## 2 Finite Element Analysis of the Electromagnetic Track Launcher  
### Finite Element Analysis Method  
#### 2.1.1 Governing Equation  
The electromagnetic launch process can be regarded as a solution problem of a three-dimensional transient eddy current field containing a moving conductor. The electromagnetic track launcher can be divided into two areas: armature and track. The track is divided into the inner track and outer track, which are parallel to each other. The armature contacts with the inner track. When the launcher is working, the excitation current flows in from the upper outer track and out from the lower inner track through the armature. An electromagnetic thrust is generated and pushes the armature to launch out. The schematic diagram of electromagnetic track launcher is shown in Figure 1, where Figure 1a is the side view of the launcher and (b) the front one. \(I\) the excitation current, \(v\) the armature velocity and the other model parameters in Figure 1 are shown in Table 1.  
Since it is not convenient to solve [PERSON]'s equations directly, electromagnetic potentials are usually introduced during finite element calculation to simplify the problem, of which the \(A\)-\(\varphi\) method is a commonly used expression, which expresses the electromagnetic field governing equations through the magnetic vector potential \(A\) and the electric scalar potentials \(\varphi\), making their numerical calculations easier to realize [14]. Ignoring the displacement current, the diffusion equations of armature and track can be deduced by using the _A-\(\varphi\)_ method as follows:  
\[\left\{ \begin{array}{l}\
abla^{2}\textbf{{A}}-\mu_{\text{d}}\sigma_{\text{d}}( \
abla\varphi+\frac{\partial\textbf{{A}}}{\partial\textbf{{A}}}-\textbf{{v}} \times\
abla\times\textbf{{A}})=\textbf{{0}}\\ \
abla^{2}\textbf{{A}}-\mu_{\text{g}}\sigma_{\text{g}}(\
abla\varphi+\frac{ \partial\textbf{{A}}}{\partial\textbf{{t}}})\;=\textbf{{0}}\end{array} \right. \tag{1}\]  
where, \(\mu_{\text{d}}\), \(\mu_{\text{g}}\) represent the armature permeability and the track one, respectively. \(\sigma_{\text{d}}\), \(\sigma_{\text{g}}\) represent the armature conductivity and the track one, respectively. When the launcher is working, the magnetic induction intensities cancel each other out in the direction \(z\) and there is only a component in the direction \(y\), so \(A\) has components only in the directions \(x\), \(z\) and \(v\) only in the direction \(x\). Therefore, ignoring \(\frac{\partial\Delta_{x}}{\partial y}\frac{\partial\Delta_{z}}{\partial y}\), \(A_{y}\), \(v_{y}\) and \(v_{z}\), Equation (1) is written as follows [11]:  
\[\left\{ \begin{array}{l}\frac{\partial^{2}A_{x}}{\partial x^{2}}+\frac{ \partial^{2}A_{x}}{\partial z^{2}}-\mu_{\text{d}}\sigma_{\text{d}}\left(\frac{ \partial\varphi}{\partial x}+\frac{\partial A_{x}}{\partial x}\right)=0\\ \frac{\partial^{2}A_{x}}{\partial x^{2}}+\frac{\partial^{2}A_{x}}{\partial z^{ 2}}-\mu_{\text{d}}\sigma_{\text{d}}\left[\frac{\partial\varphi}{\partial z}+ \frac{\partial A_{x}}{\partial t}-v_{x}\left(\frac{\partial A_{x}}{\partial z }-\frac{\partial A_{y}}{\partial x}\right)\right]=0\\ \frac{\partial^{2}A_{x}}{\partial x^{2}}+\frac{\partial^{2}A_{x}}{\partial z^{ 2}}-\mu_{\text{g}}\sigma_{\text{g}}\left(\frac{\partial\varphi}{\partial x}+ \frac{\partial A_{y}}{\partial x}\right)=0\\ \frac{\partial^{2}A_{x}}{\partial x^{2}}+\frac{\partial^{2}A_{x}}{\partial z^{ 2}}-\mu_{\text{g}}\sigma_{\text{g}}\left(\frac{\partial\varphi}{\partial x}+ \frac{\partial A_{y}}{\partial t}\right)=0\end{array} \right. \tag{2}\]  
During the working process of the launcher, the track surface temperature changes rapidly, in which the heat conduction process causes the greatest impact. By consider