingforceFyandthetwistingmomentMzarenon-zero.Determinethetractionvectorcomponentsactingonthecross-sectionatz=−and,usingsymmetryandantisymmetryasappropriate,sketchthesmallestdomainonwhichthisproblemcanbesolved.Specifythenecessaryrigidbodyconstraints.3.AssumethattheaxialforceFz,thebendingmomentMx,thetwistingmomentMzandtheshearingforceFyarenon-zero.Determinethetractionvectorcomponentsactingonthecross-sectionatz=−and,usingsymmetryandantisymmetryasappropriate,sketchthesmallestdomainonwhichthisproblemcanbesolved.Specifythenecessaryrigidbodyconstraints.Example4.3.3LetusformulatetheproblemshowninFigure4.6asaplanestressproblem.ThesolutiondomainisshowninFigure4.6(b)andtheboundaryconditionsaregiveninTable4.2.Thestress–strainlawisgivenbyEquation(3.59).InthiscasethevirtualworkofinternalstressesisB(u,v):=([D]{v})T[E][D]{u}tdxdxz12Table4.2Example4.3.3:boundaryconditions.SegmentsDescriptionConditionAB,CD,EFFreeTn=Tt=0DATractionloadingTn=T0,Tt=0BE,FCSymmetryun=Tt=0 P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoComeELASTOSTATICPROBLEMS125wheretzisthethickness,matrix[D]isadifferentialoperatormatrixandthematerialstiffnessmatrix[E]isdefinedbyEquation(3.59)⎡⎤∂0⎡⎤⎢⎢∂x1⎥⎥1ν0⎢∂⎥E⎢ν10⎥[D]:=⎢0⎥[E]:=⎣⎦⎢∂x2⎥1−ν21−ν⎣∂∂⎦002∂x2∂x1and{v}:={vv}T,{u}:={uu}T.Thevirtualworkofexternalforcesis1212F(v)=−T0v1tzds.DAThespaceE()isdefinedbyEquation(4.17).Sincealloftheprescribeddisplacementsarezero,thespaceofadmissiblefunctionsisthesameasthespaceoftestfunctions:E˜()=E0():={u|u∈E(),u=0onand}.iinBEFCThegeneralizedformulationofthisproblemis:“Findui∈E˜()suchthatB(u,v)=F(v)forallv∈E0().”Thesolutionofthisproblemexists(becausetheequilibriumconditioniissatisfied)andisuniqueuptoonerigidbodydisplacementu(2).Itcanbeshownthatthesolutioniscontinuouson¯,thereforeitispossibletosetu(2)=0bylettingu=0atapoint2(e.g.,atthepointCinFigure4.6).Thefunctionsthatlieinafiniteelementspacearealwayscontinuouson¯.Rigidbodyconstraintsareenforcedbysettingoneormorecomponentsofthedisplacementvectoratnodepoint(s)toanarbitraryvalue,usuallytozero.Thisispermissibleprovidedthattheequilibriumconditionsaresatisfied.Letusnowsolvethisproblemwiththefollowingdata:=100mm,w=40mm,tz=2mm,r0=3.5mm,b=10mm,T0=75MPa.Thematerialpropertiesare:E=70×103MPa,ν=0.35.Thesematerialpropertiesaretypicalforthealuminumalloy6061-T6,theyieldstrengthofwhichis265MPa.Thegoalofanalysisistodeterminethelocationandmagnitudeofmaximumprincipalstresswithareasonablecertaintythattherelativeerrorisnotgreaterthan2%.Tosolvethisproblembythefiniteelementmethod,subspacesforE()areconstructed.Inthisexample,a10-elementmesh,shownintheinsetinFigure4.8,wasconstructed.Thepolynomialdegreeswereuniformlydistributedandrangedfromp=1top=8(trunkspace)2.ThesolutionswereperformedwithStressCheck.Theconvergenceofthefirstprincipalstresswithrespecttothenumberofdegreesoffreedom(N)anditslocationareshowninFigure4.8.ItisseenthatthefirstprincipalstressisvirtuallyindependentofNforp≥6.TheestimateofthelimitvalueisbasedontheassumptionthattheabsolutevalueoftheerrorinσisproportionaltoN−1.12Thedefinitionfortheterm“trunkspace”isgiveninSection5.2.1inChapter5. P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoCome126GENERALIZEDFORMULATIONS280Estimatedlimit259.2MPap=5p=6p=7p=8260p=4240p=2p=3220locationofmaximum200p=1180100200300400500600700Max.valueofthefirstprincipalstress(MPa)160Numberofdegreesoffreedom(N)Figure4.8Example4.3.3:convergenceofthefirstprincipalstress.Remark4.3.3ThesolutionoftheprobleminExample4.3.3isavailableinstandardengi-neeringhandbooks,seeforexample[60],[95].Thesolutionisbasedontheassumptionthatthelengthislargeincomparisonwiththeotherdimensionssothatitdoesnotaffectthesolution.Thestressdistributioncorrespondingtothesolutionisindependentofthematerialproperties.Example4.3.4InExample4.3.2itwaspossibletoreducethesizeoftheproblembyutilizingsymmetry.Insomecasesitispossibletotakeadvantageofantisymmetry.Onaplaneofantisymmetrythetangentialcomponentsofthedisplacementvectorandthenormalstressarezero.Consider,forexample,anannularplatewithinternalradiusri,externalradiusro,loadedbyconstantshearingtractionsontheinnerandoutersurfaces,asshowninFigure4.9.Thethicknessisconstant.Anystraightlinepassingthroughtheoriginisalineofantisymmetry.Figure4.9AnnularplateforExample4.3.4. P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoComeELASTOSTATICPROBLEMS127Table4.3Example4.3.4:boundaryconditions.SegmentsDescriptionConditionAB,CDAntisymmetryTn=ut=0(o)BCTractionloadingTn=0,Tt=Tt(i)DATractionloadingTn=0,Tt=TtPointBRigidbodyconstraintu(B)=0yThereforeonecandefinethesolutiondomainastheshadedsectorABCDindicatedinFigure4.9.Thesolutiondomainisboundedbytwolinesofantisymmetry.NotethatsettingthetangentialdisplacementstozeroonboundarysegmentsABandCDdoesnotpreventrigidbodyrotationabouttheorigin.Thereforeitisnecessarytoimposearigidbodyconstraintatonepoint(e.g.,pointB)inthecircumferentialdirection.TheboundaryconditionsarelistedinTable4.3.Exercise4.3.4Considertheannularplateloadedbyconstantshearingtractionsontheinnerandoutersurfaces,showninFigure4.9.Letri=50mm,ro=100mm,Thethicknessisconstant:t=5mm.Thematerialpropertiesare:E=200×103MPa,ν=0.3.LetT(i)=zt120MPa.(o)Determinethe(constant)tractionTtactingontheoutersurfacesothatthetractionssatisfythestaticequilibriumequationsandobtainfiniteelementsolutionsusing(a)ase-quenceofuniformlyrefinedtriangularelementswithp=1;(b)asequenceofuniformlyrefinedquadrilateralelementswithp=1;and(c)oneelementusingp=1,2,...,8.PlotthecircumferentialdisplacementofpointAvs.thenumberofdegreesoffreedom.Exercise4.3.5ArectangularplateofconstantthicknessissubjectedtolinearlyvaryingnormaltractionsasshowninFigure4.10.Thisisaclassicalproblemofelasticity,represen-tativeofbeamsinpurebending.Takingadvantageofsymmetryandantisymmetry,definetheboundaryconditionsforthedomainABCD.Checkwhetherrigidbodydisplacementispossibleandspecifytheappropriateconstraint(s)ifnecessary.4.3.2TheprincipleofminimumpotentialenergyBydefinitionthepotentialenergyisthefunctional1π(u):=B(u,u)−F(u).(4.29)2Figure4.10PlateforExercise4.3.5. P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoCome128GENERALIZEDFORMULATIONSTheprincipleofminimumpotentialenergystatesthattheexactsolutionofthegeneralizedformulationbasedontheprincipleofvirtualworkistheminimizerofthepotentialenergyonthespaceofadmissiblefunctions:π(u)=minπ(u).(4.30)EXu∈E˜()TheproofgivenbyTheorem2.6.2inSection2.6isdirectlyapplicabletotheproblemofelasticity.Thistheoremisvalidevenifthedefinitionofthepotentialenergyismodifiedbyanarbitraryconstant.Specifically,referringtoEquations(4.15)and(4.16),wedefinethepotentialenergyfortheproblemofelasticityasfollows:1(u):=Cijkl(ij−αijT)(kl−αklT)dV21+kij(ui−di)(uj−dj)dS2∂s−FiuidV−TiuidS.(4.31)∂TTheadvantageofthisdefinitionoverthedefinitiongivenbyEquation(4.29)isthatinthespecialcaseswhenafreebodyissubjectedtoatemperaturechange(ij=αijT),orabodywithaspringboundaryconditionisgivenarigidbodydisplacement(ui=di),then(u)=0,whereasπ(u)=0.InthefiniteelementmethodE˜()isreplacedbyafinite-dimensionalsubspaceS˜:π(u)=minπ(u).(4.32)FEu∈S˜()InExample4.3.3asequenceofhierarchicfiniteelementspaceswasusedwiththepolynomialdegreeprangingfrom1to8.Whenasequenceofspacesishierarchic(i.e.,S1⊂S2⊂...)thenthepotentialenergyconvergesmonotonically.Exercise4.3.6Comparethedefinitionofπ(u)givenbyEquation(4.29)withthedefinitionof(u)givenbyEquation(4.31)andshowthatthetwodefinitionsdifferbyaconstant,definedasfollows:11(u)−π(u)=CααT2dV+kdddS.ijklijklijij22∂sExercise4.3.7Showthatforisotropicelasticmaterialswithν=0,(u)canbewritteninthefollowingform:211+νE(u):=λ−αT+2G−(αT)2dVkkijij2νν1+kij(ui−di)(uj−dj)dS−FiuidV−TiuidS(4.33)2∂s∂T P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoComeELASTOSTATICPROBLEMS129Figure4.11Circularholeinaninfiniteplatesubjectedtounidirectionaltension(σ∞).andverifythatifanunconstrainedelasticbodyissubjectedtoatemperaturechangeT(i.e.,ij=αTδij)then(u)=0.Example4.3.5Inthisexampleweillustratetheperformanceoftheerrorestimatorbasedonextrapolationonthebasisofatwo-dimensionalproblem,theexactsolutionofwhichisananalyticfunction.Wewillbeconcernedwiththesolutionintheneighborhoodofacircularholeinaninfiniteplatesubjectedtounidirectionaltension,shownschematicallyinFigure4.11.We“remove”thedomainABCDEshowninFigure4.11fromtheinfiniteplateandimposeonedgesBCandCDthestressdistributionoftheclassicalsolutionfortheinfiniteplate:3a233a4σx=σ∞1−cos2θ+cos4θ+cos4θ(4.34)r222r4a213a4σy=σ∞−cos2θ−cos4θ−cos4θ(4.35)r222r4a213a4τxy=σ∞−sin2θ+sin4θ+sin4θ(4.36)r222r4whereσ∞istheuniaxialstressintheplate;aistheradiusofthehole;andrandθarepolarcoordinates,definedasshowninFigure4.11.OnboundarysegmentsABandDEsymmetryconditionsareimposed.ThecirculararcEAisstressfree.Weselectb=w=4aandcomputethesolutionforplanestrainconditionswithν=0.3.Thecomponentsofthedisplacementvector,withtherigidbodydisplacementandtherigidbodyrotationtermssettozero,areσaraa3∞ux=(κ+1)cosθ+2((1+κ)cosθ+cos3θ)−2cos3θ(4.37)8Garr3σaraa3∞uy=(κ−3)sinθ+2((1−κ)sinθ+sin3θ)−2sin3θ(4.38)8Garr33See,forexample,[45]. P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoCome130GENERALIZEDFORMULATIONSwhereGisthemodulusofrigidityandκisaconstantwhichdependsonlyonPoisson’sratioν:⎧⎨3−νforplanestressκ:=1+ν(4.39)⎩3−4νforplanestrain.Becauseweknowtheexactsolution,theexactvalueofthepotentialenergycanbecomputed.Notingthatu∈E0(),EXB(u,u)=F(u)EXEXEXandtheexactvalueofthepotentialenergyis11(u)=B(u,u)−F(u)=−F(u)=−u2.(4.40)EXEXEXEXEXEXE22Therefore(u)canbedeterminedfromcomputingF(u):EXEXwbF(uEX)=(σxux+τxyuy)tzdy+(τxyux+σyuy)tzdx00σ2a2t∞z=15.3873(4.41)Ewherew=b=4awasused.ThiscomputationwasperformedwithMathematica.ThefiniteelementcomputationswereperformedwithStressCheck,usingonlytwofiniteelements.TheverticesofthefiniteelementsarelabeledABCFandCDEFinFigure4.11.LineCFisnormaltothecircle.Theloadvectorswerecomputedfromthetractionscorre-spondingtothestresscomponentsgivenbyEquations(4.34)through(4.36)using14-pointGaussianquadratureontheelementsidescorrespondingtoboundarysegmentsBCandCD.ThenumberofdegreesoffreedomN,thenormalizedpotentialenergycomputedfromthefiniteelementsolution,theestimatedandtrueratesofconvergence2β,theestimatedandtruerelativeerrorsinenergynorm(er)E,andtheeffectivityindexθaregivenforν=0.3inTable4.4.Theestimatedrelativeerrorintheenergynormwascomputedusingthebestavailableestimateoftheexactvalueofthepotentialenergy,thatis,thesequenceofpotentialenergiescomputedforp=6top=8.TheprocedureisdescribedinSection2.7.Forprangingfrom1to6theestimatedandtruerelativeerrorsareveryclose.Forp=7andp=8theestimatesarenotasaccurate.ThisisbecausetheestimateisbasedontheassumptionthattheerrorintheenergynormisproportionaltoN−β.Inthiscase,however,theerrorgoestozerofasterthanN−β,thereforetheerrorintheenergynormisoverestimated,whereastherateofconvergenceisunderestimated.ThispointisaddressedinChapter6.Exercise4.3.8Considertheproblem−u=f(x,y)on¯={x,y|0≤x≤a,0≤y≤b} P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoComeELASTOSTATICPROBLEMS131Table4.4Example4.3.5:circularholeinaninfiniteplate.Planestrain,ν=0.3.Estimatedandtruerelativeerrorsinenergynorm.Effectivityindex(θ).β(er)E(%)(u)EFEpNEst.TrueEst.Trueθσ2a2tz∞18−7.36767——20.5920.591.00220−7.547400.440.4413.8013.791.00332−7.620090.730.739.809.781.00448−7.659040.920.936.736.711.00568−7.678761.191.214.444.401.01692−7.688051.561.622.772.701.037120−7.691651.771.941.731.611.078152−7.692951.772.221.140.961.19∞∞−7.6936500wheref(x,y)=sin(mπx/a)sin(nπy/b).Theboundaryconditionisu=0on∂.Theexactsolutionofthisproblemsis1a2b2uEX=sin(mπx/a)sin(nπy/b)(4.42)π2m2b2+n2a2andthepotentialenergyoftheexactsolutionis1a3b3(uEX)=−·(4.43)8π2m2b2+n2a2Letm=3,n=2.Usingp-extensiononuniformmesheswith1,4,and16elements,tabulatetheeffectivityindicesasinTable4.4.Exercise4.3.9AnalyzethenotchedrectangularplateloadedbyashearforceVandbendingmomentsM1,M2asshowninFigure4.12.Thematerialis6061-T6aluminum:E=69.6GPa,ν=0.365,σyld=276MPa.TheplateisloadedbyashearforceVandbendingmomentsM1,M2asshown.Figure4.12NotationforExercise4.3.9. P1:OSOJWST055-04JWST055-SzaboFebruary18,20117:5PrinterName:YettoCome132GENERALIZEDFORMULATIONS1.Definethesolutiondomain.Definealldimensionsasparameters.4Forinitialvaluesusea=180mm,b=30mm,c=45mm,d=65mm,h=24mm,R=3mm,thicknesstz=4mm.Imposethenecessaryrestrictionsontheparameters(e.g.,c+d+2R0issomerealnumber.Assumefurtherthattheelementstiffnessmatrixhasbeencomputedforα=1.Howwilltheelementsofthestiffnessmatrixchangeasfunctionsofαin P1:OSOJWST055-05JWST055-SzaboFebruary16,20117:56PrinterName:YettoCome164FINITEELEMENTSPACESone,twoandthreedimensions?Assumethatintwodimensionsthethicknessisindependentofα.Exercise5.7.2RefertoEquations(3.67)through(3.71).Developanexpressionforthe(k)computationofthetermsofthestiffnessmatrix,analogoustokgivenbyEquation(5.62),ijforaxisymmetricelastostaticmodels.5.7.2LoadvectorsThecomputationofelement-levelloadvectorscorrespondingtovolumeforces,surfacetrac-tionsandthermalloadingisbasedonthecorrespondingtermsontherighthandsideofEquation(5.59).VolumeforcesComputationoftheloadvectorcorrespondingtovolumeforce{F}actingonelementkisastraightforwardapplicationofEquation(5.52):(k)Tri={Ni}{F}|Jk|dξdηdζi=1,2,...,3n.(5.64)stSurfacetractionsEvaluationoftheloadvectortermscorrespondingtosurfacetractionsdependsonthesideoftheelementonwhichthetractionsareacting.Forexample,letusassumethattractionvectorsareactingonahexahedralelementonthefaceζ=1.Inthiscasetheithtermoftheloadvectoris+1+1∂r∂rr(k)={N}T{T}×dξdηii∂ξ∂η−1−1ζ=1wheretherangeofiisthesetofindicesofshapefunctionsassociatedwiththefaceζ=1.ThermalloadingThedifferentialoperator[D]appearsinthefunctionalthatrepresentsthermalloading.There-(k)foretheexpressionforrdependsontheindexi:i(k)−1Tri=[Mβ][Jk][D]{Ni}[E]{α}T|Jk|dξdηdζi=1,2,...,3nstwhereβ=1when1≤i≤n,β=2when(n+1)≤i≤2nandβ=3when(2n+1)≤i≤3n.Thematrices[Mβ]andtheoperator{D}aredefinedinSection5.7.1.5.8ChaptersummaryAfiniteelementspaceischaracterizedbyafiniteelementmeshandthepolynomialdegreesandmappingfunctionsassignedtotheelementsofthemesh.Thepolynomialdegreesidentify P1:OSOJWST055-05JWST055-SzaboFebruary16,20117:56PrinterName:YettoComeCHAPTERSUMMARY165apolynomialspacedefinedonastandardelement.Thepolynomialspaceisspannedbybasisfunctions,calledshapefunctions.Twokindsofshapefunctions,calledLagrangeandhierarchicshapefunctions,weredescribedforquadrilateralandtriangularelements.Thefiniteelementspaceisspannedbythemappedshapefunctionssubjecttotherequisitecontinuityrequirements,discussedinSection2.5.3.Unlessthemappingsofallelementsarepolynomialfunctionsofdegreeequaltoorlessthanthepolynomialdegreeofelements,rigidbodyrotationwillnotberepresentedexactlybythefiniteelementsolution.Nevertheless,rapidconvergencetothecorrectsolutionwilloccurasthefiniteelementspaceisprogressivelyenlargedbyh-,p-,orhp-extension.ThemappingfunctionsusedinFEAmustbesuchthattheJacobiandeterminantispositiveateverypointwithintheelement. P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome6RegularityandratesofconvergenceWehaveseeninSection2.4.2(Theorem2.4.2)thatthefiniteelementsolutionminimizestheerrorintheenergynormonthespaceS˜⊂S(,,p,Q).Theerrordependsontheexactsolutionofthegeneralizedformulation,thechoiceofthefiniteelementmesh,thepolynomialdegree(s)assignedtotheelementspandthechoiceofthemappingfunctionsQ.Inthischapterweaddressthequestionofhowaprioriinformationconcerningtheexactsolutionshouldbeusedin(a)definingthespaceSand(b)interpretingtheresults.Aposteriorierrorestimationandadaptivetechniquesarebrieflydiscussedattheendofthechapter.6.1RegularityThedegreeofdifficultyassociatedwithapproximatingafunctionwithpolynomialsdependsonthesizeofitsderivatives.WehaveseenanexampleofthisinSection2.6.5wheretheerrorestimate(2.97)forapproximationwithpiecewisepolynomialsofdegree1wasshowntodependontheabsolutevalueofthesecondderivativeoftheexactsolution.Ifallderivativesofafunctionareboundedondomain¯thenthefunctionisanalyticon¯.Forexample,on¯=[0,1]thefunctionu1=sinmπx(m=1,2,...)isananalyticfunction√butu2=xandu3=xlogxarenot.However,u2andu3areanalyticon¯=[a,b]where0mthensinmπxissmoother√thansinnπx,whichissmootherthanxforanyn.Thetermsregularityandsmoothnessareusedinterchangeably.IntroductiontoFiniteElementAnalysis:Formulation,VerificationandValidation,FirstEdition.BarnaSzabóandIvoBabuška.©2011JohnWiley&Sons,Ltd.Published2011byJohnWiley&Sons,Ltd.ISBN:978-0-470-97728-6 P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome168REGULARITYANDRATESOFCONVERGENCETheregularityoftheexactsolutionsofvariousstationaryproblemsischaracterizedbythefollowinginputdata:(a)Thedomain.(b)Theboundaryconditions,namely,Dirichlet,NeumannandRobinboundarycondi-tions,or,equivalently,thedisplacements,tractionsandspringboundaryconditions.(c)Thematerialproperties.(d)Thesourceterm,thatis,theheatsourceQinEquation(3.11)andthevolumeforcesFiinEquation(3.51).Theinputdataaresaidtobeadmissibleiftheexactsolutionofthecorrespondinggener-alizedformulationliesintheenergyspace.Inthefollowingwedescribeadmissibledatafortwo-dimensionalstationaryheatconductionandelasticityproblems.Inthreedimensionsthemainpointsaresimilarbutadetaileddescriptionwouldbemorecomplicated.Thedataarecharacterizedbypiecewiseanalyticfunctions1ondomainsandsubdomainsboundedbypiecewiseanalyticcurves.2Apiecewiseanalyticcurveistheunionofasetofanalyticarcs.Theendsoftheanalyticarcsarecalledsingularpoints.Weassumethatiftwoanalyticarcshaveacommonpointthenthetwoarcsareeithercoincidentorthecommonpointisanisolatedpoint.Understandingthebehavioroftheexactsolutionintheneighborhoodofsingularpointsisimportantbecausethesolutionisanalyticateverypointthatisnotasingularpointandhenceallderivativesareboundedoutsideoftheneighborhoodofsingularpoints.Intheneighborhoodofasingularpointsomederivativesmaybeboundedbutnotallderivativesarebounded[46a].Thegoalsofcomputationinheatconduction,elasticityandsimilarproblemsusuallyincludethedeterminationoffunctionalsthatarerelatedtothefirstderivativesofthesolution.Forexample,stressisrelatedtostrainbyHooke’slawinthelineartheoryofelasticity.Inmanymathematicalmodelsthefirstderivativesoftheexactsolutionarenotfiniteincertainpoints;however,thefirstderivativescomputedfromthefiniteelementsolutionarealwaysfinite.Insuchcases,reportingthemaximumvalueoftemperaturegradients,strainsandrelateddata,suchasfluxesorstresses,computedfromthefiniteelementsolutionasapproximationstothecorrespondingfunctionalsdeterminedfromtheexactsolution,wouldbeerroneousandmisleading.Thereforeiftheresultsofcomputationaretobeinterpretedcorrectlywhenthedataofinterestarefunctionsofthederivativesthenthequestionofwhetherthederivativesoftheexactsolutionareboundedmustbeanswered.Thisquestionisrelatedtotheregularityoftheexactsolution.Thefollowingexamplesillustratesingularpointsassociatedwithdomains,boundaryconditions,materialpropertiesandsourcefunctions.Singularpointsareindicatedbyopencirclesinthefigures.Example6.1.1ThedomainshowninFigure6.1(a)consistsoftwocurves1and2.Curveisanalyticwhereasistheunionoffouranalyticarcs:=∪4,henceitis122j=12jpiecewiseanalytic.Inthiscasetherearefourgeometricsingularpoints.TheboundaryofthedomainshowninFigure6.1(b)consistsofonecurve1whichistheunionofsixanalyticarcs.Therearesevengeometricsingularpoints.Whenα=2πthearcs15and16arecoincidentandformacrack.1ThedefinitionofanalyticfunctionisgiveninAppendixA,SectionA.7.1.2ThedefinitionofanalyticcurveisgiveninAppendixA,SectionA.7.2. P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeREGULARITY169Figure6.1Typicalgeometricsingularpointsassociatedwithplanardomains.Example6.1.2Variouscommonlyoccurringboundaryconditionsareillustratedschemati-callyinFigure6.2(a).Theprescribedboundaryconditionsandtheassociatedcoefficients(e.g.,springconstants,radiationcoefficients,etc.)areanalyticfunctionsprescribedonanalyticarcs.ReferringtoFigure6.2(a),theboundaryisapiecewiseanalyticcurve,comprising15analyticarcs.ArcABrepresentsprescribedtemperatureordisplacementboundaryconditions;Figure6.2Typicalsingularpointsassociatedwith(a)boundaryconditions,(b)materialinterfaces,(c)sourceterms,(d)acombinationofmaterialinterfacesandsourceterms. P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome170REGULARITYANDRATESOFCONVERGENCEarcsBC,DE,FG,GH,HI,IJ,JK,ABNAandtheboundaryoftheellipticholerepresentzero-fluxorzero-tractionboundaryconditions;boundarysegmentCDrepresentssymmetryboundarycondition(zerofluxorzeronormaldisplacement,zeroshearingtraction);arcEFrepresentsRobinorspringboundaryconditions;andarcsKL,LMandMNrepresentprescribedfluxorprescribednormaltractions.Remark6.1.1ReferringtoFigure6.2(a),thesolutionisanalyticatpointsCandD,providedthatboundarysegmentsBCandEDareatrightanglestotheboundarysegmentCDandthematerialisisotropic,orifthematerialisorthotropicthenoneoftheprincipalmaterialaxesisalignedwiththeboundarysegmentCD.WhenboundarysegmentCDisalineofsymmetrythenwecanvisualizetheentiredomainonwhichpointsCandDarenotendpointsofanalyticarcsandhencenotsingularpoints.Example6.1.3Thematerialpropertiesareanalyticfunctionsonasetofclosedsubdomains¯⊂thatcollectivelycovertheentiredomain.Theboundariesof(m),denotedby∂(m),iiiarecalledmaterialinterfaces.Anypointin∂isapointofamaterialinterface.Thematerialproperties,represented(forexample)byκ(x,y)andc(x,y)inEquation(4.4),aredefinedseparatelyoneachclosedsubdomain¯i.Inthepointsthatlieonmaterialinterfacesx∈∂i∩∂j,i=j,theremaybeadiscontinuityinmaterialproperties,ortheirderivatives.Inthatcasethecoefficientsarenotanalyticinthosepoints;neverthelessκ(i)(x,y)∈¯andiκ(j)(x,y)∈¯areanalyticfunctions.jFigure6.2(b)isaschematicrepresentationofadomainthatistheunionofasetofsubdomainsonwhichthreedifferentmaterialproperties,representedbythedifferentshadingpatterns,weredefined.Example6.1.4Analogouslytothematerialproperties,thesourcefunctionsareanalytic(s)functionsdefinedonasetofsubdomains¯i,i=1,2,...,Ms;however,thesourcefunctions(s)(s)arenotanalyticonthesourceinterfaces(x,y)∈∂∩∂,i=j.Figure6.2(c)illustratesijacasewhereMs=2.Whenthesubdomainsonwhichsourcefunctionsareprescribedarenotcoincidentwiththesubdomainsassociatedwiththematerialpropertiesthenthepointsofintersectionoftheboundariesaresingularpoints.ThisisillustratedinFigure6.2(d).6.2ClassificationThesolutiondomain,boundaryconditions,materialpropertiesandsourcetermscharacterizetheregularityoftheexactsolutionofthemathematicalproblem.Itisusefultoclassifymathematicalproblemsintothreebroadcategories,asfollows.CategoryAAmathematicalproblemissaidtobeinCategoryAwhentherearenosingularpointsin¯.InthiscasetheexactsolutionuEXoftheproblemiscontinuouson¯andanalyticonevery¯(m)and¯(s),butdoesnothavetobeanalyticon¯.AnexampleisshowninFigure4.1iiwherethedomainistheunionofthreesubdomainstowhichthematerialpropertiesdescribed P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeCLASSIFICATION171inExample4.1.1areassigned.Theexactsolutionisnotanalyticonthematerialinterfaces,neverthelesstheproblemisinCategoryA.AprecisedefinitionofproblemsinCategoryAisbasedonEquation(A.15).Wedefinethesetofpointscthatdonotlieoninterfaces(m)(m)(s)(s)(m)(s)c:=−∪i=j∂i∩∂j−∪i=j∂i∩∂j−∪i=j∂i∩∂j.IfthereexistconstantsKandR,independentof(x,y)∈c,suchthatforanyswehave∂suEX≤KRss!k=0,1,...,s,s=1,2,...