## 4 Reliability Analysis of Carbon-Fiber Structures Based on Surrogate Sampling and Kriging Model
### Fatigue Life Prediction Method Based on Surrogate Model
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Random Variable** & **Mean Value** & **SD** & **Random Variable** & **Mean Value** & **SD** \\ \hline \(E_{11}\) & 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 5: Distribution Parameters of Subjective Uncertainty Means. was added to the initial training sample set \(T_{SS}\). The predicted value \(\Psi_{K}(\theta_{u})\) and the true response value \(\Psi(\theta_{u})\) 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 \(V(M)\), \(V(E(M|\Theta_{i}))\), and \(V(E(M|\Theta_{\sim i}))\), 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. Figure 11: Flowchart of Cyclic Calculation. Figure 12: Model Prediction Results. The prediction accuracy and the required number of samples for three different fatigue life prediction methods are shown in Table 7. 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.