## 3 Numerical Simulation  
### Modeling and Parameters  
As the TNT charges detonated in the center of the container, the distribution of the internal flow field was symmetrical. Therefore, a two-dimensional model could be used to describe the distribution evolution law. In order to ensure the calculation accuracy of the flow field inside the container, the mesh size of the solid was limited to less than 0.6 cm, and the mesh size of the air was 0.1 cm. For the convenience of calculation, the flange, bolt and other structures independent of the internal flow field were simplified in the model, and the 1/4 containment vessel model was adopted as shown in Figure 3.  
Figure 2: Time history curves of the pressure loading at poles of the different TNT charges: (**a**) 27 g; (**b**) 65 g; (**c**) 106 g and (**d**) 155 g. The dotted lines indicate the three peaks of the initial pressure stage.  
The mapping method was used to map the calculation results of the one-dimensional explosive shock wave to the two-dimensional model, which ensured the calculation accuracy while maintaining simpler calculations [17; 18; 19]. The symmetrical boundaries of the two planes, both _xz_ and _yoz_, were set, respectively. The initial volume fraction method, \"INITIAL_VOLUME_FRACTION_GEOMETRY, was used to fill the explosive at the origin of the one-dimensional air grid.  
The body and end cover of the explosion containment vessel were made of Q345R steel and the vessel wall was 2.2 cm thick. The deformation of these solids under pressure loads had little effect on the flow field [20; 21]. Therefore, the solid deformation was ignored, and \"MAT_RICID could be selected for the material model. The material density \(\rho_{0}\) was 7.85 g\(\cdot\)cm\({}^{-3}\), the [PERSON]'s modulus \(E\) was 209 GPa and the [PERSON]'s ratio \(\
u\) was 0.28.  
A high energy explosive combustion model was used to describe the explosive, with the keyword \"MAT_HIGH_EXPLOSIVE_BURN. The combined JWL (Jones-Wilkins-Lee) equation represents the pressure of the explosion product. The form is as follows [22]:  
\[p=A\bigg{(}1-\frac{\omega}{R_{1}V}\bigg{)}e^{-R_{1}V}+B\bigg{(}1-\frac{\omega} {R_{2}V}\bigg{)}e^{-R_{2}V}+\frac{\omega E}{V} \tag{1}\]  
The units of \(A\), \(B\), and \(E\) are the same as the pressure, and \(R_{1}\), \(R_{2}\), \(\omega\), and \(V\) are dimensionless values, where \(E\) is the energy density and \(V\) is the relative volume. When entering the parameters, the initial energy density \(E_{0}\) and the initial relative volume \(V_{0}\) should be determined, and the specific parameters are shown in Table 1.  
The material model of air was \"MAT_NULL, and its state equation was \"EOS_LINEAR_POLYNOMIAL. The equation can be written as follows:  
\[p=C_{0}+C_{1}\mu+C_{2}\mu^{2}+C_{3}\mu^{3}+\Big{(}C_{4}+C_{5}\mu+C_{6}\mu^{2} \Big{)}E \tag{2}\]  
\[\mu=\frac{1}{V}-1=\frac{\rho}{\rho_{0}}-1 \tag{3}\]  
Regarding air as an ideal gas, the parameters were set to C0 = C1 = C2 = C3 = C6 = 0. The polytropic exponent \(\gamma\) is usually set to 1.4, thus C4 = C5 = \(\gamma\) -1 = 0.4 [22]. The initial density \(\rho_{0}\), the initial energy density \(E_{0}\) and the initial relative volume \(V_{0}\) of air were set to 1.29 g\(\cdot\)cm\({}^{-3}\), 0.25 MPa and 1.0, respectively. The parameters are shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\rho\)/g\(\cdot\)cm\({}^{-3}\) & \(P_{\text{C}}\)/GPa & \(v_{\text{D}}\)/m\(\cdot\)s\({}^{-1}\) & \(A\)/GPa & \(B\)/GPa & \(R_{1}\) & \(R_{2}\) & \(\omega\) & \(E_{0}\)/GPa & \(V_{0}\) \\ \hline
1.50 & 17.92 & 6547.2 & 374.6 & 3.39 & 4.15 & 0.95 & 0.28 & 6.34 & 1.0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material model parameters and the JWL parameters of TNT.  
Figure 3: Two-dimensional 1/4 model of the containment vessel. The blue mesh is the air and the red mesh is the explosive containment vessel.