## 4 30 cm Model Cavity Simulation
### Post-Treatment Processes
#### 4.2.2 Cooling Process  
To enable superconducting operation of the accelerator, it undergoes a cooling process to achieve the necessary temperature for superconductivity. Using COMSOL Multiphysics, we simulated the deformation of the niobium cavity as it transitions from room temperature 293.15 K to the superconducting temperature of 4.2 K. This simulation allows us to calculate the frequency of the cavity after deformation. The simulation model includes the superconducting cavity shell, 4-vane structure, and the vacuum inside the cavity (refer to Figure 19).  
Figure 16: The simulation model of BCP processes: (**a**) initial cavity model; (**b**) BCP of 150 \(\upmu\)m; (**c**) BCP of an additional 30 \(\upmu\)m.  
Figure 17: The frequency simulated by CST MWS (the electric field values are not normalized): (**a**) initial cavity; (**b**) BCP of 150 \(\upmu\)m; (**c**) BCP of an additional 30 \(\upmu\)m.  
Figure 18: The frequency simulated by COMSOL EMW (the electric field values are not normalized): (**a**) initial cavity; (**b**) BCP of 150 \(\upmu\)m; (**c**) BCP of an additional 30 \(\upmu\)m.  
The coupling calculation between the cavity's deformation after cooling down to 4.2 K and the corresponding eigenfrequency change involves three modules in COMSOL: Solid, EMW, and Deformation Geometry (DG). Within the Solid module, we solve for the shell's deformation \(\mathbf{u}\) resulting from thermal expansion, using the linear momentum balance equation and the stress-strain relation:  
\[\
abla\mathbf{\cdot}\mathbf{S}+\mathbf{F}_{V}=0, \tag{6}\]  
\[\mathbf{S}=\mathbf{C}\mathbf{:}\mathbf{e}_{el}, \tag{7}\]  
where \(\mathbf{S}\) is the second Piola-Kirchhoff stress tensor, and \(\mathbf{F}_{V}\) is the force per unit volume. The elasticity tensor \(\mathbf{C}\) is typically defined by Young's modulus and the Poisson coefficient. In the absence of any plastic effects, the elastic strain tensor \(\mathbf{e}_{el}\) carries the temperature dependence via the thermal strain tensor \(\mathbf{e}_{th}\), as expressed by  
\[\mathbf{e}_{el}=\mathbf{e}_{tot}-\mathbf{e}_{th}, \tag{8}\]  
\[\mathbf{e}_{tot}=\frac{1}{2}\Big{[}(\
abla\mathbf{u})^{T}+\
abla\mathbf{u}\Big{]}, \tag{9}\]  
\[\mathbf{e}_{th}=\mathbf{\alpha}(T-T_{ref}), \tag{10}\]  
where \(\mathbf{e}_{tot}\) represents the total strain tensor, \(\mathbf{\alpha}\) is the coefficient of thermal expansion that characterizes the ability of the material to contract and expand due to temperature variations, \(T=4.2\) K is the environment temperature, while \(T_{ref}=293.15\)\(K\) is the reference temperature.  
The boundary of the vacuum undergoes deformation through the specified mesh displacement within the DG module. During the deformation calculation, fixed constraints are applied to both ends of the SC cavity (refer to Figure 15) to prevent rigid displacement. The material properties employed in the calculation are listed in Table 4. After deformation, the Frequency Domain EMW module is utilized to further analyze the vacuum and determine the eigenfrequency of the deformed SC cavity.  
Figure 19: The simulation model used for cooling and evacuation processes.