# Spatiotemporal Mechanical Effects of Framework-Slope Systems Under Frost Heave Conditions
## 4 Development of Coupled Multi-Field Numerical Model
### Boundary Conditions and Meshing  
To investigate the spatiotemporal evolution of temperature and moisture fields under frost heave, appropriate boundary conditions were applied to the numerical model. According to the heat-moisture coupling theory in frozen soil mechanics, thermal boundary conditions are generally classified into three types [26]. In this study, both heating and cooling processes on the lateral boundaries were considered. A first-type (Dirichlet) boundary condition was applied at the upper boundary to simulate surface temperature variation, while the bottom boundary was treated as thermally insulated to reflect adiabatic conditions.  
The moisture field was modeled as a closed system, meaning that external water sources and surface evaporation were not considered. This assumption reflects the dry  
Figure 4: Specim shear strength envelope: (**a**) natural state; (**b**) frozen state.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Working Condition** & **Cohesion (kPa)** & **Angle of Internal Friction (\({}^{\circ}\))** \\ \hline Natural state & 41.2 & 30.9 \\ Frozen state & 72.5 & 42.7 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Strength parameters of soil under different working conditions.  
winter conditions commonly observed in cold regions, where the soil system tends to be self-contained in terms of moisture redistribution.  
For the mechanical boundary conditions, the bottom boundary of the model was fixed in both vertical and horizontal directions to prevent displacement. The left and right boundaries were constrained horizontally (roller supports), allowing vertical movement but preventing lateral displacement. At the interface between the frame beams and the anchor rods, right-angle constraints were applied to simulate structural integrity and displacement compatibility.  
The finite element mesh was generated using COMSOL Multiphysics built-in mapped and free triangular meshing algorithms. To achieve high accuracy in the simulation, mesh refinement was applied. The maximum mesh element size for the general slope domain was set to 0.42 m. In regions of high-stress gradients, particularly between the frames where soil arching effects are prominent, the mesh was further refined to a minimum element size of 0.0055 m. This mesh refinement is critical to ensuring the numerical stability of the coupled thermal-hydraulic-mechanical analysis and the accurate capturing of localized deformation.  
The resulting mesh configuration is illustrated in Figure 5.  
The air temperature data used in this study were obtained from on-site temperature sensors installed at the slope location, covering the full freeze-thaw cycle of the year 2024. The measured temperature trends are presented in Figure 6.  
The temperature data employed in this study were acquired from on-site temperature sensors during 2024, as shown in Figure 6. The statistically regressed dataset presented in Figure 6 was used to analyze the thermo-mechanical characteristics of the frame-anchored slope during freeze-thaw processes. The temporal evolution of the slope temperature fields was characterized using Dirichlet formulations, achieving a determination coefficient R\({}^{2}\) of 0.97410 for the fitted temperature function, as shown in Equation (2):  
\[T=-1.1+21.2\text{sin}\bigg{(}\frac{2\pi t}{365}+\frac{\pi}{2}\bigg{)} \tag{2}\]  
Figure 5: Mesh discretization of the coupled THM model in COMSOL Multiphysics.