THE P-VERSION OF THE FINITE ELEMENT METHOD IN
Centerfor ComputationalMechanicsWashingtonUniversitySt. THEIN INCREMENTALFINITEELEMENTELASTO-PLASTICStefanM. archDecemberAssistant1993METHODANALYSIS
mplementation1.of J2 plasticity.under internal.PracticalTheReferences6.The sSummary3Application:and conclusions8pressurecold working.of an attachment9911lug.2327.28-i-
ABSTRACTWhereas the higher-order versions of the finite element method (the 1 -andhp-versions) are fairly well established as highly efficient methods for monitoringand controllingthe discretizationerror in linear problems,little has been done toexploittheir benefitsin elasto-plasticstructuralanalysis.In this paper, we discusswhich aspectsof incrementalelasto-plasticfinite elementanalysisare particularamenableto improvementsby the p-version.Thesetheoreticalconsiderationsare supportedby several numericalexperiments.First, we study an exampleforwhich an analyticalsolutionis available.It is demonstratedthat the p-versionperformsvery well even in cyclesof elasto-plasticloadingand unloading,notonly as comparedto the traditionalh-versionbut also in respectto the exactsolution.Finally,an exampleof considerablepracticalimportance- the analysisof a cold-workedlug - is presentedwhich demonstrateshow the modellingtoolsofferedby higher-orderfinite elementtechniquescan contributeto an improvedapproximationof practicalproblems.ACKNOWLEDGEMENTThe researchwork of the first authorhas been supportedby the GermanResearch FoundationDFG under grant Ho1517/1-1.This supportis gratefullyacknowledged.Both authors acknowledgepartial supportby the LyndonB. JohnsonSpace Center of the NationalAeronauticsand Space Administrationunder GrantNAG 9-622.The research work of the secondauthor has been supportedin partby the Air Force Office of ScientificResearchunderGrant F49620-93-1-0173.The authorswould like to thank ProfessorB. A. Szab5 for the suggestiontowrite this paper and for many interestingremarksand discussions.Furthermore,we would like to thank Mr. S. Prost-Domaskyfor his assistancewith using lementmethod,error
1. ,non-linearin nature.problemsis ofreal-lifeApproximateof elasto-plasticity.for ceproblemsnumericalmethodsThe last two decadessolutionsinallare difficultare requiredhaveto elasto-plasticfieldsa greatby the finiteplasticity.see, seda lot of attentionissuethelastrefinementpolynomialof the errorFEMfor eerrorfor monitoringstudiedhavebeenof the h-versionon realisticin thewas based on theof the h-versionismaterialmodeling,includingin energynorm,thethoroughly,normand eloped.In theto elastomodels,thequestionsofremaindiscretizationand newmesherrorlargelyin finitefor an efficientp-versionrefinementof theandelementFor a surveymethodbasedclassincludeelementor evaluationgiven.elementelasto-plastichaveof theThefirstcodenamedcapabilitiesand provideselasto-plasticimplementationhas been employedand 6.problemsfor elasto-plasticTo the authors'incrementalmethodsof linearbeenefficiencyauthorofFEASIBLEof the state-ofthe highererror) thanlinearelasticity.stress analysis problemshas notknowledge,the first numericalmodelsin the frameofat that timenoin [7]. However,of thetheincludingmaterialreportedof theon the displacementis referredto [6]. By all measuresof performance,superioraccuracy(with respectto the discretizationfor a largeincreasingof the h-, p-, and hp-versionswell established.of the finiteThe performanceof the p-versioninvestigateduntil very entsofhp-versionof the p- and hp-versionsh-versionfocussedmethoddegree of the finite elementapproximationis increasedto improvesolution,rather than followingthe traditionalapproachof meshformulation,the readerorder methodsprovidetheirbeenin energy(h-version).degrees.The theoreticalbeenof numericaltechniqueshaveFEM, the polynomialthe accuracyof thehasconvergence[4], [5]).decade,approximationsminimizationon the applicationof the accuracyerrorsand(see, however,Duringelement[3] for detailspracticalmodelingAlmost all researchwork on elasto-plasticfinite elementanalysistraditionalh-versionof the finite element method. The theoreticalbasisin [2];they aredeal of research(FEM).For an excellentreviewof the techniquesof elasto-plasticframeworkof the finite element method,we refer to [1].explainedengineeringbecausefor solvingwitnessedproblemsofto solvep-versionforthispresentstudyhas(cf. [7])whichincludesmateriallaws as well as multi-stagefor the onanalysis.Thisin sections5
