## 3 2.5D Finite Element Substructure Method
### Description of the 2.5D Finite Element Method  
Through the decomposition of Figure 2b, the finite domain contains the long tunnel, which can be modeled by the 2.5D finite element method. Different from 2D finite element, each 2.5D element node has three degrees of freedom. The tunnel displacement is expressed as follows:  
\[\mathbf{u}(x,y,z,t)=\int_{-\infty}^{\infty}\mathbf{N}\overline{\mathbf{u}}(y,z )\exp(-ik_{x}x)\exp(-i\omega t)\mathrm{d}\omega \tag{1}\]  
where \(\mathbf{N}\) is the shape function matrix in the 2D isoparametric finite element analysis. The \(\mathbf{u}\)-bar is the nodal displacement with three components in the frequency domain and wavenumber domain. \(i\) is imaginary. The circular frequency is \(\omega\). The wavenumber is given as \(k_{x}=\cos\theta_{\mathrm{p}}\mathrm{sin}\theta_{\mathrm{r}}\omega/c_{b}\) in the \(x\) direction, with \(c_{b}\) being the wave velocity of the elastic underlying half space.  
The dry soil layer is assumed to be an elastic solid medium whose governing equation can be written in global coordinate system \((x,y,z)\) as follows:  
\[\mathbf{L}^{\mathrm{T}}\sigma=-\omega^{2}\rho\overline{\mathbf{u}} \tag{2}\]  
where \(\rho\) is density; \(\sigma\) is stress matrix; \(\mathbf{L}\) is a differential operator; the superscript T represents the transpose of the matrix; specifically, as follows:  
\[\mathbf{L}=\begin{bmatrix}\frac{\partial}{\partial x}&0&0&\frac{\partial}{ \partial y}&0&\frac{\partial}{\partial z}\\ 0&\frac{\partial}{\partial y}&0&\frac{\partial}{\partial x}&\frac{\partial}{ \partial z}&0\\ 0&0&\frac{\partial}{\partial z}&0&\frac{\partial}{\partial y}&\frac{\partial }{\partial x}\end{bmatrix}^{\mathrm{T}} \tag{3}\]  
According to the Hooke's law, the stress-strain relationship is  
\[\sigma=\mathbf{D}\mathbf{\varepsilon} \tag{4}\]  
where \(\mathbf{\varepsilon}\) is strain. It is well known that \(\mathbf{D}\) is a 3D linear elastic constitutive matrix with Lame constants \(\lambda\) and \(G\).  
\[\mathbf{D}=\begin{bmatrix}\lambda+2G&\lambda&\lambda&0&0&0\\ \lambda&\lambda+2G&\lambda&0&0&0\\ \lambda&\lambda&\lambda+2G&0&0&0\\ 0&0&0&G&0&0\\ 0&0&0&0&G&0\\ 0&0&0&0&0&G\end{bmatrix} \tag{5}\]  
The strain-displacement relation is  
\[\mathbf{\varepsilon}=\mathbf{B}\overline{\mathbf{u}} \tag{6}\]  
where \(\mathbf{B}\) is the strain matrix.  
By replacing partial derivative \(\partial/\partial x\) with longitudinal wave number \(-ik_{x}\), the calculation equation of tunnel finite element model in 2.5D form is obtained.  
\[\mathbf{K}\overline{\mathbf{u}}-\omega^{2}\mathbf{M}\overline{\mathbf{u}}= \overline{\mathbf{F}} \tag{7}\]  
where \(\overline{\mathbf{F}}\) is the interaction force vector between the tunnel and the surrounding soil; \(\mathbf{K}\) and \(\mathbf{M}\) are the stiffness matrix and mass matrix of the 2.5D tunnel model, respectively  
\[\mathbf{K}=\iint\mathbf{B}^{T}\mathbf{D}\mathbf{B}Jd\xi d\eta \tag{8}\]\[\mathbf{M}=\iint\mathbf{N}^{T}\rho\mathbf{N}Jd\xi d\eta \tag{9}\]  
where \(J\) is the Jacobian matrix which represents \(d\xi d\eta\) area mapping to \(dxdy\).  
As shown in the Figure 2d, the 2.5D finite element equation of finite domain can be rewritten as follows:  
\[\begin{bmatrix}\mathbf{S}_{pp}&\mathbf{S}_{pq}\\ \mathbf{S}_{qp}&\mathbf{S}_{qq}\end{bmatrix}\begin{Bmatrix}\overline{\mathbf{ u}}_{p}\\ \overline{\mathbf{u}}_{q}\end{Bmatrix}\mathbf{=}\begin{Bmatrix}\mathbf{0}\\ -\overline{\mathbf{F}}_{q}\end{Bmatrix} \tag{10}\]  
with dynamic stiffness matrix  
\[\mathbf{S}=\mathbf{K}-\omega^{2}\mathbf{M} \tag{11}\]  
where subscript \(q\) represents the nodes at the interface of the soil-tunnel system and subscript \(p\) represents all structural nodes except for the structural boundary.