Research on Quantification of Structural Natural Frequency Uncertainty and Finite Element Model Updating Based on Gaussian Processes

Abstract

During bridge service, material degradation and aging occur, affecting bridge functionality. Bridge health monitoring, crucial for detecting structural damage, includes finite element model modification as a key aspect. Current finite element-based model updating techniques are computationally intensive and lack practicality. Additionally, changes in loading and material property deterioration lead to parameter uncertainty in engineering structures. To enhance computational efficiency and accommodate parameter uncertainty, this study proposes a Gaussian process model-based approach for predicting structural natural frequencies and correcting finite element models. Taking a simply supported beam structure as an example, the elastic modulus and mass density of the structure are sampled by the Sobol sequence. Then, we map the collected samples to the corresponding physical space, substitute them into the finite element model, and calculate the first three natural frequencies of the model. A Gaussian surrogate model was established for the natural frequency of the structure. By analyzing the first three natural frequencies of the simply supported beam, the elastic modulus and mass density of the structure are corrected. The error between the corrected values of elastic modulus and mass density and the calculated values of the finite element model is very small. This study demonstrates that Gaussian process models can improve calculation efficiency, fulfilling the dual objectives of predicting structural natural frequencies and adjusting model parameters.

1. Introduction

The natural frequency of bridges is a crucial dynamic response indicator that is easily testable and can reflect changes in structural stiffness. Damage inevitably results in frequency variation [1,2]. It is commonly utilized in the fields of structural model updating and damage identification [3,4,5,6]. Research has demonstrated that the natural frequency of bridge structures is solely associated with the inherent characteristics of the system, including mass distribution and stiffness. This approach holds immense significance for structural damage detection and vibration analysis of bridges [7,8]. The structural parameters of bridges unavoidably possess uncertainties; consequently, the natural frequencies of structures that rely on these parameters also exhibit uncertainties [9,10,11]. There are many reasons for the uncertainty of natural frequency, such as test noise, data defects, environmental impact, model error, and so on [12]. The sources of uncertainty are mainly divided into random uncertainty and epistemic uncertainty. Random uncertainty, such as that coming from environmental and operational conditions, cannot be reduced or eliminated through human efforts. Epistemic uncertainty, such as measurement error on the real-world structure and numerical models, can be reduced by measurement methods and better models. Uncertainty of structural natural frequencies belongs to epistemic uncertainty. If the uncertainty of the natural frequency of the structure is not considered, the true value of the structural parameters of the bridge cannot be obtained, which may lead to the structure being in a relatively dangerous state. Hence, studying the transfer of structural uncertainty parameters to the uncertainty of structural natural frequencies is of particular importance [13,14,15,16,17,18]. The methods employed to estimate the natural frequency of structures primarily include the perturbation method [19], the orthogonal polynomial method [20,21], the Monte Carlo method [22,23], and the interval analysis method [24,25]. These methods generally begin with the structural dynamic equation, as well as extracting the stiffness matrix and mass matrix of the structure and performing calculations to obtain the statistical characteristics of the natural frequency of the structure [26]. Researchers typically use finite element software to establish high-precision finite element models for calculating the natural frequencies of structures. Given the complexity of bridge structures, these finite element models may include thousands of nodes and elements, necessitating substantial computational effort with low efficiency. It is also important to consider the interaction between structural and non-structural components in buildings. To enhance computational efficiency, scholars have proposed response surface models such as polynomial models, support vector machine models, neural network models, and Gaussian process models. Owing to their superior ability to simulate complex physical systems characterized by high dimensionality and strong nonlinearity, Gaussian process models have found widespread application [27,28].

Structural health monitoring is an essential measure for ensuring structural safety. During the service life of engineering structures, various factors, such as complex structural systems, non-uniform material parameters, and environmental variability, can result in discrepancies between the ideal and actual states of the structures. The inherent uncertainties also arise from the difficulty in accurately assessing the true extent of constraint exerted by the internal and external constraints, the interaction between the structural and non-structural components, soil-structure interaction phenomena, variations in the acting loads, and so forth. In the case of bridge structures, the absence of timely structural health monitoring may lead to sudden collapse, causing significant safety incidents and extensive property damage. Therefore, conducting health monitoring of bridge structures is crucial.

