## 4 Reliability Analysis of Carbon-Fiber Structures Based on Surrogate Sampling and Kriging Model
### 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 \(\left[g(x_{1}),g(x_{2}),\cdots,g(x_{N_{X}})\right]\). 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 \(\theta_{j}(j=1,2,\cdots,N_{T})\), the corresponding output statistical characteristic values \(m_{j}(j=1,2,\cdots,N_{T})\) were calculated based on \(g(x_{t})\) and the samples \(f_{X}\left(x_{t}|\theta_{j}\right)/h_{X}\left(x_{t}|\theta^{*}\right)\) 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 \(\Psi_{K}(\Theta)\) was established, and the U-learning function values for the remaining points were calculated to update the sample points. By using \(\theta_{u}=\arg\min_{\theta\in S_{SS}}U(\theta)\), the \(\theta_{u}\) with the smallest \(U(\theta)\) \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Random Variable** & **Mean Value** & **SD** & **Random Variable** & **Mean Value** & **SD** \\ \hline \(E_{11}\) (GPa) & 105.5 & 6.33 & \(X_{C}\) (MPa) & 657 & 39.42 \\ \(E_{22}\) (GPa) & 7.2 & 0.432 & \(Y_{T}\) (MPa) & 694 & 41.64 \\ \(E_{33}\) (GPa) & 7.2 & 0.432 & \(Y_{C}\) (MPa) & 651 & 39.06 \\ \(v_{12}\) & 0.34 & 0.0204 & \(Z_{T}\) (MPa) & 75 & 4.5 \\ \(v_{13}\) & 0.34 & 0.0204 & \(Z_{C}\) (MPa) & 165 & 9.9 \\ \(v_{23}\) & 0.378 & 0.02268 & \(S_{12}\) (MPa) & 108 & 6.48 \\ \(G_{12}\) (GPa) & 3.4 & 0.204 & \(S_{13}\) (MPa) & 81.1 & 4.866 \\ \(G_{13}\) (GPa) & 3.4 & 0.204 & \(S_{23}\) (MPa) & 81.1 & 4.866 \\ \(G_{23}\) (GPa) & 2.520 & 0.1512 & \(L\) (mm) & 5 & 0.3 \\ \(X_{T}\) (MPa) & 1060 & 63.6 & \(P\) (KN) & 9.5 & 0.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Surrogate Sampling Probability Density Function of Input Variables. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Random Variable** & **Mean Value** & **SD** & **Random Variable** & **Mean Value** & **SD** \\ \hline \(E_{11}\) (GPa) & \(\mu_{E_{11}}\) & 5.275 & \(X_{C}\) (MPa) & \(\mu_{X_{C}}\) & 32.85 \\ \(E_{22}\) (GPa) & \(\mu_{E_{22}}\) & 0.36 & \(Y_{T}\) (MPa) & \(\mu_{Y_{T}}\) & 34.7 \\ \(E_{33}\) (GPa) & \(\mu_{E_{33}}\) & 0.36 & \(Y_{C}\) (MPa) & \(\mu_{Y_{C}}\) & 32.55 \\ \(v_{12}\) & \(\mu_{v_{12}}\) & 0.017 & \(Z_{T}\) (MPa) & \(\mu_{Z_{T}}\) & 3.75 \\ \(v_{13}\) & \(\mu_{v_{13}}\) & 0.017 & \(Z_{C}\) (MPa) & \(\mu_{Z_{C}}\) & 8.25 \\ \(v_{23}\) & \(\mu_{v_{23}}\) & 0.0189 & \(S_{12}\) (MPa) & \(\mu_{S_{12}}\) & 5.4 \\ \(G_{12}\) (GPa) & \(\mu_{G_{12}}\) & 0.17 & \(S_{13}\) (MPa) & \(\mu_{S_{13}}\) & 4.055 \\ \(G_{13}\) (GPa) & \(\mu_{G_{13}}\) & 0.17 & \(S_{23}\) (MPa) & \(\mu_{S_{23}}\) & 4.055 \\ \(G_{23}\) (GPa) & \(\mu_{G_{23}}\) & 0.126 & \(L\) (mm) & \(\mu_{L}\) & 0.25 \\ \(X_{T}\) (MPa) & \(\mu_{X_{T}}\) & 53 & \(P\) (KN) & \(\mu_{P}\) & 0.475 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Distribution Parameters of Input Variables.