Sensitivity Analysis of Fatigue Life for Cracked Carbon-Fiber Structures Based on Surrogate Sampling and Kriging Model under Distribution Parameter Uncertainty

Abstract

The quality and reliability of wind turbine blades, as core components of wind turbines, are crucial for the operational safety of the entire system. Carbon fiber is the primary material for wind turbine blades. However, during the manufacturing process, manual intervention inevitably introduces minor defects, which can lead to crack propagation under complex working conditions. Due to limited understanding and measurement capabilities of the input variables of structural systems, the distribution parameters of these variables often exhibit uncertainty. Therefore, it is essential to assess the impact of distribution parameter uncertainty on the fatigue performance of carbon-fiber structures with initial cracks and quickly identify the key distribution parameters affecting their reliability through global sensitivity analysis. This paper proposes a sensitivity analysis method based on surrogate sampling and the Kriging model to address the computational challenges and engineering application difficulties in distribution parameter sensitivity analysis. First, fatigue tests were conducted on carbon-fiber structures with initial cracks to study the dispersion of their fatigue life under different initial crack lengths. Next, based on the Hashin fatigue failure criterion, a simulation analysis method for the fatigue cumulative damage life of cracked carbon-fiber structures was proposed. By introducing uncertainty parameters into the simulation model, a training sample set was obtained, and a Kriging model describing the relationship between distribution parameters and fatigue life was established. Finally, an efficient input variable sampling method using the surrogate sampling probability density function was introduced, and a Sobol sensitivity analysis method based on surrogate sampling and the Kriging model was proposed. The results show that this method significantly reduces the computational burden of distribution parameter sensitivity analysis while ensuring computational accuracy.

1. Introduction

As the key components of wind turbine systems, wind turbine blades entail high costs for manufacturing, transportation, and maintenance, particularly in remote areas. Therefore, long-term structural reliability is crucial [1,2,3,4,5]. Wind turbine blades are generally manufactured using a vacuum infusion method, wherein the fiber materials pre-placed in the mold are synthesized and bonded together, as shown in Figure 1. During the actual manufacturing process of composite materials, due to significant manual intervention, defects are inevitably introduced [6,7,8,9,10]. These minor defects, under the influence of complex environments and alternating loads, gradually lead to crack propagation, severely impacting the operational safety of wind turbine blades.

Due to the complexity and large size of wind turbine blade structures, it is challenging to study crack propagation in full-size cracked blades [11,12]. This paper aims to establish and prepare cracked carbon-fiber structures for fatigue analysis to provide a basis for the application of carbon-fiber composites in wind turbine blades. The existing literature has accumulated extensive experience in the simulation analysis and experimental testing of cracked fiber composites. In simulation, Kumar et al. [13] developed a new multiphase field (MPF) model combined with the cohesive zone model (CZM), effectively simulating the interaction between intralayer and interlayer cracking in cracked fiber-reinforced composite panels, significantly improving the ability to predict the fatigue life for complex failure modes. Esfarjani et al. [14] studied the dynamic fatigue-crack propagation in cracked carbon-fiber panels under non-classical thermal shock conditions based on the Lord–Shulman theory. They used a domain-independent interaction integral method to calculate the stress intensity factor of dynamic cracks and simulated it using the extended finite element method (XFEM). Pournoori et al. [15] applied the three-dimensional Hashin failure criterion, combining UMAT and VUMAT subroutines, to analyze the mechanical behavior and fatigue failure modes of cracked glass fiber composite panels under different strain rates, demonstrating the model’s predictive capability under various loading conditions. Experimentally, Xie et al. [16] explored the crack resistance of cracked fiber-reinforced composite panels through three-point bending tests and microstructure analysis, providing quantitative evaluations of the material’s tensile strength and fracture toughness. Al-Fasih et al. [17] combined experimental and numerical methods to study the fracture behavior of cracked carbon-fiber panel beam structures under four-point loading, using different carbon-fiber composite arrangements and analyzing the fatigue failure mode of crack propagation using the J-integral method, revealing the dominant failure mechanisms of skin crack extension and its impact on the structures’ structural integrity. Pan et al. [18] analyzed the vibration characteristics of single-crack and multi-crack blade disc systems using orthogonal experiments, studying the effects of the crack depth, height, and rotational speed on their vibration, and evaluated the role of crack distribution. In summary, this paper focuses on cracked carbon-fiber structures to study the influence of material parameters, cracks, and loads on fatigue reliability.

Numerous parameters affect the fatigue reliability of carbon-fiber structures. Therefore, understanding the impact of each parameter on their structural reliability and identifying the most critical ones can provide clear guidance for designing and manufacturing wind turbine blades [19,20]. Due to the high computational cost of sensitivity analysis, most studies adopt surrogate model-based methods. Antoniadis et al. [21] employed random forests as a non-parametric method and combined them with a metamodel for efficient sensitivity analysis. Their research demonstrated that this surrogate model could handle high-dimensional data and variable interactions, using permutation-based variable importance indices to quickly identify important inputs. Lamboni et al. [22] proposed a new method combining model-dependent structure extraction with surrogate models to evaluate the individual, overall, and interactive contributions of related variables to the model output. Their study showed that this method could efficiently perform variance and derivative-based sensitivity analyses. Fei et al. [23] introduced the matrix concept to improve the computational cost and accuracy of multi-objective reliability design for complex structures, developing a vector surrogate modeling (VSM) method. Numerical examples and engineering cases indicated that VSM is feasible for simultaneous modeling and integrated reliability design, offering superior computational efficiency and accuracy in high-dimensional, nonlinear problems. Khorramian et al. [24] developed a new learning function, stopping criterion, and candidate selection method, proposing an optimized surrogate model method for sensitivity analysis. They introduced a hybrid candidate selection method and a stopping criterion based on the relative mean of the learning function. The results showed that the surrogate model has a high accuracy and efficiency in sensitivity analysis. Yang et al. [25] proposed a system-reliability design optimization method based on surrogate models, using a new active learning function to predict the system-reliability gain after adding sample points, gradually updating the surrogate model and performing sensitivity analysis. Through three numerical examples and two engineering cases, they validated the efficiency and accuracy of the surrogate model-based method.

