# Fatigue Characteristics Analysis of Carbon Fiber Laminates with Multiple Initial Cracks
## 5 Fatigue Reliability Modeling of Multi-Cracked Laminates
### Fatigue Life and Reliability Modeling  
#### 5.2.1 Fatigue Life Prediction Method Based on Surrogate Model  
Accurate determination of the fatigue life distribution of carbon fiber laminates is essential for conducting subsequent sensitivity analysis, which necessitates a robust dataset. However, obtaining such life data through experimental or simulation methods can be quite inefficient. To address this, a PSO-BP neural network is utilized in this section as a proxy model. The model takes the input parameters from finite element analysis and correlates them with the fatigue life of the laminates, establishing a link between uncertain factors such as material characteristics, crack spacing, length, and angles, with the fatigue life. Utilizing the Latin hypercube sampling method [27] as indicated in Table 6, 100 sets of data were simulated to determine their fatigue lives, which are listed in Table 8. From this dataset, 80 sets were allocated for training the model, while 20 sets served as the test samples. This division facilitates a direct comparison between the model's predictions and the outcomes of the simulation, thereby enabling a rigorous assessment of the PSO-BP neural network's forecasting accuracy. As a result, this method ensures the provision of reliable and trustworthy predictions \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Parameters** & **Average Value** & **CV** & **Distribution Type** \\ \hline \multirow{7}{*}{Material properties} & E\({}_{1}\)/Mpa & 105,500 & 0.05 & Normal distribution \\  & E\({}_{2}\)/Mpa & 7200 & 0.05 & Normal distribution \\  & E\({}_{3}\)/Mpa & 7200 & 0.05 & Normal distribution \\  & G\({}_{1}\)/Mpa & 3400 & 0.05 & Normal distribution \\  & G\({}_{2}\)/Mpa & 3400 & 0.05 & Normal distribution \\  & G\({}_{3}\)/Mpa & 2520 & 0.05 & Normal distribution \\  & v\({}_{1}\) & 0.34 & 0.05 & Normal distribution \\  & v\({}_{2}\) & 0.34 & 0.05 & Normal distribution \\  & v\({}_{3}\) & 0.378 & 0.05 & Normal distribution \\  & L\({}_{2}\)/mm & 6 & 0.05 & Normal distribution \\  & @/rad & 60 & 0.05 & Normal distribution \\ multi-cracked & L\({}_{4}\)/mm & 4 & 0.05 & Normal distribution \\  & S/mm & 2 & 0.05 & Normal distribution \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Uncertainty quantification for multi-cracked laminates. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & **E\({}_{1}\)** & **E\({}_{2}\)** & **E\({}_{3}\)** & **G\({}_{1}\)** & **G\({}_{2}\)** & **G\({}_{3}\)** & **v\({}_{1}\)** & **v\({}_{2}\)** & **v\({}_{3}\)** & **L\({}_{2}\)** & **L\({}_{1}\)** & **O** &Table 9 presents a comparative evaluation of the PSO-BP neural network against the standard BP neural network, highlighting the superior predictive capabilities of the PSO-BP model. The performance metrics for the PSO-BP model, namely the MAE, RMSE, and MAPE, are reported to be 449 cycles, 558 cycles, and 1.206%, respectively. In contrast, the corresponding values for the BP neural network model were 1298 cycles, 1943 cycles, and 3.2212%. The PSO-BP model shows a significant reduction in MAE and RMSE, indicating a marked improvement in predictive accuracy. Additionally, the coefficient of determination (R\({}^{2}\)) for the PSO-BP model was 0.99051, compared to 0.87818 for the BP model. An elevated R\({}^{2}\) value denotes a stronger alignment between the forecasts generated with the PSO-BP model and the empirical data, signifying a greater degree of predictive accuracy in contrast to the conventional BP model. Furthermore, as depicted in Figure 13, the vertical axis represents the predicted fatigue life N\({}_{\text{p}}\) of the cracked carbon fiber plate via the neural network models, while the horizontal axis represents the finite element simulation values N\({}_{\text{e}}\). It is evident from the graph that the PSO-BP neural network model provides a more advantageous and valid estimation, with its fatigue life predictions for multi-cracked carbon fiber laminates closely aligning with the values calculated via traditional methods.