STRUCTURAL MODEL UPDATING WITH UNCERTAIN MODEL PARAMETERS

Silva, T. A. N.1,2, Carvalho, A.1,3, Maia, N. M. M.2, Barbosa, J. I.1,2

1 ISEL - Instituto Superior de Engenharia de Lisboa, GI-MOSM - Grupo de Investigação em Modelação e Optimização de Sistemas Multifuncionais.

Rua Conselheiro Emídio Navarro, 1959-007 Lisboa, Portugal.

2 IDMEC - Instituto Superior Técnico, Tech. Univ. of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal.

3 CEMAPRE - ISEG, Tech. Univ. of Lisbon, Rua do Quelhas 6, 1200-781 Lisboa, Portugal.

E-mail of the corresponding author: nmaia@dem.ist.utl.pt SUMMARY: When addressing a structural model updating philosophy, it is usual to follow a deterministic path. However, neglecting the test variability and/or the uncertainty in the model parameters can lead to perfectly correlated test/model responses that are not representative of the behaviour of the structure, for instance, for a slightly different test condition. Furthermore, one cannot conclude on the model validity if only deterministic data are considered for model updating. In order to take into account the uncertainty in the modelling parameters, the authors use a Monte-Carlo method to propagate it through the model results. The exploitation of Monte-Carlo methods have been increasing in various research fields due to the continuous growth in computational capabilities, leading to its application in the resolution of problems with industrial/practical dimension. In the present work the authors address the use of a Markov chain Monte-Carlo method to propagate the uncertainty of the model parameters in the context of stochastic model updating. Numerical examples are given to assess the quality of the updated data, obtained through a modal sensitivity based updating method. KEYWORDS: Model updating; Uncertainty; Monte-Carlo.

1. INTRODUCTION

The calibration of theoretical model parameters based on experimental test data is commonly known as model updating and is widely used [1,2]. Typically the model updating process follows a deterministic approach, regardless if the method is direct or iterative. In other words, it is usual to neglect the variability of the experimental data and/or the uncertainty in the model parameters, and by this one can obtain perfectly correlated test/model responses that are not representative of the real behaviour of a certain structure, which can be the case of results obtained for slightly different test condition or for identical structures [3]. Furthermore, if one only considers the deterministic model updating case, one cannot conclude on the model validity [4], being limited to model correlation metrics. In order to take into account the uncertainty in the modelling parameters, stochastic model updating (SMU) techniques must be addressed [5]. In this context, several methods can be used to propagate the uncertainty on the model parameters values through the model [6], estimating the distribution of its responses. Here, one must highlight some of the most popular methods. Hence, the perturbation method assumes that the variability of model parameters or system responses can be modelled by a set of Gaussian variables, expanding both functions and operators in a Taylor Series [7]. This approach has been used by several authors in the context of SMU [8,9]. The exploitation of Monte-Carlo method or simulations (MCS) [10] have been increasing in various research fields due to the continuous growth of computational capabilities, leading to its application in the resolution of problems with industrial dimension. However, there are authors working on implementations of SMU

procedures which are focused on the reduction of the computational effort. Khodaparast and Mottershead [11] proposed the use of the perturbation approach in order to derive the updating expressions for the parameter mean values and its covariance matrix. Also based on the sensitivity weighted least squares method, Govers and Link [12] extended the gradient based deterministic model updating algorithm to minimize the distance between the mean values and covariance matrices of experimental and numerical data. Brehm et al. [13] studied the computational efficiency of using neural networks estimations and the Latin Hypercube sampling (LHS), which proved to be more efficient than the procedure proposed by Khodaparast et al. [8]. Fang et al. [14] proposed to decompose the SMU process into several deterministic ones, which can result in an expensive computational cost if the use of surrogated or meta-models is not addressed. In its work response surface models are used as surrogates of FE ones. Rui et al. [15] developed a SMU algorithm which uses polynomial chaos as surrogates, being the uncertain model parameters represented in terms of standard random variables. On the order hand, the available experimental data often has a limited statistical description. Hence, Faverjon et al. [16] proposed to reconstruct the experimental data, using the polynomial chaos expansion [7], and to use the Karhunen-Loève method [7] to deal with the uncertainty in the model parameters. Although the propagation of the model parameters uncertainty through its responses can be done in different ways, one has considered the use of the Metropolis-Hastings (MH) algorithm. In the present paper, two different SMU implementations, based on the eigensensitivities, are described and their performance evaluated, regarding the updated responses scatter and mean values.

