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Mechanical Systemsand

Signal ProcessingMechanical Systems and Signal Processing 20 (2006) 1674–1695

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doi:10.1016/j.

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Stochastic model updating: Part 1—theory andsimulated example

C. Maresa, J.E. Mottersheadb,�, M.I. Friswellc

aSchool of Engineering and Design, Brunel University, Uxbridge, Middlesex UB8 3PH, UKbDepartment of Engineering, University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK

cDepartment of Aerospace Engineering, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, UK

Received 22 October 2004; received in revised form 22 June 2005; accepted 26 June 2005

Available online 1 September 2005

Abstract

The usual model updating method may be considered to be deterministic since it uses measurements froma single test system to correct a nominal finite element model. There may however be variability inseemingly identical test structures and uncertainties in the finite element model. Variability in test structuresmay arise from many sources including geometric tolerances and the manufacturing process, and modellinguncertainties may result from the use of nominal material properties, ill-defined joint stiffnesses and rigidboundary conditions. In this paper, the theory of stochastic model updating using a Monte-Carlo inverseprocedure with multiple sets of experimental results is explained and then applied to the case of a simulatedthree degree-of-freedom system, which is used to fix ideas and also to illustrate some of the practicallimitations of the method. In the companion paper, stochastic model updating is applied to a benchmarkstructure using a contact finite element model that includes common uncertainties in the modelling of thespot welds.r 2005 Elsevier Ltd. All rights reserved.

Keywords: Variability; Uncertainty; Model updating

see front matter r 2005 Elsevier Ltd. All rights reserved.

ymssp.2005.06.006

ding author. Tel.: +44 0151 794 4827; fax: +44 0151 794 4848.

ress: j.e.mottershead@liverpool.ac.uk (J.E. Mottershead).

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1. Introduction

In the usual ‘deterministic’ model updating method a single finite element model is optimised byminimising the error between predicted results and test data from a single physical structure [1,2].The choice of updating parameters is an important aspect of the process and should always bejustified physically. Model uncertainties should be located and parameterised sensitively to thepredictions. Finally, the model should be validated by assessing the model quality within its rangeof operation and its robustness to modifications in the loading configuration, design changes,coupled structure analysis and different boundary conditions. But predictions based on a singlecalibration of the model parameters cannot give a measure of confidence in the capability ofnumerical simulations to represent the actual structure. The credibility of the structural modelmust combine three components: (a) an assessment of the fidelity of predictions to test data; (b) anassessment of robustness to variability, uncertainty and lack of knowledge and (c) an assessmentof prediction accuracy in situations in which the test measurements are not available. The threegoals of fidelity-to-data, robustness-to-uncertainty, and confidence-in-prediction are antagonisticand a trade-off has to be achieved [3].The parameter estimation problem can be presented within a statistical/probabilistic framework

basically in two different ways corresponding to the frequency interpretation of probability or thedegree of belief interpretation. The maximum likelihood (or maximum log likelihood) approachconsists of determining the maximum of the conditional probability of the parameters on the basisof known random output measurements; this is a frequency approach. Bayesian methods, on theother hand require an additional input not needed by the maximum likelihood method, a priorprobability distribution for the parameters, which embodies our judgement of how plausible it isthat the parameters should have certain values. The selection of the priors is an extremelycontroversial aspect of the method since it is a subjective, degree of belief, judgement. In somerespects this seems to be similar to the use of Tikhonov regularisation in deterministic modelupdating [4].Amongst the earliest papers dealing with finite element model updating, the seminal work by

Collins et al. [5] adopted a linearised sensitivity approach with the statistics of the unknownparameters determined from vibration measurements with random errors. The approach, a ‘bestlinear unbiased estimator’, can be considered equivalent to a weighted least-squares method withthe weighting matrix given by the inverse data covariance matrix. More recently, Beck and hiscolleagues [6–8] developed a model updating approach using Bayesian inference. As with the workof Collins et al. [5] the statistics of the uncertain finite element parameters are determined on thebasis of randomness in the measurements from a single test piece, i.e. randomness due tomanufacturing and material variability in a number of nominally identical test structures is notconsidered. In fact, this latter variability is very much more significant than measurement noise.The treatment of uncertainty and quantification of errors is in general a two-step process, the

first step being the identification of all uncertainty and error sources whether they originate fromthe modelling assumptions, numerical computations or physical experiments. The second step isthe assessment and propagation of the most significant uncertainties and errors through themodelling and simulation process to obtain the predicted response quantities. Mosegaard andTarantotla [9] provide an introduction and thorough discussion of Monte-Carlo samplingtechniques and their application in probabilistic parameter-estimation inverse problems. Neal [10]