The aforementioned studies offer critical methods for addressing the substantial data requirements in the sensitivity analysis of cracked carbon-fiber structures. However, when the sample size of the basic random input variables is small or the understanding of these variables is limited, the uncertainty in their distribution parameters can impact the accuracy of structural fatigue performance analysis [26]. Therefore, it is crucial to consider the impact of subjective uncertainty on structural fatigue performance. Wang et al. [27] introduced distribution parameter uncertainty and proposed a novel method for efficiently evaluating the risk sensitivity of subjective uncertainty. This method is particularly effective in handling expensive models with numerous inputs and uncertain distribution parameters, demonstrating high accuracy and efficiency through two examples. Meng et al. [28] proposed a framework based on a new finite mixture model (MM) to address the uncertainty of distribution parameters in systems. Their study showed that this framework excels in accurately and efficiently achieving uncertainty propagation under multimodal distribution random variables while avoiding the computation of higher-order moments through four examples. He et al. [29] presented a method for modeling and assessing the uncertainty sensitivity for parameter distribution reliability problems. This method quantifies uncertainty sensitivity by using the Monte Carlo (MC) method to calculate the first-order derivative of the response function with respect to the distribution parameters. Their research indicated that considering the randomness of the distribution parameters allows for the accurate sensitivity analysis of each parameter. Nannapaneni et al. [30] introduced distribution parameter uncertainty and proposed a method for estimating engineering systems’ reliability that accounts for both frequency and epistemic uncertainties. The method’s effectiveness and accuracy were validated through two engineering case studies. The above literature indicates that considering distribution parameter uncertainty can enhance the accuracy of structural fatigue reliability predictions.

In summary, the current fatigue analysis studies of carbon-fiber structures in wind turbine blades typically only consider parameter uncertainty, without accounting for the uncertainty in the distribution parameters. The global sensitivity analysis of multi-input variable distribution parameters faces the problem of excessive computational demand, significantly reducing its computational and analytical efficiency. This makes it difficult to quickly identify the critical distribution parameters affecting the analysis’s reliability and accurately predict the fatigue life of carbon-fiber structures in wind turbine blades.

To overcome these challenges, this study introduces a novel sensitivity index inspired by variance-based methods. This index specifically evaluates the influence of the distribution parameters on the fatigue life of carbon-fiber structures, offering a more comprehensive understanding of their impact. Furthermore, we propose an efficient computational approach that leverages the Kriging model to directly establish the relationship between the distribution parameters and output statistics. This method incorporates a surrogate sampling probability density function, significantly improving the efficiency of sensitivity analysis under both subjective and objective uncertainty factors.

The subsequent sections of this paper are organized as follows: Section 2 presents the theoretical foundations of this study, including the Sobol sensitivity analysis and the active learning Kriging model. Section 3 details the fatigue experiments conducted on carbon-fiber structures and the validation of the finite element simulation model. Section 4 introduces the construction and validation of the surrogate model using the ALK-SS approach. Section 5 applies the proposed method to perform a Sobol sensitivity analysis, highlighting the impact of distribution parameter uncertainty on fatigue performance. Section 6 summarizes the main findings and conclusions of this study.

2. Fatigue Life Prediction and Reliability of Cracked Carbon-Fiber Structures under Uncertain Parameters

2.1. Sensitivity Analysis of Cracked Carbon-Fiber Structures Using ALK Model and Surrogate Sampling

When the uncertainty of the distribution parameters of random variables is not considered, the performance function of the structural system can be expressed as:

$$Y=g\left(X\right)$$

(1)

where $Y$ represents the output response of the structure or system, and $X=\left[X_1,X_2,\cdot \cdot \cdot ,X_n\right]$ represents the basic random input variables of the structure, with $n$ indicating the number of basic random input variables. When the structure has both distribution parameter uncertainty and random variable uncertainty, the structural model can be expressed as:

$$Y=g\left(X,\Theta \right)$$

(2)

where the subjective variables $\Theta =\left[{\Theta }_1,{\Theta }_2,\cdot \cdot \cdot ,{\Theta }_n\right]$ can be described by the probability density function $f_{\Theta }\left(\theta \right)$. It is usually assumed that $\theta$ is the distribution parameters of the random input variable $X$ and that values for this variable are mutually independent. With the accumulation of information about the objective variables, the uncertainty of the distribution parameters can be reduced or even eliminated. The uncertainty of the distribution parameters can affect the uncertainty of the random input variables $X$ through the conditional probability density function $f_X\left(\left.x\right|\theta \right)$, which is then transmitted to the output response through the performance function of the structural system, leading to uncertainty in the output statistical characteristics. The process of the subjective uncertainty transfer of the distribution parameters is shown as follows:

$$\Theta \stackrel{f_X\left(\left.x\right|\theta \right)}{\to }X\stackrel{Y=g\left(X\right)}{\to }\left[y_1,y_2,\cdot \cdot \cdot ,y_N\right]\to \left\{\begin{array}{l}E\left(Y\right)\\ V\left(Y\right)\end{array}\right.$$

(3)

where $E\left(Y\right)$ and $V\left(Y\right)$ represent the mean of and variance in the output $Y$, respectively. This complex relationship cannot be analytically expressed, but it can be described using a “black box” model $\Psi$ [31], as shown in Figure 2.

It can be seen that there is a one-to-one correspondence between the distribution parameters and the mean of and variance in the structural system’s output response. To more intuitively describe the impact of the subjective uncertainty of the distribution parameters on the mean of and variance in the structural system’s output response, an implicit function relationship can be established between the distribution parameters and the output statistical characteristics.

$$M=\Psi \left(\Theta \right)$$

(4)

where $M$ represents the statistical characteristics of the output performance. Based on the Sobol indices, the first-order sensitivity coefficient (FSC) and total first-order sensitivity coefficient (TSC) of the distribution parameter ${\theta }_i$, with respect to $M$, are defined as:

$$S_i={\displaystyle \frac{V\left(E\left(M\left|{\Theta }_i\right.\right)\right)}{V\left(M\right)}}$$

(5)

$$S_{Ti}={\displaystyle \frac{E\left(V\left(M\left|{\Theta }_{~i}\right.\right)\right)}{V\left(M\right)}}=1-{\displaystyle \frac{V\left(E\left(M\left|{\Theta }_{~i}\right.\right)\right)}{V\left(M\right)}}$$

(6)

where ${\Theta }_{~i}$ is the set of distribution parameters excluding the single distribution parameter; ${\Theta }_i$ represents the contribution of the uncertainty of the single distribution parameter ${\Theta }_i$ to the uncertainty of the output performance statistical characteristic $M$; and $S_{Ti}$ represents the effect of the interaction between the distribution parameter ${\Theta }_i$ itself and the remaining parameter distributions ${\Theta }_{~i}$ on the uncertainty of $M$.