(6.1)∂xk∂ys−kthentheexactsolutionuEXcanbeapproximatedbyaTaylorseriesintheneighborhoodofanypointincand,furthermore,theTaylorseriesconvergesexponentially.EstimationofthesizeofthederivativesisimportantbecausetheerrorterminaTaylorseriesapproximationofdegreepdependsonthesizeofthederivativeofdegreep+1.Thisisgenerallytrueforapproximationbypolynomials.Example6.2.1Considertheproblemu=0onthedomain={x,y|(x−1−d)2+y2<1},d>0showninFigure6.3.Theboundaryconditionisxu(x,y)=(z−1)=on∂.(6.2)x2+y2Sinceu(x,y)satisfiesu=0,theexactsolutionisu(x,y).Thederivativesofuareboundedbyss+1∂u1≤Ks!·(6.3)∂xk∂ys−kdFigure6.3DefinitionofthedomainforExample6.2.1. P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome172REGULARITYANDRATESOFCONVERGENCENotethattheexactsolutionisanalyticon¯andcanbeextendedbeyond¯throughTaylorseriesexpansion.Forexample,onacirclecenteredatpointA,theradiusofwhichislessthand,theTaylorseriesconverges.ItisseenfromEquation(6.2)thatthesizeofthederivativesgrowsveryrapidlyforsmallvaluesofd.Thisinfluencesthedesignofthefiniteelementmesh,whichhastobemuchfinerintheneighborhoodofpointAthanelsewhere.ThisfeaturewillbediscussedfurtherinSection6.4.2.Remark6.2.1WehaveseeninExample6.2.1thattheexactsolutionofaprobleminCate-goryAcanhaveverylowregularity.ItisdemonstratedinExample6.2.2thatinexceptionalcasestheexactsolutionofaprobleminCategoryBcanbeananalyticfunction.CategoryBAproblemissaidtobeinCategoryBwhenthereareafinitenumberofsingularpointsin¯.Figures6.1and6.2representvariousCategoryBproblems.Theneighborhoodofasingular()point(xi,yi)isdenotedbyωianddefinedbyω()={(x,y)|(x−x)2+(y−y)2≤2}.iii()ThesolutionsofCategoryBproblemssatisfyEquation(6.1)onc−∪ω¯i,i=1,2,...,m.ThebehaviorofsolutionsinωiwillbediscussedinSection6.3.Example6.2.2TheprobleminExercise4.3.8isinCategoryB,neverthelessitsexactsolutiongivenbyEquation(4.42)1a2b2uEX(x,y)=sin(mπx/a)sin(nπy/b)π2m2b2+n2a2satisfiesEquation(6.1)inallpointsof¯:s22ks−k∂uEX≤1abmπnπ≤KRs∂xk∂ys−kπ2m2b2+n2a2abwhere221abmπnπK=andR=max,·π2m2b2+n2a2abNotethatthesolutioncanbeextendedbeyondthedomain¯={x,y|0≤x≤a,0≤y≤b}byTaylorseriesapproximationaboutanypointin¯.ThesolutionuEXoftheprobleminExercise4.3.8correspondstothesourcefunctionf=sin(mπx/a)sin(nπy/b).If,forexample,f=1weregiventhenthecorrespondingexactsolutionwouldhavesingularitiesinthecornerpointsof.Thisexampleillustratesthat P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeTHENEIGHBORHOODOFSINGULARPOINTS173whenaproblemisinCategoryBitssolutionmaystillsatisfyEquation(6.1);however,theseareexceptionalcases.CategoryCWhentheexactsolutionisnotinCategoryAorCategoryBthenitisinCategoryC.Remark6.2.2Wehavedescribedadmissibledatafortwo-dimensionalstationaryheatcon-ductionandelasticityproblems.Wehavenotedthatinthreedimensionsthedataaresimilarbutadescriptionofthecategorieswouldbemorecomplicated.Thisisbecauseinthreedimensionstherearenotonlysingularpointsbutalsosingulararcs.6.3TheneighborhoodofsingularpointsWedenotethedistancefromsingularpoint(xi,yi)byr=(x−xi)2+(y−yi)2.Theregularityofafunctionu=u(x,y)ischaracterizedbythesizeofitsderivatives.Specif-ically,∂su≤K(s,r),r>0(6.4)∂xk∂ys−kwhereKisindependentofrands.Averyimportantspecialcaseiswhen(r)isoftheform(s,r)≤rλ−ss!,λ>0,s=1,2,...(6.5)whereλisafractionalnumber,calledthedegreeofsingularity.InthefollowingwewilldiscusssuchspecialcasesinconnectionwiththeLaplaceequationandtheequationsofelasticity.6.3.1TheLaplaceequationLetusconsidersolutionsoftheLaplaceequationintheneighborhoodofacornerpoint,suchaspointBinFigure6.4(a),∂2u1∂u1∂2uu≡++=0(6.6)∂r2r∂rr2∂θ2wherer,θarepolarcoordinates.Inparticular,letusseeksolutionsintheformu=rλF(θ)withλ=0.Suchsolutionsaretypicallyassociatedwithgeometricsingularities,boundaryconditionsandintersectionsofmaterialinterfaces.SubstitutingintoEquation(6.6),wegetF+λ2F=0 P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome174REGULARITYANDRATESOFCONVERGENCEFigure6.4Reentrantcorner:notation.thegeneralsolutionofwhichisF=acosλθ+bsinλθ(6.7)wherea,barearbitraryconstants.Thereforethesolutionisoftheformu=rλ(acosλθ+bsinλθ).(6.8)Letusnowconsidertheproblemu=0onbothboundarysegmentsABandBC(i.e.,θ=±α/2):acosλα/2+bsinλα/2=0acosλα/2−bsinλα/2=0.Addingandsubtractingthetwoequations,wefindcosλα/2=0thereforeλα/2=±(2m−1)π/2,m=1,2,...(6.9)sinλα/2=0thereforeλα/2=±nπ,n=1,2,....(6.10)NotethatthevaluesofλthatsatisfyEquations(6.9)and(6.10)canbeeitherpositiveornegative.However,thefunctionu,givenbyEquation(6.8),liesintheenergyspaceonlywhenλ≥0.Therefore,havingexcludedλ=0fromconsideration,wewillbeconcernedwithλ>0only.Denoting(2m−1)π2nπλ¯:=λ:=mnααwherem,n=1,2,...,thesolutioncanbewritteninthefollowingform:∞∞λ¯λu=armcosλ¯θ+brnsinλθ,r≤r(6.11)mmnncm=1n=1wherercistheradiusofconvergenceoftheinfiniteseries. P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeTHENEIGHBORHOODOFSINGULARPOINTS175ObservethatthefirstsuminEquation(6.11)isasymmetricfunctionwithrespecttothex-axis,thesecondisanantisymmetricfunction.Ifthesolutionissymmetric(withrespecttothex-axis)thenbn=0;ifitisantisymmetricthenam=0.Letusconsiderthesymmetricpartofthesolutionandassumethata1=0andλ¯min=λ¯1=π/αisnotaninteger.Then∂su≤Krλ¯1−ss!=Krπ/α−ss!.(6.12)∂xk∂ys−kThereforethedegreeofsingularityisλ¯1=π/α.Notethatwhenα=π/kwherek=1,2,...,thepowersofrareintegersanduisananalyticfunction.Forothervaluesofαthesolutionuisnotanalyticatthecornerpointandwhenα>πthenthefirstderivativeofuwithrespecttorisinfinityatthecornerpoint,providedthata1=0.Remark6.3.1Theexpressions(6.11)and(6.12)werederivedundertheassumptionthat−u=0andu=0onthestraightboundarysegmentsthatintersectatthesingularpoint.Inthegeneralcasethatincludes−u=f(x,y),wheref(x,y)isasmoothfunction,andcurvedboundaries,wehave∂su≤K(ε)rλ¯1−ε−ss!,k=0,1,2,...,s,s=1,2,...(6.13)∂xk∂ys−kwhereε>0andK(ε)→∞asε→0.Exercise6.3.1Considerthesolutionofu=0intheneighborhoodofcornerpointBshowninFigure6.4(a),andletu=0onBCandthenormalderivative∂u/∂n=0onAB.Showthatucanbewrittenas∞(2n−1)πu=arλn(cosλθ+(−1)nsinλθ)whereλ=·nnnn2αn=1Hint:Thecondition∂u/∂n=0isequivalentto∂u/∂θ=0.Exercise6.3.2ConstructtheseriesexpansionanalogoustoEquation(6.11)fortheproblemu=0intheneighborhoodofcornerpointBshowninFigure6.4(a),giventhat∂u/∂n=0onABandBC.Exercise6.3.3Showthatu(r,θ),definedbyEquation(6.8),isnotintheenergyspacewhenλ<0.Exercise6.3.4Considerfunctionsu=rλF(θ,φ)wherer,θ,φaresphericalcoordinatescenteredonacornerpointandFisasmoothfunction.Showthat∂u/∂rissquareintegrableinthreedimensionsforλ>−1/2.6.3.2TheNavierequationsInthefollowingdiscussionitisassumedthatthematerialisisotropicandelastic,hencethematerialpropertiesarecharacterizedbythetwomaterialconstantsEandν,andthevolume P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome176REGULARITYANDRATESOFCONVERGENCEforcesarezero.Weexaminethesolutionintheneighborhoodofcornerpointswhentheintersectingedgesarestressfree.Thetreatmentofothercasesisanalogous.ThestressfieldsinplanarelasticitycanbederivedfromtheAirystressfunctiondenotedbyU(r,θ).TheAirystressfunctionsatisfiesthebiharmonicequation∂21∂1∂2∂21∂1∂2++++U=0.(6.14)∂r2r∂rr2∂θ2∂r2r∂rr2∂θ2ThecomponentsofthestresstensorinpolarcoordinatesarerelatedtoUbythefollowingformulas(see,forexample,[87]):1∂U1∂2U∂2U∂1∂Uσr=+,σθ=,τrθ=−·(6.15)r∂rr2∂θ2∂r2∂rr∂θTheCartesiancomponentsofthestresstensorare∂2U∂2U∂2Uσx=,σy=,τxy=−·(6.16)∂y2∂x2∂x∂yThestressfunctioncanbewrittenincomplexvariableform3U=(z¯ϕ(z)+χ(z))(6.17)whereϕ(z)andχ(z),calledcomplexpotentials,areanalyticfunctionsofthecomplexvariablez.Theoverbarindicatesthecomplexconjugate.Thestresscomponentsinpolarcoordinatesarerelatedtoϕ(z)andχ(z)asfollows:σ+σ=2ϕ(z)+ϕ(z)=4(ϕ(z))(6.18)rθσ−σ+2iτ=2zz¯−1zϕ(z)+χ(z).(6.19)θrrθThestresscomponentsinCartesiancoordinatesarerelatedtoϕ(z)andχ(z)asfollows:σ+σ=2ϕ(z)+ϕ(z)=4(ϕ(z))(6.20)xyσ−σ+2iτ=2z¯ϕ(z)+χ(z).(6.21)yxxyThecomponentsofthedisplacementvectorinpolarcoordinates(uptorigidbodydis-placementandrotation)arerelatedtoϕ(z)andχ(z)asfollows:2G(u+iu)=z−1/2z¯1/2κϕ(z)−zϕ(z)−χ(z)(6.22)rθandinCartesiancoordinates2G(ux+iuy)=κϕ(z)−zϕ(z)−χ(z)(6.23)3ThisformulaisattributedtoEdouardJean-BaptisteGoursat(1858–1936). P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeTHENEIGHBORHOODOFSINGULARPOINTS177whereκisgivenbyEquation(4.39).ThisisknownastheKolosov–Muskhelishvilimethod.4Detailsandderivationsareavailableinbooksonelasticity,suchas[45],[75],[87].Weareinterestedinsolutionscorrespondingtoϕ(z)=(a−ia)zλ,χ(z)=(a−ia)zλ+1,λ≥0,λ=1(6.24)1234whereai(i=1,2,3,4)andλarerealnumbers.ThecorrespondingstressfunctionisU=rλ+1(acos(λ−1)θ+asin(λ−1)θ+acos(λ+1)θ+asin(λ+1)θ).(6.25)1234Inthecaseofλ=1ϕ(z)=az−iazlogz,χ(z)=(a−ia)z2,(6.26)1234andthecorrespondingstressfunctionisU=r2(a+aθ+acos2θ+asin2θ).(6.27)1234Stress-freeedgesWerefertoFigure6.4andassumethattheboundarysegmentsABandBCarestressfree,thatis,σθ=τrθ=0atθ=±α/2.UsingEquations(6.25)and(6.15)wefindσ=rλ−1λ(λ+1)[acos(λ−1)θ+asin(λ−1)θθ12+a3cos(λ+1)θ+a4sin(λ+1)θ]τ=rλ−1λ(λ−1)[asin(λ−1)θ−acos(λ−1)θ)]rθ12+rλ−1λ(λ+1)[asin(λ+1)θ−acos(λ+1)θ].34Onsettingσθ=τrθ=0atθ=±α/2,followingstraightforwardalgebraicmanipulation,weobtaincos(λ−1)α/2cos(λ+1)α/2a1=0(6.28)−sin(λ−1)α/2sin(λ+1)α/2a3andsin(λ−1)α/2sin(λ+1)α/2a2=0(6.29)−cos(λ−1)α/2cos(λ+1)α/2a4where1−λ:=·1+λ4GuryVasilievichKolosov(1867–1936),NikolaiIvanovichMuskhelishvili(1891–1976). P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome178REGULARITYANDRATESOFCONVERGENCENotethata1anda3(resp.a2anda4)arecoefficientsofsymmetric(resp.antisymmetric)functionsinEquation(6.25).Therefore,analogouslytoEquation(6.11),U(r,θ)canbewrittenintermsofsumsofsymmetricandantisymmetricfunctions.ThesymmetricfunctionsassociatedwiththeeigenvaluesofEquation(6.28)areusuallycalledModeIeigenfunctionsandtheantisymmetricfunctionsassociatedwiththeeigenvaluesofEquation(6.29)arecalledModeIIeigenfunctions.Non-trivialsolutionsexistifeitherthedeterminantofEquation(6.28)orthedeterminantofEquation(6.29)vanishes.Thiswilloccurifeithercos(λ−1)α/2sin(λ+1)α/2+sin(λ−1)α/2cos(λ+1)α/2=0,whichcanbesimplifiedtosinλα+λsinα=0,λ=0,±1,(6.30)orsin(λ−1)α/2cos(λ+1)α/2+cos(λ−1)α/2sin(λ+1)α/2=0,whichcanbesimplifiedtosinλα−λsinα=0,λ=0,±1.(6.31)WedenotesinλαsinαQ(λα):=andq(α):=(6.32)λααanddiscusstheeigenvaluescorrespondingtothesymmetricandantisymmetriceigenfunctionsseparatelyinthefollowing.Formoredetailedtreatmentofthesubjectwereferto[39],[93].Eigenvaluescorrespondingtosymmetric(ModeI)eigenfunctionsEquation(6.30)canbewrittenasQ(λα)+q(α)=0.(6.33)ThefunctionQ(λα)isplottedontheinterval0<λα<4πinFigure6.5.TheproblemistofindtherootsofEquation(6.33)foragivenα.Intheinterval0<α<αAwhereα=2.553591(146.31◦)theline−q(α)hasnopointsincommonwithQ(λα),thereforeAtherearenorealroots.Atα=αAtheline−q(αA)istangenttoQ(λα)atpointA.Thereforethereisadoublerootatthisangle.Therearedoublerootsatα=α(1)=3.625739(207.74◦),Bα=α(2)=5.499379(315.09◦)andalsoatα=α=2.875839(164.77◦).ItisseenthatBCintheintervalαA<α<αBthereareatleasttworealandsimpleroots.Atα=πandα=2πthereareinfinitelymanyrealroots.Furthermore,atα=πallrootsareintegers.PointDcorrespondstoα=3π/2(270◦)whereλα=2.565819,henceλ=0.544484.PointEcorrespondstoα=2πwhereλα=π,henceλ=1/2.PointEalsocorrespondstoα=π,whichisaspecialcase,discussednext. P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeTHENEIGHBORHOODOFSINGULARPOINTS179EFigure6.5ThefunctionQ(λα)ontheinterval0<λα<4π.InformulatingEquation(6.33)weexcludedλ=1fromconsideration.Whenλ=1thenUisgivenbyEquation(6.27).ConsideringthesymmetrictermsonlyandusingEquation(6.15),wehaveσθ=2(a1+a3cos2θ),τrθ=2a3sin2θ.Lettingσθ(±α/2)=τrθ(±(α/2)=0,wefindthatanon-trivialsolutionexistsonlyif1cosαdet=0.0sinαThereforeα=nπ(n=1,2,...).PointEinFigure6.5representsα=π.Remark6.3.2TofindthecomplexrootsofEquation(6.33)wewriteλ=ξ+iη.ThereforeEquation(6.33)becomessin(ξα+iηα)=q(α)(6.34)ξα+iηαwhichisequivalenttothefollowingsystemoftwoequations:sinξαcoshηα=ξq(α)(6.35)cosξαsinhηα=ηq(α).(6.36)Foradetaileddescriptionwereferto[89].Eigenvaluescorrespondingtoantisymmetric(ModeII)eigenfunctionsEquation(6.31)canbewrittenasQ(λα)−q(α)=0.(6.37) P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome180REGULARITYANDRATESOFCONVERGENCENotethatλ=1triviallysatisfiesEquation(6.37)forallαandrecallthatwehaveexcludedλ=1fromconsiderationwhenweformulatedEquation(6.37).Therearenorealrootsintheinterval0<α<αwhereα=2.777068(159.11◦).ThereareatleasttworealrootsinBBtheintervalα<α<αwhereα(1)=3.463416(198.44◦)andα(2)=5.732235(328.43◦).BCCCAsinthecaseofModeI,thereareinfinitelymanyrealrootsatα=πandα=2π,andatα=πallrootsareintegers.ThereisonlyonerealrootintheintervalαC<α<αAwhereα=4.493409(257.45◦).ATheangleαAisaspecialanglewhichcorrespondstoλ=1.Toshowthis,weconsidertheantisymmetrictermsinEquation(6.27).UsingEquation(6.15),wehaveσθ=2(a2θ+a4sin2θ),τrθ=−(a2+2a4cos2θ).Lettingσθ(±α/2)=τrθ(±α/2)=0,wefindthatanon-trivialsolutionexistswhenα/2sinαdet=0.12cosαThereforeα=tanα.Intheinterval0<α<2πthereisoneroot:α=αA=4.493409(257.45◦).ThiscorrespondstopointAinFigure6.5.CracksAnimportantspecialcaseiswhenthesingularpointisacracktip,thatis,α=2π.TheAirystressfunctioncorrespondingtothesymmetricpartoftheasymptoticexpansionis∞U=a(zz¯λi+Qzλi+1),λ=i/2(6.38)siiii=1whereQi=(2−i)/(2+i)wheniisoddandQi=−1wheniiseven.TheAirystressfunctioncorrespondingtotheantisymmetricpartoftheasymptoticexpansionis∞U=b(zz¯λi+Qzλi+1),λ=i/2(6.39)aiiii=1where(·)representstheimaginarypartof(·)andQi=−1wheniisoddandQi=(2−i)/(2+i)wheniiseven.Thiscanbeverifiedbycomparingeq(6.39)withthean-tisymmetrictermsinEquation(6.25).UsingEquations(6.38),(6.39)and(6.15),(6.16),thestressdistributionintheneighborhoodofacracktipcanbedetermineduptothecoefficientsai,bi(i=1,2,...,∞).ProceduresforthedeterminationofthesecoefficientsfromfiniteelementsolutionsareoutlinedinChapter7andAppendixD.Exercise6.3.5Showthat:(a)thestresscomponentscorrespondingtothesecondtermofEquation(6.38)areσx=4a2,σy=τxy=0;and(b)thestresscomponentscorrespondingtothesecondtermofEquation(6.39)areσx=σy=τxy=0.Notethatthestressσx=4a2(constant)isusuallydenotedbyTandiscalledtheT-stress.Accordingtosomemodels,theT-stressinfluencescrackgrowth.See,forexample,[55]. P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeTHENEIGHBORHOODOFSINGULARPOINTS181Exercise6.3.6Letα=3π/2andassumethatthesolutionissymmetric.1.VerifythatthelowestpositiveeigenvalueofEquation(6.30)isλ1=0.544483737.2.RefertoEquation(6.28)andshowthattheAirystressfunctioncorrespondingtoλ1canbewrittenasU=arλ1+1(cos(λ−1)θ+Qcos(λ+1)θ)(6.40)1111wherea1isanarbitraryrealnumberandQ1=0.543075579.VerifythatEqua-tion(6.40)isequivalenttoU=a(zz¯λ1+Qzλ1+1).(6.41)11Exercise6.3.7Letα=3π/2andassumethatthesolutionissymmetric.UsingEqua-tions(6.41)and(6.16)showthatthestresscomponentscorrespondingtoλ1areσ=aλrλ1−1[(2−Q(λ+1))cos(λ−1)θ−(λ−1)cos(λ−3)θ]x1111111σ=aλrλ1−1[(2+Q(λ+1))cos(λ−1)θ+(λ−1)cos(λ−3)θ]y1111111τ=aλrλ1−1[(λ−1)sin(λ−3)θ+Q(λ+1)sin(λ−1)θ].xy1111111Theangleα=3π/2isrepresentativeoffrequentlyoccurringgeometricdetailsandthisstressfieldwillbeusedforillustratingtheconvergencecharacteristicsofthefiniteelementmethod.Exercise6.3.8Letα=3π/2andassumethatthesolutionissymmetric.Showthatthedisplacementcomponentscorrespondingtoλ1,uptorigidbodydisplacementandrotationterms,area1λu=r1[(κ−Q(λ+1))cosλθ−λcos(λ−2)θ]x111112Ga1λu=r1[(κ+Q(λ+1))sinλθ+λsin(λ−2)θ]y111112GwhereκisdefinedbyEquation(4.39).Hint:UseEquations(6.23)and(6.41).Exercise6.3.9Assumethatλisreal.Showthatthecorrespondingstressfieldisintheenergyspaceonlyifλ>0.Hint:Adisplacementfieldisintheenergyspaceifthestresscomponentsaresquareintegrable:σ2+σ2+τ2rdrdθ≤C<∞.rθrθComplexeigenvaluesWehaveseenthatinplanarelasticity,incontrasttotheLaplaceequation,λcanbecomplexanditcanbeeitherasimpleoramultipleroot.Ifλiscomplexthenitsconjugateisalsoaroot.Inthecaseofmultiplerootsspecialtreatmentisnecessarywhichisnotdiscussedhere.Wereferto[58]fordetails. P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome182REGULARITYANDRATESOFCONVERGENCEConsidertheAirystressfunctionintheformU=rλ+1F(θ).Ifλiscomplexwewriteλ=ξ+iηandF=f+ig.Therefore(ξ+1+iη)(ξ+1)(lnriη)U=r(f+ig)=re(f+ig).Writing(lnriη)(iηlnr)e=e=cos(ηlnr)+isin(ηlnr)wegetU=rξ+1[(fcos(ηlnr)−ηsin(ηlnr))+i(fsin(ηlnr)−ηcos(ηlnr))].(6.42)BoththerealandimaginarypartsofUaresolutionsofthebiharmonicequation(6.14).Sincelnr→−∞asr→0,thesinusoidaltermsoscillatewithawavelengthapproachingzero.Thesingularityischaracterizedbytheestimateofthederivatives,seeEquation(6.13),withλ1replacedbyξ1=(λ1):∂us≤K(ε)rξ1−s−εs!.(6.43)∂xk∂ys−kNotethatthisestimateisindependentoftheimaginarypart.Thelowestvaluesof(λ)Thelowestvaluesof(λ)arelistedinTable6.1forthreekindsofhomogeneousboundaryconditionsprescribedonthecorneredges:(a)thefree–freecondition:5onbothedgesσ=θτrθ=0;(b)thefixed–freecondition(planestress,ν=0.3):ononeedgeur=uθ=0,ontheTable6.1Lowestpositivevaluesof(λ)atcornerpointsforthreekindsofhomogeneousboundaryconditions.Free–freeFixed–freeFixed–fixed(s)(a)(s)(a)α(λ1)(λ1)(λ1)(λ1)(λ1)45◦5.390539.562711.304345.573282.6083190◦2.739594.808250.758352.825791.49046135◦1.885373.242810.693391.573231.16088180◦1.000002.000000.500001.000001.00000225◦0.673581.302090.405940.735540.87723270◦0.544480.908530.340320.604040.74446315◦0.505010.659700.287840.537930.60945360◦0.500000.500000.250000.500000.500005Inthecaseoffree–freeboundaryconditionstheeigenvaluesareindependentofPoisson’sratioν. P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeTHENEIGHBORHOODOFSINGULARPOINTS183otheredgeσθ=τrθ=0;(c)thefixed–fixedcondition(planestress,ν=0.3):ur=uθ=0onbothedges.Inthefree–freeandfixed–fixedcasesthecharacteristicfunctionsareeithersymmetricorantisymmetric.TheseareindicatedinTable6.1bythesuperscriptssanda,respectively.Atα=180◦thecharacteristicvaluesareintegers.NotethatifλisasolutionofEquation(6.30)orEquation(6.31)then−λisalsoasolution;however,thecorrespondingstressfieldhasfinitestrainenergyonlywhentherealpartofλisgreaterthanzero.6.3.3MaterialinterfacesSomesingularpointsassociatedwithmaterialinterfacesareillustratedinFigure6.2(b).Thesolutionintheneighborhoodofthesesingularpointsisalsocharacterizedbyeigenvaluesandeigenfunctionswhicharetypicallyoftheformu=rλF(θ)inthecaseoftheLaplaceequationandu=rλF(θ)(i=1,2)inthecaseofthetwo-dimensionalNavierequations.6iiHowever,unlikeinthecaseofhomogeneousmaterials,F(θ)andFi(θ)arepiecewiseanalyticfunctions:atanglescorrespondingtomaterialinterfacesthederivativesarediscontinuous.Itispossibletodeterminetheeigenvaluesandeigenfunctionsbyclassicalmethods;however,theproblemisnowmorecomplicatedthaninthehomogeneouscase.Itispossible(andmuchmoreconvenient)todeterminetheeigenpairsnumericallyusingamethodknownastheSteklovmethod.7TheSteklovmethodisbasedontheobservationthatonacircularcontourofradius,showninFigure6.4(b),thenormalderivativeofthefunctionu=rλF(θ)is∂u∂uλ==λλ−1F(θ)=u.(6.44)∂n∂rr=Considernowproblemswheretwoormoreisotropicmaterialsarebondedandthematerialinterfacesareplanar.8ThegeneralizedformofEquation(3.18),subjecttotheassumptionsthatQ¯=0,steadystateconditionsexist,andeitheru=0or∂u/∂n=0onABandBC,isλgradv[κ]gradudxdy=uvdsforallv∈E0()(6.45)where[κ(x,y)]isapositive-definitematrixofmaterialpropertieswhicharediscontinuousatthematerialinterfaces9andisdefinedinFigure6.4(b).Equation(6.45)isacharacteristicvalueproblem;thecharacteristicvaluesarereal.OnsolvingEquation(6.45)bythefiniteelementmethod,thecharacteristicfunctionsareapproximationsofF(θ)bythe(mapped)piecewisepolynomialfunctions.Thesizeofthenumericalcharacteristicvalueproblemcanbereducedtothenumberofdegreesoffreedomassociatedwiththecontour.6Inthethree-dimensionalNavierequations,ui=rλFi(θ,φ)atverticeswhere(r,θ,φ)aresphericalcoordinatesandui=rλFi(θ,z)alongedges,where(r,θ,z)arecylindricalcoordinates,theaxialcoordinatezbeingcoincidentwiththeedge.7VladimirAndreevichSteklov(1864–1926).8Thetreatmentofanisotropicmaterialsandcurvedinterfacesisnotconsideredhere.9Theelementsofthematrix[κ]areusuallypiecewiseconstantfunctions. P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome184REGULARITYANDRATESOFCONVERGENCE0.20-0.2Al-0.4θ-0.6-0.8CrNisteel-1.090180270360Angleθ(degrees)Figure6.6Example6.3.1:thecharacteristicfunctionF1(θ).ThemethodisapplicabletotheNavierequationsaswell.InthecaseoftheNavierequationsthecharacteristicvaluesmaybecomplex[58],[85],[94].Example6.3.1Achrome–nickelsteelplatewithcoefficientofthermalconductivitykcns=16.3W/(mK)isbondedtoapurealuminumplatewithkal=202W/(mK).Theedgesareoffset,formingacornersimilartothatshowninFigure6.2(c).Assumingthattheedgesintersectingatthereentrantcornerareperfectlyinsulated(qn=0),thefirstcharacteristicvalueisλ1=0.5238.ThecorrespondingcharacteristicfunctionF1(θ)isshowninFigure6.6.Thisfunctionisdetermineduptoanarbitrarymultiplier.Exercise6.3.10DeterminethesecondeigenvalueandthecorrespondingeigenfunctionfortheproblemofExample6.3.1.Partialanswer:λ=1.4762.102Exercise6.3.11ReferringtoFigure6.7,assumethatboundarysegmentsIBandBCaretractionfree.10ThematerialpropertiesforMaterial1(resp.Material2)areE=200GPa,1ν1=0.3(resp.E2=qE1,ν2=ν1).AssumingplanestrainconditionsandzerovolumeFigure6.7Exercise6.3.11:notation.10SolutionofthisexerciserequiresproceduresthatarenotsupportedbyStressCheck P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeTHENEIGHBORHOODOFSINGULARPOINTS185Figure6.8(a)Loadingbyastepfunction.(b)Loadingbyaconcentratedforce.forces,determinethedegreesofsingularityatpointsAandBforq=0.5,q=0.1,q=0.01andq=0.001.ThisexerciseshowsthatλmindecreasesastheratioE2/E1decreases.Partialsolution:forq=0.1,λmin=0.39608atpointA,λmin=0.80153atpointB.6.3.4ForcingfunctionsactingonboundariesTheregularityofthesolutionisinfluencedbytheforcingfunctionalso.Consider,forexample,constantnormaltractionactingalongthepositivex-axis,asshowninFigure6.8(a).Thepointwherethenormaltractionchangesfromzeroto−p0isasingularpoint.Inthiscasethestresscomponentsarefinitebutnotsinglevaluedatthesingularpoint.Thestressfunctionispr210U=−π+θ−sin2θ−π≤θ≤02π2andthestresscomponentsare1∂U1∂2Up10σr=+=−π+θ+sin2θ(6.46)r∂rr2∂θ2π2∂2Up10σθ==−π+θ−sin2θ(6.47)∂r2π2∂1∂Up0τrθ=−=(1−cos2θ).(6.48)∂rr∂θ2πObservethatalongthex-axis(θ=0)τrθ=0attheorigin,butalongthey-axis(θ=−π/2)τrθ=p0/π.Attheoriginitismulti-valued.Similarly,thesumofthenormalstressesσr+σθ=−2p0(1+θ/π)rangesfrom−2p0to0attheorigin,dependingonthedirectionfromwhichtheoriginisapproached.Itcanbeshownthatthederivativesofthedisplacement P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome186REGULARITYANDRATESOFCONVERGENCEfunctionareboundedanalogouslytoEquation(6.13):∂us≤K(ε)rλ−ε−ss!,k=0,1,2,...,s,s=1,2,...