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discusses the Markov Chain Monte-Carlo method, used for the solution of integrals arising inBayesian inference and having applications including neural networks and simulated annealing.Multiple realisations of an experiment (numerical or physical) lead to the concept of the meta-

model [11–13] and the possibility to express the distance between models and operate designmodifications based on statistical concepts as opposed to the comparison between deterministicmodels based on nominal variables. As a result of Monte-Carlo simulation, the meta-modelrepresents a source for a statistical problem description, confidence measures, correlation withexperimental data, global dependencies and selection of dominant design variables. Possibilitiesfor reanalysis and model reduction for large parameterised models are discussed by Balmes et al.[14]. Fonseca et al. [15] used a maximum likelihood method to solve the inverse problem of acantilever with a lumped mass at an uncertain position.In the preceding discussion, randomness is confined to discrete parameters whereas in practice

it may be distributed over areas or volumes, such as in the case of uncertainty in the thickness of aplate dependent upon spatial coordinate, and should therefore be properly represented by arandom field. This is a digression from the main theme of the present paper, but a very topical onebecause of the spectral stochastic finite element method (SSFEM), for the forward analysis ofrandom-field problems. Details of SSFEM are given by Ghanem and Spanos [16], Schueller [17]and Matthies and Keese [18] and Deb et al. [19]. Ghanem and Red-Horse [20] carried out avibration analysis of a space-frame with joints having distributed random material properties. Alluncertainty propagation techniques rely on large amounts of computation. The SSFEM isparticularly demanding computationally.In this article, a stochastic model updating method is described. It is supposed that multiple sets

of test data are available from many structures built in the same way from the same materials, butwith manufacturing and materials variability. A finite element model is also available containingmodelling uncertainties. Parameters are selected together with Gaussian distributions and arepropagated through the model using Monte-Carlo analysis to provide multiple sets of predictedresults. The sets of predicted results are generally non-linear functions of the randomisedparameters, which must be chosen together with their distributions using engineering judgement. Itis assumed that the randomised parameters are able to account for both the variability in physicaltest pieces and uncertainty in the model. In ‘deterministic’ model updating the sensitivities of asingle set of test data to the chosen parameters are determined and the same parameters arecorrected iteratively using an objective function based on a truncated first-order Taylor seriesexpansion. In the stochastic approach a linear model is fitted to the multiple sets of predictedresults using multivariate multiple regression thereby producing a parameter sensitivity matrix thattakes into account the complete population of randomised values for all of the parameterstogether. The distance between the mean values of the experimental data and predictions is thenminimised using the gradient-and-regression approach to obtain improved estimates of the meanvalues of the randomised parameters at each step. The knowledge of the randomised parameters isincreased iteratively and this allows an improved estimate of the co-variance matrix a posteriori.The theory is applied to a simulated three degree-of-freedom mass–spring system thereby allowingthe various practical assumptions to be tested on a simple example. Complete sets of results arepresented that allow, for example, an assessment of the consequences of wrongly chosenrandomised parameters. The stochastic model updating method is applied to a spot weldedstructure, representative of many automotive body components in the companion paper [21].

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2. Theory

In this section the theoretical details, which provide a framework for practical implemen-tation of stochastic model updating, are explained. The section is separated into two parts dealingwith (i) the inverse Monte-Carlo propagation of physical-structure variability and modeluncertainty and (ii) regression analysis used to obtain a linearised model for gradientoptimisation.

2.1. Inverse error propagation

The general representation of a mechanical system includes a set of physical parameters x

whose possible variations Dx around a nominal value x0 create a model of uncertainty:

x ¼ x0 þ Dx. (1)

These variables can be different structural parameters such as material and geometricalproperties or aspects of modelling related to the boundary conditions, etc. and their variation maybe correlated or independent.For a finite element model the global matrices are expressed as linear combinations of constant

element or substructure matrices multiplied by the variable updating parameters (termed ‘inputs’in the following analysis) applying to all the terms in the substructure stiffness (or mass) matrix.More sensitive geometric parameters are studied for the case of complex joints [22–24] or byeigenvalue decomposition of the stiffness matrix and modification of its eigenvalues andeigenvectors [25,26]. For example, a general parameterisation for the stiffness matrix may bewritten as

KðxÞ ¼X

xiKei (2)

with similar decompositions for the mass or damping matrices. The choice of updating para-meters and of variation bounds for them, which can be justified physically, is one of themost important aspects of an analysis. The description of updating-parameter variabilityby a probability distribution requires access to considerable amounts of experimental datathat are not always accessible. In the case of equivalent models where the parameters aremodel dependent (such as generic-element eigenvalues) a possible assumption which couldbe used is that of a uniform probability distribution over an interval justified by designconsiderations.In the Monte-Carlo process a random parameter vector obtained from the parameter

distribution can be used to characterise the uncertainty in the output variables of interest. Therandom output variables y may be physical quantities observed at a single location, a singlephysical quantity observed at p locations, or at p time instants, or combinations of these. Thisprocess produces a posterior predictive distribution and can be used for an inverse estimate whencompared to the experimental data obtained from measurements on the actual system.The mean value of each of the n outputs is obtained from

yi ¼1

n

Xn

k¼1

yki ¼1

nyTi en; i ¼ 1; 2; . . . ; p, (3)

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where yki is the kth term in the vector yi 2 <n�1 of observations of the ith output

en ¼ 1 1 ; . . . ; 1� �T

; eTn en ¼ n and yi ¼ y1iy2i; . . . ; yni

� �T. (4)

Similarly, the vector of mean outputs can be assembled as

y ¼1

ny1 y2 ; . . . ; yp

h iTen ¼

1

nYTen. (5)

The sample mean vector y 2 <p�1 is an unbiased estimate of the population mean vector ly. Forthe observed p outputs, the covariance matrix comprising the sample variances and covariances isdetermined according to

