## 3 Finite Element Model of Near-Ground Explosion
### Modeling Geometry and Meshing  
In order to investigate the impact of different grounds on the shock wave propagation law when the rocket exploded on the launch pad, the whole rocket is selected to carry out numerical simulation. Numerical models of four different materials are chosen, including concrete ground, soil ground, rigid ground, and air. Model 1 to Model 3 are the TNT that explodes above the rigid ground, the concrete ground and the soil ground, respectively, while Model 4 is the TNT that explodes in the air. In Model 1, air domain is a \(\Phi\) 258 m \(\times\) 40 m cylinder and the ground is rigid constraint. In Model 2 and Model 3, air domain which is a \(\Phi\) 258 m \(\times\) 50 m cylinder and ground domain which is a \(\Phi\) 258 m \(\times\) 10 m cylinder are established. Model 4 only includes air domain which is a \(\Phi\) 258 m \(\times\) 50 m cylinder. After considering the symmetry of the simulation model, a 1/4 symmetrical geometrical model is adopted. In order to reduce the calculation amount, the air domain is divided into three parts according to the radial, which are 0-30 m, 30-70 m, 70-129 m. The mesh size of the air domain is 0.58 m, 1.16 m, and 2.32 m, respectively. The TNT and the ground are divided into mesh sizes of 0.20 m and 2.32 m.  
The height of the detonation center to the ground surface is 30.668 m. The finite element model and gauges distribution are shown in Figure 2.  
It is assumed that the radius of the TNT cylinder is \(r_{0}\) (m) and the height is \(h_{0}\) (m). Therefore, the length-diameter ratio of TNT is \(h_{0}/2r_{0}\). Given the mass of TNT, \(r_{0}\), and \(h_{0}\) of TNT cylinder can be calculated by using Equation (10) which is derived from Equation (9). Unit system of cm-g-ms is adopted in the numerical model. The explosive source is located at the center of mass and a point-detonation-method is used [34].  
\[\left\{\begin{array}{l}M_{T}=\rho\cdot V\\ V=\pi r_{0}^{2}h_{0}\\ h_{0}/2r_{0}=1\end{array}\right. \tag{9}\]  
\[r=\sqrt[3]{\frac{M_{T}}{2\pi\rho}} \tag{10}\]  
Eight-node element of Solid 164 is adopted for the 3D explicit analysis. In order to prevent the element distortion in large deformation and nonlinear structural analyses, ALE algorism is used. TNT and air are modeled with ALE multi-material grids, but different grounds with Lagrangian grids. Fluid-solid coupling algorithm is used between two grids [35].  
Furthermore, the transitional displacement of the nodes normal to the symmetry planes is constrained, in order to simulate the propagation effect of shock wave in the symmetric plane. Non-reflecting boundary condition is applied to top and lateral surfaces, which allows the fluid medium to flow out in order to simulate the effect of infinite space [36]. The displacement constraint is applied in the \(z\) direction of the ground, while rigid constraint is applied on the rigid ground.  
As are shown in Figure 3, gauges in the model are set at five different heights, which are from 2 m to 10 m with an interval of 2 m. Twelve gauges are set at each height, whose \(x\) is from 0 m to 110 m, with an interval of 10 m.  
Figure 3: Distribution of gauges at different heights.  
Figure 2: Physical model and finite element model of near-ground explosion. (**a**) Physical model; (**b**) Finite element model.