(6.49)∂xk∂ys−kwithλ=λ1=1intheexampleshowninFigure6.8(a).Remark6.3.3Theregularityestimate(6.49)holdsalsowhentheloadisappliedthroughtheimpositionofspringdisplacementwhenthespringdisplacementisastepfunction[23].ConcentratedforceTheproblemofaconcentratedforceactingonanelasticbodyisoftenasourceofconceptualerrorsandthereforemeritsspecialdiscussion.ReferringtoFigure6.8(b),thelinearfunctionalcorrespondingtotheconcentratedforceF0isF(v)=−F0vy(0,0).(6.50)Thederivationofthisexpressionisanalogoustotheone-dimensionalcase,seeEquation(2.81)inSection2.5.5.However,unlikeintheone-dimensionalcase,intwoandhigherdimensionsvisnotnecessarilybounded,seeSection4.1.1.ThereforeF(v)doesnotsatisfycondition3inSectionA.3andhenceconcentratedforcesareinadmissibledatainmathematicalmodelsbasedontheprincipleofvirtualwork.AnalternativewaytoshowtheinadmissibilityofconcentratedforcesistoconsidertheAirystressfunctioncorrespondingtoaconcentratedforce:F0U=−rθcosθ−π≤θ≤0πwhichcanbefoundintextbooksonelasticity,suchasTimoshenkoandGoodier[87].Itislefttothereadertoverifythat2F0σr=sinθ(6.51)rπandσθ=τrθ=0.Onasolutiondomainthatincludestheorigin,thisstressfieldisnotsquareintegrable,hencethesolutiondoesnotlieintheenergyspace.Thereforesequencesofsolutionscorrespondingtofiniteelementspacesconstructedbytheh-,p-orhp-methodsdonothavealimitintheenergyspace.Nevertheless,concentratedforcesarewidelyusedinfiniteelementanalysiswithreason-ablygoodresults.Thisapparentcontradictionisrelatedtothefactthatforafixedmeshandpolynomialdegreetheloadvectorcorrespondingtoaconcentratedforcecanbeunderstoodasaloadvectorcomputedforsomestaticallyequivalentdistributedtraction,correspondingtowhichanexactsolutionexistsintheenergyspace.ThisisillustratedbyExample6.3.2.Althoughthestaticallyequivalenttractionsarenotuniquelydefined,thedifferencesinstressdistributionscorrespondingtostaticallyequivalenttractionsdecaywithdistance, P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeTHENEIGHBORHOODOFSINGULARPOINTS187seeRemark6.3.4.ThisisrelatedtoSaint-Venant’sprinciple.11AnillustrationisgiveninExample6.3.3.Concentratedforcesaretypicallyusedforreasonsofconvenienceinmodeling.Forexam-ple,tractionstransmittedbybodiesinelasticcontact,suchas,forexample,contactbetweentheshanksoffastenersandthefastenerholes,areoftenrepresentedbystaticallyequivalentconcentratedforces.Therationaleisthatthedetailsofthedistributionoftractionsinthevicinityofpointswheretheconcentratedforcesareapplieddonothavesignificantinfluenceonthedataofinterest.Twoassumptionsareimplied:(a)thedataofinterestarefarfromthepointsofapplicationofconcentratedforces;and(b)thereisafinitelengthorarea,suchasthediameterofafastenerholeortheareaofmechanicalcontact,andtheapproximationisunderstoodtobeanapproximationofdistributedtractionsactingoverthatfinitelengthorarea.Shouldthemeshberefinedsuchthattheelementsareofthesizeofthefinitelengthorarea,thentheconcentratedforcesmustbereplacedwiththeappropriatetractions.Remark6.3.4TherateofdecayofthedifferencesinstressdistributioncorrespondingtostaticallyequivalenttractionsisrelatedtoGreen’sfunction.Itcanbeshownthatinthescalarellipticproblemandinelasticitytherateofdecayis(d/)2wheredrepresentsthediameteroftheareaonwhichthestaticallyequivalenttractionsareappliedandisthedistance.Inthefiniteelementmethoddisapproximatelythediameterofthelargestelementthatcontainsthepointofapplicationoftheconcentratedforce.Example6.3.2Considertheeight-nodequadrilateralelementshowninFigure6.9.Forthesakeofsimplicitytheelementisdefinedsuchthatthemappingisx=ξ,y=η.ThereforetheshapefunctionsdefinedbyEquations(5.7)through(5.14)arealsothebasisfunctions.Figure6.9Eight-nodequadrilateralelement.Polynomialtractionsf2(y),f3(y),f6(y)thatareequivalenttounitforcesappliedatnodes2,3,6respectively.11AdhemarJeanClaudeBarr´edeSaint-Venant(1797–1886).´ P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome188REGULARITYANDRATESOFCONVERGENCELetusassumethataconcentratedforceF2isappliedatnode2.Inthiscasetheonlynon-zeroloadvectortermisr2=F2whichfollowsfromthedefinitionofvxontheelement:8v(x,y)=v(i)N(x,y)xxii=1wherev(i)isthevalueofv(x)atnodei.Sinceatnode2allbasisfunctionsarezerowiththexexceptionofN2,wehaveF(v)=Fv(1,−1)=Fv(2)=rv(2).22x2xTheloadvectorcorrespondingtothetractionTx(y)=F2f2(y),actingonside2,thatsatisfiestheconditions1r2=Tx(y)N2(1,y)dy=F2(6.52)−11r3=Tx(y)N3(1,y)dy=0(6.53)−11r6=Tx(y)N6(1,y)dy=0(6.54)−1isthesameastheloadvectorcorrespondingtotheconcentratedforceF2actingonnode2.ForstaticalequivalencytheresultantofTxmustbeF2:1Tx(y)dy=F2;(6.55)−1andthemomentaboutnode2mustbezero:1Tx(y)(1+y)dy=0.(6.56)−1NotingthatN2(1,y)+N3(1,y)+N6(1,y)=1,onsummingEquations(6.52)through(6.54)weseethat(6.55)issatisfied.Equation(6.56)issatisfiedby(6.53),(6.54)because2N3(1,y)+N6(1,y)=(1+y).Letusconstruct,forexample,f2(y)asalinearcombinationofN2(1,y),N3(1,y),N6(1,y).Itislefttothereadertoverifythat933f2(y)=N2(1,y)+N3(1,y)−N6(1,y)(6.57)224 P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeTHENEIGHBORHOODOFSINGULARPOINTS189satisfiesEquations(6.52)through(6.56).Thisfunctionandtheanalogousfunctionscorre-spondingtounitforcesappliedinnodes3and6393f3(y)=N2(1,y)+N3(1,y)−N6(1,y)(6.58)224339f6(y)=−N2(1,y)−N3(1,y)+N6(1,y)(6.59)448areplottedinFigure6.9.Theseresultsdemonstratethatitispossibletoconstructsmoothtractionsthatarestaticallyequivalenttotheappliedconcentratedforceandtheloadvectoristhesameastheloadvectorcorrespondingtotheconcentratedforce.Thereforethefiniteelementsolutioncanbeunderstoodasanapproximationtotheexactsolutionofaproblemwithstaticallyequivalentsmoothtractions.Thesetractionsarenotuniquelydefinedanddependontheelementanditspolynomialdegree.Remark6.3.5Itcanbeshownthatsequencesoffiniteelementsolutionsu(,p)correspond-ingtoF(v)definedinEquation(6.50)convergetotheexactsolutionoftheconcentratedforceproblemateverypointinsidethedomain.Example6.3.3LetussolvetheproblemshownFigure6.8(b).SinceABisalineofsym-metry,oursolutiondomainisthecircularsectorABCandthemagnitudeoftheconcentratedforceisF0/2.OntheboundaryBCtheradialnormalstressgivenbyEquation(6.51)isap-plied.Inordertopreventrigidbodymotioninthedirectionofthey-axis,onepointconstraintisspecified.ThefirstofthreegeometricallygradedfiniteelementmeshesusedinthisexampleisshowninFigure6.10(a).Thismeshconsistsoffourfiniteelements.Thesizeofelements1and2isapproximatelyr0,thesizeofelements3and4isαr0.Adetailofthesecondmesh,consistingofsix-elements,isshowninFigure6.10(b).HerethesizeofelementsthathaveanodeatthepointofapplicationoftheforceFisα2r.Thethirdmesh,consistingofeight00Figure6.10TheconcentratedforceproblemforExample6.3.3:(a)four-elementmesh;(b)detailofthesix-elementmesh. P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome190REGULARITYANDRATESOFCONVERGENCEFigure6.11Example6.3.3:therelativeerrorintheradialstressσralongcirculararcscenteredonpointofapplicationoftheconcentratedforce,measuredinmaximumnorm.Eight-elementmesh,p=8,α=0.15,trunkspace.elements,isnotshown;however,therefinementisanalogous:thesizeoftheelementsthathaveanodeatpointAisα3r.0Sincetheexactstressdistributionisknown,seeEquation(6.51),wecancomputetherelativeerrorinmaximumnormatanypointotherthanpointA.Fortheeight-elementmeshweconstructthreecirclescenteredonpointA.Thecirclespassthroughthemid-pointsofthenodesthatlieonABandCD,thatis,theradiiareα3α2α1=r0,2=(1+α)r0,3=(1+α)r0.222TherelativeerrorinmaximumnormalongthethreecirclesisshowninFigure6.11forα=0.15.Itcanbeseenthattheerrorislargealongthesmallestcircle(radius1)whichpassesthroughtheelementsthathaveavertexatpointA,butrapidlydecreaseswithrespecttoincreasingradius.Example6.3.4Toillustratethestrongmeshdependenceofthesolutionatthepointofapplicationofconcentratedforces,wecomputethenormalizeddisplacementcomponentatpointAdefinedbyuAEtAyu¯y:=(6.60)F0wheretisthethickness.Twosetsofanalyseswereperformed.Inthefirstsetthepolynomialdegreeofelementswasincreaseduniformlyfrom1to8oneachofthethreegeometricallygradedmeshesdescribedaboveusingthetrunkspace.Thenormalizeddisplacementcomponentu¯Aisplottedvs.theynumberofdegreesoffreedomNinFigure6.12(a).Itcanbeseenthattheabsolutevalueofu¯AmonotonicallyincreaseswithN.Thereasonforthisisthattheexactsolutionisnotintheyenergyspace.NotethatthestrainenergyisproportionaltoFuAand,sinceFisfixed,uA0y0y P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeTHENEIGHBORHOODOFSINGULARPOINTS191Figure6.12Thenormalizeddisplacementcomponentu¯Avs.thenumberofdegreesofyfreedomN.(a)Uniformp-distribution,prangingfrom1to8.(b)Onthesmallestelementspisfixedat3,uniformlyincreasedfrom1to8ontheotherelements.ThenumberofelementsisdenotedbyM.tendstoinfinityasNisincreased.Infact,intheneighborhoodofpointA,uAisproportionalytologrwhereristhedistancefrompointA.Inthesecondsetofanalysespwasfixedat3ontheelementsthathaveanodeatpointAandincreaseduniformlyfrom1to8ontheotherelements.u¯Aisplottedvs.Nonasemi-logyscaleinFigure6.12(b).Inthiscaseu¯Aconvergesbutitslimitvaluedependsonthechoiceyofmesh.Example6.3.5InExample6.3.3wesawthatthedisplacementinthepointofapplicationofaconcentratedforce,correspondingtothefiniteelementsolution,isstronglymeshdependentandtendstoinfinitywithrespecttoincreasingN.InthisexamplewedemonstratethattheconcentratedforcecorrespondingtoadisplacementimposedonanodetendstozeroasNisincreased.LetusconsideronceagainthedomainABCshowninFigure6.10.ThistimeweprescribeuA=−atpointA;symmetryboundaryconditionsonAB;andzeronormaldisplacementyonarcBC.WeareinterestedincomputingtheconcentratedforceFAfromasequenceofyfiniteelementsolutions.Weconstructasequenceofgeometricallygradedmeshesof4,6,...,16elements.ThefirsttwomeshesareshowninFigure6.10.Wefixthepolynomialdegreeatp=8,andcomputethestrainenergyUcorrespondingtoeachmesh.TheforceFAisdeterminedfromyUusingtherelationshipU=FA/2.TheresultsofcomputationareshowninFigure6.13ywherethenon-dimensionalforce,definedbyFAF¯A:=y,(6.61)yEtisplottedvs.logN.Remark6.3.6Itiscommonpracticeinfiniteelementanalysistoidealizefastenedconnec-tionsasrigidorelasticlinks;thatis,pointconstraintsthatrestricttherelativeorabsolute P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome192REGULARITYANDRATESOFCONVERGENCEFigure6.13Example6.3.5:meshdependenceoftheforcecorrespondingtodisplacementimposedonanodepoint.Geometricallygradedmeshes,p=8.valueofoneormoredisplacementcomponentsatnodepoints.Theexpectationisthatthestiffnessoftheconnectionandthedistributionofnodalforces12willbeareasonableapprox-imationofthefastenedconnectionthatthemodelissupposedtorepresent.InviewoftheforegoingdiscussionandtheresultsofExamples6.3.4and6.3.5,thestiffnessandforcedis-tributionwillbeartifactsofthediscretization.Theresultsmayappeartobeplausible,buttheappearanceofplausibilityiscausedbylargeerrorsindiscretizationmaskingtheconceptualerrorintheformulation.Engineeringdecisionsshouldnotbebasedontheexpectationthattwolargeerrorswillcanceloneanother.Exercise6.3.12ShowthatthetractionTx=f2(y),wheref2(y)isgivenbyEquation(6.57),satisfiesEquations(6.52)through(6.56).Exercise6.3.13SolvetheproblemshowninFigure6.8(a)onasemicirculardomain,usingthefiniteelementmethod.SpecifyTr=σr,Tt=τrθonthecircularsegmentwhereσr(resp.τrθ)isgivenbyEquation(6.46)(resp.Equation(6.48)).Comparethecomputedvaluesσywiththeexactvaluealongtheedgesoftheelementsthathaveavertexattheorigin.Hint:σr+σθσr−σθσy=−cos2θ−τrθsin2θ.22Exercise6.3.14ConstructthreegradedmeshesasinExample6.3.3.InpointAprescribethedisplacementuA=.SolvetheproblemshowninFigure6.8(b)onaquarter-circularysolutiondomainABC,utilizingsymmetry.6.3.5StrongandweaksingularpointsInmanyengineeringapplicationsthegoalsofcomputationinvolvedeterminationofthefirstderivativesofthesolution.Forexample,inelasticitythefirstderivativesdefinethestrain12ThedefinitionandcomputationofnodalforcesarediscussedinChapter7. P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeRATESOFCONVERGENCE193andstressstates,inheattransferthetemperaturegradientandinporousflowproblemsthepiezometricgradient.Notethatwhenλ1<1inEquation(6.49)thefirstderivativeofuEXwithrespecttorisinfinity.Hence,accordingtothelineartheoryofelasticity,thestrainandstressareinfinityatsingularpointswhereλ<1.Similarly,inheattransferandviscousflowproblemsthegradientsareinfinity.Forproblemswhereλ<1itisnotmeaningfultotrytocomputethemaximumvalueofthefirstderivatives.Thecomputedvaluesofthefirstderivativeswillbefinitebutwillconvergetoinfinityinh-,p-andhp-extensions.ThereforeatasingularpointtheexactsolutionuEXwillbecalledstronglysingularifλ<1,weaklysingularotherwise.Thisappliestosingularpointsandedgesinthreedimensionsalso.Exercise6.3.15RefertoExercise6.3.11andFigure6.7.ShowthatpointsAandBarestronglysingularforanyq=1.6.4RatesofconvergenceInthissectionsomekeytheoreticalresultsthatestablishrelationshipsbetweentheerrorintheenergynormandthenumberofdegreesoffreedomassociatedwithsequencesoffiniteelementspacesS1,S2,...⊂E()arepresented.Recallthattheerrorintheenergynormdependsonthechoiceoffiniteelementspacesandthefiniteelementspacesarecharacterizedbythemesh,thepolynomialdegreeofelementspandthemappingfunctionsQ.ThesequencesoffiniteelementspacesarehierarchicifS1⊂S2⊂S3⊂...⊂E().AprioriestimatesoftheerrorintheenergynormareavailableforsolutionsofproblemsinCategoriesA,BandC.Convergenceiseitheralgebraicorexponential.ThealgebraicestimateisoftheformkuEX−uFEE()≤β(6.62)NandtheexponentialestimateisoftheformkuEX−uFEE()≤(6.63)expγNθwhereNisthenumberofdegreesoffreedom.Theseestimatesshouldbeunderstoodtomeanthatthereexistsomepositiveconstantsk,β(resp.γandθ)thatdependonuEX,suchthattheerrorwillbeboundedbythealgebraic(resp.exponential)estimateasNisincreased.ForsufficientlylargeNthe“lessthanorequal”sign(≤)oftencanbereplacedwith“approximatelyequal”(≈).Inthosecasestheestimateiscalledtheasymptoticestimateandtherateofconvergenceiscalledtheasymptoticrateofconvergence.Inthefollowingweassumethatthefiniteelementmeshesareregular:thatis,intwodimensionsanytwoelementsmayhaveavertexincommon,anentiresideincommonornopointsincommon.Forexample,themeshinFigure6.14(a)satisfiesthisrestrictionbutthemeshinFigure6.14(b)doesnot.ThenodepointshighlightedbyopencirclesinFigure6.14(b)arecalledirregularnodes.Ifameshhasirregularnodepointsthenthemeshisirregular.Analogousrestrictionsapplyinthreedimensionswhereinaregularmeshanytwoelementsmayhaveavertex,anentireedgeoranentirefaceincommon,ornopointsincommon.Iftheseconditionsarenotsatisfiedthenthemeshisirregular.Wenotethatsomeimplementations P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome194REGULARITYANDRATESOFCONVERGENCEFigure6.14(a)Exampleofaregularmesh.(b)Exampleofanirregularmesh.Theopencirclesrepresentirregularnodepoints.permittheuseofirregularmeshes.Thissimplifiesthemeshingofcomplicateddomainsverysubstantially.Withcertainrestrictions,itispossibletoenforcecontinuityonthebasisfunctionssuchthatthebasisfunctionslieintheenergyspaceE().Weassumethattheelementsdonotbecometoodistorted,inthesensethatnoneoftheinterioranglesapproachzeroor180◦asthemeshisrefined.Wefurtherassumethatintwo-dimensionalproblemsthatareinCategoriesAandBallanalyticarcsarecoveredbyelementboundariesandallsingularpointsarenodepoints.Analogousassumptionsapplyinthreedimensions.Intheearlyimplementationsofthefiniteelementmethodthepolynomialdegreeswererestrictedtop=1orp=2only.Sequencesoffiniteelementspaceswereproducedbymeshrefinement,thatis,byreductionofthediameterofthelargestelement,usuallydenotedbyh.Subsequentlythislimitationwasremoved,allowingthecreationofsequencesoffiniteelementspacesbyincreasingthepolynomialdegreeofelements,usuallydenotedbyp,whilekeepingthemeshfixed.Todistinguishbetweenthetwoapproaches,theterms“h-version”or“h-method”and“p-version”or“p-method”gainedcurrency.Wewillconsiderthreestrategiesforconstructingsequencesoffiniteelementspaces:1.Theh-method.Thepolynomialdegreeofelementsisfixed,typicallyatsomelownumber,suchasp=1orp=2,andthenumberofelementsisincreasedsuchthatthediameterofthelargestelement,denotedbyh,isprogressivelyreduced.Thepat-ternofmeshrefinementmaybequasiuniformornon-uniform.Asequenceofmeshesisquasiuniformwhentheratioofthelargestandsmallestdiametersoftheelementsisbounded,seeEquation(2.70).Examplesofnon-uniformrefinementareshowninFigure6.14.QuasiuniformmeshesareusedforCategoryAproblems.Non-uniformmeshrefinementisusedforCategoryBproblemswithstrongrefinementintheneigh-borhoodsofsingularpoints.TheuseofquasiuniformmeshesforCategoryBproblemsgenerallyleadstopoorperformance.2.Thep-method.Themeshisfixedandthepolynomialdegreeofelements,denotedbyp,isincreasedeitheruniformlyornon-uniformly.Thesequencesoffiniteelementspacesarehierarchic,thatis,S1⊂S2⊂S3....3.Thehp-method.Themeshisrefinedandthepolynomialdegreeofelementsiscon-currentlyincreased.Thehp-methodisusedforCategoryBproblems.Withproperrefinementandp-distribution,exponentialratesofconvergenceareobtained.Thefirstmathematicalanalysisofthep-methodwaspublishedin1981[4a]. P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeRATESOFCONVERGENCE195Table6.2Asymptoticratesofconvergenceintwodimensions.TypeofextensionCategoryhphpAAlgebraicExponentialExponentialβ=p/2θ≥1/2θ≥1/2BAlgebraic,Note1AlgebraicExponentialβ=1min(p,λ)β=λθ≥1/32CAlgebraicAlgebraicNote2β>0β>0Afourthstrategy,notconsideredhere,introducesbasisfunctions,otherthanthemappedpolynomialbasisfunctionsdescribedinChapter5,torepresentsomelocalcharacteristicsoftheexactsolution.Thisisvariouslyknownasthespaceenrichmentmethod,partitionofunitymethodandmeshlessmethod.Theasymptoticratesofconvergencefortwo-dimensionalproblemsaresummarizedinTable6.2andforthree-dimensionalproblemsinTable6.3.Itisassumedthat(a)ifquadrilateralorhexahedralelementsareusedthenthestandardpolynomialspaceistheproductspaceand(b)theelementboundariesarealignedwithmaterialandsourceinterfaces.NotethatwithreferencetoTables6.2and6.3:1.Uniformorquasiuniformmeshrefinementisassumed.Inthecaseofoptimalornearlyoptimalmeshrefinement,βmax=p/2.ThisisdemonstratedinExample6.4.2.2.WhenuEXhasarecognizablestructurethenitispossibletoachievefasterthanalgebraicratesofconvergencewithhp-adaptivemethods.3.InthreedimensionsuEXcannotbecharacterizedbyasingleparameter.Nevertheless,therateofp-convergenceisatleasttwicetherateofh-convergencewhenuniformorquasiuniformmeshrefinementisused.Table6.3Asymptoticratesofconvergenceinthreedimensions.TypeofextensionCategoryhphpAAlgebraicExponentialExponentialβ=p/3θ≥1/3θ≥1/3BNote3Exponentialθ≥1/5CAlgebraicAlgebraicNote2β>0β>0 P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome196REGULARITYANDRATESOFCONVERGENCERemark6.4.1TheratesofconvergenceinTable6.2wereprovenundertheassumptionthatthefiniteelementmeshesaretriangularor,ifquadrilateral,thentheproductspaceisused.Iftrunkspacesareusedandtheelementsarenotparallelograms(i.e.,themappingisnotlinear)thentheasymptoticrateofconvergenceoftheh-versionisonlyone-half(p/4ratherthanp/2)forCategoriesAandB.Thisisvisibleonlywhenthefiniteelementsolutionsarehighlyaccurate,however.6.4.1ThechoiceoffiniteelementspacesTheaprioriestimatesprovideanimportantconceptualframeworkfortheconstructionoffiniteelementspaces.ProblemsinCategoryAReferringtoTables6.2and6.3,itisseenthatforproblemsinCategoryAexponentialratesofconvergencearepossiblewhenp-andhp-extensionsareused.Theoptimalmeshconsistsofthesmallestnumberofelementsrequiredtopartitionthesolutiondomainintotriangularandquadrilateralelementsintwodimensions,andtetrahedral,pentahedralandhexahedralelementsinthreedimensions.Wheneverpossible,quadrilateralelementsshouldbeusedintwodimensions,hexahedralelementsinthreedimensions.Whenh-extensionsareusedtheoptimalrateofconvergenceisalgebraicwithβ=p/2.Uniformornearlyuniformmeshesshouldbeused.Example6.4.1Theexactsolutionofalongcylindricaltube,madeofanelasticmaterialandsubjectedtoconstantinternaland/orexternalpressure,ischaracterizedbytheradialdisplacementfunctionur(r):11+νu=−C+2C(1−ν−2ν2)rr≤r≤r,r>0r12ioiErwhereE,νarethemodulusofelasticityandPoisson’sratio,C1,C2areconstantstobedeterminedfromtheboundaryconditions,ri,roaretheinsidandoutsideradius,respectively(see,forexample,[29]page244).Assumingthattheaxialcomponentofthedisplacementiszeroattheends,planestrainconditionsexist.InthiscasethedomainandloadingareanalyticfunctionsandtheproblemisinCategoryA.Ifthetubeismadeoftwoormorematerials,sothatthematerialboundariesareconcentriccirclesorsmoothlines,thentheproblemisstillinCategoryA;however,thematerialinterfacesmustbecoveredbyelementboundaries.Thereasonforthisisthatthesolutionisnotanalyticonthematerialinterfaces.ProblemsinCategoryBThebehaviorofthesolutionintheneighborhoodofsingularpointsintwodimensionsischaracterizedbytheestimate(6.13).Ideally,thefiniteelementspaceshouldbedesignedsuchthattheinterpolationerrorisapproximatelythesameforallelementsintheneighborhoodsofsingularpoints.Thisgoalisachievedifthefiniteelementsarelaidoutsuchthatthesizesofelementsdecreaseingeometricprogressiontowardthesingularpoint.Theoptimalgrading P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeRATESOFCONVERGENCE197Figure6.15Exampleofageometricmesh(detail).