--2Recently, the applicabilityof the p-versionfor solving elasto-plasticproblems,usingthe deformationshowsthat,wheretheprocess,bothof venstressestheoriesamenablehas beenthe deformationremainyieldveryproportionalto eachsimilarto the deformationresultstheorythe presentpaperfocussesotherin manyrestrictedthroughoutcases.do not includeto casestheHowever,unloadingon the incrementalsee, e.g.,taskselasto-plasticfiniteand accuracyof the numericalof theperformance[1], [9]). The paperof the lastic.problemand cyclicelementsolutionsthe uerealisticelasto-to a lug whichthe approximatedemonstratesthan verythatof thecold-workingaccuracyp-versionof the FEM makeswell as reliable and robust.oftenproblemsrequirementsthe analysisfiniteexpansionusedcomputationalin whichswitchingto thein purelylinearof theloadingis presented.andForare then comparedthiswiththeWe demonstrateelementcodeproblems.Theis providedin practicedemandare imposed.of suchsub-thatwith the h-version.engineeringcold-worksolutionstudyand the h-version.p-versionof practicalwhenpressureresultsalso as comparedof theundergoesmanycrudeinternalanalyticalby the p-versionthe(for anthosewe discussp-versionA numericalunderThevery s.of plasticityWe identifySubsequently,of thecylindermodelingwhichanalysis.improvementsis yas follows:we wouldto elasto-plastica thick-walledan exactFinally,thatis organizedelementanalysisIt is showncarryunloadingotherin [8]. This yis, msandveryfiniteelementwithThis exampleelementFurthermore,illustratedcomparedby engineers.a finiteareanalysisit showsconvenientwhenthat theand easyas
--32. FiniteelementOncea specificidealizationin the formsolvingfiniteapproximationmaterialelementis to nfiniteminimizingissuetheerrorselementoverthe bi-linearformfieldF( )aandminimizingoverapproximately.an approximationwith a givenbyInaimtheof theworkmodelandcorrespondsof thepresentelementof thesolutionof engineeringindependentlyfiniteprimaryto the ementmethodthe functionalenergyfiniteand mechanicalthe problem of structuralanalysis is castThe finite elementmethodis a tool forof how well the selectedintroducedbasedissuepaper,of rmulationstartsbyenergyF( )(1)subsetthe domain, of the spacef and fulfill½ B( , ) denotesthe workI-[(a)onof the potentiala finite-dimensionalswainand the geometricalissue.I-I( ) ½ B( , )overas goodis to be treatedofthemodelingTheselectedproblemas can be ysismathematicalto thehas beenof the structurehas been determined,of a purely mathematicalproblemthis well-definedcomputermodeloverf2),appliedis obtained,of functionsthe displacementthe strainof the/ ( hconditions.to somean approximationthe exactsolutionhavefiniteHere,displacementuv (f2),uex minimizesl-I(a)g(f2).Consequently,the magnitudefor an assessmentof the discretizationthe numberof degreeserror in energynormdef/errorof freedomis definedof the efficiencyof a specificin the energyN usednormin the finitefiniteelementis monitoredelementscheme,as a ncreasingmesh.As far as linearversionfor problemsIn suchcases,asymptoticallyconvergenceThebyI1 11- qn( e In theofthe(2)p-versionof thethe polynomialanalysiswhichexponential(N--- oo)degreeis concerned,havean exactconvergencewithrate of the h-versionFEMsimpleis appliedin the formof the approximationthe p-versionsolutionratesfiniteis only algebraic(i.e.,uniformlyis distinctlythat is analyticcanmeshes,(i.e., [[glle cN 4 ).whereasp-on the entiresuperiorthroughoutII llelementof a uniformto the h-the domain.beachievedthebest