S ¼1

n� 1½ y1 y2 ; . . . ; yp �

T I �1

nene

Tn

� �½ y1 y2 ; . . . ; yp �

¼1

n� 1½YTY� nyyT� ð6Þ

The sample covariance matrix S is an unbiased estimator of the population covariance R.The system may be described by a generally non-linear functional relationship between the

random input and output variables at the kth observation of p outputs and q inputs

yk ¼ fðxkÞ; yk ¼ ½yk1; yk2; . . . ; ykp�T; xk ¼ ½xk1; xk2; . . . ;xkq�

T. (7)

The reader should be aware that the vectors yi in Eq. (3) and yk in Eq. (7) are of dif-ferent dimensions and contain different information. Whereas yi is the ith column of Y, yTk is thekth row.Inverse propagation may be carried out by comparing the means of the jth-iteration output

vector yj with the desired mean output y. The gradient method yields a linearised version of Eq.(7) by a Taylor expansion about the mean vector of the sample xj at the jth linearisation point

yffi fðxjÞ þdf

dxðxjþ1 � xjÞ ¼ yj þGðxjþ1 � xjÞ; G ¼

df

dx

����x¼xj

, (8)

where xj 2 <q�1; y; yj 2 <

p�1 and the subscript k has been dropped. Eq. (8) differs from themethod of Collins et al. [5] who consider the deterministic model-updating problem of correctinga single finite element model on the basis of a single experiment. The gradient matrix G is given bythe sensitivity of the model to a small perturbation in the mean values of the inputs, xj. Thisgradient may be determined more accurately by a linear regression as detailed in Section 2.2 andapplied in Section 3.In the least-squares approach we assume that all a priori information on the outputs and inputs

may be assembled in a vectory0

x0

( )with the a priori covariance matrix,

Sy0y0 Sy0x0

Sx0y0 Sx0x0

" #having

off-diagonal sub-matrices Sy0x0¼ ST

x0y0¼ 0 so that y0 and x0 are uncorrelated.

y0

x0

( )and

Sy0y0 Sy0x0

Sx0y0 Sx0x0

" #define the parameter space of an assumed Gaussian probability density function.

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In the case of the means of the outputs and inputs, this leads to the expression

pðy; xÞ ¼ cst exp �12ðy� y0Þ

TS�1y0y0ðy� y0Þ �

12ðx� x0Þ

TS�1x0x0ðx� x0Þ

� �, (9)

where cst denotes a constant.The maximum probability solution appears when the functional

Jðy; xÞ ¼ ðy� y0ÞTS�1y0y0

ðy� y0Þ þ ðx� x0ÞTS�1x0x0

ðx� x0Þ (10)

is minimised with respect to the parameter mean vector x. This leads to an iterative least-squaressolution of the non-linear expression (7) using the linearisation (8),

xjþ1 ¼ xj þ ðGTS�1M Gþ S�1xjxj

Þ�1ðGTS�1M ðy� yjÞ � S�1xjxj

ðxj � x0ÞÞ, (11)

where the covariance Sy0y0 has been replaced by SM as will be explained shortly. This expression isdiscussed in detail by Tarantola and Valette [27] using a probabilistic approach. Friswell andMottershead [2] also obtain an equation similar to (11) by a non-probabilistic least-squarescalculation. The important point is that in the presence of multiple minima the inputs areconstrained so that the unique minimum closest to the original estimates x0 is located.Convergence of the iterative process occurs when xjþ1 ffi xj and yj ffi y.In Eqs. (9) and (10), one may minimise the output error directly (i.e., the Euclidian distance) or

the weighted output error using the inverse of the output covariance matrix (i.e., the Mahalanobisdistance [28,29]).When the inputs have a normal distribution (or log-normal) and the relation between the input/

output variables is linear or can be linearised in a region of the model space, the posteriorprobability density, Eq. (9), is Gaussian or approximately Gaussian. For more complexuncertainty models with regions where the interaction of the inputs and outputs is strongly non-linear, more general methods based on ‘the conjunction of states of information’ [9] or using theMarkov Chain Monte-Carlo method [10] may be applied.The variances and covariances of the inputs should be updated together with the mean values.

If the a priori knowledge about an input is sparse, the corresponding variance will be large andduring the updating process the resulting, a posteriori, variance should decrease as a result ofincreasing the amount of knowledge on the inputs. If the change in both an updated input and itsposterior variance are small the output data is insensitive to it and a different parameterisationshould be used.The predicted values are generally not exactly the same the true ‘observed’ values because of

experimental uncertainties and modelling errors. If the model uncertainties and observed testvariabilities have Gaussian distributions, then the modelling and experimental-error covariances,for the forward problem, will combine by addition even in the case of non-linearity due to thelinearisation in Eq. (8). Continuing to use the linearisation described above, the a posterioricovariance matrix is found to be a minimum [5] when expressed in the form

Sxjþ1xjþ1¼ Sxjxj

� SxjxjGTðSM þGSxjxj

GT�1GSxjxj

. (12)

Collins et al. [5] write the covariance matrix SM as S�j�j to show that the output error is due tomeasurement noise. In the present analysis, this matrix is denoted SM to emphasise that it includeserrors arising from both model uncertainty and test variability, Sxjxj

is the a priori covariance

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matrix of the updating parameters and G is the sensitivity matrix at each iteration. Eq. (12)assumes that the measurement and parameter errors are independent, which is true for thefirst iteration, but not for subsequent ones [30]. Covariance operators (prior and posterior)are generally not diagonal but ‘band diagonal’ and parameter covariances are not null butcorrelated [31].

