## 3 Data analysis and BP Neural Network Model Building
### Testing Results
\begin{table}
\begin{tabular}{c|c c c c c|c} \hline \multicolumn{2}{c}{**Type**} & \multicolumn{4}{c}{\(\textit{x}_{max}\)} & \multicolumn{4}{c}{\(\textit{x}_{min}\)} & \multicolumn{1}{c}{\(\textit{y}_{max}\)} & \multicolumn{1}{c}{\(\textit{y}_{min}\)} \\ \hline Gradation-13 & 0.10 & 0.46 & 0.68 & 0.47 & & 0.01 & 0.07 & 0.01 & 1 & \(-1\) \\ Gradation-16 & 0.10 & 0.36 & 0.44 & 0.61 & 0.51 & & 0.01 & 0 & 0 & 1 & \(-1\) \\ \hline \end{tabular}
\end{table} 
Table 10: The gradation normalization parameters for Gradation-13 and Gradation-16. \begin{table}
\begin{tabular}{c|c c c c|c} \hline \multicolumn{2}{c}{**Type**} & \multicolumn{4}{c}{\(\textit{Weight}_{-1}\)} & \multicolumn{4}{c}{\(\textit{Threshold}_{-1}\)} \\ \hline \multirow{4}{*}{Gradation-13} & \(-0.4316\) & \(-0.2161\) & \(-1.2462\) & \(-1.3345\) & 2.7932 \\  & \(-0.0450\) & \(-0.1616\) & \(-0.8563\) & 0.8893 & 0.3971 \\  & 0.3822 & 0.8933 & 0.2511 & \(-1.0650\) & 0.8875 \\  & 0.0969 & 0.0735 & 0.6174 & 2.5105 & 2.6779 \\  & 0.0245 & 0.0011 & \(-0.0336\) & \(-0.7609\) & \(-2.9034\) & \(-3.4968\) \\  & \(-0.0735\) & \(-0.0087\) & 0.1996 & 1.2099 & 1.3825 & 1.6782 \\ Gradation-16 & 0.2572 & 0.7488 & 0.5424 & \(-0.4109\) & \(-0.5989\) & 0.3126 \\  & \(-0.0813\) & \(-0.2895\) & \(-0.4094\) & \(-0.1188\) & \(1.8452\) & 0.9682 \\  & \(-0.3841\) & \(-0.2343\) & \(-1.0723\) & 0.8142 & 1.3661 & \(-1.1031\) \\ \hline \end{tabular}
\end{table} 
Table 12: The parameters from the input layer to hidden layer for Gradation-13 and Gradation-16. engineering, a great number of steel balls in different sizes were prepared and mixed in the same proportion as uniformly as possible, then sieved to several different size groups. The original particle size distribution and sieved results of the steel balls are shown in Table 14 and Figure 9.  
According to Table 14 and Figure 9, every sieving group was mixed with several different particle size groups uniformly. For 9.5-13.2 mm group, 13.0 mm particles were not found, so 12.303 mm balls were added to keep a continuous and real size distribution in the steel balls. Then, in order to make the results of laboratory and DEM comparable, the size of steel ball experiments' container was exactly the same as the DEM model. According to different gradations, the steel balls were mixed in a standard Marshall mold, whose diameter was 100 mm as shown in Figure 10. Because there was no filler or binder in this mixture, universal testing machine were applied to compact the mixed steel balls with a static pressure of 600 kPa. Meanwhile, particles' total volume was measured by the displacement method. On the other hand, virtual compression tests on computer and porosity prediction on BP neural network model were completed at the same time. The detailed test data is shown in the following Figure 11. \begin{table}
\begin{tabular}{c c} \hline \hline
**Sieving Group (mm)** & **Particle Size Distribution (mm)** \\ \hline
16.0–19.0 & 17, 18, 19 \\
13.2–16.0 & 14, 15, 16 \\
9.5–13.2 & 10, 11, 12, 12.303 \\
4.75–9.5 & 5, 6, 7, 8, 9 \\
2.36–4.75 & 2.5, 3, 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Particle size distribution of the steel balls. Figure 9: Steel balls with different sizes. As can be seen from the above figure, model prediction accuracy for the steel ball test was very high and model error was extremely slight. The maximum relative error of prediction was no more than 1.15%. This proved the credibility of the prediction model. Meanwhile, the results of lab tests and virtual tests were very close, which also proved that for the sphere element-based model, sufficient credible raw data could be obtained by DEM.