Parameter variability estimation using stochastic response surface model updating

Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Parameter variability estimation using stochastic response surface model updating Sheng-En Fang a,n, Qiu-Hu Zhang b, Wei-Xin Ren b a b

School of Civil Engineering, Fuzhou University, Fuzhou, Fujian Province 350108, PR China School of Civil Engineering, Hefei University of Technology, Hefei, Anhui Province 230009, PR China

a r t i c l e i n f o

abstract

Article history: Received 13 July 2013 Received in revised form 14 April 2014 Accepted 23 April 2014

From a practical point of view, uncertainties existing in structural parameters and measurements must be handled in order to provide reliable structural condition evaluations. At this moment, deterministic model updating loses its practicability and a stochastic updating procedure should be employed seeking for statistical properties of parameters and responses. Presently this topic has not been well investigated on account of its greater complexity in theoretical configuration and difficulty in inverse problem solutions after involving uncertainty analyses. Due to it, this paper attempts to develop a stochastic model updating method for parameter variability estimation. Uncertain parameters and responses are correlated through stochastic response surface models, which are actually explicit polynomial chaos expansions based on Hermite polynomials. Then by establishing a stochastic inverse problem, parameter means and standard deviations are updated in a separate and successive way. For the purposes of problem simplification and optimization efficiency, in each updating iteration stochastic response surface models are reconstructed to avoid the construction and analysis of sensitivity matrices. Meanwhile, in the interest of investigating the effects of parameter variability on responses, a parameter sensitivity analysis method has been developed based on the derivation of polynomial chaos expansions. Lastly the feasibility and reliability of the proposed methods have been validated using a numerical beam and then a set of nominally identical metal plates. After comparing with a perturbation method, it is found that the proposed method can estimate parameter variability with satisfactory accuracy and the complexity of the inverse problem can be highly reduced resulting in cost-efficient optimization. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Stochastic model updating Parameter variability Parameter sensitivity analysis Stochastic response surface models Hermite polynomials

1. Introduction For most real-world problems, engineers must deal with different kinds of uncertainties such as unknown model boundary conditions and measurement noises for right decisions [1]. At the moment deterministic analyses are no longer reliable and thus statistical techniques should participate in the solutions of such problems. In the realm of FE model updating, most existing methods possess deterministic inherence without taking uncertainty factors into account [2]. This situation motivates the involvement of probabilistic methods in model updating for the solution of an inverse problem [3]. Here a clear distinction between uncertainties and errors should be firstly made where uncertainty is defined as “a potential

n

Corresponding author. Tel.: þ86 18959162363; fax: þ 86 591 22865355. E-mail address: [email protected] (S.-E. Fang).

http://dx.doi.org/10.1016/j.ymssp.2014.04.017 0888-3270/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: S.-E. Fang, et al., Parameter variability estimation using stochastic response surface model updating, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j.ymssp.2014.04.017i

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deficiency in any phase of activity of the modeling process that is due to lack of knowledge” [4]. In practice of structural dynamic analyses, this definition can be further refined into two categories of uncertainties, namely aleatory (irreducible) and epistemic (reducible) uncertainties [5]. The former refers to parameter variability caused by manufacturing tolerances and discreteness of material properties, which cannot be avoided or eliminated. But the latter one stems from lack of knowledge (e.g., a limited number of measurements or lack of structural information), which can be reduced by increasing the relevant knowledge. It is noted that in many practical cases a strict boundary cannot be defined between the two categories. On the other hand, methods for uncertainty propagation can also be divided into three categories of probabilistic, fuzzy and interval methods [6], among which the first category is within the scope of this study. At the early stage of the research in uncertainty quantification, reducible uncertainties received close attentions [7–14]. Firstly the minimum variance methods were formulated by assuming that structural randomness stemmed merely from measurement noises and the uncertainties were described as variances [7,8]. The inverse solution sought for parameter means whose best estimates were satisfied when the variance estimates were minimized. The correlations between parameter estimates and measurement errors became an essential factor since their independence assumption would lead to an unbiased, instead of minimum variance, estimator [8]. Later treatments of such reducible uncertainties through statistical model updating also relied on Bayesian statistical frameworks. Bayesian estimation was adopted for conditioning the updating problem where relative confidence measures were established for unknown parameters leading to a more reliable algorithm [9]. Deterministic structural models were embedded within a class of probability models and the initial joint probability distributions of unknown parameters were updated using the Bayes' theorem [10]. The identification of a class of probability models, rather than one model, embodies the superiority of Bayesian methods over minimum variance methods. Meanwhile, the Marcov Chain Monte Carlo method can be applied to Bayesian model updating for estimating complex posterior probability density functions with respect to small and high dimensional problems [11,12]. However, the complexity in problem solutions, as well as the requirement for high computational costs, also restrains the applications of Bayesian updating methods to complex problems. Therefore surrogate models such as polynomial chaos expansions (PCE) and Kriging models have been utilized to alleviate the computational burden [13,14]. More recent works related to Bayesian model updating can be found in [13] where a comprehensive Bayesian approach has been developed for complex problem solutions. As to the quantification of irreducible uncertainties, stochastic model updating (SMU) procedures based on the maximum likelihood function formulated by using the Monte-Carlo (MC) or the mean-centered first-order perturbation methods have been developed by Fonseca et al. [15]. The parameter variability was quantified from experimental data by maximizing the likelihood of the measurements whose components' correlation was not considered. A similar but different MC based inverse procedure [16,17] has been developed for parameter variability quantification while the iterative nature of such methods could induce considerable computational cost. Alternatively, perturbation methods can be effectively integrated into an SMU process [18–20] for uncertainty propagation by expanding the updating equation terms with a truncated Taylor series expansion around certain parameter points. By overlooking the correlation between parameters and measurements, the calculation for second-order sensitivities [19] can be avoided resulting in a simplified procedure [20]. Perturbation based methods show their superiority over MC based methods in the aspect of computational efficiency. However, when facing complex problems they also suffer from their applicability of small uncertainties and the prerequisite of a Gaussian distribution assumption. Moreover, parameter predictions are sensitive to the initial estimates of parameters and their distribution ranges. Besides the aforementioned methods, a classic model updating technique can also be extended through an equation involving the statistical properties of parameters [21]. But the iterative inherence of the inverse solution procedure would still restrain the practical application. Hence as an alternative option, an SMU process can be divided into a series of deterministic ones and the inverse problem is solved within a deterministic framework [22]. By this means the stochastic updating efficiency can be highly improved without losing estimation accuracy and precision. This paper also focuses on quantifying parameter variability since so far the relevant research is few and thus deserves further exploration. As previously mentioned, the existing SMU methods have some drawbacks such as intensive computational costs and the complexity in the establishment and solution of stochastic inverse problems. Furthermore, sensitivity analysis during the inverse solution also increases the solution difficulty in view of appearance of an ill-conditioning problem. On account of these facts, the employment of surrogate models [22,23] and a proper updating strategy may be a solution. Under such consideration, this study correlates uncertain parameters and responses using stochastic response models (SRSMs) which are actually explicit PCEs based on Hermite polynomials [24]. During updating iterations, sensitivity computation is avoided by means of SRSM reconstruction and optimization operations can be performed directly on mathematical expressions. Another advantage consists in the fact that parameter means and variances are updated and estimated in a separate way. Namely, means representing the deterministic parts of parameters are firstly sought with unchanged (initial) variances. Then the variances representing the random parts are estimated with previously updated means that remain unchanged. Such strategy can simplify the updating procedure and simultaneously improve the estimation precision. In addition, parameter sensitivity analysis is implemented based on the derivatives of PCEs, by which the significance of uncertain parameters can be easily evaluated. Lastly the proposed methods have been validated using a numerical beam and then a set of nominally identical metal plates tested by Husain et al. [25,26] who performed a perturbation updating procedure for identifying the variability in the thicknesses and the material properties of the plates. The adoption of the same test data can compare the performance of the SRSM-based method and the perturbation-based method. Please cite this article as: S.-E. Fang, et al., Parameter variability estimation using stochastic response surface model updating, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j.ymssp.2014.04.017i

