## 2 Establishment of Prediction Model
### Simplified Tunnel Model Using VM  
(1)The VM separates the ground-foundation system  
The VM is a technique that utilizes the ground (G) as a free field, subsequently establishing nodes in the internal space occupied by the foundation (F). These nodes are the nodes of the ground-foundation dynamic interaction. The VM is based on the superposition principle of elastic theory. As shown in Figure 6, in the VM solution model, the layered foundation is first dispersed with thin-layer elements along the vertical line, and node groups are configured in the foundation with the same volume as the foundation. Each node of the node group must be located on the node plane of the divided thin-layer elements, and then the foundation-foundation system is separated. The actual stiffness matrix \(\mathbf{K}_{\mathrm{F}}\) and mass matrix \(\mathbf{M}_{\mathrm{F}}\) of the foundation are expressed as the superposition of soil discharge stiffness matrix \(\mathbf{K}_{\mathrm{F}}^{\mathrm{G}}\) and mass matrix \(\mathbf{M}_{\mathrm{F}}^{\mathrm{G}}\) and stiffness matrix \(\mathbf{K}_{\mathrm{F}}-\mathbf{K}_{\mathrm{F}}^{\mathrm{G}}\), as well as mass matrix \(\mathbf{M}_{\mathrm{F}}-\mathbf{M}_{\mathrm{F}}^{\mathrm{G}}\) in the same volume as the foundation:  
\[\mathbf{K}_{\mathrm{F}}=\mathbf{K}_{\mathrm{F}}-\mathbf{K}_{\mathrm{F}}^{ \mathrm{G}}+\mathbf{K}_{\mathrm{F}}^{\mathrm{G}} \tag{32}\]  
\[\mathbf{M}_{\mathrm{F}}=\mathbf{M}_{\mathrm{F}}-\mathbf{M}_{\mathrm{F}}^{ \mathrm{G}}+\mathbf{M}_{\mathrm{F}}^{\mathrm{G}} \tag{33}\]  
Figure 6: Calculation principle of the VM. Due to the separation of the system, the interaction force and displacement will occur in the dissolving space of each substructure, which, respectively, meet the force balance condition and displacement continuity condition. The force balance conditions can be written as follows:  
\[F_{\mathcal{C}}^{\mathrm{G}}-\widetilde{F}_{\mathcal{C}}^{\mathrm{G}}+F_{ \mathcal{C}}^{F}=0 \tag{34}\]  
where \(F_{\mathcal{C}}^{\mathrm{G}}\), \(\widetilde{F}_{\mathcal{C}}^{\mathrm{G}}\) and \(F_{\mathcal{C}}^{F}\) represent the forces of foundation soil, foundation discharge and foundation respectively. The displacement continuity condition can be expressed as follows:  
\[u_{\mathcal{C}}^{\mathrm{G}}=\widetilde{u}_{\mathcal{C}}^{\mathrm{G}}=u_{ \mathcal{C}}^{F} \tag{35}\]  
where \(u_{\mathcal{C}}^{\mathrm{G}}\), \(\widetilde{u}_{\mathcal{C}}^{\mathrm{G}}\) and \(u_{\mathcal{C}}^{F}\) represent the displacement of foundation soil, foundation discharge and foundation respectively. Based on the VM, the ground-foundation system can be divided into three parts: (1) the foundation part F, (2) free-field layered foundation G, and (3) soil discharge part with the same volume as the foundation. These three parts achieve dynamic coupling through the common nodes of foundation and foundation, that is, the ground-foundation system can be equivalent to (1) + (2) \(-\) (3). (2) Derivation of frequency response function of foundation-subsoil displacement  
The kinematic equilibrium equation of the ground-foundation system under the action of external excitation force \(\mathbf{P}\) is as follows:  
\[\left( \begin{bmatrix}\mathbf{K}_{\mathrm{F}}&\mathbf{K}_{\mathrm{FG}}\\ \mathbf{K}_{\mathrm{CF}}&\mathbf{K}_{\mathrm{G}}\end{bmatrix} -\omega^{2} \begin{bmatrix}\mathbf{M}_{\mathrm{F}}\\ \mathbf{M}_{\mathrm{G}}\end{bmatrix} \right) \begin{pmatrix}\mathbf{u}_{ \mathrm{F}}\\ \mathbf{u}_{\mathrm{G}}\end{pmatrix} = \begin{pmatrix}\mathbf{P}\\ \mathbf{0}\end{pmatrix}  \tag{36}\]  
where \(\mathbf{K}\) and \(\mathbf{M}\) are the stiffness and mass matrices, respectively, and \(\mathbf{u}\) is the displacement vector. The subscripts F and G represent the foundation and ground, respectively. According to the fundamental principle of the VM, the actual stiffness and mass matrix of the foundation are decomposed into the forms of Equations (32) and (33). Substituting Equations (32) and (33) into Equation (36), the dynamic balance equations can be obtained:  
Foundation system:  
\[\left(