Equations (5) and (6) show that directly solving the Sobol indices under distribution parameter uncertainty requires triple-nested sampling. The outer layer samples the distribution parameters based on their marginal probability density function. The middle layer samples the remaining parameters ${\Theta }_{~i}$, given each value of the distribution parameters obtained from the outer layer sampling. The inner layer further samples the input variables based on the conditional probability density function, given the samples from the middle layer. The computational cost of this method increases exponentially with the number of variables, making it impractical for actual engineering problems. Based on this, we propose an efficient computational method for determining Sobol sensitivity indices under parameter uncertainty.

The Kriging model, as an unbiased estimation model with minimal variance, combines global approximation with local random errors. It can be expressed as the sum of a parametric linear regression model and a non-parametric stochastic process [32,33,34]:

$${\Psi }_K\left(\Theta \right)=F\left(\beta ,\Theta \right)+z\left(\Theta \right)=f^T\left(\Theta \right)\beta +z\left(\Theta \right)$$

(7)

where ${\Psi }_K\left(\Theta \right)$ is the Kriging model to be determined, representing the implicit relationship between the distribution parameters and the output performance characteristics; $F\left(\beta ,\Theta \right)$ is the linear regression part; $f^T\left(\Theta \right)=\left[f_1\left(\Theta \right),f_2\left(\Theta \right),\cdots ,f_p\left(\Theta \right)\right]$ is the basis function of the distribution parameters $\Theta$; and $\beta ={\left[{\beta }_1,{\beta }_2,\cdots ,{\beta }_p\right]}^T$ is the regression coefficient, which can be estimated using known response values. $z\left(\Theta \right)$ is a normally distributed stochastic process with $N\left(0,{\sigma }^2\right)$. The covariance matrix component between any two sample points, ${\theta }_i$ and ${\theta }_j$, in the design space can be expressed as:

$$Cov\left[z\left({\theta }_i\right),z\left({\theta }_j\right)\right]={\sigma }^{2}R\left({\theta }_i,{\theta }_j\right)$$

(8)

where $R\left({\theta }_i,{\theta }_j\right)\left(i,j=1,2,\cdot \cdot \cdot ,N^T\right)$ represents the correlation function between any two sample points, ${\theta }_i$ and ${\theta }_j$, in the space, determining the simulation accuracy. $N^T$ is the number of samples in the training set, and the correlation function can be expressed as:

$$R\left({\theta }_i,{\theta }_j\right)=\exp {\displaystyle \sum _{e=1}^n\left[-{\epsilon }_e{\left({\theta }_i^e-{\theta }_j^e\right)}^2\right]}$$

(9)

where ${\theta }_i^e$ and ${\theta }_j^e$ represent the e-th components of the sample points ${\theta }_i$ and ${\theta }_j$, respectively; ${\epsilon }_e\left(e=1,2,\cdot \cdot \cdot ,n\right)$ is the unknown correlation parameters.

According to the Kriging theory, the estimates of the unknown parameters $\beta$ and ${\sigma }^2$ in the model can be expressed as:

$$\stackrel{⌢}{\beta }={\left(F^T{R}^{-1}F\right)}^{-1}{F}^T{R}^{-1}m$$

(10)

$${\stackrel{⌢}{\sigma }}^2={\displaystyle \frac{{\left(m-F\stackrel{⌢}{\beta }\right)}^T{R}^{-1}\left(m-F\stackrel{⌢}{\beta }\right)}{N^T}}$$

(11)

where $m$ is the column vector of the output statistical characteristics of the training sample data; $F$ is the $N^T\times p$ matrix composed of the regression model at $N^T$ sample points. The correlation parameters $\epsilon ={\left[{\epsilon }_1,{\epsilon }_2,\cdot \cdot \cdot ,\epsilon n\right]}^T$ can be obtained by maximizing the likelihood function, expressed as:

$$\max G\left(\epsilon \right)=-{\displaystyle \frac{N^T\ln \left({\stackrel{⌢}{\sigma }}^2\right)+\ln \left|R\right|}{2}}\left(\epsilon \ge 0\right)$$

(12)

The values of $\epsilon$ obtained by solving Equation (12) constitute the Kriging model, which is the optimal surrogate model in terms of fitting accuracy.

With the unknown parameters in the Kriging model determined, the optimal linear unbiased estimation of the output moments at the test point $\theta$ can be established:

$${\Psi }_K\left(\theta \right)=f{\left(\theta \right)}^T\stackrel{⌢}{\beta }+r^T\left(\theta \right)R^{-1}\left(m-F\stackrel{⌢}{\beta }\right)$$

(13)

where $r\left(\theta \right)$ is the correlation function vector between the training sample points and the prediction point, expressed as:

$$r^T\left(\theta \right)=\left[R\left(\theta ,{\theta }_1\right),R\left(\theta ,{\theta }_2\right),\cdots ,R\left(\theta ,{\theta }_{N^T}\right)\right]$$

(14)

Therefore, for any unknown $\theta$, the statistical characteristics of the output response can be accurately predicted using Equation (13). The prediction result $\Psi \left(\theta \right)$ follows a normal distribution, $N\left({\mu }_{{\Psi }_K}\left(\theta \right),{\sigma }_{{\Psi }_K}^2\left(\theta \right)\right)$, with the mean and variance calculated as:

$${\mu }_{{\Psi }_K}\left(\theta \right)=f{\left(\theta \right)}^T\stackrel{⌢}{\beta }+r^T\left(\theta \right)R^{-1}\left(m-F\stackrel{⌢}{\beta }\right)$$

(15)

$$\begin{array}{ll}{\sigma }_{{\Psi }_K}^2\left(\theta \right)=\hfill & {\sigma }^2\left\{1-r^T\left(\theta \right)R^{-1}r\left(\theta \right)+{\left[F^T{R}^{-1}r\left(\theta \right)-f\left(\theta \right)\right]}^T\right.·\hfill \\ & \left.{\left(F^T{R}^{-1}F\right)}^{-1}\left[F^T{R}^{-1}r\left(\theta \right)-f\left(\theta \right)\right]\right\}\hfill \end{array}$$

(16)

