## 6 Conclusions  
This study addressed the fundamental design challenge in ultra-high-pressure hydrogen solenoid valves: achieving effective sealing at low pressures while maintaining structural integrity at high pressures. We developed a novel concave contact geometry based on Hertzian contact theory that outperforms conventional rectangular designs across the entire operating pressure range. Theoretical analysis established that concave contact geometries with \(R_{p}\)/\(R_{s}\)\(>\) 1 fundamentally alter stress distribution patterns, contradicting traditional design approaches. Finite element analysis confirmed that this configuration reduces stress concentration while increasing contact pressure. The design was optimized using Latin hypercube sampling with 27 systematically distributed points to efficiently explore the three-dimensional design space. The kriging metamodel constructed from these simulations demonstrated high prediction accuracy(\(R^{2}>0.88\)) for both performance metrics. Pareto analysis identified optimal trade-offs between contact pressure and equivalent stress, revealing that an \(R_{p}\)/\(R_{s}\) ratio between 1.2 and 1.25 yields the optimal balance. The optimized geometry (\(R_{p}\) = 5.73 mm, \(R_{s}\) = 4.68 mm, \(e\) = 0.95 mm) with \(R_{p}\)/\(R_{s}\) = 1.22 achieved 23.7% higher contact pressure at 2.0 MPa and 42.7% lower maximum equivalent stress at 87.5 MPa, improving the safety factor from 0.63 to 1.10. Experimental validation quantitatively confirmed the computational predictions. The concave geometry demonstrated sealing capability at 1.7 \(\pm\) 0.2 MPa, well below the threshold specified by the AIS-195 standard (2.0 MPa), whereas the rectangular geometry exhibited sealing only at 5.6 \(\pm\) 0.7 MPa, exceeding the standard limits. This 69.6% reduction in required sealing pressure indicates significantly enhanced low-pressure performance. Post-exposure analysis after 87.5 MPa testing revealed no detectable deformation in the concave geometry specimens, while rectangular specimens exhibited permanent deformation averaging 0.0580 \(\pm\) 0.007 mm at the contact edge. These findings corroborate the stress distribution mechanism predicted by Hertzian contact theory and confirm that the optimized geometry successfully reconciles the contradictory requirements across the entire pressure range. The integrated computational-experimental methodology established in this study combines theoretical contact mechanics, systematic design optimization, and experimental validation to resolve contradictory design requirements in high-pressure applications. This research contributes to the fundamental understanding of contact mechanics by demonstrating that conventional hard-to-hard sealing paradigms can be challenged through strategic exploitation of material property asymmetry in non-conforming geometries. The soft-to-hard sealing framework developed extends beyond hydrogen applications to provide new insights for extreme-pressure mechanical systems across various engineering disciplines. The practical implications extend to broader hydrogen fuel cell vehicle commercialization, potentially contributing to enhanced safety margins and improved system reliability in hydrogen infrastructure. Future research should focus on comprehensive validation under realistic operational conditions, including extended temperature ranges, actual hydrogen exposure, and long-term durability assessment under cyclic loading conditions. Conceptualization, J.C. and H.K. ; methodology, J.C.; software, J.C.; validation, J.C. and H.K. ; formal analysis, J.C.; investigation, J.C.; resources, J.C. and H.K. ; data curation, J.C.; writing--original draft preparation, J.C.; writing--review and editing, J.C. and H.K. ; visualization, J.C.; supervision, J.C. and H.K. ; project administration, J.C. and H.K. ; funding acquisition, J.C. and H.K. All authors have read and agreed to the published version of the manuscript.