## 3 2.5D Finite Element Substructure Method
### A Zigzag-Paraxial Boundary Condition Suitable for 2.5D Finite Element Method
When the seismic wave is incident obliquely, the relationship between the stress and the displacement at \(z=z_{h}\) (as shown in Figure 5) on the underlying half space surface is as follows  
\[\tilde{\mathbf{\sigma}}_{h}=\left\{\tau_{xz}\quad\tau_{yz}\quad\sigma_{z}\right\} ^{T}=-\mathbf{A}_{h}\frac{\partial^{2}\mathbf{\bar{u}}_{h}}{\partial y^{2}}- \mathbf{B}_{h}\frac{\partial\mathbf{\bar{u}}_{h}}{\partial y}+\mathbf{C}_{h} \mathbf{\bar{u}}_{h} \tag{27}\]The matrices \(\mathbf{A}_{h}\), \(\mathbf{B}_{h}\), and \(\mathbf{C}_{h}\) as the 3 \(\times\) 3 matrices are given below  
\[\mathbf{A}_{h}=\frac{iG_{h}c_{s}}{2\omega} \begin{bmatrix}1&0&0\\ 0&(1-2\alpha)/\alpha^{3}&0\\ 0&0&(\alpha-2)/\alpha\end{bmatrix}  \tag{28a}\] \[\mathbf{B}_{h}= \begin{bmatrix}0&0&-\frac{G_{h}c_{s}k_{x}}{2\omega}\Big{(} \frac{\alpha^{3}-2\alpha^{2}}{\alpha^{3}}+1\Big{)}\\ 0&0&-\frac{G_{h}(1-2\alpha)}{\alpha}\\ \frac{G_{h}c_{s}k_{x}}{2\omega}\Big{(}\frac{\alpha^{3}-2\alpha^{2}}{\alpha^{3 }}+1\Big{)}&\frac{G_{h}(1-2\alpha)}{\alpha}&0\end{bmatrix} \] (28b) \[\mathbf{C}_{h}= \begin{bmatrix}i\omega\rho C_{s}+\frac{(G_{h}c_{s}k_{x}^{2}}{2 \omega}\big{(}\frac{\alpha^{3}-2\alpha^{2}}{\alpha^{2}}\big{)}-\frac{iG_{h}k_{ x}(1-2\alpha)}{\alpha}&0\\ \frac{iG_{h}k_{x}(1-2\alpha)}{\alpha}&i\omega\rho c_{p}+\frac{iG_{h}c_{s}k_{x}^{ 2}}{2\omega}\big{(}\frac{1-2\alpha}{\alpha^{3}}\big{)}&0\\ 0&0&i\omega\rho c_{s}+\frac{iG_{h}c_{s}k_{x}^{2}}{2\omega}\Big{]}\end{bmatrix}  \tag{28c}\]  
where the matrices subscript h indicates the half space; the P and SV wave velocities are \(c_{p}\) and \(c_{s}\), respectively; \(\alpha\) is the ratio of \(c_{s}\) to \(c_{p}\). \(G_{h}\) is the shear modulus of the half-space. The thin layer element Equation (24) is assembled by displacement and stress continuity conditions in the soil layer. According to the relationship between the global coordinate  
Figure 4: The 2.5D zigzag boundary condition. Figure 5: The 2.5D paraxial boundary condition. system and the local coordinate system, \(\partial/\partial s=\partial/\partial y\) can be obtained. Finally, combined with the relationship between the surface stress and displacement of the underlying half space Equation (27), the assembled equation of Equations (24) and (27) can be obtained  
\[-\mathbf{A}\frac{\partial^{2}\overline{\mathbf{u}}}{\partial y^{2}}-\mathbf{B }\frac{\partial\overline{\mathbf{u}}}{\partial y}+\mathbf{C}\overline{ \mathbf{u}}=0 \tag{29}\]  
with  
\[\mathbf{A}=\mathbf{B}_{ss}\oplus\mathbf{A}_{h} \tag{30}\]  
where the notation \(\oplus\) is used to describe 2.5D finite element assembly in the matrix. Taking the Fourier transform of Equation (29) into the wavenumber domain and transforming into solving the quadratic eigenvalue problem,  
\[\left(k_{y}^{2}\mathbf{A}+ik_{y}\mathbf{B}+\mathbf{C}\right)\overline{ \mathbf{u}}=\mathbf{0} \tag{31}\]  
where \(k_{y}\) is the wavenumber along the y direction and \(\overline{\mathbf{u}}\) is nodal displacement vector in the wavenumber-frequency domain. According to [PERSON] et al. [44], when the seismic wave propagates in the positive direction of the \(y\)-axis, the wave number satisfies the condition that \(\mathrm{Re}(k_{y})<0\) and \(\mathrm{Im}(k_{y})>0\). Select the corresponding \(k_{y}\) on the right boundary to superimpose the displacement mode as  
\[\overline{\mathbf{u}}=\mathbf{\Phi}\mathbf{E}_{y}\mathbf{\Gamma} \tag{32}\]  
where \(\mathbf{E}_{y}\) is a diagonal matrix and the diagonal elements are \(\exp(-ik_{y}y)\); \(\mathbf{\Phi}\) and \(\Gamma\) are the eigenvector corresponding to the selected eigenvalue and the undetermined coefficient, respectively. Taking the artificial boundary on the right as an example, the surface stress of the thin layer element is discretized.