1 Finite Element Analysis Methods - Rice University Finite Element Analysis Methods ... it is often referred to as finite element analysis (FEA). FEA is the most common tool for stress

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  • Draft12.0.Copyright2009.Allrightsreserved. 7

    1 FiniteElementAnalysisMethods

    1.1 IntroductionThefiniteelementmethod(FEM)rapidlygrewasthemostusefulnumericalanalysistoolforengineersandappliedmathematiciansbecauseofitnaturalbenefitsoverpriorapproaches.Themainadvantagesarethatitcanbeappliedtoarbitraryshapesinanynumberofdimensions.Theshapecanbemadeofanynumberofmaterials.Thematerialpropertiescanbenonhomogeneous(dependonlocation)and/oranisotropic(dependondirection).Thewaythattheshapeissupported(alsocalledfixturesorrestraints)canbequitegeneral,ascantheappliedsources(forces,pressures,heatflux,etc.).TheFEMprovidesastandardprocessforconvertinggoverningenergyprinciplesorgoverningdifferentialequationsintoasystemofmatrixequationstobesolvedforanapproximatesolution.Forlinearproblemssuchsolutionscanbeveryaccurateandquicklyobtained.Havingobtainedanapproximatesolution,theFEMprovidesadditionalstandardproceduresforfollowupcalculations(postprocessing),suchasdeterminingtheintegralofthesolution,oritsderivativesatvariouspointsintheshape.Thepostprocessingalsoyieldsimpressivecolordisplays,orgraphs,ofthesolutionanditsrelatedinformation.Today,asecondpostprocessingoftherecoveredderivativescanyielderrorestimatesthatshowwherethestudyneedsimprovement.Indeed,adaptiveproceduresallowautomaticcorrectionsandresolutionstoreachauserspecifiedlevelofaccuracy.However,veryaccurateandprettysolutionsofmodelsthatarebasedonerrorsorincorrectassumptionsarestillwrong.

    WhentheFEMisappliedtoaspecificfieldofanalysis(likestressanalysis,thermalanalysis,orvibrationanalysis)itisoftenreferredtoasfiniteelementanalysis(FEA).FEAisthemostcommontoolforstressandstructuralanalysis.Variousfieldsofstudyareoftenrelated.Forexample,distributionsofnonuniformtemperaturesinducenonobviousloadingconditionsonsolidstructuralmembers.Thus,itiscommontoconductathermalFEAtoobtaintemperatureresultsthatinturnbecomeinputdataforastressFEA.FEAcanalsoreceiveinputdatafromothertoolslikemotion(kinetics)analysissystemsandcomputationfluiddynamic(CFD)systems.

    1.2 BasicIntegralFormulationsThebasicconceptbehindtheFEMistoreplaceanycomplexshapewiththeunion(orsummation)ofalargenumberofverysimpleshapes(liketriangles)thatarecombinedtocorrectlymodeltheoriginalpart.Thesmallersimplershapesarecalledfiniteelementsbecauseeachoneoccupiesasmallbutfinitesubdomainoftheoriginalpart.Theycontrasttotheinfinitesimallysmallordifferentialelementsusedforcenturiestoderivedifferentialequations.Togiveaverysimpleexampleofthisdividingandsummingprocess,considercalculatingtheareaofthearbitraryshapeshowninFigure11(left).Ifyouknewtheequationsoftheboundingcurvesyou,intheory,couldintegratethemtoobtaintheenclosedarea.Alternatively,youcouldsplittheareaintoanenclosedsetoftriangles(covertheshapewithamesh)andsumtheareasoftheindividualtriangles:

    .