2.2. Multivariate multiple regression

In this section, it is shown how an improved gradient matrix G may be determined frommultiple finite element models with randomised parameters. A multivariate multiple regressionanalysis is carried out at each iteration in order to obtain a linearised relationship between theinput, or ‘predictive’, variables X and the output, or ‘criterion’, variables Y expressed as

Y ¼ enaT þ XBþ N; a 2 <p�1; B 2 <q�p; X 2 <n�q; Y; N 2 <n�p, (13)

aT ¼ ½ a1 a2 ; . . . ; ap �; B ¼ ½ b1 b2 ; . . . ; bn �; N ¼ ½ e1 e2 ; . . . ; ep �T, (14)

en ¼ ½ 1 1 ; . . . ; 1 �T, (15)

where the iteration index is omitted for clarity. Each row in the model describes a p� 1 outputvector as a linear function of the corresponding input vector corrected by a random deviation.The following assumptions are made in order to obtain the coefficients: (a) the linear model iscorrect with a full complement of inputs; (b) each of the observation vectors (columns) in Y hasthe same covariance matrix and (c) the observation vectors are uncorrelated with each other.These assumptions may be expressed as

EðNÞ ¼ 0; covðYÞ ¼ diagðSÞ. (16)

Thus, the observation vectors are independent and have the same covariance matrix.Application of the least-squares technique leads to the estimates a; B of the linear model whichhave the following properties: (a) the estimator B is unbiased, so that sampling the samepopulation would give the same average value for B; (b) the least squares estimates of the terms inB have minimum variance among all possible linear unbiased estimators without requiringnormality of the outputs—the Gauss–Markov theorem, and (c) all the terms in B are correlatedwith each other, the relationship of the inputs with each other affecting the relationship of the B-terms with each other and since the outputs are correlated, the terms in different columns of B arecorrelated.A model corrected for means is obtained [28,29] by

B ¼1

n

Xn

k¼1

xkxTk

!�11

n

Xn

k¼1

xkyTk

!; aT ¼

1

neTn ½Y� XB�, (17)

where xTk and yTk are rows of the matrices X and Y, respectively.The final model resulting from the multivariate multiple regression may be written in the form

Y ¼ enaTþ XB. (18)

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The assumption that the structure of the model is correctly determined (i.e. linear and nofurther inputs) is problem dependent. The method may be readily extended to allow the fitting of anon-linear model with interacting inputs, but in the present study linear models are used, which isof course analogous to the conventional deterministic model updating method using first-ordersensitivities. In practice the parameters may be neither sensitive nor independent, if ill-chosen.These issues are returned to in Section 3 where the gradient matrix G in Eqs. (11) and (12) isreplaced by B

T.

A question arises regarding the size of the sample needed to carry out multivariatemultiple regression. In the following study, this is determined by convergence of a T2 statistic.By this approach the two samples (from experiments and updated finite element models)may be tested to determine, within a confidence interval, whether or not they belong to thesame parent population, as indeed they should do if the parameterisation and updating ofthe model has been carried out perfectly. The topic of hypothesis testing is dealt with inmany standard texts on statistical inference, typically [29]. An example of how hypothesistests may be used to help in assessing the ‘quality’ of an updated model is given in the companionpaper [21].

3. Three degree-of-freedom mass–spring example

The three degree-of-freedom mass–spring system shown in Fig. 1 is used to illustrate theapplication of the method and several issues of practical concern to users are considered anddiscussed. The nominal values of the parameters for the experimental system are: mi ¼ 1:0kg(i ¼ 1, 2, 3) ki ¼ 1:0N=m (i ¼ 1,y,5) and k6 ¼ 3:0N=m. The erroneous random parameters areassumed to have Gaussian distributions with mean values, k1 ¼ k2 ¼ k5 ¼ 2:0N=m and standarddeviations, s1 ¼ s2 ¼ s5 ¼ 0:30N=m. The true mean values are the same as the nominal valuesand the standard deviations are, s1 ¼ s2 ¼ s5 ¼ 0:20N=m (20% of the nominal values). The erroris 100% for the mean and 50% for the standard deviation and an initial error of this magnitudecan be found in actual applications. It is considered that n1 ¼ 10 experimental systems aremeasured and used for reference while the analytical set consists of a ‘cloud’ of n2 ¼ 1000 samplesdetermined by convergence of the T2 statistic [29] (and therefore convergence of the confidenceellipse) which can be seen in Fig. 2. The following cases are simulated:

Case 1: Only the erroneous parameters are updated as in the ideal case of perfectly localisederrors.

Fig. 1. Three degree-of-freedom mass–spring example.

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0 200 400 600 800 1000 1200

2.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

n1+n2

T α

,p,n

1+n 2-2

Fig. 2. T2-statistic for determining the optimal sample number.

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Case 2: The experimental system contains an unknown erroneous element k6 ¼ 3:5N=m whichis not localised and therefore not included in the updating parameters, which remain the same asin Case 1.