--4Evenin thefor problemsstresses,version.theResortingevenfor thesecanbe idingbecausethe higher-ordertwo categorieswhichmethodsfinitetheloads.Thefinalthe overallin thewiththesolutionisperformanceof the exactit is worthsolutionsnotingofinterface"insidethe elements.answers.thereis alwaysa continuousHowever,Even(directlyAll containedplasticmoreAs far as uncontainedfromtheof the solution,small-strainelasto-plastichow muchtheof theplastictoflowthefall intobasedmightforsuchbothelasticare strictlyh- d consequentlythep-versionsof minimumcontinuousto be at leastproblems.includingproblemson the principletheof completelythe underlyingtype,theevenbyof an abruptelasto-plasticitycan be expectedfrom purelyhyperbolicoverlooked.continuous.the categoryis concerned,in any of thekindcausethe strainsremainwhenrelevanceis still nd the like;analysisfact is oftenproperties,and reloading,the correspondingby methodsmaterialappropriatefor problemsintroducerelationshipon the strains)problemsthan thoseellipticmultiplicitybe solvedin thedependingunsmooththeis of littlematerialnotSuch a situationstress-strainflowbetterThis importantdoesupon unloadingand, consequently,andpropertiesis moresmoothnessit does not occur in small-strainchangeelasto-plasticpropertiesare wellshapeyet to inquiremethodofthat the ometricthe smoothnesssignificantlyabove.so thatof mstrainsidealfromhas f smoothnessFurthermore,not muchandof purelya priorihand,materialpurelyan ondeterminedon the othersolving"materialthe p-of so that the propertiesby the smoothnessof the exactbeof lems.andboundaryh-cf. [6].propertieswill be determinedthanN) by performingof linearizedsteps.as thebettera sequencematerialas fastsignificantlyfor practicalfunctionof the quasi-lineartwiceoccurrate can be recoveredover to elasto-plasticitywe solvesingularitiesconvergencein these cases,do not carrywhereat leastrates(i.e.,is a non-linearbut strain-dependentof the non-linearknownmeshplasticity,as the sum of thosethe stepwiserangeFI( )analysistheoryof berthe exponentialin the pre-asymptoticfor linearincrementala finiteconvergeson a geometricallythe FEMstrainsincludeto the hp-extension,In f thepotentialproblembifurcations,the limitsFEM.energy.ofTheyOf
5course,the physicalproblemfinally emergesinto unrestrictedplasticflow. However, theFEM does not directly model the physical problem, but is only a tool for solving amathematical problem. If unrestricted plastic flow is to be analyzed, a differentmathematical model .has to be used, not just a differenttype of finiteelementapproximation.Theaccuracyimportantof the strainsin elasto-plasticcomputedanalysisfromthanthe finitein purelyelementlinearsolutionproblemsstrain law itself dependson the strains. We can expect accurate resultsthe constitutivelaw is sufficientlyaccurate.Furthermore,errors tendprolongedincrementalcomputationwith many load steps.A recent study [ 10] investigatesdirectlyfrom the finite elementpoblems.Thiscomputation[10] identifiesvarietythetechniquethatof stressrequiredconcentrationFurthermore,to achievefactorssufficientthataccuracysmooth solutions.Such global accuracylevelsdue to the slow convergenceof this method.In summary,the pointwisequality of strainsof the displacementfieldwill(a smoothnessit is riterion)smallin anare impracticalthatthep-versioninvestigationand for a wideoutperformsin the energyof the pointwisestrains,evenfor an h-versionofthecomputedfor linearelasto-plasticthe p-versionerrorsmorestress-only if the input toto accumulatein ato determinethe currentmaterialproperties.The numericalvarious patternsof convergenceand shows that in all casesthe h-version.usuallyis exactlynumericallyapproximationis evenbecauseFEMnormarefor quiteapproachwillbebeneficialfor elasto-plasticcomputations,and there is no indicationthat the p-versionmight give rise to any new numericalproblemsnot encounteredin the h-version.Ournumericalexamplesstrongly support this.