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It should be mentioned that in this study parameters and responses are assumed to follow normal probability distributions represented by the means and variances (standard deviations as well). The reason for such assumption is given in Section 4.

2. Stochastic response surface modeling In deterministic model updating, original FE models can be replaced by response surface models for the sake of alleviating the computational burden [27–29]. Response surface models are actually mathematical polynomial expressions and can be easily handled in programming and optimization. Also in SMU problems, SRSMs can be used to establish the link between uncertain parameters and responses, as will be described in detail in this section. 2.1. Basic concepts An SRSM can be regarded as an extension to a deterministic response surface model and the apparent difference lies in whether parameters are random or not [24]. Such model is actually a PCE using Hermite polynomials as its bases. The PCE is useful for uncertainty quantification and optimization problems [30,31] and thus it has also been adopted in this study. A random parameter vector x can be expressed in terms of a set of standard random variables ξ ¼ fξi gni¼ 1 having a normal distribution of Νð0; 1Þ, where n denotes the number of the parameters. Then the model order is chosen for a suitable functional form with respect to the response vector y, which is also a function of ξ y ¼ FðxÞ

with

x ¼ hðξÞ

ð1Þ

where Fð U Þ and hð UÞ represent the functions correlating y with x and x with ξ, respectively. Here ξi is independent and identically distributed. If the relationship between y and x is straightly explicit, then Eq. (1) can be solved directly. However for most engineering systems, such relationship is generally implicit and thus Eq. (1) must be approximated using analysis techniques such as regression methods. At this moment, the uncertainty of x can be defined as uðxÞ ¼ μ þ sξ and y is expressed by y ¼ f ðξ; aÞ. Then the PCE can be adopted to reveal the “y x” relationship in terms of Hermite polynomials [24] y ¼ a0 þ ∑ni1 ¼ 1 ai1 Γ 1 ðξi1 Þ þ∑ni1 ¼ 1 ∑ii12 ¼ 1 ai1 i2 Γ 2 ðξi1 ; ξi2 Þ þ∑ni1 ¼ 1 ∑ii12 ¼ 1 ∑ii23 ¼ 1 ai1 i2 i3 Γ 3 ðξi1 ; ξi2 ; ξi3 Þ þ U U U

ð2Þ

where ai is the deterministic coefficient to be estimated through regression analysis; Γ p ð U Þ denotes the multi-dimensional p-degree Hermite polynomials 1

Γ p ðξi1 ; U U U ; ξip Þ ¼ ð  1Þp e2 ξ

T

ξ

1 T ∂p e2 ξ ξ ∂ξi1 ; U U U ; ∂ξip

ð3Þ

here ξ represents the vector of p random variables fξik gpk ¼ 1 and Eq. (3) processes the orthogonality property of E½Γ p Γ q  ¼ 0

ðΓ p a Γ q Þ

ð4Þ

The approximation accuracy of Eq. (2) usually improves with the increase of p. And the number of the coefficients na is na ¼

ðn þ pÞ! n!p!

ð5Þ

Subsequently regression analysis is performed to determine ai, with which the SRSM is established. Ref. [32] provides a short review about the PCE methods for more complex problems and different approaches for determining the expansion coefficients are also compared. In practice, the model order is determined according to the precision requirement of an analysis and is also problem dependent. For many engineering problems, a second-order model is adequate and thus is the first choice for a modeler. By comparing the function coefficients and the response probability distribution estimations between a certain-order model and a higher-order model, the precision and adequacy of the lower-order model can be evaluated. 2.2. Probabilistic collocation method The unknown coefficients in Eq. (2) can be estimated by using the probabilistic collocation method which imposes that model output estimates are exact at a set of selected collocation points (CPs) being the value combinations of all parameters [24]. For the pth-order PCE, its CPs are defined as the roots of (p þ1)th-order Hermite polynomials and the CP number should at least be equal to na. “Fortunately”, the number of available CPs will exponentially increase with the increase of n and p, which is usually beyond that requirement. But a problem also appears for the selection of CPs since although a random selection is allowed, the constructed SRSMs using different combinations of CPs could be quite different. Moreover, some CPs could be outside the range of the numerical applicability of the model or not within the interest of analysts. Hence an optimal selection should be performed through obeying some basic rules [24]: (a) CPs corresponding to high probability regions have priority and the origin point (value zero) is always selected; (b) the overall distribution of selected CPs should be, as far as possible, symmetric to the origin; and (c) the number of selected CPs are around twice of na in order to reduce Please cite this article as: S.-E. Fang, et al., Parameter variability estimation using stochastic response surface model updating, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j.ymssp.2014.04.017i