Case 3: The analytical model contains a connectivity error; the spring k6 is not present. Theupdating parameters remain the same as in Case 1.

Case 4: The same as Case 1 except that all the springs are Gaussian random variables withstandard deviations si ¼ 0:20N=m for i ¼ 1,y,5 and s6 ¼ 0:6N=m.The strength of the relationship between input and output variables in the model has been

measured by the Spearman correlation matrix, a classical vector correlation based on orderranking of the terms ðxki � xiÞ and ðyki � yiÞ rather than the true values of the terms in the inputand output vectors [28,29]. In Fig. 3 it is shown that for the present example each input variablecorrelates most significantly with two of the three outputs (or target modes).For each of the four cases the outputs, i.e. the natural frequencies, of the initial and updated

model and the parameter mean and standard deviation values are presented in Tables 1–12. Theerrors in the mean and standard deviations of the parameters are calculated with respect tostatistical distribution used to generate the sample of ‘experimental’ results. The convergence plotsshowing the Euclidian distance between the analytical and experimental clouds, the frequenciesand parameters, and the scatter plots of the initial and final iterations, are presented in Figs. 4–19.

Case 1: The Euclidian distance and natural frequency errors are shown in Fig. 4 and Table 1.Convergence of the mean and standard deviation of the updated random parameters is shown inFig. 5, with initial and final values, after twelve iterations, given in Tables 2 and 3. It can be seenthat, as expected, the estimated mean and standard deviation of the experimental sample is notexactly the same as the distribution that was used to produce them. The updated parameters thenconverge upon the statistics of the experimental sample rather than the underlying distribution.

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1

2

3

1

2

3

0

0.2

0.4

0.6

0.8

1

Updating varia

bles

Modes

Fig. 3. The Spearman correlation matrix between the input and output variables for the initial cloud.

Table 1

Mean natural frequency in Hz—% errors in parentheses—(Case 1)

Mode Test Initial Final

1 0.16 0.19 (18.75) 0.16 (0.0)

2 0.32 0.38 (18.75) 0.32 (0.0)

3 0.45 0.48 (6.67) 0.45 (0.0)

Table 2

Parameter mean estimates (Case 1)

Parameter Test Initial Final

K1 1.08 2.0 (100) 1.09 (9)

K2 1.09 2.0 (100) 1.07 (7)

K5 0.96 2.0 (100) 0.95 (�5)

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The size of the experimental sample is small, but this is likely to represent the availability ofphysical systems for testing quite faithfully. Of course, if more experimental samples wereavailable it would lead to a better approximation of the mean and standard deviations of thenatural frequencies.

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Table 3

Standard deviation estimates (Case 1)

Parameter Test Initial Final

K1 0.288 0.30 (50) 0.204 (2)

K2 0.130 0.30 (50) 0.226 (13)

K5 0.241 0.30 (50) 0.181 (�9.5)

Table 4

Mean natural frequency in Hz—% errors in parentheses—(Case 2)

Mode Test Initial Final

1 0.15 0.20 (33.33) 0.17 (13.33)

2 0.31 0.40 (29.02) 0.32 (3.23)

3 0.47 0.47 (0) 0.47 (0)

Table 5

Parameter mean estimates (Case 2)

Parameter Test Initial Final

K1 0.92 2.0 (100) 2.63 (163)

K2 0.89 2.0 (100) 0.50 (�50)

K5 0.97 2.0 (100) 1.05 (5)

Table 6

Standard deviation estimates (Case 2)

Parameter Test Initial Final

K1 0.178 0.30 (50) 0.143 (�28.5)

K2 0.151 0.30 (50) 0.148 (�26)

K5 0.150 0.30 (50) 0.145 (�27.5)

Table 7

Mean natural frequency in Hz—% errors in parentheses—(Case 3)

Mode Test Initial Final

1 0.15 0.19 (26.67) 0.16 (6.67)

2 0.31 0.27 (�12.9) 0.32 (3.23)

3 0.45 0.40 (�11.11) 0.43 (�4.44)

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Table 8

Parameter mean estimates (Case 3)

Parameter Test Initial Final

K1 0.99 2.0 (100) 2.93 (193)

K2 1.02 2.0 (100) 0.50 (�50)

K5 0.97 2.0 (100) 3.00 (200)

Table 9

Standard deviation estimates (Case 3)

Parameter Test Initial Final

K1 0.179 0.30 (50) 0.235 (17.5)

K2 0.186 0.30 (50) 0.149 (�25.5)

K5 0.249 0.30 (50) 0.076 (�62)

Table 10

Mean natural frequency in Hz—% errors in parentheses—(Case 4)

Mode Test Initial Final

1 0.16 0.19 (18.75) 0.16 (0.00)

2 0.32 0.27 (�15.62) 0.32 (0.00)

3 0.44 0.40 (�9.09) 0.44 (0.00)

Table 11

Parameter mean estimates (Case 4)

Parameter Test Initial Final

K1 1.00 2.0 (100) 0.80 (�20)

K2 1.07 2.0 (100) 1.34 (34)

K5 1.08 2.0 (100) 0.93 (�7)

Table 12

Standard deviation estimates (Case 4)

Parameter Test Initial Final

K1 0.124 0.30 (50) 0.239 (19.5)

K2 0.242 0.30 (50) 0.254 (27)

K5 0.155 0.30 (50) 0.185 (�7.5)

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0 2 4 6 8 10 120

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Euclidian Distance

0 2 4 6 8 10 12

0.2

0.25

0.3

0.35

0.4

0.45

0.5Frequency variation (Hz)

IterIter

f1f2f3

Fig. 4. Euclidian distance and mean natural frequency errors (Case 1).

