## 3 Preliminary Analysis and Design Direction
### Preliminary Analysis Results and Design Direction  
Finite element analysis of the three contact geometries revealed quantifiable performance differences. At low pressure (2.0 MPa), the maximum contact pressure was 123.45 MPa for rectangular contact geometry, 148.43 MPa for concave contact geometry with \(R_{p}>R_{s}\), and 133.54 MPa for concave contact geometry with \(R_{p}<R_{s}\). At high pressure (87.5 MPa), the maximum equivalent stress measured 295.53 MPa, 240.59 MPa, and 277.91 MPa for the respective designs. Table 4 summarizes these quantitative performance comparisons across all three contact geometry configurations. These values indicate that concave contact geometry with \(R_{p}>R_{s}\) achieved 20.2% higher contact pressure at low pressure and 18.6% lower maximum equivalent stress at high pressure compared to rectangular contact geometry. Contact pressure distributions depicted in Figure 6 demonstrate distinct patterns across the three configurations. The rectangular contact geometry exhibits pressure concentration at the edge contact region. The concave contact geometry with \(R_{p}>R_{s}\) shows a contact pressure distribution with higher peak values localized at the initial contact zone. This increased pressure concentration correlates with the constraint coefficient (\(\eta_{c}=0.048\)) calculated from Equation (9), which quantifies the geometric relationship between the curved surfaces. Stress distribution patterns at high pressure (87.5 MPa), illustrated in Figure 7, demonstrate significant variations between designs. The rectangular contact geometry exhibits localized stress concentration at the contact edge, with values exceeding the compressive strength of PAI (186 MPa). The concave contact geometry with \(R_{p}>R_{s}\) displays more uniform stress distribution across the contact region. The concave contact geometry with \(R_{p}<R_{s}\) shows intermediate stress distribution characteristics. This behavior aligns with Hertzian contact theory predictions in Equation (7), which establishes that in curved contact geometries, stress increase rates diminish as load increases. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Performance Metrics** & \begin{tabular}{c} **Rectangular** \\ **(Original)** \\ \end{tabular} & \
\begin{tabular}{c} **Concave** \\ **(\(R_{p}>R_{s}\))** \\ \end{tabular} &
\begin{tabular}{c} **Concave** \\ **(\(R_{p}<R_{s}\))** \\ \end{tabular} \\ \hline Max. contact pressure @ 2.0 MPa & 123.45 & 148.43 & 133.54 \\ Max. equivalent stress @ 87.5 MPa & 295.53 & 240.59 & 277.91 \\ (MPa) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of performance metrics for different contact geometries. Deformation analysis indicates a substantial difference in material response between the plunger and seat components. The elastic modulus ratio between aluminum bronze alloy (117 GPa) and PAI (4.5 GPa) results in asymmetric deformation behavior. The strain compatibility condition in Equation (12) predicts a deformation ratio (\(\delta_{p}\)/\(\delta_{s}\)) of approximately 26, indicating that the plunger experiences significantly greater deformation than the seat. This deformation characteristic influences both contact pressure generation at low pressure and stress distribution at high pressure. The conformity coefficient (\(\gamma\) = 1.1) of the concave contact geometry with \(R_{p}>R_{s}\), as defined by Equation (10), quantifies the geometric relationship between the contacting surfaces. This value contrasts with conventional contact geometry approaches, where \(R_{p}<R_{s}\) configurations (\(\gamma\) < 1) are typical. The analysis data indicate that the \(R_{p}>R_{s}\) configuration provides a more effective balance between the opposing requirements of high contact pressure at low operating pressures and low maximum equivalent stress at high operating pressures. Based on the analysis results, the concave contact geometry with \(R_{p}>R_{s}\) demonstrates the most favorable performance metrics for ultra-high-pressure hydrogen valve applications.