√ischaracterizedbythecommonfactorq=(2−1)2≈0.17,whichisindependentofthedegreeofsingularityλmin[32].Inpracticeq=0.15canbeused.Forexample,inameshconsistingofthreeelementsonI=(0,),thenodepointsshouldbelocatedatx=0,x=q2,x=q,x=.Thesearecalledgeometricmeshes.An1234exampleofageometricmeshintwodimensionsisgiveninFigure6.15.Theidealdistributionofpolynomialdegreesisthatthelowestpolynomialdegreeisassociatedwiththesmallestelementandthepolynomialdegreesincreaselinearlyawayfromthesingularpoints.Thisisbecausetheerrorsinthevicinityofsingularpointsdependprimarilyonthesizeoftheelements,whereaserrorsassociatedwithelementsfartherfromsingularpoints,wherethesolutionissmooth,dependmainlyonthepolynomialdegreeofelements.Inpracticeauniformp-distributionisused,whichyieldsverynearlyoptimalresultsinthesensethatexponentialratesofconvergencearerealizedandtheworkpenaltyassociatedwithusingauniformratherthananoptimalpolynomialdegreedistributionisnotsubstantial.Example6.4.2LetusconsideranL-shapeddomainloadedbytractionscorrespondingtothelowesteigenvaluedeterminedinExercise6.3.6.PlanestrainandPoisson’sratioof0.3areassumed.Thereentrantedgesarestressfree,theotherboundariesareloadedbytractionscomputedfromthestressfieldgiveninExercise6.3.7.Sincetheexactsolutionisknown,itispossibletocomputetheexactvalueofthepotentialenergyfromEquation(4.40):1(uEX)=−[ux(σxnx+τxyny)+uy(τxynx+σyny)]ds2Aa2λ11=−4.15454423Ewhereux,uyarethedisplacementcomponentsgiveninExercise6.3.8;nx,nyarethecompo-nentsoftheunitnormaltotheboundary;andaisthedimensionshownintheinsetinFigure6.16.TheerrorsintheenergynormwerecomputedusingEquation(2.93).Theerrorcurvesforh-extensionusinguniformmeshes(p=2),p-extension,trunkspace,onauniformmesh, P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome198REGULARITYANDRATESOFCONVERGENCEFigure6.16Example6.4.2:errorcurvesforh-andp-extensions.Firstpublishedin[80].ReprintedwithpermissionofJohnWiley&Sons,Inc.c1991.andp-extension,trunkspace,onan18-elementgeometricmeshconstructedwiththegradingfactorq=0.15,areshowninFigure6.16.Itcanbeseenthattheasymptoticratesofconvergenceareexactlyaspredictedbytheestimate(6.62).However,whenp-extensionisusedonageometricmeshthepre-asymptoticrateisexponential,ornearlyso.ThiscanbeexplainedbyobservingthatthegeometricmeshshowninFigure6.15isover-refinedforlowpolynomialdegrees,hencethedominantsourceoferroristhatpartofthedomainwheretheexactsolutionissmoothandthustherateofconvergenceisexponential,aspredictedbytheestimate(6.63).Convergenceslowstothealgebraicrateforsmallerrors,wherethedominantsourceoferroristheimmediatevicinityofthesingularpoint.Errorcurvesforh-extensionusingradicalmeshes(p=2)andhp-extensionareshowninFigure6.17.Forpurposesofcomparison,theerrorcurveforp-extensiononan18-elementgeometricmesh,showninFigure6.16,isshowninFigure6.17also.Thenodesoftheradicalmesheswerelocatedatdistancesdkfromthesingularpointdefinedbyθkp+13dk=a,k=1,2,...,M,θ===5.51Mλ10.544whereaisthedimensionshowninFigure6.17andM=3,4,...,11isthenumberoflayersofelements.Thischoiceofθisbasedontheconsiderationthatanidealmeshisoneinwhichthe P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeRATESOFCONVERGENCE199RELATIVEERRORINENERGYNORM(%)RADICALMESH(M=3)RadialdistancearenottoscaleNUMBEROFDEGREESOFFREEDOMFigure6.17Example6.4.2:errorcurvesforh-,p-andhp-extensionsandtheradicalmeshcorrespondingtoM=3usedinh-extension(nottoscale).errorcontributionofeachelementisapproximatelythesame.Foradetaileddiscussionoftheone-dimensionalcasewereferto[70].NotethatgradingintheradialdirectionisnottoscaleinFigure6.17.Thisisbecausethegradingissostrongthatonlythefirstlayerwouldbevisibleonameshdrawntoscale.Thecircumferentialdistributionofelementsisnearlyuniform,asshowninFigure6.17.ThecircumferentialdistributionwaskeptconstantforM=3toM=8thendoubledforM=9toM=11.Thiscausestheconvergencepathtohaveasmall“kink.”Itcanbeseenthattheoptimalrateofalgebraicconvergenceβ=p/2=1.0isverynearlyrealized.Theerrorcurveforthehp-extensionwasobtainedusinggeometricmesheswiththegradingfactorq=0.15.Thepolynomialdegreewasincreaseduniformlyfromp=3top=8(trunkspace).Startingwiththe18-elementmeshshowninFigure6.16,foreachincreaseinpolynomialdegreeanewlayerofelementswasaddedintheneighborhoodofthesingularpoint.Itcanbeseenthattherateofconvergenceisstrongerthanalgebraic;infactitisexponential.Whenhighaccuracyisdesiredthenhp-extensionhastobeused.TheestimatedrelativeerrorintheenergynormwascomputedbythemethoddescribedinSection2.7fromdatageneratedbyp-extensiononthe18-elementgeometricmeshshowninFigure6.16.TheestimatedandtrueerrorsandtheeffectivityindicesarelistedinTable6.4.TheparameterβisthatinEquation(6.62).Theextrapolatedvalueofthepotentialenergywascomputedfromthelastthreepointsontheconvergencepath,correspondingtop=6top=8.Thesepointsarelocatedwheretheerrorcurveistransitioningfromthepre-asymptoticrangetotheasymptoticrange.Forthisreasontherelativeerrorsareunder-estimated;however,theeffectivityindexremainsreasonablyclosetounity. P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome200REGULARITYANDRATESOFCONVERGENCETable6.4Example6.4.2:L-shapeddomain,geometricmesh,18elements,trunkspace.Planestrain,ν=0.3.Estimatedandtruerelativeerrorsinenergynorm.Effectivityindex(θ).β(er)E(%)(u)EFEpNEst.TrueEst.TrueθA2a2λ1t1z141−3.886332——25.4225.411.002119−4.1248671.031.038.448.461.003209−4.1481211.371.363.913.930.994335−4.1526511.331.302.092.140.985497−4.1536360.990.941.421.480.966695−4.1539750.780.681.091.170.937929−4.1541390.690.600.890.990.8981199−4.1542380.690.560.750.860.87∞∞−4.1544700.540Remark6.4.2Thismodelproblemisrepresentativeoftheneighborhoodsofcorners(resp.edges)intwo-dimensional(resp.three-dimensional)solutiondomains.Althoughthephysicalobjectsbeingmodeledaretypicallyfilleted,thefilletsarefrequentlyomittedinordertofacilitatemeshing.Thereforetheproblembeingsolvedbythefiniteelementmethodisoftenwithoutfillets.Ifthefilletsareomittedthenthemathematicalmodelcannotbeusedforapproximatingthemaximumstressunlessspecialpost-processingproceduresareemployed.Remark6.4.3InFigure6.17theconvergencecurvecorrespondingtotheradicalmesh(p=2)wasobtainedwithasequencerefinementcharacterizedbythenumberoflayersofelementsaroundthesingularpointdenotedbyM.Assumingthata=1.0m,thesizeofthesmallestelementsatM=11is1.8×10−6mwhichissmallerthanthetypicalgrainsizeofmetals.Thereforetheidealizationofmaterialpropertiesasperfectlyhomogeneouscannotbejustifiedonthisscale.Nevertheless,thegoalofnumericalanalysisistofindaverifiedapproximationtotheexactsolutionofthemathematicalmodel.Theproblemofdefinitionandvalidationofamathematicalmodelisseparatefromitsnumericalapproximationwhichistheproblemunderconsiderationhere.Itwouldnotbecorrecttosayforexamplethatthesmallestelementshouldnotmesmallerthanthegrainsize.Exercise6.4.1RefertoExercise6.3.1andFigure6.4(b).Let=1.0andα=7π/4.OnABandBCusethesameboundaryconditionsasinExercise6.3.1.Onprescribethenormalflux(2k−1)πq(k)=−λλk−1(cosλθ+(−1)ksinλθ)whereλ=·nkkkk2α1.Solvethisproblemfork=1,2,3andcomparetheestimatedratesofconvergencewiththetheoreticalratesgiveninTable6.2.Discusstheresults. P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoComeRATESOFCONVERGENCE2012.PrescribetheDirichletconditionsu(k)=λk(cosλθ+(−1)ksinλθ)kkonandrepeatthesolutionsfork=1,2,3.Comparetheresultswiththepreviouslyobtaineddata.Commentonthedifferences.3.Showthattheexactvalueofthestrainenergyisgivenbyλ2λk+α/2kk2Uk=(cosλkθ+(−1)sinλkθ)dθ2−α/2henceU=0.7853981633972λ11U=2.356194490192λ22U=3.926990816992λ3.34.UsingTheorem2.6.3andtherelationshipbetweenthepotentialenergyandthestrainenergyoftheexactsolution,computetheeffectivityindices.Exercise6.4.2UsingTable6.2,estimatetheasymptoticrateofconvergenceintheenergynormfortheproblemshowninFigure6.8(a)forh-andp-extensions.Constructacoarsemeshonasemicirculardomainandusep-extension,thenletp=2andrefinethemeshuniformly.AsinExercise6.3.13,specifyTr=σr,Tt=τrθonthesemicircularboundary.Useplanestrainandν=0.3.Comparetheobservedratesofconvergencewiththepredictedones.Hint:MakeuseofTheorem2.6.3.Theexactvalueofthepotentialenergyis−0.37572208p2R2t/E0whereRistheradiusofthesemicirculardomainandtisthethickness.ProblemsinCategoryCWhentheexactsolutiondoesnothavearecognizablestructure,asinhighlyheterogeneousmedia,thenh-extensionswithuniformornearlyuniformmeshrefinementshouldbeused.6.4.2UsesofaprioriinformationUnderstandingtherelationshipbetweeninputdataandtheregularityoftheexactsolutionisveryimportantinfiniteelementanalysisfortworeasons.First,inconstructinganinitialfiniteelementmesh,whethermanuallyorbyautomaticmeshgenerators,thepresenceofsingularpoints,singulararcsandboundarylayersshouldbetakenintoaccount.Second,ifthegoalofthecomputationistodeterminethemaximalstress,strain,flux,etc.,thenthedegreeofsingularityλ(seeEquation(6.13))mustbegreaterthanoneintheregionwherethemaximumissought.TheaccuracyofthefiniteelementsolutiondependsonthesmoothnessoftheexactsolutionandthefiniteelementspaceS˜(,,p,Q).Allimplementationsoffiniteelementanalysisimposelimitationsonthemaximumpolynomialdegreeandthechoiceofmappingfunctions.Thereforeanimportanttaskistoconstructafiniteelementmeshsuchthatthe P1:OSOJWST055-06JWST055-SzaboFebruary16,20118:0PrinterName:YettoCome202REGULARITYANDRATESOFCONVERGENCEdesiredaccuracycanbeachievedatp≤pmaxwherepmaxisthemaximump-levelallowedbytheimplementation.Inthefollowingwefocusonthegoaltoobtainafiniteelementsolutionthathasareasonablysmallerrorintheenergynorm.Theproblemisformulatedasfollows:givenatoleranceofrelativeerrorintheenergynormτ,designameshsuchthattherelativeerrorintheenergynormiswithintheallowedtoleranceatsomevalueofpthatislessthanorequaltopmax.ProblemsinCategoryAForproblemsinCategoryAtherulesofmeshconstructionareasfollows:1.Thematerialandsourceterminterfacescoincidewiththeboundariesofelements.2.Themeshissuchthat:(a)Themaximumratioofthelengthofanytwosidesofanelementislessthanabout6.5.(b)Oncurvedsidestheratioofthelengthofanelementsideandtheradiusofcurvatureofthesideislessthanabout0.5.(c)Thevertexanglesαisatisfythecondition0.1π<αi<0.9π.(d)Theprescribedboundaryconditionsareinterpolatedbythetraceofthefiniteelementspacesuchthattheinterpolationerrorisapproximatelyequalamongtheelementsthatlieontheboundaryatp≈pmax/2and,furthermore,theinterpolationerrorinmaximumnormislessthanaboutτ/10.Whenthemeshsatisfiesrules1through2(c)thentherateofconvergenceofthep-versionisexponentialforproblemsinCategoryA,asindicatedinTable6.1,hencerapidconvergenceisachieved.Thisshouldnotbeinterpretedtomeanthatthecomputationaleffortisnecessarilysmall,becausetheconstantinEquation(6.63)canbeverylargeandentryintotheasymp-toticrangemayoccuronlyatmuchhighervaluesofpthanthemaximumallowedbytheimplementation(pmax).Rule2(d)ismadenecessarybythefactthatpcannotexceedpmax.Inthefollowingexampleweillustrateanapplicationofrule2(d)andoutlinetheunderlyingconsiderations.Example6.4.3Inthefollowingweconsiderthemodelproblemu=0on={x,y|−10,βi>0,φi(θ)areanalyticorpiecewiseanalyticfunctions,Aiarecoefficientsthatdependonf,calledfluxintensityfactors,andρ0istheradiusofconvergence.Theregularityofthesolutionintheneighborhoodofsingularpointsischaracterizedbythesmallestλi(correspondingtonon-zeroAi)denotedbyλ.Intwo-andthree-dimensionalelasticitysimilar,althoughmorecomplicated,expressionscanbewrittenandλimaybecomplex.Hereλisunderstoodtobetherealpartofthesmallestfractionalcharacteristicvalue.Solutionswhereλ<1atoneormorepointsaresaidtobestronglysingular,otherwiseweaklysingular.Whenthesolutionisstronglysingularthefirst(andhigher)derivativesareinfiniteatthesingularpoint.Thefiniteelementmeshshouldbelaidoutsothattheelementsinthevicinityofstronglysingularpointsarethesmallest.Iftheoptionofusingp-extensionsisprovidedbythefiniteelementanalysissoftwarethenmeshgradingshouldbeingeometricprogressiontowardstronglysingularpointswithacommonfactorofapproximately0.15. P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoCome7ComputationandverificationofdataFollowingassemblyandsolution(outlinedinSections2.5.6and2.5.8),thefiniteelementsolutionisstoredintheformofdatasetsthatcontainthecoefficientsoftheshapefunc-tions,themappingfunctionsandindicesthatidentifythepolynomialspaceassociatedwitheachelement.Someofthedataofinterest,suchastemperature,displacement,flux,strain,stress,canbecomputedfromthefiniteelementsolutioneitherbydirectorindirectmethods,whileothers,suchasstressintensityfactors,canbecomputedbyindirectmethodsonly.Inthischapterthetechniquesusedforthecomputationandverificationofengineeringdataaredescribed.7.1ComputationofthesolutionanditsfirstderivativesIfoneisinterestedinthevalueofthesolutionatapoint(x0,y0,z0)thenthedomainhastobesearchedtoidentifytheelementinwhichthatpointlies.Supposethatthepointliesinthekthelement.Thenextstepistofindthestandardcoordinates(ξ0,η0,ζ0)fromthemappingfunctionsx=Q(k)(ξ,η,ζ),y=Q(k)(ξ,η,ζ),z=Q(k)(ξ,η,ζ).(7.1)0x0000y0000z000Unlessthemappingofthekthelementhappenstobelinear,theinverseofthemappingfunctionisnotknownexplicitly.Thereforethisstepinvolvesarootfindingprocedure,suchastheNewton–Raphsonmethod.Ifthepointisavertexorliesonanedge,orinthreedimensionsonaface,thenitmaybesharedbymorethanoneelement.Thenextstepistolookuptheparametersthatidentifythestandardspaceassociatedwiththeelementandthecomputedcoefficientsofthebasisfunctions.Withthisinformationthesolutionanditsderivativescanbecomputed.Forexample,letusassumethatthesolutionisaIntroductiontoFiniteElementAnalysis:Formulation,VerificationandValidation,FirstEdition.BarnaSzabóandIvoBabuška.©2011JohnWiley&Sons,Ltd.Published2011byJohnWiley&Sons,Ltd.ISBN:978-0-470-97728-6 P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoCome216COMPUTATIONANDVERIFICATIONOFDATAscalarfunctionandthestandardspaceSp,q((q))isassociatedwiththekthelement,denotedstbyk.Thenthefiniteelementsolutionatpoint(x0,y0)∈kis:n(k)uFE(x0,y0)=aiNi(ξ0,η0)(7.2)i=1wheren=(p+1)(q+1)isthenumberofshapefunctionsthatspanSp,q((q)),N(ξ,η)aresti(k)theshapefunctionsandaarethecorrespondingcoefficients.WhenthesolutionisavectorifunctiontheneachcomponentofuFEisintheformofEquation(7.2).ComputationofthefirstderivativesofuFEatthepoint(x0,y0)involvescomputationoftheinverseoftheJacobianmatrixatthecorrespondingpoint(ξ0,η0)andmultiplyingthederivativesofthefiniteelementsolutionwithrespecttothestandardcoordinates.ReferringtoEquation(5.58),⎧⎫⎡⎤−1⎧⎫⎪⎪∂uFE⎪⎪∂x∂y⎪⎪∂Ni⎪⎪⎪⎨∂x⎪⎬⎢∂ξ∂ξ⎥n⎪⎨∂ξ⎪⎬⎢⎥(k)=⎢⎥ai(7.3)⎪⎪⎪⎩∂uFE⎪⎪⎪⎭⎣∂x∂y⎦i=1⎪⎪⎪⎩∂Ni⎪⎪⎪⎭∂y(x0,y0)∂η∂η(ξ0,η0)∂η(ξ0,η0)wherex=Q(k)(ξ,η),y=Q(k)(ξ,η).Thefluxvector(resp.stresstensor)iscomputedbyxymultiplyingthetemperaturegradient(resp.thestraintensor)bythethermalconductivitymatrix(resp.materialstiffnessmatrix).ThetransformationofvectorsandtensorsisdescribedinAppendixC.Thederivativesofthefiniteelementsolutionarediscontinuousalonginter-elementbound-aries.Therefore,ifthepointselectedfortheevaluationoffluxes,stresses,etc.,isanodepoint,orapointonaninter-elementboundary,thenthecomputedvaluedependsontheelementselectedforthecomputation.Thedegreeofdiscontinuityinthenormalandshearingstressesorthenormalfluxcomponentatinter-elementboundariesisanindicatorofthequalityoftheapproximation.Inimplementationsoftheh-versionthederivativesaretypicallyevaluatedattheintegrationpointsandareinterpolatedovertheelements.Ingraphicaldisplaysintheformofcontourplots,discontinuitiesofthederivativesatelementboundariesareoftenmaskedbymeansofsmoothingthecontourlinesthroughaveraging.Inthep-versionthestandardelementissubdividedsoastoproduceauniformgrid,calledadisplaygrid,andthesolutionanditsderivativesareevaluatedatthegridpoints.Sincethestandardcoordinatesofthegridpointsareknown,inversemappingisnotinvolved.Thequalityofcontourplotsdependsonthequalityofdatabeingdisplayedandonthefinenessofthedisplaygrid.Searchingforamaximumorminimumvaluealsoinvolvessearchingonauniformgriddefinedonthestandardelements.Thefinenessofthegrid,andhencethenumberofpointssearchedfortheminimumormaximum,iscontrolledbyaparameter.Inconventionalimple-mentationsoftheh-versionthesearchgridistypicallydefinedbytheintegrationpointsorthenodepoints.Exercise7.1.1Considertwoplaneelasticelementsthathaveacommonedge.Assumethatdifferentmaterialpropertieswereassignedtotheelements.Showthatthenormalandshearing P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoComeNODALFORCES217stressescorrespondingtotheexactsolutionhavetobethesamealongthecommonedge.Hint:Considerequilibriuminthecoordinatesystemdefinedinthenormalandtangentialdirections.7.2NodalforcesRecallthedefinitionofnodalforces{f(k)}inSection2.5.9{f(k)}=[K(k)]{a(k)}−{r¯(k)},k=1,2,...,M()(7.4)where[K(k)]isthestiffnessmatrix,{a(k)}isthesolutionvectorand{r¯(k)}istheloadvectorcorrespondingtovolumeforcesandthermalloadsactingonelementk.7.2.1Nodalforcesintheh-versionWhensolvingproblemsinelasticityusingfiniteelementanalysisbasedontheh-version,nodalforcesaretreatedinthesamewayasconcentratedforcesaretreatedinstatics.Typicalusesofnodalforcesare:(a)isolatingsomeregionofinterestfromalargerstructureandtreatingtheisolatedregionasifitwereafreebodyheldinequilibriumbythenodalforces;and(b)computingstressresultants.Theunderlyingassumptionisthatnodalforcesreliablyrepresenttheloadpath,thatis,thedistributionofinternalforcesinastaticallyindeterminatestructure.Thisassumptionisusuallyjustifiedbytheargumentthatnodalforcessatisfytheequationsofstaticequilibriumforanyelementorgroupofelements.Thefollowingdiscussionwillshowthatsatisfactionofequilibriumisrelatedtotherankdeficiencyofunconstrainedstiffnessmatrices.Consequently,equilibriumofnodalforcesshouldnotbeinterpretedasanindicatorofthequalityofthefiniteelementsolutionanddoesnotguaranteethatthenodalforcesarereliableapproximationsoftheinternalforcesinastaticallyindeterminatestructure.Ontheotherhand,nodalforcesareusefulforthecomputationofstressresultants.Letusassume,forexample,thatkisaneight-nodeisoparametricquadrilateralelement.Thenumberofdegreesoffreedomperfield,denotedbyn,is8.ThenotationisshowninFigure7.1.Inexpandednotationtheelementsofthenodalforcevector{f(k)}aref(k,i)=f(k),f(k,i)=f(k),i=1,2,...,n.(7.5)xiyn+iSimilarly,theelementsofthevector{r¯}arewrittenasfollows:r¯(k,i)=r¯(k),r¯(k,i)=r¯(k),i=1,2,...,n.(7.6)xiyn+iOnadaptingthenotationusedinEquation(5.62)toproblemsofplanarelasticitywecanwrite2n(k,i)−1T(k)(k)(k,i)fx=[M1][Jk]{D}Ni[E]{εj}aj|Jk|dξdη−r¯x(7.7)(q)stj=1 P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoCome218COMPUTATIONANDVERIFICATIONOFDATAFigure7.1Nodalforcesassociatedwiththeeight-nodequadrilateralelement:notation.and2n(k,i)−1T(k)(k)(k,i)fy=[M2][Jk]{D}Ni[E]{εj}aj|Jk|dξdη−r¯y(7.8)(q)stj=1where[Jk]istheJacobianmatrixand⎧⎫∂⎡⎤⎡⎤⎪⎪⎪⎪1000⎪⎨∂ξ⎪⎬[M1]=⎣00⎦,[M2]=⎣01⎦,{D}=0110⎪⎪∂⎪⎪⎪⎩⎪⎭∂ηand[M][J]−1{D}Nforj=1,2,...,n(k)1kj{ε}=j[M][J]−1{D}Nforj=n+1,n+2,...,2n.2kj−nTheelementsoftheloadvectorcorrespondingtobodyforcesandthermalloadsareTr¯(k,i)=NF|J|dξdη+[M][J]−1{D}N[E]{α}T|J|dξdη(7.9)xixk1kik(q)(q)stst P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoComeNODALFORCES219andTr¯(k,i)=NF|J|dξdη+[M][J]−1{D}N[E]{α}T|J|dξdη.(7.10)yiyk2kik(q)(q)ststTheequationsofstaticequilibriumarenf(k,i)+Fdxdy=0(7.11)xxi=1knf(k,i)+Fdxdy=0(7.12)yyi=1kandnXf(k,i)−Yf(k,i)+(xF−yF)dxdy=0(7.13)iyixyxi=1kwhereXi,Yiarethecoordinatesoftheithnode.SatisfactionofEquation(7.11)andEquation(7.12)followsfromthefactthatnnNi(ξ,η)=1,therefore{D}Ni(ξ,η)=0.i=1i=1SatisfactionofEquation(7.13)followsfromthemappingfunctions(5.40)and(5.41)andthefactthatinfinitesimalrigidbodyrotationsdonotcausestrain:nT∂∂[M][J]−1{D}YN≡0y={001}(7.14)1kii∂x∂yi=1nT∂∂[M][J]−1{D}XN≡0x={001}.(7.15)2kii∂y∂xi=1OnsubstitutingEquations(7.14)and(7.15)intotheexpressionsforf(k,i),f(k,i),r¯(k,i)andxyxr¯(k,i),Equation(7.13)isobtained.yNotethattheconditionsforstaticequilibriumaresatisfiedindependentlyof{a(k)}.There-fore,equilibriumofnodalforcesisunrelatedtothefiniteelementsolution.Exercise7.2.1Considertheproblemofheatconductionintwodimensions.AssumethatthenodalfluxeswerecomputedanalogouslytoEquation(7.4).Definethetermwhichisanalogousto{r¯}andshowthatthesumofnodalfluxesplustheintegralofthesourcetermoveranelementiszero.Useaneight-nodequadrilateralelementandasix-nodetriangularelementtoillustratethispoint. P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoCome220COMPUTATIONANDVERIFICATIONOFDATAFigure7.2Example7.2.1:notation.7.2.2Nodalforcesinthep-versionWehaveseeninthecaseoftheeight-nodequadrilateralelementthatequilibriumofnodalforceswasrelatedtothefactsthatthesumoftheshapefunctionsisunityandthefunctionsxandycouldbeexpressedaslinearcombinationsoftheshapefunctions.WhenthehierarchicshapefunctionsbasedontheintegralsofLegendrepolynomialsareused,suchasthoseillustratedinFigure5.4,thenthesumofthefirstfourshapefunctionsisunityindependentlyofp.Therefore,intheequilibriumequations(7.11)through(7.13),n=4.Hereweconsideronlyisoparametricandsubparametricmappings.OthermappingsinvolveadditionalconsiderationswhicharediscussedinSection5.4.4.Example7.2.1Arectangulardomainrepresentinganelasticbodyofconstantthicknessissubjectedtotheboundaryconditionsun=0,ut=δonboundarysegmentsBCandDAshowninFigure7.2(a).Thesubscriptsnandtrefertothenormalandtangentdirectionsrespectively.BoundarysegmentsABandCDaretractionfree.Wesolvethisasaplanestressproblemusingoneelement,representedbytheshadedpartofthedomain,andproductspaces.Theantisymmetryconditionisappliedatx=/2.ThenodenumberingisshowninFigure7.2(b).Sinceonlyoneelementisused,thesu-perscriptthatidentifiestheelementnumberisdropped.Letting=1000mm,d=50mm,b=20mm,E=200GPA,ν=0.295,δ=5mm,thecomputedvaluesofthenodalforcesareTable7.1Example7.2.1:nodalforces(kN).ThenotationisshowninFigure7.2(b).Productspaces.pf(1)f(2)f(3)f(4)f(1)f(2)f(3)f(4)xxxxyyyy1−1971.30.0000.0001971.3−98.56498.56498.564−98.5642−64.6780.0000.00064.678−3.2343.2343.234−3.2343−50.5460.0000.00050.546−2.5272.5272.527−2.5274−50.1900.0000.00050.190−2.5092.5092.509−2.5095−50.0100.0000.00050.010−2.5002.5002.500−2.5006−49.9070.0000.00049.907−2.4952.4952.495−2.4957−49.8430.0000.00049.843−2.4922.4922.492−2.4928−49.8020.0000.00049.802−2.4902.4902.490−2.490 P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoComeNODALFORCES221Figure7.3Example7.2.1:thesmallestsolutiondomain.asgiveninTable7.1.Inthisexample{r¯}=0.Thereforethenodalforceswerecomputedfrom{f}=[K]{a}.TheresultsshowninTable7.1indicatethatequilibriumofnodalforcesissat-isfiedateveryp-level,independentlyoftheaccuracyofthefiniteelementsolution.ThisproblemcouldhavebeensolvedontheshadeddomainshowninFigure7.3,inwhichcasetheantisymmetryconditionwouldhavebeenprescribedonthex-axis.UsingthenotationshowninFigure7.3(b),thecomputednodalforcesaregiveninTable7.2forp=8.Itcanbeseenthatthenodalforcesonceagainsatisfytheconditionofstaticequilibrium.Exercise7.2.2ThenodalforcesgiveninTable7.1(resp.Table7.2)werecomputedontheshadeddomainshowninFigure7.2(a)(resp.Figure7.3(a)).ShowthatthestressresultantsactingonboundarysegmentsBCandDAarethesame.Hint:SketchandlabelthevaluesofthenodalforcesontheelementshowninFigure7.3(b)anditsantisymmetricpair.7.2.3NodalforcesandstressresultantsThenodalforcesarerelatedtotheextractionfunctionsforstressresultants.WeillustratethisonthebasisofExample7.2.1.Wehave([D]{v})T[E][D]{u}bdxdy=(T(FE)v+T(FE)v)bds(7.16)FExxyy∂whereisthedomainoftheelementshowninFigure7.2(b)andbisthethickness.Ifweareinterestedintheshearforceactingonthesidebetweennodes2and3,denotedbyV2,3,thenweselectvx=0onandvyasmoothfunctionofsuchthatvy=1onthesidebetweennodes2and3,andvy=0onthesidebetweennodes4and1.Specifically,ifweselectvy=N2(ξ,η)+N3(ξ,η)thenwehavenode32nV=T(FE)bdy=(k+k)a=f(2)+f(3)(7.17)2,3yn+2,jn+3,jjyynode2j=1wherenisthenumberofdegreesoffreedomperfield.InExample7.2.1productspaceswereused,thereforen=(p+1)2.Table7.2Example7.2.1:nodalforces(kN).SolutionontheshadeddomainshowninFigure7.3(a).ThenotationisshowninFigure7.3(b).Productspace.pf(1)f(2)f(3)f(4)f(1)f(2)f(3)f(4)xxxxyyyy8−12.454−37.3530.00049.8064.9221.1441.346−7.412 P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoCome222COMPUTATIONANDVERIFICATIONOFDATAExercise7.2.3SolvetheproblemofExample7.2.1andcomputethestressresultantsV4,1andM4,1onthesidebetweennodes4and1bydirectintegration.Keeprefiningthemeshuntilsatisfactoryconvergenceisobserved.ComparetheresultswiththosecomputedfromthenodalforcesinTable7.1.Bydefinition+d/2M4,1=Txybdy.