## 4 Numerical Modeling
### Geometric Model and Parameters  
To facilitate this study, several simplifications were adopted in the model. The hypotheses are as follows:  
1. The soil was treated as a layered and homogeneous medium, and certain factors such as groundwater level were not taken into account. 2. The deformation of the rammer during the dynamic compaction process was disregarded. 3. The geostatic stress and deformation were not measured, as the primary focus of this study was on the dynamic response of the Yangtze River dike under dynamic compaction. 4. A completely fixed boundary condition was assumed at the bottom boundaries, while the peripheral area of the foundation model was assumed to satisfy an infinite element boundary condition by using nonreflecting boundary. The dimensions of the numerical models were deliberately specified to measure 200 m in length, 200 m in width, and 13 m in height. The slope of the Yangtze River dike was \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**No.** & **Soil Layer** & **Layer Thickness** & **Density** & **Poisson’s Ratio** & **Friction** & **Cohesion** & **Young’s** \\  & & **(m)** & **(kg/m\({}^{3}\))** & **Internal** & **Angle (\({}^{5}\))** & **(kPa)** & **Modulus (MPa)** \\ \hline
1 & plain fill & 2 & \(1.9\times 10^{3}\) & 0.35 & 15.0 & 5.1 & 2 \\
2 & clay & 2 & \(1.87\times 10^{3}\) & 0.30 & 15.6 & 10.3 & 3 \\
3 & silty clay & 9 & \(1.85\times 10^{3}\) & 0.30 & 17.4 & 9.2 & 3 \\
4 & Yangtze River Dike & - & \(2\times 10^{3}\) & 0.25 & 20.1 & 29.8 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Physical and mechanical properties of the soils in the field test site. simulated with a width of 25 m and a height of 15 m, and the location of the dynamic compaction treatment was set at a distance of 40 m from the toe of the dike. The adoption of a large-sized model was aimed at minimizing the disturbance of vibration waves in the boundaries of the model, thereby enhancing the accuracy of the simulation results. Top view and sectional illustrations of the model are presented in Figures 9 and 10, respectively. The relationship between the length of the element size and the wavelength is illustrated in Equation (1), where \(\Delta\)x represents the maximum size of the element and \(\lambda_{\mathrm{T}}\) denotes the wavelength corresponding to the cutoff frequency of interest. As Rayleigh waves generated by dynamic compaction typically have a wavelength larger than 12 m, the element size was set at 0.5 m. The element type used is C3D8R, which is a solid element employing reduced integration technique. The total number of elements is 4,772,986 for the foundation soil. The loading process consists of two steps: the first step involves establishing the geostress equilibrium, and the second step is the actual loading. The time integration algorithm used is an implicit integration method. The damping ratio is set to 5%. To ensure accurate results, a time step of less than 0.01 s is recommended for dynamic compaction vibration analysis. In this study, a time step of 0.005 s was used. Figure 11 shows the mesh of the numerical model. \[\Delta\mathrm{x}\leq\bigg{(}\frac{1}{6}\sim\frac{1}{12}\bigg{)}\lambda_{ \mathrm{T}} \tag{1}\]  
Figure 10: The cutaway view of the numerical model. Figure 9: The top view of the numerical model. Currently, scholars usually use two methods to simulate the impact of dynamic compaction on soils in finite element models: one is a simplified model that approximates the stress process between the compaction hammer and soil as a triangle or sinusoidal waveform; the other treats the hammer as a rigid body that makes contact with the soil surface at a certain velocity. This study chooses the latter method because it better aligns with practical applications. Specifically, the compaction hammer's velocity upon contacting the soil is determined based on its fall distance, and the type of contact between the hammer and soil is established.