ORI GI NALPAPERStructuralDynamicModelUpdatingTechniques:AStateoftheArtReviewShankarSehgal1HarmeshKumar1Received:21March2015 / Accepted:27March2015CIMNE, Barcelona, Spain2015Abstract This paper presents a review of structuraldynamic model updatingtechniques. Starting witha tu-torial introductionof basic concepts of model updating,thepaperreviewsdirectanditerativetechniquesofmodelupdatingalongwiththeirapplicationstoreallifesystems.The main objective of this paper is toreviewthe mostwidelyappliedmodel updatingtechniques sothat begin-nersaswellaspractisingengineerscanappreciate,chooseandthen utilize the most suitable model updating tech-niquefor their customizedapplication. Another objectiveistohighlight thecurrent issues, applications andobser-vations for further advancements in the eld of modelupdating.1 IntroductionUse of thin and light-weight products in modern daymachines and structures is increasing day by day.Therefore better dynamic testing and analysis tools arebecoming the urgent need of hour. In the automotive,aircraft andspaceshipengines, there is anever existingdemandof attainingbetter fuel economy; whichcanbemet toa goodextent byusingthinproducts as well aswiththeuseoflight weight materialssuchasaluminiumandplasticscompositesinsteadoftheconventionallyusedheavyweight materialssuchassteels. Particularlyinthecaseofsatellites, somepartsaresothinthat theycangetcollapsedjust duetotheirownweight iftestedundertheeffect ofgravity. Thinandlight weight productshavelotmore tendencies to vibrate than their thick and heavyweight counterparts. Excessivevibrationscanevenresultinpre-maturefailureofproducts, forexample, whetheritis the suspensionof anautomobile, wingof anaircraft,the printed-circuit-board installed in a spaceship, bladesof anair-cooler, or thecompact-discof acomputer etc.Ontheotherhand, consumersoftodaysworlddesirefornon-vibrating and silent functioning of such products.Thus it becomes veryimportant for engineers tounder-stand the vibration behavior of structures through theirdynamicanalysis. Dynamicanalysis aims at understand-ing, evaluating, analyzingandmodifying(ifrequired)thestructural dynamicbehavior whichcanberepresentedbymany terms such as natural frequencies, eigenvalues,eigenvectors, dampingratios, FrequencyResponseFunc-tions (FRFs) etc. Thedynamicanalysisof structurescanbe done through either experimental route or by usingtheoretical approach[77, 33].Thetheoreticalrouteinvolvestheformationofanana-lyticalmodelofthesystemeitherusingaclassicalmethod[46] or through Finite Element (FE) method [91]. Theapplication of classical method is generally limited tosimple systems only, whileFE method is preferred for reallifecomplexsystems. HoweverFEmethodisnot abletopredictthedynamicresponsesofstructureswithcompleteaccuracyduetothepresenceof certainerrors intheFEmodel. Sucherrors are inherent inanFEmodel due tofollowingreasons:1. Faultyboundaryconditions2. Incorrectvaluesofmaterialproperties3. Discretizationofcontinuumorapoorqualitymesh4. Difcultyinmodelingcomplexreallifeshapes&HarmeshKumarharmesh@pu.ac.in1DepartmentofMechanicalEngineering, UniversityInstituteofEngineeringandTechnology, PanjabUniversity,Chandigarh160014,India1 3ArchComputatMethodsEngDOI10.1007/s11831-015-9150-35. Assumptions for simplicationpurposesuchas con-sidering damping on a linear basis instead of non-linear6. Incorrectmodelingofjoints7. UseofroundingoffmethodsincomputationsThusthereisaneedtocorrect anFEmodelsothat itsvibration behavior matches with the actual dynamic re-sponse obtainedexperimentallyas showninFig.1. Theprocedure usedtoupdate the model is calledFEmodelupdating (FEMU) [39]. In FEMU techniques, the ex-perimental andFEresponsesarerst comparedandcor-related so as to ensure that the FE model underconsideration qualies for further updating procedure and aconceptuallynewFEmodel isnot required. Anumberofgraphicalaswellasnumericalcomparisonandcorrelationtechniquesareusedforthispurpose. Furthertoavoidtheincompatibilityinthesizes of experimental andFEdatasetstheuseofsizecompatibilitytechniquesisrequired.InFEMU, experimental resultsareconsideredastargetsandtheinputsofFEmodelareadjustedinsuchawaythattheoutputs of FEmodel haveabetter matchwiththeir ex-perimentalcounterparts.Anumber of researchers have proposed the variousstructuralmodelupdatingtechniquesbothbyclassicalap-proach and FE route. While going through the available lit-erature on this area, a need is felt to summarize all the resultsandconclusionsmadebydifferentresearchers. Therefore,this paper is an attempt to provide a reviewof major researchactivities carried out in the model updating eld.Inthispaper,rstthebasictheoreticalissuesrelatedtoFEMU procedure have been discussed. After which, a stateof the art reviewof direct and iterative techniques ofFEMUhas been presented. Application areas of FEMUhavealsobeendiscussed. Thenal part ofthepaperdis-cusses the current problems and future directions forFEMUrelatedresearch.2 FiniteElementModelUpdating(FEMU)Procedure2.1 FiniteElementMethodInFEmethod, acomplexcontinuousregionofastructureis discretized into simple geometric shapes called niteelements. Figure2 shows a cantilever beamof length910mm, width 50mm and thickness 5mm. The cantileverbeamisdividedinto60niteelements. Eachelementhastwo nodes. At each node, two degrees of freedomaremeasured, out of whichoneis thedisplacement iny-di-rectionandtheotheristherotationaboutz-axis.Boththedegrees of freedom of node number 1 are fullyconstrained.Theniteelements canbeaxial elements, torqueele-ments, beambending elements, thin plate bending ele-ments, thick plate bending elements, etc. Approximateresults of displacement shapes and stress elds can beobtainedfor such nite elements usingshape functions.Continuityacross element boundaries canbemaintainedusingeither displacement or energyapproaches. Thedis-placement approach makes use of equilibrium, com-patibility and the constitutive laws; while the energyapproachisbasedontheprincipal ofvirtual work. Inen-ergy approach, the internal work is equated to externalwork. For dynamic analysis purpose, each element needs tobe expressed in formof elemental mass, stiffness anddamping matrices. Equations(1) and (2) represent theelementalmassandstiffnessmatricesforabeamelement.me qAa1057822a2713a22a8a213a6a22713a7822a136a222a8a2266437751ke EI2a333a33a3a4a23a2a233a33a3a2a23a4a2266437752whereme, q,A,a,ke,EandIarerespectivelytheelementmass matrix, density, area of cross-section of element, half-length of element, element stiffness matrix, Youngsmodulusofelasticityandmoment ofinertiaofcross-sec-tion of beamelement. Subsequently, the individual ele-ments are assembled to formtheir global counterparts,whichare alsojointlyknownas systemmatrices. Thesesystem matrices along with certain boundary conditions areusedtoformulateasetofgoverningequations, whicharethenprocessedonacomputer toevaluatedynamicchar-acteristicsofthesystem.Fromabove discussion it can be concluded that thedynamicresponseofthestructuredependsuponanumberFig.1 Comparison of experimental, initial analytical andupdatedanalyticalresultsS.Sehgal, H.Kumar1 3of parameters. These parameters are called input pa-rameters whichaffect the dynamic response parameters.These input parameters may be belonging to material,structural, niteelement orcomputational categories. Thevarious input parameters are assembled in the formofcauseandeffectdiagramandareshowninFig. 3.Dynamic response is generally measured in terms ofeigenvalues, eigenvectors, and FRFs. The frequency atwhich a structure naturally vibrates once it is set intomotion is called its natural frequency. Square of the naturalfrequencyis knownas eigenvalue. Eigenvector or modeshapeof aparticular modedenesthedisplacement con-gurationofstructureat correspondingnatural frequency.Firstsixmodeshapesofacantileverbeamstructureareasdrawn in Fig.4. Extreme and intermediate positions ofdifferentpointsofacantileverbeamaredrawninFig. 