## 2 Establishment of Prediction Model
### Simplified Tunnel Model Using VM
This results in the formation of a complex and multi-factor communication system composed of the train-track-tunnel-soil layer. Therefore, when investigating the environmental vibration of underground rail transit, the study of the overall vibration propagation mechanism is typically divided into the study of the key problems in each subsystem of the propagation path. The TLM-VM-PML method adopted in this paper can be used to study the vibration transmission characteristics of the tunnel foundation-subsoil system. The dynamic responses at different locations of the surface can be obtained by inputting the foundation-subsoil displacement frequency response function derived above. The frequency response function of foundation soil displacement derived from TLM-VM-PML is applied to the solution of the dynamic response of foundation soil around the tunnel foundation. Firstly, it is necessary to conduct an appropriate simplification of the tunnel-soil model. In the existing research, tunnel structures were simplified as equivalent beams embedded in layered soil 3. Subsequently, the tunnel vibration is considered to be the source propagating to soil layers. Considering that the TLM is difficult to disperse vertically, the following simplified assumptions are made in the tunnel-soil system:  
1. Euler-Bernoulli beams in three-dimensional space are used to simulate the tunnel structure. Each node has six degrees of freedom, which encompass the translation in three directions of spatial coordinates and rotation around three coordinate axes. 2. The foundation soil is composed of several horizontal thin-layer elements, each of which has the same soil layer properties. 3. The foundation soil and tunnel structure are in close contact through the interaction node group with no relative displacement. Based on the above assumptions, the tunnel-soil model is developed using the proposed TLM-VM-PML method. As shown in Figure 7, the TLM is used to simulate the layered soil. The elastic layer of the soil is divided into \(N\) thin-layer elements. The PML boundary conditions are used to simulate the elastic half-space below the elastic layers. The tunnel structure is simulated by the Euler-Bernoulli beam. The tunnel structure is set as a circular shield tunnel with outer diameter \(D\), inner diameter \(d\), wall thickness \(\delta\), elastic modulus \(E_{\mathrm{p}}\), mass density \(\rho_{\mathrm{p}}\), Poisson ratio \(v_{\mathrm{p}}\), and rail buried depth \(H\). Accordingto the fundamental principle of the VM, the tunnel-soil interaction node group is set in the inner space of the foundation soil occupied by the tunnel structure. The depth of the node group is equivalent to the buried depth of the tunnel rail. The space coordinate system is established with the ground above the excitation point in the tunnel as the origin. The surface isometric observation points are set along the \(x\) axis with an interval of \(r\). The research idea for solving the dynamic response of the model is illustrated in Figure 8. Firstly, the tunnel foundation is divided into elements to form a tunnel-soil dynamic interaction node group, which is located on the node surface of thin-layer elements. The spatial position coordinates of each node are calculated. Then, the basic solution of frequency-space domain displacement under point loads derived in Section 2.1 is introduced to form the dynamic flexibility matrix \(\mathbf{A}(\mathrm{i}\omega)\) of the common node groups of tunnel and foundation soil. The Euler-Bernoulli beam element is used to simulate the tunnel and its same-volume soil discharge. The impedance matrix \(\mathbf{S}(\mathrm{i}\omega)\) of the tunnel is obtained by calculating the stiffness matrix and mass matrix as well as the soil discharge and combining the impedance function Equation (40). Finally, the dynamic interaction force of the node group is obtained, and the displacement at any point of the foundational soil around the tunnel foundation can be calculated by combining the Green's function of the TLM.