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

Iter

Parameter variation (N/m)

k1k2k5

0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Iter

STD variation (N/m)

k1k2k5

Fig. 5. Parameter mean and standard deviation (Case 1).

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Figs. 6 and 7 show how the scatter ellipse of predicted results converges upon the experimentalscatter ellipse in the plane of the first two natural frequencies. The ellipses shown are for twostandard deviations. The cloud of yellow points (very light grey in greyscale) gives the predictednatural frequencies with the corresponding ellipse shown in green. The experimental points and

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0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26

0.3

0.32

0.34

0.36

0.38

0.4

0.42

f1

f2

Iter# 1 analyticexperiment

0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.210.27

0.28

0.29

0.3

0.31

0.32

0.33

0.34

0.35

f1f2

Iter# 12

analyticexperiment

Fig. 6. Initial and final scatter ellipses (Case 1).

0.14 0.145 0.15 0.155 0.16 0.165 0.17 0.175 0.18

0.435

0.44

0.445

0.45

0.455

0.46

0.465

f1

f3

Iter# 12

0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35

0.42

0.43

0.44

0.45

0.46

0.47

0.48

f2

f3

Iter# 12

analyticexperiment

analyticexperiment

Fig. 7. Final scatter ellipses (Case 1).

C. Mares et al. / Mechanical Systems and Signal Processing 20 (2006) 1674–1695 1687

ellipse are shown in red (dark grey in greyscale). It can be seen that after 12 iterations the twoscatter ellipses in Fig. 6 overlay each other quite closely. They have similar orientations but thereis a noticeable difference in the sizes of the ellipses. There are a number of possible explanations:(i) The experimental ellipse is estimated from only ten samples and as can be seen from Table 3 itdiffers from the underlying distribution that the experimental samples were drawn from (withsi ¼ 0:2). In fact the cloud of analytical samples converge better onto the underlying distribution

ARTICLE IN PRESS

0 1 2 3 4 5 6 7 8 9 100.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Iter

Euclidian Distance

0 1 2 3 4 5 6 7 8 9 10

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Iter

Frequency variation (Hz)

f1f2f3

Fig. 8. Euclidian distance and mean natural frequency errors (Case 2).

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

Iter

Parameter variation (N/m)

k1k2k5

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Iter

STD variation (N/m)

k1k2k5

Fig. 9. Parameter mean and standard deviation (Case 2).

C. Mares et al. / Mechanical Systems and Signal Processing 20 (2006) 1674–16951688

than onto the ellipse from the ten experiments. (ii) The covariances are determined a posterioriand do not appear in the objective function. (iii) The covariance correction is based on a linearisedgradient at each step in the design space and depends upon the quality of the estimates afteraveraging in Eq. (17). (iv) The gradient method may converge to a local minimum. The number ofsamples in the experimental population is determined according to convergence of the T2 statistic.A population of 1000 was taken but whether or not convergence has been achieved is a subjectivejudgement.

ARTICLE IN PRESS

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26

0.3

0.32

0.34

0.36

0.38

0.4

0.42

f1

f2

Iter #1

analyticexperiment

0.14 0.15 0.16 0.17 0.18 0.19

0.295

0.3

0.305

0.31

0.315

0.32

0.325

0.33

0.335

0.34

0.345

f1f2

Iter# 10

analyticexperiment

Fig. 10. Initial and final scatter ellipses (Case 2).

0.145 0.15 0.155 0.16 0.165 0.17 0.175 0.18 0.185

0.46

0.465

0.47

0.475

0.48

0.485

0.49

f1

f3

Iter# 10

analyticexperiment

0.295 0.3 0.305 0.31 0.315 0.32 0.325 0.33 0.335 0.34 0.3450.455

0.46

0.465

0.47

0.475

0.48

0.485

0.49

0.495

f2

f3

Iter# 10

analyticexperiment

Fig. 11. Final scatter ellipses (Case 2).

C. Mares et al. / Mechanical Systems and Signal Processing 20 (2006) 1674–1695 1689

Case 2: The case of an erroneous parameter, not included in the random updated parameters, isseen to result in converged natural frequencies, one of which remains significantly in error. Theestimated means of the randomised parameters are converged, two being improved whilst theother moves further away from the test data. The standard deviations are improved. Thisinformation is conveyed in detail in Figs. 8–11 and Tables 4–6.

Case 3: In the case of an absent spring, k6, the natural frequencies converge closely, the meansof the parameters converge but with significant errors, and two of the estimated standard

ARTICLE IN PRESS

0 1 2 3 4 5 6 7 8 9 100.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Iter

Euclidian Distance

0 1 2 3 4 5 6 7 8 9 10

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Iter

Frequency variation (Hz)

f1f2f3

Fig. 12. Euclidian distance and mean natural frequency errors (Case 3).