The finite element model updating technique is a crucial method for bridge health monitoring, aiming to derive structural parameter information based on measured real responses of the structure [29,30,31,32,33]. The structural parameter information obtained through finite element model updating reflects, to some extent, the true condition of the structure during service. This information is essential for timely detection and warning of potential hazards during the operation of a structure. The natural frequency of bridge structures is an easily measurable true structural response [10,19] and is commonly used for model updating. However, there is inherent uncertainty in the natural frequency, as the structural parameters determining this frequency also possess uncertainties. Therefore, the finite element model updating technique needs to consider this uncertainty. The Gaussian process model considers that the uncertainty of the natural frequency is related to the distribution of the conditional probability density function under the assumed model [34,35]. The Gaussian process model not only provides predicted values at certain points but also quantitatively predicts the variance of these values, making it a suitable approach for incorporating uncertainty.

This paper presents a method for analyzing the natural frequencies of simply supported beam structures using a Gaussian process-based simplified calculation. This method can predict the mean and standard deviation of the first three natural frequencies of the structure. The Gaussian process model is utilized to modify the elastic modulus of the damaged element as well as the elastic modulus and mass density of the undamaged element. It has been demonstrated that Gaussian processes can improve the calculation efficiency and achieve the goal of updating structural models.

2. Gaussian Process Regression Theory

2.1. Basic Theory

Unlike conventional regression models, the Gaussian process regression model is nonparametric and capable of accurately simulating the input-output relationships of complex systems while also providing statistical characteristics of the regression values. If the random process $\{f(x_1),f(x_2),\dots ,f(x_{n-1}),f(x_n)\}$ is a Gaussian process, then any finite term in it is a joint Gaussian distribution, denoted as follows:

$$f(x)~N(\mu ,C)$$

(1)

where

$$\begin{gathered} f(x)={\left[f(x_1),f(x_2),\dots ,f(x_{n-1}),f(x_n)\right]}^T \\ \mu ={[\mu (x_1),\mu (x_2),\dots ,\mu (x_{n-1}),\mu (x_n)]}^T \end{gathered}$$

C is the covariance matrix, which is a nth-order square matrix.

There are n training sample sets ${(x_1,x_2,\dots ,x_{n-1},x_n)}^T$ and corresponding outputs ${(y_1,y_2,\dots ,y_{n-1},y_n)}^T$. $x_i(i=1,2,\dots ,n)$ is an m-dimensional vector, denoted as $x_i^m$. According to the Gaussian process assumption, ${(y_1,y_2,\dots ,y_{n-1},y_n)}^T$ and $y^{\ast }$ are joint Gaussian distributions.

$$\left[\begin{array}{c}Y\\ y^{\ast }\end{array}\right]~N\left(\left[\begin{array}{c}\mu \left(X\right)\\ \mu \left(x^{\ast }\right)\end{array}\right],\left[\begin{array}{cc}C(X,X)& C(X,x^{\ast })\\ C(x^{\ast },X)& C\left(x^{\ast },x^{\ast }\right)\end{array}\right]\right)$$

(2)

It can be obtained from the conditional probability density formula.

$$p(y^{\ast }|Y)=\frac{p(Y,y^{\ast })}{p(Y)}$$

(3)

In Formula (3), $p(•)$ represents the probability density function.

2.2. Deduction of Regression Values

To derive the probability density of $y^{\ast }|Y$, denote $\mu \left(X\right)={\mu }_1$, $\mu \left(x^{\ast }\right)={\mu }_2$, and $C=\left[\begin{array}{cc}C\left(X,X\right)& C\left(X,x^{\ast }\right)\\ C\left(x^{\ast },X\right)& C\left(x^{\ast },x^{\ast }\right)\end{array}\right]=\left[\begin{array}{cc}{C}_{11}& C_{12}\\ C_{21}& C_{22}\end{array}\right]$.

We change Formula (2) to Formula (4) below:

$$\left[\begin{array}{c}Y\\ y^{\ast }\end{array}\right]~N\left(\left[\begin{array}{c}{\mu }_1\\ {\mu }_2\end{array}\right],\left[\begin{array}{cc}{C}_{11}& C_{12}\\ C_{21}& C_{22}\end{array}\right]\right)$$

(4)

Formula (5) can be obtained from matrix theory:

$$C=\left[\begin{array}{cc}{C}_{11}& C_{12}\\ C_{21}& C_{22}\end{array}\right]=\left[\begin{array}{cc}{I}_n& 0\\ C_{21}{C_{11}}^{-1}& I\end{array}\right]\left[\begin{array}{cc}{C}_{11}& 0\\ 0& D\end{array}\right]\left[\begin{array}{cc}{I}_n& {C_{11}}^{-1}{C}_{12}\\ 0& I\end{array}\right]$$

(5)

