## 4 Validation of Numerical Model
### Mesh Sensitivity Study  
A mesh sensitivity study was achieved based on the h-refinement strategy [20], in essence, numerical calculations were considered converged when the results obtained using a mesh size by a factor of two changed by less than 10%. Three cell sizes of 100 mm, 200 mm, and 400 mm were considered. The overpressures were monitored at forty-two locations, G1 to G42, in the main tunnel, as shown in Figure 4. Monitoring locations G1 to G7 were placed at the entrance of the main tunnel, where these locations were equally spaced horizontally and vertically; the spacings were set to 1.925 m and 2.2 m, respectively. The inner width and height of the main tunnel were 7.7 m and 6.6 m, respectively. Other monitoring locations were made similarly at spacings of 5 m along the longitudinal direction, as shown in Figure 2b. Figure 4: Monitoring locations inside the protective tunnel. (**a**) Entrance of main tunnel; (**b**) Along the main tunnel. Figure 5 shows the results of overpressure histories calculated at forty-two monitoring locations for the mesh size of 100 mm. For monitoring locations, G1 to G7, the maximum overpressures were observed at the first peaks, namely near the shock front at approximately \(t=0.13\) s, as shown in Figure 5a. The overpressures thereafter fluctuate continuously due to repeated shock-wave reflections by the inside walls and diffractions at the corners. Similar observations for other monitoring locations were made, as shown in Figure 3b-f. The overpressure histories simulated for the cell sizes of 200 mm and 400 mm were similar to those for the 100-mm cell size, and these are not presented in this study. Figure 5: Overpressure histories for monitoring locations G1 to G42 and mesh size = 100 mm. (**a**) G1 to G7; (**b**) G8 to G14; (**c**) G15 to G21; (**d**) G22 to G28; (**e**) G29 to G35; (**f**) G36 to G42. Figure 6 shows the propagation of blast waves simulated using the cell size of 100 mm for the protective tunnel, where the upper and lower bounds in the overpressure range were fixed as 180 kPa and 48 kPa, respectively, to efficiently show the overpressure distribution as the blast waves propagate. The ground burst-induced blast waves enter the external tunnel entrance and propagate to the main tunnel through the entry space. Complex effects, including reflections and diffractions, were simulated effectively, while the blast waves propagate inside the tunnel. The diffractions with a vortex formed are observed at the first corner at \(t=51\) ms and the second at \(t=113\) ms, as shown in Figure 4a,b, respectively. Mach stems are also identified distinctly where reflected blast waves merge with incident shocks at \(t=161\) ms and 211 ms, as shown in Figure 4c,d, respectively. The results of the mesh sensitivity study for cell sizes of 100 mm, 200 mm, and 400 mm are shown in Figure 7, where the values of the overpressures at the center on the ground at each distance from the entrance of the main tunnel, namely monitoring locations G2, G9, G16, G23, G30, and G37 (distances = 0 m, 5 m, 10 m, 15 m, 20 m, and 25 m, respectively), were used to evaluate the mesh sensitivity. For monitoring location G2 (distance = 0 m), the overall behaviors of the overpressures over time were similar for the three cell sizes. The maximum overpressures calculated for cell sizes of 100 mm and 200 mm were similar, with differences of less than 10%, whereas those for the 400-mm mesh appeared somewhat smaller. This was also observed for the other monitoring locations. The maximum overpressures, \(P_{M}\), obtained in Figure 7 by varying the distance from the main tunnel entrance for cell sizes of 100 mm, 200 mm, and 400 mm are shown in Figure 8, and the differences in the maximum overpressures for cell sizes of 100 mm and 200 mm and those of 200 mm and 400 mm are shown in Figure 9. As shown in this figure, the results for cell sizes of 200 mm and 400 mm do not converge, with differences ranging from approximately 10% to 24%.