A highly efficient probabilistic framework of finite element model updating in the presence of measurement noise/uncertainty using intelligent inference is presented. This framework uses incomplete modal measurement information as input and is built upon the Bayesian inference approach. To alleviate the computational cost, Metropolis–Hastings Markov chain Monte Carlo (MH MCMC) is adopted to reduce the size of samples required for repeated finite element modal analyses. Since adopting such a sampling technique in Bayesian model updating usually yields a sparse posterior probability density function (PDF) over the reduced parametric space, Gaussian process (GP) is then incorporated in order to enrich analysis results that can lead to a comprehensive posterior PDF. The PDF obtained with densely distributed data points allows us to find the most optimal model parameters with high fidelity. To facilitate the entire model updating process with automation, the algorithm is implemented under ansys Parametric Design Language (apdl) in ansys environment. The effectiveness of the new framework is demonstrated via systematic case studies.

## Introduction

Finite element analyses are widely used to predict the structural dynamic responses. The result from a finite element analysis, however, oftentimes is different from the experimental measurement from an actual structure. This discrepancy is due to a number of factors, ranging from the noise in measurement, normal variation of the structure, to the error in the finite element model itself. In the context of structural dynamic analysis, there has been growing interest in updating the finite element model based on vibration responses measured, such as natural frequencies and mode shapes, to facilitate robust design, vibration control, and structural health monitoring [1,2]. The updating of deterministic model using modal information has been well practiced [35]. Typically, such model updating is conducted through formulating an inverse problem based upon the difference between current model prediction and the corresponding response measurement under the same operating condition. Although this inverse problem can be solved by means of iteratively minimizing the response difference, the result may still be not accurate since the inversion of the sensitivity matrix may be close to ill-conditioned which, fundamentally, is caused by the insufficient amount of input information. That is, the measurements are generally incomplete for practical problems [6,7]. The issue is further compounded by the uncertainties and normal variations mentioned above [810].

Ideally, model updating should be conducted in the probabilistic sense, i.e., treating model parameters to be updated as random variables with means and variances. This can reveal the underlying properties of structures under the inevitable uncertainties and variations [11]. Several types of probabilistic approaches have been explored for this purpose. Soize presented a nonparametric probabilistic approach based on random matrix theory to model the structural uncertainties and estimate the dispersion parameters [12]. Khodaparast et al. developed a perturbation scheme to analyze the statistical moments of updated parameters from measured variability in structural modal responses [13]. It is worth noting that the Bayesian inference-type of methods has recently attracted significant attention due to some intrinsic advantages [1416]. The essence of Bayesian inference is to establish a probabilistic model to correct the prior beliefs based on the evidences. It starts from specifying the model parameters with prior information in the form of PDF, which may be viewed as imposing soft physical constraints to enable a unique and stable solution. Then, by introducing measured response data, the assumed prior PDF is updated to the so-called posterior PDF that will then be analyzed to yield the optimal model parameters. This actually avoids the abovementioned drawback of matrix inversion, since the Bayesian model updating is facilitated through finite element forward analyses under certain model parameter sample. Moreover, it is built upon rigorous probabilistic framework that can directly incorporate various sources of uncertainties in the model parameters to be updated. In recent years, the Bayesian inference has seen increasing usage in a variety of engineering problems. Beck and Katafygiotis first formulated a Bayesian probabilistic procedure for structural model updating and validated its underlying idea by case illustration on a two degrees-of-freedom (2DOF) linear planar shear building [14]. Following that, a number of studies have expanded the formulation to tackle more complicated problems in finite element model updating [17]. In the context of system identification, this type of methods has also been applied to the diagnosis of the fatigue crack growth and the model class selection [15,16].

Although the Bayesian inference approach possesses promising features, its application to large-scale complex systems faces challenges. One obstacle is the high computational cost. In general, Monte Carlo-based analysis is adopted to acquire the dynamic responses of parameterized model samples, which may lead to prohibitive computational cost when the dimension of the finite element model is high. In some special cases, the computational issue can be resolved by asymptotically approximating the posterior PDF with the so-called most probable value [18]. This technique, however, is only valid for simplified cases [19]. For large-scale problems, one school of thought is to alleviate the computational cost of every single run in the Monte Carlo analysis by employing order-reduced model [2023]. Another school of thought is to build certain surrogate models to mimic the behavior of the original finite element model by creating generic input–output relations, such as response surface models [24,25], artificial neural networks [26], and Kriging predictor [27]. One issue with these approaches is that the error of response prediction, e.g., the modal truncation error in the order-reduced approach, may become considerable when compared with the actual response deviation between the measurement and finite element model prediction. Indeed, a lot of researchers have looked into the sampling procedure and suggested the Markov chain Monte Carlo (MCMC) for analysis acceleration [2830]. A Markov chain that contains a reduced number of samples is generated using, for example, the Metropolis–Hastings (MH) algorithm and the importance sampling technique. When applying the MH MCMC, the proposal PDF needs to be defined with proper variance, which fundamentally determines the random sampling property over the entire parametric space. The efficiency and accuracy of MCMC depend heavily on the selection of such proposal PDF. When the posterior PDF is peaked, the peaked region of posterior PDF will never be reached if the proposal PDF is too wide (i.e., with large variance). On the other hand, the Markov chain travels very slowly before reaching the peaked region if the proposal PDF is too narrow [29,31,32]. Even though high efficiency may be reached with wide proposal PDF, the posterior PDF obtained may not be informative since it only contains sparsely distributed data points and most of them are even out of the peaked region. Without the essential information, the model parameters identified based on such posterior PDF may not be considered optimal. Apparently, to improve the identification accuracy, the enrichment of sparse posterior PDF is required, which has not been addressed yet in related studies.

The objective of this research is to tackle the fundamental issue of computational cost mentioned above to enable the direct application of Bayesian inference-based probabilistic model updating in the finite element models of practical structures. Specifically, we aim at structures with large scale and high dimensionality whereas only a severely incomplete set of modal information can be obtained as input. We adopt the MCMC procedure to conduct the Bayesian model updating. While it inevitably results in a sparse posterior PDF, utilizing the distributed data points as training data we then employ GP emulator [33,34] to enrich the posterior PDF. This is realized by extending the distribution over the reduced parameterized sample space to that over the full parameterized sample space. The comprehensive posterior PDF obtained generally is well represented by its statistical moments, i.e., mean and standard deviation due to its dense data point distribution. As such, the identification of uncertain parameters can be significantly improved. Meanwhile, to truly unleash the power of Bayesian probabilistic model updating for practical applications, we incorporate all the algorithm development into the ansys environment through apdl programming, i.e., the entire procedure is realized within ansys by directly utilizing the finite element mesh and the built-in solvers. This allows us to take full advantage of the efficiency and robustness of commercial solvers, and at the same time yields quantitative understanding of the computational cost involved in analyzing practical structures based on a commercial package.