In the formula, $I_n$ and $I$ represent the unit matrix, D represents the Schuler complement of $C_{11}$, and its value is $C_{22}-C_{21}{C_{11}}^{-1}{C}_{12}$. Then,

$$C^{-1}=\left[\begin{array}{cc}{I}_n& -{C_{11}}^{-1}{C}_{12}\\ 0& I\end{array}\right]\left[\begin{array}{cc}{C_{11}}^{-1}& 0\\ 0& D^{-1}\end{array}\right]\left[\begin{array}{cc}{I}_n& 0\\ -C_{21}{C_{11}}^{-1}& I\end{array}\right]$$

(6)

$$|C^{-1}|=\det \left(\left[\begin{array}{cc}{I}_n& -{C_{11}}^{-1}{C}_{12}\\ 0& I\end{array}\right]\left[\begin{array}{cc}{C_{11}}^{-1}& 0\\ 0& D^{-1}\end{array}\right]\left[\begin{array}{cc}{I}_n& 0\\ -C_{21}{C_{11}}^{-1}& I\end{array}\right]\right)=\frac{1}{|C_{11}|\cdot |D|}$$

(7)

The joint probability density of n-dimensional normal random variables is the following:

$$p(x_1,x_2,\dots ,x_n)=\frac{1}{{(2\pi )}^{1/2}|C|}\exp \{-\frac{1}{2}{(X-\mu )}^T{C}^{-1}(X-\mu )\}$$

(8)

Formula (9) can be obtained from Formula (4), (5), (6), and (8).

$$p(Y)=\frac{1}{(2\pi {)}^{n/2}|C_{11}|}\exp \{-\frac{1}{2}{\left(Y-{\mu }_1\right)}^T{C_{11}}^{-1}\left(Y-{\mu }_1\right)\}$$

(9)

$$\begin{array}{ll}p(Y,y^{\ast })& =\frac{1}{{(2\pi )}^{(n+1)/2}|C{|}^{1/2}}\exp \left(-\frac{1}{2}{\left[\begin{array}{c}Y-{\mu }_1\\ y^{\ast }-{\mu }_2\end{array}\right]}^T{C}^{-1}\left[\begin{array}{c}Y-{\mu }_1\\ y^{\ast }-{\mu }_2\end{array}\right]\right)\\ & =\frac{1}{{(2\pi )}^{\mathrm{n}/2}|C_{11}|}·\frac{1}{{(2\pi )}^{1/2}|D|}\exp \{-\frac{1}{2}[(Y-{\mu }_1{)}^T{C_{11}}^{-1}(Y-{\mu }_1)\\ & +[(y^{\ast }-{\mu }_2)-C_{21}{C_{11}}^{-1}(Y-{\mu }_1){]}^T{D}^{-1}[(y^{\ast }-{\mu }_2)-C_{21}{C_{11}}^{-1}(Y-{\mu }_1)]]\}\end{array}$$

(10)

According to Formulas (3), (7), (9), and (10), the following can be concluded:

$$\begin{array}{ll}p(y^{\ast }|Y)=& \frac{1}{{(2\pi )}^{1/2}|D|}\exp \{-\frac{1}{2}[(y^{\ast }-{\mu }_2)-C_{21}{C_{11}}^{-1}(Y-{\mu }_1){]}^T{D}^{-1}\\ & [(y^{\ast }-{\mu }_2)-C_{21}{C_{11}}^{-1}(Y-{\mu }_1)]\}\end{array}$$

(11)

Therefore, the regression value $y^{\ast }$ follows a normal distribution as follows:

$$(y^{\ast }|Y)~N(\mu ,{\sigma }^2)$$

(12)

In Formula (12),

$$\mu ={\mu }_2+C_{21}{C_{11}}^{-1}(Y-{\mu }_1)$$

(13)

2.3. Selection of the Mean Function and Covariance Function

The mean function and covariance function are crucial elements of a Gaussian process regression model because they directly impact the model’s regression performance. Hence, it is essential to carefully select these functions. Due to limited prior knowledge regarding the overall trend of potential functions and for model simplification purposes, a zero-mean function is typically adopted as the mean function. Considering that the square exponential function ensures smoothness and infinite differentiability of the fitted function, the covariance function in this document employs the square exponential function. Formula (13) can be simplified to Formula (14):

$$\mu =C_{21}{C_{11}}^{-1}Y$$

(14)

The Gaussian kernel is shown in Formula (15):

$$c_{ij}={{\omega }_0}^2\exp \left(-\frac{1}{2}\underset{k=1}{\overset{m}{{\displaystyle \mathsf{\Sigma }}}}{\left(\frac{x_i^k-x_j^k}{{\omega }_k}\right)}^2\right)$$

(15)

In the formula, $x_i^k\left(x_j^k\right)$ represents the k-th element of the m-dimensional input and $W=\{{\omega }_0,{\omega }_1,{\omega }_2,\dots ,{\omega }_m\}$ is called the hyperparameter.