−d/2Thisexerciseshowsthatintegrationofstressesismuchlessefficientthanextraction.ThisisbecausethepresenceofsingularitiesatpointsAandDinfluencestheaccuracyofthenumericalintegration.7.3VerificationofcomputeddataHavingperformedafiniteelementanalysis,itisnecessarytodeterminewhetherthedataofinterestcomputedfromthefiniteelementsolutionaresufficientlyaccurateforthepurposesoftheanalysis.Theaccuracyofthedatadependsonthechoiceofthefiniteelementspaceandthemethodbywhichthedataarecomputed.WehaveseenasimpleillustrationofthisinExample2.5.8.Onlydirectmethodsofcomputationwillbeconsideredinthissection.Inmanypracticalproblemsthedataofinterestlieinsomesmallsubdomainofthesolutiondomain.Forexample,wemaybeinterestedinthestressesinthevicinityofafastenerholeinalargeplate.Theneighborhoodofthefastenerholeistheregionofprimaryinterest,therestoftheplateistheregionofsecondaryinterest.Errorsinthecomputeddatamaybecausedbyinsufficientdiscretizationintheregionofprimaryorsecondaryinterest,orboth.Errorscausedbyinsufficientdiscretizationoftheregionofsecondaryinterestarecalledpollutionerrors.Sincetheexactsolutionisindependentoftheparametersthatcharacterizethefiniteele-mentspace,verificationinvolvesstepstoascertainthat(a)thecorrectinputdatawereusedintheanalysisand(b)thedataofinterestaresubstantiallyindependentoftheparametersthatcharacterizethefiniteelementspace.Thisinvolvesobtainingtwoormoresolutionscorre-spondingtoasequenceoffiniteelementspacesandexaminationoffeedbackinformation,thatis,informationgeneratedfromfiniteelementsolutions.Finiteelementspacesgener-atedbyp-extensionarehierarchicwhereasfiniteelementspacesgeneratedinh-extensions,withthesequenceofmeshescreatedbymeshgenerators,aretypicallynothierarchic.Therecommendedstepsareasfollows:1.Displaythesolutiongraphicallyandcheckwhetherthesolutionisreasonable.Forexample,plottingthedeformedconfigurationwiththescalefactorsetto1providesinformationonwhetheralargeerroroccurredinspecifyingtheloadingandconstraintconditionsormaterialproperties.2.Estimatetherelativeerrorintheenergynormanditsrateofconvergence.Theesti-matedrelativeerrorinenergynormisausefulindicatoroftheoverallqualityofthesolution,roughlyequivalenttoestimatingtheroot-mean-squareerrorinstresses[80].Theestimatedrateofconvergenceisanindicatorofwhethertherateofchangeoferrorisconsistentwiththeasymptoticratefortheproblemclass.SubstantialdeviationsfromthevaluesgiveninTable6.2andTable6.3typicallyindicateerrorsintheinputdataorinthefiniteelementmesh.Forexample,someelementsmaybehighlydistorted. P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoComeVERIFICATIONOFCOMPUTEDDATA2233.Checkforthepresenceofjumpdiscontinuitiesinstressesandfluxesinregionswherethestressesorfluxesarelarge.Thenormalfluxandthenormalandshearingstresscomponentsatinter-elementboundariesmustbecontinuous.Inregionswherethestressesorfluxesaresmall,somediscontinuityisgenerallyacceptable.Jumpdiscon-tinuitiesinstressesandfluxesbetweentheregionsofprimaryandsecondaryinterestareindicationsofpollutionerrors.4.Showthatthedataofinterestaresubstantiallyindependentofthemeshand/orthepolynomialdegreeofelements.Thesestepsareillustratedbythefollowingexample.Example7.3.1LetusconsidertheproblemofExample4.3.5.ReferringtothenotationinFigure4.11andFigure7.4,weleta=1,b/w=1.5andσ∞=1.TheboundaryconditionsarethesameasinExample4.3.5.ThemaximumtensilestressoccursatpointEanditsexactvalueisσx=3.Thegoaloftheanalysisistoapproximatethemaximumtensilestressbythefiniteelementmethodandtoverifythattheerrorofapproximationissmall.Verificationproceduresandtheuseoffeedbackinformationareillustratedforthreevaluesofthelengthparameterbinthefollowing.Theregionofprimaryinterestistheneighborhoodofthecircularhole.Theinitialmeshislaidoutsothatthesizeofelementsintheneighborhoodoftheholeisapproximatelythesizeoftheradiusofthehole,asindicatedinFigure7.4.SincethisproblembelongsinCategoryAweexpectexponentialconvergenceinenergynormwhenp-extensionisused.UsingthemethodofSection2.7,theestimatedrelativeerrorsinenergynormwerecomputedandplottedinFigure7.5forthreevaluesofthelengthparameterb,indicatedinthefigure.Alsoplottedarethetrueerrors,indicatedbythedashedlines.Forb=10theestimatedandtrueerrorsaresoclosethattheycannotbedistinguishedonthisdiagram.√Notethattherelativeerrorintheenergynormisplottedonalogarithmicscalevs.N.ThereasonforthisisseeninEquation(6.63)andTable6.2.Sinceexponentialconvergenceisexpected,therelativeerrorcurveshouldappearasastraightlineinthisdiagram.Asseeninthefigure,thisisindeedthecaseandtheestimatedrelativeerrorsarereassuringlysmall.ThefirststepintheverificationprocessindicatesthatthesequenceoffiniteelementspacesFigure7.4Four-elementmesh,b/w=1.5. P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoCome224COMPUTATIONANDVERIFICATIONOFDATARelativeerror(%)Figure7.5Four-elementmesh,b/w=1.5.Solidlines:estimatedrelativeerrorsinenergynorm,p-extension,trunkspace.Dashedlines:trueerrors.generatedbyuniformp-extensionandthemeshshowninFigure7.5yieldssmallrelativeerrorsintheenergynormattheexpectedrateofconvergenceforthethreevaluesofb.ThecomputedvaluesofσxatpointEareshowninFigure7.6(a).Itcanbeseenthatforb=10thevaluesofσxconvergestronglytoalimitingvalueof3.0whichisthecorrectlimit.Thisdoesnothappenforb=100andb=1000,however.Thevalueofσxchangessubstantiallywithpintherangep=1,2,...,8allowedbytheimplementation.Thisindicatesthatasbisincreased,thefour-elementmeshbecomesinadequatefromthepointofviewofcomputingandverifyingσxwhenthemaximumvalueofpislimitedto8.Thesituationwouldbe)ExσComputedvaluesof(Figure7.6Four-elementmesh,b/w=1.5.(a)ComputedvaluesofσxinpointE.(b)Computedvaluesofσxintheneighborhoodoftheholeforb=100,p=8trunkspace. P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoComeVERIFICATIONOFCOMPUTEDDATA225differentif,forexample,thedisplacementofpointCweretobecomputed.ItislefttothereadertodemonstratethisinExercise7.3.1.Thisexampledemonstratesthatitisnotsufficienttohaveasmallrelativeerrorintheenergynormifthegoalistocomputethemaximumstressor,moregenerally,anydatacomputedfromthefirstderivatives.Inthisexamplethesizeoftheregionofprimaryinterestisfixedbutthesizeoftheregionofsecondaryinterestincreaseswithb.ReferringtoEquations(4.34)to(4.36)weseethatthestressvarieslike1/r2and1/r4intheneighborhoodofthehole,butfarfromtheholethestressisverynearlyconstant.Thecenteroftheholeisasingularpointthatliesoutsideof,butcloseto,thesolutiondomainanditsinfluenceonthestressdistributiondecaysinproportionto1/r2.Whenb=10theestimatedrelativeerrorintheenergynormandtheerrorin(σx)Earebothsmall,indicatingthatthefiniteelementmeshisproperlydesignedforapproximatingtheexactsolutionanditsfirstderivativesintherangep=1,2,...,8.Whenb=100orb=1000theerrorintheenergynormissmallbuttheerrorin(σx)Eislarge.Theproblemisthat,giventherestrictionp≤8imposedbytheimplementation,theelementsintheregionofsecondaryinterestaretoolarge.Thepollutionerrorcausedbyinadequatemeshingismanifestedbythelargejumpinσxattheboundarybetweentheinnerandouterlayersofelements,asshowninFigure7.6(b).Ontheotherhand,thevalueofthepotentialenergyisdominatedbythenearlyconstantstressawayfromtheholeandthelargelocalerrorinthesmallneighborhoodoftheholeisinsignificantincomparison.Inproblemslikethisthefiniteelementmeshshouldbegradedingeometricprogressionwithreferencetothecenterofthehole,withacommonfactorofapproximately0.15.TheexactsolutionofthisproblemisinCategoryA;however,whenthesizeoftheholeissmallinrelationtotherestofthedomainthenthemeshshouldbelaidoutasifitwereinCategoryBinordertoobtainagoodapproximationofthefirstderivativesovertheentiredomain.Remark7.3.1ItisclearlyvisibleinFigure7.6that(σx)Ehasnotconvergedtoalimitingvalueforb=100andb=1000.Itispossibletoconstructmeshessuchthatitwouldappearthatthedatahasconvergedbuttheapparentlimitiswrong.Thereforeitisnotsufficienttotestforindicationsofconvergenceofthedataofinterest,itisalsonecessarytocheckforpollutionerrors.Remark7.3.2TherearesomeinterestingfeaturesdemonstratedinFig7.5:(a)Thedistortionofthelargeelementsincreaseswithincreasingb,thereforeonewouldexpectanincreaseinerror.However,therelativeerrordecreases.Thisisduetothefactthattheenergynormofthesolutionovertheentiredomainincreasesfasterwithrespecttobthantheabsoluteerror.Usingtheabsoluteerrorintheenergynormatp=8,b=10forreference,theabsoluteerroratp=8is22.0timesgreateratb=100and38.5timesgreateratb=1000.(b)Therateofconvergence(theslopeofthedashedline)isdecreasingwithincreasingvaluesofb.(c)Theerrorestimatoroverestimatestheerrorforb=100andunderestimatesitforb=1000. P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoCome226COMPUTATIONANDVERIFICATIONOFDATAExercise7.3.1Usingthefour-elementmeshshowninFigure7.4,computeandverifythevalueofthedisplacementofpointCforb=100andb=1000fortheproblemofExam-ple7.3.1.Letν=0.3andassumeplanestrainconditions.ReportthevalueofuxG/(aσ∞)anditsrelativeerror.SeeEquation(4.37).Exercise7.3.2SolvetheproblemofExample7.3.1usingb=1000andageometricallygradedmesh.ComputeandverifythevalueofσxatpointE.Exercise7.3.3SolvetheproblemofExample7.3.1usingb=1000andasequenceofautomaticallygeneratedmeshescomprisingsix-nodetriangles.ComputeandverifythevalueofσxatpointE.Example7.3.2ConsidertheL-shapeddomainproblemofExample6.4.2.Theexactsolutionhasastrongsingularityatthereentrantcorner.Thereforeitwouldnotbemeaningfultocomputethemaximumstressontheentiredomain.However,ifweexcludetheelementsthatsharethesingularvertexthenthestresseswillconvergeontherestofthedomain.ReferringtotheresultsofExercise6.3.6andthedefinitionofstressinvariantsinAp-pendixC,SectionC.2,thefirststressinvariantcorrespondingtotheexactsolutionforplanestrainisσ+σ+σ=4(1+ν)Aλrλ1−1xyz11whereA1isarbitraryandλ1=0.544483737.Therelativeerrorinthefirststressinvariantcorrespondingtothe18-elementgeometricallygradedmeshshownintheinsetofFigure6.16,p-extensionandtrunkspace,isplottedinFigure7.7.Theresultsindicatestrongconvergence.Figure7.7TheL-shapeddomainproblemofExample6.4.2.Relativeerrorsinthefirststressinvariantoutsideoftheinnermostlayerofelements. P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoComeVERIFICATIONOFCOMPUTEDDATA227Figure7.8TheGirkmannproblem:notation.Firstpublishedin[81].ReproducedwithkindpermissionfromSpringerScience+BusinessMediac2010.Exercise7.3.4SolvetheproblemofExample7.3.2usingasequenceofautomaticallygeneratedmeshescomprisingsix-nodetriangles.Plottherelativeerrorinthefirststressinvariantforr>0.0225awhereristheradialdistancefromthereentrantcornerandaisthedimensionshownintheinsetofFigure6.16.Exercise7.3.5Asphericalshellofthicknessh=0.06m,crownradiusRc=15.00m,isconnectedtoastiffeningringatthemeridionalangleα=2π/9(40◦).Thedimensionsoftheringarea=0.60m,b=0.50m.Theradiusofthemid-surfaceofthesphericalshellisRm=Rc/sinα.ThenotationisshowninFigure7.8.Thez-axisistheaxisofrotationalsymmetry.Theshellismadeofreinforcedconcrete,assumedtobehomogeneous,isotropicandlinearlyelastic,withYoung’smodulusE=20.59GPaandPoisson’sratioν=0.Considergravityloadingonly.Theequivalent(homogenized)unitweightofthematerialcomposedoftheshellandthecladdingis32.69kN/m3.AssumethatuniformnormalpressurepisactingatthebaseABofthestiffeningring.Theresultantofpequalstheweightofthestructure.1.FindtheshearingforceQαinkN/munitsandthebendingmomentMαinNm/munitsactingatthejunctionbetweenthesphericalshellandthestiffeningring.2.Determinethelocation(meridionalangle)andthemagnitudeofthemaximumbendingmomentintheshell.3.Verifythattheresultsareaccuratetowithin5%.ThisproblemwasfirstdiscussedbyGirkmann1[30]andsubsequentlybyTimoshenkoandWoinowsky-Krieger[88].Solutionsbyclassicalmethodsarepresentedinbothreferences.In1KarlGirkmann(1890–1959). P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoCome228COMPUTATIONANDVERIFICATIONOFDATAFigure7.9NotationforExercise7.3.6.theclassicalsolutionsthestiffeningringwasassumedtobeweightless.Solutionsbytheh-andp-versionsofthefiniteelementmethodarediscussedin[81].Exercise7.3.6Anannularaluminumplatewithinnerradiusrsandouterradiusrs+b,constantthicknessda,wasjoinedbyshrinkfittingtoastainlesssteelshaft.Thecon-figurationisshowninFigure7.9.Denotethemechanicalpropertiesofaluminumasfollows:modulusofelasticity,Ea;modulusofrigidity,Ga;massdensity,a;coeffi-cientofthermalexpansion,αa;andthecorrespondingmechanicalpropertiesofstainlesssteelasEs,Gs,s,αs.Lets=80mm,rs=17.5mm,da=15mm,b=150mm,Ea=72.0×103MPa,G=28.0×103MPa,=2800kg/m3,α=23.6×10−6/K,E=aaas190×103MPa,G=75.0×103MPa,=7920kg/m3,α=17.3×10−6/K.sssConsiderthefollowingconditions:(a)Theshaftandthealuminumplatewereheatedto220◦C,theshaftwasinsertedandthentheassemblywascooledto20◦C.Assumethatat220◦Ctheclearancebetweentheshaftandtheplatewaszeroandα>α.(b)Theassemblyasisspinningaboutthez-axisatanangularvelocityω.Estimatethevalueofωatwhichthemembraneforceinthealuminumplate,Fr,isapproximatelyzeroatr=rs.Bydefinitionda/2Fr=σrdz.−da/2Specifyωinunitsofcyclespersecond(hertz).Notethatinordertohaveconsistentunits,kg/m3mustbeconvertedtoNs2/mm4.7.4FluxandstressintensityfactorsInthissection,proceduresforthecomputationofthecoefficientsoftheasymptoticexpansionsfromfiniteelementsolutionsarediscussed.ThesecoefficientsarecalledfluxorstressintensityfactorsdependingonwhetherwespeakoftheLaplaceortheNavierequation.Thealgorithmisbasedontwokeyresults:(a)theexistenceofapath-independentintegraland(b)theorthogonalityofthecharacteristicfunctions.TheprocedureisillustratedonthebasisoftheLaplaceequation.7.4.1TheLaplaceequationConsideratwo-dimensionaldomainwithboundary.ForanytwofunctionsinE()wehaveuvdxdy=(∇u·n)vds−(∇v·n)uds+vudxdy P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoComeFLUXANDSTRESSINTENSITYFACTORS229Figure7.10Definitionof.wherewehaveappliedthedivergencetheoremtwice.WhenuandvsatisfytheLaplaceequation,thatis,u=0andv=0,thenthisequationsbecomes:(∇u·n)vds=(∇v·n)uds(7.18)whichisapplicabletoandanysubdomainof.Path-independentintegralNowconsiderasubdomainintheneighborhoodofacornerpoint,shownastheshadedregioninFigure7.10.Assumethateitheru=0or∇u·n=0andeitherv=0or∇v·n=0onand.ThenEquation(7.18)becomes24(∇u·n)vds+(∇u·n)vds=(∇v·n)uds+(∇v·n)uds1313whichisequivalentto(∇u·n)vds−(∇v·n)uds=−(∇u·n)vds+(∇v·n)uds.1133Observethatintegrationalongisclockwiseaboutthecornerpointwhereasintegration1alongiscounterclockwise.Reversingthedirectionofintegrationalongsothatboth31integralsarecounterclockwiseaboutthecornerpoint,wefindthatthetwointegralsareequal,andsinceisarbitrary,wemayselectanarbitrarycounterclockwisepathandtheintegralexpressionI:=−(∇u·n)vds+(∇v·n)uds(7.19)willbepathindependent. P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoCome230COMPUTATIONANDVERIFICATIONOFDATAOrthogonalityLetu=rλiφ(θ),v=rλjφ(θ)and=whereisacircularpathofradius,centeredijonthecornerpoint.Thenonwehave∂u∇u·n==λλi−1φ(θ)ii∂rr=∂v∇v·n==λλj−1φ(θ).jj∂rr=Usingds=dθ,Equation(7.19)canbewrittenas+α/2I=(λ−λ)λi+λjφ(θ)φ(θ)dθ.jiij−α/2SinceIispathindependent,theintegralexpressionmustbezerowhenλj=±λi.Notethatsincedoesnotincludethecornerpoint,solutionscorrespondingtothenegativecharacteristicvaluesareintheenergyspace.Inthefollowingwewilldenote+α/2+α/2−−Cij:=φi(θ)φj(θ)dθandCij:=φi(θ)φj(θ)dθ(7.20)−α/2−α/2−whereφj(θ)isthecharacteristicfunctioncorrespondingto−λj.Thecharacteristicfunctions−areorthogonalinthesensethatCij=0whenφj=φiandφj=φi.Remark7.4.1InthecaseoftheLaplaceoperatorallcharacteristicvaluesarerealandsimple.Exercise7.4.1ShowthatfortheasymptoticexpressiongivenbyEquation(6.11)α/2ifi=jCij=0ifi=j.ExtractionofAkUsingtheorthogonalitypropertyofthecharacteristicfunctions,itispossibletoextractthecoefficientsfromthefiniteelementsolution.Letusconsidertheasymptoticexpansion∞u=Arλiφ(θ).(7.21)EXiii=1SupposethatweareinterestedincomputingAk.Wethendefinetheextractionfunctionwkasw:=r−λkφ−(θ)kk P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoComeFLUXANDSTRESSINTENSITYFACTORS231whichsatisfiestheLaplaceequationandtheboundaryconditionsandevaluatethepath-independentintegralonacircularpathcenteredonthecornerpoint:I(uEX,wk)=−(∇uEX·n)wkds+(∇wk·n)uEXds.Itisnowlefttothereadertoshowthat,utilizingtheorthogonalitypropertyofthecharacteristicfunctions,wehave1Ak=−−I(uEX,wk).(7.22)2CkkλkInthefiniteelementmethoduEXisreplacedbyuFEtoobtainanapproximatevalueforAk.Thismethodofcomputingthecoefficients,calledthecontourintegralmethod,isveryefficient,asillustratedbythefollowingexample.Example7.4.1WerefertotheresultsofExercise6.3.1andconstructamodelproblemonthedomainshowninFigure7.11(a)sothattheexactsolutionisalinearcombinationofthefirsttwotermsoftheasymptoticexpansion:u=arλ1(cosλθ−sinλθ)+arλ2(cosλθ+sinλθ)EX111222φ1(θ)φ2(θ)whereλ=π/(2α),λ=3π/(2α).Letα=7π/4=315◦.Theboundaryconditionsare:12∂u/∂n=0onAB;u=0onBCandonAC;thatis,thefluxcorrespondingtouEXisspecified:q=−aλrλ1−1(cosλθ−sinλθ)−aλrλ2−1(cosλθ+sinλθ).n11112222Figure7.11DomainforExample7.4.1. P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoCome232COMPUTATIONANDVERIFICATIONOFDATAWenormalizethecharacteristicfunctionsasfollows.Letθibetheanglewheretheabsolutevalueofφi(θ)ismaximum:|φi(θi)|=max|φi(θ)|whereIα:={θ|−α/2≤θ≤+α/2}.θ∈IαThenormalizedcharacteristicfunctionsaredefinedby:ϕi(θ):=φi(θ)/φi(θi).(7.23)Thereforethemaximumvalueofϕi(θ)onIαisunity.Thefunctionsϕ1(θ)andϕ2(θ)areshowninFigure7.11(b).Ifweleta1=1.0,a2=0then,usinga16-elementmeshwithonelayerofgeometricallygradedelementsaroundthecornerpoint,trunkspace,theestimatedrelativeerrorintheenergynormis17.21%atp=8.Usingthemethodofextractiondescribedinthissection,thecomputedvalueofthecoefficientofthefirstnormalizedcharacteristicfunctionisA1=1.348.Itsexactvalueis1.414,thatis,therelativeerroris4.66%.Exercise7.4.2FortheproblemdescribedinExample7.4.1,leta1=0anda2=1.0.Deter-minetheapproximatevalueofthecoefficientofthesecondnormalizedcharacteristicfunctionA2andestimatetherelativeerror.Theexactvalueis−1.414.7.4.2PlanarelasticityTheanalysisofcornerpointsinplanarelasticityisanalogoustothatoftheLaplaceequation,butmuchmorecomplicated.Complicationsarisefromthefactthatinplanarelasticitythecharacteristicvaluesmaybecomplex,andnotalloftherealrootsaresimple.Furthermore,therearecorneranglesthatrequirespecialtreatment[58].Fulltreatmentofthissubjectisbeyondthescopeofthistext.Onlyoneimportantspecialcase,crack-tipsingularities,willbediscussed.Forcracks(α=2π)Equations(6.30)and(6.31)reducetooneequation:nsin2λπ=0,thereforeλn=±,n=1,2,3,...2andhenceallrootsarerealandsimple.Thegoalistocomputethecoefficientsofthefirsttermsofthesymmetric(ModeI)andantisymmetric(ModeII)expansions,seeEquations(7.24)through(7.26).InlinearelasticfracturemechanicsitiscustomarywritetheCartesiancomponentsofthecorrespondingModeIstresstensorinthefollowingform:KIθθ3θ3/2σx=√cos1−sinsin+T+O(r)(7.24)2πr222KIθθ3θ3/2σy=√cos1+sinsin+O(r)(7.25)2πr222KIθθ3θ3/2τxy=√sincoscos+O(r)(7.26)2πr222 P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoComeFLUXANDSTRESSINTENSITYFACTORS233where−π≤θ≤π,Tisaconstant,calledtheT-stress,seeExercise6.3.5.TheconstantKIiscalledtheModeIstressintensityfactor.Theantisymmetric(ModeII)stresstensorcomponentsareusuallywritteninthefollowingform:KIIθθ3θ3/2σx=−√sin2+coscos+O(r)(7.27)2πr222KIIθθ3θ3/2σy=√sincoscos+O(r)(7.28)2πr222KIIθθ3θ3/2τxy=√cos1−sinsin+O(r)(7.29)2πr222whereKIIiscalledtheModeIIstressintensityfactor.ComputationofstressintensityfactorsInthissectionthecomputationofstressintensityfactorsfromfiniteelementsolutionsbythecontourintegralmethod(CIM)isoutlined.TheCIMfortheNavierequations,analogouslytotheCIMfortheLaplaceequationoutlinedinSection7.4.1,isbasedontheexistenceofapath-independentintegralandtheorthogonalityofeigenfunctions.DetailedderivationofthecontourintegralmethodfortheNavierequationsisavailablein[79],[80].WewillconsidertheNavierequationintwodimensionsundertheassumptionthatthevolumeforcesarezeroandthetemperatureandthicknessareconstants.Thepath-independentcontourintegral,analogoustoEquation(7.19),isI:=(T(u)w+T(u)w)ds−(T(w)u+T(w)u)ds(7.30)xxyyxxyywherethesuperscriptsu(resp.w)representstheexactsolution(resp.atestfunctionthatsatisfiestheNavierequationandtheboundaryconditionsontheedgesthatintersectatthesingularpoint),andisanarbitrarycontourthatbeginsatoneedgeandrunsinacounterclockwisedirectiontotheother,asshowninFigure7.10.Wewillassumethatisacircularcontour,theradiusofwhichisarbitrary,andwewillbeinterestedinthecomputationofKI,KIIandT.TheprocedureisoutlinedinAppendixD.Underspecialconditionsthestressintensityfactorscanbedeterminedfromtheenergyreleaserate.TheprocedureforcomputingstressintensityfactorsfromtheenergyreleaserateisdescribedinAppendixD.Example7.4.2LetusconsidertheproblemdescribedinExercise4.3.9andassumethatacrack1.0mmlongdevelopedatthelocationofthemaximumtensilestress.Thecrackisorientednormaltotheboundary.ThegoalofthecomputationistodeterminethestressintensityfactorsKI,KIIandtheT-stress.Thelocationofthemaximumtensilestress,foundfromthesolutionofExercise4.3.9,isindicatedinFigure7.12(b)whereα=12.8◦.Acrack1mmlongisintroducedandamesh,suchasthemeshshownFigure7.12(a),isgenerated.Themeshdependsonuser-specifiedsettingsforthemeshgenerator.UponsolvingthisproblemandextractingKI,KIIandTby P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoCome234COMPUTATIONANDVERIFICATIONOFDATAFigure7.12Thefiniteelementmeshforthenotchedplatewithacrack.Numberofelements:627.thecontourintegralmethod,theresultsshowninTable7.3areobtained.Itcanbeseenthatthecomputeddatadonotchangesignificantlyaspisincreasedfrom2to8.Exercise7.4.3RefertoAppendixD.2.VerifytheaccuracyofK2+K2inTable7.3byIIIcomputingtheenergyreleaserateG.Hint:UsethecentraldifferenceformulatoestimateG:(a+a)−(a−a)G≈−·2aLeta=0.01aandcheckthesensitivityofGtothechoiceofa.Whathappensifaistoosmallortoolarge?Exercise7.4.4EstimatethecriticalcracklengthfortheprobleminExample7.4.2.AssumethatthelocationandorientationofthecrackisthesameasinExample7.4.2andthefractureTable7.3Example7.4.2:computedvalues√ofstressintensityfactors(MPamm).pNKIKIIT22669260.27.35−40.7835885259.77.32−39.47410355259.87.37−39.25516079259.97.35−39.15623057260.07.34−39.10731289260.17.33−39.07840775260.17.32−39.04 P1:OSOJWST055-07JWST055-SzaboFebruary16,20118:2PrinterName:YettoComeCHAPTERSUMMARY235√toughnessofthematerialis900MPamm.Inthisexercisefracturetoughnessistreatedasafixednumber.Inrealityfracturetoughnessisarandomvariablethathasalargedispersion.Thereforethecriticalcracklengthisafunctionofarandomvariable.7.5ChaptersummaryAttheendofthesolutionprocessthecoefficientsoftheshapefunction,themappingandthematerialpropertiesareavailableattheelementlevel.Thedataofinterestarecomputedfromthisinformationeitherbydirectorindirectmethods.Inordertomeettherequirementsofverificationitisnecessarytoshowthattheerrorsinthedataofinterestdonotexceedstatedtolerances.Inpracticalproblemstheexactsolutionistypicallyunknownanditisnotpossibletodeterminetheerrorsofapproximationwithprecision.Itispossible,however,toshowthatthenecessaryconditionsaresatisfiedfortheerrorsinthedataofinteresttobesmall.Errorestimationisbasedonaprioriknowledgethatthedataofinterestcorrespondingtotheexactsolutionarefiniteandindependentofthediscretizationparameter.Thereforeanecessaryconditionfortheerrortobesmallisthatthecomputeddatashouldexhibitconvergencetoalimitvalueasthenumberofdegreesoffreedomisincreased.Aneffi-cientandrobustwaytoachievethisistouseproperlydesignedmeshesandincreasethepolynomialdegree. P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoCome8Whatshouldbecomputedandwhy?Uptothispointwehavebeenconcernedwiththeformulationofmathematicalmodels,theirnumericalsolutionbythefiniteelementmethod,computationofdatafromthenumericalsolutionandtheirverification.Inthischapterweconsiderthequestionofwhatshouldbecomputedandwhy.Thisquestionhastobeaddressedintheprocessofconceptualization.Becausethesubjectisverylargeanddiverse,wewillconsiderthequestionwithreferencetoaspecificproblemclassonly:mathematicalmodelsformulatedforthepurposeofestimatingtheservicelifeofstructuralandmechanicalcomponentssubjectedtocyclicloadingofvariableamplitude.Wewillbeconcernedwithcomponentsmadeofmetalsonly.Theabilitytomakereliablepredictionsoftheservicelifeofsafety-criticalmechanicalsystemsisobviouslyofgreatimportanceintheformulationandjustificationofdecisionsconcerningdesign,certificationandmaintenance.Aninterestingaspectisthatcyclicloadingcausesdamageaccumulationinthematerialthroughdislocationsfollowedbytheformation,coalescenceandpropagationofcracks.Math-ematicalmodelsbasedoncontinuummechanics,whereitisassumedthatthedisplacementfieldisacontinuousfunction,donotaccountforthesephenomena.Nevertheless,modelsofdamageaccumulationaretypicallybasedonthelineartheoryofelasticityorsmall-strainplasticity.Suchmodels,coupledwithexperimentation,havebeendemonstratedtobeuse-fulforpredictingtheformationandpropagationofcracks.Themainpointsareoutlinedinthischapter.Anotherinterestingandhighlyrelevantaspectofthisproblemclassisthatvastamountsofexperimentaldatahavebeencollectedovermanyyears,yetproperinterpretationofthedataremainsanopenquestion.Experimentalmeasurementsofmetalfatiguetypicallyhavesubstantialstatisticaldispersion.Thismakestheseparationofuncertaintiesassociatedwithmodelingassumptions(epistemicuncertainties)andstatistical(aleatory)uncertaintiesachallengingproblem.TheconceptualframeworkofverificationandvalidationoutlinedinIntroductiontoFiniteElementAnalysis:Formulation,VerificationandValidation,FirstEdition.BarnaSzabóandIvoBabuška.