The Use of Component Mode Synthesis Techniques for Large Finite Element ModelUpdating Using Dynamic Response Data

Hector A. Jensen, Eduardo Millas, Danilo KusanovicDepartment of Civil Engineering, Santa Maria University, Valparaiso, Chile

email: hector.jensen@usm.cl

ABSTRACT: This paper introduces a methodology that integrates a model reduction technique into a finite element model updatingformulation. A Bayesian model updating approach based on a stochastic simulation method is considered in the present work.Stochastic simulation techniques require a large number of finite element model re-analyses to be performed over the space ofmodel parameters. Substructure coupling techniques for dynamic analysis are proposed to reduce the computational cost involvedin the dynamic re-analyses. The effectiveness of the proposed strategy is demonstrated with an identification application for a finiteelement building model using simulated seismic response data.

KEY WORDS: Bayesian updating; Component mode synthesis; Finite element; Model updating; Transitional Markov ChainMonte Carlo method.

1 INTRODUCTION

Model updating using measured system response has a widerange of applications in areas such as structural responseprediction, structural control, structural health monitoring, andreliability and risk assessment [1], [2], [3], [4], [5]. Fora proper assessment of the updated model all uncertaintiesinvolved in the problem should be considered. In this context afully probabilistic Bayesian model updating approach providesa robust and rigorous framework for model updating due to itsability to characterize modeling uncertainties associated withthe underlying structural system [6]. An efficient methodcalled transitional Markov chain Monte Carlo is implementedin this work for Bayesian model updating [7]. This stochasticsimulation method requires the solution of a large number offinite element dynamic re-analyses over the space of modelparameters. Thus, the computational demands depend on thenumber of finite element re-analyses and the time requiredfor performing each dynamic finite element analysis. Thepresent work proposes to use an efficient model reductiontechnique to alleviate the computational burden involved inthe implementation of a Bayesian technique for finite elementmodel updating. Specifically, a class of model reductiontechniques known as substructure coupling for dynamic analysisis considered here [8].

The organization of the paper is as follows. The mathematicalbackground of the substructure coupling technique for dynamicanalysis is outlined in Section 2. Basic aspects of Bayesian finiteelement model updating using dynamic data are presented inSection 3. The integration of the model reduction techniquewith the Bayesian model updating approach is discussed inSection 4. The effectiveness of the proposed scheme, in termsof computational efficiency and accuracy, is demonstrated withan identification application for a finite element building modelusing simulated seismic response data.

2 MODEL REDUCTION TECHNIQUE

A model reduction technique called substructure couplingor component mode synthesis is considered in this work[8]. Sub-structuring involves dividing the structure into anumber of substructures obtaining reduced-order models of thesubstructures and then assembling a reduced-order model of theentire structure.

2.1 Basic Equations

In the present formulation it is assumed that the structuralsystem satisfies the following equation of motion

Mu(t)+Cu(t)+Ku(t) = f(t) (1)

where M, C, and K are the N ×N mass, damping and stiffnessmatrices of the finite element model, respectively, u(t) is thevector of dynamic displacements, and f(t) is the excitationvector. The first step of the model reduction technique is thedefinition of a number of substructure modes. In order to definethe set of substructure modes, the following partitioned form ofthe mass matrix Ms ∈ Rns×ns

and stiffness matrix Ks ∈ Rns×nsof

the substructure s,s = 1, ...,Ns is considered

Ms =

[Ms

ii Msib

Msbi Ms

bb

], Ks =

[Ks

ii Ksib

Ksbi Ks

bb

](2)

where the indices i and b are sets containing the internaland boundary degrees of freedom of the superstructure s,respectively. The boundary degrees of freedom include onlythose that are common with the boundary degrees of freedomof adjacent substructures, while the internal degrees of freedomare not shared with any adjacent substructure. In this frameworkall boundary coordinates are kept as one set us

b(t) ∈ Rnsb and the

internal coordinates in the set usi (t) ∈ Rns

i . The correspondingequation of motion of the undamped substructure s can bewritten as

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

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Msus(t)+Ksus(t) = fs(t) (3)

where us(t)T =< usi (t)

T ,usb(t)

T >∈ Rnsis the displacement

vector (physical coordinates) of dimension ns = nsi + ns

b andthe vector fs(t) contains the externally applied forces as wellas the reaction forces on the substructure due to its connectionto adjacent substructures at boundary degrees of freedom.

