## 3 2.5D Finite Element Substructure Method
### A Zigzag-Paraxial Boundary Condition Suitable for 2.5D Finite Element Method
^{T} \tag{17c}\]  
Thus, Equation (14) can be rewritten as  
\[\mathbf{A}_{ss}\frac{\partial^{2}\tilde{\mathbf{u}}}{\partial s^{2}}+\left( \mathbf{A}_{st}+\mathbf{A}_{st}^{T}\right)\frac{\partial^{2}\tilde{\mathbf{u }}}{\partial s\partial t}+\mathbf{A}_{tt}\frac{\partial^{2}\tilde{\mathbf{u}} }{\partial t^{2}}-ik_{x}\left[(\mathbf{A}_{rs}+\mathbf{A}_{rs}^{T})\frac{ \partial\tilde{\mathbf{u}}}{\partial s}+(\mathbf{A}_{rt}+\mathbf{A}_{rt}^{T} )\frac{\partial\tilde{\mathbf{u}}}{\partial t}\right]-k_{x}^{2}\mathbf{A}_{ rr}\tilde{\mathbf{u}}+\omega^{2}\rho\tilde{\mathbf{u}}=0 \tag{18}\]  
with the coefficient matrices of the Equation (18) being  
\[ \begin{cases}\mathbf{A}_{ss}=\mathbf{L}_{s}^{T}\mathbf{D}\mathbf{L}_{s}\\ \mathbf{A}_{st}=\mathbf{L}_{s}^{T}\mathbf{D}\mathbf{L}_{t}\\ \mathbf{A}_{tt}=\mathbf{L}_{t}^{T}\mathbf{D}\mathbf{L}_{t}\\ \mathbf{A}_{rs}=\mathbf{L}_{r}^{T}\mathbf{D}\mathbf{L}_{s}\\ \mathbf{A}_{rt}=\mathbf{L}_{r}^{T}\mathbf{D}\mathbf{L}_{t}\\ \mathbf{A}_{rr}=\mathbf{L}_{r}^{T}\mathbf{D}\mathbf{L}_{r}\end{cases}  \tag{19}\]By discretizing Equation (18) along the \(s\) direction, the displacement vector of any thin layer element can be expressed:  
\[\mathbf{\bar{u}}=\mathbf{N}_{e}\mathbf{\overline{u}} \tag{20}\]  
with the nodal shape function matrix \(\mathbf{N}_{e}\) as  
\[\mathbf{N}_{e}= \begin{bmatrix}N_{1}&0&0&N_{2}&0&0\\ 0&N_{1}&0&0&N_{2}&0\\ 0&0&N_{1}&0&0&N_{2}\end{bmatrix}  \tag{21}\]  
with  
\[N_{1}=\frac{s_{2}-s}{s_{2}-s_{1}} \tag{22a}\] \[N_{2}=\frac{s-s_{1}}{s_{2}-s_{1}} \tag{22b}\]  
In the frequency domain, the discrete value of displacement \(\mathbf{\overline{u}}\) at nodes are prescribed as follows:  
\[\mathbf{\overline{u}}=\left\{\mathbf{\overline{u}}_{1}^{T}\quad\mathbf{ \overline{u}}_{2}^{T}\right\}^{T}=\left\{u_{1x}\quad u_{1y}\quad u_{1z}\quad u _{2x}\quad u_{2y}\quad u_{2z}\right\}^{T} \tag{23}\]  
Equation (20) is taken into Equation (18). The weak form of the equivalent integral is obtained by using the definition of the stress on the surface. \[-\mathbf{B}_{ss}\frac{\partial^{2}\mathbf{\overline{u}}}{\partial s^{2}}- \left(\mathbf{B}_{st}-\mathbf{B}_{st}^{T}\right)\frac{\partial\mathbf{ \overline{u}}}{\partial s}-(\mathbf{B}_{rs}+\mathbf{B}_{rs}^{T})\frac{\partial \mathbf{\bar{u}}}{\partial s}-(\mathbf{B}_{rt}-\mathbf{B}_{rt}^{T}+\mathbf{B}_ {rr}-\mathbf{B}_{tt}+\omega^{2}\mathbf{M})\mathbf{\overline{u}}=\sigma_{z} \tag{24}\]  
with coefficient matrices of the equation  
\[ \begin{cases}\mathbf{B}_{ss}=n_{z}\int\mathbf{N}^{T}\mathbf{A}_{ss} \mathbf{N}dt\\ \mathbf{B}_{st}=n_{z}\int\mathbf{N}^{T}\mathbf{A}_{st}\frac{\partial\mathbf{N }}{\partial t}dt\\ \mathbf{B}_{tt}=n_{z}\int\frac{\partial\mathbf{N}^{T}}{\partial t}\mathbf{A}_{ tt}\frac{\partial\mathbf{N}}{\partial t}dt\\ \mathbf{B}_{rs}=-ik_{x}n_{z}\int\mathbf{N}^{T}\mathbf{A}_{rs}\mathbf{N}dt\\ \mathbf{B}_{rt}=-ik_{x}n_{z}\int\mathbf{N}^{T}\mathbf{A}_{rt}\frac{\partial \mathbf{N}}{\partial t}dt\\ \mathbf{B}_{rr}=-k_{x}^{2}n_{z}\int\mathbf{N}^{T}\mathbf{A}_{rr}\mathbf{N}dt\\ \mathbf{M}=\rho n_{z}\int\mathbf{N}^{T}\mathbf{N}dt\end{cases}  \tag{25}\]  
The stress vector on the right side of the Equation (24) can be expressed as  
\[\sigma_{z}=\left\{-n_{z}\mathbf{L}_{t}^{T}\sigma_{1}\quad n_{z}\mathbf{L}_{t} ^{T}\sigma_{2}\right\}^{T}=\left\{-\tau_{xz}\quad-\tau_{yz}\quad-\sigma_{z} \quad\tau_{xz}\quad\tau_{yz}\quad\sigma_{z}\right\}^{T} \tag{26}\]  
As shown in Figure 4, the zigzag boundary changes from a red surface to a curve with three degrees of freedom at each node. A zigzag boundary condition suitable for 2.5D finite element method is derived in the local coordinate system. Due to the zigzag boundary is limited to rigid base, the 2.5D paraxial boundary condition in the underlying half space can improve the boundary accuracy. The paraxial boundary condition is a paraxial approximation of the exact dynamic stiffness matrix.