The rest of the paper is organized as follows: In Sec. 2, the general formulation of finite element model updating at the presence of uncertainty is outlined first. Following that, a Bayesian inference-based model updating framework integrated with a computationally accelerating scheme, i.e., MCMC, is formulated. The GP that aims at improving the updating accuracy is outlined. Section 3 provides case studies on a mock-up wind turbine with detailed parametric investigations regarding model updating performance. Section 4 gives the concluding remarks.

## Methodology Formulation

### Problem Definition.

The dynamic equation of a linear structural system is generally given as
$Mx¨+Cx˙+Kx=F$
(1)
where M, C, and K are, respectively, the mass, damping, and stiffness matrices, x is the N-dimensional displacement vector, and F is the external excitation. We assume that the structure is lightly damped with proportional damping, and therefore, we focus our attention of model updating on the mass and stiffness matrices only. For simplicity and without loss of generality, we assume that the mass and stiffness matrices of the actual structure are represented as [35]
$M̃(θ)=M+∑i=p+1qθiMe,i$
(2a)
$K̃(θ)=K+∑i=1pθiKe,i$
(2b)
In the above equations, M and K represent the nominal mass and stiffness matrices to be updated, and $Ke,i$ and $Me,i$ are the elemental stiffness and mass matrices associated with those finite elements whose properties are subject to updating. The total number of elements to be updated is q. The changes of elemental properties are related to the variation of certain structural properties, i.e., material property, geometry, or boundary conditions that are characterized by a q-dimensional parameter vector $θ$. In this research, we assume that the model updating, i.e., identification of $θ$, is facilitated by using modal measurement information. Correspondingly, the eigenvalue problem of the finite element model of the actual structure is
$(K̃(θ)−ωi2(θ)M̃(θ))φ̃i(θ)=0$
(3)

where $ωi2(θ)$ and $φ̃i(θ)$ are, respectively, the ith eigenvalue and eigenvector. Since the mass and stiffness matrices are functions of $θ$, the eigenvalues and eigenvectors are functions of $θ$ as well.

In order to identify the unknown parameter vector $θ$ to update the nominal model, we need to incorporate the measurement of natural frequencies and mode shapes into the analysis. In practice, however, the measured modal information is usually severely incomplete. First, normally, only the lower-order natural frequencies and mode shapes can be realistically measured due to constraints in actuation power and sensing sensitivity in practice. Moreover, while the number of DOFs of the finite element model of a structure can be very high, in most cases only a small number of sensors can be installed on the structure to measure the modal amplitudes at the corresponding small number of DOFs [6,7], and the motions at rotational DOFs cannot be directly measured at all. In what follows, we assume that we are able to acquire the following information from measurement:
$Λ=[ω12,ω22,...,ωn2]$
(4a)
$Φ=[φ1,φ2,...,φn]$
(4b)
$φi=[φi,1, φi,2,....φi,m]T$
(4c)

where $Λ$ denotes a vector that consists of n measured eigenvalues (i.e., the squares of measured natural frequencies), $Φ$ is the associated mode shape matrix with m measured DOFs, and each mode shape is described as $φi$ in which only the amplitudes at m DOFs are available. Both n and m are much smaller than N, the total number of DOFs of the structures. Our objective is to use the above equations together with the measurement information to identify the unknown vector $θ$. Specifically, due to the inevitable uncertainties such as model variations and measurement noise, $θ$ is a random vector. We need to identify the mean and variance of every component, $θi$ ($i=1,…,q$), of $θ$.

### Bayesian Inference Framework for Model Updating.

The underlying idea of Bayesian inference can be fully represented by the following Bayes' rule:
$p(θ|D)=p(D|θ)p(θ)∫p(D|θ)p(θ)dθ$
(5)

When applying Bayes' theorem for structural model updating [14], the hypothesis $θ$ is interpreted as the vector of model parameters that need to be identified. D denotes the evidence, which in this study is the measured modal information shown in Eqs. (4a)(4c). The prior PDF $p(θ)$ represents the initial distribution of model parameters $θ$ built upon the empirical knowledge of engineers. Without explicit understanding of the target problem, this term can be simply defined as a standard statistical distribution, such as normal and uniform distributions. The posterior PDF $p(θ|D)$ indicates the updated distribution of model parameters $θ$ conditional on the prior PDF and measurement data D. The likelihood PDF $p(D|θ)$ is used to evaluate the agreement between the measurement and the response prediction of the model parameterized by $θ$.

The key procedure to derive the likelihood PDF is to establish the probabilistic relationship between the measurement data and the model response prediction in the presence of uncertainty. In this research, one type of uncertainty, the measurement error, is taken into account. The measurement error in practice is due to ambient noise that can be modeled as a normal distribution with zero mean. Hence, the measurement can be expressed as
$D=g(θact)+ε$
(6)
where g represents the operator of model response under actual parameter vector $θact$, and $ε$ is the measurement error. We can also formulate the likelihood PDF as a normal distribution of model parameters with respect to the measurement data D [36]. Specifically, for the jth eigenvalue $ωj2$ measured, we can have
$p(ωj2|θ)=bje−(ωj2−ω̃j2(θ))22aj$
(7)
where $aj$ is the variance of $ωj2$, $bj$ is a constant associated with the variance $aj$, and $ω̃j2(θ)$ is the jth eigenvalue obtained from the model with parameters $θ$. Equation (7) indicates that larger PDF value will be produced when $ω̃j2(θ)$ is closer to the measured eigenvalue $ωj2$, based on which the parameter candidate $θ$ can be quantitatively screened in a probabilistic manner. Similarly, for the jth measured incomplete mode shape $φj$, we can have
$p(φj|θ)=dje−(φj−sΓφ̃j(θ))TCj−1(φj−sΓφ̃j(θ))2$
(8)
where $Cj$ is the covariance matrix of $φj$, $dj$ is a constant associated with the variance $Cj$, and $φ̃j(θ)$ is the jth mode obtained from the model with parameters $θ$. Care should be taken for mode shape comparison shown in Eq. (8), since the mode shape measurement information is generally incomplete and not exactly mass-normalized. Here, we left-multiply the numerically predicted mode shape $φ̃j(θ)$ with a Boolean matrix $Γ$ to pick out the modal components that correspond to the DOFs at which the sensors are placed (i.e., at which the modal amplitudes are measured). Moreover, we use a scalar s to calibrate the measured mode $φj$ with respect to the numerically predicted mode $φ̃j(θ)$ which is defined as
$s=〈φj,Γφ̃j(θ)〉||Γφ̃j(θ)||2$
(9)

where $〈.,.〉$ is the inner product of two vectors, and ||.|| is the Euclidean norm. It is worth nothing that when another type of uncertainty, e.g., modeling uncertainty is introduced, the discrepancy between the measurement and the nominal response can be considered as the direct summation of the measurement uncertainty and the response prediction variation due to modeling uncertainty, since each type of uncertainty is simply assumed to be statistically independent. In such case, the variance parameters $aj$ and $Cj$ shown in Eqs. (7) and (8) need to be adjusted based on the cumulative uncertainty level.

