# A Novel Method of Calibrating Micro-Scale Parameters of PFC Model and Experimental Validation
## 4 Experimental Design and Analysis  
### Experimental Scheme and Results  
As stated above, three influencing factors and three response variables were considered in this study. Hence, a three-level-three-factor model was established using Box-Behnken design. According to the laboratory test results, the defined levels of each factor are given in Table 2. To make statistical calculation simple, dimensionless processing was conducted on these factors according to  
\[X_{i}=\left(x_{i}-x_{0}\right)/\Delta x_{i} \tag{10}\]  
where \(X_{i}\) and \(x_{i}\) (\(i=1\), \(2\), and \(3\)) denote the dimensionless coded value and actual value of influencing factor, respectively, \(x_{0}\) is the actual value of the factor at the center point, and \(\Delta x_{i}\) means the step change value of the influencing factor.  
According to the Box-Behnken design scheme, a total of 17 groups of uniaxial compression simulations were carried out by the PFC software. Table 3 shows the simulation results. Accordingly, different models, including the linear, 2 FI (Two-Factor Interaction), quadratic, and cubic models were employed to fit the second-order response surface equation. Lack-of-fit test results indicate that the  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Influencing Factors**} & \multirow{2}{*}{**Codes**} & \multicolumn{3}{c}{**Level**} & \multirow{2}{*}{\(\Delta x_{i}\)} \\ \cline{3-3} \cline{5-6}  & & **Low (-1)** & & **Center (0)** & **High (1)** \\ \hline \(A\)/MPa & \(X_{1}\) & 50 & 100 & 150 & 50 \\ \(B\)/GPa & \(X_{2}\) & 20 & 40 & 60 & 20 \\ C & \(X_{3}\) & 1 & 2 & 3 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Ranges of influencing factors.  
Figure 2: Technical route of determination of micro-scale parameters using Box-Behnken response surface methodology and desirability function optimization (BB-DFO) method.  
quadratic model fits well the nonlinear relationship between the independent and dependent variables (see Table 4). It is noted that the cubic model and higher are aliased. The fitted equations of the response variables with respect to the actual factors are given as  
\[\dot{\vec{y}}_{1}=-6.95+1.00A+0.46B+9.63C+0.07 AC+0.15 BC-0.01B^{2}-4.42C^{2}\quad (R^{2}=0.9995) \tag{11}\]  
\[\dot{\vec{y}}_{2}=10.76+1.64B-11.46C-0.23 BC+2.78C^{2}\quad(R^{2}=0.9994) \tag{12}\]  
\[\dot{\vec{y}}_{3}=0.04+0.14C-0.02C^{2}\quad(R^{2}=0.9998) \tag{13}\]  
Equations (11)-(13) show that all the correlation coefficients are greater than 0.99, suggesting that the fitted response surface functions are reliable. Also, the predicted values of the response variables are found to be extremely agreeable with the simulated results (Figure 3). The average prediction errors of the uniaxial compressive strength, [PERSON]'s modulus, and [PERSON]'s ratio reach 0.70%, 0.85%, and 0.21%, respectively.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**No.** & **A/MPa** & **B/GPa** & **C** & **y\({}_{1}\)/MPa** & **y\({}_{2}\)/GPa** & **y\({}_{3}\)** \\ \hline  
\hline \end{tabular}
\end{table}
Table 3: Box–Behnken design scheme and results.  
Figure 3: Comparison of predicted and actual values of response variables.