    Now,youhavesomechoicesforthetypeoftriangles.Youcouldpickstraightsided(linear)triangles,orquadratictriangles(withedgesthatareparabolas),orcubictriangles,etc.Theareaofastraightsidedtriangleisasimplealgebraicexpression.Theareaofacurvedtriangleisrelativelyeasytocalculatebynumericalintegration,butiscomputationallymoreexpensivetoobtainthanthatforthelineartriangle.ThefirsttwotrianglemeshchoicesareshowninFigure11foralargeelementsize.Clearly,thesimplestraightsidedtriangularmesh(ontheleft)approximatestheareaveryclosely,butatthesametimeintroducesgeometricerrorsalongtheboundary.Theboundarygeometricerrorinalineartrianglemeshresultsfromreplacinga

  • FEAConcepts:SWSimulationOverview J.E.Akin

    Draft12.0.Copyright2009.Allrightsreserved. 8

    boundarycurvebyaseriesofstraightlinesegments.Thatgeometricboundaryerrorcanbereducedtoanydesiredlevelbyincreasingthenumberoflineartriangles.Butthatdecisionincreasesthenumberofcalculationsandmakesyoutradeoffgeometricaccuracyversusthetotalnumberofrequiredareacalculationsandsummations.

    Areaisascalar,soitmakessensetobeabletosimplysumitspartstodeterminethetotalvalue,asshownabove.Othertopics,likekineticenergyorstrainenergy,canbesummedinthesamefashion.Indeed,theveryfirstapplicationsofFEAtostructureswasbasedonminimizingtheenergystoredisalinearelasticmaterial.TheFEMalwaysinvolvessometypeofgoverningintegralstatement.Thatintegrationisalsoconvertedtothesumoftheintegralsovereachelementinthemesh.Evenifyoustartwithagoverningdifferentialequation,itgetsconvertedtoanequivalentintegralformulationbyoneofthemethodsofweightedresiduals(MWR).Thetwomostcommonmethods,forFEA,aretheGalerkinMethodandtheMethodofLeastSquaresFigure.

    Figure11Anareacrudelymeshedwithlinearandquadratictriangles

    Youmaythinkthatthegeometricboundaryerrorcitedforthelineartrianglesiseliminatedbychoosingtousethemeshofcurvedquadratictriangles(ontheright).Theparabolasegmentspassthroughthreepointslyingexactlyontheboundarycurve,butcandegeneratetostraightlinesintheinterior.(Tospeedplottingofsmallelements,mostsystemsdrawalltheparabolasastwostraightlinesegments,asontherightinFigure11.)Thus,theboundaryshapeerrorisindeedreduced,attheexpenseofmorecomplicatedareacalculations,butitisnoteliminated.Somegeometricerrorremainsbecausemostengineeringcurvesarecirculararcs,splines,ornurbs(nonuniformrationalBsplines)andthusarenotmatchedbyaparabola.Themostcommonwaytoreducemeshgeometricerroristosimplyusesmallerelements,likeFigure12shows.ThedefaultelementchoiceinSWSimulationisthequadraticelement.Othersystemsofferawiderrangeofedgepolynomialdegree(e.g.cubic),aswellasothershapeslikequadrilateralsorrectangles.Inthreedimensionalsolidapplicationssomesystemsofferdozensofchoicesfortheedgedegreepolynomialorder,andshapesincludinghexahedral,wedges,andtetrahedralelements.Hexahedralelementsaregenerallymoreaccurate,butcanbemorechallengingtomeshautomatically.Tetrahedralelementscanmatchhexahedralelementperformancebyusingmore(smaller)elements,andtetrahedralelementsaremucheasiertomeshautomatically.SWSimulationusesonlytetrahedralelementsforsolidstudies.

    AnexampleofthesmalltwodimensionalgeometricboundaryerrorduetodifferentcurvedshapesisseeninFigure13whereacirculararcandaparabolapassthroughthesamethreepoints.(Anewmethod,calledisogeometricanalysis,canessentiallyeliminateallgeometricerrors,butitintroducesnewapproximationsinotherstudystages,suchasintherestraintconditions.)

  • FEAConcepts:SWSimulationOverview J.E.Akin

    Draft12.0.Copyright2009.Allrightsreserved. 9

    Figure12Meshrefinementquicklyreducesgeometricboundaryerrorsforlinear(left)orquadraticelements

    Figure13Linearorparabolicelementsneverexactlymatchcircularshapes

    1.3 StagesofAnalysisandTheirUncertaintiesAFEAalwaysinvolvesanumberofuncertaintiesthatimpacttheaccuracyorreliabilityofeachstageofaFEAanditsresults.Thebook,BuildingBetterProductswithFiniteElementAnalysisbyAdamsandAskenazi[1]givesanoutstandingdetaileddescriptionofmostoftherealworlduncertaintiesassociatedwithsolidmechanicsFEA.Allengineersconductingstressstudiesshouldreadit.ThatbookalsopointsouthowpoorsolidmodelingskillscanadverselyaffecttheabilitytoconstructmeshesforanytypeofFEA.Here,themostimportantFEAuncertaintiesarehighlighted.

