## 4 Numerical Modeling
### Validation of the Numerical Model  
Figure 12 illustrated the vertical vibration acceleration contour map of the dynamic compaction site obtained by finite element analysis at different times. It can be observed that at 0.2 s, the dynamic compaction has caused vibration in the soil within a radius of 20 m, and at 0.5 s, the toe of the dike is also affected and vibrates, followed by the propagation of the vibration wave to the top of the dike. Therefore, the required signals for the study would be collected within the simulation time, including the PGA at the top of the dike. The energy propagation of the dynamic compaction wave weakens with the distance from the compaction point, and the surface vibration acceleration of the soil gradually decreases with the distance from the compaction point. These are consistent with the performance of the field tests, and quantitative validation is described in the following paragraphs. The validity of numerical simulation results was assessed by comparing them with on-site detection data. In the simulation, detection points were established every 5 m along the perpendicular line from the compacted point to the dike. [PERSON] et al. [28] found the distance decay effect of the peak ground acceleration by summarizing a large amount of dynamic compaction field data. The vibration decay law can be fitted by the following negative exponential power function, as expressed in Equation (2). \[a\;or\;v=\mathrm{k}r^{-\beta} \tag{2}\]  
where \(\beta\) is the attenuation coefficient, k is the equivalency coefficient, \(a\) is the peak vibration acceleration (cm/s\({}^{2}\)) of the measuring point, \(v\) is the peak vibration velocity (mm/s) of the measuring point, and \(r\) is the horizontal distance from the tamping point to measuring point. Table 7 displays the values of fitted functions' parameters. Figure 13 displayed the comparison between numerical simulation results and field measurements. In order to quantify the differ  
Figure 11: Mesh generation of the numerical model. ence between the numerical simulation and field test results, the relative difference between the fitted curves was calculated by Equation (3) and the results are displayed in Figure 14.  
\[D(r)=[F(r)-M(r)]/F(5)\times 100\% \tag{3}\]  
where \(F(r)\) is the fitted function of field test results, \(M(r)\) is the fitted function of simulation results, \(D(r)\) is the relevant difference between numerical simulation fitted curve and field test fitted curve, and \(r\) is the horizontal distance from the tamping point to measuring point. As observed in Figure 13, the numerical simulation performed by ABAQUS can accurately simulate the propagation characteristics of dynamic compaction vibration effects on the soil surface and reflect the attenuation laws of PGA and PGV in both the vertical and horizontal directions. The attenuation laws were characterized by negative power function curves, which are in close agreement with the field measurement results. As shown in Figure 14, the majority of the relevant differences are within 10%. However, there are still some discrepancies between the numerical calculations and the field measurements, especially for PGVs between 5 and 15 m. This is mainly due to the fact that the propagation of vibration acceleration and velocity on the surface is significantly affected by the geological conditions of the site. Although the numerical simulation used the same soil mechanics parameters as the field, the soil was simplified as layered homogeneous materials in the analysis, while the field site is complex with uneven soil distribution and uneven surface. In non-uniform soil layers, the dynamic compaction waves will undergo a large amount of reflection, dissipation, and refraction, leading to a faster attenuation rate than the ideal case, resulting in an overestimation of the numerical simulation results. However, these differences do not affect the study of the phenomenon and the laws involved.