As the probability of concerned model parameters is updated by taking into account both the measured natural frequencies and the measured modes, we can write the likelihood PDF in a multiplication form, which is due to the direct summation of involved response contributions within the exponential function (Eqs. (7) and (8)). In the case that a number of datasets are involved, the likelihood PDF will further become
$p(D|θ)=∏i=1r∏j=1np(φi,j|θ)p(ωi,j2|θ)$
(10)

where r is the number of measured datasets, and n is the number of measured modes in each dataset. $ωi,j2$ and $φi,j$ represent, respectively, the jth eigenvalue and the associated mode in the ith measured dataset.

As measurement error exists, each measured dataset differs from another. We thus combine the multiple measurements to minimize the negative impact of measurement error to model updating. Substituting Eq. (10) into Eq. (5) yields the posterior PDF, which can eventually reveal the underlying characteristics of the actual structured identified. The general framework of Bayesian inference for model updating is shown in Fig. 1(a). As can be seen, no matrix inversion is involved in this process.

Fig. 1
Fig. 1
Close modal

### MCMC Sampling Enhancement to Expedite Bayesian Model Updating.

The computational framework of Bayesian inference involves the calculation of the denominator and the numerator shown in Eq. (5). Take the denominator as an example. The integral can be numerically evaluated as
$∫p(D|θ)p(θ)dθ≈1N∑i=1Np(D|θi)$
(11)

where N is the number of model samples $θ$ parameterized from the prior PDF $p(θ)$. According to the law of large number, the larger the N is, the better the accuracy one will achieve. Increasing N, however, will substantially increase the computational cost, since the modal information predictions $ω̃j2(θ)$ and $φ̃j(θ)$ required in the likelihood PDF have to be calculated by sampling-based Monte Carlo simulation (e.g., with N repeated runs). When the dimension of the finite element model is high, each single run of the eigenvalue problem is already costly.

To mitigate the computational cost issue here, we adopt the MH MCMC for model parameter sampling [29,37,38]. The fundamental idea of MCMC is that it can generate a chain including a smaller number of model parameter samples, which can directly reduce the number of runs of finite element analysis. The general procedures of MH MCMC are summarized as follows [39]:

With $θt*$ at time t, the aim is to generate the next chain value $θt+1*$.

• (a)

Proposal step: Sample “candidate” Z from the proposal distribution, $Z∼q(z|θt*)$. Note that $q(z|θt*)$ is the proposal distribution of z depending on the deterministic parameter $θt*$.

• (b)

Acceptance step: With probability $α(Z,θt*)=min(1,{[p(D|Z)p(Z)q(θt*|Z)]/[p(D|θt*)p(θ¯t)q(Z|θt*)]})$

Generate U from a uniform (0, 1) distribution

if $α(Z,θt*)>U$, we set $θt+1*=Z$ (i.e., acceptance)

else $θt+1*=θt*$ (i.e., rejection).

Thereafter, the integral can be alternatively evaluated as
$∫p(D|θ)p(θ)dθ≈1M∑i=1Mp(D|θi*)$
(12)

where M is the length of Markov chain that typically includes a much smaller number of model samples $θ*$ (MN). While the posterior PDF can be obtained by calculating Eq. (5) based on the reduced model sample input, it contains a sparsely distributed dataset.

### GP Emulation to Improve Updating Accuracy.

The MCMC sampling may be able to identify the normalization constant (i.e., the denominator in Eq. (5)) by estimating the mean of likelihood PDF. However, the global maximum of the numerator $p(D|θ)p(θ)$ may be difficult to reach, due to the limitation of the sampling technique itself, e.g., the selections of proposal PDF and its variance [29,32]. Meanwhile, the small number of sampled candidates cannot densely span over the entire parametric space especially when the high-dimensional model updating problem is considered. Therefore, the most optimal model parameters corresponding to that global maximum in many cases cannot be found based on such sparse posterior PDF.

To enhance the model updating fidelity with the reduced sample size, in this research we propose to enrich the posterior PDF over the original parametric space $θ$ by using the sparse data input and output that can be considered as training data in statistical inference analysis. It is worth mentioning that the reason we consider the data from MCMC rather than from random sampling as training data lies in that the data from MCMC are more representative since they are generally distributed at the neighborhood of the peak, i.e., highest probability point with stationary probability that is in accordance with the underlying principle of MH sampling technique. Usually, such training data points selected in the probabilistic distribution are able to well capture the peak through inference emulation due to their location nearness [40]. In this situation, GP is a promising inference tool to emulate the relation between model parameter input and the corresponding posterior probability output. The simplicity and computability of GP have made it effective in a number of applications. One of the early applications was the use of GP as a means of emulating complex simulation [33,34]. This approach has been further extended to the field of structural dynamics [4143]. Specifically, a GP is a collection of random variables of which any finite number has a multivariate normal distribution. A GP can be interpreted as a distribution over a function $f(θ)$ that allows the random variables to be the output of $f(θ)$ at an input $θ$ [40]. In this case, $f(θ)$ denotes the posterior PDF $p(θ|D)$. In general, a GP is fully specified by a mean function and a covariance function, which are defined below [44]
$E[f(θi)]=m(θi)$
(13a)
$E[(f(θi)−m(θi))(f(θj)−m(θj))]=k(θi,θj)$
(13b)

where m(.) and k(.) represent, respectively, the mean and covariance functions of this GP. There exist a wide range of commonly used mean and covariance functions in literature [40]. For a finite-dimension matrix $Θ$ comprising of inputs $θi$ as its rows, the corresponding mean vector and covariance matrix can be denoted as $μ(Θ)$ and $Σ(Θ,Θ)$.

The process of updating a GP with data is equivalent to conditioning the distribution at observed input–output relation. Let a reduced dataset be comprised of $θ*$ and an output vector $f(θ*)$ be generated from MCMC. Let a set of to-be-predicted target output be comprised of the original sampled data $θ$ and the corresponding vector of unknown outputs be $f(θ)$. Together, as well as individually, the data and target points constitute a finite set, and thus can be partitioned as shown in the following multivariate distribution [40]:
$f(θ)|θ,f(θ*),θ*∼N(u+Σ*TΣ**−1(f(θ*)−u*),Σ−Σ*TΣ**−1Σ*)$
(14)

where u and $Σ$ are, respectively, the mean vector and the covariance matrix over the unobserved/original parametric space.

