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1、Chapter7ErrAnalysisTherearevarioussourcesoferrsinfiniteelementanalysis.Erranalysishasbecomeindispensableintoday’sengineeringapplicationoffiniteelementsoftware.Inthischapterwewilldiscussthematurebasisoferranalysisflinears

2、taticanalysisusingfiniteelements.1SourcesoferrTheerroffiniteelementresultsmayariseatanyproceduresoffiniteelementanalysis.Agoodmatureuseroffiniteelementmethodshouldbeawareoftheallpossibleerrsourcesinthefiniteelementanalys

3、is.Generallypossiblemajsourcesoferrcanbeputintothefollowingcategies1)ModelingerrItreferstothedifferencebetweenaphysicalsystemitsmathematicalmodel.Itshouldbekeptatanacceptabledegree.Otherwisethemathematicalmodelneedberevi

4、sed.Usuallysimplificationsareintroducedintothemathematicalmodel.Inthemodelfinstanceprobablysmallholesothergeometricirregularitiesinthestructureareneglectedloadsaresimplifiedboundaryconditionsareidealizedasbytreatingacomp

5、onentwithrelativelylargestiffnessasrigid.Quiteoftenatwodimensionalproblemisstudiedigningitsthreedimensionalacteristicastaticanalysisisperfmedigningthedynamicacter.2)DiscretizationerrThemathematicalmodelhasinfinitenumbero

6、fdofbutfinitedofareusedinfiniteelementanalysis.Thefiniteelementsolutionisaffectedbythenumberofelementsinthemodelthedofpernodethenumericalintegrationruleetc.Actuallythiskindoferrhasbeendiscussedatthefirstsectionoflastchap

7、ter.3)TruncationerrroundingerrItreferstolossofinfmationduetotruncationroundingofnumberstofitafinitecomputerwdlength.4)AccumulationerrThiskindoferrariseswhenglobalequationsaresolvedrepeatedlyasinnonlineardynamicproblems.T

8、hetermnumericalerristhecombinedresultoftruncation(rounding)erraccumulationerr.Inthischapteremphasisismainlyplacedondiscretizationerrnumericalerr.Theaboveclassificationexcludespossibleusererrbugs.Usererrreferstomistakesma

9、deusersafteraneffectivemodelhasbeenestablished.Bugsareprogrammingerrsembeddedinfiniteelementanalysissoftware.Someofthemaredebuggedduringthepreprocessingoffinite223smallchangeofoccurs.Physicallythisimpliesthatleftsegmenth

10、asvirtuallynoconstraintsontherightsegment.Analertshouldbesoundedinthesolutionprocessindicatingthatthecoefficientmatrixisalmostsingular.1kIntersectionregion(6.2b)(6.2a)u1u2(a)Wellconditioned()12kk?Intersectionregion(6.2b)

11、(6.2a)u1u2(b)Illconditioned()21kk?Fig.7.2ExamplefmatrixconditionAccdingtotheclassificationinthelastsectionthisisribedtotruncationerr.Mathematicallyithastodowiththeconditionnumberofthecoefficientmatrix.Theillconditionedsi

12、tuationintheaboveexamplecanbesummarizedasthecasewhereastiffmemberisconnectedtoamuchmeflexiblemember.AfurtherplaneframeexampleisgiveninFig.7.3.Theaxialstiffnessofthehizontalmemberismuchlargerthanthesidewaystiffnessofthetw

13、ocolumns.Asaresultthecomputedvaluesofthehizontaldisplacementsatthetworigidjointsmaylacksufficientaccuracysincetheybotharesmallquantities.Itwouldbeevenwseifweusethedifferenceofthetwosmallquantitiestocomputetheaxialstressi

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