The original Kriging surrogate model cannot balance accuracy and efficiency. This paper improves the fitting accuracy of the Kriging surrogate model by considering the standard deviation of the estimated value and using the U-learning function to enhance the prediction accuracy and efficiency of the Kriging model. The criterion for selecting indicators from the candidate sample pool is given as follows [35]:

$$U\left(\theta \right)={\displaystyle \frac{\left|{\Psi }_K\left(\theta \right)\right|}{{\sigma }_{{\Psi }_K}\left(\theta \right)}}$$

(17)

In the candidate sample pool, points with larger standard deviations, i.e., smaller $U$ values, should be added to the training sample set to update the Kriging model. Referring to the convergence condition in the literature [36], when Equation (18) is satisfied, the estimated value of the output performance characteristic is considered acceptable, and the Kriging surrogate model stops updating.

$$\left|{\displaystyle \frac{{\Psi }_K\left({\theta }_u\right)-\Psi \left({\theta }_u\right)}{\Psi \left({\theta }_u\right)}}\right|\ge 2$$

(18)

where ${\Psi }_K\left({\theta }_u\right)$ is the predicted value of the surrogate model at the sample point ${\theta }_u$ with the smallest $U$ value in the sample pool, and $\Psi \left({\theta }_u\right)$ is the actual response value at that sample point. When the relative error at the point that contributes the most to the fitting accuracy exceeds the set threshold, the model update terminates.

The sensitivity analysis method based on the active learning Kriging model reduces the computational cost of solving sensitivity indices to the cost of constructing the surrogate model, significantly alleviating the computational burden. Since both the distribution parameters and input variables have uncertainties, constructing the surrogate model still requires a double-loop to obtain training samples, with the computational cost being dependent on the distribution parameter sample value in each loop. To address this issue, this paper further introduces the surrogate sampling probability density function (SS-PDF) into the construction process of the Kriging model for distribution parameters and output statistical moments.

According to the theory proposed by Li [37], assuming a fixed distribution parameter ${\theta }^{*}$ is given for the model input variable $X$, samples drawn based on its probability density function can cover the entire variation space of input variable $X$ when the distribution parameter $\Theta$ changes. Therefore, the SS-PDF $h_X\left(\left.x\right|{\theta }^{*}\right)$ with the distribution parameter fixed at ${\theta }^{*}$ can be used for sampling the input variables, simplifying the double-loop sampling in the process of solving the output statistical characteristics to a single-layer calculation.

Considering $M$ as the mean of the model output response, its value can be calculated using the following formula:

$$\begin{array}{ll}M=& E_X\left(\left.Y\right|\Theta \right)={\displaystyle \underset{R^n}{\int }g\left(x\right)f_X\left(\left.x\right|\theta \right)dx}=\\ & {\displaystyle \underset{R^n}{\int }g\left(x\right)\frac{f_X\left(\left.x\right|\theta \right)}{h_X\left(\left.x\right|{\theta }^{*}\right)}{h}_X\left(\left.x\right|{\theta }^{*}\right)dx}=\\ & E_{X^{*}}\left(g\left(x\right){\displaystyle \frac{f_X\left(\left.x\right|\theta \right)}{h_X\left(\left.x\right|{\theta }^{*}\right)}}\right)\end{array}$$

(19)

When $M$ is the variance in the model output response, according to $V\left(Y\right)=E\left(Y^2\right)-E^2\left(Y\right)$. Its value can be calculated using the following formula:

$$\begin{array}{ll}M=& E_X\left(\left.Y^2\right|\Theta \right)-E_X^2\left(\left.Y\right|\Theta \right)=\\ & {\displaystyle \underset{R^n}{\int }{g}^2\left(x\right)f_X\left(\left.x\right|\theta \right)dx}-{\left({\displaystyle \underset{R^n}{\int }g\left(x\right)f_X\left(\left.x\right|\theta \right)dx}\right)}^2=\\ & {\displaystyle \underset{R^n}{\int }{g}^2\left(x\right)\frac{f_X\left(\left.x\right|\theta \right)}{h_X\left(\left.x\right|{\theta }^{*}\right)}{h}_X\left(\left.x\right|{\theta }^{*}\right)dx}-\\ & {\left({\displaystyle \underset{R^n}{\int }g\left(x\right)\frac{f_X\left(\left.x\right|\theta \right)}{h_X\left(\left.x\right|{\theta }^{*}\right)}{h}_X\left(\left.x\right|{\theta }^{*}\right)dx}\right)}^2=\\ & E_{X^{*}}\left(g^2\left(x\right){\displaystyle \frac{f_X\left(\left.x\right|\theta \right)}{h_X\left(\left.x\right|{\theta }^{*}\right)}}\right)-E_{X^{*}}^2\left(g\left(x\right){\displaystyle \frac{f_X\left(\left.x\right|\theta \right)}{h_X\left(\left.x\right|{\theta }^{*}\right)}}\right)\end{array}$$

(20)

where $R^n$ is the $n$-dimensional sample space of the input variable $X$, and $E_{X^{*}}\left(\cdot \right)$ is the mean determined by the SS-PDF $h_X\left(\left.x\right|{\theta }^{*}\right)$.

Equations (19) and (20) indicate that, by introducing the surrogate sampling probability density function to estimate the output statistical characteristics of the uncertain distribution parameters, only a set of input variable $X$ samples generated according to $h_X\left(\left.x\right|{\theta }^{*}\right)$ is needed. When the outer distribution parameters change, this set of samples can be reused in the inner loop, thereby reducing the number of simulations.

