# Fracture Threshold Analysis and Parameter Matching of Cut-Out Induced Bolts for Subway Couplers
## 4 Fracture Threshold Optimization
### Result Validation  
#### 4.3.1 Simulation Validation  
In simulation validation, we reconstructed the cut-out induced bolts model based on the optimization results, ensuring that the rest of the model and boundary conditions remained unchanged, and then rebuilt the simulation model. Through computational simulation, we obtained the simulation results of the optimized bolts. Figure 24 shows the  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Maximum Iterations** & **Minimum Iterations** & **Population Size** & **Mutation Rate** & **Crossover Rate** \\ \hline
50 & 25 & 60 & 0.01 & 0.5 \\ \hline \end{tabular}
\end{table}
Table 3: GA parameters.  
Figure 23: The optimal fracture force of each population in the genetic algorithm.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Maximum Iterations** & **Minimum Iterations** & **Population Size** & **Mutation Rate** & **Crossover Rate** \\ \hline
50 & 25 & 60 & 0.01 & 0.5 \\ \hline \end{tabular}
\end{table}
Table 3: GA parameters.  
timing diagram of the coupler collision obtained from the simulation model calculation. The force-time curve obtained from the simulation calculation is also shown in Figure 25. The collision was considered to start when the coupler head was fully in contact, at which point T = 0 ms. When T = 10 ms, the buffer stroke was fully consumed, and the expansion tube was about to enter the stable crushing stage; when T = 49 ms, according to the force-time curve, it can be observed that the expansion tube was fully consumed at this point, and the coupler was about to enter the rigid phase, with the contact force beginning to rise; when T = 53.15 ms, based on the element failure situation reported in the simulation calculation, it can be judged that the bolts were pulled apart at this time; at T = 54 ms, the bolts can be observed flying out from the timing diagram.  
The stress variation of each bolt is shown in Figure 26. When T = 0 ms, the coupler head was fully coupled, and due to the presence of preload, the maximum stress of the four bolts was concentrated in the cut-out zone, within the range of 516.84-519.25 MPa. When T = 10 ms, the expansion tube was about to enter the stable crushing stage, and due to the influence of inertial effects, the bolt stress experienced a short rise and slight deformation, with the maximum stress of the four bolts ranging from 1002.92 to 1096.05 MPa. When T = 30 ms, the expansion tube entered the stable crushing stage, with a relatively stable  
Figure 24: Simulation calculation time sequence diagram.  
Figure 25: The force–time curve obtained by simulation calculation.  
force value. The maximum stress was slightly lower than at T = 10 ms and remained stable. During this stage, the bolts did not undergo further deformation, with the maximum stress of the four bolts ranging from 807.65 to 877.45 MPa. When T = 53 ms, the expansion tube was fully consumed, and the entire coupler entered the rigid phase, with the force value rising. The bolt experienced a sudden increase in force, further deformation, and was about to break, with the maximum stress of the bolts ranging from 1050.79 to 1144.54 MPa. When T = 55 ms, the bolts broke, and the bolts stress was unloaded.  
We compared the fracture force obtained from the simulation model calculation with the predicted values from the fitting model. The surrogate model predicted the fracture force for the bolt group to be 1894.13 kN, while the simulation model calculated the fracture force for the bolt group to be 1855.59 kN, a difference of approximately 2.08%. The error was less than 5%, which proves the effectiveness of the fitting model from another perspective.