## 5 Conclusions  
The DEM-FDM coupled simulation of pullout tests was performed on two types of geogrids embedded within a ballast sample. This method was not only able to reproduce the deformation characteristic of the geogrid using the continuous elements in FLAC3D but could also investigate the micro-mechanical behavior of ballast particles using PFC3D. The simulation results show that the DEM-FDM coupled method can well predict the relationship between pullout force and displacement. Compared with the DEM geogrid model with bonded particles, the development of FDM geogrid stress and deformation were more accurately simulated. For the ballast particles with a particle size of 40 mm and the same geogrid reinforcement area, the triaxial geogrid with an aperture of 75 mm shows better pullout resistance than the biaxial geogrid with an aperture of 65 mm, which is consistent with the experimental results. Most of the geogrid deformation and particle movement occurred mainly near the geogrid reinforcing zone, where a strong interaction forms between the geogrid and particles. Moreover, the biaxial geogrid reinforced ballast samples compacted at the initial 10 mm and then expanded; meanwhile, the triaxial geogrid reinforced ballast samples compacted up to a displacement of about 18 mm and then expanded. The extent of dilation in the triaxial geogrid reinforced sample was larger than that in the biaxial geogrid sample. Furthermore, the interlocking effect was the largest at about 70 mm on both sides of the geogrid. The triaxial geogrid showed much larger front wall resistance than the biaxial geogrid in the fully confined zone. The particles near the geogrid reinforced area produced larger displacement and rotation. The greater the displacement and rotation of the ballast, the stronger the interlocking between the geogrid and the ballast. The particle movement near the triaxial geogrid was larger than that of the  
Figure 14: Average vertical displacement of surcharge block.  
Figure 13: Vertical displacement of surcharge block after pullout: (**a**) biaxial geogrid and (**b**) triaxial geogrid.  
biaxial geogrid. Moreover, the dilation in the triaxial geogrid reinforced sample was larger than that in the biaxial geogrid sample.  
Conceptualization, J.F. and C.C.; methodology, J.L. and C.C.; software, J.L. and R.R.; validation, J.F. and J.L.; formal analysis, C.C. and R.R.; investigation, J.L. and C.C.; resources, R.R.; data curation, J.L.; writing--original draft preparation, J.L.; writing--review and editing, J.F. and C.C.; visualization, J.L.; supervision, C.C. and R.R.; project administration, J.F.; funding acquisition, C.C. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The research described in this paper was financially supported by the Science and Technology Planning Project of Hunan Province, China (No. 2019 RS2050), the National Natural Science Foundation of China (No. 51708438), Key Research and Development Program in Hunan Province (2022 SK2099) and the Science and Technology Project of the Hunan Provincial Department of Water Resources (XSKJ2021000-23).  
The authors declare no conflict of interest.