## 3 Hybrid Technique Implementation
### BEM Implementation for a Poroelastic Displacement Field  
Based on the weighted residual methodology, Equations (1) and (2) can be written as  
\[\int_{R}\big{(}\sigma_{ij,j}+U_{i}\big{)}\ u_{i}^{*}\ dR=0 \tag{24}\]  
\[\int_{R}\left(q_{i,i}+\dot{\xi}_{i}-\mathbb{C}_{i}\right)\ p_{i}^{*}\ dR=0 \tag{25}\]  
where  
\[\sigma_{ij,j}=(x+1)^{m}\left[C_{ijkl}\ u_{k,lj}-A\delta_{ij}p_{,j}-\beta_{ij} \left(T_{j}+\tau_{1}\dot{\Gamma}_{j}\right)\right]++\frac{m}{x+1}\sigma_{ij}\]  
\[q_{i,i}=-\overline{k}(x+1)^{m}\left(p_{,il}+\rho_{,j}\ddot{u}_{i,i}+\frac{ \partial_{0}+\phi\phi\rho_{,j}}{\phi}\ddot{v}_{i,j}\right)+\frac{m}{x+1}q_{i}\]in which \(U_{i}=\rho F_{i}-\rho\ddot{u}_{i}-\phi\rho x\ddot{v}_{i}\), and \(u^{*}_{i}\) and \(p^{*}_{i}\) are weighting functions. For the first terms of (24) and (25), integration by parts yields  
\[-\int_{R}\sigma_{ij}\;u^{*}_{i,j}\;dR+\int_{R}U_{i}\;u^{*}_{i}\;dR=-\int_{S_{2}} \lambda_{i}\;u^{*}_{i}\;dS \tag{26}\]  
\[-\int_{R}q\;p^{*}_{i,j}\;dR+\int_{R}\dot{\zeta}_{i}\;p^{*}_{i}\;dR-\int_{R} \mathbb{C}_{i}\;p^{*}_{i}\;dR=-\int_{S_{4}}L_{i}\;p^{*}_{i}\;dS \tag{27}\]  
According to [PERSON] [29], we can write  
\[\int_{R}\sigma^{*}_{ij,j}\;\;u_{i}\;dR=-\int_{S}u^{*}_{i}\;\lambda_{i}\;dS- \int_{S}p^{*}_{i}\;L_{i}\;dS+\int_{S}\lambda^{*}_{i}\;u_{i}\;\;dS+\int_{S}L^{* }_{i}\;p_{i}\;\;dS \tag{28}\]  
which may be written as  
\[C^{n}\mathrm{q}^{n}=-\int_{S}\mathrm{p}^{*}\;\mathrm{q}dS+\int_{S}\mathrm{q} ^{*}\;\mathrm{p}dS+\int_{S}\mathrm{q}^{*}\;p\;\;dS+\int_{S}\mathrm{p}^{*}\; \frac{\partial p}{\partial n}\;\;dS \tag{29}\]  
in which  
\[C^{n}= \begin{bmatrix}C_{11}&C_{12}\\ C_{21}&C_{22}\end{bmatrix} ,\;\mathrm{q}^{*}= \begin{bmatrix}u^{*}_{11}&u^{*}_{ 12}&\omega^{*}_{13}\\ u^{*}_{21}&u^{*}_{22}&\omega^{*}_{23}\\ u^{*}_{31}&u^{*}_{32}&\omega^{*}_{33}\end{bmatrix} ,\;\mathrm{p}^{*}= \begin{bmatrix} \lambda^{*}_{11}&\lambda^{*}_{12}&\mu^{*}_{13}\\ \lambda^{*}_{21}&\lambda^{*}_{22}&\mu^{*}_{23}\\ \lambda^{*}_{31}&\lambda^{*}_{32}&\mu^{*}_{33}\end{bmatrix} \]  
\[\mathrm{q}= \begin{bmatrix}u_{1}\\ u_{2}\\ \omega_{3}\end{bmatrix} ,\;\mathrm{p}= \begin{bmatrix}\lambda_{1}\\ \lambda_{2}\\ \mu_{3}\end{bmatrix} ,\;\mathrm{a}^{*}= \begin{bmatrix}\mathrm{a}^{*}_{1}\\ \mathrm{a}^{*}_{2}\\ 0\end{bmatrix} ,\;\mathrm{b}^{*}= \begin{bmatrix}\mathrm{b}^{*}_{1}\\ \mathrm{b}^{*}_{2}\\ 0\end{bmatrix} \]  
Now, we introduce the following definitions:  
\[\mathrm{q}=\psi\;\mathrm{q}^{j},\;\mathrm{p}=\psi\;\mathrm{p}^{j},\;p=\psi_{0} \;p^{j},\;\frac{\partial p}{\partial n}=\psi_{0}\left(\frac{\partial p}{ \partial n}\right)^{j} \tag{30}\]  
Substituting (30) into (29) yields  
\[C^{n}\mathrm{q}^{n}=\sum_{j=1}^{N_{\mathrm{c}}}\left[-\int_{\Gamma_{j}}\mathrm{ p}^{*}\psi\;d\Gamma\right]\mathrm{q}^{j}+\sum_{j=1}^{N_{\mathrm{c}}}\left[ \int_{\Gamma_{j}}\mathrm{q}^{*}\psi\;d\Gamma\right]\mathrm{p}^{j}+\sum_{j=1}^{ N_{\mathrm{c}}}\left[\int_{\Gamma_{j}}\mathrm{a}^{*}\psi_{0}\;d\Gamma \right]p^{j}+\sum_{j=1}^{N_{\mathrm{c}}}\left[\int_{\Gamma_{j}}\mathrm{b}^{*} \psi_{0}\;d\Gamma\right]\left(\frac{\partial p}{\partial n}\right)^{j} \tag{31}\]  
which may be expressed as  
\[C^{i}\mathrm{q}^{i}=-\sum_{j=1}^{N_{\mathrm{c}}}\widehat{\mathbb{H}}^{ij} \mathrm{q}^{j}+\sum_{j=1}^{N_{\mathrm{c}}}\widehat{\mathbb{G}}^{ij}\mathrm{p} ^{j}+\sum_{j=1}^{N_{\mathrm{c}}}\widehat{\mathbb{A}}^{ij}p^{j}+\sum_{j=1}^{N_{ \mathrm{c}}}\widehat{\mathbb{G}}^{ij}\left(\frac{\partial p}{\partial n}\right) ^{j} \tag{32}\]  
in which  
\[\mathbb{H}^{ij}= \begin{cases}\widehat{\mathbb{H}}^{ij}&if\;\;i\
eq j\\ \widehat{\mathbb{H}}^{ij}+C^{i}&if\;\;i=j\end{cases}