5.FRF(i,j)isdenedasratio ofharmonicresponseatlo-cation i and the harmonic excitation force at location j. Ifthemeasuredresponseis intheformdisplacement, thencorresponding FRF is called as receptance (or admittance ordynamiccomplianceor dynamicexibility) FRF. Other-wise, velocity or acceleration response signals can be used toproducemobilityoracceleranceFRFrespectively. Accel-eranceFRFis alsosometimes writtenas inertanceFRF.FurtheranyFRFiscalledasapoint ortransient FRFde-pendingupon whether the response and force are measured atsame or different locations respectively. Point and transientReceptance FRF have been drawn in Fig.6a, b respectively.2.2 ExperimentalMethodInFEMU, theexperimental results canbeobtained by twoways: simulated experimental results and real lifeexperimental results. Formertypeofresultsaregeneratedby varying some parameters of the FE model therebyproducingaperturbedFEmodel,whosesimulationresultsare assumed as experimental results and are surely differentfromthedynamicresultsof original FEmodel. Real lifeexperimental results can be generated through modaltesting.Arepresentativeexperimental set-upforniteelementmodelupdatingofastructureisshowninFig.7.Oneendof thetest structurecanbexedtoanon-vibratingbasethroughwelding, boltingorrivetingetc. therebyresultingingroundedsupport condition. Test structurecanalsobesupported on very soft springs or light elastic bands, whichleads toafreesupport condition. Insuchsituation, it isimportant toattachthesuspensionascloseaspossibletonodal points of the particular mode. Another methodofsupporting the structure is in situ type, where the teststructure is connected to some other component whichpresents a non-rigidattachment. Input tothe test set-upgenerallyincludesashortimpulseor, asinewaveexcita-tionover afrequencyrangeof interest, or awhitenoise.Thisforceinput isprovidedbyeitherashaker, oranim-pulse hammer thereby resulting in forced vibration and freevibrationcasesrespectively. Structurecanalsobeexcitedbysuddenlyreleasingit fromadeformedposition. Selec-tion of any particular type of shaker depends upon itsfrequencyrange, acceleration, rms(root-mean-square)ve-locity, rms displacement, maximumload, operatingtem-perature range, power supply, overall dimensions andmass. Impulse hammer is selected based upon its forcerange, frequencyrange, sensitivity, timeconstant, operat-ingtemperaturerange, powersupply, mass, diameterandlength of head. Output of the experimental set-up isFig.2 FE model of a cantileverbeamstructureFig.3 CauseandeffectdiagramFig.4 Modeshapesofcantileverbeamstructure.aFirst,secondandthirdmodeshapes,bFourth,fthandsixthmodeshapesStructuralDynamicModelUpdatingTechniques:AStateoftheArtReview1 3generally in the form of a function of acceleration, which ismeasured at a number of points using sensors like ac-celerometers etc. Selection of such sensors is generallybasedupontheircharacteristicssuchasmass,accelerationrange, thresholdacceleration, sensitivity, frequencyrange,and temperature sensitivity, operatingtemperature rangeand power supply. Signal fromaccelerometer is thenconditionedusingasignal conditioner, whichis selecteddependinguponitssensorexcitationcurrent, sensorexci-tation voltage, frequency range, output voltage, gain,power supply, operating temperature range, overall di-mensions and mass. Conditioned signal is then passed on toFast FourierTransformanalyzerssothat thetimedomainsignal canbe convertedtofrequencydomain. Suchfre-quency domaindata is thenanalyzed ona computer toidentify the experimental dynamic response of the teststructure. Experimental dynamic response is then com-paredwithits FEcounterpart toexamineanyerrors andalsotoascertainsufcient degreeof correlationbetweenthetwoasdiscussedinnextsection.2.3 ComparisonandCorrelationTechniquesBefore applyinganyFEMUtechnique, the experimentaland FEdata sets need to be compared so as to ensureFig.5 Extreme and intermediate positions drawn using solid anddashed lines respectively. a First mode, b second mode, c third mode,dfourthmode,efthmode,fsixthmodeFig.6 Receptance FRF. a PointFRF,btransientFRFAccelerometer Impact HammerForce Signal Acceleration Signal Data Acquisition System FRF Data Modal Data Updated FE Model Modal Analysis System FEMU System Fig.7 Representativeexperimentalset-upforFEMUS.Sehgal, H.Kumar1 3existence of some correlationbetweenexperimental andFE responses and also to determine that whether it is worthto update the proposed FEmodel or a completely newmodel is required. These techniques include comparison ofFRFs, natural frequenciesandmodeshapes(Modal ScaleFactor (MSF), Modal Assurance Criterion (MAC), Nor-malised MAC and Coordinate MAC (COMAC) etc.).Thesetechniquesarediscussedas:-2.3.1 ComparisonofFRFsAsimplemethodof comparingexperimental andFEre-sults is to plot experimental and FE FRFs on a single graphas shown in Fig. 8. A visual comparison of FRFs is done tond out presence of any correlation between the ex-perimental andFEresults. For example, Fig. 8ashowsaclearcorrelationbetweentheexperimentalandFEresults.ThisisthecasewhereFEmodel updatingisrequiredtoincrease thecorrelationfurther, while Fig. 8bshows thecase of total mismatch between the measured and predictedresults. Under suchconditions, insteadof FEmodel up-dating, aconceptuallynewFEmodelisrequired.2.3.2 ComparisonofNaturalFrequenciesIt isimportantto comparethenaturalfrequencies obtainedfromFEanalysisof thestructurewiththeonesacquiredthrough modal testing as shown in Table1. If the per-centageerrorsaresmall, onecanupdatetheFEmodelofthestructureinorder tominimizesucherrors. But if theerrorsareverylarge, say, morethan200%, thenitisnotadvised to update the model rather a new formulation of theFEmodelisrequired.2.3.3 ComparisonofModeshapesModeshapeportraysthepatternorcongurationinwhichthestructurevibratesat anyparticular natural frequency.Modeshapesareinherent propertiesofastructure. Thesedo not depend on the forces or loads acting on the structure.Thesewill changeif thematerial properties (mass, stiff-ness, damping properties), or boundary conditions (sup-ports) of the structure change. A simple method forcomparing mode shapes is through overlaying technique asshowninFig.8. Bythismethod, thedifferencebetweenexperimental andFEmodeshapeisdrawnonagraph. Inthis method the difference between two correspondingmode shapes should approach zero. Main limitation of suchgraphicalmethodsisthattheyarenotsosupportiveifthecorrelationprocessistobeautomated. Forautomaticcor-relationofmodes, weneedsomequantitative(numerical)measureof correlation, whichcanbeeasilyimplementedthroughacomputer program. OnesuchmeasureisMSF,whichisslopeofthebest straight linethroughthemodesplottedonanxygraphwithexperimental modeononeaxisandanalytical onother[2]. DesiredvalueofMSFisunity for good correlation. MSF between experimental andanalyticalmodeshapescanbeobtainedusingEqs. (3)and(4);where{[X}iand{[A}jrepresenttheithexperimentalandjthanalytical modeshaperespectively; whilesuper-script T denotes the transpose of the correspondingvector.MSF ;Xf gi; ;Af gj ;Xf gTi;Af gj;Af gTj;Af gj3MSF ;Af gj; ;Xf gi ;Af gTj;Xf gi;Xf gTi;Xf gi4MaindrawbackofMSFisthat it doesnot provideanyinformation regarding the scatter of the xy plots. To avoidthisproblem, it isrecommendedtouseMAC. Thiscrite-rionis alsoknownas modeshapecorrelationcoefcient[72]. ItcanbecalculatedusingEq.(5).MAC ;Xf gi; ;Af gj ;Xf gTi;Af gj

2;Xf gTi;Xf gi ;Af gTj;Af gj5MAC is a measure of scatter of points from the straight linecorrelation. MAC value equal to unity means perfectFig.8 Comparison of FRF plots. a Good correlation, b poorcorrelationStructuralDynamicModelUpdatingTechniques:AStateoftheArtReview1 3correlation, whileaMACvalueequal tozeromeans nocorrelationbetweentwomodes.Figure9ashowsatypicalMACplot betweenperfectlycorrelatedexperimental andFE modes. It is seen that all the diagonal elements are equaltounityandoff-diagonal elements arenil, whichmeansperfect correlation between experimental and FE modes. Apoor correlationcaseisshowninFig. 9b, wherenoneofthediagonal element is unity; andoff-diagonal elementsarealsonotzero.OnemajorlimitationofMACisthat it cannot identifythesystematicdeviations,whichcanbeovercometosomeextent bycombiningMACwiththebest t straight lineplots described earlier. Another limitation of MAC is that itis not a true orthogonalitycheckbecause of absence ofmass or stiffness matrix in the formula of MAC. ThislimitationcanalsobeavoidedbyusingnormalizedMAC(NMAC) [72]. The NMACis alsocalledas normalizedcrossorthogonalityasrepresentedbyEq.(6).NMAC ;Xf gi; ;Af gj ;Xf gTiW ;Af gj

2;Xf gTiW ;Xf gi ;Af gTjW ;Af gj 6NMACincludes a weighting matrix W , which can bereplacedbyeithermassmatrixorstiffnessmatrix.NMAChas a limitation of not describing the spatial distribution ofcorrelation, for which purpose, one can use CoordinateMAC(COMAC)asgiveninEq.(7)[70]as:-COMACk PLl1;X kl ;A kl