63. NumericalaspectsIn the presentof J2 plasticitypaper,to problemsfrom metalJ2 flow theory,usingwe are dealingplasticity.the vonwith the applicationThe computationsMises yield law.displacementsand small strains. The incrementalstarts with the followingassumptions:The total strainincrementcan be decomposedof the p-versionof the FEMare based on the ideal lationof the von Misesinto a purelyelasticyieldand a purelylawplasticpart:dl dlzel deptWe assumethe engineeringThe yield criterionof the deviatoricstressJ2 denotesdefinitionis independenttensor:of the strainof the hydrostaticand of the third invariant(4)the secondof the material.Duringpressure- %.invariantof the deviatoricNo strainhardeninga plasticloadingincrement,denotesthestresstensoris assumed.are permissible.Once the stress path reachesin that point upon further loading.givenvector.2F stress(3)Only stressF 0 in a point,the stressandstate( y thestatesplasticis confineduniaxialsuchstrainsto theyieldthat F 0will developyieldsurfaceby eq. (4):In equationtranspose.(5),The directionof the plastic Here,aUsingHooke'sdstressincrementflow is givenvectorby (normalityandthesuperscriptrule)L- dJis a proportionalitylaw for the elastic&el D-lc/c,t indicates(6)factorand r denotesthestressesin vectorialnotation.part of the strains,(7)
--7whereDis the linear-elasticto determinedathe elasto-plastic D.,&correspondingmaterialthematerial tangential- F--F F--1OFstrainforof the tangentduringattentionde.and onfiniteelementespeciallyThisHowever,the mostrulesincompressibleplastic(8)in a itionof deformation(notehavethatconcretea directWithvolumetricschemeofon thelawof errorwhichis, however,a viewto oftheto the one presentedin [14].returnmethodhas beensmallaccuratestepsspecificallytheimpactof the constitutivesufficientlythatissuescheme.on the efficiencythis aspect,plasticthat it followsnohastechnique.providedto determineor a similarA comprehensivebe consideredelasto-plasticprogramintegrationto the 13],theon the integrationpoint levelis to be used for that. Whereasimplicit scheme similartraditionaltangent-radialchallengingis createdelementmethodhas to includemechancisdefineshas only an influencediscretizationwithin rocksimple[12],Misesyield criterionwithouthardeningintegrationof the constitutivelaw.theFormulato the accuratein the FEMpresent study favor a fully generalHowever,at present,onlytheimplemented.of the(cf. [11],assuranceof thematrixdedicatedthe last two decadesJ2 flow(8)a Newton-Raphsonstiffnesshas been(6) can be combined'is neededthe non-linearsolver, the integrationaccuracyof the final results.control/F tincrementmatrix(3), (5), andit can be used to integratethe constitutive.lawelement method in case an explicit algorithmthe correctnessMuchformulasincrementDep. This relationshipstiffnessFurthermore,of the finitestressD-to a totalmatrixmatrix,arerelatedstrainsforvontheto the usecorrespondform equationscomponent):for theusedofto an(4) and (6) thatStartingfromacompressibleelastic behavior,the materialbecomesprogressivelymore incompressibleduring the elasto-plasticdeformationprocess as the amountof plastic strain increasesascomparedto the elastictraditionalh-versionThe problemandvariousintegration"strain.This will eventuallyof the finiteof incompressibilityschemeshaveor by introducingToday,most approachesidea, entailingall themethods.elementlockingbeenmethodlead to a severeif the displacementin J2 plasticitydevisedan independentto avoidhas alreadytheapproximationuse the mixed finite elementdisadvantagesand herstudiedbyof the tin theis used.in [ 15],"reducedpressure.the latterto mixed
