## 3 Validation of Proposed CEDM  
### Model Test and Numerical Simulation  
Figure 3 shows the laboratory test design and the finite element model (FEM) for DC. As shown in Figure 3a,b, the testing equipment consisted of a loading device, a model box filled with soil sample, a measurement cell, and a designed data-acquisition and analysis system. The impact load was applied by raising the drop hammer to a certain height and letting it fall freely. The drop hammer was a cylinder with a diameter of 135 mm, a height of 14.5 mm and a weight of 16.8 kg. The model box was made of wooden plate with a thickness of 2 cm. The height of the soil foundation was 120 cm, and the horizontal dimensions were 80 cm by 80 cm. Soil was dried, crushed, sieved, and filled to the model box layer by layer. Tiny earth pressure gauges were placed in the designated location to measure the stress distribution, as shown in Figure 3a,b. Figure 3: Laboratory test design and numerical simulation for dynamic compaction: (**a**) front view and (**b**) top view of model test; and (**c**) 2D axisymmetric FEM. Figure 2: Solution procedure for solving ellipsoidal parameters using trial-and-error method. Based on the designed model test and considerations of symmetry, a 2D axisymmetric FEM was established based on the ABAQUS software [31] to reproduce the field case, as shown in Figure 3c. The width and depth of the soil foundation were set to 40 cm and 80 cm, respectively, while the tamper radius, height, and weight were set to 67.5 mm, 14.5 mm, and 4.2 kg, respectively. Following the method proposed by [PERSON] et al. [13], the applied impact loading was simulated by exerting an initial velocity on the tamper to impact the ground surface. The friction effect was simulated by defining the tangential contact between the tamper and the soil or between soil particles as proportional to the normal force using the Coulomb friction law, allowing the interfaces to separate from each other when the shear stress exceeded a critical value (frictional resistance). In this study, a four-node quadrilateral element (CPE4) with reduced integration was adopted to perform the mesh generation. To avoid a sudden change in mesh density and improve computational efficiency, the tamper was divided into 435 fine grids, while a gradient grid division scheme was applied to the soil foundation, as illustrated in Figure 3c. It can be seen that, starting from the fine grid near the tamping point, the grid size gradually increases as the distance from the tamping point increases. Consequently, 14,400 elements for soil foundation were generated. The side boundaries were fixed against the horizontal movement while the bottom was fixed in both horizontal and vertical directions. Because cap models [13; 22; 23; 24; 25] have been widely used to simulate the dynamic behavior of soil under DC, a cap model [25] with a shear yield surface and an elliptical volumetric hardening cap was adopted to simulate the DC process, as shown in Figure 4.  
The shear failure surface is represented by  
\[F_{s}=q-p\tan\varphi-c=0 \tag{31}\]  
where \(c\) and \(\varphi\) are the Drucker-Prager model parameters. The expression for the hardening cap is as follows:  
\[F_{c}=\sqrt{\left(p-p_{a}\right)^{2}+\left(\chi q\right)^{2}}-\chi(c+p_{a}\tan \varphi)=0 \tag{32}\]  
where \(p\) and \(q\) are the principal and deviator stresses, respectively; \(\chi\) is the parameter controlling the shape of the cap; and \(p_{a}\) is the intersection point of shear failure surface and cap, which may be expressed as follows:  
\[p_{a}=\frac{p_{b}-\chi c}{1+\chi\tan\varphi} \tag{33}\]  
Figure 4: Yield surface of the cap model in \(p\)-\(q\) plane. where \(p_{b}\) is the isotropic yield stress related to the plastic volumetric strain (\(\varepsilon_{\rm v}^{p}\)), which may be expressed as follows:  
\[p_{a}=p_{0}-\frac{1}{A}\ln\left(1-\frac{\varepsilon_{\rm v}^{p}}{B}\right) \tag{34}\]  
where \(p_{0}\) is the in situ mean effective stress; and \(A\) are \(B\) the volumetric hardening coefficients.