## 2 Numerical Calculation Model
### One-Way Fluid-Solid Coupling Numerical Simulation Method  
In fluid-solid coupling calculations, the fluid domain is solved based on the mass conservation (continuity equation) and the conservation of momentum, and the solid structure is computed according to the governing equation for solid deformation. The continuity equation is [8]:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho v)=0 \tag{1}\]  
where \(\rho\) is the density of the cooling medium, and \(v\) is the velocity vector. The Conservation of momentum equation [9] is:  
\[\frac{\partial}{\partial t}(\rho\mathbf{v})+\
abla\cdot(\rho\mathbf{v}\mathbf{v})=-\
abla p +\
abla(\mathbf{\tau})+\rho\mathbf{g}+\mathbf{F} \tag{2}\]  
where \(\rho\) is the density of the fluid, \(t\) is the time, \(\mathbf{v}\) is the velocity vector, \(p\) is the pressure, \(\mathbf{\tau}\) is the stress tensor, \(\mathbf{g}\) is the gravitational acceleration vector, and \(\mathbf{F}\) is the external force vector. During the phase-change simulation, the widely employed Lee model was utilized, which is currently the most extensively applied phase-change model. The Lee model [10] of continuity equation is:  
\[\frac{\partial}{\partial t}(\alpha_{v}\rho_{v})+\
abla\cdot(\alpha_{v}\rho_{v }\upsilon_{v})=m \tag{3}\]  
where \(\alpha\) is the volume fraction of vapor, \(\rho\) is the vapor density, and \(\mathbf{v}\) is the velocity vector; the subscript \"\(\upsilon\)\" is the vapor phase, \(m\) is the mass transfer source term. The equation for solving the mass transfer source term in the Lee model for the phase change process is:  
\[m=\left\{ \begin{array}{l}r_{e}\times\alpha_{l}\rho_{l}\frac{T_{l}-T_{sat}}{ T_{sat}},T\geq T_{sat}\\ r_{c}\times\alpha_{v}\rho_{v}\frac{T_{sat}-T_{l}}{T_{sat}},T<T_{sat}\end{array} \right. \tag{4}\]  
where subscript \"\(l\)\" represents the liquid phase, \(T\) represents temperature, and \(T_{sat}\) is the saturation temperature of liquid nitrogen. \(r\) is the phase change mass transfer coefficient, and subscripts \"\(e\)\" and \"\(c\)\" are the vaporization and condensation processes, respectively. The governing equation for solid deformation is:  
\[\rho_{s}+\tilde{d}_{s}=\
abla\sigma_{s}+f_{s} \tag{5}\]  
where \(\rho_{s}\) is the density of the solid material, \(\tilde{d}_{s}\) is the local acceleration vector, \(\sigma_{s}\) is the Cauchy stress tensor, and \(f_{s}\) is the volume force tensor. The vibration analysis in fluid-solid coupling calculations involves the equivalent of displacement and stress. Of course, in this model that includes heat transfer, it is also necessary to adhere to the principles of temperature and heat flux equivalent. At the fluid-solid coupling interface, some of the physical quantities of the fluid and the solid should act as boundary conditions for each other, namely: \begin{table}
\begin{tabular}{c c} \hline
**Case** & **Number of Elements** \\ \hline
1 & 116,958 \\
2 & 496,368 \\
3 & 971,722 \\
4 & 1,701,574 \\
5 & 4,260,446 \\ \hline \end{tabular}
\end{table} 
Table 1: The number of elements corresponding to each case in the mesh independence test. \[\left\{ \begin{aligned} \text{d}_{\text{f}}&=\text{d}_{\text{s}}\\ \mathbf{n}\cdot\mathbf{\tau}_{\text{f}}&=\mathbf{n}\cdot\mathbf{\tau}_{ \text{s}}\\ \text{T}_{\text{f}}&=\text{T}_{\text{s}}\\ \text{Q}_{\text{f}}&=\text{Q}_{\text{s}}\end{aligned} \right. \tag{6}\]  
where \(f\) and \(s\) represent the fluid side and solid side near the fluid-solid coupling interface, respectively; \(d\) is displacement, \(\tau\) is stress, \(\mathbf{n}\cdot\mathbf{\tau}\) is the normal stress at the fluid-solid coupling interface, \(T\) is temperature, and \(Q\) is heat flux. The fluid-solid coupling method can be categorized into two types: two-way fluid-solid coupling and one-way fluid-solid coupling. Two-way fluid-solid coupling is typically employed when both the fluid and structural domains undergo significant deformations, where changes in one physical domain significantly impact the geometric dimensions and shape of the other domain, particularly at the interface [11].