--8On the otherextensivelyhand,the problemfor the p-versionof lockingof the finiteOurnumericalexperimentsto incompressibilitymethoddemonstratein the p-versionhas(see [16], [17],when the polynomialhave to be introducedwilldoes not occurdueelementbeen shown that no locking effects occurNo special precautionsor mixed methodsincompressibility studiedand it hasdegree p is greaterto obtain accurateclearlyanalysisbeen[18]),thatthan 4.results.lockingof the elastoplasticduetoproblem.4. ImplementationFEMIn many details, the implementationof an elasto-plasticcode is identicalto a standardh-versionimplementation.only brieflyFirst,the main featureswe carrystartingavailableout a purelythat the errorof the exactenergyThroughoutthe presentTo avoidtheWhentheerrorstheerrorby the plasticin energyestimatereasonableTherefore,accuracyfor a purelythelesstensora verydifferentproductp-extension,degreeon thereliableestimatep-levels(cf. [6]).trialpurelylinearTherefunctionssolutionof curvedsolutionhasuser-definedtheseenverye.g.,will be introducedof the errorrestrictiveas a rule10% of whatexpectthat,the convergencecomputations,non-linearbut we ratherin the systemto imposeto aboutathatalreadydownwe esentwill slowreasonablemapping,boundariesis no guaranteeWeIn our practicalof the linearcurvedas the linearaccurate.effectsit seemsgeometricalin the presenceare not originallysolution.elasticevenstarts.law, and thoseon the linearof these,threeto describeof accuracybewhich[6])elementsorderaccuracyof thumb,we wouldweconsideranalysis.The next step is to determinewill start to yield. The remainingeachfromby approximate(seefortoeffectsmaterialset the rm.requirementsa uniformThe maximumpolynomialenergyis monitored.Basedof hierarchicallargewill be of the sameincompressibilityconvergesby extrapolationmethodthe non-linearnonlinearnormand performthe spacein the linearwe can use onTherefore,solutionin energystudy,elasticused for the presentapproximation).of the straincan be obtainedused (productuselinearat p l(piecewiselinearis p 8. The convergenceassumptionbeenof the implementationmateriallaw in a p-versionTherefore,we summarizethe Newton-Raphsonat which fractionof the total imposedload the systemload is divided into a number of load increments,and inmethodwith a consistenttangentpredictoris applied.The Newton-Raphsoniterationsare stopped when both the relative magnitudeof residualforce vector and of the change in the incrementaldisplacementsin either the Euclideanorthemaximumcheckfor elasticnormare smallerunloadingthanis performeda prescribedin everythreshold.integrationIn eachpoint.loadincrement,a
"9Finally, stressresultsareavailableonly in the integrationpointssincethe incrementalelasto-plasticlaw has been evaluatedonly there. To obtain stressresults elsewhere,aleast-squaresapproximationfor eachof the componentsof the stressvector is computedfor eachelementindividually. For this least-squaresfit, we employ the trial functionscorrespondingto the polynomial level usedin the finite elementapproximationof thedisplacements.5. The thick-walledFirst,walledtubewe studytubeunloadingunderunderan exampleinternal- includinginternalfor whichpressure.elasto-plasticpressurean analyticalTheanalyticalre-loadingsolutionis available,thethick-solutionfor boththe loadingandin the reversedirection- has beengiventhein[19]. It is one of the few elasto-plastictwo-dimensionalproblemsfor which an analyticalsolutioneven for the unloadingis availabe,and thereforewe use it as a basis for theevaluationof the numericalHowever,tube mayRichqualitythis problembe usedet al. [20]is also of considerableas a modelsuggestof the h- and p-versionspracticalfor the cold-workingto use the analyticaloversizedholeand taperedmandrel,to be cold-worked.regionof residualsolutionUponstresses.5.1 AnalyticalTheas wellexample,the pressurizedengineering.of a widewe will studysleeve,of the mandrel,Thismethodas duringsolutionfor the