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

Iter

Parameter variation (N/m)

k1k2k5

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Iter

STD variation (N/m)

k1k2k5

Fig. 13. Parameter mean and standard deviation (Case 3).

C. Mares et al. / Mechanical Systems and Signal Processing 20 (2006) 1674–16951690

deviations are improved. The connectivity error has a stronger effect on the estimated parametersthan an erroneous parameter, as in Case 2. Figs. 12–15 and Tables 7–9 give full details.

Case 4: When all the stiffnesses are randomised the mean values of natural frequencies convergeexactly in this example and the estimated means and standard deviations of the parameters are

ARTICLE IN PRESS

0.12 0.14 0.16 0.18 0.2 0.22 0.240.24

0.25

0.26

0.27

0.28

0.29

0.3

0.31

0.32

0.33

0.34

f1

f2

Iter #1

analyticexperiment

0.13 0.14 0.15 0.16 0.17 0.18 0.19

0.29

0.295

0.3

0.305

0.31

0.315

0.32

0.325

0.33

0.335

0.34

f1f2

Iter# 10

analyticexperiment

Fig. 14. Initial and final scatter ellipses (Case 3).

0.145 0.15 0.155 0.16 0.165 0.17 0.175 0.18

0.425

0.43

0.435

0.44

0.445

0.45

0.455

f1

f3

Iter# 10

analyticexperiment

0.29 0.295 0.3 0.305 0.31 0.315 0.32 0.325 0.33 0.335 0.34

0.42

0.425

0.43

0.435

0.44

0.445

0.45

0.455

0.46

f2

f3

Iter# 10

analyticexperiment

Fig. 15. Final scatter ellipses (Case 3).

C. Mares et al. / Mechanical Systems and Signal Processing 20 (2006) 1674–1695 1691

improved. The scatter plot shows that the ellipses are superimposed but with different dimensionsand orientations. The differences in shape of the experimental and analytical clouds, in their finalconfigurations, shows that the two models come from different families with different covariancematrices. Although in this case the mean natural frequencies are estimated exactly becauseadjusting the statistics of the chosen updating parameters compensates for the randomness in theother parameters, it is not possible to say that this will happen for other problems of unrecognisedrandom variability. Details are given in Figs. 16–18 and Tables 10–12.

ARTICLE IN PRESS

0 1 2 3 4 5 60

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Iter

Euclidian Distance

0 1 2 3 4 5 6

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Iter

Frequency variation (Hz)

f1f2f3

Fig. 16. Euclidian distance and mean natural frequency errors (Case 4).

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

Iter

Parameter variation (N/m)

k1k2k5

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Iter

STD variation (N/m)

k1k2k5

Fig. 17. Parameter mean and standard deviation (Case 4).

C. Mares et al. / Mechanical Systems and Signal Processing 20 (2006) 1674–16951692

4. Conclusions

A stochastic model updating method using inverse Monte-Carlo propagation of physicalstructure variability and model uncertainty together with multivariate multiple regression for

ARTICLE IN PRESS

0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.260.3

0.32

0.34

0.36

0.38

0.4

0.42

f1

f2

Iter #1

analyticexperiment

0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20.29

0.3

0.31

0.32

0.33

0.34

0.35

f1f2

Iter #6 analyticexperiment

Fig. 18. Initial and final scatter ellipses (Case 4).

0.1 0.12 0.14 0.16 0.18 0.2 0.22

0.4

0.42

0.44

0.46

0.48

0.5

f1

f3

Iter #6

analyticexperiment

0.26 0.28 0.3 0.32 0.34 0.36 0.38

0.4

0.42

0.44

0.46

0.48

0.5

f2

f3

Iter #6

analyticexperiment

Fig. 19. Final scatter ellipses (Case 4).

C. Mares et al. / Mechanical Systems and Signal Processing 20 (2006) 1674–1695 1693

optimisation by the gradient method is described in detail. A T2 statistic is used to determine thesize of the analytical sample. Simulated examples are used to demonstrate how the method may beapplied, and include various cases of ill-parameterisation typical of what might happen in a realapplication. Such a real application is studied in the companion paper. Although the initial errorsin the analytical model were large, when the structure of the model was correct, the updated modelwas very good. The examples discussed demonstrate that the structure of the model (theparameterisation used to describe the model dynamics) is the most important aspect of the modelupdating process.

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Acknowledgements

The research reported in this paper is supported by Engineering and Physical Sciences ResearchCouncil (EPSRC) Grants GR/R26818 and GR/R34936.

References

[1] J.E. Mottershead, M.I. Friswell, Model updating in structural dynamics: a survey, Journal of Sound and Vibration

162 (2) (1993) 347–375.

[2] M.I. Friswell, J.E. Mottershead, Finite Element Model Updating in Structural Dynamics, Kluwer Academic Press,

Dordrecht, 1995.

[3] F.M. Hemez, S.W. Doebling, M.C. Anderson, A brief tutorial on verification and validation. in: Proceedings of the

22nd International Modal Analysis Conference, Dearborn, USA, 2004.

[4] H. Ahmadian, J.E. Mottershead, M.I. Friswell, Regularisation methods for finite element model updating,

Mechanical Systems and Signal Processing 12 (1) (1998) 47–64.

