## 4 Design Optimization
### Comparative Performance of Optimized Design  
Comparative analysis of the optimized concave contact geometry against conventional rectangular contact geometry revealed significant performance improvements.  
Figure 11 illustrates the contact pressure distribution at 2.0 MPa. The rectangular contact geometry shown in Figure 11a exhibits concentration primarily at the edge rounding region, while the concave contact geometry presented in Figure 11b demonstrates higher pressure with more effective distribution along the contact interface.  
Figure 11: Contact pressure distribution comparison at 2.0 MPa: (**a**) rectangular contact geometry; (**b**) concave contact geometry.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & \(R_{p}\) (mm) & \(R_{s}\) (mm) & \(E\) (mm) & \(R_{p}\)/\(R_{s}\) (MPa) & **Max. Contact Pressure** (MPa) & **Max. Equivalent Stress** (MPa) \\ \hline Metamodel & & & & & \\ FEA & 5.73 & 4.68 & 0.95 & 1.22 & 147.30 & 173.04 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Comparison of predicted and FEA results for optimal design.  
Figure 12 displays the equivalent stress distribution at 87.5 MPa. The rectangular contact geometry illustrated in Figure 12a shows concentrated stress at the contact edge exceeding the PAI material's compressive strength (186 MPa). The concave contact geometry shown in Figure 12b distributes stress more uniformly, maintaining values below the compressive strength threshold.  
Table 10 quantifies these performance differences. At 2.0 MPa, the concave contact geometry generates a maximum contact pressure of 152.69 MPa, 23.7% higher than the rectangular contact geometry's 123.45 MPa. At 87.5 MPa, the concave contact geometry limits maximum equivalent stress to 169.42 MPa, 42.7% lower than the rectangular contact geometry's 295.53 MPa.  
The safety factor analysis further confirms this improvement. The concave contact geometry achieves a safety factor of 1.10, operating below the PAI material's compressive limit, while the rectangular contact geometry's safety factor of 0.63 indicates operation beyond compressive limits. This represents a 74.6% improvement in structural safety.  
These improvements directly address the dual requirements for ultra-high-pressure hydrogen valves: effective low-pressure sealing and high-pressure structural integrity. The optimized concave contact geometry with an \(R_{p}\)/\(R_{s}\) ratio of 1.22 implements Hertzian contact principles to achieve this balance.  
The stress reduction mechanism corresponds to the nonlinear expansion of contact area with increasing load as described in Equation (11). The conformity coefficient (\(\gamma\) = 1.22) creates optimal geometric constraint conditions that distribute stress while maintaining adequate contact pressure.  
These results validate our optimization methodology and demonstrate that appropriate application of contact mechanics principles can resolve the contradictory requirements in ultra-high-pressure applications.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Performance Metrics** & **Rectangular (MPa)** & **Concave (MPa)** & **Improvement (\%)** \\ \hline Max. contact pressure @ 2.0 MPa (MPa) & 123.45 & 152.69 & 23.7 \\ Max. equivalent stress @ 87.5 MPa (MPa) & 295.53 & 169.42 & 42.7 \\ Safety Factor & 0.63 & 1.10 & 74.6 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Performance comparison between rectangular and concave contact geometries.  
Figure 12: Equivalent stress distribution comparison at 87.5 MPa: (**a**) rectangular contact geometry; (**b**) concave contact geometry.