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the influence of a single point and thus to present a stable estimation of coefficients. Then after all the CPs are determined, the corresponding responses can be computed through FE analysis in order to form the samples. Lastly regression analysis is performed on all the samples for a robust estimation of the unknown coefficients ai. It is noted that once the model order of an SRSM is predefined, then the CPs remain unchanged. This feature benefits the reconstruction of the SRSM since the reselection of CPs is unnecessary. 2.3. Evaluation of an SRSM and estimation of response probability distributions Like other surrogate models, the accuracy and adequacy of a pth-order SRSM should be evaluated before any usage. Such evaluation can be implemented by comparing the model with the (p þ1)th-order model. If the estimated probability density functions (pdfs) of responses by the two models are in a satisfactory agreement, then the pth-order model can be viewed as an accurate one. Otherwise the (p þ1)th-order model should be compared with the (p þ2)th-order model until the pdfs errors arrive at a predefined tolerance. Practically, a visualized comparison can be firstly made by comparing the estimated coefficients of two different models. If the two pairs of coefficients are very similar, then the lower-order model can be regarded as adequate. Based on the adequate model, the probability distributions of responses can be estimated by firstly performing MC sampling on ξ ¼ fξ1 ; ξ2 ; ⋯; ξn g [24]. Then parameter samples are obtained according to x ¼ hðξÞ. Subsequently, the response samples are numerically computed based on the parameter samples. Finally the probability distributions of responses can be statistically estimated on the basis of the response samples. It can be seen that the estimation precision is affected by the number of samples, which indicates that an adequate estimation generally requires a large number of samples resulting in considerable computational expenses. This drawback will considerably slow down the SMU process since in each iteration the MC sampling and the response estimation should be repeated. Due to it, a direct calculation method has been developed in this study where a second-order model is used as an example. According to Eq. (2), one can obtain the expectation and variance of a response y, EðyÞ and VðyÞ given as n

i1

n

EðyÞ ¼ a0 ΕðΓ 0 Þ þ ∑ ai1 Ε½Γ 1 ðξi1 Þ þ ∑ i1 ¼ 1

∑ ai1 i2 Ε½Γ 2 ðξi1 ; ξi2 Þ

i1 ¼ 1 i2 ¼ 1

n

¼ a0 Εð1Þ þ ∑ ai1 Eðξi1 Þ þ i1 ¼ 1

n

i1  1

n

∑

i1 ¼ i2 ¼ 1

ai1 i2 Eðξ2i1  1Þ þ ∑

∑ ai1 i2 Εðξi1 ξi2 Þ

i1 ¼ 2 i2 ¼ 1

ð6Þ

¼ a0 VðyÞ ¼ Eðy2 Þ ½EðyÞ2 "

n

¼E

a0 Γ 0 þ ∑ ai1 Γ 1 ðξi1 Þ þ ∑ i1 ¼ 1

"

#

i1

n

∑ ai1 i2 Γ 2 ðξi1 ; ξi2 Þ

i1 ¼ 1 i2 ¼ 1

n

 a0 Γ 0 þ ∑ ai1 Γ 1 ðξi1 Þ þ ∑ i1 ¼ 1

#!

i1

n

∑ ai1 i2 Γ 2 ðξi1 ; ξi2 Þ

i1 ¼ 1 i2 ¼ 1

a20

ð7Þ

Then using the orthogonality property of Hermite polynomials (Eq. (4)), one has n

n

i1

n

VðyÞ ¼ E a20 þ ∑ ai1 ξi1  ∑ ai1 ξi1 þ ∑ i1 ¼ 1

n

¼ ∑

i1 ¼ 1

a2i1

þE

i1 ¼ 1 n

∑

n

∑

i1 ¼ i2 ¼ 1

¼ ∑ a2i1 þ 2  i1 ¼ 1

n

i1 ¼ i2 ¼ 1

n

¼ ∑ a2i1 þ E

i1 ¼ 1

"

i1 ¼ 1 i2 ¼ 1

ai1 i2 ðξ2i1

n

 1Þ þ ∑ !

i1 ¼ i2 ¼ 1

n

a2i1 i2 þ ∑

#

i1  1

∑ ai1 i2 ξi1 ξi2 

i1  1

n

∑

i1  1

i1

!

∑ ai1 i2 Γ 2 ðξi1 ; ξi2 Þ a20

i1 ¼ 1 i2 ¼ 1

i1 ¼ 2 i2 ¼ 1

a2i1 i2 ðξ2i1 1Þ2 þE

n

∑

n

∑ ai1 i2 Γ 2 ðξi1 ; ξi2 Þ ∑ "

n

∑

i1 ¼ i2 ¼ 1

!

ai1 i2 ðξ2i1

n

 1Þ þ ∑

i1  1

#!

∑ ai1 i2 ξi1 ξi2

i1 ¼ 2 i2 ¼ 1

∑ a2i1 i2 ξ2i1 ξ22

i1 ¼ 2 i2 ¼ 1

∑ a2i1 i2

ð8Þ

i1 ¼ 2 i2 ¼ 1

It can be seen from Eqs. (6) and (8) that the means and variance of the response can be easily calculated on the basis of the model coefficients. By this means, the uncertainty propagation from parameters to responses can be efficiently achieved. 3. Parameter sensitivity analysis Selection of updating parameters is essential for a successful model updating process. The selected parameters should represent certain physical quantities (such as stiffness) of a structure whose responses should be sensitive to the parameter variations. Such selection is often implemented by means of experiential judgment or sensitivity analysis using mathematical and statistical techniques. Experiential judgment highly depends on analysts' personal experience and parameters selected by different analysts could be quite different. Thus it is particularly unreliable for complex structures. Sensitivity analysis [33,34], in a common sense, makes a perturbation to a certain design point of a parameter and the Please cite this article as: S.-E. Fang, et al., Parameter variability estimation using stochastic response surface model updating, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j.ymssp.2014.04.017i

