## 5 Numerical Simulation
### Shock Chamber Model  
The numerical model utilized a mix of Eulerian and Lagrangian methods. The explosives' stemming material and air were represented in the mesh, while the steel cylinder was depicted in the Lagrange mesh. The combined Eulerian algorithm enabled interaction between the blasting shock wave and the shock chamber. The Lagrangian element representing the shock chamber was linked within the part (air) to maintain an explosive gas flow a ratio of 4:1 between the Lagrangian and Eulerian grids. Additionally a friction coefficient of 0.6 was set between the sand stemming material near the orifice and the shock chamber wall in the blast hole.  
In this research, explicit hydrodynamic simulation tools like AUTODYN played a role with the use of a coupled Euler-Lagrange approach. This method leverages the ability of an interface to intersect with a fixed Eulerian grid in any direction. The interactions between Eulerian cells and the Lagrangian interface assist in establishing a pressure distribution  
Figure 13: Images from the 5000 fps high-speed camera used for the analysis. Red: Only sand, green: STF + sand, dark blue: STF + spike plug + sand, and light blue: STF + twist plug + sand.  
along the boundaries of elements, as shown in Figure 14. The design of the shock chamber model involved creating a three 3D structure within a Lagrangian framework comprising 80,000 elements, as detailed in Figure 15.  
In studying the materials inside the blast hole, such as the explosive substances, we used the method with 360,000 elements. Additionally, the 3D analysis used a mix of Euler-Lagrange techniques. This method involved using a mesh resolution of 2 mm by 2 mm by 10 mm for elements representing the conditions around the explosion area.  
The Jones-Wilkins-Lee model was used, and trinitrotoluene (TNT) was set as the explosive charge (Table 6). TNT properties were provided in the AUTOUDYN program. For the air conditioning, the internal energy was input as 2.068 \(\times\) 10\({}^{5}\) J/kg, corresponding to the standard atmospheric conditions.  
An outflow condition was established around the air region to allow gas to escape, and a boundary condition was set at the edges of the model, surrounding the shock chamber.  
To prevent deformation of the shock chamber, a low-power, non-explosive chemical expansion agent called NRC was utilized. In this model, Equation (8) introduced the TNT equivalent, which was then applied to the model. Taking into account NRC's heat of combustion at 3452 kcal/kg and an explosion yield factor of 0.2, it was determined that 30 g of NRC is equivalent to 18 g of TNT. In this context, \(W\) signifies the TNT equivalent in kilograms, \(\mu\) denotes the explosion yield factor, \(M\) represents the mass of the explosive in kilograms, and \(E_{c}\) stands for the heat of combustion of the explosive in kcal/kg.  
\[W=\frac{\mu\times M\times E_{c}}{1140}. \tag{8}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**A (GPa)** & **B (GPa)** & **R1** & **R2** & **w** \\ \hline
371.2 & 3.231 & 4.15 & 0.95 & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Chapman–Jouguet (C-J) and Jones–Wilkins–Lee (JWL) characteristic values (TNT).  
Figure 14: Schematic of the coupling Lagrangian–Eulerian meshes.  
Figure 15: AUTODYN numerical model of the shock chamber model.