2PLl1 ;X 2klPLl1 ;A 2kl7COMACshouldbecalculatedonlyafterndingLnum-berofthecorrelatedmodepairsthroughMACorNMAC.COMACvaluecloseto1indicatesgoodcorrelationataparticularcoordinatesayk. TheCOMACplotsshowinggood and poor correlation are drawn in Fig. 10a, brespectively.MACor itsvariantsconsider modeshapematchingatonly discrete coordinates. If one wants to consider themodeshapecorrelationonacontinuousbasis, thenimageprocessingandpatternrecognitionbasedZernikemomentdescriptorcanbeused[107]. Further, ascanninglaservi-brometercanalsobeusedfor3Ddigitalimagecorrelation[48].Aftercomparisonandcorrelatingtheexperimental andFEdatasets, next stepistomakethesecompatiblewitheachothersothatFEMUcanbeperformed.Compatibilitytechniqueshavebeenexplainedinnextsubsection.2.4 SizeCompatibilityTechniquesMeasureddegreesoffreedomaregenerallyfarlesserthantheir FE counterparts due to limited number of sensors usedinexperimental set-up. Moreover some degrees of free-domsareverydifculttobemeasured(e.g. rotationalde-grees of freedomor those corresponding to the pointswhich are physically inaccessible). Thus the size of FEFig.9 ModescorrelationusingMAC.aGoodcorrelation,bpoorcorrelationTable1 ComparisonofnaturalfrequenciesModeno. Experimental(Hz) FE(Hz) Differenceinfrequencies(%)1 24.1 24.3 0.82 76.1 73.1 3.93 119.3 116.1 2.74 158.0 153.9 2.65 184.1 180.7 1.86 211.6 200.5 5.2S.Sehgal, H.Kumar1 3modal matricesisgenerallynot compatiblewiththeirex-perimental counterparts, whileinFEMUone-to-onecor-respondence between the two results sets is required; whichcanbeachievedbyeitherexpandingtheexperimental re-sults or by reducing the FE model as explained in followingsections.2.4.1 CoordinateExpansionMeasured data sets can be expanded so that they are of thesame size as their FE counterparts. Such a process is calledas coordinateexpansion. Asimplemethodof coordinateexpansionistosubstitutetheunmeasuredcoordinatesbytheir FE counterparts. This method is computationally veryefcient. Maindisadvantageof thismethodisthat some-times it leads to erroneous solutions during FEMU. Toavoidthisproblem, sometransformationmatrixbasedco-ordinate expansion should be done as used in Eq.(8);whereNandnarethesizesofexpandedandmeasuredeigen-vectorsrespectively.;expanded N1T Nn;measuredf gn18For an undamped system, the transformation matrix T canbeobtainedbywritingthegoverningequationsofthesys-teminpartitionedmatrices formusingKidders method[60].InKiddersmethod,sub-matricesarerelatedtomea-suredandunmeasuredeigenvectors.Fora dampedsystem,partitionedmatricesbasedmethodcanbeextendedtodealwithcomplexmeasuredmodes [47]. Another methodofcoordinateexpansionisbasedontheconceptthatunmea-sured eigenvectors can be expressed as a linear combinationof measured eigenvectors [90]. System equivalent reductionand expansion processes (SEREP) [89], Curve tting [110]and FE eigenvectors in conjunction with MAC matrix basedmethodsarealsousedincoordinateexpansion[71].2.4.2 ModelReductionModel reduction process is basically the inverse of theexpansionprocess. ThisprocessisgenerallyappliedtoananalyticalmodelsothatthesizeoftheFEmodelmatricescanbereducedandbrought closer totheir experimentalcounterparts. Asimple methodof model reductionis toeliminatethosedegreesof freedom(DOF) whicharenotavailable in experimental data. Model thus obtained isknown as reduced model. Main drawback of this method isthat mass and stiffness terms related to the eliminated DOFarecompletelylostandnowherecompensated[33].Another method of model reduction is to produce acondensedmodelwhichwillrepresenttheentirestructurecompletelybutapproximately.Inthismethodacondensedmodel is obtainedbytransformingtheoriginal model asgiven in Eq.(9); where n and N are the sizes of reducedandniteelementmodelbasedeigen-vectorsrespectively.;reducedf gn1T nN;FEf gN19One method of obtaining condensed modelis through staticreduction [45], in which size of systemmatrices is reduced byneglectinginertia andstiffness terms associatedwithun-measured DOFs. This method is generallyknown as Guyanreduction method. Main drawback of static reduction methodis that it does not reproduce any of the eigenvalues or eigen-vectors of theoriginal full FEmodel. Another methodofmodelreductionisimprovedreducedsystem(IRS)methodwhich can closely reproduce the eigenvalues and eigenvectorsof the original FEmodel [88, 40]. If reduced model is requiredto have eigenvalues and eigenvectors exactly same as that oforiginal FE model then SEREP should be used [89, 97].3 ReviewofFEMUTechniquesTheFEMUtechniquescanbebroadlyclassiedintotwocategories:- direct(non-iterative)techniquesand iterativetechniquesDirect techniques can provide the solution to modelupdating problemin just a single step; hence these arecomputationally very efcient, and divergence relatedFig.10 ModescorrelationusingCOMAC.aGoodcorrelation,bpoorcorrelationStructuralDynamicModelUpdatingTechniques:AStateoftheArtReview1 3problems do not occur. Another important feature of directtechniques is that theyreproduce the measureddata ex-actly. Therefore measurement noise andspurious modesare also reproduced by such techniques. Thereby, thesetechniques requiredverygoodqualitymodal testingandanalysisprocedures. Direct techniquescomputeaclosed-formsolution for systemmatrices using the structuralequations of motion and orthogonality properties of modes.These techniques are also called matrix techniques, be-causesuchtechniques foundthesolutionintheformofupdatedsystemmatricesbysolvingaset ofmatrixequa-tions. Themaindrawbackofdirect techniquesisthat up-datedmass andstiffness matricesmaynot besymmetricandpositivedenite. It becomes verydifcult tounder-standsuchkindofsystemmatricesonaphysicalbasis.Dynamic response of FEmodel of anystructure is afunction of a number of parameters, as explained inSect. 2.1. The iterative techniques compute updated valuesof material and structural parameters of FE model in such away that during each iteration mismatch between ex-perimental andFEresponseisreduced. Thesetechniquesarealsocalledasgradientbasedtechniques. Iterationsarestopped when the values of updating parameters stopconverging or the error function is reduced totolerablelevel. Error functionisgenerallyanon-linear functionofexperimental and FE responses such as eigenvalues,eigenvectors or FRFs. Iterativetechniques result inonlysymmetricandpositivedeniteupdatedsystemmatrices,whichcanbeeasilyunderstoodonaphysical basis. Butthesetechniquesrequireanumberofiterationsbeforear-riving at the nal result. Thus iterative techniques arecomputationally less efcient and divergence relatedproblemscanalsoariseduringiterations. Thetechnologyand research developments reported for both direct anditerative techniques are discussed in following sub-sections.3.1 DirectTechniquesofFEMUSomeresearchershavedevelopeddirecttechniqueswhichconsider adjustment of elementsof systemmatricesonamathematical basis rather than a physical basis. Afewresearchershavealsoworkedfordevelopmentofsuchdi-rect techniques, which can consider adjustment of physicalproperties of systemmatrices. The direct techniques arefurtherclassiedintotwocategories, viz. matrixelementsadjustment baseddirect techniquesandphysical propertyadjustmentbaseddirecttechniques.Foremost, in 1978, Baruch and Bar-Itzhack developedadirect method of FEMU by assuming FE mass matrix to becorrect [11]. Theyupdated the FEeigenvectors andFEstiffnessmatrixbyminimizingtheweightednormofdif-ferencebetweenmeasuredandFEeigenvectorssubjecttoorthogonalityconstraints. Duringsametime, Baruchalsodeveloped a direct method in which the FE stiffness matrixandFEeigenvectors are updatedinsucha manner thatsome weighted norm of the difference between the updatedandanalytical stiffness matrices is minimizedusingLa-grange multipliers [10]. In this method, the FE mass matrixisassumedcorrect andhencenot updated. Laterin1979,Berman adopted the mathematical approach of Baruch [10]and proposed a direct method for updating the mass matrix[14].Thismethodalsoincludedanadditionalconstrainttopreserve the symmetryof updatedmatrices. These tech-niques[11, 10, 14]arealsocalledastechniquesofrefer-ence because one of the three quantities (eigenvectors,mass and stiffness matrices) is assumed to be reference andtheothertwoareupdated.In1983,BermanandNagyproposedanothermethodinwhich the basic approach of the method of Baruch [10] andthemethodofBerman[14]werecombinedtoupdatebothmassandstiffnessmatricesof thesysteminasequentialmanner[15]. MethodofBermanandNagyhasbeenusedbyvarious researchers ina number of applications. Ap-plicationofthismethodwasalsostudiedbyModaketal.[83] for dynamic design of a xedxed beamand anF-structure. Baisetal.[9]appliedthedirecttechniquesofBaruch[10] andalsoof BermanandNagy[15] for dy-namicdesignofadrillingmachine. DhandoleandModak[30] performedtheFEmodel updatingof vibro-acousticcavitiesusing the direct method of Berman and Nagy [15].Theyinvestigatedasimulatedexampleof atwodimen-sional rectangular cavity with a exible surface havingstructuralmodelingerrorsrelatedtothematerialproperty,geometryandboundaryconditions; underincompleteandnoisy simulated experimental data. Effectiveness of themethod was compared with an iterative method on thebasis of accuracyof predictionof vibro-acoustic naturalfrequencies and the frequency responses both inside as wellas outsidethefrequencyrangetakenduringupdating. Itwas concludedinthestudythat thedirect techniquere-sultedinanaccuratepredictionofthevibro-acousticnat-ural frequencies and the response inside the frequencyrange considered during model updating. However, beyondtheupdatingfrequencyrange,thepredictionsbasedonthedirect updated vibro-acoustic models are not as accurate asgivenbytheiterativetechnique. Further thedirect tech-niqueseemstobequiteversatilecomparedtoitsiterativecounterpartfordealingwithcomplexcavities. Thisisduetothereasonthat for complexcavities, completeknowl-edgeof modelinginaccuracies andselectionof updatingparametersbecomesverydifcult. Recently, Modak[81]also developed a direct vibro-acoustic model updatingtechnique using modal test data. This technique can beused for updating the vibro-acoustic FEmodel of suchsystems that involve an elastic structure enclosing aS.Sehgal, H.Kumar1 3medium, likeair. Byusingthistechnique, massandstiff-nessmatrixof