2.2. Sensitivity Index Analysis Process Based on the Kriging Model and Surrogate Sampling

Based on the above derivation, the flowchart for solving the sensitivity indices $S_i$ and $S_{Ti}$ using the ALK-SS method is shown in Figure 3. The specific steps are as follows:

• Generate $N_K$ samples in the distribution parameter sample space according to the marginal probability density function $f_{\theta }\left(\theta \right)$, forming a sample pool $S_{SS}=\left[{\theta }_1,{\theta }_2,\cdots ,{\theta }_{N_K}\right]$;
• Extract $N_X$ distribution parameter samples $\left[x_1,x_2,\cdots ,x_{N_X}\right]$ according to the surrogate sampling probability density function $h_X\left(\left.x\right|{\theta }^{*}\right)$, and substitute them into the simulation model to obtain the corresponding output statistical characteristic values $\left[g\left(x_1\right),g\left(x_2\right),\cdot \cdot \cdot ,g\left(x_{N_X}\right)\right]$;
• Randomly extract $N_T$ distribution parameter samples $\left[{\theta }_1,{\theta }_2,\cdots ,{\theta }_{N_T}\right]$ from the sample pool $S_{SS}$. For each distribution parameter sample point ${\theta }_j\left(j=1,2,\cdots ,N_T\right)$, calculate the corresponding output statistical characteristic values $m_j\left(j=1,2,\cdots ,N_T\right)$ based on $g\left(x_t\right)$ and $f_X\left(\left.x_t\right|{\theta }_j\right)/h_X\left(\left.x_t\right|{\theta }^{*}\right)$ according to Equations (19) and (20), forming the initial training sample set $T_{SS}=\left[{\theta }_j,m_j\right]$;
• Establish the initial Kriging surrogate model ${\Psi }_K\left(\Theta \right)$ based on the current $T_{SS}$;
• Select the update sample point ${\theta }_u$ from $S_{SS}$ as:

  $${\theta }_u=\arg \underset{\theta \in S_{SS}}{\min }U\left(\theta \right)$$

  (21)

  Calculate the predicted value ${\Psi }_K\left({\theta }_u\right)$ and the true response value $\Psi \left({\theta }_u\right)$. Stop the active learning process and proceed to step 6 when Equation (21) is satisfied; otherwise, add $\left[{\theta }_u,\Psi \left({\theta }_u\right)\right]$ to the training set $T_{SS}$ and return to step 4;
• Randomly extract $N_{\Theta }$ test distribution parameter samples $\left[{\theta }_1,{\theta }_2,\cdots ,{\theta }_{N_{\Theta }}\right]$ from the sample pool $S_{SS}$, and substitute them into the Kriging model to obtain the corresponding output performance characteristic value samples $\left[m_1,m_2,\cdots ,m_{N_{\Theta }}\right]$, from which the total variance $V\left(M\right)$ can be calculated;
• Fix the distribution parameter ${\Theta }_{~i}$ at its sample point ${\theta }_i^j\left(j=1,2,\cdots ,N_{\Theta }\right)$. Combine it with other distribution parameter samples ${\theta }_{~i}^k\left(k=1,2,\cdots ,N_{\Theta }\right)$ and, based on the established Kriging model, solve the conditional output statistical characteristic values $\left.M\right|\left({\theta }_i^j,{\theta }_{~i}^k\right)\left(k=1,2,\cdots ,N_{\Theta }\right)$ to obtain their conditional mean $E\left(\left.M\right|{\theta }_i^j\right)$. Traverse $j\left(j=1,2,\cdots ,N_{\Theta }\right)$ to obtain $N_{\Theta }$ sample values of $E\left(\left.M\right|{\Theta }_i\right)$ under the $i$-th dimension distribution parameter conditions $\left[E\left(\left.m\right|{\theta }_i^1\right),E\left(\left.m\right|{\theta }_i^2\right),\cdots ,E\left(\left.m\right|{\theta }_i^{N_{\Theta }}\right)\right]$. Calculate the variance in the conditional expectation of the output performance characteristic values $V\left(E\left(\left.M\right|{\Theta }_i\right)\right)$, and use Equation (5) to obtain the primary sensitivity index $S_i$ of the distribution parameter ${\Theta }_i\left(i=1,2,\cdots ,n\right)$;
• Fix the distribution parameter ${\Theta }_{~i}$ at its sample point ${\theta }_{~i}^j\left(j=1,2,\cdots ,N_{\Theta }\right)$. Combine it with other distribution parameter samples ${\theta }_i^k\left(k=1,2,\cdots ,N_{\Theta }\right)$ and, based on the established Kriging model, solve the conditional output statistical characteristic values $\left.M\right|\left({\theta }_{~i}^j,{\theta }_i^k\right)\left(k=1,2,\cdots ,N_{\Theta }\right)$ to obtain its conditional mean $E\left(\left.M\right|{\theta }_{~i}^j\right)$. Traverse $j\left(j=1,2,\cdots ,N_{\Theta }\right)$ to obtain $N_{\Theta }$ sample values of $E\left(\left.M\right|{\Theta }_{~i}\right)$ under the $i$-th dimension distribution parameter conditions $\left[E\left(\left.m\right|{\theta }_{~i}^1\right),E\left(\left.m\right|{\theta }_{~i}^2\right),\cdots ,E\left(\left.m\right|{\theta }_{~i}^{N_{\Theta }}\right)\right]$. Calculate the variance in the conditional expectation of the output performance characteristic values $V\left(E\left(\left.M\right|{\Theta }_{~i}\right)\right)$, and use Equation (6) to obtain the total sensitivity index $S_{Ti}$ of the distribution parameter ${\theta }_i\left(i=1,2,\cdots ,n\right)$.

The ALK-SS method proposed in this paper introduces the SS-PDF, making the computational model independent of the actual distribution parameters and avoiding the complex process of cyclic sampling of the input variables $X$. This further improves the efficiency of parameter uncertainty sensitivity analysis.

3. Experiments and Simulation

3.1. Experimental Method

In this study, composite carbon-fiber material (T800H-6K/803A) with dimensions of 230 mm × 25 mm × 1.5 mm was selected as the experimental specimen, in accordance with GB/T 3354-2014. The dispersion of carbon fibers within the matrix can significantly impact the material’s mechanical properties [38,39]. To minimize the influence of fiber dispersion on the experimental results, the preparation process for all samples was kept consistent, with the process parameters being strictly controlled to ensure uniform fiber dispersion. The structure and dimensions are shown in Figure 4. The properties of the unidirectional ply provided for the study are listed in

Table 1. Material Properties for T800H-6K/803A Composite Laminates.

Material Properties	Value
Young’s modulus	E11 = 105.5 GPa; E22 = E33 = 7.2 GPa
Poisson’s ratio	v12 = v13 = 0.34; v23 = 0.378
Shear modulus	G12 = G13 = 3.4 GPa; G23 = 2.520 GPa
Strength	XT = 1060 MPa; XC = 657 MPa; YT = 694 MPa; YC = 651 MPa; ZT = 75 MPa; ZC = 165 MPa
Fracture energy	S12 = 108 MPa; S13 = S23 = 81.1 MPa

. The specimens were laminated in a [0/90]₄ lay-up sequence.