2. THEORETICAL BACKGROUND

2.1. Numerical model A structure can be numerically modelled considering a N degree-of-freedom (DOF) system, for which the dynamic equilibrium equation is: ( ) ( ) ( ) ( )t t t t+ + =M x Cx K x f&& & (1)

for a viscously damped structure. In steady-state conditions, for a harmonic excitation (( ) i tt eω=f F ), if one

assumes that the system damping is negligible, (1) can be recast as:

( )2ω− =K M X F (2)

For a free vibration condition, (2) forms the eigenproblem of (3) from where both eigenfrequencies (2

nω ) and

eigenmodes ( nΦ ) can be extracted.

( )2

n nω− =K M Φ 0 (3)

2.2. Deterministic model updating In a model updating process, it is assumed that exist a numerical model which parameters must be calibrated in order to approximate the experimentally obtained response model. Hence, one can establish an error between the responses of the numerical model and the experimental ones, both in terms of the eigenfrequencies:

( ) ( )2 2 2

X A nn nω ω ω= ∆− (4)

and, in terms of the eigenmodes:

( ) ( )X A nn n∆− =Φ Φ Φ (5)

where the subscripts X and A correspond to experimental and numerical quantities, respectively. The objective of the model updating process is to minimize the difference between the numerical and experimental responses and it can be achieved considering the eigensensitivities to the model parameters [17]. In order to compute the referred sensitivity, one can expand the errors of (4) and (5) in a Taylor series that is function of the model parameters selected for updating, the updating parameters. Considering a 1st order approximation, the expressions to compute the errors of (4) and (5) are defined as a Taylor series truncated with respect to the PN updating parameters, as:

( )2

1

2P

A nn i

i

N

i

pp

ωω

=

∂≈

∂∆ ∑ (6)

and

( )

1

PN

i

A nn i

i

pp=

∆∂

≈∂∑Φ

Φ (7)

For m N≤ measured eigensolutions, after mode pairing, one can compute the updating parameters vector p that

minimizes the error vector ε for a given sensitivity matrix S [18]. Hence, in an iterative scheme, the updated pis obtained with reference to the parameters of the previous current iteration [17], as: 1 ii i i+ = +p Tεp (8)

with

T2 T 2 T

1 1i m mω ω∆ ∆ ∆= ∆ ε Φ ΦL (9)

and

( ) 1T T

i i i p iε ε

−= +T S W S W S W (10)

where εW and pW are weighting matrices [17]. Note that, for ε =W I and p =W 0 , iT is the Moore-Penrose

pseudo-inverse.

2.3. Stochastic model updating In the context of SMU, taking into account that the uncertainty on the modelling parameters and the test variability [19], one can recast (8) as:

( )( )1 1i i i ii i i i++ ∆ ∆+ = + + + ∆+∆p p p p εT εT (11)

where it is considered that iε has a random perturbation around its mean value (i i i= ∆+ε ε ε ). Applying the

perturbation approach of [19,20], one can define: 1i i i i+ = +p T εp (12)

and, considering that the parameter and experimental eigensolutions vectors are uncorrelated,

( ) ( ) ( ) ( )

( ) ( )( )1

T

1

T

Cov ; Cov ; Cov ; Cov ;

Cov ; Cov ;

i i i i i i i

X X i

i i i

i i i

+ + = + −∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆

∆ ∆ + ∆ ∆+

p p p p z Tp z p

z z z z

T

T T (13)