The first step in the GP emulation is to define the suitable mean and covariance functions. Following what is suggested by Xia and Tang [43], we define the mean function as
$m(θ)=h(θ)β$
(15)
where $h(.)$ is a vector of the known functions with respect to $θ$, and $β$ is a vector of unknown coefficient to be optimally identified later. Furthermore, the covariance function shown in Eq. (13b) is selected as the squared exponential function
$k(θi,θj)=σf2e−b(θi−θj)T(θi−θj)+σn$
(16)

where $ψ={σf,b}$ is a set of hyper parameters. $ψ$ together with $β$ can be optimized by maximizing the marginal likelihood. Typically, we assume that there is no error between two arbitrary observations on the covariance value, and thus choose $σn=0$ [33]. For this particular covariance function, it is found that the covariance is almost unity between variables whose corresponding inputs are very close, and decreases as their distance in the input space increases. This indicates that simulator runs with close inputs will have similar outputs [40]. The enhanced framework incorporated with GP enrichment is illustrated in Fig. 1(b).

## Implementation and Case Studies

In this section, we present case analysis demonstration and parametric investigation. The analysis is carried out in ansys environment under apdl programming.

### Illustrative Case.

The illustrative case is based on a mock-up wind turbine structure shown in Fig. 2. Wind turbines operate under harsh conditions and require robust design and maintenance [45]. They have huge size and are generally subject to various variations especially during modal testing measurements. The Young's modulus, mass density, and Poisson's ratio of this structure (including both the tower and the blades) are initially assumed to be, respectively, $2.06×1011 Pa$, $7.85×103 kg/m3$, and 0.3. The geometry parameters are provided in Fig. 3. The blade roots are fixed to a shaft whose two ends are fixed to the nacelle that connects with the tower. The finite element mesh for this case illustration is generated by the “smart” meshing function in ansys, yielding a mesh with 188,210 solid elements and 108,321 DOFs. The local mesh density is automatically adjusted based upon the local model geometry. Convergence analysis has been carried out to verify that the mesh density is sufficient for this particular structure. Our goal here is to use sensor data to update the baseline model. Without loss of generality, we further assume that there exist two parameters that are different from the initial assumptions, i.e., the Young's modulus of the tower and the Young's modulus of the blades. We let such differences be denoted as $θ1$ and $θ2$, respectively, which need to be identified/updated.

Fig. 2
Fig. 2
Close modal
Fig. 3
Fig. 3
Close modal

The incompleteness of measurement indeed renders the system identification an underdetermined problem from the conventional inverse analysis standpoint. Here, we assume that we have seven uniaxial sensors/accelerometers to extract the modal information. Furthermore, we assume that the first four x-direction bending modes of the system are measured. The placement of the sensors and the measurement patterns of bending modal modes are given in Figs. 4 and 5.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal

### apdl-Based Implementation.

One effort in this research is to incorporate the Bayesian inference framework with commercial finite element package, so the algorithms developed can be fully implemented to practical applications. Here, we adopt ansys as the computational platform and develop apdl code to automatically manage the model parameter definition, mesh generation, modal analysis, data postprocessing, and extraction. apdl stands for ansys Parametric Design Language. It allows one to perform analysis based on parameters (variables) defined. It features programming functionalities, e.g., repeating commands or macros, do-loops, and scalar, vector, and matrix operations [46]. In this research, the modal property evaluation, which is an essential part of the Bayesian model updating framework developed (Fig. 1(a)), is executed following the procedures shown in Fig. 6. The apdl pseudocode with comments is given in the Appendix.

Fig. 6
Fig. 6
Close modal

### Model Updating Demonstration and Efficiency/Accuracy Comparison.

This section reports a representative model updating case based on the algorithms presented. For demonstration, we let the actual model parameters (Young's modulus difference ratios) be $θact=[θ1,θ2]=[−0.1056,0.0736]$, i.e., the actual Young's modulus of the tower is 89.44% of the nominal value, and that of the blades is 107.36% of the nominal value. In this study, we utilize simulated sensor data as input for model updating. As mentioned, the modal information of the first four x-direction bending modes is extracted by the sensors. This allows us to alter certain parameters concerning the measurements for systematic investigation. We let the simulated sensor data be subjected to measurement noise. The simulated sensor data are obtained by multiplying the finite element response of the nominal model with random error under specified standard deviation. In particular, here we assume that measured natural frequencies are subjected to 1% measurement error, i.e., error with 1% standard deviation. For mode shape information, only the response amplitudes at the DOFs corresponding to the seven sensors are measured, and each datum is subjected to 3% standard deviation measurement error. Lager error in mode shape measurement is adopted, because mode shapes are oftentimes subjected to more severe measurement uncertainty. In this case, three sets of measurements are employed.

Assuming we know this model investigated is subjected to moderate structural parameter variation, we thus can simply define the prior PDF as a normal distribution with zero mean and 10% standard deviation of the nominal model parameters of the baseline finite element model, as shown in Fig. 7. It is worth noting that the results of following different scenarios are computed based on the same parameters defined previously.

Fig. 7
Fig. 7
Close modal

#### Bayesian Model Updating—Baseline Results.

We first use a full-scale Monte Carlo analysis to represent comprehensively the updated posterior PDF, which serves as a baseline for the subsequent investigations. In this case, 10,000 samples that are parameterized from the prior PDF are defined. Following the procedures outlined in Fig. 1(a), we can obtain the posterior PDF as shown in Fig. 8. This posterior PDF that is well represented by sufficient data points resembles the normal distribution, where the distribution mean and standard deviation reveal the fundamental properties of the updated probabilistic model. In particular, the mean values indicate the sampled model parameters with the highest probability of being actual model parameters. Here, the mean values of model parameters obtained are $μ(θ)=[−0.1135,0.0650]$, showing a good agreement with the actual model parameters $θact=[−0.1056,0.0736]$. The relative errors of model parameters are $[7.48%,11.68%]$. In terms of Young's modules values, the relative errors are $[0.79%, 0.86%]$. The standard deviations $σ(θ)=[0.0138 , 0.0164]$ are small, which indicates that the model updating has high confidence level. Hence, the effectiveness of Bayesian inference for modal updating is verified.

Fig. 8
Fig. 8
Close modal

As mentioned, the characterization of posterior PDF using the basic idea of Bayesian inference is computationally expensive, since it requires carrying out brute-force Monte Carlo analysis of large-scale finite element model. On a personal computer with Intel E5620 2.4 GHz (two processors), based on the ansys platform with the apdl command, the model updating using the full-scale Monte Carlo analysis (i.e., 10,000 samples) takes about 11 hrs.

