## 3 Soil Parameter Discrete Element Calibration
### Discrete Element Parameter Calibration and Result Analysis
#### 3.3.4 Box-Behnken Test Design and Analysis of Results
Response surface analysis plots were made for \(X_{6}\) and \(X_{7}\), as shown in Figure 8a-d. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Source**} & \multicolumn{2}{c}{**3\% Moisture Content**} & \multicolumn{2}{c}{**6\% Moisture Content**} & \multicolumn{2}{c}{**9\% Moisture Content**} & \multicolumn{2}{c}{**12\% Moisture Content**} \\ \cline{2-9}  & \(p\)**-Value** & **Significance** & \(p\)**-Value** & **Significance** & \(p\)**-Value** & **Significance** & \(p\)**-Value** & **Significance** \\ \hline Models & 0.0092 & ** & 0.006 & ** & 0.0036 & ** & 0.0065 & ** \\ \(X_{4}\) & 0.4593 & & 0.4435 & & 0.3511 & & 0.3246 & \\ \(X_{6}\) & 0.0027 & ** & 0.0027 & ** & 0.0047 & ** & 0.6462 & \\ \(X_{7}\) & 0.0012 & ** & 0.0014 & ** & 0.0019 & ** & 0.2138 & \\ \(X_{4}\times X_{6}\) & 0.7215 & & 0.4642 & & 0.3723 & & 0.3452 & \\ \(X_{4}\times X_{7}\) & 0.6353 & & 0.62 & & 0.5421 & & 0.5182 & \\ \(X_{6}\times X_{7}\) & 0.0385 & ** & 0.0075 & ** & 0.0015 & ** & 0.0003 & *** \\ \(X_{4}\)\({}^{2}\) & 0.4837 & & 0.9241 & & 0.2809 & & 0.1086 & \\ \(X_{6}\)\({}^{2}\) & 0.2402 & & 0.1431 & & 0.0489 & ** & 0.1895 & \\ \(X_{7}\)\({}^{2}\) & 0.0281 & ** & 0.0061 & ** & 0.0018 & ** & 0.0034 & ** \\ Lack of Fit & 0.0728 & Not significant & 0.1107 & Not significant & 0.1672 & Not significant & 0.1882 & Not significant \\ \hline R\({}^{2}\) & 0.9499 & & 0.9583 & & 0.9661 & & 0.9568 \\ Adeq Precision & & 11.8888 & & 13.2918 & & 14.0464 & & 11.8414 \\ \hline \hline \end{tabular}  
**Statistically significant difference; *** Extremely statistically significant difference.  
\end{table} 
Table 10: Analysis of variance of Box-Behnken experimental regression model under different water content conditions. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{2}{c}{**Factors**} & \multicolumn{2}{c}{**Rest Angle**} & \multicolumn{2}{c}{**Error Rate (\({}^{\circ}\))**} \\ \cline{2-9}  & \(X_{4}\) & \(X_{6}\) & \(X_{7}\) & (\({}^{\circ}\)) & **3\%** & **6\%** & **9\%** & **12\%** \\ \hline
1 & 0(0.45) & \(-1\) (0.15) & 1 (0.625) & 42.65 & 0.3017 & 0.2003 & 0.1255 & 0.0279 \\
2 & \(-1\) (0.25) & 1 (0.35) & 0 (0.375) & 51.36 & 0.5678 & 0.4456 & 0.3556 & 0.1708 \\
3 & 0 & 1 & 1 & 68.57 & 1.0932 & 0.9300 & 0.8098 & 0.5631 \\
4 & 0 & 0 (0.25) & 0 & 39.01 & 0.1907 & 0.0979 & 0.0295 & 0.1108 \\
5 & 0 & \(-1\) & \(-1\) (0.125) & 24.28 & 0.2590 & 0.3168 & 0.3593 & 0.4467 \\
6 & 0 & 1 & \(-1\) & 45.37 & 0.3849 & 0.2769 & 0.1974 & 0.0342 \\
7 & 1 (0.65) & \(-1\) & 0 & 33.50 & 0.0227 & 0.0570 & 0.1157 & 0.2363 \\
8 & 1 & 1 & 0 & 47.39 & 0.4465 & 0.3337 & 0.2506 & 0.0802 \\
9 & \(-1\) & 0 & \(-1\) & 32.50 & 0.0080 & 0.0854 & 0.1423 & 0.2592 \\
10 & 1 & 0 & \(-1\) & 33.06 & 0.0093 & 0.0694 & 0.1274 & 0.2463 \\
11 & 0 & 0 & 0 & 41.67 & 0.2720 & 0.1728 & 0.0997 & 0.0502 \\
12 & 1 & 0 & 1 & 55.43 & 0.6919 & 0.5600 & 0.4628 & 0.2634 \\
13 & \(-1\) & 0 & 1 & 59.33 & 0.8110 & 0.6698 & 0.5658 & 0.3523 \\
14 & \(-1\) & \(-1\) & 0 & 34.53 & 0.0540 & 0.0282 & 0.0887 & 0.2129 \\ \hline \hline \end{tabular}  
\end{table} 
Table 9: Box-Behnken experimental design and results. Excluding the insignificant factors, the quadratic polynomial equation between significant factors and rest angle error rate under different water content conditions was obtained as:  
\[Y_{3\%}=0.237+0.2319X_{6}+0.2796X_{7}+0.1664X_{6}X_{7}+0.19X_{7}{}^{2} \tag{10}\]  
\[Y_{6\%}=0.1405+0.173X_{6}+0.201X_{7}+0.1924X_{6}X_{7}+0.2102X_{7}{}^{2} \tag{11}\]  
\[Y_{9\%}=0.0695+0.1155X_{6}+0.1422X_{7}+0.2116X_{6}X_{7}+0.0908{}K_{6}{}^{2}+0. 2127X_{7}{}^{2} \tag{12}\]  
\[Y_{12\%}=0.0763+0.2369X_{6}X_{7}+0.1485X_{7}{}^{2} \tag{13}\]  
Combining Equations (10)-(13), the limiting equation that can be solved was obtained as Equation (14):  
\[\left\{ \begin{array}{l}\min Y_{3\%},Y_{6\%},Y_{9\%},Y_{12\%}\\ s.t.\left\{\begin{array}{l}X_{4}=0.45\\ -1\leq X_{6}\leq 1\\ -1\leq X_{7}\leq 1\end{array}\right.\end{array} \right.