2.3.1. Uncertainty propagation As mentioned, in SMU the uncertainty of the model parameters, as well as the uncertainty of the experimental results, can be propagated using MCS. In the last decades we have witness an increasingly use of the MCS to simulate complex distributions [10]. The general idea behind these methods is to generate a large set of configurations and to get from it the expected behaviour of some quantity related to a given random phenomenon. As the addressed problem is a multivariate simulation one uses a Markov chain Monte-Carlo (MCMC) method to propagate the uncertainty in the parameters. In fact, one uses the MH algorithm to sample configurations according to the numerical model under study. These samples are simulated exploring the space of variables using the mechanism of Markov chains, i.e., giving more weight to the most probable. Regardless of the initial value, the chain will converge to a Markov invariant distribution that one wants to simulate [21]. The Metropolis algorithm was first introduced by Metropolis et al [22] and generalized by Hastings [23], resulting in the MH algorithm. This algorithm is a versatile, efficient and powerful simulation technique and one of the most common in the available MCMC methods. The MH algorithm borrows, from the well-known Acceptance-Rejection method, the idea of generating candidates that are either accepted or rejected. However, if rejected, it retains the value before rejection takes place [24]. In order to assess the quality of the updated model responses, one uses the Mahalanobis distance, a feature metric that measures the similitude of two data sets by evaluating the distance between the mean value of a reference data set and each point of the data set to be compared with, considering the covariance matrix of the reference one. If the covariance of the reference data set is neglected, i.e. the identity matrix, one reduces the Mahalanobis distance to the Euclidean one [25].

2.3.2. Numerical implementations Based in the work of Khodaparast [19], one has implemented a SMU procedure, here presented as case 1. Note that the test variability is addressed and the reference for the updating process are its eigenfrequencies mean values and covariance matrix.

Case 1 steps: i) Knowing the experimental eigenfrequencies mean values and covariance matrix (reference values), one

gives an initial guess for the scatter of each updating parameter, establishing its mean values and covariance matrix (at this point a diagonal matrix based only on the parameters variances);

ii) Propagate the updating parameters uncertainty using the Metropolis-Hastings algorithm; iii) Compute the mean value and covariance matrix of the updating parameters point cloud; iv) Build the model for the mean updating parameters and compute the eigensensitivities for the updating

parameters; v) Update the mean values and covariance matrix of the updating parameters, using (12) and (13); vi) Repeat steps ii)-v) until convergence, based on the Mahalanobis distance of the reference and model

scatter, is achieved. The case 2 implementation can be described as a deterministic model updating process that is computed for each observation of the MCMC sample. Case 2 steps:

i) Knowing the experimental eigenfrequencies mean values and covariance matrix (reference values), give an initial guess for the scatter of each updating parameter, establishing its mean values and covariance matrix (at this point a diagonal matrix based only on the parameters variances);

ii) Propagate the updating parameters uncertainty using the Metropolis-Hastings algorithm; iii) Build the model for each observation of the parameters sample and compute the eigensensitivities for

the updating parameters; iv) Update each one of the built models, using (12), considering the mean value to be the updating

parameters vector for each observation; v) For the updated cloud, compute the mean value and covariance matrix of the model responses; vi) Repeat steps iii)-v) until convergence, based on the Mahalanobis distance of the reference and model

scatter, is achieved.

3. NUMERICAL EXAMPLE

The numerical example here presented was presented by Mares et al. [26] and several authors used it as benchmark example. It consists in a MDOF lumped system with 3 DOF (Figure 1) with �� � 1kg (� � 1,2,3) and, �� � 1N/m (� � 1,… ,5) and �� � 3N/m. As reference model one used �� � 10 simulated experimental sets of results, randomly generated assuming a normal distribution with ��� � 1N/m and ��� � 0.2N/m for � � �1,2,5�. For the numerical model, one used the Metropolis-Hastings algorithm to propagate the uncertainty in the modelling parameters, for an initial ��� � 2N/m and ��� � 0.3N/m for � � �1,2,5�, using a sample of �� � 1000 observations. Note that the selected updating parameters were assumed to be only the erroneous ones. Taking as reference the same simulated experimental data for both model updating implementations, cases 1 and 2, results were obtained and its discussion is presented on the following section.

