## 3 Soil Parameter Discrete Element Calibration
### Discrete Element Parameter Calibration and Result Analysis
#### 3.3.4 Box-Behnken Test Design and Analysis of Results  
The range of Box-Behnken test factor values can be determined using the steepest climb test, with the level of each parameter of number 6 as high (+1), number 2 as low (\(-1\)), and number 4 as intermediate (0). Three center points were used in the test process to evaluate the error of the test results, and 15 trials were required. The test procedure and results are shown in Table 9. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Source** & **DF** & **Standardized Effect** & **MS** & **Impact Rate** & _p-Value_ & **Significance** \\ \hline Models & 8 & & 613.49 & 96.66\% & \textless{}0.001 & *** \\ X1 & 1 & 0.3722 & 3.91 & 0.08\% & 0.723 & \\ X2 & 1 & 0.6005 & 10.18 & 0.20\% & 0.57 & \\ X3 & 1 & 1.3233 & 49.44 & 0.97\% & 0.234 & \\ X4 & 1 & 5.0889 & 731.13 & 14.40\% & 0.007 & ** \\ X5 & 1 & 0.5114 & 7.38 & 0.15\% & 0.627 & \\ X6 & 1 & 8.0859 & 1845.89 & 36.36\% & \textless{}0.001 & *** \\ X7 & 1 & 8.7801 & 2176.4 & 42.87\% & \textless{}0.001 & *** \\ Bend & 1 & & 83.59 & 1.65\% & 0.136 & \\ Error & 6 & & & 28.23 & 3.34\% & \\ Total & 14 & & & 100.00\% & & \\ \hline \hline \end{tabular}  
* Statistically significant difference; *** Extremely statistically significant difference.  
\end{table} 
Table 7: Analysis of Plackett–Burman test results. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**No**} & \multicolumn{3}{c}{**Factors**} & \multicolumn{3}{c}{**Resting Angle**} & \multicolumn{3}{c}{**Error Rate (\({}^{\circ}\))**} \\ \cline{2-9}  & \(X_{4}\) & \(X_{6}\) & \(X_{7}\) & (\({}^{\circ}\)) & **3\%** & **6\%** & **9\%** & **12\%** \\ \hline
1 & 0.15 & 0.10 & 0 & 15.22 & 0.5354 & 0.5716 & 0.5984 & 0.6530 \\
2 & 0.25 & 0.15 & 0.125 & 31.72 & 0.0317 & 0.1072 & 0.1630 & 0.2769 \\
3 & 0.35 & 0.20 & 0.250 & 41.76 & 0.2748 & 0.1754 & 0.1020 & 0.0480 \\
4 & 0.45 & 0.25 & 0.375 & 39.96 & 0.2198 & 0.1247 & 0.0545 & 0.0890 \\
5 & 0.55 & 0.30 & 0.500 & 42.61 & 0.3007 & 0.1993 & 0.1244 & 0.0286 \\
6 & 0.65 & 0.35 & 0.625 & 45.26 & 0.3816 & 0.2739 & 0.1944 & 0.0318 \\
7 & 0.75 & 0.40 & 0.750 & 68.95 & 1.1047 & 0.9407 & 0.8196 & 0.5718 \\ \hline \hline \end{tabular}  
* Statistically significant difference; *** Extremely statistically significant difference.  
\end{table} 
Table 8: Steepest climbing test arrangement and results. For the Box-Behnken test, an ANOVA was conducted for four water content conditions, and the results are shown in Table 10. It can be seen that the \(p\)-values of the four water content models were \(p_{3\%}\) = 0.0092, \(p_{6\%}\) = 0.006, \(p_{9\%}\) = 0.0036, and \(p_{12\%}\) = 0.0065, all less than 0.01, indicating high significance, and the fitted R\({}^{2}\) was R\({}^{2}\)\({}_{3\%}\) = 0.9499, R\({}^{2}\)\({}_{6\%}\) = 0.9583, R\({}^{2}\)\({}_{9\%}\) = 0.9661, and R\({}^{2}\)\({}_{12\%}\) = 0.9568. The model correlation was very high, the signal-to-noise ratio measured by Adeq Precision was greater than 4, the model was less disturbed by external factors, the misfit term was not significant (lack of fit \(>\) 0.05), and the model credibility was high. For soil with 3 and 6% water content, the rolling friction coefficient \(X_{6}\) and JKR coefficient \(X_{7}\) were significant and the recovery coefficient \(X_{4}\) became insignificant, mainly due to the change in the Plackett-Burman and Box-Behnken test values, where the recovery coefficient was not a decisive parameter in changing the test results within the interval discussed for the Box-Behnken test. Examining the interaction between factors revealed that the rolling friction and JKR coefficient interaction \(X_{6}\times X_{7}\) and JKR coefficient squared \(X_{7}\) were significant; for soil with 9% water content, in addition to \(X_{6}\), \(X_{7}\), \(X_{6}\times X_{7}\), and \(X_{7}\)\({}^{2}\), the rolling friction coefficient squared \(X_{6}\)\({}^{2}\) was significant; for soil with 12% water content only \(X_{6}\times X_{7}\) and \(X_{7}\)\({}^{2}\) were significant.