## 3 Results and Discussion
### Average Strain Energy Density from Axial to Radial Compression  
Similar to Figure 8, Figure 9 shows the average strain energy density as a function of the axial compression and the radial compression. The strain energy density represents the energy stored in an object per unit volume. When external forces are applied to an object, the energy given to the object from the outside is stored as potential energy inside the object. Since the strain energy density is not evenly distributed in an object, the mean value per unit volume was used for analysis in this study. In general, the average strain energy density increased exponentially with increasing axial compression, and then linearly with increasing radial compression. However, the strain energy density showed a different behavior compared to the contact pressure during radial compression. As shown in Figure 9, types 1 and 3 showed the largest difference in the strain energy density at TR = 0.5, unlike the contact pressure in Figure 8, and the difference between the two values tended to decrease gradually as TR increased. Of all cases, type 3 at TR = 0.25 showed the largest increase in strain energy density. For example, at an internal pressure of 10 MPa, the strain energy density of type 3 at TR = 0.25 is 1.75 MJ/m\({}^{3}\) in Figure 9a. During radial compression, the strain energy density of type 3 tended to decrease gradually as TR increased to 1.00 and 1.50. In contrast, in the case of type 1, the maximum values were in the range of 0.9-1.0 MJ/m\({}^{3}\) at TR = 0.25-1.00, showing no large changes in strain energy density during radial compression. As mentioned above, the strain energy density can be divided into dilatational and distortional strain energies. The distortional strain energy density is related to the von [PERSON] failure criterion. In other words, a ductile material could fail if the distortional strain energy density exceeds a critical value, which, in the present case, can vary according to the design criteria, such as the safety factor, yield stress, or ultimate tensile strength. Here, under the four TR conditions, type 1 shows the lowest values of strain energy density for all cases, which implies the lowest risk of material failure. Figure 9: Changes in average strain energy density stored in the UMS model with axial and radial compression. The yellow area indicates axial compression from 10% to 40% of the compression ratio during assembly. The blue area represents radial compression from 2.5 to 10.0 MPa of the internal pressure during operation: (**a**) TR = 0.25; (**b**) TR = 0.50; (**c**) TR = 1.00; and (**d**) TR = 1.50. In this study, critical features such as equivalent stress, gap distance, and contact pressure were determined during both assembly and internal pressurization. It was confirmed that axial compression during the assembly process plays an important role in an exponential increase of critical values related to the reliable sealing capacity of the UMS system. Radial compression during operation caused linear changes in the above values. The factors required for the sealing system to maintain its features are no failure and no detachment. These could be predicted through the equivalent stress distribution, gap distance, contact pressure, and strain energy density. Therefore, the major conditions for determining the stability of a UMS system can be suggested as follows:  
* the smallest stress distribution;
* the narrowest and smallest detachment region at low TR;
* the most stable values of contact pressure and strain energy density for changes in TR, and
* no deformation in the insulating spacer even at high TR. Considering these conditions in general, type 1 with its symmetric open grooves showed the best results. However, type 1 is limited by the rapid concentration of stress on the open groove. To solve this problem, the change in geometry near the open groove should be applied more smoothly through chamfering, etc.