#### Model Updating Through Fast MCMC Sampling Integrated With GP Emulation.

We implement fast sampling in the Bayesian inference framework. Here, without loss of generality, we use MCMC to reduce the number of parameterized samples to be evaluated from 10,000 to 1000. That is, the time needed to generate samples is reduced dramatically by 90%. As a result, the time for the entire model updating process is reduced from 11 hrs to 1.15 hrs based on the same computational platform. The sparse posterior PDF obtained by Bayesian inference with MCMC results in the most optimal parameters $θopt=[−0.1241,0.0520]$, with relative errors $[17.52%,29.35%]$ as compared with the actual model parameter $θact$. In terms of Young's modulus values, the relative errors are $[1.85%,2.16%]$.

We then implement the enhanced algorithm that combines GP and MCMC. Utilizing the reduced samples and the respective posterior probability values as training data points, we conduct GP emulation to enrich the sparse posterior PDF over the entire parametric space (i.e., the original 10,000 model samples). The posterior PDF obtained is shown in Fig. 9. To facilitate the result comparison, we present two-dimensional distribution of eight identified model parameters with the highest probability values from, respectively, the sparse and the enriched posterior PDFs in Fig. 10. The norms/distances of model parameters identified from the sparse posterior PDF mostly are larger than those from the enriched posterior PDF, indicating a higher average error that is represented by the average radius (Fig. 10). Specifically, for the most optimal parameters, we can obtain more accurate results $θoptGP=[−0.1127,0.0645]$ with reduced relative errors $[6.72%,12.36%]$ as compared with the MCMC result. In terms of Young's modulus values, the relative errors are $[0.71%,0.91%]$. It is also found that the width of such enriched posterior PDF is quite narrow, with standard deviations $σ(θ)=[0.0142 , 0.0170]$, which means that the enhanced inference algorithm possesses good robustness. The results are generally consistent with the baseline results obtained with 10,000 samples. The identification results of all three scenarios are compared in Table 1.

Fig. 9
Fig. 9
Close modal
Fig. 10
Fig. 10
Close modal
Table 1

Comparison of identification results under different approaches

Identified model parameter ($θ$)Error (%)Identified Young's modulus ($1011 Pa$)Error (%)
Full MC Bayesian$[−0.1135,0.0650]$$[7.48,11.68]$$[1.826,2.194]$$[0.79,0.86]$
MCMC + Bayesian$[−0.1241,0.0520]$$[17.52,29.35]$$[1.804,2.167]$$[1.85,2.16]$
MCMC + GP + Bayesian$[−0.1127,0.0645]$$[6.72,12.36]$$[1.828,2.193]$$[0.71,0.91]$
Identified model parameter ($θ$)Error (%)Identified Young's modulus ($1011 Pa$)Error (%)
Full MC Bayesian$[−0.1135,0.0650]$$[7.48,11.68]$$[1.826,2.194]$$[0.79,0.86]$
MCMC + Bayesian$[−0.1241,0.0520]$$[17.52,29.35]$$[1.804,2.167]$$[1.85,2.16]$
MCMC + GP + Bayesian$[−0.1127,0.0645]$$[6.72,12.36]$$[1.828,2.193]$$[0.71,0.91]$

Here, it is worth mentioning that the accuracy of GP emulation depends on the number of training data points. The prediction accuracy of GP generally increases when more training data points are involved. It, however, increases the computational cost of Monte Carlo-based finite element analysis. In practice, the number of training data points needs to be carefully selected with the tradeoff between analysis efficiency and accuracy. To illustrate this tradeoff tendency, different numbers of data points for GP emulations are employed to investigate the computational efficiency and accuracy, and the results are plotted in Fig. 11. It can be observed that the integration of GP enables high-fidelity updating as compared to the cases solely utilizing small number of Monte Carlo analysis for Bayesian model updating.

Fig. 11
Fig. 11
Close modal

### Parametric Investigations on Input Data.

Besides the computational efficiency, the fidelity of Bayesian model updating is one major concern. Intuitively, a number of parameters may affect the model updating results. In what follows, we evaluate the influences from the number of datasets, the measurement error level, and sensor configuration (i.e., the number and placement of sensors and the modal information measured).

#### Number of Measured Datasets.

From Eq. (10), the number of measured datasets impacts considerably the profile of the posterior PDF. Mathematically, larger number of measured datasets will reduce the width of the posterior PDF, thereby increasing the distribution confidence level. To visualize this trend, the mean error and standard deviation of the posterior PDF versus the number of measured datasets are calculated and compared in Fig. 12. The result indeed confirms the tendency mentioned above, in which both the mean error and standard deviation decrease as the number of measured datasets increases. For example, as the number of measured datasets reaches 20, the mean error and the standard deviation of identified Young's modulus of tower become 0.39% and 0.0042.

Fig. 12
Fig. 12
Close modal

#### Measurement Error Level.

Similarly, we evaluate the model updating fidelity with respect to the level of measurement error. Without loss of generality, throughout this research we assume that the measurement errors of sensors are independent of each other. From the statistics standpoint, this assumption yields a diagonal covariance matrix of the measured mode shapes. Here, we employ three different error levels, i.e., 0.5%, 1%, and 2% for both natural frequency and mode shape measurements, and the results are compared in Fig. 13. It can be observed that the mean error and the standard deviation of the posterior PDF increase monotonically as the level of measurement error increases. For example, when the measurement level reaches 2%, the mean error and the standard deviation of identified Young's modulus of tower become 2.32% and 0.0098, respectively. Apparently, the accuracy of the measurement data plays a very important role in model updating.

Fig. 13
Fig. 13
Close modal

#### Sensor Configuration and Modal Sensitivity Influence.

The quality of the measurement data depends heavily on the distribution of sensors, i.e., the number of sensors used and the locations of the sensors. Moreover, the locations of the sensors have to do with which vibration modes can be reliably measured. Here, we configure a number of sensor scenarios that correspond to different combinations of the aforementioned factors. While the number of sensors available may be different, all sensors are numbered the same way as shown in Fig. 4.

• Configuration 1: seven sensors and four x-direction modes (baseline)

• Configuration 2: four sensors (3–6) and four x-direction modes

• Configuration 3: two sensors (5, 6) and four x-direction modes

• Configuration 4: two sensors (6, 7) and four x-direction modes

• Configuration 5: two sensors (3, 4) and four x-direction modes

• Configuration 6: one sensor (5) and four x-direction natural frequencies only (without mode shapes)

• Configuration 7: seven sensors and one mode (the first mode)

• Configuration 8: seven sensors and one mode (the third mode)

• Configuration 9: seven sensors and one mode (the fourth mode)

For consistency, all other parameters remain to be the same. For example, all the measurements are subject to 1% error. Figures 14 and 15 show, respectively, the mean errors and the standard deviations of the posterior PDFs obtained under these sensor configurations. The tendency of mean errors is quite similar with that of the standard deviations. By comparing the results, we can find that first three configurations are more desirable. The reason is straightforward for configurations 1 and 2, as they can extract relatively more modal information. Although configuration 3 yields similar amount of modal information with configuration 4, it leads to a better identification performance. This difference in performance is due to the difference in mode shape patterns they capture. Particularly, the mode shapes extracted by configuration 3 consist of motions of both the tower and the blades. As such, the measurement can better reflect the property change of both the tower and the blades. In comparison, configuration 4 cannot capture the vibration pattern of blades, which degrades the identification accuracy. Likewise, configuration 8 leads to smaller mean error and smaller standard deviation than configurations 7 and 9. This can be explained from the sensitivity standpoint. That is, the third natural frequency measured (under configuration 8) is more sensitive to the model parameters than the first and fourth natural frequencies measured (under configurations 7 and 9).