2.2 Fixed-Interface and Constraint Normal Modes

The fixed-interface normal modes are obtained by restraining allboundary degrees of freedom and solving the eigenproblem

KsiiΦΦΦ

sii = Ms

iiΦΦΦsiiΛΛΛ

sii (4)

where the matrix ΦΦΦsii contains the complete set of ns

i fixed-interface normal modes, and ΛΛΛs

ii is the corresponding matrixcontaining the eigenvalues. The fixed-interface normal modesare normalized with respect to the mass matrix Ms

ii satisfying

ΦΦΦsii

T MsiiΦΦΦ

sii = Is

ii , ΦΦΦsii

T KsiiΦΦΦ

sii = ΛΛΛs

ii (5)

where Isii is the identity matrix. On the other hand the constraint

modes are defined as the static deformation of a structure whena unit displacement is applied to one coordinate of a specifiedset of constraint coordinates while the remaining coordinates ofthat set are restrained, and the remaining degrees of freedomof the structure are force-free. In this context the interfaceconstraint modes are obtained by setting a unit displacement onthe boundary coordinates us

b(t) and zero forces in the internaldegrees of freedom. The set of interface constraint-modes ΨΨΨs

c isgiven by

ΨΨΨsc =

[ΨΨΨs

ibIs

bb

]=

[−Ks

ii−1Ks

ibIs

bb

](6)

where ΨΨΨsib ∈ Rns

i×nsb is the interior partition of the constraint-

mode matrix and Isbb ∈ Rns

b×nsb is the identity matrix.

2.3 Craig-Bampton Method

The Craig-Bampton method is used in the present formulationto define a set of generalize coordinates [9]. This methodemploys a combination of fixed-interface normal modes andinterface constraint modes to define the following displacementtransformation

us(t) ={

usi (t)

usb(t)

}=

[ΦΦΦs

ik ΨΨΨsib

0sbk Is

bb

]{vs

k(t)vs

b(t)

}= ΨΨΨs vs(t)

(7)where ΦΦΦs

ik ∈ Rnsi×ns

k is the interior partition of the matrix ΦΦΦsii

of the nsk kept fixed-interface normal modes (ns

k ≤ nsi ), vs(t)

represents the substructure generalized coordinates composedby the modal coordinates vs

k(t) of the kept fixed-interface normalmodes and the boundary coordinates vs

b(t) = usb(t), ΨΨΨs ∈ Rns×ns

is the Craig-Bampton transformation matrix with ns = nsk +

nsb and all other terms have been previously defined. The

substructure mass matrix Ms ∈ Rns×nsand stiffness matrix Ks ∈

Rns×nsin generalized coordinates vs(t) are given by

Ms= ΨΨΨsT MsΨΨΨs , Ks

= ΨΨΨsT KsΨΨΨs (8)

Next, the vector of generalized coordinates for all the Nssubstructures

v(t)T =< v1(t)T , ...,vNs(t)T > ∈ Rnv (9)

where nv = ∑Nss=1 ns is introduced. Based on this vector,

a new vector q(t) that contains the independent generalizedcoordinates consisting of the fixed-interface modal coordinatesvs

k(t) for each substructure and the physical coordinatesvl

b(t), l = 1, ...,Nb at the Nb interfaces is defined as

q(t)T =< v1k(t)

T , ...,vNsk (t)T ,v1

b(t)T , ...,vNb

b (t)T > ∈ Rnq (10)

where nq = ∑Nss=1 ns

k +∑Nbl=1 nl

b, and nlb is the number of degrees

of freedom at the interface l (l = 1, ...,Nb). These two vectorsare related by the transformation

v(t) = Tq(t) (11)

where the matrix T ∈ Rnv×nq is a matrix of zeros and onesthat couples the independent generalized coordinates q(t) ofthe reduced system with the generalized coordinates of eachsubstructure. The assembled mass matrix M ∈ Rnq×nq and thestiffness matrix K ∈ Rnq×nq for the independent reduced set q(t)of generalized coordinates take the form

M = TT

M1 0 0

0. . . 0

0 0 MNs

T , (12)

K = TT

K1 0 0

0. . . 0

0 0 KNs

T (13)

where the matrices Msand Ks

,s = 1, ...,Ns are defined in Eq.(8).