    ThetypicalstagesofaFEAstudyarelistedbelow:

    1. Constructthepart(s)inasolidmodeler.Itissurprisinglyeasytoaccidentallybuildflawedmodelswithtinylines,tinysurfacesortinyinteriorvoids.Thepartwilllookfine,exceptwithextremezooms,butitmayfailtomesh.MostsystemshavecheckingroutinesthatcanfindandrepairsuchproblemsbeforeyoumoveontoaFEAstudy.Sometimesyoumayhavetoexportapart,andthenimportitbackwithanewnamebecauseimportedpartsareusuallysubjectedtomoretimeconsumingchecksthannativeparts.Whenmultiplepartsformanassembly,alwaysmeshandstudytheindividualpartsbeforestudyingtheassembly.Trytoplanaheadandintroducesplitlinesintotheparttoaidinmating

  • FEAConcepts:SWSimulationOverview J.E.Akin

    Draft12.0.Copyright2009.Allrightsreserved. 10

    assembliesandtolocateloadregionsandrestraint(orfixtureorsupport)regions.Today,constructionofapartisprobablythemostreliablestageofanystudy.

    2. Defeaturethesolidpartmodelformeshing.Thesolidpartmaycontainfeatures,likearaisedlogo,thatarenotnecessarytomanufacturethepart,orrequiredforanaccurateanalysisstudy.Theycanbeomittedfromthesolidusedintheanalysisstudy.Thatisarelativeeasyoperationsupportedbymostsolidmodelers(suchasthesuppressoptioninSW)tohelpmakesmallerandfastermeshes.However,ithasthepotentialforintroducingserious,ifnotfatal,errorsinafollowingengineeringstudy.Thisisareliablemodelingprocess,butitsapplicationrequiresengineeringjudgment.Forexample,removingsmallradiusinteriorfilletscangreatlyreducesthenumberofelementsandsimplifiesthemeshgeneration.But,thatcreatessharpreentrantcornersthatcanyieldfalseinfinitestresses.Thosefalsehighstressregionsmaycauseyoutooverlookotherareasoftruehighstresslevels.Smallholesleadtomanysmallelements(andlongruntimes).Theyalsocausestressconcentrationsthatraisethelocalstresslevelsbyafactorofthree.Thedecisiontodefeaturethemdependsonwheretheyarelocatedinthepart.Iftheylieinahighstressregionyoumustkeepthem.Butdefeaturingthemisallowedifyouknowtheyoccurinalowstressregion.Suchdecisionsarecomplicatedbecausemostpartshavemultiplepossibleloadingconditions(loadcases)andalowstressregionforoneloadcasemaybecomeahighstressregionforanotherloadcase.

    3. Combinemultiplepartsintoanassembly.Again,thisiswellautomatedandreliablefromthegeometricpointofviewandassemblieslookasexpected.However,geometricmatingofpartinterfacesisverydifferentfordefiningtheirphysical(displacement,ortemperature)mating.Thephysicalmatingchoicesareoftenunclearandtheengineermayhavetomakearangeofassumptions,studyeach,anddeterminetheworstcaseresult.Havingtousephysicalcontactsmakesthelinearproblemrequireiterativesolutionsthattakealongtimetorunandmightfailtoconverge.

    4. Selecttheelementtype.SomeFEAsystemshaveahugenumberofavailableelementtypes(withunderlyingtheoreticalrestrictions).TheSolidWorkssystemhasonlythefundamentaltypesofelements.Namely,trusselements(bars),frameelements(beams),thinshells(orflatplates),thickshells,andsolids.Thesystemselectstheelementtype(beginningin2009)basedont

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