## 2 Discrete Element Parameter Model and Parameter Calibration of Soil-Soil Interaction
### Simulation Test
#### 2.5.3 Box-Behnken Design Test  
Based on the results of the steepest ascent experiment, a Box-Behnken experiment was conducted using Design-Expert to investigate the three significant parameters. The experiments were designed with three factors and three levels within the selected range of the steepest ascent results. The experimental results are shown in Table 7.  
The experimental results were analyzed using the software Design-Expert, yielding a quadratic regression model as depicted in Table 8. From the table, it is evident that the surface energy (G) and the soil-steel static friction coefficient (E) have a highly significant impact on the soil heap angle (\(p<0.01\)). The regression model itself has a significance level of \(p<0.01\), indicating an extremely significant relationship between the heap angle and the regression equation obtained. Furthermore, the lack-of-fit term has a \(p\)-value of 0.1287, which is greater than 0.05, suggesting that the proportion of abnormal errors in the actual fit of the regression equation is small, indicating a good fit. The coefficient of variation (CV) for this experiment is 0.81%, indicating good reli \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{3}{c}{**Test Factors**} & \multicolumn{2}{c}{**Angle of**} & \multirow{2}{*}{**Relative Error/\%**} \\ \cline{2-2} \cline{4-5}  & **G** & **A** & **E** & & **Accumulation (\({}^{\circ}\))** & \\ \hline
1 & 3.50 & 0.35 & 0.50 & 34.38 & 3.29 \\
2 & 5.80 & 0.5 & 0.60 & 36.10 & 1.55 \\
3 & 8.20 & 0.65 & 0.70 & 37.25 & 4.78 \\
4 & 10.5 & 0.80 & 0.80 & 39.67 & 11.59 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Scheme and results of steepest ascent test. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{3}{c}{**Test Parameter**} & \multirow{2}{*}{**Repose Angle**} \\ \cline{2-2} \cline{4-5}  & **G** & **A** & **E** & **0/(\({}^{\circ}\))** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 7: Scheme and results of Box–Behnken Design. coefficient \(\text{R}^{2}_{\text{pre}}\) is 0.8419, the coefficient of determination \(\text{R}^{2}\) is 0.9369, and the adjusted coefficient of determination \(\text{R}^{2}_{\text{adj}}\) is 0.8558. This suggests that 85.58% of the variation in response values comes from the selected factors, with a difference of less than 0.2 between the predictive and determination coefficients. The adequacy precision (Adeq Precision) is 11.905, typically desired to be greater than 4, indicating good precision of the regression model. The regression equation is as follows. \[ \begin{split}&Angle=\\ & 35.29+0.76G+0.2875A+0.3975E\\ &-0.2175 Ga+0.3125 GE+0.0575 AE\\ &+0.4833G^{2}+0.2533A^{2}+0.1433E^{2}\end{split}  \tag{6}\]  
By plotting perturbation diagrams based on the existing quadratic polynomial model, as shown in Figure 5, it can be observed that, with the increase in levels of B:A and C:E, there is a gradual increase in the heap angle. The relationship between A:G and the heap angle exhibits a trend of slow increase followed by rapid increase. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Source of Variance**} & \multicolumn{2}{c}{**Mean Sum of**} & **Degree of** & **Mean** & \multirow{2}{*}{**\(p\)-Value**} \\  & **Squares** & **Freedom** & **Square** & \\ \hline Model & 8.59 & 9 & 0.9549 & 0.0020 \\ G & 4.62 & 1 & 4.62 & 0.0001 \\ A & 0.6612 & 1 & 0.6612 & 0.0255 \\ E & 1.26 & 1 & 1.26 & 0.0058 \\ GA & 0.1892 & 1 & 0.1892 & 0.1740 \\ GE & 0.3906 & 1 & 0.3906 & 0.0662 \\ AE & 0.0132 & 1 & 0.0132 & 0.7011 \\ G\({}^{2}\) & 0.9833 & 1 & 0.9833 & 0.0107 \\ A\({}^{2}\) & 0.2700 & 1 & 0.2700 & 0.1136 \\ E\({}^{2}\) & 0.0864 & 1 & 0.0864 & 0.3406 \\ Residual & 0.5786 & 7 & 0.0827 & \\ Lack of Fit & 0.4190 & 3 & 0.1397 & 0.1287 \\ Pure Error & 0.1595 & 4 & 0.0399 & \\ Cor Total & 9.17 & 16 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: ANOVA of Box–Behnken Designer quadratic model. Figure 5: Perturbation diagram.