Figure 1 – Theoretical MDOF system under study.

4. RESULTS AND DISCUSSION

The results of Figures 2-4 and Table 1 were obtained for the numerical implementation of case 1. For this particular updating strategy, one can observe that the simulated experimental, reference, eigenfrequencies are replicated by the updated model, with an absolute error below 0.5% (Table 1 and Figure 2.c). However, in terms

of the model parameters, the updated mean values and standard deviations do not converged exactly to the reference ones (Figures 2.a and 2.b). Regarding the scatter clouds of the model responses, the Mahalanobis distance of the updated responses cloud in relation to the centre of the reference one seems to converge to a minimum value (Figure 2.d). Although, if one observes the difference, computed for consecutive iterations, of the distance of the model responses cloud for (Figure 2.e), it can be seen that there are iterations on which the cloud is pushed way from the reference centre due to the random behaviour of the MH algorithm, both in terms of centre and scatter of the model responses. Note that the present results were obtained for a MCMC sample without burn-in observations, but no difference on the Mahalanobis distance behaviour is observed if burn-in was considered.

Table 1 – Reference model and numerical model, initial and updated, results for the mean values and standard deviations of the updating parameters and mean values of the eigenfrequencies (case 1).

Reference - � Initial (error %) - �! Updated (error %) - �! ��""" [N/m] 1.010 1.944 (92.5) 1.082 (7.18)

��""" [N/m] 1.048 2.009 (91.66) 0.945 (-9.85) �#""" [N/m] 1.104 1.978 (79.15) 1.143 (3.57)

��$ [N/m] 0.230 0.271 (17.55) 0.094 (-58.99)

��% [N/m] 0.163 0.267 (63.65) 0.088 (-46.16)

��& [N/m] 0.175 0.287 (64.43) 0.140 (-19.63)

'� [Hz] 0.1604 0.2022 (26.1) 0.1597 (-0.4) '� [Hz] 0.3256 0.3949 (21.29) 0.3256 (0.02) '( [Hz] 0.452 0.4811 (6.43) 0.4535 (0.34)

a)

d)

e)

b)

c)

Figure 2 – Convergence plots (case 1): a) mean value of the updating parameters; b) standard deviation of the updating parameters; c) mean eigenfrequencies; d) Mahalanobis distance; e) variation of the Mahalanobis distance.

(solid line: numerical model results; dash-dotted line: reference model values)

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Figure 3 – Frequency scatter plots (case 1): a) '�vs '� view; b) '�vs '( view; and, c) '�vs '( view.

Figure 4 – Receptance plot (case 1): thicker lines obtained for the mean parameters of the reference and the updated models.

On Figure 3, one can evaluate the scatter of the eigenfrequencies computed for the initial and the updated model parameters, and it is possible to compare it with the scatter cloud for the reference data. It is evident that mean values of the updated model responses are now very close to the reference ones (Figure 2.c). Moreover, one must highlight the fact that the response scatter is now smaller that the obtained for the reference data, which is important in terms of the confidence in the updated results (Figure 3). Likewise, if one computes the receptance functions of the model and compares their scatter with the receptance scatter for the reference data, one can conclude on the receptance dispersion (Figure 4). Note that the scatter of the receptance function around the 2nd eigenfrequency is higher than the one observed around the other two. Although it is aligned with the test scatter, one can conclude on the impact of the uncertainty in the model parameters on a certain frequency range, and not only on the eigenfrequencies. Regarding the numerical implementation referred as case 2, one presents the results of Figures 5-8 and Table 2.

Table 2 – Reference model and numerical model, initial and updated, results for the mean values and standard deviations of the updating parameters and mean values of the eigenfrequencies (case 2).

