## 1 Introduction
### Sensitivity Analysis of Cracked Carbon-Fiber Structures Using ALK Model and Surrogate Sampling
\begin{array}{rl}\sigma_{\Psi_{K}}^{2}(\theta)=&\sigma^{2}\Big{\{}1-r^{T}( \theta)R^{-1}r(\theta)+\left[F^{T}R^{-1}r(\theta)-f(\theta)\right]^{T}.\\ &\left(F^{T}R^{-1}F\right)^{-1}\left[F^{T}R^{-1}r(\theta)-f(\theta)\right] \Big{\}}\end{array}  \tag{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]:  
\[U(\theta)=\frac{\left\lvert\Psi_{K}(\theta)\right\rvert}{\sigma_{\Psi_{K}}( \theta)} \tag{17}\]  
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. \[\left|\frac{\Psi_{K}(\theta_{u})-\Psi(\theta_{u})}{\Psi(\theta_{u})}\right|\geq 2 \tag{18}\]  
where \(\Psi_{K}(\theta_{u})\) is the predicted value of the surrogate model at the sample point \(\theta_{u}\) with the smallest \(U\) value in the sample pool, and \(\Psi(\theta_{u})\) 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 [PERSON] [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 \(h_{X}(x|\theta^{*})\) 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:  
\[ \begin{array}{ll}M=&E_{X}(Y|\Theta)=\int\limits_{R^{n}}g(x)f_{X}(x|\theta)dx =\\ &\int\limits_{R^{n}}g(x)\frac{f_{X}(x|\theta)}{h_{X}(x|\theta^{*})}h_{X}(x| \theta^{*})dx=\\ &E_{X^{*}}\bigg{(}g(x)\frac{f_{X}(x|\theta)}{h_{X}(x|\theta^{*})}\bigg{)} \end{array}  \tag{19}\]  
When \(M\) is the variance in the model output response, according to \(V(Y)=E\big{(}Y^{2}\big{)}-E^{2}(Y)\). Its value can be calculated using the following formula:  
\[