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analysis results are reliable only under low-level perturbations. On the other hand, variance analysis within a statistical framework is capable of providing reliable predictions of parameter significance [22,28,29]. This study presents a sensitivity analysis strategy by means of differentiating Eq. (2) for the partial derivatives of y with respect to ξ. Then by calculating the expectations of these derivatives, a parameter sensitivity matrix S is formed in which the significance of each parameter is expressed by the corresponding element. For example, suppose a second-order SRSM containing three parameters x ¼ fx1 ; x2 ; x3 g, the response y can be written as y ¼ a0 þ a1 ξ1 þ a2 ξ2 þa3 ξ3 þa11 ðξ1 2  1Þ þa22 ðξ2 2  1Þ þ a33 ðξ3 2  1Þ þ a12 ξ1 ξ2 þ a13 ξ1 ξ3 þ a23 ξ2 ξ3 The partial derivatives of y with respect to ξi (i¼1, 2, 3) are given as ∂y ¼ ða1 þ 2a11 ξ1 þ a12 ξ2 þa13 ξ3 Þ; ∂ξ1 ∂y ¼ ða2 þ 2a22 ξ2 þ a12 ξ1 þa23 ξ3 Þ; ∂ξ2 ∂y ¼ ða3 þ 2a33 ξ3 þ a13 ξ1 þa23 ξ2 Þ; ∂ξ3 with their expectations     ∂y ∂y E ¼ a1 ; E ¼ a2 ; ∂ξ1 ∂ξ2

 E

 ∂y ¼ a3 ∂ξ3

On the other hand, the partial derivatives of y with respect to x ¼ fx1 ; x2 ; x3 g are ∂y ∂y ∂ξ1 1 ¼ ¼ ða1 þ2a11 ξ1 þa12 ξ2 þ a13 ξ3 Þ; ∂x1 ∂ξ1 ∂x1 s1 ∂y ∂y ∂ξ2 1 ¼ ¼ ða2 þ2a22 ξ2 þa12 ξ1 þ a23 ξ3 Þ; ∂x2 ∂ξ2 ∂x2 s2 ∂y ∂y ∂ξ3 1 ¼ ¼ ða3 þ2a33 ξ3 þa13 ξ1 þ a23 ξ2 Þ; ∂x3 ∂ξ3 ∂x3 s3 where si (i ¼1, 2, 3) denotes the standard deviation of parameter xi. Then the expectations of these derivatives are       ∂y a1 ∂y a2 ∂y a3 ¼ ; E ¼ ; E ¼ E ∂x1 ∂x2 ∂x3 s1 s2 s3 Since the probability distributions of x ¼ fx1 ; x2 ; x3 g are already known, the expectation of ∂y=∂xi is actually a constant (say bi) due to the fact that the expectation is calculated based on the mean of xi and is independent of the mathematical expression between y and xi. Therefore, one has   ∂y a1 ¼ b1 -a1 ¼ b1 s1 ; ¼ E ∂x1 s1   ∂y a2 E ¼ b2 -a2 ¼ b2 s2 ; ¼ ∂x2 s2   ∂y a3 E ¼ b3 -a3 ¼ b3 s3 ; ¼ ∂x3 s3 where b ¼ fb1 ; b2 ; b3 g takes into account the effects of the means of x. After involving the standard deviations of x, a ¼ fa1 ; a2 ; a3 g simultaneously allows for the means and the variability of parameters. Hence, E½∂y=∂ξi  ¼ ai embodies response sensitivity to the mean (deterministic part) and the variability (stochastic part) of parameter xi. Finally, for m responses y ¼ fy1 ; y2 ; ⋯ym g with respect to n parameters, S can be established as follows:   1 0 0  1 E ∂y1 =∂ξ1 … E ∂y1 =∂ξn a11 … a1n B C B ⋮ ⋮ ⋱ ⋮ ⋱ ⋮ C S¼@ ð9Þ A¼@ A   ⋯ E½∂ym =∂ξn  E ∂ym =∂ξ1 am1 ⋯ amn where aij represents the sensitivity of response yi to parameter xj . And aij is in nature the coefficient of an individual term ξj in the PCE of yi. For practical applications, analysts can select parameters first by personal experience. Then a sensitivity analysis should be performed in order to validate the previous selection. If the response is not sensitive to the variations of certain parameters, then these parameters must be neglected or considered to be deterministic in the subsequent analysis for the sake of model simplification. Please cite this article as: S.-E. Fang, et al., Parameter variability estimation using stochastic response surface model updating, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j.ymssp.2014.04.017i

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4. The stochastic model updating procedure In this study an SMU procedure has been developed and its primary merits lie in two aspects: (a) SRSMs are used as surrogates to represent the relationship between uncertain parameters and responses. And in each optimization iteration these models are reconstructed according to the newly predicted probabilistic properties of parameters, which avoids the construction and analysis of sensitivity matrices and thus facilitates the convergence process. By these means a fast and efficient optimization process can be achieved. (b) The means and standard deviations of parameters are updated in a separate but successive way. The separate updating may reduce the number of optimization variables in an individual updating process, which simplifies the inverse problem and facilitates the convergence. Otherwise the simultaneous updating of means and standard deviations will increase the optimization complexity since they could affect each other due to e.g., their different levels in relative errors (see Eqs. (10) and (11)). Meanwhile, the first updating of means actually determines the deterministic part of parameters and thus controls the “base” of the updating procedure. Then the subsequent updating of standard deviations embodies the random part of parameters. Generally speaking, the separate updating scheme will benefit the optimization convergence and also improve the estimation of parameter variability. To be more specific, the developed SMU procedure can be implemented in five steps. (1) The probability distributions of parameters and responses are assumed according to known information or actual measurements. For most engineering problems, a clear probability inference of structural parameters usually requires a large volume of experimental data, which is often impractical due to expense consideration or experimental limitations. Thus a normal (Gaussian) distribution is popularly adopted without losing the generality, which is, under such circumstance, more appropriate than other distributions. This is due to the facts that (a) normal distributions are often found in engineering problems when the data collection is adequate and (b) in many cases a detailed description of probability distributions is not required since only the means and variances are sought [18]. To say the least, in case of nonnormal distributions, transformation techniques such as the Rosenblatt transformation [35] can be employed in order to obtain normal distributions. (2) An initial SRSM with respect to each response is constructed using the probabilistic collocation method and the regression method introduced in Section 2. And the model adequacy is evaluated through the comparison with a higher-order SRSM. (3) The sensitivity analysis is performed to screen the initially selected parameters and significant parameters whose changes induce clear variations of the responses are kept for subsequent updating. (4) The inverse optimization starts from this step. Firstly the means of the analytical responses are calculated using the current SRSMs and then the objective function is formed containing the relative errors of means between analytical and