Fig. 14
Fig. 14
Close modal
Fig. 15
Fig. 15
Close modal

Indeed, the sensitivity of the modal information with respect to the model parameters to be updated plays a very important role, which is analyzed in detail in what follows. Our goal of the forward sensitivity analysis is to acquire the modal response trend versus the model parameters which are densely sampled in the prespecified ranges using Monte Carlo evaluation. Based on the response distribution obtained, intuitively the sensitivity of modal response with respect to the model parameters can be reflected [4749]. We first analyze the relations of the natural frequencies with respect to the two model parameters (i.e., the Young's modulus of the tower and that of the blades). Figures 16 and 17 show the projection views. We can observe that the first and second natural frequencies are both quite sensitive to the first model parameter, and the fourth natural frequency appears to be irrelevant to the first model parameter but is highly sensitive to the second model parameter. Meanwhile, the third natural frequency is sensitive to both model parameters, which is the underlying reason that configuration 8 leads to better performance than configurations 7 and 9.

Fig. 16
Fig. 16
Close modal
Fig. 17
Fig. 17
Close modal
Care should be taken when we take mode shape sensitivity into consideration. Here, the mode shape sensitivity is evaluated by the mode shape difference with respect to the model parameter change. To facilitate the sensitivity analysis and direct comparison, here we employ the mode assurance criterion (MAC) that is widely used in structural dynamic analysis to quantify the difference of mode shapes [5052]
$MACi(r)(θ)=|ϕi(r)(θ)Tϕi(r)(θ⌢)|2(ϕi(r)(θ)Tϕi(r)(θ))(ϕi(r)(θ⌢)Tϕi(r)(θ⌢))$
(17)
where subscript i indicates the ith mode order, superscript r indicates the sensor configuration index, and $ϕi(r)(θ⌢)$ is the ith mode shape of the structure with actual model parameter $θ⌢$ that is extracted by sensors with the rth configuration. With MAC of individual mode as basis, we can further define an aggravated MAC that represents the combinatorial effect of mode shape change/difference as
$MAC(r)(θ)=∏i=1nMACi(r)(θ)$
(18)
In the subsequent analysis, we consider sensor configurations 1–5, because other configurations do not involve the mode shapes. Furthermore, as configuration 1 yields the most amount of information, it is used as the reference. We then define a performance index for configurations 2–5 as
$L(r)(θ)=MAC(r)(θ)−MAC(1)(θ) (r=2,…,5)$
(19)

The distributions of this mode shape performance index versus the model parameters are calculated and then plotted in Figs. 18 and 19. It can be observed that for configurations 2 and 3, the mode shape performance index remains to be almost constant over the entire parametric range. In fact, the performance index values are close to zero, which indicates that as far as mode shapes are concerned, configurations 2 and 3 yield similar results as configuration 1. One may then conclude that, in the case that only two sensors are used, these two configurations are good choices in terms of measurement. In both configurations, sensor location 5 is at the tip of a blade, where the response amplitude is contributed from both the tower and the blade. In comparison, configurations 4 and 5 yield less than satisfactory identification performance, primarily due to the fact that the mode shape performance indices under these two sensor configurations no longer keep constant (i.e., more like a random distribution) with respect to concerned model parameters. In other words, the mode shape responses of configurations 4 and 5 mostly deviate from the desirable mode shape response of configuration 1 over the entire parametric space.

Fig. 18
Fig. 18
Close modal
Fig. 19
Fig. 19
Close modal

## Conclusions

This paper presents an efficient probabilistic model updating framework using intelligent inference based on incomplete modal measurement information. MH MCMC is adopted to reduce the number of samples needed. To tackle the issue of MCMC that the posterior PDF obtained may not be informative since it only contains sparsely distributed data points, GP is then incorporated into the analysis as an emulator to enrich the sampled data. Case studies on a mock-up wind turbine using simulated data as input demonstrate that accurate model updating can be achieved with significantly reduced computational cost. While more measurement data and lower measurement error generally lead to better model updating performance, it is found that sensor configuration plays a very important role. Forward sensitivity analysis can provide vital information to guide the proper selection of sensor number/locations as well as the order of modes to be measured. The model updating framework is implemented in ansys environment with apdl code.

## Acknowledgment

This research is supported by National Science Foundation under Grant No. CMMI-0900275.

### Appendix: apdl Pseudocode

To assist interested readers to apply the algorithm developed in this research toward practical applications, an apdl pseudo-code that can be implemented in ANSYS environment is provided below.

/PREP7

ET,1,SOLID185

BLOCK,0,LENGTH1,0,WIDTH1,0,THICKNESS1

BLOCK,0,LENGTH2,0,WIDTH2,0,THICKNESS2

MPTEMP,,,,,,, Model Geometry, define material and element types well as generate mesh

MPTEMP,1,0

MPTEMP,EX,1,YOUNGM

….

FINISH

*DIM,UNSAMP,NUMSAMP,DEMSAMP

(2F8.4) Read sampled model parameters from external txt file

*DO,J,1, NUMSAMP

*DO,JJ,1, DEMSAMP

/PREP 7

MPTEMP,,,,,

……

MPDATA,EX,JJ+1 Change the Young's modulus of element at specified location range – model parameter definition

MPDATA,EX,JJ+1, YOUNGM *(1-UNSAMP(J,JJ))

ESEL,S,CENT,Z,UNCARE(JJ,5), UNCARE(JJ,6),

ESEL,R,CENT,X,UNCARE(JJ,1), UNCARE(JJ,2),

ESEL,R,CENT,Y,UNCARE(JJ,3), UNCARE(JJ,4),

EMODIF,ALL,MAT,JJ+1,

/FINISH

*ENDDO

….

/SOLU Modal analysis and data post-processing

SOLVE

….

FINISH

….

*ENDDO

*MWRITE,FREQUNCE,output_freq,TXT,,JIK,NUMORD, NUMSAMP

(6F11.5) Data export for Bayesian inference

*MWRITE,MODEUNCE,output_mode,TXT,,JIK,NUMORD, NUMSAMP *NUMSENS

(6F11.5)

Note: variables with bold and italic fonts are defined by user.