©2011JohnWiley&Sons,Ltd.Published2011byJohnWiley&Sons,Ltd.ISBN:978-0-470-97728-6 P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoCome238WHATSHOULDBECOMPUTEDANDWHY?Chapter1andthemethodsavailableforestimatingandcontrollingtheerrorsofdiscretizationoutlinedinChapters2through7providemeansforthere-evaluationofexistingdataandimprovementofmathematicalmodelsformulatedforthepredictionofdamageaccumulationcausedbycyclicloading.8.1BasicassumptionsMathematicalmodelsconstructedforthepredictionofdamageaccumulationcausedbycyclicloadingarebasedonthefollowingassumptions:1.Thereexistoneormorefunctionals,computablefromthesolutionofmathematicalmodelsbasedoninfinitesimalstrain,small-deformationtheory,thatcanbecorrelatedwithcrackinitiationeventsandcrackpropagationrateswithsufficientaccuracytosuitthepurposesofengineeringdecision-making.2.Thereexistoneormoreproceduressuitableforthegeneralizationoftheresultsoffatigueexperimentsperformedunderaparticularcyclicloading,characterizedbyameanvalueandconstantamplitude,tocyclicloadingcharacterizedbyarbitrarymeanvalueandconstantamplitude.3.Thereexistoneormoreproceduressuitableforcorrelatingdamageaccumulationwithvariableamplitudecyclicloading.Eachoftheseassumptionsimpliestheexistenceofoneormoremathematicalmodels.InthefollowingwewillbeconcernedprimarilywiththeformulationofmodelspertainingtoAssumption1.Thesemodelsdifferbythedefinitionoffunctionalsthatarecorrelatedwithdamageaccumulationevents.Thefunctionalsarecalleddrivingforcesofdamageaccumu-lationordriversofdamageaccumulation(DDA).Assumptions2and3willbediscussedonlybriefly.8.2Conceptualization:driversofdamageaccumulationConsiderthemodelproblemofasolidbodyofconstantthicknesswithanotchofradius≥0.AschematicillustrationisshowninFigure8.1.Thesolutiondomainisdenotedby.Weassumethatoutsideofthecloseneighborhoodofthenotch,denotedby,theassumptionsofsmall-strain,small-deformationmodelsofcontinuummechanicsarejustified.Inhigh-cyclefatigue,thatis,whenthenumberofcyclestofailureisgreaterthan104,theassumptionsofthelineartheoryofelasticityaregenerallyconsideredtobeapplicableon−.Inlowcyclefatigue,inelasticdeformationmayoccuron−.Thedomain,boundedbyand,isdividedintotwoparts.Theimmediateneigh-SSborhoodofthenotch,boundedbyandPZandindicatedasthehatchedregioninFigure8.1,iscalledtheprocesszoneanddenotedbyPZ.Cracksnucleateandpropagateintheprocesszoneandthereforetheassumptionsofcontinuummechanicsarenotapplicablethere.Inthedomain−,boundedby,and,theassumptionsofcontinuummechanicsPZPZSSareapplicable,butthesmall-strainassumptionsarenot.Inhigh-cyclefatiguethesizeoftheprocesszoneistypicallysmall,oftheorderofthesizeofarepresentativevolumeelement(RVE)whichisthesizeofthesmallestbodythatcanbemodeledbythemethodsofcontinuummechanics. P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoComeCONCEPTUALIZATION:DRIVERSOFDAMAGEACCUMULATION239Figure8.1Notchedplate:notation.Letusassumethatamathematicalmodelwasformulatedthataccountsforthephysicalprocessesleadingtotheformationofdiscontinuitiesandcracksintheprocesszoneandthesolutionofthismodelisavailableovertheentiredomain.WedenotethissolutionbyuPZ(x,t)={ux(x,t)uy(x,t)}PZwherexisthepositionvectorandtistime.Wedenotethesolutionofthecontinuummechanicsproblembasedonsmall-straintheorybyuSS(x,t)={ux(x,t)uy(x,t)}SSwherethesubscriptisareminderthatthissolutionisbasedonsmall-straintheory.ThejustificationforusinguSSforthepredictionofdamageaccumulationrestsontheassumptionthatthereissomedomain,theboundaryofwhichisrepresentedbyinSSFigure8.1,outsideofwhichthedifferencebetweenuPZanduSSisnegligible:u−u≈0on−.(8.1)PZSSmaxInotherwords,itwouldbepossibletocomputeuonfromubyprescribinguPZSSSSonSS.Inprincipleitispossibletoformulateamathematicalproblemthatwouldmodeltheformationofvoidsandcracks.InpracticethisisnotfeasiblebecausesuchamodelwouldbeverycomplicatedandrequireinformationaboutthephysicalpropertiesofthematerialonlengthscalessmallerthantheRVE.Thesepropertiesaredifficulttoobtainandtheirstatisticaldispersionislarge.ThereforesubstantialuncertaintieswouldbeintroducedintomathematicalmodelsofphysicalphenomenaonlengthscalessmallerthantheRVE.Inordertoavoidthisproblem,weassumethatdamageaccumulationisrelatedtoandcanbepredictedfromfunctionalscomputablefromuSS.Inotherwords,failureinitiationandcrackpropagationeventswhicharerelatedtothesolutionofthehighlynonlinearprobleminsideofPZcanbedeterminedfromuSSeventhoughtheassumptionsofthesmall-straincontinuumtheoryarenotapplicableinsideofSS.ThisinvolvesthecorrelationoffunctionalscomputedfromuSSwithfailureinitiationeventsobservedinphysicalexperiments.Thisistheconceptualbasisofclassicalmodelsoffatigueaswellasmodelsofcrackpropagationdiscussedinthefollowingsections. P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoCome240WHATSHOULDBECOMPUTEDANDWHY?Figure8.2TypicalS–Ncurvesfor7050-T7451aluminumplates,basedonReference[43].8.3ClassicalmodelsofmetalfatigueThefirstinvestigationsoffatigueinmetalsweremotivatedbyoccurrencesofunexpectedfailureinrailcaraxlesinthemid-1800s.Railcaraxlesareroundbeamssubjectedtopurebendingbetweenthebearings.Astheaxlerotates,thestressescausedbythebendingmomentvarysinusoidally,eachrotationcorrespondingtoonecycle.Onexaminingthefailuresurfacesitwasfoundthatcracksoriginatedatsomesmallimperfectionatthesurfaceandthenpropagatedatanacceleratingrateuntiltheaxlebroke.Anumberofexperimentswereconductedinwhichsimilarpatternsoffailurewereobserved.Themaximumnormalstress,computedbythebendingformula,wasplottedagainstthenumberofcyclesatwhichthespecimensfailed.Thereissubstantialscatterinthedatapointsinfatigueexperiments.CurvesfittedtothedatapointsarecalledWohlercurves¨1ormorecommonlyS–NcurveswhereSrepresentsthemaximumcyclicstressσmaxandNrepresentsthenumberofcyclestofailure.TypicalS–Ncurvesfor7050-T7451aluminum,basedon[43],areshowninFigure8.2whereR,calledthestressratioorcycleratio,istheratiooftheminimumreferencestressorloadtothemaximumreferencestressorload.WhenR=−1theminimumreferencestressiscompressiveandhasthesameabsolutevalueasthemaximumreferencestress.Bydefinition,thefatiguestrengthisthecurvecorrespondingtoR=−1.Thespecimenswere7.62mm(0.3inch)diameterroundbars.Thesurfaceconditionofthebarsisnotspecified.1AugustWohler(1819–1914).¨ P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoComeCLASSICALMODELSOFMETALFATIGUE241Axialloadingwasappliedat13.3Hz.Itcanbeseenthatthevarianceofloglife(log10N)tendstoincreasewithdecreasingstresslevel.Theempiricalformulagivenin[43],convertedtoSIunits,islogN=9.73−3.24log(0.1450σ(1−R)0.63−106.9)(8.2)1010maxwhereσ>106.9(1−R)−0.63isinunitsofMPa.Thisformula,constructedbynonlinearmaxregression,canbeunderstoodasanapproximationofthemeanoftherandomvariablelog10N(σmax,R)whichwillbedenotedbyμ.Thestandarddeviationinloglifeisdefinedbym(i)2i=1(log10Ni(σmax,Ri)−μi)s=(8.3)m−1wheremisthenumberofdatapoints.BasedonthedatashowninFigure8.2,s=0.471.IfitisassumedthatlogN(σ(i),R)hasnormalprobabilitydensitysoN≡exp(logN)has10imaxii10ilog-normalprobabilitydensity.Whenthemaximumstressinpolishedspecimensofsteelandtitaniumsubjectedtosinusoidalloading(R=−1)isbelowacertainvalue,thenithasbeentraditionallyassumedthatthespecimenswillnotfailunderanynumberofcycles.Thisvalueiscalledtheendurancelimitorfatiguelimit.Somematerials,suchasaluminum,donothaveafatiguelimit.Forthosematerialsthefatiguestrengthat107or5×107cyclesistypicallyreportedasthefatiguelimit.Aroughestimateofthefatiguelimitisone-halfoftheultimatetensilestrength.Theassumptionthatafatiguelimitexistswasmadeonthebasisofpracticalconsiderations.Ittakes28hourstoreach107cycleswithaconventionaltestingmachineoperatingat100Hz.Thereforemuchtestingcouldnotbeperformedbeyond107cycles.Itisnowpossibletoreach1010cyclesinlessthenaweekbymeansofpiezoelectricfatiguetestingmachinesoperatingat20kHz.Basedonfatiguedatacollectedinthegigacyclerange,theexistenceofafatiguelimitisinquestion[17].Theimplicationisthatirreversiblechangesoccurinthematerialevenatlowstresslevels,verylikelyintheformofdislocations.Thesechangesgraduallyaccumulateandeventuallyresultintheformationofcracks.Intheclassicaltreatmentoffatiguethemaximumnormalstressormaximumshearingstresswasassumedtobethedrivingforcefordamageaccumulation.Thisshouldbeunderstoodinthecontextofconceptualization:theinvestigatorshadinmindtheobjectiveofpredictingtheservicelifeofmechanicalcomponentsinwhicheitherthenormalorshearingstresswasdominantand,furthermore,thestressdistributioninthevicinityofthestressmaximacouldberepresentedbysmoothfunctions.Theyhadattheirdisposalsimpleformulasthatcorrelateknownforcesandmomentswithnormalandshearingstresses.Thereforethegoalofcomputationwastodeterminethemaximumstressatnotches,fillets,holes,etc.,whichwasthencorrelatedwithcalibrationdataintheformofS–Ncurvestopredictthenumberofcyclesatwhichfatiguefailurewouldbeexpectedtooccur.Example8.3.1WeconsiderashaftofdiameterDwithaU-notch.TheshaftissubjectedtobendingmomentM.ThenotationisshowninFigure8.3. P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoCome242WHATSHOULDBECOMPUTEDANDWHY?Figure8.3ShaftofdiameterDwithU-notch:notation.ThemaximumstressisthenominalstressmultipliedbythegeometricstressconcentrationfactorKtσmax=Ktσnom(8.4)whereinthisexampleσnomiscomputedbytheclassicalbendingformula32Mσnom=.(8.5)π(D−2h)3Ktdependsontheratiosh/randh/D.FormulasforKtcanbefound,forexample,in[95],Table17.1,entry15.Theseformulasarebasedondatacollectedfromtheliteraturefittedwithempiricalcurves.Itisstatedthat“overthemajorityoftherangesspecifiedbythevariables,thecurvesfitthedatapointswithinmuchlessthan5%.”However,theaccuracyofthedataintheliteraturehadnotbeenrecorded.Forexample,theformulaforKtin[95],Table17.1,entry15,forthecaseh=ris232h2h2hKt=3.04−7.236+9.375−4.179.(8.6)DDDExercise8.3.1FortheproblemdescribedinExample8.3.1letD=30mmandh=r=5mm.DetermineKtbyfiniteelementanalysisandverifythatitisaccuratetowithin5%.CompareyourresultwithKtcomputedfromEquation(8.6).Definition8.3.1ThegeometricstressconcentrationfactordenotedbyKtistheratioofthemaximumstressinthevicinityofanotch,hole,fillet,screwthreadorotherfeaturethatcauseslocallyincreasedstresstothenominalstress.Asthenameimplies,Ktcanbedeterminedfromthegeometricaldescriptionofthepartandthetypeofloading.Itdoesnotdependonmaterialproperties.Definition8.3.2Nominalstress,definedformachineelementssubjectedtotension,bendingandtorsion,isunderstoodinmachinedesigntobethemaximumnormalorshearingstressatanotch,fillet,holeorotherstressrisercomputedbyformulasbasedontheassumptionthatthestraindistributionoverthecrosssectionisalinearfunction.ThiswasillustratedinExample8.3.1.Becausethisdefinitioncannotbegeneralizedtoarbitrarydomains,wewillunderstandnominalstresstomeanthemaximumstressthatwouldexistatthelocationofanotchifthenotchwerenotpresent,unlessotherwisestated. P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoComeCLASSICALMODELSOFMETALFATIGUE2438.3.1ModelsofdamageaccumulationThecalibrationdatapresentedintheformofS–NcurvesorequationssuchasEquation(8.2)arecollectedunderlaboratoryconditionsusingconstantamplitudeloadingatfixedRra-tios.However,mechanicalandstructuralcomponentsinservicearesubjectedtovariableamplitudeloadingrepresentedbycomplexload–timefunctions,calledloadspectra.Forex-ample,theloadspectraofrotorcraftcomponentsarecharacterizedbyblocksofhighR-ratio,low-amplitudecyclesinterspersedwithlow-valueminimathatcorrespondtothestart–stopcycle[41].Severalmodelsdevisedtoaccountfortheaccumulationoffatiguedamagehavebeenproposedforthepredictionoffatiguelifeofmechanicalandstructuralcomponentssubjectedtovariableamplitudeloading.Forexample,Osgood[54]summarized15suchmodelsdividedintothreecategories:modelsoflinearcumulativedamage,modelsofnonlinearcumulativedamageandothermodels.Althoughoftencriticizedforitsshortcomings,themodelmostcommonlyusedisknownasMiner’srule,whichfallsintothecategoryoflinearcumulativedamagemodels.Miner’sruleisbasedontheassumptionthatifncyclesofconstantamplitudeloadcharacterizedby(σmax,R)areimposedandtheS–NdataindicatethatfatiguefailurewouldoccuratNcycles,thenthefractionoffatiguelifeexpendedwouldben/N≤1.Furthermore,whenmultipleloadcyclesniwithcorrespondinglimitvaluesNiareimposedandthereareKsuchcycles,thenthefractionoffatiguelifeexpended(F)isKniF=≤1.(8.7)Nii=1SinceN=N(σ(i),R)isarandomvariable,Fisafunctionofrandomvariables.Typicaliimaxigoalsofcomputationare:(a)toestimatetheprobabilityProb(F1.Sincethesequenceofloadingisappliedtoonespec-imenorpart,therandomvariablesarefullycorrelatedandthereforetheforegoingprocedureforcomputingthecumulativedistributionfunctionforFisthesameasforM=1withtheexceptionthatEquation(8.10)isreplacedbyKMnKn(M)iiFj==M=MFj.(8.11)NiNii=1i=1ThereforetheabscissasinFigure8.4andFigure8.5shouldbeunderstoodasfatiguelifeexpendedperblockofloading.TheprobabilitythatfailurewillnotoccurwhenMblocksofloadingareappliedcanbereadfromFigure8.5asfollows:lettingF=1/M,theordinateofthecumulativedistributionfunctionatthispointisthedesiredprobability,thatis,Prob(F(M)<1).Figure8.5Example8.3.2:empiricalcumulativedistributionfunctiongeneratedbytheMonteCarlomethodin1000iterationsforM=1. P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoCome246WHATSHOULDBECOMPUTEDANDWHY?Forexample,whenM=3theordinateatF=1/3readfromFigure8.5is0.81,thereforetheprobabilitythatfailurewillnotoccurisestimatedtobe81%.Remark8.3.1TheonlysourceofuncertaintyconsideredinExample8.3.2wasthealeatoryuncertaintyoffailureeventsoftestcouponsundercyclicloading.Wedidnotaddressepis-temicuncertaintyassociatedwiththeinterpretationoftheoutcomeoffatigueexperimentsontestcoupons.Ratherweacceptedtheinterpretationpresentedin[43]andweassumedthattheprobabilitydensityfunctionoflog10N(σmax,R)isnormal,themeanisrepresentedbyEquation(8.2)andthestandarddeviationisindependentofthecycleratio.Manyotherinterpretationsarepossible.Theseinterpretationsarebasedonindividualbeliefandjudgmentandarethereforesubjective.AnotherimportantassumptionwasthatMiner’sruleisasufficientlyaccuratepredictorofdamageaccumulation.Uncertaintiesassociatedwithmaterialproperties,boundaryconditionsanderrorsinnumericalapproximationwereneglected,implyingtheassumptionthatthedomi-nantuncertaintyisthealeatoryuncertaintyinthefatiguedata.Ofcourse,theeffectsofthevari-ousassumptionsontheprobabilityoffailurecanbeexaminedthroughvirtualexperimentation.Exercise8.3.2SolvetheproblemofExample8.3.2usingJ=102andJ=104forMCMConeblockofloading.Comparethepredictionsofprobabilityoffailure.Exercise8.3.3InExample8.3.2thestandarddeviations=0.471wasassumedtobeindependentofR.SupposethatuponexaminingthedatashowninFigure8.2anexpertrecommendsusings=s(R)=0.5106+0.1814R.SolvetheproblemofExample8.3.2foroneblockusingtherecommendeds(R)valuesandcomparethepredictionsofprobabilityoffailure.Exercise8.3.4InExample8.3.2thestressesshowninTable8.1wereassumedtobeexact.Supposethatthestresseswereunderestimatedby10%.Predicttheprobabilityoffailureforoneblockwhentheunderestimatedvaluesofσ(i)areused.max8.3.2NotchsensitivityItwasobservedthatwhennotches,filletsandholesofsmallradiusarepresentthentheS–Ncurvesarenotsatisfactorypredictorsofcrackinitiationevents.ReferringtoFigure1.2,thisshouldbeunderstoodtomeanthatmodelsbasedontheassumptionthatthemaximumstressisthedrivingforcefordamageaccumulationcouldnotadequatelypredictthefatiguelifeofmechanicalcomponentswithnotchesofsmallradiusandthereforehadtoberejected.Severalalternativemodelshavebeenproposedofwhichwementiontwointhefollowing:1.Neuber2[49]proposedthatthedrivingforceshouldbethemaximumnormalorshearingstressaveragedoveracharacteristicmaterial-dependentdistance.HeintroducedthenotionofeffectivestressconcentrationfactorKe.BydefinitionKe=q(Kt−1)+1(8.12)2HeinzNeuber(1906–1989). P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoComeCLASSICALMODELSOFMETALFATIGUE247whereqiscalledthenotchsensitivityindex,definedbyNeuberasfollows:1πq:=√,f:=,≥0>0(8.13)1+f/π−ωwhereisthenotchradius,isanexperimentallydeterminedmaterialconstant(inlengthunits)and0≤ω<πiscalledtheflankangle.Thelowerlimitof,denotedby0,imposesarestrictionontherangeofvalidityofEquation(8.12).InthenotationusedinFigure8.1,ω=2π−α.Theparameterhasbeencorrelatedwiththeultimatetensilestress(UTS).Forexample,KuhnandHardrath[40]foundthatforsteelswithUTSrangingbetween345and1725MPatheestimatedrangeofis430to0.9μmrespectively.Theyreportedthatscatterinincreaseswithdecreasing.2.Peterson3[60]retainedtheformalismofEquation(8.12)butproposedanalternativedefinitionforthenotchsensitivityindexwhichwedenotebyq¯:1q¯:=,≥0>0(8.14)1+α/whereαisanexperimentallydeterminedmaterialconstantand0isalowerboundonthenotchradius.PetersongaveapproximatevaluesforαforsteelsasafunctionoftheirUTS:forUTSrangingbetween345and1725MPatheestimatedrangeofαis380to33μmrespectively.Theexperimentallydeterminedvaluesofq¯foraluminumandsteelpublishedin[60]indicatethat0isgreaterthanapproximatelyα/4.ThedifferencesbetweenKevaluesbasedonthenotchsensitivityindicesproposedbyNeuberandPetersonarenotlarge.Forlow-strengthsteels(UTSabout400MPa)KecomputedusingNeuber’sdefinitionisabout9%lowerthanKecomputedfromPeterson’sdefinition.Forhigh-strengthsteelsthedifferencesarenegligiblysmall[69].NeuberandPetersonintroducedmaterial-dependentparameters.Itisofcoursepossibletoimprovethefittingofcalibrationdatawiththeuseofadditionalparameters.However,calibrationdoesnotdistinguishbetweenaleatoricandepistemicuncertainties.Therefore,makingsuccessfulpredictionswithinorclosetotherangeofparametersforwhichthemodelwascalibratedisnotvalidation.Remark8.3.2TheASTMgrainsizenumber4ofconventionallyheat-treatedAF1410steelisapproximately10.8[90].Thiscorrespondstoanaveragegrainsizeofapproximately8.5μm.Theestimateofgivenin[40]issmallerthanthisgrainsize.Remark8.3.3NeuberandPetersonwereconcernedwiththefatiguestrengthofnotchedmachineelements.Theyassumedthatthereisaclearlydefinednotchofradius>0andthemaximumstressdependson.Therearemanypracticalproblemswherethemaximumstressisnotrelatedtoanotchradius,however.Consider,forexample,theT-junctionoftwopipesofidenticalcross-sectionsasshowninFigure8.6.Theintersectionoftheexternalsurfacesisfilleted.Thefilletisidealizedasa“rollingballfillet,”thatis,thefilletsurfacewouldbe3RudolphEarlPeterson(1901–1982).4ASTMStandardE11296(2004). P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoCome248WHATSHOULDBECOMPUTEDANDWHY?Figure8.6T-junctionoftwopipesofidenticaldimensions.incontactwithaballofradiusrf>0rolledalongtheintersectioncurvewhilemaintainingcontactwiththeexternalsurfaces.Theintersectionoftheinternalsurfacesischamfered.Thechamferradiusisrc≥0.SupposethattheT-junctionissubjectedtointernalpressurep.Themaximumstresswilloccuralongtheintersectionoftheinternalsurfacesanddependonlyweaklyonrc.Remark8.3.4Averagingstressesoveramaterial-dependentdistancefornotchedbars,shaftsandbeamscanbeunderstoodalsoasaveragingoveranareaorvolume.ThereforeNeuber’sconceptualizationadmitsalternativeinterpretationsongeneraldomains.Exercise8.3.5EstimatethemaximumprincipalstressintheT-junctionshowninFigure8.6andverifythattheerrorisnotgreaterthan5%.AssumethattheoutsidediameterofbothpipesisDo=105.0mm,theinsidediameterisDi=95.0mmandthecenterlinesintersectatrightangles.Theradiusoftherollingballfilletis8.0mmandPoisson’sratiois0.3.Performthecomputationsfortwochamferradii:rc=0andrc=5.0mm.TheT-junctionisloadedbyinternalpressurep=1.0MPaandcorrespondingnormaltractionsT=pD2/(D2−D2)0n0ioiactingonthecross-sections.Selectlengthdimensionsforthepipessuchthattheeffectsofthosedimensionsonthedataofinterestarenegligible.8.3.3ThetheoryofcriticaldistancesThetheoryofcriticaldistances(TCD)isbasedontheassumptionthatthereisamaterialpropertycalledcriticaldistance.AccordingtoTaylor[86],TCDcoversagroupofmethods,basedontheassumptionthatbrittlefailureandfatiguefailurearepredictablefromtwomaterialparameters:acriticaldistancedandacriticalstressσ0.Inparticular,theline,areaandvolumemethodsdescribedin[86]differbywhetherthestressisaveragedoveraline,areaorvolume,respectively.Fromtheperspectiveofvalidation,eachofthesemethodscanbeunderstoodasamodelofdrivingforcefordamageaccumulation.Sincethesedefinitionsofdrivingforcedonotdependonafilletorholecharacterizedbyaradius,theyareapplicabletoproblemssuchastheT-jointproblemshowninFigure8.6andwherethefilletradiusiszero,forexample,cracks.Example8.3.3Letusconsideraconicalfeaturemachinedintoapolishedsteelplate.Thecross-sectionisshowninFigure8.7(a).Thethicknessoftheplateis3.020mm,andthe P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoComeCLASSICALMODELSOFMETALFATIGUE249Figure8.7(a)Cross-sectionofaconicalfeature:notation.