start Assuming normal probability distributions on structural parameters and responses Initializing means and standard deviations of parameters constructing stochastic response surface models

probabilistic collocation method & regression method

calculating means and standard deviations of responses constructing objective functions establishing inverse problem and updating parameter means and standard deviations separately No

optimization converges? Yes

outputting results and calculating means and standard deviations of responses end Fig. 1. Flowchart of stochastic model updating procedure.

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measured responses m

F 1 ðμx Þ ¼ ∑

i¼1

μsrsm  μexp i i exp μi

!2 ð10Þ

and μexp denote the ith response means predicted by the SRSM and estimated from experimental where μsrsm i i measurements, respectively. It is noted that in this step, the standard deviations of parameters remain unchanged. In each iteration, all the SRSMs are reconstructed using the updated means of parameters. And the updating process continues until the relative errors fall into predefined tolerances. (5) The last updated means are assigned to the subsequent updating process for estimating standard deviations. In this step the means remain unchanged. The updating process is similar to that in step 4 except that the objective function now contains the relative errors of standard deviations between analytical and measured responses x

m

F 2 ðs Þ ¼ ∑

i¼1

ssrsm  sexp i i exp si

!2 ð11Þ

and sexp represent the ith response standard deviations predicted by the SRSM and estimated from where ssrsm i i experimental measurements, respectively. It is noted that standard deviations, which are the square roots of variances, are adopted due to the consideration that standard deviations directly stand for parameter variability in the same unit (e.g., Hz, but for variance, the unit is Hz2). And it would be easier for the least square optimization. Then with the updated means and standard deviations of all the parameters, one obtains the final SRSMs based on which the means and standard deviations of all the responses can be calculated. A single objective optimization strategy has been adopted and the dimensionless expression guarantees equal updating of each term in Eqs. (10) and (11). Finally, the constrained nonlinear optimization algorithm fmincon provided by Matlabs [36] has been employed, which attempts to find a constrained minimum of a scalar function having several variables. In this study, the upper and lower bounds of parameters are set as the constraints. min FðxÞ x

such that

lb r x r ub

ð12Þ

where x denotes the parameter constrained within the lower bound lb and the upper bound ub. The flowchart of the proposed SMU procedure is illustrated in Fig. 1. It is mentioned that in general complete covariance matrices should be involved in Eq. (11), but in this analysis only the diagonal elements of the response covariance matrix are considered since (a) the use of the most straight-forward objective function is also an option [18] giving a simplified optimization problem; (b) in some cases response correlations caused by parameter variability are not strong and thus do not significantly affect the updating predictions; (c) the uncertainties due to measurement errors or different modal extraction techniques are out of the scope of this study. On the other hand, complete covariance matrices can also be adopted which produces a more complex updating process [16,21,25,26] and thus leads to a computationally intensive SMU problem.

5. Numerical validation A numerical simply-supported beam (Fig. 2) has been firstly used for validation. The beam model had 30 identical finite elements with the cross-section dimension of 25 cm  25 cm, the elastic modulus (E) of 30 GPa, the density (D) of 2400 kg/m3 and Poisson ratio (P) of 0.2. The geometric parameter was represented by the area moment of inertia (I). And the first four flexural frequencies were taken as the responses. The beam parameters and the frequencies were assumed to follow normal probability distributions. The initial means of the parameters were given by the above nominal values and the corresponding frequency means (μf ) were 44.4, 176.1, 390.9 and 682.1 Hz computed by FE modal analysis.

5.1. Parameter sensitivity analysis To illustrate the frequency sensitivity to the parameter variations, four parameters; E, I, D and P were investigated by performing the sensitivity analysis procedure described in Section 3. Each parameter had an identical variability of 1% variation to the nominal value in the interest of equally evaluating the significance of all the parameters. Fig. 3(a) shows that E, I and D present very similar significance to the frequencies. But the influence of P does not appear. This observation agrees well with the theoretical expression for the flexural frequencies of a simply-supported uniform beam, which is already finite element 10 cm

300 cm

Fig. 2. A numerical simply-supported beam.

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Significance Contribution (%)

50

E D

40

I P

30 20 10 0

f1

f2

f3

f4

Frequency

Significance Contribution (%)

80

E

70

I

D

60 50 40 30 20 10 0

f1

f2

f3

f4

Frequency Fig. 3. Parameter sensitivity analysis of the beam: (a) equal variability and (b) unequal variability.

known as fn ¼

rffiffiffiffiffi EI 2L2 m

n2 π

ð13Þ

where m denotes the linear mass in kg/m related to the parameter D; L is the beam length and n defines the mode order. It can be seen that E, I and D have equal status in the expression indicating their equal significance contributions to the frequencies. Meanwhile, Eq. (13) does not embody any effect of P. The analysis results are very similar to those in [22] which utilizes the statistical analysis of variance (ANOVA) method. It should be mentioned that the ANOVA method follows a quite different clue which is based on variance analyses and thus parameter significance is determined by the statistical F-test comparison. Different to the ANOVA, our method utilizes the first-order derivation analysis of the parameter-response expression, which could be easier to handle and is more cost-efficient. Subsequently considering the fact that usually there is more variability in geometric properties due to manufacturing tolerances than that in material properties, the variability of E, I and D was defined as 1%, 2% and 1% respectivelyfor another comparison. The variability proportions are 1/2 (E/I, D/I) and 1 (E/D). At this moment P was not considered due to its insignificance to the flexural frequencies. Fig. 3(b) demonstrates that the significance contribution of I increases up to around 50% for all the four frequencies, while that of E and D decreases to around 25%. It is found that the significance proportions are 1/2 (E/I, D/I) and 1 (E/D), which is equal to the variability proportions. But in this case the proportions by the ANOVA method are in a square relationship [22]. This is due to the fact that in this analysis the sensitivity matrix is derived from the expectations of the derivations of Eq. (2) and parameter significance is embodied by standard deviations (see Section 3). But in ANOVA parameter significance is represented by variances being the square of standard deviations. This also reveals that the two parameter sensitivity analysis methods are inherently different. As a conclusion, the reliability and accuracy of the proposed sensitivity analysis method have been initially proved. 5.2. Stochastic model updating of uncertain parameters In this section, the “true” parameter variability (standard deviations) was defined as 1%, 2% and 1% variations to the nominal values (means) for E, I and D respectively. Accordingly the “measured” standard deviations sf for all the four frequencies are 0.54, 2.11, 4.60 and 7.83 Hz, respectively. It is noted that in practical cases, sf should be obtained by repetitive measurements on a set of nominally identical structures. Then at the beginning of updating, the initial parameter variability Please cite this article as: S.-E. Fang, et al., Parameter variability estimation using stochastic response surface model updating, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j.ymssp.2014.04.017i