boththestructural aswell astheacousticparts of the model can be updated by preserving theirsymmetry.ThemethodofBermanandNagy[15]wasfurtherex-tendedbyCeasar[19]byincludingadditional constraintsrelatedtopreservationofthetotalmassofsystemandtheinterfaceforces. Matrixperturbationconcept wasusedbyChenet al. [26] tosimultaneouslyupdatemassandstiff-ness matrices of the system. Baseduponrst order ap-proximation Sidhu and Ewins [99] developed a directmethod using error matrix approach. This method wasuseful for only small modeling errors. Kabe [57] suggestedadirect methodwhichcouldretainthebandwidthofsys-temmatricesbyidentifyingthenull elementsof originalsystemmatrices and constraining themto remain zeroduringFEMU.Further, a range of direct updating techniques wereformulatedbyCaesar[20]consideringdifferent objectivefunctions,constraintsandwhethermass matrixorstiffnessmatrixisupdatedrst.Later, Lim[73]proposedaFEMUmethod in which sub-matrix-scaling factors are used asupdatingparameters. Duetotheuseofsub-matrix-scalingfactors number of unknowns is reduced appreciably. SmithandBeattie suggested a quasi-Newton methodfor stiffnessupdating which preserve structural connectivity and canalso handle noisy modal data [101, 13]. Application of thismethodwasstudiedlaterbyRamamurti andRao[95]byusing different nite elements. They updated the FEmodelsof arectangular platewithholesusing3-Dplateelements; acraneusing3-Dbeamelementsandarectan-gular plateusingbrickelements. Bucher andBarun[17]developedmodaldatabaseddirect methodhavingtheca-pabilitytodeal withpartiallyknownexperimental eigen-solutions. Themethodwasveriedbyapplyingit tonu-merical examplesofaclampedbeamandaserial spring-mass system. Farhat andHemez[37] proposedaFEMUmethodusinganelement-by-element sensitivitymethod-ology and demonstrated the potential of the method byusingseveral simulationexamples. Later, Friswell et al.[41] developed a method for simultaneous updating ofdampingandstiffnessmatricesassumingthemassmatrixto be correct. Singular value decomposition and matrixapproximationtechniquebasedmethodwas proposedbyXiamin [111]. This method is particularly useful if themismatchbetweenexperimentalandFEresponsesisverylarge. Recently, Carvalhoet al. [21] developedamethodwhichcanidentifyandprevent thereproductionofspuri-ous modes into updated results. The direct techniquesdiscussed so far mainly aimat removing the mismatchbetween experimental and FE responses, without botheringfor anyadjustment of physical properties of systemma-trices. Presently research is also oriented towardsdevelopment of direct techniques whichaimat reducingthemismatchbetweenexperimental andFEresponsesbyadjustingthephysicalpropertiesoftheFEmodel.In2007, Huet al.[49]proposedthecross-model-cross-modemethod(CMCM) inwhichadjustment of physicalpropertiesofthesystemmatricesiscarriedout toupdatethemass andstiffness matrices simultaneously. Satisfac-toryperformanceofthemethodwasdemonstratedbyap-plyingitfor simulatedexamplesofa shearbuilding modeland a three-dimensional frame structural model. During thedemonstration, somepreset error coefcients wereintro-duced into the FE model of the structures. Then the CMCMmethod was used to estimate the correction factors ofFEMU. Main disadvantage of the method is that it requiresthemeasurement ofspatiallycompletemodal data, whichisgenerallynotpossibleforlarge-scale reallifestructures.Further, in2011, substructureenergyapproach(SEA)based physical property adjustment type direct method wasdevelopedbyFangetal.[34].Mainbenetofthismethodis that it can handle spatially incomplete experimentalmodaldataalsoasopposedtotheCMCMmethod.Inthismethodthe complete systemis dividedintoseveral subsystems. Instead of updating the complete system, only thecriticalsubsystemsareidentiedandupdatedusingasetoflinearsimultaneousequationsdeducedfromtheenergyfunctionalofsubstructuremodelsandsubstructuremodes.For validationpurpose, theyappliedthe methodfor up-datingthemodelsofamass-springsystem, atwo-dimen-sional and a three-dimensional lattice structure usingsimulated experimental data. Results obtained by Fanget al., for numericalexample ofa three-dimensional latticestructureshowedthatthemethodworkedsatisfactorilyforFEMUpurpose. Jacquelinet al. [53] developeda directprobabilisticmodel updatingtechniquethat takesintoac-count theuncertaintiesrelatedtoexperimental resultsbyusingtherandommatrixapproach. Recently, Jianget al.[56] reducedthe model updating process to the problem ofthe best approximation type. This technique is successfullyusedfornumericalexamplesofundampedsystems. How-ever thetechniqueneedstobetestedfurther for dampedsimulated systems as well as for actual experimentalresults.Major contributions related to direct techniques ofFEMU have been presented in a brief chronological formatinTable2.3.2 IterativeTechniquesofFEMUAlgorithmfor iterativetechniques of FEMUis drawninFig. 11.Insuchtechniquesanerrorfunctionisminimizediteratively to nd the updating parameters or correctionfactors. These techniquesoriginated in1974,whenCollinset al. proposed the eigendata sensitivity based iterativeStructuralDynamicModelUpdatingTechniques:AStateoftheArtReview1 3method also known as inverse eigensensitivity method(IESM) [27]. Later, Chen and Garba [25] used matrixperturbation technique for iteratively computing theeigensolutionandeigendatasensitivities.Thismethodwasfurther improvedbyKimet al. [62] byincludingsecondordersensitivities.FurthertheconvergenceofthismethodwasimprovedbyLinetal.[75]by employingboththeFEandtheexperimentalmodaldataforevaluatingsensitivitycoefcients.Thisalsohelpedinapplicationofthemethodforcasesoflargeerrors.InIESM, modaldatasuchaseigenvalues, eigenvectorsand damping ratios are used to forman error function.Modal data is obtained through modal analysis of measuredFRFs.Ifmodalanalysisisnotdonecarefullythentheex-tracted modal data maycontaincertainerrors, which isfurthertransmittedtotheresultsofFEMU. Toavoidthisproblem one can use the measured FRFs directly for modelupdatingusingResponseFunctionMethod(RFM)aspro-posedbyLinandEwins[76]. RFMdoesnot requireanymodal extraction to be performed on measured data therebyeliminatingthe chances of errors due tomodal analysisbeingtransmittedtoupdatedresults. ApplicationofRFMfordynamicdesignofaxedxedbeamandanF-struc-turewasdiscussedbyModaket al. [83]. EffectivenessofIESM and RFM has been compared by Imregun et al. [52],Modaket al. [84]. Forincompleteexperimental datacase(where experimental eigenvector is not completely known)withnonoise, RFMworksbetterthanIESM, whilelatterperformsbetterinthepresenceofnoiseparticularlywhenthe updating range covers a greater number of modes.Moreover, inRFM, ifthenumberandlocationof testingfrequenciesarenotselectedproperlythenthemethodmaynot beabletoconvergeasreportedbyModaket al. [84]basedupontheir comparative studyusingsimulatedex-perimentaldata.WhileformulatingthebasicRFM, LinandEwins[76]hadnottakenintoaccountdamping.Later,Aroraetal.[3,4] proposedtwotechniques ofextendingthe basic RFM inTable2 MajorcontributionsindirecttechniquesofFEMUReferences MajorContributionBaruchandBar-Itzhack[11] UpdatingofstiffnessmatrixBerman[14] UpdatingofmassmatrixBermanandNagy[15] Updatingstiffness, massmatricesinsequentialmannerCeasar[19] ImprovementofBermanandNagymethodbypreservationoftotalmassandinterfaceforcesChenetal. [26] UseofmatrixperturbationconceptforsimultaneousupdatingofmassandstiffnessmatricesSidhuandEwins[99] ErrormatrixapproachbaseddirectupdatingtechniquetohandlesmallmodelingerrorsKabe[57] ConstraintonretainingbandwidthofsystemmatricesCaesar[20] FormulationofanumberofdirecttechniquesLim[73] Sub-matrixscalingfactorsbasedupdatingtechniqueSmithandBeattie[101] UpdatingstiffnessmatrixwithxstructuralconnectivityBucherandBraun[17] UpdatingforincompleteexperimentalmodaldataFarhatandHemez[37] Element-by-elementsensitivitymethodologybasedupdatingtechniqueFriswelletal. [41] SimultaneousupdatingofdampingandstiffnessmatricesRamamurtiandRao[95] FEMUusingdifferentniteelementsXiamin[111] MatrixapproximationtechniquebasedFEMUmethodModaketal. [83] DynamicdesignofxedxedbeamandF-structureusingthemethodofBermanandNagyBaisetal. [9] DynamicdesignofdrillingmachineusingmethodofBaruchandalsoBermanandNagyCarvalho[21] FEMUmethodwhichcanidentifyandpreventthereproductionofspuriousmodesintoupdatedresultsHuetal. [49] Physicalpropertyadjustmentbasedcross-model-cross-modemethodDhandoleandModak[30] FEMUofvibro-acousticcavitiesusingthemethodofBermanandNagyDhandoleandModak[31] ApplicationofmethodofBermanandNagyinvibro-acusticseldFangetal. [34] SubstructureenergyapproachbasedphysicalpropertyadjustmenttypedirectmethodJacquelinetal. [53] DevelopmentofprobabilisticmodelupdatingJiangetal. [56] DevelopmentofdirectupdatingtechniqueusingbestapproximationmethodModak[81] UpdatingofvibroacousticniteelementmodelsusingmodaltestdataS.Sehgal, H.Kumar1 3ordertowidentheapplicabilityofthemethodbyconsid-eringdamping. First proposedtechniquewastocombineRFMwithdampingidenticationmethodof Pilkey[92].This technique involved