2.4 System Response

The dynamic response of the finite element model of the originalsystem is obtained by modal solution in the present formulation.In this approach it is assumed that the dynamic response can berepresented by a linear combination of the mode shapes as

u(t) = ϒϒϒηηη(t) (14)

where u(t) is the displacement vector of the original structure, ϒϒϒis the matrix of mode shapes associated with the eigen-problemof the undamped equation of motion of the original system, andηηη(t) is the vector of modal response functions. The naturalfrequencies of the original unreduced model ωr,r = 1, ...m,where m is the number of modes considered are obtained bysolving the eigen-problem of the reduced-order system model

(K−ω2r M)υυυqr = 0 , r = 1, ...,m (15)

where υυυqr,r = 1, ...,m are the mode shapes of the reduced-ordersystem. Introducing a constant matrix T ∈ RN×nu (nu = ∑Ns

s=1 ns)to map the vector

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u(t)T =< u1(t)T , ...,uNs(t)T >∈ Rnu (16)

of the physical coordinates for all substructures to theindependent physical coordinates u(t) of the original structure,the physical mode shapes υυυr of the structure can be recoveredfrom the mode shapes υυυqr as

υυυr = TΨΨΨTυυυqr , r = 1, ...,m (17)

where the matrix ΨΨΨ ∈ Rnu×nv is a block diagonal matrix definedin terms the Craig-Bampton transformation matrices of allsubstructures, that is, ΨΨΨ = blockdiag(ΨΨΨ1, ...,ΨΨΨNs).

3 BAYESIAN FINITE ELEMENT MODEL UPDATING

3.1 Problem Formulation

Consider a parameterized finite element model class M of astructural system by a set of model parameters θθθ ∈ Θ ⊂ Rnp .The plausibility of each model within a class M based on dataD is quantified by the updated joint probability density functionp(θθθ |M,D) (posterior probability density function). By Bayes’Theorem [6], [10] the posterior probability density function ofθθθ is given by

p(θθθ |M,D) =p(D|M,θθθ) p(θθθ |M)

p(D|M)(18)

where p(D|M) is the normalizing constant which makes theprobability volume under the posterior probability densityfunction equal to unity, p(D|M,θθθ) is the likelihood functionbased on the predictive probability density function for theresponse given by model class M, and p(θθθ |M) is the priorprobability density function selected for the model class M.In what follows it is assumed that D contains input dynamicdata and output responses from measurements on the system.Specifically let un(t j,θθθ) denotes the output at time t j at the nth

observed degree of freedom predicted by the proposed structuralmodel, and u∗n(t j) denotes the corresponding measured output.The prediction and measurement errors

en(t j,θθθ) = u∗n(t j)−un(t j,θθθ) , n = 1, ...,No , j = 1, ...,Nt (19)

where No denotes the number of observed degrees of freedomand Nt denotes the length of the discrete time history data, aremodeled as independent and identically distributed Gaussianvariables with zero mean [3], [11]. Using the above probabilitymodel for the prediction error it can be shown that the likelihoodfunction p(D|M,θθθ) can be expressed in terms of a measure-of-fit function J(θθθ |M,D) between the measured response andthe model response at the measured degrees of freedom. Suchfunction is given by [11], [12]

J(θθθ |M,D) =1

NtNo

No

∑n=1

Nt

∑j=1

[u∗n(t j)−un(t j,θθθ)]2 (20)

For a large number of available data (NtN0 is large) it has beenfound that the most probable model parameters are obtained byminimizing J(θθθ |M,D) over all parameters in Θ that it dependson [12]. Under the previous assumption the posterior probability

density function p(θθθ |M,D) is in general concentrated in theneighborhood of a lower dimensional manifold in the parameterspace [13]. In general the problem of estimating the modelparameters is potentially ill-posed, that is, there may be morethan one solution. To solve this problem a simulation-based Bayesian model updating technique is adopted in thisstudy. Such technique, which efficiently generates samplesasymptotically distributed as the posterior probability densityfunction, is described in the following section.

