## 2 Research Method
### Numerical Model
#### 2.1.3 Validation of Numerical Results  
The acceleration dynamic response value obtained by numerical simulation is compared with the field measured results, in order to verify the reliability of the numerical simulation method (material characteristics, train load, boundary conditions, etc.) adopted in the cross-tunnel. According to the reference [30], the test tunnel site is a double-line standard tunnel of China's high-speed railway trunk lines. The internal height of the tunnel is 8.78 m in height and 12.6 m in width. CRTS II slab ballastless track is used. DASP-V10 engineering platform software was used for data acquisition. The acceleration sensor is INV9828. Two acceleration sensors are arranged at each measuring point to collect the transverse and vertical vibration acceleration of tunnel lining. Figure 3 is the comparison between field tests and corresponding numerical models.  
The same finite element numerical model is established, according to the actual engineering geological conditions on site, as shown in Figure 3. Through on-site investigation, the parameters of the surrounding rock are basically the same as those of rock level V. The tunnel is about 20 m deep. The numerical simulation method presented in this paper is adopted. For example, the surrounding rock adopts the elastoplastic model and Mohr-Coulomb yield criterion, the initial and the secondary support structure adopt the elastic model, the boundary is set as the three-dimensional viscoelastic artificial boundary, and the dynamic load of high-speed train is selected according to Equation (1). The train speed is 300 Km/h. The comparison of vibration acceleration between numerical simulation and field measurement at each measuring point is shown in Figure 4.  
According to the Figure 4, the numerical simulation results are basically consistent with the field measured values, with the maximum relative error of 15% and the same variation trend. It shows that the numerical simulation method adopted in this paper is highly reliability and can be applied to the study of the dynamic response of cross-tunnel.  
The dynamic response value of tunnel lining structure subjected to train load, including displacement and tensile stress, obtained through the corresponding field and model tests, are shown in Table 3. As can be seen from the table, the displacement response of the inverted arch structure of the tunnel is 2.3-14 mm, and the maximum tensile stress is 0.14-0.31 MPa. In the numerical calculation results of this paper, the maximum displacement of the inverted arch is 1-6 mm, and the maximum tensile stress is 0.1-0.6 mpa. The results show that the numerical calculation results are close to the field measurements and model test results, and the dynamic response of the cross tunnel is within a reasonable range.  
Figure 4: Comparison of vibration acceleration between numerical simulation and field measurement: (**a**) Transverse acceleration; (**b**) Vertical acceleration.  
Figure 3: Comparison between field tests and corresponding numerical models: (**a**) Field measurement; (**b**) Numerical model; (**c**) Local model grid; (**d**) Schematic diagram of measuring point layout.