## References

1.
Sinha
,
J. K.
, and
Friswell
,
M. I.
,
2002
, “
Model Updating: A Tool for Reliable Modeling, Design Modification and Diagnosis
,”
Shock Vib. Dig.
,
34
(
1
), pp.
27
35
.
2.
Fang
,
S. E.
,
Perera
,
R.
, and
De Roeck
,
G.
,
2008
, “
Damage Identification of a Reinforced Concrete Frame by Finite Element Model Updating Using Damage Parameterization
,”
J. Sound Vib.
,
313
(
3–5
), pp.
544
559
.
3.
Teughels
,
A.
, and
DeRoeck
,
G.
,
2004
, “
Structural Damage Identification of the Highway Bridge Z24 by FE Model Updating
,”
J. Sound Vib.
,
278
(
3
), pp.
589
610
.
4.
Wu
,
J. R.
, and
Li
,
Q. S.
,
2004
, “
Finite Element Model Updating for a High-Rise Structure Based on Ambient Vibration Measurements
,”
Eng. Struct.
,
26
(
7
), pp.
979
990
.
5.
Jaishi
,
B.
, and
Ren
,
W. X.
,
2005
, “
Structural Finite Element Model Updating Using Ambient Vibration Test Results
,”
J. Struct. Eng.
,
131
(
4
), pp.
617
628
.
6.
Rahai
,
A.
,
,
F.
, and
Esfandiari
,
A.
,
2007
, “
Damage Assessment of Structure Using Incomplete Measured Mode Shapes
,”
Struct. Control Health Monit.
,
14
(
5
), pp.
808
829
.
7.
Zhou
,
K.
, and
Tang
,
J.
,
2016
, “
Rapid Identification of Properties of Column-Supported Bridge-Type Structure by Using Vibratory Response
,”
J. Vib. Control
,
22
(
5
), pp.
1415
1430
.
8.
Friswell
,
M. I.
,
Garvey
,
S. D.
, and
Penny
,
J. E. T.
,
1995
, “
Model Reduction Using Dynamic and Iterated IRS Techniques
,”
J. Sound Vib.
,
186
(
2
), pp.
311
323
.
9.
Chen
,
H.-P.
,
2010
, “
Mode Shape Expansion Using Perturbed Force Approach
,”
J. Sound Vib.
,
329
(
8
), pp.
1177
1190
.
10.
Liu
,
F.
,
2011
, “
Direct Mode-Shape Expansion of a Spatially Incomplete Measured Mode by a Hybrid-Vector Modification
,”
J. Sound Vib.
,
330
(
18–19
), pp.
4633
4645
.
11.
Patelli
,
E.
,
Murat Panayirci
,
H.
,
Broggi
,
M.
,
Goller
,
B.
,
Beaurepaire
,
P.
,
,
H. J.
, and
Schueller
,
G. I.
,
2012
, “
General Purpose Software for Efficient Uncertainty Management of Large Finite Element Models
,”
Finite Elem. Anal. Des.
,
51
, pp.
31
48
.
12.
Soize
,
C.
,
2005
, “
Random Matrix Theory for Modeling Uncertainties in Computational Mechanics
,”
Comput. Methods Appl. Mech. Eng.
,
194
(
12–16
), pp.
1333
1366
.
13.
Khodaparast
,
H.
,
,
J.
, and
Friswell
,
M.
,
2008
, “
Perturbation Methods for the Estimation of Parameter Variability in Stochastic Model Updating
,”
Mech. Syst. Signal Process.
,
22
(
8
), pp.
1751
1753
.
14.
Beck
,
J. L.
, and
Katafygiotis
,
L. S.
,
1998
, “
Updating Models and Their Uncertainties. I: Bayesian Statistical Framework
,”
J. Eng. Mech.
,
124
(
4
), pp.
455
461
.
15.
Mthembu
,
L.
,
Marwala
,
T.
,
Friswell
,
M. I.
, and
,
S.
,
2011
, “
Model Selection in Finite Element Model Updating Using the Bayesian Evidence Statistic
,”
Mech. Syst. Signal Process.
,
25
(
7
), pp.
2399
2412
.
16.
Zarate
,
B. A.
,
Caicedo
,
J. M.
,
Yu
,
J.
, and
Ziehl
,
P.
,
2012
, “
Bayesian Model Updating and Prognosis of Fatigue Crack Growth
,”
Eng. Struct.
,
45
, pp.
53
61
.
17.
Becker
,
W.
,
Oakley
,
J. E.
,
Surace
,
C.
,
Gill
,
P.
,
Rowson
,
J.
, and
Worden
,
K.
,
2012
, “
Bayesian Sensitivity Analysis of a Nonlinear Finite Element Model
,”
Mech. Syst. Signal Process.
,
32
, pp.
18
31
.
18.
Simoen
,
E.
,
,
C.
, and
Lombaert
,
G.
,
2013
, “
On Prediction Error Correlation in Bayesian Model Updating
,”
J. Sound Vib.
,
332
(
18
), pp.
4136
4152
.
19.
Ching
,
J.
, and
Beck
,
J. L.
,
2004
, “
New Bayesian Model Updating Algorithm Applied to a Structural Health Monitoring Benchmark
,”
Struct. Health Monit.
,
3
(
4
), pp.
313
332
.
20.
Castanier
,
M. P.
,
Tan
,
Y.-C.
, and
Pierre
,
C.
,
2001
, “
Characteristic Constraint Modes for Component Mode Synthesis
,”
AIAA J.
,
39
(
6
), pp.
1182
1187
.
21.
,
C.
, and
,
D. C.
,
2013
, “
Component Mode Synthesis Techniques for Finite Element Model Updating
,”
Comput. Struct.
,
126
(
1
), pp.
15
28
.
22.
Liu
,
Y.
,
Li
,
Y.
,
Wang
,
D.
, and
Zhang
,
S.
,
2014
, “
Model Updating of Complex Structures Using the Combination of Component Mode Synthesis and Kriging Predictor
,”
Sci. World J.
,
2014
, p.
476219
.
23.
Zhou
,
K.
,
Liang
,
G.
, and
Tang
,
J.
,
2016
, “
Component Mode Synthesis Order-Reduction for Dynamic Analysis of Structure Modeled With NURBS Finite Element
,”
ASME J. Vib. Acoust.
,
138
(
2
), p.
021016
.
24.
Myers
,
R. H.
, and
Montgomery
,
D. C.
,
2009
,
Response Surface Methodology: Process and Product Optimization Using Designed Experiments
, 3rd ed.,
Wiley
,
New York
.
25.
Fang
,
S. E.
,
Ren
,
W. X.
, and
Perera
,
R.
,
2012
, “
A Stochastic Model Updating Method for Parameter Variability Quantification Based on Response Surface Models and Monte Carlo Simulation