(b)Contoursofthefirstprincipalstress,quartersymmetry.dimensionsoftheconicalfeatureareR1=1.016mm,R2=0.330mm,rf=0.391,H=0.889mm.Poisson’sratiois0.280.Theplateissubjectedtotensioninthexdirection.ThecontoursofthefirstprincipalstressareshowninFigure8.7(b).Itcanbeseenthatthemaximumstressisnotsignificantlyinfluencedbythefillet,thereforeNeuber’sorPeterson’smodelisnotapplicableinthiscase.OntheotherhandTCDisapplicable.Exercise8.3.6Considertheproblemofacircularholeinaninfiniteplatesubjectedtounidirectionaltension.ThenotationisshowninFigure4.11andtheclassicalsolutionforσxisgivenbyEquation(4.34).ThegeometricstressconcentrationfactorisKt=σmax/σ∞=3whereσmax=σx(a,±π/2).(a)Showthat1a+d1σ(d):=σ(r,π/2)dr=σ1+2+O(d/a)≈Kσxx∞e∞da1+d/awhereKe=q¯(Kt−1)+1istheeffectivestressconcentrationfactorandq¯isPeter-son’snotchsensitivityindexgivenbyEquation(8.14)withα=dand=a.(b)Computetherelativeerrorσ(d)−Kσxe∞er=100(d)σxford/a=0.1.TheresultsofthisexerciseshowthattheeffectivestressconcentrationfactordefinedbyPetersoncanbeviewedasanapproximationtothelineorareamethodofTCD. P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoCome250WHATSHOULDBECOMPUTEDANDWHY?8.4LinearelasticfracturemechanicsLinearelasticfracturemechanics(LEFM)isbasedontheassumptionthatthedrivingforceforcrackpropagationundercyclicloadingistheamplitudeofthestressintensityfactor.Paris’law5[59]establishesarelationshipbetweencrackgrowthrateandtheamplitudeofthestressintensityfactoratcycleratioR=0:da=C(K)m(8.15)dNwhereaisthecracklength,Nisthenumberofcycles,CandmarematerialconstantsandKistheamplitudeofthestressintensityfactor.TheconstantCisinunitsof(MPa)−mm1−m/2orequivalent,theconstantmisdimensionless.ThisisaphenomenologicalrepresentationofexperimentalobservationsschematicallyillustratedinFigure8.8.Equation(8.15)isapplicableintherangeofKwherethegrowthratecurveplottedonalog–logscalecanbewellapproximatedbyastraightline.ThisrangeiscalledRegionIIortheParisKregion.RegionIiscalledthethresholdregionandRegionIIIiscalledthestabletearingcrackgrowthregion[69].KisusuallyunderstoodtomeanKI.ThethresholdvalueofKisindicatedby(K)th.At(K)ththecracklengthincrementpercycleisoftheorderofmagnitudeoftheinteratomicspacingofthematerial.Whetherthereexistsathresholdvalue(K)thbelowwhichcrackgrowthwillnotoccurisnotasettledquestion.Theavailableexperimentaldatahavealargescatterinthisregion.Figure8.8CrackgrowthrateasafunctionofKforR=0.5PaulC.Paris(1930–). P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoComeLINEARELASTICFRACTUREMECHANICS251ThereareseveralmodelsformulatedtoaccountfortheeffectsofthecycleratioR.Forexample,theWalkercorrection(fromtheAFGROWManual)ismK(1−R)n−1forR≥0da=Cm(8.16)dN|K|(1−R)n−1forR<0maxwhere0(KI)min.LetKI≡(KI)max−(KI)minanddenoteby(KI)ththethresholdvalueofKI.Inotherwords,acrackwillnot P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoComeDRIVINGFORCESFORDAMAGEACCUMULATION253propagatewhentheamplitudeoftheaveragestresssatisfiesthefollowingcondition:1d(K)2Ith1/2σd≤√dx+O(d)≈(KI)th.(8.18)d02πxπdEquatingσdtotheendurancelimitσ0atafixedRvalue,wehavethefollowingestimateforthecriticaldistance:22(KI)thd≈.(8.19)πσ0RIf(KI)thandσ0arematerialpropertiesthendisamaterialpropertyalso.Onehastobearinmind,however,thattheendurancelimitvarieswiththesizeofspecimens:whenthesizeincreases,theendurancelimitdecreases.Thiscontradictstheassumptionthatσ0isamaterialproperty.Thereforedisamaterialpropertyonlyif(KI)thvarieswithsizeinthesamewayasσ0.Duetolargestatisticaldispersioninthedataitisnotpossibletoobtainaccurateestimatesfor(KI)thandσ0.Itisunlikelythatthepropositionthatdisamaterialpropertycanbevalidatedwithrespecttoreasonablecriteriaforrejection.Example8.5.1ThethresholdstressintensityfactorforconventionallyprocessedAF1410steel7isapproximately16.0MPaanditsendurancelimitisapproximately950MPa.ThereforefromEquation(8.19)wehaved=180μm.Thisestimateofdismuchlargerthantheestimatesgivenin[40]and[60].Exercise8.5.1Assumethattheerrorin(KI)this±15%andtheerrorinσ0is±10%.Estimatetherangeoferrorforthecriticaldistanced.8.6DrivingforcesfordamageaccumulationIntheforegoingsectionswesummarizedwidelyusedmethodsforthepredictionoftheformationandpropagationofcrackscausedbycyclicloading.Therearemanysuccess-fulapplicationsofthesemethods;however,thoseapplicationstypicallyinvolvegeometricconfigurationsandloadingconditionssimilartothoseusedincalibration.Makingreliablepredictionsforgeometricconfigurationsandloadingconditionsthatareverydifferentfromthoseusedincalibration,suchascorrosiondefectsandverysmallcracks,remainsachal-lengingproblem.Thisindicatesthatepistemicandaleatoryuncertaintiesaremixedinthecalibrationdata.Thereforetheproblemofidentificationofthedrivingforcefordamageaccumulationremainsanopenquestion.Manyplausibleconceptualizationsarepossible.Wenowhavecomputationalmethodsatourdisposalwhichwerenotavailableatthetimewhentheclassicalmodelsoffatigueandcrackpropagationwereformulated.Wealsohaveawell-developedconceptualframeworkforvalidation.Thereforenewpossibilitiesexistfortheinterpretationandgeneralizationof7Thisisahigh-strengthsteelusedinmanysafety-criticalaerospaceapplications.Itsultimatetensilestrengthisapproximately1670MPa. P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoCome254WHATSHOULDBECOMPUTEDANDWHY?existingnewfatiguedata.Forexample,thefollowingfamilyofmodelsdoesnotrequiretheassumptionthatamaterial-dependentcriticaldistanceexists.LetGdefineafamilyofpossibledrivingforcesintermsofsomefunctionalF(uSS)>0andaconditionC:G(F,u,C,T)=F(u(x,t),T)dV,x∈R3(8.20)SSSSCwhereuSSisthesolutionofacontinuummechanicsproblembasedonsmall-straintheoryandTrepresentstemperature.ThedomainofintegrationCdependsontheconditionbywhichCisdefined.Formulationofthisconditionispartoftheconceptualizationprocess.Forexample,onemaydefineFtobethevonMisesstressandCasafunctionofthefirstprincipalstressσ1:C={x|σ1>βσyldwhenF(t)=Fmax}(8.21)whereσyldistheyieldstress,0<β≤1isadimensionlessparameterandF(t)istheload–timehistory.Ofcourse,thedefinitionofCmaydependonthedefinitionofF.ThechoiceofCasafunctionofstressorstrainisrelatedtotheexperimentallyobservedfactthatthelargerthevolumesubjectedtoelevatedstressorstrain,thelowertheendurancelimitandhencethelikelihoodoffailureisgreater.Ingeneral,thesolutionuSS(x,t)isnotknown;onlyanapproximationtouSS,whichwillbedenotedbyuFE(x,t),isknown.ReplacementofuSSbyuFEispermissibleonlywhenitcanbeguaranteedthat|G(F,uSS,C,T)−G(F,uFE,C,T)|≤τ|G(F,uSS,C,T)|(8.22)whereτisaspecifiedtolerance.Numericalaccuracyisessentialbecause,unlesstheaccuracyofthecomputeddataisknown,itisnotmeaningfultocompareexperimentalobservationswithpredictionsbasedonamathematicalmodel.ThispointwasdiscussedinSection1.2.2.Example8.6.1LetusconsidertheconicalfeaturedescribedinExample8.3.3andassumethatF=1andletQ=σ1/σyldwhereσ1isthefirstprincipalstressandσyld=1517MPaistheyieldstress.Letusassumefurtherthatauniaxialstressof1379MPaisappliedtotheplate.Inthiscasethedrivingforceisthevolumeofthematerialwhereσ1>ασyld.Notethatthisdefinitionofdrivingforceatleastqualitativelyaccountsfortheexperimentallyobservedfactthatthelargerthevolumeofmaterialexposedtoelevatedstress,thelowertheendurancelimit.Thevolumecorrespondingtoα=0.95isillustratedinFigure8.11.Theplanesnormaltothex-andy-axesareplanesofsymmetry.8.7CyclecountingInordertoestimatedamageaccumulationbymeansofMiner’srule,orsomeothermodelofcumulativedamage,itisnecessarytoconvertload–timehistoriesintosetsofstressreversalscharacterizedbypairsof(σmax,R).Standardpracticesforcyclecountinginfatigueanalysis P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoComeVALIDATION255Figure8.11Example8.6.1:overthedarkgreyregionσ1≥0.95σyld,quartersymmetry.aredescribedinanASTMstandard.8Discussionofcyclecountingmethodsisbeyondthescopeofthisbook,excepttonotethatthedefinitionofacycledependsonthemethodusedforcyclecountingandthereforethechoiceofcountingmethodwillinfluenceestimatesofdamageaccumulation.8.8ValidationWehaveseenthatthemathematicalmodelusedforpredictingtheprobabilityoffailurecausedbycyclicloadingisbasedonanumberofassumptionsinthefollowingcategories:1.Definitionofdrivingforce.Themostwidelyusedclassicaldefinitionsandnewpossi-bilitieswereoutlined.2.Statisticalcharacterizationofcalibrationdata.ThiswasdiscussedinExample8.3.2.3.Choiceofthemodelofcumulativedamage.WediscussedMiner’sruleonlybutnotedthatseveralothermodelsexist.4.Themethodofcyclecounting.Thepurposeofvalidationistotestwhetheramodelmeetsnecessaryconditionsforacceptance.Theoutcomeofvalidationexperimentsisevaluatedwithreferencetooneormore8ASTME1049-85(Reapproved2005),StandardPracticesforCycleCountinginFatigueAnalysis. P1:OSOJWST055-08JWST055-SzaboFebruary18,20117:7PrinterName:YettoCome256WHATSHOULDBECOMPUTEDANDWHY?Figure8.12PredictedprobabilityofthenumberofrepeatedloadingsequencestofailureMbasedonExample8.3.2.metricsandthecorrespondingcriteriawhichareformulatedtakingintoaccounttheintendeduseofthemodel.LetusconsideronceagaintheproblemofExample8.3.2.Supposethatvalidationexperi-mentsareplannedinwhichthesequenceofloadingshowninTable8.1willberepeateduntilfailureoccurs.Weareaskedtomakeapredictionontheoutcomeoftheexperiment.WedenotethenumberoftimesthesequenceisrepeatedbyMand,usingtheprocedureout-linedinExample8.3.2,estimatethecumulativedistributionfunctionF(x)=Prob(M1.(9.60)Theparameterqisfixed,andp≥qisincreaseduntilconvergenceisrealized.Theanisotropic(h)productspaceonstisdefinedbySppq((h)):=span(ξkηζm,(ξ,η,ζ)∈(h),prststk,=0,1,2,...,p,m=0,1,2...,q).(9.61)Thedefinitionofanisotropicspacesonthestandardpentahedralelementisanalogous.Advantagesanddisadvantagesofhierarchicshellandthin-solidmodelsTheadvantagesofthin-solidformulationsovershellformulationsarethattheyareeasiertoimplementandcontinuitywithotherbodies,suchasstiffeners,iseasiertoenforce.Thedisadvantagesarethatthin-solidformulationscannotbeappliedtolaminatedshellsunlesseachlaminaisexplicitlymodeled;thenumberoffieldfunctionsmustbethesameforeachdisplacementcomponent;andthedownwardextensionofthefamilyofhierarchicmodelstosatisfytherequirementofasymptoticconsistencycannotbeapplied.Theanisotropicspaceswithq=1aresimilarbutnotequivalenttotheReissner–MindlinplatemodelortheNaghdishellmodel.Exercise9.3.1Themid-surfaceofahyperboloidalshellisgivenbyx2y2z2R2+−=1,−L≤z≤L,α2:=tRt2Rt2(αL)2R2−Rt2cwhereRtisthethroatradiusandRcisthecrownradius.LetRt=1.0m,L=1.0mandα=1.Denotethethicknessoftheshellbyd.AssumealsothatthematerialiselasticwithE=2.0×105mPa,ν=0.3.Assumealsothatanormalpressurep=pcos2θisactingon0theinsidesurfaceoftheshellwhereθistheanglemeasuredfromthepositivex-axisasshowninFigure9.11.Letp0=20.0kPa.Theedgeaty=−Lisfixed,thatis,alldisplacementcomponentsarezero,andtheedgeaty=Lisfree.1.ConstructasolidmodeloftheshellwithddefinedasaparameterandconstructameshsimilartothatshowninFigure9.11.ppq(h)2.UsingtheanisotropicproductspacesSpr(st)andtheanisotropictrunkspacesppq(h)Str(st),investigatetheratesofconvergenceintheenergynormford=0.01mandd=0.001mforq=1,2,3.Thisexampleshowstheeffectsoflocking:therelativeerrorissubstantiallylargerford=0.001mthanford=0.01m,however,theestimatedasymptoticratesofconvergenceare P1:OSOJWST055-09JWST055-SzaboFebruary16,20118:5PrinterName:YettoCome288BEAMS,PLATESANDSHELLSFigure9.11Hyperboloidalshell.closeto1.0.Thestrongboundarylayeratthefixededgeisclearlyvisiblewhenthedeformedshapeisplotted.9.4TheOakRidgeexperimentsInaninvestigation,performedatOakRidgeNationalLaboratory(ORNL)inthe1970s,theresultsoffiniteelementanalysiswerecomparedwiththeresultsofphysicalexperiments[22],[34].Themotivationwasthatprovenelasticstressanalysismethodswerenotyetdevelopedforvariouscommonlyusedshellconfigurationsinnuclearpowerplantsandconsequentlyreliabledesigninformationwasnotyetavailable.Aprojectwasundertakeninwhichfourinstrumentedtestarticles,eachsubjectedto13loadcases,weretestedandanalyzedwiththeaimofassessingtheutilityofnumericalsimulationmethodsavailableatthattime.IntheterminologyofChapter1,asetofvalidationexperimentswasperformed.Inthefollowingtheprocessesofconceptualization,verificationandvalidationarediscussedwithreferencetoonetestarticleandoneloadcaseonly.9.4.1DescriptionFourtestarticlesweremanufacturedandinstrumentedwithgreatcare.Onlythefirsttestarticlewillbediscussedinthefollowing.Thistestarticlewasmadebyweldingtwocarbonsteelpipesandthencarefullymachiningittotheintendedtestdimensions.Inordertoreduceresidualstresses,thetestarticlewasannealedseveraltimesinthemachiningprocess.TheexperimentalarrangementisshowninFigure9.12.Thehorizontalpartiscalledthecylinder,theverticalpartiscalledthenozzle.Thelengthofthecylinderwas39.0in(991mm).Thelengthofthenozzle,measuredfromthepointofintersectionofthecenterlineofthenozzlewiththecenterlineofthecylinder,was19.5in(495mm).Theoutsidediameterofthecylinder(resp.nozzle)was10in(254mm)(resp.5.0in(127mm)).Theintendedwallthicknessofthecylinder(resp.nozzle)was0.1in(2.54mm)(resp.0.05in(1.27mm)). P1:OSOJWST055-09JWST055-SzaboFebruary16,20118:5PrinterName:YettoComeTHEOAKRIDGEEXPERIMENTS289Figure9.12Experimentalarrangement.Previouslypublishedin[83].ReproducedwithpermissionfromtheJournalofAppliedMechanics,anASMEpublicationc2005.Thetestarticlewasinstrumentedusing322three-gaugefoilrosettes9bondedontheinsideandoutsidesurfacesbyepoxyadhesiveandcured.Thegaugesintherosetteswerearrangedina-pattern(i.e.,thedirectionsofstrainmeasurementswere120◦apart).Fordetailswereferto[22],[34].AsseeninFigure9.12,therightendofthecylinderwasrigidlyclampedtoaheavyflatplateboltedtoaframe.Smallflangesweremachinedintotheendsofthecylinderandnozzletosupportthesealandtheclampingforces.Heavyloadingfixtureswereattachedontheoppositeendofthecylinderandontheendofthenozzletoprovidesealandseatingfortheapplicationofforces.Atotalof13loadcasesthatincludedpressureloadingandaxialforces,shearforces,bendingmomentsandtwistingmomentswasinvestigated.Theforcesandmomentswereappliedtothecylinderandthenozzlebyhydraulicramsactingthroughloadcells.Thepressureloadingwasappliedbymeansofahydraulicfluid.Inordertocompensatefortheweightofthehydraulicfluidandtheendfixtures,abalancingforcewasap-pliedtothefixtureatthefreeendofthecylinderthroughacablethatisvisibleinFigure9.12.Forall13loadcasestheloadwasappliedinincrementsof20%ofthefullload,thendecreasedtozeroagainin20%decrements.Onlyoneoftheloadingcases,thepres-sureload,isdiscussedinthefollowing.Themaximumvalueofthepressurewas50.0psi(344.8kPa).9Micro-MeasurementstypeEA-06-030YB-120,optionSE. P1:OSOJWST055-09JWST055-SzaboFebruary16,20118:5PrinterName:YettoCome290BEAMS,PLATESANDSHELLS9.4.2ConceptualizationForstructuraldetails,suchasthetestarticlediscussedhereandloadedinaccordancewiththeORNLtestplan,mathematicalmodelsbasedonthetheoryofelasticityaretypicallyassumed.Thepossibilityofinelasticdeformationatthejunctionoftheshellscannotbeexcluded,however.Elsewhere,simplificationssuchastheNaghdiorNovozhilov–Koiterassumptionsmaybepossible.Keepinginmindthatuncertaintiesexistingeometricdescription,materialproperties,loadingandconstraints,thedevelopmentofaplanfortheassessmentoftheeffectsofuncertaintiesonthedataofinterestispartoftheconceptualizationprocess.VirtualexperimentationAnimportanttoolfortheassessmentoftheinfluenceofvariousmodelingassumptionsonthedataofinterestisvirtualexperimentation.InthisprojectthedataofinterestwerethevonMisesstressesinthestraingaugelocations.Thefollowingquestionscanbeansweredbyvirtualexperimentation:1.Aretheequationsofthelineartheoryofelasticityadequateforrepresentingtheresponseofthetestarticletothegivensetofloadingconditions?2.Whatisthesizeoftheintersectionregionwherethefullythree-dimensionalrepresen-tationshouldberetained?3.Whatisthesimplestshellmodelthatiscapableofapproximatingthestrainreadingsinthegivenlocations?4.Whataretheeffectsofvariationsininputdata?Utilizingsymmetry,one-halfofthesolutiondomainisconsidered.Thefiniteelementmeshwasconstructedsuchthatthesizeoftheintersectionregioniscontrolledbyparameters.Thenozzleandtheshellwerepartitionedintohexahedralelementsandtheintersectionregionwaspartitionedintohexahedralandpentahedralelements.ThefiniteelementmeshisshowninFigure9.13.TheparametersthatcharacterizetheintersectionregionaredenotedbydsanddninFigure9.13(b).ThemeshlayoutisolatesthelineofintersectionoftheoutersurfacesoftheshellforthereasonsdiscussedinconnectionwiththeL-shapeddomaininChapter6.Thesolutionisstronglysingularalongthisline.Themeshlayoutalsoanticipatestheexistenceofboundarylayersattheendsoftheshellandthenozzle.Uniformpressureloadingwasappliedontheinsidesurface.Oneendoftheshellwasfixed,theoppositeendoftheshellandtheendofthenozzlewereclosedbyendplates(notshowninFigure9.13).ThestraingaugelocationsontheinsideandoutsidesurfacesnearesttotheintersectionarelabeledC,D,E,FinFigure9.13(b).Theresultsofvirtualexperimentshaveshownthat(a)theinfluenceofasmallplasticzonealongthesingularlineattheintersectionhasnegligiblysmalleffectonthedataofinterest,(b)geometricnonlinearitiescanbeneglected,(c)thedataofinterestarenotsensitivetothesizeoftheintersectionregionand(d)anisotropictrunkspacesdefinedbyEquation(9.60)yieldconsistentresultsforanyqvalue. P1:OSOJWST055-09JWST055-SzaboFebruary16,20118:5PrinterName:YettoComeTHEOAKRIDGEEXPERIMENTS291NozzleIntersectionregionShellyxzFigure9.13(a)Finiteelementmesh.(b)Detailattheintersection.Previouslypublishedin[83].ReproducedwithpermissionfromtheJournalofAppliedMechanics,anASMEpublicationc2005.9.4.3VerificationVerificationisaprocesscomprisingthestepsdescribedinSection7.3.Oneofthestepsistoshowthatthedataofinterestaresubstantiallyindependentofthechoiceofdiscretization.ThediscretizationandverificationusedintheoriginalORNLinvestigationandintheinvestigationreportedin[83]aresummarizedinthissection.Itwasconcludedinbothcasesthattheerrorsofdiscretizationwerenegligiblysmallincomparisonwithothererrorsinbothcases.VerificationintheORNLinvestigationThetreatmentofshellmodelsbythefiniteelementmethodwasinitsveryearlystagesofdevelopmentatthetimeoftheORNLinvestigation.Althoughtheinvestigatorswereawareofsomecontemporaryworkoncurvedshellelements,shellswerecommonlyapproximatedbyassembliesofflatplateandmembraneelementsandmostoftheavailableexperiencewaswiththoseelements.Forthisreasontheinvestigatorsdecidedtouseaspatialassemblyflatplateelementsforthepurposeofanalyzingtheshellintersectionproblem[22].TheHCTtriangularelement,describedinSection9.2.3,waschosenfortheapproximationofthedisplacementcomponentnormaltothemid-surfaceoftheshell.FourHCTtriangleswereassembledtoproduceonefour-sidedelementwithfivenodes.Thetangential(membrane)componentsofthedisplacementvectorwereapproximatedbyasimilarassemblyoftriangles.Thesevectorcomponentswereapproximatedovereachcomponenttrianglebyquadraticpolynomialsconstrainedsothatthebasisfunctionswerelinearovertheexternaledges.Thisisknownastheconstrainedlinearstraintriangle(CLST).Aquadrilateralmembraneelementhastwodegreesoffreedomateachofitsfivenodes[22].Thefivenodesofeachfour-sidedelementwerelocatedonthemid-surfaceoftheshell.Thereforetheassembledtriangleswerenotco-planarandhencethereisathirdrotation P1:OSOJWST055-09JWST055-SzaboFebruary16,20118:5PrinterName:YettoCome292BEAMS,PLATESANDSHELLSFigure9.14ThefiniteelementmeshusedintheORNLinvestigation.Previouslypublishedin[83].ReproducedwithpermissionfromtheJournalofAppliedMechanics,anASMEpublicationc2005.component,notpresentintheconstituenttriangles.TheusualtreatmentisthatateachnodetherotationcomponentsaretransformedintoaCartesiancoordinatesystem,theoriginofwhichliesontheshellsurfaceatthenodeandoneaxisiscoincidentwiththenormaltothesurface.Therotationcomponentinthedirectionofthenormalisthensettozero.Thiscancausevariousproblems,however.Foradiscussiononthispointwereferto[22],[34].ThefiniteelementspaceusedintheORNLinvestigationisthespanoftheassembledbasisfunctionsonthemeshoffour-sidedelementsshowninFigure9.14.Verificationwasbasedoncomparisonofresultsusingtwomeshes.ItwasconcludedthattheresultswerenotsensitivetocoarseningthemeshshowninFigure9.14.Verificationbasedonp-extensionThemeshusedintheinvestigationreportedin[83]isshowninFigure9.13.p-Extensionwasusedtodemonstratethatthecomputeddataaresubstantiallyindependentofthediscretization.ThisisillustratedinFigure9.15wherethevonMisesstressisplottedvs.thenumberofdegreesoffreedomNonasemi-logscaleforpointBshowninFigure9.13(b).Theconvergenceofstraindatacomputedattheothergaugelocationsconsideredhereissimilar. P1:OSOJWST055-09JWST055-SzaboFebruary16,20118:5PrinterName:YettoComeTHEOAKRIDGEEXPERIMENTS29318312016110251001467p=849012Est.limit:14.44ksi(99.57MPa)801070Equivalentstress[ksi]60Equivalentstress[MPa]850p=16345101010NFigure9.15ConvergenceofthevonMisesstressatpointBshowninFigure9.13(b).Previ-ouslypublishedin[83].ReproducedwithpermissionfromtheJournalofAppliedMechanics,anASMEpublicationc2005.9.4.4Validation:comparisonofpredictedandobserveddataIncomparingdatacomputedfrommathematicalmodelswithphysicalmeasurements,itisnecessarytorecognizethedifferencesbetweenthemathematicalproblembeingsolvedandthephysicalsystembeingmodeled.ThesedifferencesareenumeratedinthefollowinginrelationtotheORNLtestarticlediscussedhere.Differencesbetweenthemathematicalmodelandthephysicalreality1.Geometricvariations.TheORNLinvestigatorswerecarefultominimizeerrorsinmanufacturingthetestarticle.However,somevariationsinwallthicknessandotherdimensionsoccurred.Thefollowingquotationisfromreference[22]:“Acarefuldimensionalinspectionofthemachinedmodelindicatedthat,despitethecaretakeninmachining,therewerewallthicknessvariationsinbothnozzleandcylinderwiththenozzlethicknessbeingasmuchas15%greater(0.007to0.008inchescomparedwiththenominal0.050inches)inthefourthquadrantthaninthesecond.”Inthemathematicalmodelitisassumedthattheshellandthenozzlearedefinedbyperfectcylindricalsurfacesandconstantwallthickness.Itwasintendedtomanufacturetheintersectionwithzerofilletradius.Inreality,themillingtoolleavessomefillet.Inthemathematicalmodelthefilletradiusiszero.Variationsinthicknessdaffectthedistributionofmembraneandbendingforceswithin P1:OSOJWST055-09JWST055-SzaboFebruary16,20118:5PrinterName:YettoCome294BEAMS,PLATESANDSHELLStheshellandtherelationshipsbetweenstressesandthemembraneforcesandbendingmoments.Thestressescorrespondingtomembraneforcesareproportionalto1/dwhereasthestressescorrespondingtobendingareproportionalto1/d2.Thetestarticlehadsmallflangesattheends.Themathematicalmodeldoesnotaccountfortheflanges.2.Variationsinmaterialproperties.Theinvestigatorsassumedthatmaterialpropertiesarethenominalelasticconstantsofcarbonsteel.ModulusofelasticityE=30×106psi(207GPa);Poisson’sratioν=0.3.InrealitythemodulusofelasticityandPoisson’sratioarestochasticfunctionsofthespatialvariablesandtheiraveragevaluesmaydifferfromthenominalvaluesbyafewpercent.TherearenodataonthevariationofthemodulusofelasticityandPoisson’sratiowithinthetestarticle,butitisreasonabletoexpectthatthemeanvalueofE(resp.