## 1 Introduction
### 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:  
\[Y=g(X) \tag{1}\]  
where \(Y\) represents the output response of the structure or system, and \(X=[X_{1},X_{2},\cdots,X_{n}]\) 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:  
\[Y=g(X,\Theta) \tag{2}\]  
where the subjective variables \(\Theta=[\Theta_{1},\Theta_{2},\cdots,\Theta_{n}]\) can be described by the probability density function \(f_{\Theta}(\theta)\). 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 \(f_{X}(x|\theta)\), 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:  
\[\Theta\xrightarrow{f_{X}(x|\theta)}X\xrightarrow{Y=g(X)}[y_{1},y_{2},\cdots,y _{N}]\rightarrow\left\{ \begin{array}{l}E(Y)\\ V(Y)\end{array} \right. \tag{3}\]  
where \(E(Y)\) and \(V(Y)\) 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  
Figure 2: Schematic Diagram of Distribution Parameter Uncertainty Transfer. implicit function relationship can be established between the distribution parameters and the output statistical characteristics. \[M=\Psi(\Theta) \tag{4}\]  
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:  
\[S_{i}=\frac{V(E(M|\Theta_{i}))}{V(M)} \tag{5}\]  
\[S_{Ti}=\frac{E(V(M|\Theta_{\sim i}))}{V(M)}=1-\frac{V(E(M|\Theta_{\sim i}))}{V (M)} \tag{6}\]  
where \(\Theta_{\sim i}\) 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 \(\Theta_{\sim i}\) 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 \(\Theta_{\sim i}\), given each value of the distribution parameters obtained from the outer layer sampling.