## 4 Finite Element Analysis Method and Results
### Frost-Heave Scenario  
In the frost-heave scenario, after setting the initial temperature of the entire model to 3 \({}^{\circ}\)C, the Fourier boundary condition (\(-5\)\({}^{\circ}\)C; 5 W/m\({}^{2}\) K) is applied [35] to the top surface of the model, and analysis is performed until the temperature of the entire model converges to \(-5\)\({}^{\circ}\)C.  
Among the measurement points of the model, the time for the temperature to converge to \(-5\)\({}^{\circ}\)C in \"S-1-\", which experiences the minimum effect from the thermosiphon, is approximately 700 h in CASE_H--the case in which the thermosiphon is not utilized. CASE_HA, CASE_HB, and CASE_HC (scenarios featuring thermosiphons) demonstrate temperature convergence after 590, 440, and 360 h, respectively. Notably, as the number of thermosiphons increases, the temperature converges to \(-5\)\({}^{\circ}\)C at a faster rate, as shown in Figure 10.  
Figure 10 also illustrates the graph of the temperature decrease with respect to time measured at x-direction measurement points No. 1 and No. 6 corresponding to z-direction measurement points A, B, and C. For CASE_H, the temperature trends corresponding to the depth measurement points of No. 1 and No. 6 are approximately identical. However, for CASE_HA and CASE_HB, the temperature of measurement point No. 6 decreases faster than that for No. 1, because measurement point No. 6 is greatly affected by the thermosiphon, thereby yielding this trend. However, for CASE_HC, the temperature trends of measurement points No. 1 and No. 6 are similar to those of CASE_H because the effect of the thermosiphon increases at No. 1; this increase is attributed to the thermosiphon placement, as depicted in Figure 10d. The temperature trend of the overall measurement point indicates that CASE_HC, featuring the maximal number of thermosiphons, has a much steeper temperature drop than that observed for CASE_H. These time-dependent temperature trends manifest as the trends corresponding to the unfrozen water content and pore ice ratio, as explained in next figure. Figure 11 presents a comparison of the results of unfrozen water content and pore ice ratio that are measured at the measurement points \"S-1-B\" and \"S-6-B\" for all cases. As the temperature of the measurement point converges to \(-5\)\({}^{\circ}\)C, the unfrozen water content and pore ice ratio converge to 0.160 and 0.357, respectively. Similar to the temperature trend observed in Figure 10, for CASE_HA and CASE_HB, the measurement point \"S-6-B\" has a steeper slope compared to that of \"S-1-B\"; this disparity in steepness demonstrates the effect of the thermosiphon. Additionally, according to the number of thermosiphons applied, CASE_HC converges first, i.e., at the fastest rate, and CASE_H converges last, i.e., at the slowest rate. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{3}{c}{Parameters for elastic–plastic constitutive model} \\ \hline Poisson’s ratio & 0.3 \\ \hline \(\lambda\), \(\kappa\), \(\mathrm{M}\), \(\beta\) & 0.35, 0.07, 0.7, 0.18 \\ \hline \(\alpha_{1}\), \(\alpha_{2}\) & 0.2, 0.4 \\ \hline \(p_{0}\), \(p_{r}\) (kPa) & 80, 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._Figure 11: Comparison of results corresponding to points “S-1-B” and “S-6-B”: (**a**) unfrozen water content and (**b**) pore ice ratio. Figure 10: Temperature variations at measurement points with respect to time: (**a**) CASE_H, (**b**) CASE_HA, (**c**) CASE_HB, and (**d**) CASE_HC. As the temperature of the soil decreases to a certain state, the frost-heave progress ceases at different times and yields the heave amount for each case. Figure 12 indicates the frost-heave trends of the soil surface along the width corresponding to No. 1 and No. 6 of CASE_H and CASE_HA with respect to time. In Figure 12a,b, the frost heave converges at approximately 240 h in CASE_H. In addition, the widths of the soil surface corresponding to No. 1 and No. 6 exhibit almost the same convergence times and heave amounts.