## 3 Simulation of BBT
### Algorithm and Material Parameters  
The Lagrange algorithm is the most widely used method in LS-DYNA [32; 33; 34; 35]. The material of the Lagrange algorithm is attached to the grid, and the flow of each particle is tracked, so the grid and the material deform together. Therefore, it is easy to determine the time history in the Lagrange algorithm, and it is also easy to determine the material and structure interface and the internal stress-strain state. Therefore, the dynamic behavior of solid materials can be well simulated. In the Euler algorithm, the computational grid is fixed in space and cannot move with the object, but the material moves relative to the grid, so the grid will not be distorted. Therefore, the Euler algorithm has an advantage in dealing with large deformation problems. The arbitrary Lagrange-Eulerian (ALE) algorithm takes advantage of the Lagrange algorithms and Euler. Explosives and air adopt the ALE algorithm, which can avoid the abnormal solution caused by the excessive deformation of explosives and air elements in the calculation process. The energy-gathering device uses the Lagrange algorithm. The explosive material model uses MAT_HIGH_EXPLO_SIVE_BURN. The JWL equation of state is used to describe the pressure-volume relationship of the detonation product after the explosive is detonated in the model, as Equation (11) [36]:  
\[p=A\bigg{(}1-\frac{\omega}{VR_{1}}\bigg{)}e^{-R_{1}V}+B\bigg{(}1-\frac{\omega }{VR_{2}}\bigg{)}e^{-R_{2}V}+\frac{\omega E_{0}}{V} \tag{11}\]  
where: \(p\) is the pressure of the detonation product, \(A\), \(B\), \(R_{1}\), \(R_{2}\) and \(\omega\) are the material constants determined by experiments, \(V\) is the relative volume of the detonation product, and \(E_{0}\) is the initial internal energy density of the detonation product. The explosive parameters and the parameters of the JWL equation of state are shown in Table 1[37]. In the engineering site, the energy-gathering device is made of PVC material. Under the action of detonation pressure, the dynamic mechanical response of the detonation pressure is very complicated. In order to simplify the calculation, the energy-gathering device is calculated using MAT_STEINBERG material and Gruneisen equation of state, which defines the pressure of the compressed material as Equation (12), and the parameters of the equation of state are shown in Table 2[38]. \[p=\frac{\rho_{0}C^{2}\mu\big{[}1+\big{(}1-\frac{20}{2}\big{)}\mu-\frac{a}{2}\mu ^{2}\big{]}}{\bigg{[}1-(S_{1}-1)\mu-S_{2}\frac{\mu^{2}}{\mu+1}-S_{3}\frac{\mu^ {3}}{(\mu+1)^{2}}\bigg{]}}+(\gamma_{0}+a\mu)E \tag{12}\]  
For expanded materials as, \(p=p_{0}C^{2}\mu+(\gamma_{0}+a\mu)E\). Where \(C\) is the curve intercept, \(S_{1}\), \(S_{2}\), and \(S_{3}\) are the slope coefficients; \(\gamma_{0}\) is the unitless Gruneisen coefficient; \(a\) is the correction coefficient; \(E\) is the initial internal energy; \(\mu\) is the volume parameter; \(\rho_{0}\) is the material density. Since the energy-gathering device does not fit the wall of the blasthole completely, an air field is required. Regarding air as an ideal gas, use MAT_NULL material and \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameter** & \(A\)**/GPa** & \(B\)**/GPa** & \(R_{1}\) & \(R_{2}\) & \(\omega\) & \(V\) & \(E_{0}\)**/GPa** \\ \hline Value & 326 & 5.81 & 5.81 & 1.56 & 0.57 & 1 & 2.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the JWL equation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & \(C\)**/(cm/\(\mu\)s)** & \(S_{1}\) & \(S_{2}\) & \(S_{3}\) & \(\gamma_{0}\) & \(a\) & \(E\) \\ \hline Value & 0.387 & 1.82 & \(-\)0.075 & \(-\)0.043 & 1.58 & 0.41 & 2.67 \(\times\) 10\({}^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Energy-gathering device parameters. LINEAR_POLYNOMIAL state equation to express as shown Equation (13).