Reference - � Initial (error %) - �! Updated (error %) - �! ��""" [N/m] 1,010 2.066 (104.59) 1.092 (8.13) ��""" [N/m] 1,048 2.142 (104.41) 0.970 (-7.44) �#""" [N/m] 1,104 1.921 (74.02) 1.120 (1.45)

��$ [N/m] 0,230 0.344 (49.62) 0.142 (-38.41) ��% [N/m] 0,163 0.234 (43.91) 0.132 (-19.07) ��& [N/m] 0,175 0.326 (86.65) 0.031 (-82.01)

'� [Hz] 0,1604 0.2068 (28.92) 0.1605 (0.07) '� [Hz] 0,3256 0.3966 (21.81) 0.3253 (-0.09) '( [Hz] 0,4520 0.4815 (6.53) 0.4533 (0.28)

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Figure 5 – Convergence plots (case 2): a) mean value of the updating parameters; b) standard deviation of the updating parameters; c) mean eigenfrequencies; d) Mahalanobis distance; e) variation of the Mahalanobis distance.

(solid line: numerical model results; dash-dotted line: mean results for the reference model)

a)

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c)

Figure 6 – Frequency scatter plots (case 2): a) '�vs '� view; b) '�vs '( view; and, c) '�vs '( view.

a)

b)

c)

Figure 7 – Frequency scatter plots (case 2 - detailed view of Figure 6): a) f�vs f� view; b) f�vs f( view; and, c) f�vs f( view.

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Figure 8 – Receptance plot (case 2): thicker lines obtained for the mean parameters of the reference and the updated models.

Whereas the general comments on the convergence of the model parameters and eigenfrequencies for the case 1 also applies to the case 2, there are particular details that deserve to the addressed. As specified, the numerical implementation of case 2 is driven only by the reference scatter centre. Hence, it is expected that the model responses scatter cloud tends to be concentrated on this particular point. In fact this is verified through the results of Figures 6 and 7. The results of Figure 7 must be emphasised. Note that the shape of the updated scatter cloud was obtained every time one runs the updating procedure, and one can observe that all the points in the cloud try to reach the centre of the reference one. The reduced dispersion of the updated responses is also patent on Figure 8, where a much smaller scatter for the computed receptance functions is obtained for the case 2, in comparison with the ones of case 1 (Figure 4). From the receptance scatter, considering the values of the updated parameters of Table 2, apparently the scatter of �# has a stronger impact on the scatter of the model responses, as one can observe from the concentrated receptance scatter plot of Figure 8. Note that �# is the only updating parameter that links two DOFs (�� and �(). On the side of the updated cloud convergence, one must emphasise the smoothness of both Mahalanobis distance and its difference between consecutive iterations (Figures 5.d and 5.e), which indicates that the evolution of the updated cloud through the iterations is always convergent.

5. CONCLUSION

The present work addresses the use of the Metropolis-Hastings algorithm to propagate the uncertainty of the model parameters in the context of stochastic model updating. Two different stochastic model updating implementations, based on the eigensensitivities, are described and their performance evaluated, regarding the updated responses scatter and mean values. From the obtained results, one can conclude that both implementations lead to updated results which eigenfrequencies converged to the reference ones. However, more than just reproduce the mean values of the reference data, one can see that the scatter of the updated results is contained inside the reference one, indicating that the updated model will not generate any result outside the reference scatter, or validation domain. Though, the case 1 implementation presents a slightly drawback, regarding the instable convergence of the Mahalanobis distance due to the random character of the Metropolis-Hastings algorithm. After this exploratory work, one is concerned to evaluate the impact of the use of statistical inference as a criterion for the updating process. On the other hand, stochastic model updating methods are usually focused on the minimization of the distance between reference and model scatter clouds centres. However, for experimental data that is often obtained from data with limited statistical, is it the best approach? Hence, one must try to weight each experimental response, instead of following its mean values.

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ACKNOWLEDGEMENTS

The current investigation had the support of national funds through the Portuguese Foundation for Science and Technology, FCT, under the project PEst/OE/EME/LA0022/2011 and the PhD grant SFRH/BD/44696/2008.

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