Figure 5 illustrates part of the experimental process for the cracked carbon-fiber structure. Initially, a static tensile test was conducted until the carbon-fiber structure completely fractured, obtaining the maximum stress value that the specimen could withstand. Subsequently, 60% of the maximum stress value from the static tensile test was used for the fatigue test. The loading method was a constant amplitude sinusoidal load with a frequency of 30 Hz and a stress ratio of 0.1. Under a constant load P, the fatigue life and its dispersion for different crack lengths L of the carbon-fiber structure are shown in

Table 2. Fatigue Test Results for Different Initial Crack Lengths.

L/mm	P/KN	Experimental Result/105					Mean Value/105	SD/105	CV
		1	2	3	4	5
4	9.5	5.84	6.21	6.41	6.02	6.11	6.11684	0.21	0.04
5		3.85	3.93	4.37	4.11	4.21	4.09344	0.21	0.05
6		3.74	3.35	3.83	3.71	3.85	3.69575	0.18	0.03
7		2.71	3.06	2.86	2.92	2.97	2.90287	0.13	0.05
8		0.54	0.48	0.47	0.32	0.38	0.43478	0.09	0.02

, and the trend of the fatigue life variations for different crack lengths L is depicted in Figure 6.

Table 2 and Figure 6 illustrate the trend of the fatigue life changes as the crack length increases. It is clearly observed that, as the crack length L increases, the fatigue life significantly decreases and the rate of decrease gradually increases. This indicates that crack growth has a substantial negative impact on the fatigue life of the material. The coefficients of variation are all within 10%, indicating that the fatigue life of the carbon-fiber structure exhibits a certain degree of dispersion due to the uncertainty in the carbon-fiber material parameters. This dispersion reflects the influence of the internal structure of the material and external load uncertainty on the fatigue life. Additionally, as the crack length L increases, the standard deviation (SD) shows a decreasing trend. This suggests that the larger the crack length, the more significant the impact of the crack on the fatigue life, resulting in reduced dispersion of the fatigue life. Further analysis suggests that, when the crack length is relatively small, the random effects of factors such as the microstructure of the composite material on the fatigue life are more significant, leading to greater dispersion in the fatigue life. As the crack length increases to a certain extent, the dominant role of the crack becomes more pronounced, and the fatigue life is primarily determined by the crack propagation rate, leading to reduced dispersion in the fatigue life.

3.2. Simulation Method

Based on finite element simulation, this paper predicts the fatigue life of cracked carbon-fiber structures by writing a UMAT material subroutine, providing a model for the uncertainty analysis of their fatigue life. The main computational process of the UMAT subroutine is as follows:

• At the start of the increment step, the ABAQUS main program transfers the initial material properties, stress, strain, strain increment, and state variables to the UMAT;
• The UMAT subroutine calculates the stress at the end of the increment step based on the user-provided material constitutive model and updates the stress, strain, and state variables accordingly;
• At the end of the increment step, the UMAT transfers the stress, strain, and state variables back to the main program and provides the Jacobian matrix of the constitutive relationship to the ABAQUS main program to form the global stiffness matrix of the structure and determine whether the system has reached equilibrium;
• If the system is not balanced, the iteration is considered non-convergent, and the next iteration is performed until convergence is achieved, at which point the next increment step calculation begins.

The three-dimensional modeling was performed using solid elements selected based on the design dimensions. The simulation of composite laminate typically employs a layer-by-layer approach, cutting the carbon-fiber structure into eight layers along the thickness direction, with each layer representing a single element. For each layer, the material properties are identical except for the ply angle. The model and its boundary conditions are shown in Figure 7, where the red area indicates the pre-existing crack. The specific settings of the model include: one end fully fixed and the other end subjected to a periodic sinusoidal fatigue load. The UMAT subroutine is used to perform stress analysis, fatigue damage assessment, material parameter degradation, and fatigue life prediction of the carbon-fiber structure, as illustrated in the flowchart in Figure 8.

Using the above fatigue simulation analysis method, the fatigue life of carbon-fiber structures with different initial crack lengths under a fatigue load of 9.5 KN was simulated. As shown in Figure 9, with the increasing number of load cycles, the extent of the crack propagation continues to expand, and the stress level at the critical points also increases.

3.3. Comparison and Discussion

The data of the initial sample set directly affect the accuracy of the Sobol sensitivity analysis method based on the ALK-SS model, so it is essential to ensure the accuracy of the simulation model. The experimental results of carbon-fiber structures with different crack lengths L were compared with the simulation results, as shown in

Table 3. Fatigue Life of Bonded Structures at Different Thicknesses.

L/mm	P/KN	Experimental Result/105	Simulation Result/105	Relative Error/%
4	9.5	6.11684	6.11661	−1.64
5		4.09344	4.17888	3.17
6		3.69575	3.89040	6.09
7		2.90287	2.75049	−0.87
8		0.43478	0.45769	0.36

and Figure 10.

According to the data in Table 3 and Figure 10, the relative error between the simulated fatigue life results and the experimental results of the cracked carbon-fiber structures is within 10%. This indicates that the simulation model can accurately predict fatigue life and is highly consistent with the actual experimental data. Therefore, the finite element simulation model prediction method not only generates reliable response values for the training sample set but also provides an accurate data foundation for subsequent sensitivity analysis. This further verifies the effectiveness of the simulation model in predicting fatigue life and ensures the reliability of the analysis results.

4. Reliability Analysis of Carbon-Fiber Structures Based on Surrogate Sampling and Kriging Model

4.1. Uncertainty Analysis

In studying the fatigue reliability of composite carbon-fiber structures with initial cracks, uncertainty is a key factor. The dual randomness of the material parameters, load, and initial crack length significantly affects the overall performance of and prediction accuracy for carbon-fiber structures. Therefore, it is necessary to quantify and explore the impact of these uncertainties on the performance of carbon-fiber structures. This paper considers the distribution parameter uncertainty of 20 input variables and investigates its impact on the reliability of cracked carbon-fiber structures. Additionally, it identifies important distribution parameters through global sensitivity analysis. All input variables follow independent normal distributions [40], and their distribution parameters are shown in

Table 4. Distribution Parameters of Input Variables.