a two step procedure in which rststepinvolvedupdatingofonlymassandstiffnessmatriceswhile damping matrix was obtained in second step by usingthemassandstiffnessmatricesfoundinrststep.SecondproposedtechniquebyAroraet al. [4] wastoconsider themodel parameters andsystemmatrices inacomplex form so as to deal with complex modes of dampedstructures. Latertheycomparedthesetwotechniquesandfound that complex parameter based FEMUtechniquegives better results than the one with dampingidentication [5]. Arora et al. [6] also worked on the use ofcomplex parameters based RFMof FEMUfor dynamicdesignpurposes. Arora[7]alsocomparedtheaccuracyofbasic RFM[76] against the direct method BernamandNagy[15] andfoundthat theperformanceofbasicRFMwassuperiortothedirectmethodintermsoftheaccuracyinpredictionof FRFs. Recently, in2012, thebasicRFM[76]wasfurtherextendedbyPradhanandModak[93]todevelopanormalRFMthatwasbasedontheestimatesofnormal FRFscomputedbyusingonlystiffnessandmassmatrices. Normal RFMwas later used by Pradhan andModak[94] for dampingmatrixidenticationusingFRFexperimentaldata.Fig.11 AlgorithmforiterativetechniquesofFEMUStructuralDynamicModelUpdatingTechniques:AStateoftheArtReview1 3In 2000, Modak et al. [82] developed a constrainedoptimizationbasedFEMUtechnique. Theyevaluatedtheperformance of the said technique by applying it to a xedxedbeamstructure. Thistechniquewascomputationallymore intensive than IESMbut was able to address thedifcultythat might ariseduetolargedifferencebetweenthesensitivities of natural frequencies andmodeshapes.Later, theyalsousedtheconstrainedoptimizationbasedFEMU for dynamic design of xedxed beam andF-structure[85]. Useof thistechniquefor better FEfor-mulationof acousticcavitieswasdonebyDhandoleandModak[32].Neuralnetwork(NN)methodhasalsobeenappliedforFEMU by Atalla and Inman [8] for FE model updating of aexible frame. In this method, a NN is trained andvalidatedusingFEresponses. Thereafter thisNNcanbeemployedtoobtainupdatedphysicalparametersbytakingexperimental responsesasinputs. OncetheNNmodel isproperlytrained, theNNbasedcalculationsarerelativelyfast compared to conventional optimization techniquesregardlessofthecomplexityofthereallifestructure. NNbasedmodel updatingmethodisalsorobust tothenoisepresentthere[67].Mainlimitationofthismethodisthatitrequiresalargenumberoftrainingdataqp(wherepisthe number of updating parameters and q is the number oflevelsor valueswhichanyupdatingparameter cantake).However this problemcanbecircumventedbyusinganorthogonal arraymethodinwhichthenumberoftrainingdatasetsarereducedtop(q-1)?1only[23,16].Becauseoftheexperimentallimitations, ifonlynaturalfrequencies (or eigenvalues)aremeasured thenFEMU canalsobe performedbyusingreducedorder characteristicpolynomial(ROCP)basedmethodasproposedbyLi[69].Li applied the ROCP based method for FE model updatingof abeamstructureelasticallyconstrainedat oneend. Inthismethodapolynomial isdenedintermsofthemea-suredeigenvalues, FEeigenvalues, FEeigenvectors andupdatingparameters. Assumingthemeasurednatural fre-quenciestobetherootsof thepolynomial, aset of non-linear equationsarederived. Thisset of non-linear equa-tions is further solved for calculating the values of theupdatingparameters.Sometimes, it becomes difcult tomeasure FRFs ac-curately due to small size or delicate nature of test structure[like a hard disk drive (HDD)]. Under such conditionsModel UpdatingusingBase Excitation(MUBE) methodcan be applied as proposed by Lin and Zhu [74]. They usedMUBE method for FEMU of a cantilever beam and a trussstructure, and showed that the method works satisfactorily.Inthis method, base of the structure is excitedwithanunknownforceinputusinganelectricshakerandthedis-placement outputmeasuredfromthetestset-upis usedformodel updating. This method is of great use whenexcitation force is not known or difcult to measure.Jamshidi and Ashory [55] compared and found that MUBEgives better updating results than RFMfor incompleteexperimentaldatawithmeasurementnoise.Further theproblemof model updatingfor areal lifestructure is quite complex due to presence of non-linearity,damping, measurement errorsandlargenumberofupdat-ing parameters. Under such conditions, the traditionalmodel updating techniques may fail to converge or may getstuck in local minima rather than nding a global minimumof theerror function. Inthat kindof situations, it is ad-visabletousesuchoptimizationtechniqueswhichareca-pable of nding the global optimumresults even for acomplicated optimization problem. Fewsuch promisingtechniquesareSimulatedAnnealing(SA), GeneticAlgo-rithms (GA) and Particle SwarmOptimization (PSO).LevinandLieven[68]usedandcomparedSAandGAforFEMUandobservedthat SAbasedFEMUmethodper-forms better than its GA based counterpart. Success of bothSAandGAbasedFEMUtechniques depends uponap-propriatechoice of updatingparameters. If updatingpa-rameters are not selected properly, then both thesetechniques give unsatisfactoryresults. Marwala [79] ap-pliedthe PSOmethodinthe eldof FEMU. PSOis apopulationbasedstochasticsearchalgorithmderivedfromsocial-psychological behaviorofbiological entities(birds,sh, ants, etc.) when they are foraging for resources (food).It is particularly useful if the number of updating pa-rametersisverylarge. Further, Mthembuet al. [87]usedPSOforselectingthebestmodelfromamongstanumberofupdatedstructuralmodels.Kwon and Lin [64] suggested a robust FEMU techniqueusing the concept of Taguchi method and were able toobtaingoodFEMUresults. Theyusedbothfrequencyaswell asmodal dataforformulationoftheobjectivefunc-tion. This technique was reported to be robust againstvarious noises because the parameters were updated insuchawaythatsignaltonoiseratioismaximized.It is to be mentioned here that due to their iterative nature,theiterativetechniquesseemtobecomputationallyinef-cient, particularly, if appliedtoFEmodels of largesizestructures. Working on these directions Guo and Zhang [44]suggestedtheuseofresponsesurfacemethodology(RSM)based FEMU. In this technique, FE calculations are not re-quiredat eachiterationstep, therebymakingit computa-tionallyefcient. Inthismethod, n-dimensional responsesurfaces are created by taking updating parameters as inputsanddynamicFEresponsesasoutputs. Updatedvaluesofmodel parameters are found by using the response surfacesandthemeasuredresponsesbyminimizingtheerrorfunc-tion. RSM based FEMU is reported to be as accurate as thesensitivity based method, and, moreover it is more robust andcomputationally efcient than the sensitivity based FEMU.S.Sehgal, H.Kumar1 3Later, Ren and Chen [96] applied RSM as well as sensitivitybasedFEMUtechniquestoaprecast concretebridgeandcompared the convergence of objective function by both thetechniques.Theirstudyshowedthatforagivennumberofiterations,theobjectivefunctionofFEMUisreducedtoalower level by using the RSM based FEMU technique thanitssensitivitybasedcounterpart. IntraditionalRSMbasedFEMUtechnique, the experimentally measured signals wererst transformed to one or more response parameters such asnatural frequencies. This reducedthe informationinthetraining data used for developing response surface models. Inorder to circumvent this problem, time domain based resultswere used by Shahidi and Pakzad [98] for RSMbased FEMUtechnique. Time domainbasedtechnique was helpful inextracting more information frommeasured signals andcompensatefortheerrorpresentinthemeta-models.Ef-ciency of RSMbased FEMUtechnique was further increasedby Chakraborty and Sen [22] whilst developing an adaptiveRSMbasedFEMUtechniquebyreplacingtheleastsquaremethod with the moving least square method.It isimportant tomentionthat FEmodel ofareal lifestructurecontainsalargenumberofparameters. Ifalltheparameters are taken as updating parameters, then themodel updatingproblembecomes toocomplexandtimeconsuming. Alargenumber of updatingparameters alsoresult in ill-conditioning of the problemor trapping inmanylocalminima[61,39].Toidentifyandselectonlyafewimportant updating parameters, Fissette et al. [38]proposed a force balance method. This method can be usedfor error location and then updating parameters are selectedfromonlythe regions of modelingerrors. Waters [108]proposedamodiedforcebalancemethodinwhichit isassumedthat theregions of responseerrors arenot nec-essarily an indication of modeling error. Recently, Kim andPark [61] developed an automated parameter selectionprocedure. This is a twophase method, wherein, duringrst phaseanupdatingparameter isassignedtoeacher-roneous FE. Then two neighboring updating parameters aremerged if their sensitivity (of response w.r.t. parameters) isof same sign thereby reducing the number of updatingparameters. Thisisrepeateduntil all theneighboringup-datingparametershaveoppositesignofsensitivity. Ifthenumber of updating parameters is acceptable then theprocedurecanbestopped; otherwiseonecanmoveontothesecondphase. Insecondphasethosetwoneighboringparameters of opposite sign are found and grouped togetherwhich will result in least reduction in total sensitivity. Thismethodwasvalidatedbyapplyingit totheFEmodel ofcoverofaHDDhaving1115FEsand6732dofs.KimandPark used the force balance method of Fissette and Ibrahimforerrorlocalizationpurposeandfoundthatoutofatotalof1115FEsonly628FEscontainedthemodelingerrors.Aftertheapplicationofrststageofautomaticparameterselection procedure, Kim and Park successfully reduced thenumberofupdatingparametersto150;whichwasfurtherdrasticallyreducedto20usingthesecondphaseof theirsuggestedprocedure.Moreover the updating parameters are generally thephysicalparameterssuchas thickness,density, modulus ofelasticity,damping,Poissonsratioetc.ofstructure.Theseparametersdonothaveasinglediscretevaluethroughoutthebodyofareal lifeobject. Spatial distributionofsuchparameters was considered by Adhikari and Friswell [1] byexpressingtheupdatingparametersasspatiallycorrelatedrandomelds.If a number of identical test structures are taken, all maynot have same value of a particular material parameter viz.modulus of elasticity, density etc. This variability inseemingly identical test pieces may arise due to manysourcessuchasgeometrictolerances, manufacturingpro-cessesetc. Thuseachmaterial parameterhasastochasticnaturewithameanvalueandavariance. Suchstochasticnatureofmaterialparameterswastakenintoaccountdur-ingFEMUbyMaresetal.