## 3 Dual Boundary Element Method (DBEM)
### Inner Solution
\begin{array}{l}\left[(U_{ij})_{\Gamma^{R}\Gamma^{R}}\quad(U_{ij})_{\Gamma^{ R}\Gamma^{+}}\quad(U_{ij})_{\Gamma^{R}\Gamma^{-}}\right]\\ \left((U_{ij})_{\Gamma^{+}\Gamma^{R}}\quad(U_{ij})_{\Gamma^{+}\Gamma^{+}}\quad (U_{ij})_{\Gamma^{+}\Gamma^{-}}\right]\\ \left(L_{ij}\right)_{\Gamma^{-}\Gamma^{R}}\quad(U_{ij})_{\Gamma^{-}\Gamma^{+}} \quad(U_{ij})_{\Gamma^{-}\Gamma^{-}}\end{array} \left[ \begin{array}{c}( \frac{\partial\phi_{j}}{\partial n})_{\Gamma^{R}}\\ (\frac{\partial\phi_{j}}{\partial n})_{\Gamma^{+}\Gamma^{-}}\\ \left(\frac{\partial\phi_{j}}{\partial n}\right)_{\Gamma^{-}}\end{array}  \right]=\left[ \begin{array}{c}(T_{ij})_{\Gamma^{R}\Gamma^{R}}\quad(T_{ij})_{ \Gamma^{R}\Gamma^{+}}\quad(T_{ij})_{\Gamma^{R}\Gamma^{-}}\\ (T_{ij})_{\Gamma^{+}\Gamma^{+}}\quad(T_{ij})_{\Gamma^{+}\Gamma^{-}}\\ (M_{ij})_{\Gamma^{-}\Gamma^{-}}\end{array} \right]\left[ \begin{array}{c}( \phi_{j})_{\Gamma^{R}}\\ (\phi_{j}^{+})_{\Gamma^{+}}\\ (\phi_{j}^{-})_{\Gamma^{-}}\end{array} \right]\end{array} \tag{19}\]  
In Equation (19), there are four types of the integral of the fundamental solution as described below:  
\[ \begin{array}{l}\left[\mathbf{U}\right]=\left(U_{ij}\right)=\int\limits_{ \Gamma_{j}}\left(\ln r\right)d\Gamma(Q),\ \left[\mathbf{T}\right]=\left(T_{ij}\right)=\int\limits_{\Gamma_{j}}\frac{ \partial}{\partial n_{Q}}(\ln r)d\Gamma(Q)-\pi\delta_{ij},\\ \left[\mathbf{L}\right]=\left(L_{ij}\right)=\int\limits_{\Gamma_{j}}\frac{ \partial}{\partial n_{P}}(\ln r)\,d\Gamma(Q)+\pi\delta_{ij},\ \left[\mathbf{M}\right]=\left(M_{ij}\right)=\int\limits_{ \Gamma_{j}}\frac{\partial^{2}}{\partial n_{P}\partial n_{Q}}(\ln r)d\Gamma(Q). \end{array}  \tag{20}\]  
For \(i\
eq j\), the Gaussian quadrature numerical integration can readily perform the above integrals with no singularity. Whereas for \(i=j\), the singularity makes the integrals difficult and in that case, the integrals can be calculated analytically. \[ \begin{array}{l}\left(U_{ij}\right)=l_{j}\left[\ln\left(\frac{l_{j}}{2} \right)-1\right],\ \left(T_{ij}\right)=-\pi,\\ \left(L_{ij}\right)=\pi,\ \left(M_{ij}\right)=\frac{4}{l_{j}}.\end{array}  \tag{21}\]  
where \(l_{j}\) is the length of the \(j\)th element. By substituting boundary conditions given by Equations (9), (11), and (16) into Equation (19), we obtain the following two different algebraic equations depending on which boundary \(P(x,z)\) belongs to. \[\begin{array}{l}\left  
\[ \begin{array}{l}\left([\mathbf{M}]-\frac{\omega^{2}}{\beta}[\mathbf{L}]\right)( \phi)^{(k)}_{\Gamma^{+}}+([\mathbf{M}]-ik[\mathbf{L}])(\phi)^{(k)}_{\Gamma^{M} }+[\mathbf{M}](\phi)^{(k)}_{\Gamma^{R}}+[\mathbf{M}](\phi)^{(k)}_{\Gamma^{W}}+ \left([\mathbf{M}]-\frac{i\omega[\mathbf{L}]}{(\beta)^{(k-3/2)}}\right)(\phi^{ +})^{(k)}_{\Gamma^{+}}\\ +\frac{i\omega[\mathbf{L}]}{(\beta)^{(k-3/2)}}(\phi^{-})^{(k)}_{\Gamma^{+}}+ \left([\mathbf{M}]-\frac{i\omega[\mathbf{L}]}{(\beta)^{(k-3/2)}}\right)(\phi^{ -})^{(k)}_{\Gamma^{-}}+\frac{i\omega[\mathbf{L}]}{(\beta)^{(k-3/2)}}(\phi^{+}) ^{(k)}_{\Gamma^{-}}=-\frac{2\gamma gA}{\omega}[\mathbf{L}]\left(\frac{\cosh(k +h)}{\cosh\Omega h}\right)_{\Gamma^{M}}\\ \\ \end{array}  \tag{23}\]  
\[ \begin{array}{l}P\in\Gamma^{-}\\ \text{where }(\beta)^{(k-3/2)}=\frac{4C_{D}}{3\pi}\left|\frac{\partial\phi^{+}}{ \partial\pi}\right|^{(k-3/2)}.\\ \end{array} \]  
The superscript \((k)\) in Equations (22) and (23) represents the current iteration step and the iteration step \((k-3/2)\) on \((\beta)^{(k-3/2)}\) means the average of \((k-2)\) and \((k-1)\). By incorporating the iterative scheme, the unknown velocity potentials and normal velocities on the boundaries are calculated repeatedly until a desired accuracy is reached. When applying [PERSON]'s model, \(\beta\) is replaced by \(\frac{2\gamma u}{kb}\), then the solutions can be obtained without iterative calculations.