## Materials and Methods
### Numerical Analysis Procedure  
The bulk modulus of the UMS was considered to be 10 GPa with a Poisson's ratio of 0.499. Subsequently, its [PERSON]'s modulus and [PERSON]'s parameters were derived. For the pipe pups and the retainer ring, made from ASTM A596-F65, [PERSON]'s modulus and Poisson's ratio were 201 GPa and 0.3, respectively. The insulating spacer, ASTM D709-G10, was chosen with [PERSON]'s modulus of 17.2 GPa and Poisson's ratio of 0.36. All the materials were assumed to be homogeneous and isotropic.  
Figure 3: U-shaped sealing assembly process: (**a**) insertion of UMS; (**b**) axial compression due to displacement in assembly; and (**c**) radial compression due to internal pressure in operation.  
The analysis was performed in two stationary steps. Firstly, axial compressive displacement was applied to examine the nonlinear hyperelastic behavior of the UMS under axial compression during assembly. The constrained displacement condition was applied, and the UMS was compressed gradually at 10%, 20%, 30%, and 40% of the thickness of the UMS leg, \(t_{leg}\). As a result, the pipe pups were displaced by 1.5, 3.0, 4.5, and 6.0 mm, respectively. Secondly, internal pressure was applied to the already-compressed UMS to examine the nonlinear hyperelastic behavior of the UMS under radial compression during operation. The pressure was increased gradually to 2.5, 5.0, 7.5, and 10 MPa, respectively. To achieve more rigorous and accurate solutions, an adaptive meshing technique was applied.  
Numerical techniques to solve nearly incompressible hyperelastic finite-element models are known to be quite complicated because the Poisson's ratio \(\
u\) is close to 0.500. Thus, in the present analysis, a parametric continuation solving technique was applied to improve the convergence of the results during axial and radial compressions due to the imposed displacement during assembly and internal pressure in operation, respectively. Contact conditions were applied to the inner surface between the UMS and insulating spacer, the outer surface between the UMS and one pipe pup, the outer surface between the UMS and the other pipe pup, and the outer surface between the UMS and retainer ring.  
The friction between the UMS and other parts, including the pipe pups, retainer ring, and insulating spacer, was applied as contact boundary conditions. Because of the deformable-to-rigid contact conditions between the UMS and other components, it was important yet difficult to analyze the forces transferred across the contact area. We consider both normal and shear stresses transmitted across the contact surfaces. An isotropic spring foundation with spring constant \(k_{tot}\) was applied to stabilize the unconstrained contacts, which were gradually eliminated during auxiliary pressure sweeps.  
To achieve smooth and precise results, quadrilateral and second-order Lagrangian mesh elements were used to discretize the axisymmetric geometry. A mesh optimization procedure was conducted until the smallest change in the errors became less than 0.1% as the results converged. The minimum and average element qualities to define the mesh distribution were maintained over 0.200 and 0.700, respectively. The results of the FEA are presented in terms of equivalent stress, contact pressure, gap distance, and strain energy density. The correlations among the results are also analyzed.