## 3 Numerical Simulation
### Establishment and Validation of the Numerical Model  
In the model, the slope of the trapezoidal ditch was 1:0.33, the width of the bottom of the ditch was 1.5 m, the height to earth up was 2.1 m, the bottom of the ditch was made of a 200 mm thick cushion (A1), and the supporting angle of the foundation of the bottom was 120\({}^{\circ}\). A2 was the backfill on the side of the pipe, A3 was the backfill on the top of the pipe, and A4 was the soil on both sides of A3. To prevent the reflection of vibration energy by fixed boundary in Abaqus dynamic analysis, which may cause errors, the model boundary adopted an infinite element boundary to simulate the far-field infinite soil [27; 28]. The size of the model was set to 8 m \(\times\) 9 m \(\times\) 20 m, as shown in Figure 10. According to the study by [PERSON] et al. [29], the soil outside the three-fold pipe diameter has little influence on the soil of the flexible pipe, so the size used in the model is reasonable. In order to improve the calculation accuracy, the backfill in the trench was divided into smaller elements. Before determining the element size, several comparisons were made. The results show that when the element size of backfill soil was less than 0.04 m \(\times\) 0.04 m \(\times\) 0.1 m, the simulation results were almost not affected by the element's size. Therefore, the maximum size of the backfill element was set to 0.03 m \(\times\) 0.03 m \(\times\) 0.07 m. The constitutive relation of soil adopted the classical Mohr-Coulomb constitutive. According to the compaction degree and physical properties of the soil at different positions, the material models such as Young's modulus, internal friction angle, and cohesion were obtained by referring to the relevant specifications [24; 26], as shown in Table 4. The implicit method was used in the model. Figure 10: _Cont._  
To improve calculation accuracy and efficiency, the HDPE double-walled corrugated pipe was converted into a three-dimensional straight wall pipeline with the same ring stiffness and outer diameter. The wall thickness of the straight wall pipeline can be calculated using Formula (6) [30]. The equivalent straight wall pipe model parameters are shown in Table 5. The constitutive model of the pipe adopted the classical Mohr-Coulomb constitutive. \[PS=\frac{EI}{0.149^{D}} \tag{6}\]  
where \(PS\) is the pipe stiffness, \(E\) is the elastic modulus of pipe, \(I\) is the inertia moment of the cross-section of the pipeline, and \(r\) is the mean radius of the pipe. The vibration load of the vibratory roller is primarily generated by the front wheel vibration of the roller, and the rear vehicle as the static load had a negligible effect on the internal pressure of the subgrade. Therefore, in this numerical simulation, the vibratory roller was simplified into a wheel that can apply vibration load. The dynamic force, \(F_{st}\), between the vibratory roller and the soil can be obtained by the two-degree-of-freedom \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Types of Backfill Soil**} & \multirow{2}{*}{\(\rho\) (kg/m\({}^{3}\))} & \multicolumn{2}{c}{**Modulus of Compression**} & \multirow{2}{*}{\(\upsilon\)} & \multicolumn{2}{c}{**Angle of**} & \multirow{2}{*}{**Force of**} \\  & & **(MPa)** & & & **Internal Friction (\({}^{\circ}\))** & **Cohesion (kPa)** \\ \hline Soil-rock mixture & 2100 & 72 & 0.17 & - & - \\ Graded broken stone & 2300 & 120 & 0.35 & 38 & 1 \\ A4 & 1600 & 7 & 0.3 & 35 & 15 \\ A3 & 1600 & 5 & 0.3 & 26 & 18 \\ A2 & 1600 & 7 & 0.25 & 26 & 16 \\ A1 & 1690 & 10 & 0.25 & 48 & 10 \\ Undisturbed soil & 2000 & 30 & 0.3 & 28 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Physical parameters of backfill. Figure 10: Finite element model: (**a**) cross-section of finite element model and (**b**) longitudinal section of finite element model.