2.4. Hyperparameter Estimation

Hyperparameter estimation is the most complex and critical step in developing Gaussian process regression models. If the hyperparameter estimation is not accurate enough, it directly affects the model’s accuracy. This paper employs maximum likelihood estimation to determine the optimal hyperparameter and the likelihood function is expressed in Formula (16):

$$L\left({\omega }_0,{\omega }_1,\dots ,{\omega }_m\right)=\frac{1}{{(2\pi )}^{n/2}|C_{11}{|}^{1/2}}\exp (-\frac{1}{2}{Y}^T{C_{11}}^{-1}Y)$$

(16)

By taking the logarithm, the following can be obtained:

$$\ln L=-\frac{1}{2}\ln |C_{11}|-\frac{n}{2}\ln (2\pi )-\frac{1}{2}{Y}^T{C_{11}}^{-1}Y$$

(17)

Let $\frac{\partial \ln L}{\partial W}=0$ and iterate using multiple sets of initial points. When ln L is at its maximum, the corresponding parameter is a hyperparameter.

3. Certainty Analysis of the Natural Frequency of a Simply Supported Beam

This study focuses on the propagation of structural uncertain parameters to structural frequency uncertainty. Based on the Gaussian process model, the uncertainty in the nature frequency of a simply supported beam model is analyzed.

3.1. Finite Element Model of a Simply Supported Beam

Based on ABAQUS 6.14 software, the finite element model of a simply supported beam is established, as shown in Figure 1. The simply supported beam model adopts a solid element, the span is 4.8 m, the section size is 0.2 × 0.25 m, and the two ends of the beam are hinged. The material is reinforced concrete. The elastic modulus of the material is E = 32 GPa, the mass density is ρ = 2500 kg/m³, and the moment of inertia is 2.604 × 10⁻⁴ m⁴. The first three modes calculated using the finite element software ABAQUS are presented in Figure 2.

3.2. Prepare the Training Sample Set

The mass density ρ and elastic modulus E of the simply supported beam are taken as variable parameters, and the material parameters are assumed to obey a normal distribution. Considering that the sampling interval is larger than the interval $\mu \pm 3\sigma$, the sampling intervals for the mass density and elastic modulus are set to [2200, 2800] kg/m³ and [29,35] GPa, respectively. To maintain good uniformity of the sampling parameters in higher dimensions, the Sobol sequence is employed to sample the input parameters. The mass density and elastic modulus samples obtained from the Sobol sequence are subsequently substituted into the finite element model, and the corresponding first three natural frequencies are calculated to generate the training samples. The d-dimensional array generated by the Sobol sequence conforms to the uniform distribution of the unit hypercube [0, 1]; thus, the direct array produced by the Sobol sequence needs to be mapped to the corresponding physical parameter space before being applied to the ABAQUS finite element software. The mapping method is described in Formula (18):

$$y=a+\frac{x-0}{1-0}\cdot (b-a)=(b-a)x+a$$

(18)

In the formula, a and b represent the upper and lower limits, respectively, of the corresponding physical parameter range, x is the value to be mapped, and y is the value after mapping.

The generated variable parameters are the mass density ρ, the elastic modulus E, and the training sample set with 20 sampling times. The training sample set is shown in

Table 1. Training sample set for the simply supported beam model.

Sampling Frequency	ρ (×102 Kg/m3)	E (GPa)	First Natural Frequency (Hz)	Second Order Natural Frequency (Hz)	The Third Natural Frequency (Hz)
1	22.196	33.056	18.763	72.017	137.790
2	27.950	29.420	15.774	60.543	115.841
3	23.943	29.396	17.036	65.388	125.113
4	25.917	33.866	17.575	67.458	129.072
5	23.113	30.717	17.724	68.030	130.161
6	26.576	33.299	17.210	66.056	126.390
7	24.685	31.457	17.356	66.617	127.461
8	25.414	30.314	16.792	64.451	123.322
9	22.463	34.640	19.093	73.281	140.211
10	27.347	33.008	16.892	64.833	124.052
11	23.521	32.523	18.079	69.393	132.772
12	26.280	31.299	16.779	64.402	123.221
13	23.147	34.998	18.905	72.563	138.843
14	26.890	32.708	16.957	65.083	124.530
15	24.327	31.226	17.419	66.858	127.921
16	25.272	30.183	16.802	64.491	123.390
17	22.053	31.183	18.282	70.172	134.261
18	27.792	30.635	16.142	61.956	118.541
19	24.249	34.416	18.317	70.303	134.512
20	25.940	33.423	17.452	66.986	128.173

.

