## 4 Algorithm for Developing Virtual Specimens  
### Mapping Area Calculation for Two-Dimensional Specimens  
When it comes to the virtual aggregate specimens, several assumptions were made before the simulations as follows:  
* 1 The differences of the densities among particles were not obvious, so the same density value of \(\rho_{1}\) can be assigned to all the virtual particles in the simulations;
* 2 The virtual specimens consist of particles and voids only without any water.  
The virtual specimen of SMA-13 in the penetration test was then developed, and the gradations of SMA-13 are shown in Table 3. To improve the simulation efficiency, only the coarse aggregates were developed despite the fine particles.  
The particle numbers of different grades in the two-dimensional simulation were determined based on the mapping area, as the following equations shown:  
\[M=\rho\times\frac{\pi\times D^{2}}{4}\times h \tag{8}\]  
where \(M\) is the total mass of the specimen in three dimensions, g; \(\rho\) is the density of the specimen, g/cm\({}^{3}\); \(D\) is the diameter of the specimen, cm; \(h\) is the height of the specimen, cm.  
When it comes to the two-dimensional specimen, the mass can be converted as follows:  
\[m=\frac{M}{V}\times S=\frac{V\times\rho}{V}\times D\times h=Dh\rho \tag{9}\]  
where \(m\) is the converted mass of the specimen in two dimensions, g; \(V\) is the total volume of the specimen in three dimensions, cm\({}^{3}\); \(S\) is the area of the specimen in two dimensions, cm\({}^{2}\); the others are the same as the former.  
Thus, the required area for generating the total virtual particles could be calculated based on Equation (10).  
\[S_{total}=\frac{m}{\rho_{1}}=\frac{Dh\rho}{\rho_{1}} \tag{10}\]  
Figure 6: Various virtual particles of different morphological characteristics.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Gradation** & \multicolumn{8}{c}{**Passing Ratio (\%) for Different Sieving Sizes (mm)**} \\ \hline  & 16 & 13.2 & 9.5 & 4.75 & 2.36 & 1.18 & 0.6 & 0.3 & 0.15 & 0.075 \\ SMA-13 & 100 & 95 & 79.5 & 48.9 & 36.7 & 25.8 & 16.7 & 9.7 & 7.1 & 6.5 \\ \hline \end{tabular}
\end{table}
Table 3: Gradation for SMA-13.  
where \(S_{total}\) is the required area for generating the total virtual particles, cm\({}^{2}\); \(\rho_{1}\) is the density of all the coarse aggregates, which was assumed to be the same value in the simulations, g/cm\({}^{3}\); the others are the same as the former.  
Based on the assumption that there is no water within the specimen, the following relation can be given:  
\[\rho=\frac{\rho_{1}\times V\times(1-K)}{V}=\rho_{1}\times(1-K) \tag{11}\]  
where \(K\) is the void content of the total specimen in three dimensions, %; the others are the same as the former.  
The mapping area of different grades of particles can be developed in the end as follows:  
\[\begin{split} S_{total}=& Dh\ \times(1-K)\\ S_{i}=S_{total}\times P_{i}=& Dh\times(1-K) \times P_{i}\ \ \ i=4.75,\ 9.5,\ 13.2,\ 16\end{split} \tag{12}\]  
where \(S_{4.75}\), \(S_{9.5}\), \(S_{13.2}\), \(S_{16}\) are the mapping areas of the 4.75-, 9.5-, 13.2-, 16-mm virtual particles, respectively, cm\({}^{2}\); \(P_{4.75}\), \(P_{9.5}\), \(P_{13.2}\), \(P_{16}\) are the proportions of the 4.75-, 9.5-, 13.2-, 16-mm virtual particles to the total mass, respectively; the others are the same as the former.