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of E, I and D was assumed to be 2%, 4% and 2% respectively according to which the initial SRSM of each frequency was constructed based on 20 CPs (a double of the coefficient number of a second-order model), as illustrated in Fig. 4. A secondorder SRSM was chosen and compared with the third-order model in order to evaluate the adequacy of the former model (only the model expressions for the first frequency are presented as an example). (1) The second-order models f 1 ¼ 44:4018 þ 0:4441ξ1 þ 0:8837ξ2  0:4443ξ3 0:0022ðξ21  1Þ 0:0090ðξ22  1Þ þ 0:0067ðξ23 1Þ þ 0:0088ξ1 ξ2 0:0088ξ2 ξ3  0:0044ξ1 ξ3 (2) The third-order models f 1 ¼ 44:4018 þ 0:4441ξ1 þ 0:8837ξ2  0:4443ξ3 0:0022ðξ1 2  1Þ  0:0090ðξ2 2  1Þ þ 0:0067ðξ3 2  1Þ þ 0:0088ξ1 ξ2 0:0088ξ2 ξ3  0:0044ξ1 ξ3 þ10  2  ½0:0022ðξ1 3 3ξ1 Þ þ 0:0179ðξ2 3 3ξ2 Þ  0:0111ðξ3 3  3ξ3 Þ  0:0044ξ2 ðξ1 2 1Þ  0:0090ξ1 ðξ2 2  1Þ þ 0:0022ξ3 ðξ1 2  1Þ þ 0:0067ξ1 ðξ3 2 1Þ þ 0:0132ξ2 ðξ3 2 1Þ þ 0:0090ξ3 ðξ2 2  1Þ  0:0088ξ1 ξ2 ξ3  It can be seen that the coefficients of the two models are identical in the second-order terms. Simultaneously the frequency pdf estimations by the two models are also exactly the same (Table 1). Therefore, the adequacy and accuracy of the second-order SRSM were validated. Subsequently the probability distributions of the analytical frequencies can be calculated using the SRSMs and then combined with the “actual” frequency distributions to form the objective functions for establishing a stochastic inverse problem. Here it is noted that the frequency probability distributions predicted by the initial SRSMs are different to the “measured” distributions because the initial variability assumption of E, I and D are double of the “actual” values. Thus the SMU procedure (Section 4) should be implemented in order to update the initial model and to predict the “actual” means and standard deviations of the parameters.

(0,0,0)

collocation points Cube length is

2 3

Fig. 4. Collocation points for a second-order SRSM having three parameters.

Table 1 Comparison of frequency pdf estimations between initial second- and third-order SRSMs (beam, Hz). f1

f2

f3

f4

Second-order model Mean Standard deviation

44.40 1.0844

176.11 4.2536

390.83 9.2725

681.97 15.8013

Third-order model Mean Standard deviation

44.40 1.0844

176.11 4.2536

390.83 9.2725

681.97 15.8013

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5.3. Results and discussions It is observed from Table 2 that the updated means are very close to their nominal values and the predicted parameter variability is almost identical to the “true” values. Therefore, the feasibility and accuracy of the proposed method has been proved. It is also noted that compared with the existing SMU methods [16–20], the proposed method avoids the construction of sensitivity matrices and the implementation procedure is more simple. Meanwhile, the proposed method is more cost-efficient than Monte-Carlo based SMU methods since only a few iterations are required. Then in order to further validate the method, the measured frequencies from some metal plates tested in the laboratory [25] have been used for the experimental validation. 6. Experimental validation Thirty-three nominally identical metal plates tested in [25] were adopted for further validation. The nominal geometric dimensions of each rectangular plate were 564 mm (length)  110 mm (width)  1.45 mm (thickness). And the nominal material properties were Young's modulus of 210 GPa, the shear modulus of 83 GPa and the mass density of 7860 kg/m3. All the plates were tested under a free-free boundary condition and the first five frequencies were measured. In order to avoid other uncertainty sources such as changing boundary conditions as greatly as possible, the plates were tested under an identical condition. For example, the identical suspension positions and the same testing devices were taken. Finally based on the assumption of normal distributions, the frequency means, variances and standard deviations were statistically estimated and listed in Table 3. It can be seen that the frequency variations are very small (the maximum magnitude is 0.74% for mode 3) implying the small parameter variability. Then after investigation, the geometric thickness (t), Young's modulus (E) and the shear modulus (G) were chosen for SMU under the same assumption of normal probability distributions. The FE model of the plates was established using shell elements (Fig. 5). The first five frequencies were numerically computed and used for the objective functions. It can be seen from Fig. 5 that modes 1, 2 and 4 are the first three flexural modes, while modes 3 and 5 are the torsional modes. Two case studies have been investigated including the updating of thickness parameters and material parameters separately. In the first case, the plate model was equally divided into three portions: t1, t2 and t3 along the plate length. In the second case, two parameters E and G of the entire plate were considered. 6.1. Parameter sensitivity analysis The frequency sensitivity to the parameter variations was firstly analyzed where the variability of t, E and G were all defined as 1%. Fig. 6(a) presents the sensitivity evaluations of t1, t2 and t3. It is observed that for the first flexural and torsional modes (modes 1 and 3), t2 dominates the vibrational deformation of the entire plate while the two side portions, t1 and t3, show little influence. However, for the other 3 modes (modes 2, 4 and 5), the domination of t2 sharply decreases whereas that of t1 and t3 significantly increases with the increase of the mode order. For the second flexural mode, the domination of t1, t2 and t3 is in the same level. Moreover, it is found that the significance of t1 and t3 is always identical. This observation is rational because t1 and t3 are two symmetric portions of the plate. It also in turn proves the validity of the proposed sensitivity analysis method. Then the sensitivity analysis was also performed on the mixture of geometric and material properties (t, E and G), where t denotes the thickness of the entire plate. Fig. 6(b) demonstrates that t dominates all the 5 modes indicating that thickness provides much greater contribution to the bending and torsional stiffness of the plate than E and G. Meanwhile, E controls only the flexural vibrations (modes 1, 2 and 4) while G is important for the two torsional modes (modes 3 and 5). Table 2 Parameter variability estimations of the beam.

