## 4 Conclusions  
In this study, we developed a confocal ellipsoidal densification model (CEDM) for estimating compacted zones with different degrees of density due to dynamic compaction. In addition, a self-designed laboratory test and a 2D axisymmetric FEM were performed to investigate the DC process. Based on the obtained results, the applicability of the proposed CEDM was verified. The following conclusions were drawn:  
1. A CEDM composed of HCZ and WCZ was proposed for describing the subarea characteristics of improvement ranges of soil foundations under DC. Based on the confocal assumption of HCZ and WCZ ellipses, a mass balance equation considering variations in soil dry density in different compacted zones was established. Six available parameters, including radius of tamper, crater depth, initial dry density and its corresponding friction angle, and maximum dry density and its corresponding friction angle, were selected to solve the HCZ and WCZ ellipsoidal parameters (center point, major and minor axes). 2. The development pattern of the crater depth obtained by FEM simulations was consistent with the measured results, and the relative errors between the simulated and measured values under different tamping times were within \(\pm\)10%. In addition, the same attenuation laws for peak dynamic stresses in the vertical and horizontal directions were observed. The simulated peak dynamic stresses at specific positions were close to the measured values, and their relative errors remained consistently below 15%. These results demonstrated that the established FEM exhibits reliable capability for simulating the DC process and can be further used for verifying the applicability of the proposed CEDM. 3. Because the edge of the densification ellipsoid was assumed to pass through the corner point of the tamper or crater, HCZ and WCZ ellipses tended to underestimate improvement in lateral extent in this area. However, the overall compacted zones corresponding to initial and maximum dry densities after the first, third, fifth, and seventh tampings obtained from simulations were well fitted by the predicted HCZand WCZ ellipses, with fitting degrees of over 90%. These results indicated the feasibility of the proposed CEDM for estimating improvement ranges of soil foundations caused by DC. 4. With increases in tamping times, the HCZ ellipse moved down in the vertical direction without volumetric expansion, while the WCZ ellipse expanded along both depth and lateral directions. Specifically, as the number of tampings increased from one to seven, the HCZ ellipse moved downward by 7 cm, while the major and minor axes remained unchanged. At the same time, the center point of WCZ ellipse moved downward by 10 cm, while the major axes increased from 20 cm to 23.5 cm and the minor axes increased from 12 cm to 14.5 cm. These findings indicated that successive tamping mainly contributed to WCZ expansion, with an average area expansion rate of approximately 7%. The soil was treated as homogeneous and isotropic in the proposed CEDM. However, idealized elliptical densification zones and confocal assumptions considering symmetry do not hold true in real-world scenarios. In future research, we will investigate the flexibility of the elliptical model for estimating dynamic compaction effects in real-world scenarios, focusing on irregular or heterogeneous soil conditions through application of piecewise treatment or a spatial interpolation technique. Conceptualization, H.S., F.Z., X.L., K.S. and F.X. ; methodology, H.S. and F.Z. ; software, H.S. and F.Z. ; validation, H.S., F.Z., X.L., K.S. and F.X. ; formal analysis, H.S. and F.Z. ; investigation, H.S., F.Z., X.L., K.S. and F.X. ; resources, H.S., F.Z., X.L., K.S. and F.X. ; data curation, H.S., F.Z., X.L. and K.S. ; writing--original draft preparation, H.S. and F.Z. ; writing--review and editing, H.S., F.Z., X.L., K.S. and F.X. ; visualization, H.S. and F.Z. All authors have read and agreed to the published version of the manuscript.