3.2 Simulation-Based Approach

An efficient method called transitional Markov chain MonteCarlo is implemented in this work for Bayesian model updating[7]. Validation calculations have shown the effectiveness ofthis approach in a series of practical Bayesian model updatingproblems [14]. The method can be applied to a wide range ofcases including high-dimensional posterior probability densityfunctions, multimodal distributions, peaked probability densityfunctions, and probability density functions with flat regions.The method iteratively proceeds from the prior to the posteriordistribution. It starts with the generation of samples from theprior distribution in order to populate the space in which also themost probable region of the posterior distribution lies. For thispurpose a number of non-normalized intermediate distributionsp j(θθθ |M,D), j = 1, ...,J, are defined as

p j(θθθ |M,D) ∝ p(D|M,θθθ)α j p(θθθ |M) (21)

where the parameter α j increases monotonically with j suchthat α0 = 0 and αJ = 1. The parameter α j can be interpretedas the percentage of the total information provided by thedynamic data which is incorporated in the jth iteration of theupdating procedure.The first step ( j = 0) corresponds to theprior distribution and in the last stage ( j = J) the samples aregenerated from the posterior distribution. The idea is to choosethe values of exponents α j in such a way that the change of theshape between two adjacent intermediate distributions be small.This small change of the shape makes it possible to efficientlyobtain samples from p j+1(θθθ |M,D) based on the samples fromp j(θθθ |M,D). The value of the parameter α j+1 is chosen suchthat the coefficient of variation for {p(D|M,θθθ k

j)α j+1−α j ,k =

1, ...,N j} is equal to some prescribed target value. The upperindex k = 1, ...,N j in the previous expression denotes the samplenumber in the jth iteration step (θθθ k

j,k = 1, ...,N j). Once theparameter α j+1 has been determined the samples are obtainedby generating Markov chains where the lead samples areselected from the distribution p j(θθθ |M,D). Each sample of thecurrent stage is generated by applying the Metropolis-Hastingsalgorithm [15]. The lead sample of the Markov chain is asample from the previous step that is selected according to theprobability equal to its normalized weight

w(θθθ kj) =

w(θθθ kj)

∑N jl=1 w(θθθ l

j)(22)

where w(θθθ kj) represents the plausibility weight which is given

by

w(θθθ kj) = p(D|M,θθθ k

j)α j+1−α j (23)

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For a detailed implementation of the transitional Markovchain Monte Carlo method, including proofs concerning thestatistical properties of the estimators the reader is referred to[7], [16].

4 MODEL UPDATING PROCESS

In the present formulation it is assumed that the stiffness matrixof the original system depends linearly on the model parametersθθθ . This is the case encountered in many practical applicationssuch as model updating, structural optimization and damagedetection techniques. This linear dependence implies that at thesubstructure level the stiffness matrix as well as its partitionsadmit a similar characterization. In particular the following twocases are considered here. In the first case it is assumed thatthe stiffness matrix of a substructure s does not depend on themodel parameters. In this case the stiffness matrix is writtenas Ks = Ks

0. The corresponding normal and constraint modesare computed once for the corresponding model. In the secondcase it is assumed that the stiffness matrix of a substructure sdepends only on one model parameter, say θ j ( j-th componentof θθθ ). Specifically, let S j be the set of substructures that dependson θ j. Then the stiffness matrix of a substructure s ∈ S j can bewritten as Ks = Ksθ j, where the matrix Ks is independent ofθ j. Of course the partitions of the stiffness matrix Ks admitthe same parametrization. From this representation it is easy toshow that the eigenvalues and eigenvectors associated with thefixed-interface normal modes are given by

ΛΛΛsii = ΛΛΛs

iiθ j , ΦΦΦsii = ΦΦΦs

ii (24)

where the matrices ΛΛΛsii and ΦΦΦs

ii are the solution of the eigen-problem

KsiiΦΦΦ

sii = Ms

iiΦΦΦsiiΛΛΛ

sii (25)

where the matrices ΦΦΦsii and ΛΛΛs

ii are independent of θ j. Inaddition, the interface constraint modes are also independent ofθ j since

ΨΨΨsib =−Ks

ii−1Ks

ib =−Ksii−1Ks

ib (26)

Therefore a single substructure analysis is required to providethe exact estimate of the normal and constraint modes for anyvalue of the modal parameter θ j [17]. Based on the previousresults it is simple to verify that the reduced stiffness matrix ofthe substructure s, considering the full set of interface degreesof freedom, takes the form

Ks= ˆKsθ j , ˆKs = ΨΨΨsT KsΨΨΨs (27)

Thus, it is clear that the reduced matrix ˆKs is a constantmatrix independent of the model parameters θθθ . Consequentlythe stiffness matrix of the Craig-Bampton reduced system canbe written as

