# Fracture Threshold Analysis and Parameter Matching of Cut-Out Induced Bolts for Subway Couplers
## 3 Effect of Cut-Out Zone Structural Parameters
### The Establishment of the Surrogate Model  
Four methods, Least Squares Regression (LSR), Moving Least Squares Method (MLSM), HyperKriging (HK), and Radial Basis Function (RBF), were used to fit the fracture force with respect to the two independent variables (groove depth and groove width). The LSR method minimizes the sum of the squares of the differences between the fitted values and the original values, with its objective shown in Equation (8). The MLSM is based on the same principle as the Least Squares method, but the weighting coefficients for the sampling points are functions of the distance between the sample points and the evaluation points. The HK method generates an interpolation model based on the existing sampling points. The RBF method uses the linear superposition of several basis functions (including linear functions, cubic functions, thin-plate spline functions, Gaussian functions, etc.) for fitting. Both the RBF method and the HK method typically pass through the original data more accurately. \[\text{min}E\ =\ \sum_{i\,=\,1}^{n}\left(f_{i}^{predicted}-f_{i}\right)^{2} \tag{8}\]  
In the equation, \(n\) represents the number of calculations, \(f_{i}\) represents the original value, and \(f_{i}^{predicted}\) represents the fitted value. To characterize the prediction accuracy of the fitted model, the coefficient of determination (R\({}^{2}\)) and the relative average absolute error (RAAE) are commonly used, with their calculation formulas shown in Equations (9) and (10):  
Figure 19: The area–time and force–time curve of bolts breaking after the expansion tube operation (the 57 th set of bolt conditions in the DOE). \[R^{2} = 1-\frac{\sum\limits_{i=1}^{n}\left[y_{i}-\hat{y}_{i}\right]^{2}}{ \sum\limits_{i=1}^{n}\left[y_{i}-\overline{y}\right]^{2}} \tag{9}\] \[RAAE = \frac{\frac{\frac{1}{n}\sum\limits_{i=1}^{n}\left[abs(y_{i}-\hat{ y}_{i})\right]}{\sqrt{\left(\overline{n}\sum\limits_{i=1}^{n}\left[y_{i}- \overline{y}\right]^{2}\right)}}} \tag{10}\]  
where \(y_{i}\) represents the actual data value, \(\hat{y}_{i}\) represents the model prediction value, \(\overline{y}\) represents the average of the actual data and \(n\) represents the number of data points. The closer \(R^{2}\) is to 1 and the lower RAAE is, the better the fitted model fits the original data. As shown in Table 2, the MLSM has the highest coefficient of determination and the lowest relative average absolute error in fitting the fracture force, both of which are superior to the other three methods. Therefore, the MLSM is selected for fitting the fracture force. After the selection of the fitting method, the response diagrams between the depth and width of the cut-out zone and the fracture time and fracture force were drawn separately. In Figure 20, the response surface between the depth and width of the cut-out zone and the fracture force after fitting is shown. As evident from Figure 20, the fracture force of the cut-out induced bolts exhibits significant variation within specific regions, with values predominantly clustering at two distinct force levels: 1200 kN and 1800 kN. The depth of the cut-out zone emerges as the primary determinant influencing the dynamic mechanical performance of the cut-out induced bolts, as evidenced by the fracture force response surface showing pronounced variations along the depth dimension. In contrast, the width of the cut-out zone demonstrates comparatively minor influence on the dynamic mechanical properties, with the fracture force response surface displaying only marginal changes along the width dimension. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & \(\mathbf{R^{2}}\) & **RAAE** \\ \hline LSR & 0.8577 & 0.2411 \\ MLSM & 0.9849 & 0.0727 \\ HK & 0.9036 & 0.1235 \\ RBF & 0.9753 & 0.0840 \\ LSR & 0.8577 & 0.2411 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Certainty coefficients of different fitting methods. Figure 20: The response surface of fracture force.