Mean Standard deviation

E

I

D

0.9997 0.0099

0.9997 0.0198

0.9992 0.0099

Note: The means and standard deviations have been normalized to the nominal values of parameters.

Table 3 Statistical features of measured frequencies of the plates (Hz). Frequency

f1

f2

f3

f4

f5

Mean Standard deviation Variance

24.12 0.113 0.013

66.92 0.252 0.063

77.65 0.571 0.326

131.97 0.424 0.180

158.80 0.974 0.949

Note: The data are taken from Ref. [25].

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FE Model

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

11

Fig. 5. FE model and mode shapes of the plates.

Significance contribution (%)

100 t1 t2 t3

90 80 70 60 50 40 30 20 10 0

f1

f2

f3

f4

f5

Frequency

Significance Contribution (%)

100 t E G

90 80 70 60 50 40 30 20 10 0

f1

f2

f3

f4

f5

Frequency Fig. 6. Parameter sensitivity analysis of the plates: (a) geometric parameters and (b) geometric and material parameters.

As a conclusion, it is found that the significance of parameter uncertainties can be well evaluated using the proposed sensitivity analysis method. And the effects of different uncertain parameters on the dynamic properties of a structure can also be investigated in detail.

6.2. Stochastic model updating of thicknesses The thickness variability was firstly sought and three parameters (t1, t2, and t3) were updated. With respect to the five frequencies, five second-order SRSMs were constructed through the regression of 20 CPs and then compared with the Please cite this article as: S.-E. Fang, et al., Parameter variability estimation using stochastic response surface model updating, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j.ymssp.2014.04.017i

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third-order models for the purpose of model adequacy checking. The expression for the first frequency is given here as an example. (1) The second-order models f 1 ¼ 24:2680 þ0:0137ξ1 þ 0:2154ξ2 þ 0:0137ξ3  0:0006ðξ21  1Þ  0:0010ðξ22  1Þ  0:0006ðξ23  1Þ þ 0:0010ξ1 ξ2 þ 0:0010ξ2 ξ3 þ 0:0010ξ1 ξ3 (2) The third-order models f 1 ¼ 24:2680 þ0:0137ξ1 þ 0:2154ξ2 þ 0:0137ξ3  0:0006ðξ21  1Þ  0:0010ðξ22  1Þ  0:0006ðξ23  1Þ þ 0:0010ξ1 ξ2 þ 0:0010ξ2 ξ3 þ 0:0010ξ1 ξ3 þ 10  2 ½0:0009ðξ1 3  3ξ1 Þ  0:0003ðξ2 3  3ξ2 Þ þ 0:0009ðξ3 3  3ξ3 Þ 0:0019ξ2 ðξ1 2  1Þ þ0:0009ξ1 ðξ2 2  1Þ 0:0003ξ3 ðξ1 2  1Þ  0:0003ξ1 ðξ3 2  1Þ 0:0019ξ2 ðξ3 2  1Þ þ0:0009ξ3 ðξ2 2  1Þ þ0:0010ξ1 ξ2 ξ3  It can be seen that in the aspect of the first- and second-order terms, the coefficients of the two models are exactly identical. Simultaneously the frequency pdf estimations by the two models were also compared (Table 4a) and it is found that they are exactly the same with very little difference in the standard deviations. Therefore, the adequacy and accuracy of the second-order SRSM have been proved. Subsequently the analytical frequency distributions can be calculated using the SRSMs and then combined with the measured distributions to form the objective functions for establishing a stochastic inverse problem. The SMU was implemented using the procedure described in Section 4. And the means and standard deviations of t1, t2 and t3 were updated in a successive way with their values listed in Table 5a. Compared with the updating results in [25,22], the estimations of the symmetric portions t1 and t3 are also symmetric, namely μt1 ¼ μt3 and st1 ¼ st3 . The means estimated by three different methods are similar while some difference appears in the estimations of the standard deviations. The standard deviations estimated by the proposed method and [22] seem more rational since for such small metal plates, the variability of the three portions should be close. Then by assigning the updated means and standard deviations of the parameters to the SRSMs, the corresponding frequency probability properties can be Table 4a Comparison of frequency pdf estimations between initial second- and third-order SRSMs (plate: thickness, Hz). f1

f2

f3

f4

f5

Second-order model Mean Standard deviation

24.27 0.21627

67.20 0.38850

76.56 0.66371

132.41 0.76911

156.72 1.02216

Third-order model Mean Standard deviation

24.27 0.21627

67.20 0.38852

76.56 0.66373

132.41 0.76910

156.72 1.02216

Table 5a Variability quantification of plate thicknesses (mm). Prediction

Husain et al. [25]

Mean Variance (  10  4)

Fang et al. [22]

Proposed method

t1

t2

t3

t1

t2

t3

t1

t2

t3

1.4528 1.290

1.4493 0.535

1.4528 1.290

1.4605 0.627

1.4552 0.770

1.4605 0.627

1.4552 0.974

1.4524 0.747

1.4552 0.974

Table 6a Measured and estimated variances of frequencies using thicknesses as parameters. Mode

Measured

Husain et al. [25]

Errors (%)

Fang et al. [22]

Errors (%)

Proposed method

Errors (%)