Random Variable	Mean Value	SD	Random Variable	Mean Value	SD
E11 (GPa)	${\mu }_{E_{11}}$	5.275	XC (MPa)	${\mu }_{X_C}$	32.85
E22 (GPa)	${\mu }_{E_{22}}$	0.36	YT (MPa)	${\mu }_{Y_T}$	34.7
E33 (GPa)	${\mu }_{E_{33}}$	0.36	YC (MPa)	${\mu }_{Y_C}$	32.55
v12	${\mu }_{{\nu }_{12}}$	0.017	ZT (MPa)	${\mu }_{Z_T}$	3.75
v13	${\mu }_{{\nu }_{13}}$	0.017	ZC (MPa)	${\mu }_{Z_C}$	8.25
v23	${\mu }_{{\nu }_{23}}$	0.0189	S12 (MPa)	${\mu }_{S_{12}}$	5.4
G12 (GPa)	${\mu }_{G_{12}}$	0.17	S13 (MPa)	${\mu }_{S_{13}}$	4.055
G13 (GPa)	${\mu }_{G_{13}}$	0.17	S23 (MPa)	${\mu }_{S_{23}}$	4.055
G23 (GPa)	${\mu }_{G_{23}}$	0.126	L (mm)	${\mu }_L$	0.25
XT (MPa)	${\mu }_{X_T}$	53	P (KN)	${\mu }_P$	0.475

. It is assumed that the means of all input variables have subjective uncertainty and follow the normal distribution shown in

Table 5. Distribution Parameters of Subjective Uncertainty Means.

Random Variable	Mean Value	SD	Random Variable	Mean Value	SD
E11	105.5	0.05	XC	657	0.05
E22	7.2		YT	694
E33	7.2		YC	651
v12	0.34		ZT	75
v13	0.34		ZC	165
v23	0.378		S12	108
G12	3.4		S13	81.1
G13	3.4		S23	81.1
G23	2.520		L	5
XT	1060		P	9.5

.

According to the principle that the SS-PDF should cover the entire range of input variables, the SS-PDF of the selected input variables follows the normal distribution shown in

Table 6. Surrogate Sampling Probability Density Function of Input Variables.

Random Variable	Mean Value	SD	Random Variable	Mean Value	SD
E11 (GPa)	105.5	6.33	XC (MPa)	657	39.42
E22 (GPa)	7.2	0.432	YT (MPa)	694	41.64
E33 (GPa)	7.2	0.432	YC (MPa)	651	39.06
v12	0.34	0.0204	ZT (MPa)	75	4.5
v13	0.34	0.0204	ZC (MPa)	165	9.9
v23	0.378	0.02268	S12 (MPa)	108	6.48
G12 (GPa)	3.4	0.204	S13 (MPa)	81.1	4.866
G13 (GPa)	3.4	0.204	S23 (MPa)	81.1	4.866
G23 (GPa)	2.520	0.1512	L (mm)	5	0.3
XT (MPa)	1060	63.6	P (KN)	9.5	0.57

.

4.2. Fatigue Life Prediction Method Based on Surrogate Model

For this study, we selected $N_X={10}^4$ sets of distribution parameter samples $\left[x_1,x_2,\cdots ,x_{N_X}\right]$, which were input into the finite element simulation model established in Section 3.2. A Python program was written to implement automatic cyclic simulation, yielding the corresponding output statistical characteristic values $\left[g\left(x_1\right),g\left(x_2\right),\cdot \cdot \cdot ,g\left(x_{N_X}\right)\right]$. The flowchart for the automatic cyclic simulation is shown in Figure 11.

We randomly selected $N_T=30$ sets of distribution parameter samples $\left[{\theta }_1,{\theta }_2,\cdots ,{\theta }_{N_T}\right]$. For each distribution parameter sample point ${\theta }_j\left(j=1,2,\cdots ,N_T\right)$, the corresponding output statistical characteristic values $m_j\left(j=1,2,\cdots ,N_T\right)$ were calculated based on $g\left(x_t\right)$ and the samples $f_X\left(\left.x_t\right|{\theta }_j\right)/h_X\left(\left.x_t\right|{\theta }^{*}\right)$ according to Equations (19) and (20), forming the initial training sample set $T_{SS}=\left[{\theta }_j,m_j\right]$. The initial Kriging surrogate model ${\Psi }_K\left(\Theta \right)$ was established, and the U-learning function values for the remaining points were calculated to update the sample points. By using ${\theta }_u=\arg \underset{\theta \in S_{SS}}{\min }U\left(\theta \right)$, the ${\theta }_u$ with the smallest $U\left(\theta \right)$ was added to the initial training sample set $T_{SS}$. The predicted value ${\Psi }_K\left({\theta }_u\right)$ and the true response value $\Psi \left({\theta }_u\right)$ for this point were calculated. The active learning process stops when Equation (21) is satisfied. Finally, an ALK model based on the SS-PDF was established to accurately predict the failure probability of cracked carbon-fiber structures. We randomly selected $N_{\Theta }=2000$ sets of distribution-parameter samples $\left[{\theta }_1,{\theta }_2,\cdots ,{\theta }_{N_{\Theta }}\right]$, which were input into the Kriging model to obtain the corresponding output performance characteristic value samples $\left[m_1,m_2,\cdots ,m_{N_{\Theta }}\right]$. From these samples, we calculated the total variance $V\left(M\right)$, $V\left(E\left(\left.M\right|{\Theta }_i\right)\right)$, and $V\left(E\left(\left.M\right|{\Theta }_{~i}\right)\right)$, and then used Equations (5) and (6) to compute the main sensitivity indices $S_i$ and the total sensitivity indices $S_{Ti}$, respectively.

The fatigue life prediction analysis of cracked carbon-fiber structures was conducted using both the ALK model and the ALK-SS model. The results of 1000 sets of prediction data for the ALK model and the ALK-SS model are shown in Figure 12.

The comparison results of some predicted data are shown in Figure 13. The vertical axis represents the predicted fatigue life values of the cracked carbon-fiber structure by the surrogate models, while the horizontal axis represents the finite element simulation analysis values. It can be seen that the predicted life values by both the ALK model and the ALK-SS model fall within a 1.5-fold error range. However, the fatigue life values predicted by the ALK-SS model are more concentrated along the equality line, with an error of only 0.373%, whereas the error for the ALK model is 1.768%. In summary, the ALK-SS model is more accurate and is better-suited for predicting the fatigue life of cracked carbon-fiber structures.

The prediction accuracy and the required number of samples for three different fatigue life prediction methods are shown in

Table 7. Comparison of Different Life-Prediction Methods.