ν)iswithinabout2%(resp.5%)ofthenominalvalue.Therefore,systematicaswellasrandomerrorsarepresentinmakingcomparisonsbetweenmeasuredstrainsandstrainscomputedfromthemathematicalmodel.Therelationshipbetweenstressandstraininthetestarticlewillbecomenonlinearinlocationswherethestraincorrespondingtotheproportionallimitisexceeded.3.Differencesinconstraintconditions.Rigidendfixtureswereattachedtothefreeendofthecylinderandthenozzle.DetailsontheendfixturesarenotgivenintheORNLreport;however,theinvestigatorsassumedthattheendfixturesweresufficientlyrigidtoconstraintheendsofthecylinderandnozzlesoastomaintaintheendsasplanecircles[22].InthemathematicalmodelconstructedbytheORNLinvestigators,theendfixtureswererepresentedbyendplates.Intheinvestigationdescribedin[83]thefixtureattachedtothenozzlewasalsorepresentedbyanendplate;however,atthefreeendofthecylindertheradialandtangentialdisplacementcomponentsweresettozero.4.Differencesinloadingconditions.Thetestarticlewasloadedthroughhydraulicramsactingontheendfixtures.Theaccuracyoftheappliedloadandhencetheaccuracyofthestressresultantsdependsontheaccuracyoftheloadcells.Thetransferoftheloadthroughtheendfixtureswasthroughmechanicalcontact.Theprecisedistributionofthetractionsactingontheendsisnotknown.InthemathematicalmodelusedintheORNLinvestigation,theappliedloadswererepresentedbynodalforces.Intheinvestigationdescribedin[83]thepressureloadingwasrepresentedbyconstantnormaltraction.Itcanbeseenthatevenunderverycarefullycontrolledexperimentalconditionssomedegreeofuncertaintyconcerningthephysicalsystemispresent.Someoftheseuncertaintiescanbereduced,otherseithercannotbereducedormaynotbefeasibletoreduce.Forexample,themeanvalueoftheelasticconstantscanbedeterminedbycoupontests.Thedimensionsofthetestarticlecanbemeasuredwithhighaccuracy.Ontheotherhand,itwouldbeverydifficulttodeterminethedistributionofthetractionsorconstraintconditionsimposedbytheendfixtures.Inaddition,somedegreeofuncertaintyisassociatedwiththeinstrumentsemployedinmakingtheobservationsandtheeffectsoftheenvironmentontheinstruments.Inviewoftheseuncertaintiesonecannotexpectveryclosecorrelationbetweencomputedandexperimentaldata.ThelargestuncertaintiesintheORNLexperimentsarerelatedtothedifficultiesassociatedwithmanufacturingthin-walledobjectstotighttolerancesand,possibly,uncertaintiesinthemathematicalrepresentationoftheconstraintconditions. P1:OSOJWST055-09JWST055-SzaboFebruary16,20118:5PrinterName:YettoComeTHEOAKRIDGEEXPERIMENTS295Table9.1ThevonMisesstress(psi)inthegaugelocationsC,D,E,FshowninFigure9.13(b).Fullythree-dimensionalmodel.Trunkspace.Thepointsarelocatedintheplaneofsymmetryonthefixed-endsideofthecylinder(1psi=6.895kPa).pNPt.CPt.DPt.EPt.F36095112251195913729165894108281246111680167291968751777412455121581764419441627497125351208917237190057405611254212021171081910485753012544119821709418903Expt.15569145541698113100Diff.(%)24.121.5−0.7−30.7PredictionsandobservationsThecomputedvonMisesstressinthegaugelocationsC,D,E,FshowninFigure9.13(b)andthevonMisesstresscomputedfromtheexperimentaldataaregiveninTable9.1.Thecomputeddataweredeterminedusingthefullythree-dimensionalmodel,trunkspace,withprangingfrom3to8.Itcanbeseenthatthestressdataconvergestrongly,buttheerrorsbetweentheexperimentalmeasurementsandpredictionsarelarge.Assumingthatthecriterionforrejectionwassetat20%errorinthepredictedvonMisesstress,thenthemathematicalmodelhastoberejected.Oneinstanceofexceedingthetolerancesetinthedesignofvalidationexperimentsissufficientforrejection.Whenamodelfailstomeetthecriteriasetforvalidationthenthenextproblemistoidentifypossiblereasonsforrejectingthemodel.Virtualexperimentationprovidesaveryeffectivetoolforsuchanalyses.Significantsensitivitytovariationsinwallthicknesswasfoundtoexist,whichleadstotheconclusionthatthelikelyreasonsforrejectionaretheuncertaintiesintheinputdata.Thereforeamodelthataccountsfortheeffectsofuncertaintiesinthegeometricdescriptionoftheshellsonthedataofinteresthastobeconsidered.Owingtolackofsufficientstatisticalinformation,uncertaintyquantification(UQ)ofteninvolvessomerelianceonexpertopinion.9.4.5DiscussionTheORNLinvestigationhighlightssomeofthedifficultiesandlimitationsofexperimentalvalidationofmathematicalmodelsforthin-shellproblems.Theexperimentaldataaredom-inatedbyuncertaintiescausedbydifficultiesassociatedwiththefabricationofthin-walledobjectstoexactingtolerances.Increasingthewallthicknesswouldreduceerrorscausedbymanufacturingtolerancesbutthenthethin-shellmodelmightnotbeapplicable.Incorrelationwithexperimentalobservationsthepredictionsofamathematicalmodelarecomparedwithdatathatareeitherobservabledirectlyorcanbecomputedfromobservabledata.Inthemodelproblemdiscussedhere,thestraincomponentsinsurfacepointslocatedinthevicinityoftheshellintersectionweremeasuredandthevonMisesstresseswerereportedinthegaugelocations. P1:OSOJWST055-09JWST055-SzaboFebruary16,20118:5PrinterName:YettoCome296BEAMS,PLATESANDSHELLSInmanycasesthedataofinterestarenotobservable.Forexample,onemaybeinterestedinthemaximumvalueoftheintegralofthenormalstressoversomesmallarea,afunctionalthatcannotbeobserved.Therefore,evenifamathematicalmodelwereshowntobesuccessfulinpredictingcertainmeasureddata,itmightnotbesuitableforcomputingotherdataofinterest.Virtualexperimentationisaveryusefultoolfortheevaluationoftheeffectsofuncertaintiesintheinputdataonthedataofinterest.Thevalidityofamathematicalmodelcannotbeestablishedbyexperimentalcorrelation.Thepurposeofvalidationexperimentsistodeterminewhethercertainnecessaryconditionsaremet.Exercise9.4.1Discusstherelativeimportanceofcomputationalandphysicalexperimenta-tioninrelationtotheORNLinvestigationoftheshellintersectionproblem.9.5ChaptersummaryInmanycasesitisadvantageoustomakecertainaprioriassumptionsconcerningthemodeofdeformationofanelasticbodyandusedimensionallyreducedmodelsinsteadoffullythree-dimensionalmodels.Whetheradimensionallyreducedmodelshouldbeusedinaparticularcasedependsonthegoalsofcomputationandtherequiredaccuracy.Generallyspeaking,dimensionallyreducedmodelsarewellsuitedforstructuralanalysiswherethegoalsofcomputationaretodeterminestructuralstiffness,displacementsandnaturalfrequenciesbutarenotwellsuitedforstrengthanalysisofstructuralconnectionsandotherdetails.Theaccuracyofthedataofinterestdependsnotonlyonthediscretizationused,butalsoonhowwelltheexactsolutionofadimensionallyreducedmodelapproximatestheexactsolutionofthecorrespondingfullythree-dimensionalmodel.Thedifferencesbetweenthedataofinterestdeterminedfromtheexactsolutionofadimensionallyreducedmodelandthecorrespondingfullythree-dimensionalmodelareerrorsofidealization,whichwerediscussedinSection1.1andillustratedinFigure1.1.Thehierarchicviewofmodelsprovidesaconceptualframeworkforthecontrolofmodelingerrors.Asthethicknessisreduced,theexactsolutionsofthehierarchicbeam,plateandshellmodelsconverge,respectively,totheexactsolutionoftheBernoulli–Eulerbeammodel,theKirchhoffplatemodelandtheNovozhilov–Koitershellmodel.Thesemodels,unlikethehierarchicmodels,requireboththedisplacementfunctionsandtheirfirstderivativestobecontinuous.ThereforetheexactsolutionsofthemodelsinthehierarchicfamilythatlieinC0()convergetoasolutionthatliesinC1().UnlessthepolynomialdegreeissufficientlyhightosatisfyorwellapproximateC1()continuity,h-convergencewillbesloworpossiblynon-existent.Ontheotherhand,p-convergencewilloccur,butentryintotheasymptoticrangewilloccuronlywhenp≥4. P1:OSOJWST055-10JWST055-SzaboFebruary16,201110:13PrinterName:YettoCome10NonlinearmodelsItwasnotedinChapter1andinSection3.7thattheformulationofamathematicalmodelinvolvesmakingvariousrestrictiveassumptionsandthereforeanymathematicalmodelshouldalwaysbeviewedasaspecialcaseofamorecomprehensivemodel.Inordertotestwhetherthoseassumptionsarereasonableinthecontextofaparticularapplication,itisnecessarytoperformvirtualexperimentation.Thecomputationaltoolsusedforvirtualexperimentationmusthavethecapabilitytosolvenonlinearproblems.Thesubjectofformulationandnumericaltreatmentofnonlinearmodelsisverylarge.Abriefintroductiontothisimportantsubjectispresentedinthischapterwithemphasisonthealgorithmicaspects.Foradditionaldiscussionanddetailswerefertootherbookssuchas[66],[72].10.1HeatconductionMathematicalmodelsofheatconductionofteninvolveradiationheattransferandtheco-efficientsofheatconductionaretypicallyfunctionsofthetemperature.Theformulationofmathematicalmodelsthataccountforthesephenomenaisoutlinedinthefollowing.10.1.1RadiationWhentwobodiesexchangeheatbyradiationthenthefluxisproportionaltothedifferenceofthefourthpoweroftheirabsolutetemperatures:q=κff(u4−u4)(10.1)nsRIntroductiontoFiniteElementAnalysis:Formulation,VerificationandValidation,FirstEdition.BarnaSzabóandIvoBabuška.©2011JohnWiley&Sons,Ltd.Published2011byJohnWiley&Sons,Ltd.ISBN:978-0-470-97728-6 P1:OSOJWST055-10JWST055-SzaboFebruary16,201110:13PrinterName:YettoCome298NONLINEARMODELSwhereu,uRaretheabsolutetemperaturesoftheradiatingbodies,κ=5.699×10−8W/(m2K4)istheStefan–Boltzmannconstant,10≤f≤1istheradiationshapefactorsand00isincreased.Ifλisinthepointspectrumthentherearefunctionsu¯iwhichsatisfyEquation(10.17)forω=0.Thatis,thenaturalfrequencyiszero.Ifσistensile(positive)thenthestructuralstiffnessisincreasedasλisincreased.SeeijExercise10.2.7.Remark10.2.1Usingtheproceduresofvariationalcalculus,thestrongformoftheequationsofequilibriumisfoundtobeσ¯+F¯+(σ0u¯)=0(10.18)ij,jikji,k,jwhere¯σij:=Cijkl(¯kl−T¯αkl).Thecorrespondingnaturalboundaryconditionsare:(¯σ+σ0u¯)n=T¯on∂(10.19)ijkji,kjiT P1:OSOJWST055-10JWST055-SzaboFebruary16,201110:13PrinterName:YettoComeSOLIDMECHANICS305and(¯σ+σ0u¯)n=k(δ¯−u¯)on∂.(10.20)ijkji,kjijjjsExample10.2.1Consideranelasticcolumnofuniformcross-section,areaA,momentofinertiaI,lengthandmodulusofelasticityE.Thecentroidalaxisofthecolumncoincideswiththex-axis.AcompressiveaxialforcePisapplied,henceσ0=−P/A.Thedisplacement111fieldisassumedtobeofthefollowingform:dwu¯1=−x2,u¯2=w(x1),u¯3=0.(10.21)dx1Thereforetheonlynon-zerostraincomponentisd2w11=2x2(10.22)dx1andthetermσ0uucanbewrittenasijα,iα,j22Pd2wdwσ0u¯u¯=−x2+.(10.23)ijα,iα,jA22dxdx11Inthiscasethestrainenergycanbewritteninthefollowingform:2221PdwPdwU=E−I2dx1−dx1(10.24)20Adx120dx1whereIisthemomentofinertiaofthecross-section:I:=x2dxdx.(10.25)223AThetermP/AcanbeneglectedinrelationtothemodulusofelasticityEin(10.24).Thisexam-pleillustratesthatdimensionallyreducedmodelscanbederivedfromthethree-dimensionalformulation.Exercise10.2.5NeglectingthetermP/Ain(10.24),determinetheapproximatevalueofPatwhichacolumnfixedatbothendswillbuckle.Useonefiniteelementandp=4.Reporttherelativeerror.Hint:RefertoEquation(9.27)andthedefinitionN5(ξ)=ψ4(ξ)whereψj(ξ)(j=4,5,...)isgivenbyEquation(9.26).Theexactvalueofthecriticalforceis4π2EI/2.Exercise10.2.6Showthattheworkdonebyσ0duetothelinearstraintermsisexactlyijcanceledbytheworkdonebyF0,T0andδ0inthesenseofEquation(10.9).iiiExercise10.2.7Considera50mm×50mmsquareplateofthickness1.0mm.Thematerialpropertiesare:E=6.96×104MPa;ν=0.365;=2.71×10−9Ns2/mm4.Theplateis P1:OSOJWST055-10JWST055-SzaboFebruary16,201110:13PrinterName:YettoCome306NONLINEARMODELSfixedononeedge,loadedbyanormaltractionTnontheoppositeedgeandsimplysupportedontheotheredges(softsimplesupport).DeterminethefirstnaturalfrequencycorrespondingtoTn=−50MPa,Tn=0andTn=50MPa.(Partialanswer:ForTn=−50MPathefirstnaturalfrequencyis578.6hertz.)Exercise10.2.8Ifthemulti-spanbeamdescribedinExercise9.1.9hadbeenpre-stressedbyanaxialforcesuchthataconstantpositiveinitialstressσ0actedonthebeam,wouldtherotationatsupportBbelargerorsmallerthanifthebeamhadnotbeenpre-stressed?Explain.10.2.3PlasticityTheformulationofmathematicalmodelsbasedontheincrementalanddeformationaltheoriesofplasticityispersentedinthefollowing.Intheincrementaltheory,asthenameimplies,arelationshipbetweentheincrementinthestraintensorandthecorrespondingincrementinthestresstensorisdefined.Inthedeformationaltheorythestraintensor(ratherthantheincrementinthestraintensor)isrelatedtothestresstensor.Itisassumedthatthestraincomponentsaresufficientlysmalltojustifysmallstrainrepresentationandtheplasticdeformationiscontained,thatis,theplasticzoneissurroundedbyanelasticzone.Uncontainedplasticflow,asinmetalformingprocesses,isnotwithinthescopeofthefollowingdiscussion.Theformulationofplasticdeformationisbasedonthreefundamentalrelationships:(a)yieldcriterion;(b)flowrule;and(c)hardeningrule.WewillusethevonMisesyieldcriterion4andanassociativeflowruleknownasthePrandtl–Reussflowrule.5Foracomprehensivediscussionwereferto[72].NotationThestressdeviatortensorisdefinedby1σ˜ij:=σij−σkkδij.(10.26)3ThesecondinvariantofthestressdeviatortensorisdenotedbyJ2andisdefinedby1J2:=σ˜ijσ˜ij21=(σ−σ)2+(σ−σ)2+(σ−σ)2+6(σ+σ+σ).1122223333111223313Inauniaxialtestσistheonlynon-zerostresscomponent.ThereforeJ=2σ2/3.The11211equivalentstress,oftencalledthevonMisesstress,isdefinedsuchthatinthespecialcaseofuniaxialstressitisequaltotheuniaxialstress:1σ¯=(σ−σ)2+(σ−σ)2+(σ−σ)2+6(σ+σ+σ).(10.27)11222233331112233124RichardvonMises(1883–1953).5LudwigPrandtl(1875–1953),EndreReuss(1900–1968). P1:OSOJWST055-10JWST055-SzaboFebruary16,201110:13PrinterName:YettoComeSOLIDMECHANICS307Figure10.2Typicaluniaxialstress–straincurve.Wewillinterpretuniaxialstress–straindiagramsasarelationshipbetweentheequivalentstressandtheequivalentstrain.Theelastic(resp.plastic)strainswillbeindicatedbythesuperscripte(resp.p).Thethreeprincipalstrainsaredenotedby1,2,3.Theequivalentelasticstrainisdefinedby√2¯e:=(e−e)2+(e−e)2+(e−e)2(10.28)1223312(1+ν)whereνisPoisson’sratio.ThedefinitionoftheequivalentplasticstrainfollowsdirectlyfromEquation(10.28)bysettingν=1/2:√¯p:=2(p−p)2+(p−p)2+(p−p)2.(10.29)1223313Atypicaluniaxialstress–straincurveisshowninFigure10.2Exercise10.2.9ShowthatthefirstderivativesofJ2withrespecttoσx,σy,σzandτxyareequalto˜σx,˜σy,˜σzand˜τxyrespectively.AssumptionsTheassumptionsonwhichtheformulationofthemathematicalproblemofplasticityisbasedaredescribedinthefollowing:1.Confinedplasticdeformation.Thestraincomponentsaremuchsmallerthanunityonthesolutiondomainanditsboundary,andthedeformationsaresmallinthesensethatequilibriumequationswrittenfortheundeformedconfigurationareessentiallythesameastheequilibriumequationswrittenforthedeformedconfiguration. P1:OSOJWST055-10JWST055-SzaboFebruary16,201110:13PrinterName:YettoCome308NONLINEARMODELS2.Decompositionofstrain.Anincrementinthetotalstrainisthesumoftheincrementintheelasticstrainandtheincrementintheplasticstrain:epdij=dij+dij.(10.30)Wewillnotconsiderthermalstrainhere.3.Yieldcriterion.WedefineF(σ,¯p):=σ¯−H(¯p).(10.31)ijWhenF<0thenthematerialiselastic.PlasticdeformationmayoccuronlywhenF=0.AnystressstateforwhichF>0isinadmissible.Thisisknownastheconsistencycondition.Therefore,inplasticdeformation∂FdF=0:dσ−Hd¯p=0.(10.32)ij∂σij4.Flowrule.ThePrandtl–Reussflowrulestatesthatp∂Fpd=d¯.(10.33)ij∂σijIncrementalstress–strainrelationshipAnincrementinstressisproportionaltotheelasticstrain:epdσij=Cijkldkl=Cijkl(dkl−dkl).SubstitutingEquation(10.33)weget∂Fdσ=Cd−Cd¯p.(10.34)ijijklklijkl∂σklUsing(10.32)weobtain∂F∂F∂F∂FHd¯p=dσ=Cd−Cd¯p(10.35)ijijklklpqrs∂σij∂σij∂σpq∂σrswherethedummyindicesweresuitablyrenamed.FromEquation(10.35)anexpressionford¯pisobtained:∂FCijklp∂σijd¯=dkl.∂F∂FH+Cpqrs∂σpq∂σrs P1:OSOJWST055-10JWST055-SzaboFebruary16,201110:13PrinterName:YettoComeSOLIDMECHANICS309OnsubstitutingintoEquation(10.34)wegettheincrementalelastic–plasticstress-strainrelationship:⎛⎞∂F∂F⎜Cijmn∂σ∂σCuvkl⎟dσ=⎜C−mnuv⎟d.(10.36)ij⎝ijkl∂F∂F⎠klH+Cpqrs∂σpq∂σrsThebracketedexpressionin(10.36)iswelldefinedforelastic–perfectlyplasticmaterials(i.e.,materialsforwhichH=0).Thecomputationsinvolvethefollowing.Giventhecurrentstressstateσij,computedσij=Cijkldklcorrespondingtoanincrementintheappliedload.IneachintegrationpointcomputeF(σij+dσij).IfF(σij+dσij)≤0thennothingfurtherneedstobedone.IfF(σij+dσij)>0thenrecomputedσijusingEquation(10.36).RepeattheprocessuntilF(σij+dσij)≈0.Theprocessisstartedbyalinearanalysis.Analternativealgorithm,knownasthereturnmappingalgorithm,hasbeenusedbyseveralinvestigators.Fordetailswereferto[72].ThedeformationtheoryofplasticityInthedeformationtheoryofplasticityitisassumedthattheplasticstraintensorisproportionaltothestressdeviatortensor.ReferringtoFigure10.2,σ¯¯e+¯p=EswhereEsisthesecantmodulus.SincetheelasticpartofthestrainisrelatedtothestressbyHooke’slawσ¯¯e=Ewehave11¯p=−σ.¯(10.37)EsEInauniaxialstressstate˜σ=2¯σ/3andhenceEquation(10.37)canbewrittenas311¯p=−σ.˜2EsEThisisgeneralizedtop311ij=−σ˜ij.(10.38)2EsE P1:OSOJWST055-10JWST055-SzaboFebruary16,201110:13PrinterName:YettoCome310NONLINEARMODELSForexample,inplanarproblems,⎧⎫⎧⎫p⎪⎪⎨x⎪⎪⎬⎪⎪⎨σ˜x⎪⎪⎬p311yσ˜yp=−·(10.39)⎪⎪z⎪⎪2EsE⎪⎪σ˜z⎪⎪⎩p⎭⎩⎭xyτ˜xyExample10.2.2Inthisexampleanalgorithmbasedonthedeformationtheoryofplasticityisformulatedforplanestress.Inthedeformationtheoryofplasticitytheelastic–plasticcompliancematrixisthematrix[C]whichestablishestherelationshipbetweenthetotalstrainandstress:{}=[C]{σ}.Theelastic–plasticmaterialstiffnessmatrix[Eep]istheinverseoftheelastic–plasticcom-pliancematrix.Usingthedefinitionofthestressdeviatorandtherelationshipbetweentheplasticstrainanddeviatoricstress(10.39),wehave⎧⎫⎡⎤⎧⎫⎨p⎬2/3−1/30⎨σ⎬x311xp=−⎣−1/32/30⎦σ.yy⎩p⎭2EsE⎩⎭γxy002τxyUsing{}={e}+{p},arelationshipisobtainedbetweenthetotalstraincomponentsandthestresstensor:⎧⎫⎛⎡⎤⎡⎤⎞⎧⎫⎨x⎬1−ν01−1/20⎨σx⎬1E−Esy=⎝⎣−ν10⎦+⎣−1/210⎦⎠σy·⎩⎭EEsE⎩⎭γxy002(1+ν)003τxyThesolutionisobtainedbyiteration:foreachintegrationpointtheequivalentstressandstrainarecomputed.Fromthestress–strainrelationshipthesecantmodulusiscomputedandtheappropriatematerialstiffnessmatrix[Eep]isevaluated.Theprocessiscontinueduntiltheequivalentstressandstraindonotdeviatefromtheuniaxialstress–straincurvebymorethanapre-settolerance(usually1%orless).Aninterestingbenchmarkstudyontheperformanceoftheh-andp-versionsofthefiniteelementmethodispresentedin[27].Exercise10.2.10Showthatinuniaxialtensionorcompression,¯e=¯e,¯p=¯pand11σ¯=σ1.Exercise10.2.11DeriveEquation(10.28)byspecializingtherootmeansquareofthedifferencesofprincipalstrainstotheone-dimensionalcasesothat¯e=¯e.110.2.4MechanicalcontactMathematicalmodelsofmechanicalcontactbetweensolidbodiesinvolvenon-linearboundaryconditionswrittenintermsofagapfunctiong=g(s,t)≥0wheresandtaresurfaceparameters:Wheneverg=0,thenormalandshearngtractionsincorrespondingpointsmust P1:OSOJWST055-10JWST055-SzaboFebruary16,201110:13PrinterName:YettoComeSOLIDMECHANICS311Figure10.3Exampleofmechanicalcontact:notationhaveequalvalueandoppositesense.Wheng>0thenthetractionsarezero.Theconditiong<0isnotallowedbecauseitwouldcorrespondtopenetration.Inmanypracticalproblemscontactingbodiesarelubricatedandthereforeshearingtrac-tionsarenegligiblysmallincomparisonwiththenormaltractions.Ontheotherhand,shearingtractionsindryfrictionstronglyinfluencewear,seeforexample[56].Inthefollowingexampleaprocedureforsolvingelasticcontactproblemsisillustratedinaone-dimensionalsetting.Example10.2.3Letusconsidertheproblemofcontactbetweentwoelasticbars.ThenotationisshowninFig.10.3.Weassumethattheaxialstiffness(AE)iandthespringcoefficientci(i=1,2)areconstantsforeachbarandbar2isfixedontheright.Thegapbetweenthebarsisg=g0−U2+U3whereg0istheinitialgap.ThegoalistodeterminethecontactforceFcasafunctionoftheappliedforceF.Thedifferentialequationsforthebarsare:−(AE)u+cu=0,i=1,2(10.40)iiiiandthecorrespondingsolutionsare:"ui(x)=aicoshλix+bisinhλixwhereλi=ci/(AE)i.(10.41)Weassociatealocalcoordinatesystemwitheachbar,suchthatx=0attheleftend.ThereforeU1=u1(0),U2=u1(1)andU3=u2(0).Tothebarontheleftweapplytheboundaryconditions(AE)u(0)=−F,(AE)u()=−F11111candobtainthesolutionFcoshλ11u1(x)=coshλ1x−sinhλ1x(AE)1λ1sinhλ11Fccoshλ1x−·(10.42)(AE)1λ1sinhλ11Tothebarontherightweapplytheboundaryconditions(AE)u(0)=−F,u()=022c22 P1:OSOJWST055-10JWST055-SzaboFebruary16,201110:13PrinterName:YettoCome312NONLINEARMODELSandobtainthesolution:Fcsinhλ22u2(x)=coshλ2x−sinhλ2x.(10.43)(AE)2λ2coshλ22Givenaninitialgapg0>0,theforceneededtoclosethegap,denotedbyF0,canbecomputedfromEquation(10.42)bysettingu1(1)=g0andFc=0.Thisyields:F0=(AE)1λ1g0sinhλ11.(10.44)ForanyF>F0acontactforceFc>0willdevelopandtheconditiong=0mustbesatisfied.LetusnowassumethatF=αF0wasappliedwhereα>1andhenceg<0whichviolatesthecontactcondition.FromEquations(10.42)and(10.44)wegetFccoshλ11U2≡u1(1)=αg0−(10.45)(AE)1λ1sinhλ11andfromEquation(10.43)wegetFcsinhλ22U3≡u2(0)=·(10.46)(AE)2λ2coshλ22Thereforethegapisg=g0−U2+U3=g0−αg0+FcQ(10.47)where1coshλ111sinhλ22Q:=+·(10.48)(AE)1λ1sinhλ11(AE)2λ2coshλ22Lettingg=0weget⎧⎨(α−1)g0forα≥1Fc=Q(10.49)⎩0forα<1.InthisexampleitwaspossibletofindFcintwosteps.Inthefirststeptheforceneededtoclosethegap,denotedbyF0,wasdetermined.InthesecondstepthecontactforceFcwasdeterminedforF=αF0whereα≥1.Intwoandthreedimensionstheproblemismorecomplicatedbecauseitisnecessarytodeterminethecontactsurfaceswhichdependonthecontactforce,thematerialpropertiesandthegeometricattributesofthecontactingbodies.Achallengingaspectofthenumericalsolutionofcontactproblemsisthatsmallerrorsinthenumericalapproximationofthecontactingsurfaceshavesubstantialeffectsonthecomputedvaluesofthetractions.ThemethodmostcommonlyusedforthenumericaltreatmentofcontactconditionsisknownastheaugmentedLagrangianmethod,seeforexample[57],[73],[92]. P1:OSOJWST055-10JWST055-SzaboFebruary16,201110:13PrinterName:YettoComeCHAPTERSUMMARY31310.3ChaptersummaryIntheprocessofconceptualizationoneusuallystartswithlinearmodelsbearinginmindtherestrictiveassumptionsincorporatedinthosemodels.Inordertoproperlyformulateamathematicalmodelfortherepresentationofsomephysicalreality,itisnecessarytoconsidertheeffectsofthoseassumptionsand,whennecessary,removeormodifythem.Virtualex-perimentationisaveryusefultoolfortheassessmentoftheeffectsofmodelingassumptionsonthedataofinterest.Theconceptofmodelhierarchiesandtheabilitytoseamlesslypassfromlineartononlinearanalysesarethepracticalprerequisitestoestimatingandcontrol-lingmodelingerrors.Inthischapterthealgorithmicaspectsofnonlinearformulationswereoutlined.