## 2 Numerical Simulation  
### Vehicle-Track-Subgrade Coupled Model  
A refined nonlinear analysis model of the vehicle-track-subgrade system was established using the finite element software ABAQUS ([https://software.3 ds.com/](https://software.3 ds.com/)). During the vehicle modeling, the main structure was simplified into three parts: the carbody, bogie, and wheelset, which were connected through the primary suspension and secondary suspension. The simplified dynamic model of the vehicle [36] is shown in Figure 1. The vehicle model assumes complete symmetry on both sides and does not consider the effect of eccentric forces. The primary suspension and secondary suspension were simplified to undergo completely linear deformations, modeled using connector elements. The dimensions of the vehicle components were modeled on a 1:1 scale, referencing the CRH3 train. The axle load of the vehicle was simulated as a concentrated force applied at the vehicle's center of gravity, with the interactions between components set to transfer forces downward. The wheel tread adopts the wear-type tread. The main parameters of the train vehicle model are shown in Table 1.  
Figure 1: Simplified diagram of the vehicle model. This study focused on the CRTS II slab ballastless track, simulating structures including the rail, fasteners, track slab, CA mortar layer, and supporting layer [37]. The fasteners were simulated using Cartesian connectors, with their calculation parameters shown in Table 2. The rail model was based on the geometric cross-section of a domestic 60 kg/m rail. To replicate the actual conditions of the rail, the track irregularity spectrum was applied to the model using random process theory. According to the fitting formulas for track irregularities specified in the High-speed Railway Ballastless Track Irregularity Spectrum, various irregularities such as gauge irregularity, alignment irregularity, and vertical irregularity were incorporated into the track. Damping materials were defined using Rayleigh damping, considering both mass damping and material damping, with the damping expression as follows:  
\[C=\alpha M+\beta K \tag{1}\]  
where \(\alpha\) is the mass damping coefficient, \(\beta\) is the stiffness damping coefficient. The subgrade section included the subgrade surface layer, the subgrade bottom layer, and the lower embankment. The subgrade width was set to 8.6 m, with a slope ratio of 1:1.5. The foundation section included the surface soil layer and bedrock, with the soil cavity located within the surface soil layer directly beneath the subgrade. To improve the accuracy of the model, the mesh density was increased in the subgrade structure and soil cavity areas. The irregularly shaped soil cavity was discretized using C3D4 elements, while the subgrade foundation and track structure, which have more regular shapes, were divided into C3D8R elements. Although the vehicle model is not included in the dynamic response analysis, the vehicle body was meshed using R3D4 (4-node 3D rigid) and R3D3 (3-node 3D rigid) elements for other structural analyses. Binding constraints are set between the layers of the track slab and between the track and the subgrade surface layer. In total, the track structure contained 42,108 elements and 77,223 nodes, the soil cavity structure had 1671 elements, and the subgrade structure consisted of 293,004 elements. In the analysis, the rail and track structures were modeled as elastic materials, while the subgrade and foundation soils were represented using the Mohr-Coulomb ideal elastic-plastic constitutive model. This approach accounted for the plastic deformation of the roadbed structure and the geotechnical body, allowing the model to effectively simulate subgrade stability and potential collapse under varying load conditions. The parameters of the track-subgrade model are shown in Table 3.