## 2 Materials and Methods
### Finite Element Simulation Model of Hydro-Pneumatic Spring Seal Based on ABAQUS
#### 2.3.4 Solver Settings  
The analysis was divided into three steps:  
1. Solving for the contact between the seal rings;  
During assembly, there is an overlap between the O-ring and the end cap groove, as well as between the step seal ring and the O-ring. The software automatically finds the contact surfaces using the interference adjustment command in the interaction module of ABAQUS software.  
2. Fixing the end cap and compressing the seal rings with the piston rod;  
The compression ratios of the entire seal ring are chosen as 10%, 15%, 20%, 25%, and 30%. The installation gap between the piston rod and the end cap is set to 0.2 mm. The calculation formula of the compressibility \(W\) is as follows:  
\[W=(d_{0}-h)/d_{0}*100\% \tag{2}\]  
where \(d_{0}\) is the initial radial length of the seal before compression, mm; \(h\) is the radial length of the compressed rubber parts, mm. The compression ratio of the seal ring is controlled by changing the depth of the seal groove arranged on the end cap. The displacement of the piston rod and the depth of the seal groove for different compression ratios are shown in Table 1.  
3. The cyclic pressure shock of high-pressure hydraulic oil on the seal ring.  
The pressure infiltration module (penetration pressure) in ABAQUS software can replace complex fluid-structure interaction models to study the fluid loading conditions between two surfaces under the impact of high-pressure liquid. During the reciprocating sealing process, the addition of high-pressure hydraulic oil on one side of the seal ring exerts a force on the seal ring, ultimately transforming into contact pressure at the sealing lip. In finite element analysis, there are generally two methods for applying fluid pressure. The specified boundary method is to pre-set the fluid pressure on the sealing surface before applying the pressure load. However, this method does not automatically adjust the pressure loading surface with the deformation of the seal ring, which can lead to errors in the analysis. The pressure penetration method: this method involves setting up contact pairs for fluid penetration, specifying the starting point for loading, and defining the pressure values to apply. It allows the nodes where fluid pressure is applied to adjust  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Solution** & 1 & 2 & 3 & 4 & 5 \\ \hline Compression ratio & 10\% & 15\% & 20\% & 25\% & 30\% \\ Displacement (mm) & 1.63 & 2.095 & 2.56 & 3.025 & 3.49 \\ Sealing groove depth (mm) & 8.17 & 7.705 & 7.24 & 6.775 & 6.31 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Piston rod displacement and seal groove depth for different compression ratios.  
Figure 7: Pressure penetration schematic.  
automatically with the deformation of the contacting bodies, providing a more accurate simulation of the seal ring's deformation process [20].  
To realistically simulate the seal ring's response to pressure shock caused by high-pressure hydraulic oil in the working conditions of the hydro-pneumatic spring, a simulated fluid pressure with a frequency of 2 Hz was applied to the seal ring. This analysis aimed to assess the sealing performance between the seal ring and the end cap and piston rod interface under cyclic pressure shock conditions. As shown in Figure 8, the pressure shock cycle included pressure values ranging from 5 to 30 MPa with a 5 MPa increment; the peak value of pressure value is \"P\".