[78]usingasimulatedexampleandobtainedquitesatisfactoryresults. Inthismethod, themeanvaluewas representedbythecentreof thescatterellipse while the size andorientationof the ellipse wasdetermined by variance of test data. The updated analyticalscatter ellipse overlays quite closely the experimentalscatter ellipse. SameworkwasthenextendedbyMotter-sheadet al. [86] by carrying out stochastic FEMU for a setofphysicalstructures.The stochastic FEMU based approach is based onprobabilistic models, which require large number of tests tobe conducted and also the large volumes of test data; whichdemands for highexperimental as well as computationalefforts. Inorder toget ridof thelargequantities of testdata, onecanoptfortheapproachthat wasusedbyKho-daparast et al. [59] for performing interval FEMU in whichranges or distributions of updatingparameters arefoundrather than just one true value of any updating parameter.Table3 presents the major technological developments andadvancements in the eld of iterative techniques of FEMU.4 ApplicationsofFEMUTechniquesFEMUiswidelyapplicableintheeldofbetterFEmodelformulations, damageanalysis of structures, non-destruc-tive characterization of material properties and also fordynamic design purposes. With the aim of better FE modelformulations, FEMUhasbeenusedinanumberofappli-cations such as aircrafts, satellites, automobiles, nuclearpower plants, rotor bearing systems, laser spot welds,bridges, dams, multi-storeybuildings, steel frames, trussstructures, acoustics, etc. It also nds applications inStructuralDynamicModelUpdatingTechniques:AStateoftheArtReview1 3Table3 MajorcontributionsiniterativetechniquesofFEMUReferences ContributionCollinsetal. [27] Developmentofinverseeigensensitivitymethod(IESM)ChenandGarba[25] UseofmatrixperturbationtechniqueforimprovedcomputationofsensitivitiesofIESMKimetal. [62] ImprovementofIESMbyIncludingsecondordersensitivitiesFissetteetal. [38] SelectionofupdatingparametersusingforcebalancemethodforFEMULinandEwins[76] Developmentofresponsefunctionmethod(RFM)Lammensetal. [65] ApplicationofFEMUinstructuraloptimizationofatennisracketImregunetal. [52] ComparisonofIESMandRFMbasedFEMUtechniquesLinetal. [75] ImprovementofconvergenceofIESMWaters[108] DevelopmentofmodiedforcebalancemethodforerrorlocalizationinFEMULevinandLieven[68] DevelopmentandcomparisonofsimulatedannealingandgeneticalgorithmbasedFEMUtechniquesAtallaandInman[8] DevelopmentofneuralnetworksbasedFEMUtechniqueLevinandLieven[67] RobustFEMUtechniqueusingneuralnetworksCunhaandPiranda[28] ApplicationofFEMUinidenticationofelasticconstantsofcompositematerialsWahabetal. [106] ApplicationofFEMUindamageanalysisofreinforcedconcretebeamsModaketal. [82] DevelopmentofconstrainedoptimizationbasedFEMUtechniqueChangetal. [23] ImprovementofneuralnetworkbasedFEMUtechniqueusingorthogonalarraymethodLi[69] Reducedordercharacteristicpolynomial(ROCP)methodSinhaetal. [100] ApplicationofFEMUindynamicdesignofspacerlocationsoftubesofnuclearpowerplantsModaketal. [83] ApplicationofRFMfordynamicdesignofaxedxedbeamandanF-structureModaketal. [84] ComparativestudyofRFMandIESMButkewitschandSteffen[18]ApplicationofFEMUindesignsynthesisofaheavytrucksideguardBaisetal. [9] ApplicationofiterativemethodfordynamicdesignofadrillingmachineGuoandZhang[44] Developmentofresponsesurfacemethod(RSM)basedFEMUtechniqueKwonandLin[63] DevelopmentoffrequencyselectionmethodforFRFbasedFEMUHwangandKim[51] ImprovementofcomputationalaspectofFEMUbaseddamagedetectiontechniqueTeughelsandRoeck[103] ApplicationofFEMUindamageidenticationofhighwaybridgeModaketal. [85] ApplicationofconstrainedoptimizationbasedFEMUtechniquefordynamicdesignofxedxedbeamandF-structureMarwala[79] DevelopmentofparticleswarmoptimizationbasedFEMUtechniqueKwonandLin[64] DevelopmentofTaguchibasedFEMUtechniqueVepsaetal. [105] ApplicationofFEMUtoafeedwaterpipelineofanuclearpowerplantZhouandFarquhar[113] ApplicationofFEMUinndingtheinvivomaterialpropertyofwheatstemMottersheadetal. [86] ApplicationofstochasticFEMUtechniquetoasetofphysicalstructuresDecouvreuretal. [29] StudyofeffectofdispersionerroronacousticFEMUJaishiandRen[54] ModalexibilityresidualbasedFEMUtechniqueLinandZhu[74] Developmentofmodelupdatingusingbaseexcitation(MUBE)methodGondhalekaretal. [43] ApplicationofFEMUfordynamicdesignofacomplexautomobilestructureKimandPark[61] DevelopmentofautomatedparameterselectionprocedureforFEMUZapicoetal. [112] ApplicationofneuralnetworksbasedFEMUtoasteelframeJamshidiandAshory[55] ComparativestudyofMUBEandRFMbasedFEMUtechniquesAroraetal. [3] DevelopmentofdampedFEMUtechniquebycombiningRFMwithadampingidenticationmethodAroraetal. [4] DevelopmentofdampedFEMUtechniquebyimplementingRFMusingcomplexparameterstechniqueAroraetal. [5] ComparativestudyofRFMbaseddampedFEMUtechniquesAroraetal. [6] DynamicdesignofF-structureusingcomplexparameterstechniquebasedRFMHusainetal. [50] ApplicationofFEMUtolaserspotweldsFangandPerera[35] ApplicationofRSMbasedFEMUindamageanalysisofabridgeDhandoleandModak[30] ApplicationofdirectmethodofBermanandNagyforvibro-acousticFEMUWeberetal. [109] ConsistentregularizationofnonlinearFEMUindamageidenticationS.Sehgal, H.Kumar1 3damage analysis of rotor blades of helicopters, bridges,multi-storey framed concrete buildings, reinforced concreteframes, reinforcedconcrete beams, etc. It has alsobeenused successfully for non-destructive characterization ofelastic constants of composites, longitudinal moduli ofliving wheat, structural damping, etc. FEMU has also beenused for dynamic design of automobiles, drilling machines,spacers of nuclear power plants, F-structures, beams andtennis racket etc. The owchart showninFig. 12 alsorepresents the applications of FEMU in a number ofdisciplines.5 ResearchIssuesinFEMUThere are conictingopinions of various researchers re-garding use of objective functions and test structure used inFEMU. Future research needs to be directed towardsstandardizing a benchmark objective function and teststructure,sothatthesecanbeusedworldwidebydifferentresearchers for comparing the effectiveness of differentFEMUtechniques.It is also a well known fact that damping is present in allreal life structures. But still there are many model updatingtechniques in which damping effects of a real life structurehavebeenneglected[8],69, 68, 96, 104, 80, 102]. Suchtechniquesneedtobeinvestigatedsoastoincreasetheirabilitytodealwithdampedreallifesystems.ExistingFEMUtechniquesaimtowardsmappingonlythe dynamic behavior of the structure. More efforts arerequiredfordevelopingsuchFEMUtechniqueswhichcanpredict accurately the dynamic as well as static behavior ofstructures.More research is required in the development of realisticupdated mass matrices by combining the FEMU techniqueswith digital image processing and x-ray/ultrasonic im-ages. 3-D digital image processing has been used formode-correlation purpose, but its use for objective functionformulationstillawaitsmoreefforts.Fuzzy logic, neural networks and their combinationhavebeenused, for model formulationpurpose, inmanysuchelds where systemparameters are not understoodproperly. FEMU is a similar problem where the elements oflargescalesystemmatricesarenot knownexactly. Thusresearchcanalsobe directedtowards the applicationofsuchblack-boxbasedtechniquesinFEMU.FEMUhas been used to infer the in vivo materialproperties of crops. Still, more research efforts are requiredin the direction of application of FEMU for better modelingandcharacterizationofbones, eshandplantleavesetc.6 ConclusionsInthispaperareviewofanumberofdirect anditerativetechniques of FEMU has been presented. Direct techniquesnd little applications in industry due to the fact thatTable3 continuedReferences