Method	${\mathit{R}}^2$	Sample Size
FEA	—	2000+
ALK model	0.98232	97
ALK-SS model	0.99627	43

. In traditional methods, N sets of sample points need to be input into the finite element program for simulation to obtain the predicted fatigue life. Increasing the number of samples can improve the accuracy of the uncertainty analysis in life prediction, but it also increases the corresponding computational cost. Typically, the fatigue life prediction results of the carbon-fiber structures from the simulation analysis method are used as a reference. As shown in Table 7, the correlation coefficient $R^2$ for the ALK-SS model is closer to 1. In comparison, the ALK model has a lower $R^2$, which clearly indicates that the ALK-SS model has a significant advantage in prediction accuracy. Additionally, the ALK-SS model requires 43 samples to construct its predictions, while the ALK model requires 97 samples. This indicates that the ALK-SS model can provide a higher accuracy performance with a smaller number of samples.

4.3. Fatigue Reliability Modeling and Assessment

Given that the ALK-SS model demonstrates smaller errors and requires fewer sample quantities compared to the simulation analysis and ALK model, with a correlation coefficient as high as 0.99627, the fatigue life data obtained from this model are used for statistical analysis. The histogram of fatigue life obtained from the ALK-SS model is shown in Figure 14, and it is fitted using the Weibull, Normal, Gamma, and Lognormal distribution functions. The goodness-of-fit evaluation results are shown in

Table 8. Determination Coefficients of Life Frequency Fitting Curves.

Distribution Type	Weibull	Normal	Gamma	Lognormal
$R^2$	0.9879	0.9632	0.9261	0.9301

.

From Table 8, it can be seen that the Weibull distribution function fits the fatigue life of cracked carbon-fiber structures the best, providing a more accurate description of the fatigue life frequency distribution and its failure probability. The cumulative failure distribution function expression for the fatigue life of cracked carbon-fiber structures is given by:

$$F\left(N\right)=1-e^{-{\left({\displaystyle \frac{N}{490072}}\right)}^{6.29255}}$$

(22)

The reliability expression is:

$$R\left(N\right)=e^{-{\left({\displaystyle \frac{N}{490072}}\right)}^{6.29255}}$$

(23)

5. Sensitivity Analysis of Carbon-Fiber Structures Based on Surrogate Sampling and Kriging Model

From the perspective of the number of function calls in the computational process, given that the distribution parameter dimension b = 20, the input variable sample size $N_X={10}^4$, and the test distribution parameter sample size $N_{\Theta }=2000$, the pure ALK model method requires 97 sets of distribution parameter sample points to establish the relationship between the distribution parameters and the output performance characteristics. This quickly provides accurate sensitivity index results, but it requires repeated sampling of the input variables in the inner loop. In contrast, the ALK-SS method only needs 43 sets of input variable samples in the loop, significantly reducing the usage of the simulation model and demonstrating markedly higher efficiency. Figure 15 presents a bar chart of the sensitivity index results for the output fatigue life. The values of the main sensitivity index and the total sensitivity index for the same characteristic are not significantly different, indicating that the interaction between the distribution parameters of the variables has little impact on the fatigue life of the cracked carbon-fiber structure.

The results show that the sensitivity index values for the parameters ${\mu }_L$ and ${\mu }_P$ are significantly higher than those for other parameters. This indicates that, among the 20 parameters of the cracked carbon-fiber structures, the subjective uncertainty parameters of the initial crack length L and the load P have a crucial impact on the fatigue life. It suggests that, in the manufacturing and usage process of carbon-fiber structures, the accuracy of the initial crack length L and the load P should be well-controlled to reduce the uncertainty of distribution parameters and thereby lower the level of structural failure. Additionally, the subjective parameters ${\mu }_{E_{11}}$, ${\mu }_{X_T}$, and ${\mu }_{S_{12}}$ have a certain impact on the fatigue life of carbon-fiber structures, while the subjective parameters ${\mu }_{Z_T}$, ${\mu }_{Y_T}$, and ${\mu }_{G_{12}}$ have a slight impact on the carbon-fiber structure. Among the material parameter distribution parameters, this pattern was observed.

6. Conclusions

In practical conditions, the distribution parameters of structural system input variables cannot be accurately determined. During the laying and molding process of carbon-fiber composites, extensive manual operations and complex working environments lead to uncertainties in material parameters, initial cracks, and loads. These uncertainties significantly affect the operational safety of wind turbine blades. Therefore, it is necessary to consider the impact of distribution parameter uncertainty on their structural fatigue life. To overcome the high computational cost of directly solving sensitivity indices, an ALK-SS computational method based on surrogate models and Kriging models is proposed. The main conclusions are as follows:

• Finite element simulations were conducted on cracked carbon-fiber laminates with a [0/90]₄ ply sequence using simulation software and a UMAT subroutine. The results showed that stress concentration at the crack tip is the key reason for the failure of the carbon-fiber laminate. Comparative experiments indicated that the error in fatigue life prediction obtained from the simulation model is within 10%, demonstrating that the finite element simulation model can generate reliable response values for the training sample set, providing an accurate data foundation for subsequent sensitivity analysis.
• By obtaining the true response values of the training samples from the simulation model, a fatigue life prediction model for cracked carbon-fiber structures based on the active learning Kriging (ALK) model was established. Combining surrogate sampling and the Kriging model to predict fatigue life data, the reliability function expression of cracked carbon-fiber structures was derived, and the reliability modeling and assessment of the fatigue life of cracked carbon-fiber structures were performed.
• Compared with traditional experimental and simulation methods, the ALK model improves the computational efficiency by introducing an active learning function while ensuring accuracy, overcoming the high computational cost of directly solving indices. The ALK method can establish a Kriging surrogate model between the distribution parameters and the output statistical characteristics using a small number of distribution parameter sample points, reducing the number of function calls. Based on this framework, the ALK-SS method further introduces the SS-PDF, improving the utilization rate of inner-layer input variable samples and significantly reducing the computational load of parameter sensitivity analysis while ensuring computational accuracy. As discussed in Appendix A, the proposed method not only incorporates uncertainties in both parameters and distributions but also significantly reduces computational costs by using Kriging with proxy sampling, making it more suitable for large-scale applications compared to previous works. Despite the advantages of this study, incorporating a broader variety of comparison models to further enhance the robustness of the conclusions, improve the fitting accuracy, and reduce the computational workload is a direction for future research.