3.3. Establishment of the Gaussian Process Model

Given that we need to analyze the first three natural frequencies of the structure and that each natural frequency has a different relationship with the structural parameters, a Gaussian process model is established for each of the first three natural frequencies. The accuracy of each regression is tested separately.

Formula (19) shows that the following relationship holds between the structural parameters and the output:

$$f=C_{21}{C_{11}}^{-1}Y$$

(19)

In the above equation,  $f$ is the natural frequency to be predicted.

When the training sample is input, the elements of the covariance matrix become functions of the hyperparameters. Therefore, the process of establishing the Gaussian process regression model involves searching for optimal hyperparameters. The quality of these hyperparameters significantly impacts the accuracy of the model regression. The maximum likelihood estimation method is utilized to solve for the hyperparameters. By taking the first-order natural frequency analysis as an example, a Gaussian response surface model is established. The results of three sets of optimal hyperparameters searched by MATLAB2024 are shown below:

$$\left[\begin{array}{l}{\omega }_0\\ {\omega }_1\\ {\omega }_2\end{array}\right]=\left[\begin{array}{l}12.1779\\ 9.1004\\ 6.5945\end{array}\right]\left[\begin{array}{l}{\omega }_0\\ {\omega }_1\\ {\omega }_2\end{array}\right]=\left[\begin{array}{l}34.6032\\ 9.8167\\ 9.0316\end{array}\right]\left[\begin{array}{l}{\omega }_0\\ {\omega }_1\\ {\omega }_2\end{array}\right]=\left[\begin{array}{l}42.2132\\ 6.7300\\ 5.3848\end{array}\right]$$

Before the Gaussian process model is used for regression, the accuracy of the model must be verified. A test set with a sample size of 10 was used to verify the accuracy of the Gaussian process model. The test samples are shown in

Table 2. Test set for simply supported beam models.

Sampling Frequency	ρ (102 Kg/m3)	E (GPa)	First Natural Frequency (Hz)	Second Natural Frequency (Hz)	Third Natural Frequency (Hz)
1	25.1721	31.2183	17.171	65.907	126.100
2	22.2782	31.3822	18.482	70.940	135.729
3	27.5624	30.7631	16.343	62.727	120.017
4	24.4813	34.2102	18.174	69.754	133.462
5	25.9395	33.4231	17.452	66.986	128.171
6	27.4755	31.3259	16.417	63.001	120.561
7	23.8378	34.7391	18.560	71.236	136.303
8	26.3841	34.5374	17.591	67.517	129.182
9	23.3941	32.7542	18.192	69.872	133.601
10	27.5421	32.9424	16.458	63.216	120.864

.

It can be seen from

Table 3. Validation results of the Gaussian process model.

Sampling Frequency	First Natural Frequency (Hz)		Second Natural Frequency (Hz)		Third Natural Frequency (Hz)
	Prediction Value (Finite Element)	Prediction Value (Gaussian Process)	Prediction Value (Finite Element)	Prediction Value (Gaussian Process)	Prediction Value (Finite Element)	Prediction Value (Gaussian Process)
1	17.171	17.171	65.907	65.907	126.100	126.100
2	18.482	18.482	70.940	70.940	135.729	135.729
3	16.343	16.343	62.727	62.727	120.017	120.016
4	18.174	18.174	69.754	69.754	133.462	133.460
5	17.452	17.452	66.986	66.986	128.171	128.170
6	16.417	16.418	63.011	63.009	120.563	120.576
7	18.560	18.561	71.236	67.512	136.301	136.311
8	17.591	17.580	67.517	67.512	129.182	129.192
9	18.192	18.192	69.872	69.830	133.603	133.594
10	16.458	16.458	63.216	63.188	120.864	120.858

that the first three natural frequencies predicted by the Gaussian process are very close to the natural frequencies predicted by the finite element method. The established Gaussian process model can be used to calculate the natural frequency of the simply supported beam.

3.4. Uncertainty Quantification of the Structural Dynamic Response

3.4.1. Generation of Calculation Samples

Assuming that the structural parameters follow a joint Gaussian distribution, MATLAB is used to generate calculation samples of this two-dimensional Gaussian distribution (mass density and elastic modulus), as shown in Figure 3.

3.4.2. Uncertainty Quantification

Prior to using the Gaussian process regression model to predict the natural frequency of the structure, all calculation samples must fall within the range of ±5% to eliminate samples outside the uncertainty interval. The first three statistical results of the natural frequency, based on 5000 sampling times, are presented in

Table 4. First three frequency quantization results for a simply supported beam model.

	First Natural Frequency	Second Natural Frequency	Third Natural Frequency
Mean value (Hz)	17.3988	66.7635	127.7688
Standard deviation	0.2641	1.0237	1.9755

.