ContributionMthembuetal. [87] ApplicationofparticleswarmoptimizationinselectionofupdatedFEmodelsRenandChen[96] ApplicationofRSMbasedFEMUtechniquetoaconcretebridgeAdhikariandFriswell[1] DevelopmentofspatiallydistributedFEMUtechniqueusingspatiallycorrelatedrandomeldsDhandoleandModak([31] Comparativestudyofvibro-acousticFEMUtechniquesChellinietal. [24] ApplicationofFEMUindamageanalysisofasteelconcretecompositeframeArora[7] ComparativestudyofFEMUtechniquesLepoittevinandKress[66] UseofFEMUformodaldampingpredictionKhodaparastetal. [59] DevelopmentofintervalFEMUtechniqueusingtheKrigingpredictorMottersheadetal. [86] ApplicationofsensitivitymethodforFEMUofhelicopterairframeGolleretal. [42] ApplicationofstochasticFEMUforcomplexaerospacestructuresBayraktaretal. [12] ApplicationofFEMUtoarchdamsystemsKhanmirzaetal. [58] ApplicationofFEMUtomulti-storeyshearbuildingsFangandPerera[36] ApplicationofRSMbasedFEMUindamageidenticationofreinforcedconcreteframeDhandoleandModak[32] AcousticFEMUusingconstrainedoptimizationtechniquePradhanandModak[93] DevelopmentofnormalRFMPradhanandModak[94] ApplicationofRFMindampingmatrixidenticationShahidiandPakzad[98] DevelopmentoftimedomainresultsbasedmodelupdatingChakrabortyandSen[22] DevelopmentofmovingleastsquaremethodbasedmodelupdatingStructuralDynamicModelUpdatingTechniques:AStateoftheArtReview1 3updatedFEmodels of most of thedirect techniques aredifculttobeunderstoodonaphysicalbasis.Whileafewphysical propertyadjustmentsbaseddirect techniquesarealso now available in literature; their application to real lifesystemsneedstobeexploredfurther. Ontheother hand,iterative techniques have received considerable attentionfromindustrial applicationpoint of viewandhavebeenapplied successfully in a number of elds such as aircrafts,automobiles, satellites, machinetools andcivil structuresetc. Further anumber of future researchdirections havebeenhighlightedwhichcanbeusedfor further advance-mentsintheeldofmodelupdating.References1. Adhikari S, Friswell MI (2010) Distributedparameter modelupdating using the Karhunen-Loeve expansion. Mech SystSignalProcess24:3263392. AllemangRJ, BrownDL(1982) Acorrelationcoefcient formodal vector analysis. Proceedings of the rst internationalmodalanalysisconference. Orlando, USA,pp1101163. Arora V, Singh SP, Kundra TK (2009) Finite element model up-dating with damping identication. J Sound Vib 324:111111234. Arora V, Singh SP, Kundra TK (2009) Damped model updatingusingcomplexupdatingparameters. JSoundVib320:4384515. AroraV, SinghSP, KundraTK(2009) ComparativestudyofdampedFE modelupdating methods. Mech Syst Signal Process23:211321296. AroraV, SinghSP, KundraTK(2009)DampedFEmodel up-datingusingcomplexupdatingparameters:itsusefordynamicdesign. JSoundVib324:3503647. Arora V(2011) Comparative study of nite element modelupdatingmethods. JVibControl17(13):202320298. AtallaMJ, InmanDJ(1998) Onmodel updatingusingneuralnetworks. MechSystSignalProcess12(1):1351619. BaisRS, GuptaAK, NakraBC, KundraTK(2004)Studiesindynamic design of drilling machine using updated nite elementmodels. MechMachTheory39:1307132010. BaruchM(1978) Optimizationprocedure tocorrect stiffnessandexibilitymatricesusingvibrationtests. AmInstAeronautstronautJ16(11):1208121011. Baruch M, Bar-Itzhack IY (1978) Optimal weightedorthogonalizationof measuredmodes. AmInst Aeronaut stro-nautJ16(4):34635112. Bayraktar A, SevimB, Altunisik AC(2011) Finite elementmodel updatingeffects onnonlinear seismicresponseof archdamreservoirfoundation systems. Finite Elem Anal Des47:8597Fig.12 ApplicationsofFEMUS.Sehgal, H.Kumar1 313. BeattieCA, SmithSW(1992)Optimalmatrixapproximantsinstructuralidentication. JOptimTheoryAppl74(1):235614. Berman A (1979) Mass matrix correction using an incomplete setof measured modes. Am Inst Aeronaut stronaut J 17:1147114815. BermanA, NagyEJ(1983)Improvement ofalargeanalyticalmodel using test data. Am Inst Aeronaut stronaut J 21:1168117316. BestereldDH, Bestereld-MichnaC, BestereldGH, Bester-eld-SacreM(1995)Total qualitymanagement. Prentice-Hall,EnglewoodCliffs, NJ, USA17. Bucher I, Braun S (1993) The structural modication inverse prob-lem: an exact solution. Mech Syst Signal Process 7(3):21723818. ButkewitschS, SteffenVJ (2002) Shape optimization, modelupdating and empirical modeling applied to the design synthesisofaheavytrucksideguard. IntJSolidsStruct39:4747477119. Caesar B(1986) Updateandidenticationof dynamicmathe-matical models. Proceedingsof thefourthinternational modalanalysisconference. LosAngeles, USA,pp39440120. Caesar B(1987) Updating systemmatrices using modal testdata, In: proceedings of thefthinternational modal analysisconference, London, UK, pp.45345921. CarvalhoJ, DattaBN, GuptaA, LagadapatiM(2007)Adirectmethodformodelupdatingwithincompletemeasureddataandwithout spurious modes. Mech Syst Signal Process21:2715273122. ChakrabortyS, SenA(2014)Adaptiveresponsesurfacebasedefcient niteelement model updating. FiniteElemAnal Des80:334023. ChangCC, ChangTYP, XuYG(2000) Adaptiveneural net-works for model updating of structures. Smart Mater Struct9:596824. ChelliniG,RoeckGD,NardiniL,SalvatoreW(2010)Damageanalysisof asteel-concretecompositeframebyniteelementmodelupdating.JConstrSteelRes66:39841125. Chen JC, Garba JA (1980) Analytical model improvement usingmodaltestresults. AmInstAeronautstronautJ18(6):68469026. ChenJC,KuoCP,GarbaJA(1983)Directstructuralparameteridentication by modal test results. Proceedings of the 24thstructuraldynamicsandmaterialsconference.California,USA,pp444927. Collins JD, Hart GC, HasselmanTK, KennedyB(1974) Sta-tisticalidenticationofstructures. AmInstAeronautstronautJ12(2):18519028. Cunha J, Piranda J (1999) Application of model updatingtechniques in dynamics for the identication of elastic constantsofcompositematerials. ComposBEng30:798529. DecouvreurV,BelED,BouillardP(2006)Ontheeffectofthedispersion error when updating acoustic models. ComputMethodsApplMechEng195:39440530. Dhandole SD, Modak SV (2009) Simulated studies in FE modelupdating with application to vibro-acoustic analysis of thecavities, In: Proceedings of the third international conference onintegrity, reliabilityandfailure, Porto/Portugal31. Dhandole SD, Modak SV (2010) A comparative study ofmethodologiesfor vibro-acousticFEmodel updatingof cavitiesusingsimulateddata. IntJMechMaterDes6:274332. Dhandole S, Modak SV(2012) Aconstrained optimizationbased method for acoustic nite element model updating ofcavitiesusingpressureresponse.ApplMathModel36:39941333. Ewins DJ (2000) Modal testing: theory, practice and application,2ndedn.ResearchStudiesPressLimited, England, UK34. FangH, WangTJ, ChenX(2011) Model updatingof latticestructures: a substructure energyapproach. MechSyst SignalProcess25(5):1469148435. Fang SE, Perera R(2009) Aresponse surface methodologybased damage identication technique. Smart Mater Struct18:11436. FangSE, PereraR(2011) Damageidenticationbyresponsesurface basedmodel updatingusingD-optimal design. MechSystSignalProcess25:71773337. Farhat C, HemezFM(1993) Updatingniteelement dynamicmodels usinganelement-by-element sensitivitymethodology.AmInstAeronautstronautJ31(9):1702171138. FissetteE, StavrinidisC, IbrahimS(1988) Error locationandupdatingof analytical dynamic models usingaforcebalancemethod. Proceedings of thesixthinternational modal analysisconference. Orlando, USA,pp1063107039. Friswell MI, MottersheadJE(1995) Finiteelement model up-dating in structural dynamics. Kluwer Academic Publishers,Dordrecht, TheNetherlands40. Friswell MI, GarveySD, PennyJET(1995) Model reductionusing dynamic and iterated IRS techniques. J Sound Vib186(2):31132341. Friswell MI, Inman DJ, Pilkey DF (1998) The direct updating ofdampingandstiffness matrices. AmInst Aeronaut stronaut J36(3):49149342. GollerB,BroggiM,CalviA,SchuellerGI(2011)Astochasticmodel updating technique for complex aerospace structures.FiniteElemAnalDes47:73975243. GondhalekarAC, KundraTK, ModakSV(2007)Dynamicde-sign of a complex automobile structure via nite element modelupdating. Proceedingsofthe25thinternational modal analysisconference. Orlando, USA,pp161944. Guo Q, Zhang L (2004) Finite element model updating based onresponsesurfacemethodology. Proceedingsof the22ndinter-national modal analysis conference. Michigan, USA,pp1990199745. GuyanRJ(1965)Reductionof stiffnessandmassmatrices.AmInstAeronautstronautJ3(2):38038646. HartogJPD(1934)Mechanical vibrations. McGraw-Hill BookCompanyInc,USA47. He J, Ewins DJ (1991) Compatibility of measured and predictedvibrationmodesinmodelimprovementstudies. AmInstAero-nautstronautJ29(5):79880348. HelfrickMN, Niezrecki C, AvitabileP, Schmidt T(2011) 3Ddigital imagecorrelationmethodsfor full-eldvibrationmea-surement. MechSystSignalProcess25:91792749. HuSLJ,LiH,WangS(2007)Cross-modelcross-modemethodformodelupdating.MechSystSignalProcess21:1690170350. HusainNA, Khodaparast HH, SnaylamA, JamesS, SharpM,DeardenG, OuyangH(2009)Modaltestingandniteelementmodelupdatingoflaserspotwelds. JPhysConfSer181:1851. Hwang HY, Kim C (2004) Damage detection in structures usinga few frequency response measurements. J Sound Vib 