1 2 3 4 5

0.013 0.063 0.326 0.180 0.949

0.01 0.06 0.13 0.24 0.28

23.1 4.8 60.1 33.3 70.5

0.021 0.141 0.207 0.544 0.742

61.5 123.8 36.5 202.2 21.8

0.017 0.065 0.157 0.259 0.482

30.8 3.2 51.8 41.7 49.2

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calculated. As to the estimations of the means, the average errors for the five frequencies are 0.82% and 0.87% before and after updating, which indicates very accurate FE modeling and updating processes. The corresponding errors from [25] are 1.47% and 1.49% respectively. Meanwhile for the variance estimations (Table 6a), all the three methods produce relatively large errors. But the proposed method gives the most balanced estimations with an average error of 35.3%, which are 38.4% in [25] and 89.2% in [22]. The estimations with respect to the flexural modes are much better than those in [22] and the estimations with respect to the torsional modes are better than those in [25]. It should be mentioned that in SMU accurate estimations of parameter variances are still a pending problem which needs further research. Fig. 7(a) depicts the scatter plots of the measured and estimated frequencies which do not overlap indicating unsatisfactory convergence, as has also been found in [25,22]. Due to it, in the next section the material parameters were adopted for updating. 6.3. Stochastic model updating of Young's and shear moduli In order to investigate the variability influence of material parameters, Young's modulus and the shear modulus of the entire plate were also updated. Five second-order SRSMs were constructed based on 12 CPs and then compared with the corresponding third-order models. The coefficients of the second- and third-order models were compared and the expression for the first frequency is given here as an example. (1) The second-order models f 1 ¼ 24:2701 þ 0:1254ξ1  0:0041ξ2 þ 10  3 ½  0:2097ðξ21  1Þ þ0:1142ðξ22  1Þ 0:2079ξ1 ξ2  (2) The third-order models f 1 ¼ 24:2701 þ 0:1254ξ1  0:0041ξ2 þ 10  3 ½  0:2097ðξ21  1Þ þ0:1142ðξ22  1Þ 0:2079ξ1 ξ2 þ 0:0015ðξ31 3ξ1 Þ  0:0015ðξ32  3ξ2 Þ 0:0012ξ2 ðξ21 1Þ þ 0:0028ξ1 ðξ22 1Þ It is seen that the coefficient predictions are very stable for the two models. Then the frequency pdf estimations were also compared (Table 4b) and it is found that only very little difference appears in the aspect of the standard deviations. Therefore, the adequacy of the second-order SRSMs for SMU has been proved. Table 5b demonstrates that the estimated

Fig. 7. Scatter plots of plate frequencies after stochastic model updating using (a) thicknesses as parameters and (b) material properties as parameters.

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Table 4b Comparison of frequency pdf estimations between initial second- and third-order SRSMs (plate: Young's and shear moduli, Hz). f1

f2

f3

f4

f5

Second-order model Mean Standard deviation

24.27 0.12551

67.21 0.37408

76.56 0.37103

132.41 0.79663

156.73 0.69739

Third-order model Mean Standard deviation

24.27 0.12551

67.21 0.37406

76.56 0.37104

132.41 0.79654

156.73 0.69734

Table 5b Variability quantification of plate material properties (GPa). Prediction

Husain et al. [25]

Mean Variance

Fang et al. [22]

Proposed method

E

G

E

G

E

G

209.6 2.6

83.8 1.5

209.2 4.6

84.9 1.2

208.8 1.84

85.5 1.49

Table 6b Measured and estimated variances of frequencies using material properties as parameters. Mode

Measured

Husain et al. [25]

Errors (%)

Fang et al. [22]

Errors (%)

Proposed method

Errors (%)

1 2 3 4 5

0.013 0.063 0.326 0.180 0.949

0.01 0.06 0.28 0.22 1.09

23.1 4.8 14.1 22.2 14.9

0.015 0.114 0.239 0.437 0.957

15.4 81.0 26.7 142.8 0.8

0.007 0.058 0.290 0.267 1.013

46.2 7.9 11.0 48.3 6.7

parameter means are very close to those in [25,22]. And the variances estimated by the proposed method are also close to those in [25]. Meanwhile after updating, the average errors of the five frequencies improved from 0.82% to 0.12%, which indicates very accurate FE modeling and updating processes. Subsequently the frequency variances were checked (Table 6b) and it is observed that better estimations were obtained after compared with the results in Table 6a. And the average error of 24% by the proposed method is much better than that of 53% given in [22]. Fig. 7(b) illustrates the satisfactory convergence of the predicted frequencies to the measured ones. Hence, one can conclude that the material parameters are suitable parameters for this case study, as has also been found in [25,22]. 7. Conclusions A stochastic model updating method based on SRSMs has been developed in the interest of estimating parameter variability of structures. The FE model containing uncertain parameters is replaced by the SRSMs for fast calculation of the probability distributions of structural responses, as well as for cost-efficient solutions of SMU problems. The SRSMs are updated and reconstructed during optimization iterations, by which means the analysis of stochastic sensitivity matrices is avoided. Then the complexity of an SMU problem is reduced and the computational efficiency is considerably improved. On the other hand, the means and standard deviations of parameters are updated and estimated in a separate and successive way. By this means estimation of parameter variability can be effectively implemented. Meanwhile, a new sensitivity analysis method based on SRSMs has also been proposed and the effects of parameter variability on responses can be well investigated. Finally, the proposed methods have been verified firstly against a numerical beam and then against a set of experimental metal plates. The analysis results have been compared with two existing SMU methods and the feasibility and reliability of the proposed methods have been confirmed.

Acknowledgments The research is supported by the National Natural Science Foundation of China (Grant no. 51108090) and also by the Natural Science Foundation of Fujian Province, China (Grant no. 2011J05129). The financial supports from the Scientific Research Foundation for the Returned Overseas Chinese Scholars (Grant no. LXKQ201201) and the Cultivation Project of Outstanding Youth Researchers in Universities of Fujian Province (Grant no. JA12020) are also acknowledged. Please cite this article as: S.-E. Fang, et al., Parameter variability estimation using stochastic response surface model updating, Mech. Syst. Signal Process. (2014), http://dx.doi.org/10.1016/j.ymssp.2014.04.017i

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