Figure 4 illustrates the prediction results and fluctuation ranges of the first three natural frequencies. The x-axis represents the number of samples, while the y-axis represents the natural frequency.

Table 5. Training sample set.

Sampling Frequency	E2 (GPa)	E4 (GPa)	E9 (GPa)	E* (GPa)	ρ (102 Kg/m3)	First Natural Frequency (Hz)	Second Natural Frequency (Hz)	Third Natural Frequency (Hz)
1	27.2023	25.8051	17.5670	31.8729	25.9390	16.155	63.928	120.964
2	17.5377	21.9375	30.7791	31.5757	24.7328	17.067	63.665	120.495
3	29.6737	22.0295	20.0047	32.7476	25.6848	16.526	64.406	123.501
4	23.8525	30.5884	16.0390	31.3275	26.0233	15.936	63.762	118.980
5	26.4158	18.9016	16.2206	33.0163	23.9286	16.718	65.661	125.746
6	19.8829	26.7479	30.5993	31.1181	25.8985	16.706	63.075	119.047
7	32.8117	18.1561	20.1196	33.5409	24.1205	17.095	66.118	128.542
8	22.0164	23.3501	32.9871	32.5300	25.3936	17.190	64.496	123.084
9	28.8885	32.7953	26.1926	30.8855	23.8388	17.331	67.206	127.578
10	16.9035	25.4949	25.0960	33.4889	24.0029	17.563	66.523	124.295
11	30.6576	27.6807	27.4336	31.9076	23.8981	17.520	67.449	129.100
12	22.8009	17.4835	20.3149	31.1701	24.2425	16.548	63.279	121.783
13	25.5322	33.1038	20.5505	33.5708	24.2900	17.381	68.468	128.251
14	18.8986	27.0092	29.2578	31.7865	24.3018	17.340	65.520	123.176
15	31.6762	19.8062	28.8367	31.1530	24.7124	16.934	64.172	125.048
16	21.4762	22.6671	28.7161	30.4921	24.6411	16.846	63.618	121.282
17	27.8002	19.7078	27.6925	31.4187	25.8173	16.562	62.726	121.603
18	18.1738	18.4981	22.2077	31.1921	25.4552	16.274	61.573	117.189
19	29.8177	32.0284	19.1462	32.9457	24.9396	16.925	67.258	126.768
20	24.7056	26.4121	20.2486	30.7303	24.7766	16.520	64.473	121.894
21	26.5507	16.4547	31.2794	31.1871	26.2351	16.372	61.037	119.612
22	19.3796	18.6914	23.1853	30.7731	26.0896	16.063	60.766	116.008
23	33.4207	31.3786	32.6864	33.4111	25.8618	17.423	66.813	128.061
24	22.5581	24.4242	22.2253	31.2447	26.0319	16.295	62.806	118.959
25	28.6331	16.4075	32.2912	30.7562	25.6198	16.514	61.576	120.974
26	16.2742	17.3984	17.9721	31.8297	25.6481	15.983	60.991	115.480
27	30.8652	32.7281	30.2812	32.1927	25.7346	17.133	65.963	125.782
28	23.4265	31.6006	20.3914	33.2162	25.5281	16.844	66.096	123.614
29	24.9696	24.9104	24.1412	31.3725	24.1546	17.076	65.679	125.007
30	18.4042	30.9490	32.8131	33.1818	24.1272	17.830	67.347	125.735

and Figure 4 clearly show that the mean value of the first natural frequency of the structure is 17.3988, with a standard deviation of 0.2641. The mean value of the second natural frequency is 66.7635, with a standard deviation of 1.0237. Similarly, the mean value of the third natural frequency is 127.7688, with a standard deviation of 1.9755. The predicted values of the structural natural frequencies fluctuate around the mean values. As the number of samples increases, the statistical characteristics of the predicted structural frequencies become more apparent. The calculation results show that the Gaussian process regression model has a high accuracy and small error in predicting the natural frequency of the structure. The statistical characteristics of the predicted natural frequency values are in line with the actual situation of the structure.

4. Finite Element Model Updating of a Simply Supported Beam

4.1. Finite Element Model of a Simply Supported Beam

To conduct research on finite element model updating, the simply supported beam in Figure 1 is divided into 16 regions, each of which is 0.3 m in length, as shown in Figure 5. Each region has different material parameters. Each region is composed of multiple solid elements.

In this paper, the stiffness reduction method is utilized to simulate beam damage. As illustrated in Figure 6, the elastic modulus of regions 2, 4, and 9 is reduced by 10%, 20%, and 30%, respectively. The dynamic response of the simply supported beam after damage is considered the test value, and the dynamic response of the undamaged simply supported beam is considered the initial value.