270:11452. ImregunM, EwinsDJ, HagiwaraI, IchikawaTA(1994)Com-parison of sensitivity and response function based updatingtechniques. Proceedings of 12th international modal analysisconference. Hawaii, USA, pp1390140053. JacquelinE, AdhikariS,FriswellMI(2012)Asecond-momentapproachfor direct probabilistic model updatinginstructuraldynamics. MechSystSignalProcess29:26228354. Jaishi B, RenWX(2006) Damagedetectionbyniteelementmodel updatingusingmodal exibilityresidual. J SoundVib290:36938755. JamshidiE, Ashory MR(2009)Comparativestudy of RFMandmodel updating method using base excitation test data. Pro-ceedings of the27thinternational modal analysis conference.Orlando, USA, pp2399240956. Jiang J, Dai H, Yuan Y (2013) A symmetric generalized inverseeigenvalue problemin structural dynamics model updating.LinearAlgebraAppl439:1350136357. KabeAM(1985)Stiffnessmatrixadjustmentusingmodedata.AmInstAeronautstronautJ23(9):14311436StructuralDynamicModelUpdatingTechniques:AStateoftheArtReview1 358. Khanmirza E, Khaji N, Majd VJ (2011) Model updating ofmultistory shear buildings for simultaneous identication ofmass, stiffnessanddampingmatricesusingtwodifferent soft-computingmethods.ExpertSystAppl38:5320532959. Khodaparast HH, MottersheadJE, BadcockKJ(2011)Intervalmodel updatingwithirreducibleuncertaintyusingtheKrigingpredictor. MechSystSignalProcess25:1204122660. KidderRL(1973)Reductionofstructuralfrequencyequations.AmInstAeronautstronautJ11(6):89290261. KimGH, ParkYS(2008) Anautomatedparameter selectionprocedure for nite-element model updating and its applications.JSoundVib309:77879362. KimKO, AndersonWJ, SandstromRE(1983) Non-linear in-verseperturbationmethodindynamicanalysis. AmInst Aero-nautstronautJ21(9):1310131663. Kwon KS, Lin RM (2004) Frequency selection method for FRF-basedmodelupdating. JSoundVib278:28530664. Kwon KS, Lin RM (2005) Robust nite element model updatingusingTaguchimethod. JSoundVib280:779965. Lammens S, Heylen W, Sas P (1992) Model updating andstructural optimization of a tennis racket. Mech Syst SignalProcess6(5):46146866. Lepoittevin G, Kress G (2011) Finite element model updating ofvibrating structures under free-free boundary conditions formodal damping prediction. Mech Syst Signal Process 25:2203221867. LevinRI, LievenNAJ (1998) Dynamic nite element modelupdatingusingneuralnetworks.JSoundVib210(5):59360768. LevinRI, LievenNAJ (1998) Dynamic nite element modelupdating using simulated annealing and genetic algorithms.MechSystSignalProcess12(1):9112069. Li WL(2002) Anewmethod for structural model updating and jointstiffness identication. Mech Syst Signal Process 16(1):15516770. Lieven NAJ, Ewins DJ (1988) Spatial correlation of modeshapes, the coordinate modal assurance criterion (COMAC).Proceedings of the sixthinternational modal analysis confer-ence. Orlando, USA,pp69069571. Lieven NAJ, Ewins DJ (1990) Expansion of modal data forcorrelation. Proceedingsoftheeighthinternational modal ana-lysisconference. Orlando, USA, pp60560972. LievenNAJ,WatersTP(1994)Errorlocationusingnormalizedcross orthogonality. Proceedings of the 12th international modalanalysisconference. Honolulu, Hawaii,pp76176473. LimTW(1990) Submatrixapproach tostiffness matrix cor-rection using modal test data. AmInst Aeronaut stronaut J28(6):1123113074. Lin RM, Zhu J (2007) Finite element model updatingusingvibration test data under base excitation. J Sound Vib303:59661375. LinRM, LimMK, DuH(1995)Improvedinverseeigensensi-tivitymethodfor structural analytical model updating. TransASMEJVibAcoust117(2):19219876. Lin RM, Ewins DJ (1990) Model updating using FRFdata.Proceedingsof 15thinternational modal analysisseminar. KULeuven, Belgium, pp14116277. Maia NMM, Silva JM(1997) Theoretical and experimentalmodalanalysis. ResearchStudiesPressLimited, England, UK78. MaresC,MottersheadJE,FriswellMI(2006)Stochasticmodelupdating: part 1- theory and simulated example. Mech SystSignalProcess20:1674169579. MarwalaT(2005)Finiteelementmodelupdatingusingparticleswarmoptimization. IntJEngSimul6(2):253080. Marwala T, Sibisi S (2005) Finite element model updating usingBayesian framework and modal properties. J Aircr 42(1):27527881. Modak SV (2014) Direct matrix updating of vibroacoustic niteelementmodelsusingmodaltestdata. AIAAJ52(7):1782. Modak SV, Kundra TK, Nakra BC (2000) Model updating usingconstrainedoptimization. MechResCommun27(5):54355183. Modak SV, Kundra TK, Nakra BC (2002a) Prediction ofdynamiccharacteristicsusingupdatedniteelement models. JSoundVib254(3):44746784. ModakSV, KundraTK, NakraBC(2002b)Comparativestudyofmodelupdatingmethodsusingsimulatedexperimental data.ComputStruct80:43744785. ModakSV, KundraTK, NakraBC(2005)Studiesindynamicdesignusingupdatedmodels. JSoundVib281:94396486. MottersheadJE, LinkM, Friswell MI (2011) The sensitivitymethodinniteelementmodelupdating:atutorial. MechSystSignal Process 25:22752296. doi:10.1016/j.ymssp.2010.10.01287. MthembuL,MarwalaT,FriswellMI,AdhikariS(2010)Finiteelement model selection using particle swarmoptimization.Proceedings of the 28th international modal analysis conference.Florida, USA,pp415288. OChallan JC(1989) Aprocedure for an improved reducedsystem(IRS) model. Proceedings of theseventhinternationalmodalanalysisconference. LasVegas,USA, pp172189. OChallanJC,AvitabileP,RiemerR(1989)Systemequivalentreductionandexpansionprocesses. Proceedingsoftheseventhinternational modal analysis conference. Las Vegas, USA,pp293790. OChallan JC, LiewIW, Avitabile P, Madden R(1986) Anefcient methodof determiningrotational degreesof freedomfrom analytical and experimental modal data. Proceedings of thefourth international modal analysis conference. Los Angeles,USA,pp505891. Petyt M (1998) Introduction to nite element vibration analysis.CambridgeUniversityPress, Cambridge, UK92. Pilkey DF(1998) Computation of damping matrix for niteelementmodelupdating, Ph.D. thesis, VirginiaPolytechnicIn-stituteandStateUniversity, Virginia93. PradhanS, ModakV(2012)Normalresponsefunctionmethodfor mass andstiffness matrix updatingusingcomplexFRFs.MechSystSignalProcess32:23225094. Pradhan S, Modak V(2012) Amethod for damping matrixidenticationusingfrequencyresponsedata. MechSystSignalProcess33:698295. Ramamurti V, Rao MH (1998) Studies on model updating usingdifferent nite elements, the editor. J Sound Vib 211(3):50751296. RenWX, ChenHB(2010) Finiteelement model updatinginstructuraldynamicsbyusingtheresponsesurfacemethod.EngStruct32:2455246597. Sastry CVS, Mahapatra DR, Gopalakrishnan S, Ramamurthy TS(2003) An iterative system equivalent reduction expansionprocessforextractionofhighfrequencyresponsefromreducedorder niteelement model. Comput MethodsAppl MechEng192:1821184098. Shahidi S, Pakzad S (2014) Generalized response surface modelupdatingusingtimedomaindata, ASCEJournal of StructuralEngineering140(8)99. Sidhu J, Ewins DJ (1984) Correlation of nite element andmodal test studies of apractical structure. Proceedings of thesecondinternationalmodalanalysisconference.Orlando,USA,pp756762100. Sinha JK, Mujumdar PM, Moorthy RIKA(2000) parameteridenticationtechniquefor detectionof spacer locationsinanassembly of two coaxial exible tubes. Nucl Eng Des196:139151101. SmithSW, Beattie CA(1991) Secant-methodadjustment forstructuralmodels. AmInstAeronautstronautJ29(1):119126S.Sehgal, H.Kumar1 3102. SteenackersG,DevriendtC,GuillaumeP(2007)Ontheuseoftransmissibilitymeasurements for niteelement model updat-ing. JSoundVib303:707722103. TeughelsA,RoeckGD(2004)StructuraldamageidenticationofthehighwaybridgeZ24byFEmodelupdating.JSoundVib278:589610104. TuZ,LuY(2008)FEmodelupdatingusingarticialboundaryconditionswithgeneticalgorithms. ComputStruct86:714727105. Vepsa A, Haapaniemi H, LuukkanenP, Nurkkala P, Saaren-heimo A (2005) Application of nite element model updating toafeedwaterpipelineofanuclear powerplant. Nucl EngDes235:18491865106. Wahab MMA, Roeck GD, Peeters B (1999) Parameterization ofdamageinreinforcedconcretestructuresusingmodelupdating.JSoundVib228(4):717730107. WangW, MottersheadJE, MaresC(2009)Mode-shaperecog-nition and nite element model updating using the Zernikemomentdescriptor. MechSystSignalProcess23:20882112108. Waters TP (1998) A modied force balance method for locatingerrors in dynamic nite element models. Mech Syst SignalProcess12(2):309317109. Weber B, Paultre P, Proulx J (2009) Consistent regularization ofnonlinearmodelupdatingfordamageidentication.MechSystSignalProcess23:19651985110. WilliamsEJ, GreenJSA(1990) spatial curvettingtechniquefor estimating rotationaldegrees of freedom. Proceedings of theeighthinternational modal analysisconference. Orlando, USA,pp376381111. XiaminZ(1999)Abest matrixapproximationmethodforup-datingtheanalyticalmodel. JSoundVib223(5):759774112. ZapicoJL, BuelgaAG, GonzalezMP, AlonsoR(2008)Finiteelement model updating of a small steel frame using neuralnetworks. SmartMaterStruct17:111113. ZhouJ, FarquharT(2005)Wheatstemmoduliinvivoviaref-erencebasismodelupdating. JSoundVib285:11091122StructuralDynamicModelUpdatingTechniques:AStateoftheArtReview1 3