## 2 Numerical Simulation of Armature Electromagnetic Force
### Finite Element Model  
The typical C-shaped armature and the 30 mm \(\times\) 30 mm rectangular caliber electromagnetic railgun are taken as the research objects. Based on the COMSOL finite element simulation software, the armature electromagnetic force is simulated and analyzed. The armature and rail model parameters are shown in Table 1.  
The electromagnetic railgun geometric model and the mesh segmentation result are shown in Figure 1. Among them, the air region and some rail regions are not drawn. Considering the distribution characteristics of the electromagnetic field, the mesh of the rail is divided into cuboid shapes that gradually thicken from the inside to the outside and the mesh of the C-shaped armature is divided into hexahedral shapes with relatively regular regularities through the operations of \"mapping\", \"distribution\", \"sweeping\", \"free quadrilateral grid\", and \"size\". The air area is divided using a \"free tetrahedral\" grid. Through the above meshing operation, the average element mass of the armature and rail areas is 0.9171, and the average element mass of the entire model area is 0.9119. The closer this value is to 1, the better the mesh quality.  
In order to observe the change in electromagnetic force more clearly with the excitation current, the trapezoidal excitation current is used closely to the project (constant current at the maximum current), as shown in Figure 2. The excitation current flows from the lower face of rail A in Figure 1, flows through the armature, and then flows out of the lower face of rail B.  
\begin{table}
\begin{tabular}{c c c} \hline
**Model Parameters** & **Symbol** & **Values** \\ \hline Rail length/mm & \(l_{\mathrm{r}}\) & 3500.00 \\ Rail height/mm & \(h_{\mathrm{r}}\) & 40.00 \\ Rail width/mm & \(w_{\mathrm{r}}\) & 20.00 \\ Rail conductivity/(S/m) & \(\sigma_{\mathrm{r}}\) & \(3.45\times 10^{7}\) \\ Rail permeability/(H/m) & \(\mu_{\mathrm{r}}\) & \(4\pi\times 10^{-7}\) \\ Armature length/mm & \(l_{a}\) & 50.00 \\ Armature height/mm & \(h_{a}\) & 28.00 \\ Armature width/mm & \(w_{a}\) & 30.00 \\ Armature conductivity/(S/m) & \(\sigma_{a}\) & \(2.20\times 10^{7}\) \\ Armature permeability/(H/m) & \(\mu_{a}\) & \(4\pi\times 10^{-7}\) \\ \hline \end{tabular}
\end{table}
Table 1: Parameters of armature and rail model.  
Figure 1: Electromagnetic railgun geometric model and the mesh segmentation result.  
The material properties of the electromagnetic railgun are defined in the finite element simulation software based on Table 1. In this paper, the controlled diffusion equation method is used to numerically simulate the electromagnetic railgun, and the specific setting method is as follows: Setting the conditions for the physics fields of the \"current\" and \"magnetic field\", and then adding velocity terms by modifying the electromagnetic field control equation. The armature velocity is calculated using the physics field of \"global ordinary differential and differential algebraic equations\", and then so are the values of the relevant parameters in the kinetic equation, as shown in Table 2. During transient solving, we use the backward difference scheme in time. In the backward difference scheme, the equations are evaluated at the current time; therefore, it is an implicit scheme.