4.2. Selection of Parameters to Be Corrected

In the context of updating the structural finite element model, changes in the material parameters for each element can result in changes in the structural dynamic response. However, considering the change in material parameters for each element in practical engineering applications where the structures are complex and the number of elements in the model is large is not realistic due to the significant computational cost involved. To improve calculation efficiency while ensuring accuracy, it is common practice to use the material parameters of damaged or vulnerable parts of the engineering structure as variable parameters to modify the finite element model. In addition to considering damaged elements, the uncertainty of undamaged elements or other parameters, such as mass density, should also be considered in the finite element model updating process. Therefore, in this paper, the elastic modulus (E₂, E₄, and E₉) of regions 2, 4, and 9 and the elastic modulus E* and mass density $\rho$ of the undamaged element are used as the parameters to be corrected in the finite element model.

4.3. Prepare the Training Sample Set

In this paper, the uncertainty range of the unit parameters is determined by changing ±5% on the basis of the nominal value. The sampling intervals of regions 2, 4, and 9 consider both damage and self-uncertainty, while the sampling intervals E* and $\rho$ consider only self-uncertainty. The specific sampling interval is designed as follows.

$$\begin{array}{l}{E}_2=[0.5E,1.05E]=[16 \mathrm{GPa},33.6 \mathrm{GPa}]\\ E_4=[0.5E,1.05E]=[16 \mathrm{GPa},33.6 \mathrm{GPa}]\\ E_9=[0.5E,1.05E]=[16 \mathrm{GPa},33.6 \mathrm{GPa}]\\ E^{\ast }=[0.95E,1.05E]=[30.4 \mathrm{GPa},33.6 \mathrm{GPa}]\\ \rho =[0.95\rho ,1.05\rho ]=[2375 {\mathrm{Kg}/\mathrm{m}}^3,2625 {\mathrm{Kg}/\mathrm{m}}^3]\end{array}$$

To maintain uniformity in sampling parameters across high dimensions, a Sobol sequence sampling method is adopted in this study. The training sample set consists of 5 sample parameters and 30 sampling instances, as shown in Table 5.

4.4. Establishment of the Gaussian Process Model

The maximum likelihood estimation method is used to search for the three sets of hyperparameters of the Gaussian process model, as shown in Formula (20).

$$\begin{array}{ccc}\left[\begin{array}{l}{\omega }_0\\ {\omega }_1\\ {\omega }_2\\ {\omega }_3\\ {\omega }_4\\ {\omega }_5\end{array}\right]=\left[\begin{array}{l}22.2409\\ 34.3012\\ 34.3692\\ 35.2377\\ 24.8127\\ 21.0682\end{array}\right]& \left[\begin{array}{l}{\omega }_0\\ {\omega }_1\\ {\omega }_2\\ {\omega }_3\\ {\omega }_4\\ {\omega }_5\end{array}\right]=\left[\begin{array}{l}22.2409\\ 34.3012\\ 34.3692\\ 35.2377\\ 24.8127\\ 21.0682\end{array}\right]& \left[\begin{array}{l}{\omega }_0\\ {\omega }_1\\ {\omega }_2\\ {\omega }_3\\ {\omega }_4\\ {\omega }_5\end{array}\right]=\left[\begin{array}{l}44.9978\\ 50.9396\\ 51.3454\\ 53.2361\\ 44.7894\\ 38.9147\end{array}\right]\end{array}$$

(20)

Before using the Gaussian process model for regression, its accuracy needs to be validated. A test set consisting of 10 samples is utilized to assess the accuracy of the model. The test samples, based on the finite element model of the damaged structure, are presented in

Table 6. Correction results of the finite element model of the damaged structure.

Sampling Frequency	E2 (GPa)	E4 (GPa)	E9 (GPa)	E* (GPa)	ρ (102 Kg/m3)	First Natural Frequency (Hz)	Second Natural Frequency (Hz)	Third Natural Frequency (Hz)
1	27.1812	25.4831	17.4988	31.7789	25.9248	16.098	63.986	121.012
2	17.6521	21.9542	30.8954	31.7851	24.6988	17.189	63.851	120.502
3	29.5841	22.0212	19.9988	32.6898	25.7922	16.512	64.398	123.496
4	23.9682	30.6212	16.1211	31.3522	26.0121	16.002	63.811	119.002
5	27.1812	25.4831	17.4988	31.7789	25.9248	16.098	63.986	121.012
6	28.7627	33.0349	19.9499	33.0040	25.6559	16.771	66.428	125.057
7	17.9770	18.1957	30.6718	33.3882	24.9277	17.241	63.832	122.109
8	30.3688	18.1093	17.7471	30.7089	26.0680	15.737	61.333	118.593
9	23.9780	27.0065	18.8913	33.5704	25.7254	16.658	65.453	123.032
10	24.8057	23.0404	18.7572	31.8289	23.9476	16.842	65.793	124.936

, while the corresponding test results based on the Gaussian process model are presented in

Table 7. The correction results of the Gaussian process model and finite element model.

