## 2 Numerical Simulation
### Validation  
The field monitoring location was at the DK820+955 to DK821+200 subgrade section of the Shanghai-Kunming high-speed railway in Guizhou. The vibration monitoring device used was the 291-2 vibration picker developed by the China Earthquake Administration, as depicted in Figure 3. This equipment has an acceleration resolution of 1 \(\times\) 10\({}^{-5}\) m/s\({}^{2}\), with a maximum measurable vibration acceleration of 40 m/s\({}^{2}\). The data-acquisition system, designed by the Beijing Oriental Institute of Vibration and Noise Technology, was paired with DASP-V11 software ([[http://www.coiniv.com/home](http://www.coiniv.com/home)]([http://www.coiniv.com/home](http://www.coiniv.com/home))) for data processing.  
Four positions were selected at 5-m intervals to monitor the vibration response of the surrounding foundation environment during train passage. A schematic diagram of the distribution of the monitoring points is shown in Figure 4.  
Figure 3: Equipment used for field monitoring: (**a**) Model 291-2 vibration picker, (**b**) Data-acquisition instrument.  
The two data groups from measuring point 1, which exhibited the smallest amplitude errors between the measured and simulated results and the highest curve fit, were analyzed using one-way ANOVA with a significance level \(\alpha\) of 0.05. The significance value \(p\) of 0.064 \(>\alpha\) indicated no significant difference between the two groups, confirming the accuracy of the model at measuring point 1. Additionally, the vibration propagation farther from the subgrade had minimal impact on the stability of the soil hole and subgrade, with the measuring points closer to the subgrade showing smaller error values. This demonstrates that the model could be reliably applied to the dynamic response analysis of soil within the subgrade. As illustrated in Figures 5 and 6, both the field monitoring and numerical simulation of vertical vibration acceleration waveforms showed significant variations in vibration acceleration. The peak values in the numerical simulation were slightly higher than those from field monitoring, with errors within 20 mm/s\({}^{2}\) and error rates of 11.18%, 9.96%, 9.99%, and 8.27%, all within an acceptable range. The smaller vibration acceleration observed during field monitoring could be attributed to various factors, including the complexity of site conditions, the presence of vegetation, and the varying soil-damping properties at different locations, which affected the dissipation of vibration energy through the ground. In the numerical simulation, we made certain necessary simplifications, such as treating the infinite foundation as finite elements with viscoelastic boundaries and not accounting for soil property differences. These simplifications helped reduce computational complexity while still capturing the essential dynamic response of the system. Additionally, the train speed during field monitoring was 239 km/h, slightly lower than the simulated speed of 250 km/h. These factors contributed to the higher vibration acceleration in the simulation results. Overall, the small error between the simulation and field monitoring confirmed the model's accuracy and suitability for dynamic response analysis. Future work could refine the model by incorporating variable soil-damping properties and more precise train speed conditions.