## 4 Finite Element Analysis Method and Results
### Frost-Heave Scenario
However, in CASE_HA in which thermosiphons are utilized, the results reveal a slightly different trend, as shown in Figure 12c,d. For CASE_HA, the convergence time for the frost heave of the soil surface is 180 h, which implies that the rate of convergence is faster than that for CASE_H. In addition, the frost-heave magnitude of the soil surface is different along the widths at No. 1 and No. 6. These results are clearly attributable to the effect of the thermosiphons. Note that the frost-heave amount of the surface where the thermosiphon is embedded at No. 6 is smaller than that at No. 1, which is located farther away from the thermosiphon. Similarly, CASE_HB and CASE_HC in which the numbers of thermosiphons are further increased also exhibit comparable frost-heave tendencies. The reduction rates of the frost heave magnitudes corresponding to CASE_HA, CASE_HB, and CASE_HC are 10.8, 36.2, and 55.6%, respectively; this result clearly demonstrates that as the number of thermosiphons increases, the inhibiting effect on the frost heave increases accordingly. Similar to the frost-heave tendency of the soil surface, the upward displacement tendency of the pipeline is also investigated. Figure 13 graphically demonstrates the displacement of the pipeline as time progresses. The time for the convergence of the pipeline displacement is remarkably similar to that for the frost heave of the soil surface. For CASE_H, it converges at approximately 240 h. Moreover, for CASE_HA, CASE_HB, and CASE_HC, it converges at 180, 140, and 100 h, respectively. The maximum displacement of the pipeline is measured at the measurement point \"P-6-90\", and the displacement of the pipeline in CASE_H is approximately 3.2 mm. The displacements of the pipeline in  
Figure 12: Frost heave along the width at different times: (**a**) CASE_H (No. 1), (**b**) CASE_H (No. 6), (**c**) CASE_HA (No. 1), and (**d**) CASE_HA (No. 6). CASE_HA, CASE_HB, and CASE_HC with thermosiphons applied are 3.07, 1.93, and 1.19 mm, respectively. Compared to CASE_H, it is confirmed that the displacement of the pipeline in CASE_HA, CASE_HB, and CASE_HC is reduced by 4.28, 39.77, and 62.71%, respectively. The axial stress of the pipeline increases as the displacement behavior of the pipeline increases, and when the behavior of the pipeline converges, the axial stress of the pipeline converges in tandem. Figure 14 illustrates the axial stresses of all cases measured at the measurement points that are set according to the x-direction of the pipeline. The pipeline behavior according to the frost heave of the soil exhibits the largest axial tension stress (positive stress) at the measurement point \"P-6-0\" set at the top of the pipeline. Additionally, the largest axial compression stress (negative stress) is observed at the measurement point \"P-6-180\" set at the bottom of the pipeline. The magnitude of the axial stress for the pipeline decreases according to the number of thermosiphons applied, which is similar to the displacement of the pipeline. Compared to the maximum axial stress (tension) of CASE_H measured at the measurement point \"P-6-0\" in Figure 14a, CASE_HA, CASE_HB, and CASE_HC exhibit axial stress reduction  
Figure 14: Axial stress along the length for all cases: (**a**) top line of pipeline and (**b**) bottom line of pipeline. Figure 13: Pipeline displacement along the length at different times: (**a**) CASE_H and (**b**) CASE_HC. rates of 1.06, 39.78, and 63.75%, respectively. Furthermore, Figure 14b, featuring the results of the maximum axial stress (compression), also depicts similar results. The axial stress (compression) reduction rates compared to CASE_H are 0.34, 37.39, and 61.92% in CASE_HA, CASE_HB, and CASE_HC, respectively.