## 1 Introduction
### Sensitivity Analysis of Cracked Carbon-Fiber Structures Using ALK Model and Surrogate 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]:  
\[\Psi_{K}(\Theta)=F(\beta,\Theta)+z(\Theta)=f^{T}(\Theta)\beta+z(\Theta) \tag{7}\]  
where \(\Psi_{K}(\Theta)\) is the Kriging model to be determined, representing the implicit relationship between the distribution parameters and the output performance characteristics; \(F(\beta,\Theta)\) is the linear regression part; \(f^{T}(\Theta)=\left[f_{1}(\Theta),f_{2}(\Theta),\cdots,f_{p}(\Theta)\right]\) 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. \(z(\Theta)\) is a normally distributed stochastic process with \(N\big{(}0,\sigma^{2}\big{)}\). The covariance matrix component between any two sample points, \(\theta_{i}\) and \(\theta_{j}\), in the design space can be expressed as:  
\[Cov\big{[}z(\theta_{i}),z(\theta_{j})\big{]}=\sigma^{2}R\big{(}\theta_{i}, \theta_{j}\big{)} \tag{8}\]  
where \(R\big{(}\theta_{i},\theta_{j}\big{)}\left(i,j=1,2,\cdots,N^{T}\right)\) 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:  
\[R\big{(}\theta_{i},\theta_{j}\big{)}=\exp\sum_{e=1}^{n}\left[-\varepsilon_{e} \Big{(}\theta_{i}^{e}-\theta_{j}^{e}\Big{)}^{2}\right] \tag{9}\]  
where \(\theta_{i}^{e}\) and \(\theta_{j}^{e}\) represent the e-th components of the sample points \(\theta_{i}\) and \(\theta_{j}\), respectively; \(\varepsilon_{e}(e=1,2,\cdots,n)\) 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:  
\[\widehat{\beta}=\left(F^{T}R^{-1}F\right)^{-1}F^{T}R^{-1}m \tag{10}\]  
\[\widehat{\sigma}^{2}=\frac{\left(m-F\widehat{\beta}\right)^{T}R^{-1}\left(m-F \widehat{\beta}\right)}{N^{T}} \tag{11}\]  
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 \(\varepsilon=\left[\varepsilon_{1},\varepsilon_{2},\cdot\cdot\cdot,\varepsilon n \right]^{T}\) can be obtained by maximizing the likelihood function, expressed as:  
Figure 2:0  
The values of \(\varepsilon\) 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:  
Figure 2:1  
where \(r(\theta)\) is the correlation function vector between the training sample points and the prediction point, expressed as:  
Figure 2:2  
Therefore, for any unknown \(\theta\), the statistical characteristics of the output response can be accurately predicted using Equation (13). The prediction result \(\Psi(\theta)\) follows a normal distribution, \(N\left(\mu_{\Psi_{K}}(\theta),\sigma_{\Psi_{K}}^{2}(\theta)\right)\), with the mean and variance calculated as:  
Figure 2:3  
\[