Sampling Frequency	First Natural Frequency (FEM)	First Natural Frequency (Gaussian)	Second Natural Frequency (FEM)	Second Natural Frequency (Gaussian)	Third Natural Frequency (FEM)	Third Natural Frequency (Gaussian)
1	16.098	16.099	63.986	63.980	121.012	121.018
2	17.189	17.178	63.851	63.832	120.502	120.508
3	16.512	16.498	64.398	64.401	123.496	123.498
4	16.002	16.012	63.811	63.815	119.002	119.012
5	16.098	16.099	63.986	63.98	121.012	121.018
6	16.771	16.773	66.428	66.431	125.057	120.061
7	17.241	17.269	63.832	63.830	122.109	122.170
8	15.737	15.782	61.333	61.468	118.593	118.750
9	16.658	16.686	65.453	65.566	123.032	123.224
10	16.842	16.826	65.793	65.692	124.936	124.702

. Table 7 shows that the difference between the values predicted by the Gaussian process model and the actual values from the finite element model for the damaged structure is very small. This confirms that the established Gaussian process model can indeed be used for finite element model updating.

4.5. Construction of the Objective Function

The objective function constructed by the first three natural frequencies is shown in Formula (21).

$$f(x)=\left\{\begin{array}{l}{f}_1(x)=\left|f_1-P_1\right|\\ f_2(x)=\left|f_2-P_2\right|\\ f_3(x)=\left|f_3-P_3\right|\end{array}\right.$$

(21)

In the formula, f₁, f₂, and f₃ represent the values of the first three natural frequencies based on the Gaussian process, and P₁, P₂, and P₃ represent the values of the first three natural frequencies based on the FEM.

To facilitate the calculation, Formula (21) is changed to Formula (22).

$$f_{\ast }(x)={f_1}^2(x)+{f_2}^2(x)+{f_3}^2(x)$$

(22)

In the formula, $f_{\ast }(x)$ is the optimized objective function. The MATLAB 2024 software is used to optimize the calculation and obtain the corresponding input parameters. To avoid the search results falling into a local optimum, this paper uses multiple sets of initial points for iterative search and takes a set of parameters with the smallest objective function.

4.6. Correction of the Results

Table 8. Model parameter correction results.

Method of Calculation	E2 (GPa)	E4 (GPa)	E9 (GPa)	E* (No Damage) (GPa)	ρ (102 Kg/m3)
Finite element method	25.146	24.495	20.803	32.189	25.468
Gaussian process	25.004	24.651	20.712	32.259	25.324
Relative error %	0.564	0.637	0.438	0.219	0.564

Relative error = (the value of the Gaussian process − the value of the finite element method)/the value of the finite element method.

illustrates that the discrepancy between the parameter correction results obtained from the Gaussian process model and those derived from the FEM is negligible. The relative errors for the elastic modulus values E₂, E₄, E₉, and E* are 0.564%, 0.637%, 0.438%, and 0.219%, respectively. Similarly, the relative error for the mass density ρ is 0.564%. Consequently, the Gaussian process model can be used to effectively substitute for complex finite element modeling to achieve structural parameter correction for simply supported beams.

5. Conclusions

To enhance computational efficiency, this study proposes a method for correcting structural model parameters based on the Gaussian process model.

(1)

A method is proposed for predicting the natural frequencies of structures based on the Gaussian process. This method predicts the mean and standard deviation of the first three natural frequencies of a simply supported beam. The predicted values for the first three natural frequencies closely align with the results obtained from finite element model calculations, underscoring the applicability of the Gaussian process for predicting natural frequencies in simply supported beam structures.

(2)

An objective function that employs the first three natural frequencies as independent variables is formulated. Subsequently, the Gaussian process model is utilized to modify the elastic modulus of the damaged element, as well as the elastic modulus and mass density of the undamaged element. The Gaussian process model demonstrates remarkable computational efficiency, with negligible relative errors between its values and those obtained from the finite element method.

(3)

This article did not discuss the uncertainty of constraint. The polluted data in the real world—environmental and operational variability—are also not considered in this paper. In future studies, we will investigate the uncertainty of the degree of constraint and environmental and operational variability.