# Microscopic Numerical Analysis of Sand Liquefaction Under Subway Train Load
## 3 Numerical Modeling and Microscopic Parameter Calibration
### Numerical Simulation Calculation Process  
When generating numerical samples, if particles are generated according to the actual particle size, the resulting particles can reach millions or even tens of millions. Due to the current performance of computers, the more particles there are, the greater the impact on computational efficiency. In order to perform simulation calculations quickly and efficiently, a reasonable method is to use as few particles as possible without affecting the numerical model results. There are two main methods to reduce the number of particles. One is to generate numerical samples with the same size as indoor tests, and then moderately increase the average particle size of the numerical samples to achieve the goal of reducing the total number of particles; the second is to construct numerical samples using actual particle size, while moderately reducing their size to achieve a reduction in the number of particles. It should be noted that a small number of particles may have a certain impact on the simulation results, so it is necessary to control the number of particles appropriately while ensuring computational efficiency. [PERSON] et al. [45] conducted research on this issue and found that when the ratio of the appearance size of the numerical sample to the median particle size inside is around 30 or exceeds 30, the change in particle number no longer has a significant impact on the accuracy of the simulation, and the size effect can be ignored. [PERSON] et al. [46] used the particle radius expansion method to obtain the required porosity to match the macroscopic response of real materials. Therefore, this article refers to the research results to reasonably enlarge and shrink the particle size or model size.  
A three-dimensional, cylindrical, numerical specimen (70 \(\times\) 140 mm) was generated, with the particle size distribution based on the results of laboratory particle analysis. The average particle diameter was scaled up to reduce the total number of particles, ensuring computational feasibility while maintaining the gradation characteristics of the tested sand. Basic physical parameters were set, including a particle density of 2640 kg/m\({}^{3}\) and a contact stiffness of 3 \(\times\) 10\({}^{8}\) N/m.  
A target confining pressure (100-200 kPa) was applied via a servo system to achieve isobaric consolidation. The specimen was considered stable when the volume change rate became less than 5 mm\({}^{3}\) per 5 min, ensuring uniform initial stress distribution before cyclic loading.  
Using the trial-and-error method, the friction coefficient (1.7) and damping coefficient (0.7) were adjusted to match the simulated stress-strain curves with laboratory triaxial compression test data. This calibration process ensured that the numerical model accurately reproduced the macro-mechanical behavior of the sand.  
Biased sinusoidal waves were used to simulate subway train vibration loads, with frequencies ranging from 0.5 to 2.5 Hz and dynamic stress amplitudes of 10-20 kPa. A comparative analysis was conducted under seismic loading conditions (60 kPa bidirectional waves). The constant volume method was employed to simulate undrained conditions, with lateral wall velocities controlled to maintain a fixed specimen volume throughout the test (Figure 6).  
Input data comprised dynamically tracked mesoscopic parameters (coordination number, porosity, and contact force chains) and macroscopic responses (excess pore pressure and axial strain). Output data included liquefaction cycle counts and stress-strain hysteresis curves for quantitative validation against experimental results, thereby enabling a rigorous analysis of deviations attributable to model simplifications (e.g., limitations of the linear contact model).