[5] J.D. Collins, G.C. Hart, T.K. Hasselman, B. Kennedy, Statistical identification of structures, AIAA Journal 2

(1974) 185–190.

[6] J.L. Beck, L.S. Katafygiotis, Updating models and their uncertainties. I: Bayesian statistical framework, Journal of

Engineering Mechanics 124 (4) (1998) 455–461.

[7] L.S. Katafygiotis, J.L. Beck, Updating models and their uncertainties. II: Model identifiability, Journal of

Engineering Mechanics 124 (4) (1998) 463–467.

[8] M.W. Vanick, J.L. Beck, S.K. Au, Bayesian probabilistic approach to structural health monitoring, Journal of

Engineering Mechanics 126 (2000) 738–745.

[9] K. Mosegaard, A. Tarantola, Probabilistic approach to inverse problems, International Handbook of Earthquake

and Engineering Seismology (Part A), Academic Press, New York, 2002, pp. 237–267.

[10] R.M. Neal, Probabilistic inference using Markov chain Monte Carlo methods, Technical Report CRG-TR-93-1,

Department of Computer Science, University of Toronto, 1993.

[11] J. Marczyk, Meta-computing and computational stochastic mechanics, in: J. Marczyk (Ed.), Computational

Stochastic Mechanics in a Meta-computing Perspective, CIMNE, Barcelona, 1997.

[12] J. Marczyk, Principles of Simulation-based Computer-aided Engineering, FIM, Barcelona, 1999.

[13] I. Doltsinis, F. Rau, M. Werner, Analysis of random systems, in: I. Doltsinis (Ed.), Stochastic Analysis of

Multivariate Systems in Computational Mechanics and Engineering, CIMNE, Barcelona, 1999, pp. 9–159.

[14] E. Balmes, D. Ravary, D. Langlais, Uncertainty propagation in modal analysis. in: Proceedings of the 22nd

International Modal Analysis Conference, Dearborn, USA, 2004.

[15] J.R. Fonseca, M.I. Friswell, J.E. Mottershead, A.W. Lees, Uncertainty identification by the maximum likelihood

method, Journal of Sound and Vibration, in press, doi:10.1016/j.jsv.2005.07.006.

[16] R.G. Ghanem, P.D. Spanos, Stochastic Finite Elements: A Spectral Approach, Dover, New York, 2003.

[17] G.I. Schueller, A state-of-the-art report on computational stochastic mechanics, Probabilistic Engineering

Mechanics 12 (4) (1997) 197–321.

[18] H.G. Matthies, A. Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential

equations, Computer Methods in Applied Mechanical Engineering 194 (2005) 1295–1331.

[19] M.K. Deb, I. Babuska, J.T. Oden, Solution of stochastic partial differential equations using Galerkin finite element

techniques, Computer Methods in Applied Mechanical Engineering 190 (2001) 6359–6372.

[20] R. Ghanem, J.R. Red-Horse, Modal properties of a space-frame with localised system uncertainties, in:

Proceedings of the 8th ASCE Speciality Conference on Probalistic Mechanics and Structural Reliability,

PMC2000-001, 2000.

[21] J.E. Mottershead, C. Mares, S. James, M.I. Friswell, Stochastic model updating: part 2—application to a set of

physical structures, Mechanical Systems and Signal Processing, submitted for publication, doi:10.1016/

j.ymssp.2005.06.007.

ARTICLE IN PRESS

C. Mares et al. / Mechanical Systems and Signal Processing 20 (2006) 1674–1695 1695

[22] J.E. Mottershead, M.I. Friswell, G.H. Ng, J.A. Brandon, Geometric parameters for finite element updating of

joints and constraints, Mechanical Systems and Signal Processing 10 (1996) 171–182.

[23] J.E. Mottershead, C. Mares, M.I. Friswell, S. James, Selection and updating of parameters for an aluminium

space-frame model, Mechanical Systems and Signal Processing 14 (6) (2000) 923–944.

[24] C. Mares, J.E. Mottershead, M.I. Friswell, Results obtained by minimizing natural-frequency errors and using

physical reasoning, Mechanical Systems and Signal Processing 17 (1) (2003) 39–46.

[25] G.M. Gladwell, H. Ahmadian, Generic elements for finite element updating, Mechanical Systems and Signal

Processing 9 (1996) 601–614.

[26] H. Ahmadian, G.M. Gladwell, F. Ismail, Parameter strategies in finite element updating, Journal of Vibration and

Acoustics 119 (1997) 37–45.

[27] A. Tarantola, B. Valette, Generalised nonlinear inverse problems solved using the least squares criterion, Reviews

of Geophysics and Space Physics 20 (2) (1982) 219–232.

[28] A. Rencher, Methods of Multivariate Analysis, Wiley, New York, 1995.

[29] A. Rencher, Methods of Statistical Inference and Applications, Wiley, New York, 1998.

[30] M.I. Friswell, The adjustment of structural parameters using a minimum variance estimator, Mechanical Systems

and Signal Processing 3 (2) (1989) 143–155.

[31] G. Backus, F. Gilbert, Uniqueness in the inversion of inaccurate gross earth data, Philosophical Transactions,

Royal Society of London, Series A 226 (1970) 123–192.