,”
Mech. Syst. Signal Process.
,
33
, pp.
83
96
.
26.
Anderson
,
J.
,
1995
,
An Introduction to Neural Network
,
MIT Press
,
Cambridge, MA
.
27.
Gao
,
H. Y.
,
Guo
,
X. L.
,
Ouyang
,
H. J.
, and
Han
,
F.
,
2013
, “
Crack Identification of Cantilever Plates Based on a Kriging Surrogate Model
,”
ASME J. Vib. Acoust.
,
135
(
5
), p.
051012
.
28.
Beck
,
J. L.
,
Au
,
S. K.
, and
Vanik
,
M. W.
,
2001
, “
Monitoring Structural Health Using a Probabilistic Measure
,”
Comput. Aided Civ. Infrastruct. Eng.
,
16
(
1
), pp.
1
11
.
29.
Ching
,
J. Y.
, and
Chen
,
Y. C.
,
2007
, “
Transitional Markov Chain Monte Carlo Method for Bayesian Model Updating, Model Class Selection, and Model Averaging
,”
J. Eng. Mech.
,
133
(
7
), pp.
816
832
.
30.
,
W. M.
,
2010
,
Understanding Computational Bayesian Statistics
,
Wiley
,
Hoboken, NJ
.
31.
Gilks
,
W. R.
,
Richardson
,
S.
, and
Spiegelhalter
,
D.
,
1996
,
Markov Chain Monte Carlo in Practice
,
Chapman and Hall
,
London
.
32.
Miao
,
F.
, and
Ghosn
,
M.
,
2011
, “
Modified Subset Simulation Method for Reliability Analysis of Structural Systems
,”
Struct. Saf.
,
33
(
4–5
), pp.
251
260
.
33.
Kennedy
,
M. C.
, and
O'Hagan
,
A.
,
2000
, “
Predicting the Output From a Complex Computer Code When Fast Approximation are Available
,”
Biometrika
,
87
(
1
), pp.
1
13
.
34.
O'Hagan
,
A.
,
2006
, “
Bayesian Analysis of Computer Code Outputs: A Tutorial
,”
Reliab. Eng. Syst. Saf.
,
91
(
10-11
), pp.
1290
1300
.
35.
Zhao
,
J.
,
Tang
,
J.
, and
Wang
,
K. W.
,
2008
, “
Enhanced Statistical Damage Identification Using Frequency-Shift Information With Tunable Piezoelectric Transducer Circuitry
,”
Smart Mater. Struct.
,
17
(
6
), p.
065003
.
36.
Vanik
,
M. W.
,
Beck
,
J. L.
, and
Au
,
S. K.
,
2000
, “
Bayesian Probabilistic Approach to Structural Health Monitoring
,”
J. Eng. Mech.
,
126
(
7
), pp.
738
745
.
37.
Brooks
,
S. P.
,
1998
, “
Markov Chain Monte Carlo Method and Its Application
,”
J. R. Stat. Soc.
,
47
(
1
), pp.
69
100
.
38.
Dostert
,
P.
,
Efendiev
,
Y.
, and
Hou
,
T. Y.
,
2008
, “
Multiscale Finite Element Methods for Stochastic Porous Media Flow Equations and Application to Uncertainty Quantification
,”
Comput. Methods Appl. Mech. Eng.
,
197
(
43–44
), pp.
3445
3455
.
39.
Martinez
,
W.
, and
Martinez
,
A.
,
2007
,
Computational Statistics Handbook With MATLAB
, 2nd ed.,
Taylor & Francis Group
,
Boca Raton, FL
.
40.
Rasmussen
,
C. E.
, and
Williams
,
C. K. I.
,
2006
,
Gaussian Process for Machine Learning
,
MIT Press
,
Cambridge, MA
.
41.
DiazDelaO
,
F. A.
, and
,
S.
,
2010
, “
Structural Dynamic Analysis Using Gaussian Process Emulators
,”
Eng. Comput.
,
27
(
5
), pp.
580
605
.
42.
DiazDelaO
,
F. A.
, and
,
S.
,
2012
, “
Bayesian Assimilation of Multi-Fidelity Finite Element Models
,”
Comput. Struct.
,
92–93
, pp.
206
215
.
43.
Xia
,
Z.
, and
Tang
,
J.
,
2013
, “
Characterization of Dynamic Response of Structures With Uncertainty by Using Gaussian Processes
,”
ASME J. Vib. Acoust.
,
135
(
5
), p.
051006
.
44.
Seeger
,
M.
,
2004
, “
Gaussian Processes for Machine Learning
,”
Int. J. Neural Syst.
,
14
(
2
), pp.
69
106
.
45.
Ciang
,
C. C.
,
Lee
,
J.-R.
, and
Bang
,
H.-J.
,
2008
, “
Structural Health Monitoring for a Wind Turbine System: A Review of Damage Detection Methods
,”
Meas. Sci. Technol.
,
19
(
12
), p.
122001
.
46.
ANSYS
,
2011
, “
ANSYS Parametric Design Language Guide 14.0
,” ANSYS, Inc., Canonsburg, PA.
47.
Melchers
,
R. E.
, and
Ahammed
,
M.
,
2004
, “
A Fast Approximate Method for Parameter Sensitivity Estimation in Monte Carlo Structural Reliability
,”
Comput. Struct.
,
82
(
1
), pp.
55
61
.
48.
,
A. T.
,
Molinari
,
M.
,
Bursi
,
O. S.
, and
Friswell
,
M. I.
,
2011
, “
Finite Element Model Updating of a Semi-Rigid Moment Resisting Structure
,”
Struct. Control Health Monit.
,
18
(
2
), pp.
149
168
.
49.
Zhou
,
K.
, and
Tang
,
J.
,
2015
, “
Reducing Dynamic Response Variation Using NURBS Finite Element-Based Geometry Perturbation
,”
ASME J. Vib. Acoust.
,
137
(
6
), p.
061008
.
50.
Allemang
,
R. J.
,
2003
, “
The Modal Assurance Criterion—Twenty Years of Use and Abuse
,”
Sound Vib.
,
37
(
8
), pp.
14
21
.
51.
Banerjee
,
S.
,
Ricci
,
F.
,
Monaco
,
E.
, and
Mai
,
A.
,
2009
, “
A Wave Propagation and Vibration-Based Approach for Damage Identification in Structural Components
,”
J. Sound Vib.
,
322
(
1–2
), pp.
167
183
.
52.
Brehm
,
M.
,
Zabei
,
V.
, and
Bucher
,
C.
,
2010
, “
An Automatic Mode Pairing Strategy Using an Enhanced Modal Assurance Criterion Based on Modal Strain Energies
,”
J. Sound Vib.
,
329
(
25
), pp.
5375
5392
.