## 2 Finite Element Model
### Model Establishment  
Using ANSYS software, a banded pressure vessel (Figure 1) was used as a prototype to carry out numerical simulation calculations on the welding process of the double-layer hoop structure. The inner diameter of the structure is R. The welding joint adopts a V-shaped groove, and the angle of the weld groove is 40\({}^{\circ}\). The thickness of the weld (the thickness of the outer plate) is b, and the weld width is \(\delta\). Based on the actual welding of the layered pressure vessels, displacement constraints were set at the bottom and left side of the hoop structure model. Set the interlayer friction coefficient to 0.6 to simulate the interlayer friction. The coupling element SOLID5 with temperature and displacement degrees of freedom was selected to perform thermal-structural coupling analysis of the welding process by the direct method. In the element mesh division, considering that there are significant temperature and stress gradients in the weld area and the near-weld area, the mesh was densified so that the mesh size of the weld and its surrounding area is not more than 2 mm. The mesh division results are shown in Figure 2. The gray mesh area in the figure is the weld joint. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Temperature** & **Elastic Modulus before Yield** & **Yield Stress** & **Elastic Modulus after Yielding** \\ \(T\) (\({}^{\circ}\)C) & \(E\) (GPa)** & \(\sigma\) (MPa) & \(E^{\prime}\) (GPa) \\ \hline
20 & 203 & 356 & 20.3 \\
100 & 194 & 321 & 19.4 \\
200 & 188 & 321 & 18.8 \\
300 & 186 & 312 & 18.6 \\
400 & 174 & 283 & 17.4 \\
500 & 161 & 249 & 16.1 \\
600 & 136 & 170 & 13.6 \\
1000 & 10 & 11 & 1 \\
1500 & 10 & 5 & 1 \\
2000 & 1 & 0.5 & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the constitutive relation of bilinear dynamic elastic–plastic. Figure 1: Schematic diagram of a bandaged pressure vessel. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Temperature** & **Density** & **Enthalpy** & **Thermal** & **Thermal Expansion** & **Specific Heat** & **Poisson’s** \\ \(T\) (\({}^{\circ}\)C) & \(\rho\) (kg\(\cdot\)m\({}^{-3}\)) & \(h\) (J\(\cdot\)m\({}^{-3}\)) & **Conductivity** & **Coefficient** & **Capacity** & **Ratio** \\  & & & \(\Lambda\) (W\(\cdot\)m\({}^{-2}\)\(\cdot\)K\({}^{-1}\)) & **\(\alpha\) (W\(\cdot\)m\({}^{-2}\)\(\cdot\)K\({}^{-1}\))** & **C (J\(\cdot\)K\({}_{g}\)\({}^{\circ}\)C\({}^{-1}\))** & **\(\mu\)** \\ \hline
0 & 7800  
In the welding simulation, the inner and outer surfaces of the model were convective boundary conditions. The convection coefficient is 10 W-m\({}^{-2}\).K\({}^{-1}\), and the emissivity is 0.85. The initial ambient temperature was 20 \({}^{\circ}\)C. In the simulation process, the welding seam is divided into several layers, so that the depth of each layer is very small, and the Gaussian heat source can simulate this situation well [34]. So, in the thermal analysis, it is assumed that the heat input of the welding arc meets the Gaussian distribution. The expression of heat flow distribution is  
\[q(r)=q_{m}e^{(-Kr^{2})}, \tag{1}\]  
where \(q(r)\) is the heat flow density at the center of the heat source, \(q_{m}\) is the maximum heat flow density at the center of the heat source, and \(K\) is the heat concentration factor. In actual welding, the base metal will melt, change from solid to liquid, and absorb energy, while the molten pool part will solidify, change from liquid to solid, and emit heat. Therefore, the latent heat of phase transition also has an influence on the temperature change, which must be considered when calculating the temperature change in the crystallization temperature range. Otherwise, it will cause a large deviation in the calculation result. For the treatment of latent heat of solid-liquid phase change, there are mainly equivalent specific heat methods: the enthalpy method and the temperature rise method. Among them, the equivalent specific heat method is relatively simple and practical and has high calculation efficiency.