K = K0 +Nθ

∑j=1

ˆK jθ j (28)

where Nθ is the number of independent model parameters andthe matrices K0 and ˆK j, j = 1, ...,Nθ are defined as

K0 = TT

K1

0δ10 0 0

0. . . 0

0 0 KNs0 δNs0

T , (29)

ˆK j = TT

ˆK1δ1 j 0 0

0. . . 0

0 0 ˆKNsδNs j

T (30)

where δs0 = 1 if the substructure s does not depend on the modelparameters θθθ and δs0 = 0 otherwise, δs j = 1 if the substructure sdepends on the parameter θ j and δs j = 0 otherwise, and all otherterms have been previously defined. Note that the assembledmatrices ˆK j, j = 1, ...,Nθ are independent of the value of θθθ andtherefore these matrices are computed and assembled once. Thisresults in substantially savings since there is no need to definethese matrices during the identification process.

5 NUMERICAL EXAMPLE

5.1 Problem Description

The structural system shown in Figure (1) consists of a tenfloors three-dimensional reinforced concrete building. Materialproperties of the reinforced concrete structure have beenassumed as follows: Young’s modulus E = 2.34× 1010 N/m2;Poisson ratio ν = 0.3; and mass density ρ = 2500 kg/m3.The height of each floor is 3.5 m leading to a total heightof 35.0 m for the structure. The floors are modeled withshell elements with a thickness of 0.3 m and beam elementsof rectangular cross section of dimension 0.3m × 0.6m fromfloors 1 to 5 and 0.25m × 0.5m from floors 6 to 10. Eachfloor is supported by 48 columns of rectangular cross sectionof dimension 0.8m × 0.9m. The corresponding finite elementmodel has approximately 40,000 degrees of freedom. A 2% ofcritical damping for the modal damping ratios is introduced inthe model.

y

z

x

Figure 1. Isometric view of the finite element model.

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For the identification application considered in this examplethe structure is divided into a number of substructures and it isassumed that a stiffness reduction is concentrated in one or moresubstructures producing changes in the stiffness characteristicsof those substructures. In particular the structural model issubdivided into six substructures. Substructures 1, 3 and 5 arecomposed by the column elements of the first, second and thirdfloor, respectively. Substructures 2 and 4 correspond to the slabsand beam elements of the first and second floor, respectively, andsubstructure 6 contains the upper floors structural components(columns, beams and slabs). For illustration purposes a modelclass M is introduced to monitor the updated process, which isdefined in terms of a stiffness reduction of 20% of the nominalstiffness value in the x direction of the columns of the firstfloor. The model class M contains two parameters (θ1,θ2)associated with the stiffness of the columns of the first andsecond floor, respectively. The model class is characterizedfrom the unchanged or nominal structure (θi = 1, i = 1,2)corresponding to the reduced-order model.

The identification process is based on simulated data. Tothis end the original unreduced finite element model is excitedhorizontally (in the x direction) with the Santa Lucia ground-motion record recorded during the 2010 Chilean earthquake.The input ground acceleration time history is shown in Figure(2). It corresponds to a ground motion of moderate intensity.The measured response is simulated by first calculatingthe absolute acceleration response of the modified originalunreduced structure at floors 1, 2, 3 and 4 (in the x direction)and then adding Gaussian discrete white noise with standarddeviation equal to 10% of the root-mean square value ofthe corresponding absolute acceleration time histories. Theresponses are computed at the center of mass of the floors. Onehundred seconds of data with sampling interval ∆t = 0.05 s areused, given a total of Nt = 2000 data points.

0 10 20 30 40 50 60 70 80 90 100−0.5

0

0.5

acce

lera

tion

[g]

time [s]

Figure 2. Santa Lucia ground-motion record (2010 Chileanearthquake)

The simulated response data provides the data for the modelupdating process. The actual implementation of this processis carried out by using a reduced-order system model which isdefined as follows. For each substructure of the finite elementmodel it is selected to retain all fixed-interface normal modesthat have frequency less than a given cut-off frequency. The cut-off frequency is set proportional to the 12th modal frequencyof the original unreduced finite element model. Validationcalculations show that retaining a total of 365 internal degreesof freedom for all substructures is adequate in the contextof this application. In fact, with this number of generalizedcoordinates the fractional error (in percentage) between the

modal frequencies using the complete finite element model andthe modal frequencies computed using the reduced-order modelfalls bellow 0.1% for the lowest 12 modes. Then, a total of 365interior degrees of freedom, corresponding to the fixed interfacegeneralized coordinates, out of 39,136 of the original modelare retained for the all substructures. The number of interfacedegrees of freedom is equal to 864 in this case. The totalnumber of degrees of freedom of the reduced model representsa 97% reduction with respect to the unreduced model. Thus, asignificant reduction in the number of generalized coordinates isobtained with respect to the number of the degrees of freedomof the original unreduced finite element model.