problemrangehow wellservice,is pulledthe holeis generallyof the tubegiven in [19] and applied to the cold-workingare adopted in the analysis: elastic-perfectlyplasticplane strain situation von Mises incompressibilityexistsfor the analysismethod.ofthisthroughis surroundedusedto increasefor holesthefatiguetheby aat essureprocessby [20]. The followingplasticzoneshas beenassumptionsmodelyield conditionof bothfor compressibleIn the applicationsincee.g., in aircraftwith a lubricatedthe removaltime of aircraft manufacture,structuralcomponents.elementis usedto producefavorablestressesthatconcentrationaround a hole. In this method,ansometimescompressiveinterestof holes,cold-workedholes, and in our practical applicationsolution is suited for a cold-workedlug.Cold-workingis a processwhichsignificantlyreduce the effects of stressof the finitethe elasticelasticto cold-working,and(no analyticalexactdeformation).the mandrelis assumedto remainpurelyelastic.solution
--lONotation.Considera tube with an renceI ldstress6y.ratiobyandby E b, its Poissonthe radii of the mandrelthe mandrela and an extemalvbDenoteand the hole is calledinto the tube,a plasticzonetheits radiusb, Young'smodulusby a I.existsp is created.the mandrel(lY[21n-91- -qt' [a 2(Yy,b- m ,thenthe entireThis(9)and the hole can be shownto be 2]P ff P betweenThefor the equation, and the pressureofinterference.of radiusradius is defined by the followingimplicit equation(if no solutionthe interferenceis simply too small to create a plastic zone):radius(10) cylinderbecomesplastic,andthis caseis of no interestfor us.The circumferentialGo stressesaroundthe hole are givenair pP21nr l [(11)O'yUponremoval2Gy [are givenby:a2by:p r b.oftheno reversemandrel,yieldingthetubeoccurs,undergoesand the If
--1121np l -I 0-21nP- aaSrSp1- -Jr2(b2-a2)J(12)----- Y {[ - .Pff- ]-[21nP l-- 2]a2(r2 b2)02.aOtherwise,compressiveimplicit equation:yielding21n- -a - 1 PRIn this case,poccursp rSb.r2(b2-a2)Jin the zonea r OR, whereOR is givenby the- O.(13)2Cythe circumferentialstressis:oy{[21o2 1 ]21n r 1 P 'l[04b2JJ a r -Pnp2(Y0 (LLb2 r jjp Jf l4 LLThisproblembecausearound5.2 NumericalTheWe study7jjp r b.anddeformationtheoriesof plasticityare proportional.is not correctwhichthe hole after unloading.describedof the FEM.the capabilitiesComparison2rendersThe correctthe stressessolutionin the compressiveis foundin [19].Examplesproblemthe h-version22r2 ]and the stressesthat eq. (6) in [20]zonep. .llfor both the incrementalit is axisymmetricNoteplasticis suitableb2(14)p. r pin the previousThe finiteof the p-versionsolutiona tube withan internalprocess.is solvedsolutionsby the p-versionto this problemwillas welldemonstrateand its accuracy.with the exactto a cold-workingsectionelementA planeradiusstraina l.5insituationand an outeris assumed.radiusA pressureb 4.0insubjectedof P 62,000psias
--12is appliedto bothdeterminetheinnersidethe correspondingboth bodies.Thus,of theinterferencean interferencedefinedby Ep 10,000,000psi,We select this value of sare takenare baseddisplacementsdifferentplasticonlyare constantalongmeshesonly in one simplehaveonep-versionof themandrel.Wedisplacementspropertiesif v--0.5von Misesquarteryieldof theThe materialmaterial.exactly.TheAll finiteofof theandexactdisplacement-The nalyzed.For each of the two versionsTheimposingDfaceas the sum of the radial, and Vb 0.3.analyzed.mesh,outerincompressiblethe perimeter.beentheis obtained.be appliedto be Eb 30,000,000psisymmetry,andthe uniaxialyield stressgy 58,000psi,Poissonratio in order to be close to theelasticallycannoton an 0.03399intube are"Vp 0.49.basedtubepurelyelasticextremelymandrelhigh accuracyTheradialof the FEM,hasbeenstudiedrequirements.CV yFigure1: MeshesIn Fig.uniform1, themeshof the tube.tn -Pused in the p-versionon A-D,boundarywherebiggestrespectively.elementfor theconditionsof the tube undercylinderand a gradedt,, denotesThe p-levelis decreasingFor the h-version,elements,usedof two elementsThetraction-free.energy normmeshesanalysisin themeshwe use hown.boundaryelementsradialA meshWeuseafor the representationuntil the estimatedin thethan the smallest.pressure.us 0 on edgeThe otherof 70 8-nodedgradedareof six elementsis increasedon each elementbelow 0.1%.was ten timesp-versionare Uy 0 on edgethe balandandareerrorin140 8-nodedsuchthattheof 300 8-noded
13elements was also analyzed to make sure thatthe mesh of 140 elements was fine enoughin the circumferentialdirection.The three differenth-versionmeshes arc shown in Fig. 2.The resultsobtained with the 300-element meshare practicallyidenticalto those of the140-element mesh, so only the lattermcsh isused in our cxaml les.70 elements140 elements300 clcm.1:;igur 2: Meshes used in the h-versionanalysisof the tube under internalpressureFurthermore, wealso analyzed the meshwith 70 elements, employing9-nodedelements instead of the 8-noded elements. The trialfunctionspace associated with the 9noded elements corresponds to theproduct space atp 2 in the p-version.The h-version computations were carried out with the commercialsolver ADINA[21]. The ADINAfiniteelementsystem offersa specialpressure-displacement element
14for incompressibleelementmaterials,basedon the mixedfiniteelementformulation[22].Tiffswas used in the analysis.5400044000u)Q.u340O0U) - --24000E-n(.1140004OOO1.5FigureFEM2.03: CircumferentialanalysisIn Fig.loadingnoded--e p-Version 2 elements. [" EI---U p-Version 6 elements.iZ ---AAnalytJcaJ sol.Istressand exact3, the2.53.0Radial Distance from Hole Center.in pressurizedtube,loadinganalyticalcase is comparedelementsand thewithexactsolutionof theelastic-plasticinterface.determinedfromthe h-versionHowever,elementswiththethe p-versionsolutionarecorrespondingthe p-versionThe p-version8-nodedTheseresultsResultsof p-versionsolution.displayedstress(c 0)fortheThe h-versionsolutionwith 8in Fig. 4. Note the very goodin the whole range, exceptthis location,the h-versionthe qualityof the resultsissolutionis much smootherand followsthe exactmerely producessmoothedstress results close to theIt is clearlythe ementbetweentile exact and the finite elementsolutionsat the interfacebetweenthe elasticand plasticzones. Atsolutiouswith the 8-nodedelementstend to ionable,solution more3.5(in)visiblesolutions,thatthe elasto-plasticbut not so wellfromconsiderablywhenradiusthe(p) can eselementshavearc displayedan additional"internalin Fig. 5. No oscillationswemode"employtrialare presentthe9-nodedfunction.anymore,Theeven
15in the coarser mesh. The computer time for this analysis is somewhat greamr than for the8-noded elements. It is interesting that, even in the h-version, obviously some degree of"p-extension"and switching to a different space of trial functions is required to capturethe solution of this problem properly.54oo0!44000 Je 1tgl340O0w /}me- DE240o0O( --eB---B1400040001.52.02.5Radial DistanceAdina- 70 elements.Adina - 140 elements.Analytical sol.3.03.54.0from Hole Center. (in)Figure 4: Circumferentialstress in pressurizedtube, loadingFEM analysis with 8-noded elements and exact solution.case. Resultsof h-version
63OOO020OO01OOOO!-I 3.03.54.0Radial Distance from Hole Center. (in)HgureFEM5: CircumferentialanalysisUpon removingthe last stage
framework of the finite element method, we refer to [1]. Almost all research work on elasto-plastic finite element analysis was based on the traditional h-version of the finite element method. The theoretical basis of the h-version is explained i
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