5.2 Results

The model updating is performed using the transitional Markovchain Monte Carlo method with 1000 samples per stage. Theprior probability density function for the model parametersθi, i = 1,2 are independent uniform distributions defined overthe interval [0.5,1.5]. The reference structure (unchanged) ischaracterized in terms of the model parameters with valuesequal to θi = 1, i = 1,2. Figures (3) and (4) show the histogramsdefined by the posterior samples of the model parameters. Inaddition, the values of the nominal system parameters are alsoindicated in the figures.

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

50

100

150

θ1

Frequency

Nominal Model

Parameter

Figure 3. Posterior histogram of model parameter θ1. Meanestimate: θ1 = 0.79

It is seen that the agreement between the actual system and themodel characterized by the posterior samples is excellent. Themodel parameter θ1 is distributed around the value θ1 = 0.8. Infact the mean estimate of this parameter is equal to θ1 = 0.79.On the other hand it is observed that the parameter related tothe stiffness of the columns of the second floor is distributedaround its actual value θ2 = 1.0. This is reasonable since thesecolumns are unchanged. To get insight into the identificationprocess Figures (5-7) show how the samples in the θ1−θ2 spaceconverge. From the different steps or stages of the transitionalMarkov chain Monte Carlo method it is observed that the datais strongly correlated along a certain direction in the parameterspace. Such correlation shows that an increase in the stiffnessof the columns of the first floor is compensated by a decreasein the stiffness of the columns of the second floor during the

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0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

50

100

150

θ2

Frequency

Nominal Model

Parameter

Figure 4. Posterior histogram of model parameter θ2. Meanestimate: θ2 = 1.03

identification process, which is consequent from a structuralpoint of view. Thus, all points along that direction correspondto structural models that have almost the same response at themeasured degrees of freedom.

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

θ1

θ2

prior

Figure 5. Plots of the samples in the θ1 − θ2 space. Priordistribution

The number of finite element runs required for theidentification process depends on the number of transitionalMarkov chain Monte Carlo stages which in this case is equalto 8. The total computational time for the identification usingthe original unreduced finite element model is expected to be ofthe order of 3 days. In contrast, for the reduced-order modelthe computational demand is reduced to less than 4 hours.Thus, a drastic reduction in computational efforts is achievedwithout compromising the predictive capability of the proposedidentification methodology.

6 CONCLUSIONS

A methodology that integrates a model reduction technique intoa finite element model updating formulation using dynamicresponse data has been presented. In particular, a method basedon fixed-interface normal component modes plus interfaceconstraint modes is considered in this work. In general, the

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

θ1

θ2

step 4

Figure 6. Plots of the samples in the θ1−θ2 space. Fourth stageof the transitional Markov chain Monte Carlo method

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

θ1

θ2

step 8

Figure 7. Plots of the samples in the θ1−θ2 space. Eighth stageof the transitional Markov chain Monte Carlo method

method produces highly accurate models with relatively fewcomponent modes. The finite element model updating is carriedout by using a simulation-based Bayesian model updatingtechnique. Specifically, the transitional Markov chain MonteCarlo method is implemented in the present formulation. Itis demonstrated that the fixed-interface normal mode of eachcomponent and the characteristic interface modes are computedonly once from a reference finite element model. In thismanner the re-assembling of the reduced-order system matricesfrom components and interfaces modes is avoided during theupdating process. Results show that the computational effortfor updating the reduced-order model is decreased drastically bytwo or three orders of magnitude with respect to the unreducedmodel, that is, the full finite element model. Furthermore,the drastic reduction in computational efforts is achievedwithout compromising the predictive capability of the proposedidentification methodology.

ACKNOWLEDGMENTS

The research reported here was supported in part by CONICYT(National Commission for Scientific and Technological Re-

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search) under grant number 1110061. This support is gratefullyacknowledged by the authors.

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