## 3 Numerical Results and Discussion
### 3D-FR-NWT with Radiation Boundary Conditions
\[u(x,y)=\frac{m_{0}}{1}\frac{m_{0}}{\cos q}\ [\ 1\ \ \ \cos(\frac{\left|y \ \ \ L_{fsy}\right|}{L_{damp}}\ \theta)\ ]\ on\ S_{A1} \tag{20}\]  
\[u(x,y)=\frac{m_{0}}{1}\frac{m_{0}}{\cos q}\ [\ 1\ \ \ \cos(\frac{\sqrt{\left(x \ \ \ L_{fsy}\right)^{2}+\left(y \ \ \ L_{fsy}\right)^{2}}}{\frac{L_{damp}}{q}}\ \theta)\ ]\ on\ S_{A2} \tag{21}\]  
\[u(x,y)=\frac{m_{0}}{1}\frac{m_{0}}{\cos q}\ [1\ \ \ \cos(\frac{\left|x \ \ \ L_{fsy}\right|}{L_{damp}}\ \theta)\ ]\ on\ S_{A3} \tag{22}\]  
where \((x,y)\) and \(\left(L_{fsy},L_{fsy}\right)\) denote the x and y coordinates of collocation nodes in ADZ and the starting points of ADZ, respectively. \(L_{damp}\) is the length of ADZ and \(m_{0}\) is the target value of the damping coefficient. The optimal angle (\(q\)) is \(\rho/2\). Figure 9 compares the results of applying two different types of damping terms. Shape 3 ramp function and one wavelength-damping zone were applied. In the time-domain result of [14], both the \(h\ \ \ f\)-type and \(f_{n}\ \ \ h\)-type damping schemes exhibited stable damping performance when the damping zone was 1.5 times the wavelength. On the other hand, in FR-NWT, the damping effect was sufficient, even when the length of the damping zone was approximately one wavelength, and there was little difference according to the damping term type. Briefly, the \(f_{n}\ \ \ h\)-type showed better damping performance in time-domain analysis, but there was no significant difference in the frequency-domain problem. The \(f_{n}\ \ \ h\)-type was judged to be slightly better, but there was little difference between the two damping types. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Damping Type** & **Free Surface Boundary Condition** \\ \hline \(h\ \ f\) type & \(\frac{\varphi_{F}}{\varphi_{F}^{2}}+g\frac{\varphi_{F}}{\varphi_{2}}+m_{1} \frac{\varphi_{F}}{\varphi_{1}}+m_{2}\,F=0\) \\ \(f_{n}\ \ \ h\) type & \(\frac{\varphi_{F}}{\varphi_{F}^{2}}+g\frac{\varphi_{F}}{\varphi_{2}}+m_{1} \frac{\varphi_{F}}{\varphi_{1}}+m_{2}\frac{\varphi_{F}}{\varphi_Figure 8: Longitudinal (x direction) distribution of the radiated wave amplitudes for three different shapes of ramp functions with the \(f_{n}\)\(h\)-type damping method (\(w=1.5\) rad/s, \(m_{0}=2.0\), \(L_{damp}/l=1.0\)). Figure 7: Various shapes of the ramp function for the damping coefficients: shape 1 (\(q=p\)), shape 2 (\(q=3p/4\)), shape 3 (\(q=p/2\)). Figure 9: Comparison of the damping effects for different damping terms (\(h\)\(f\) type, \(f_{n}\)\(h\) type): (**a**) radiation wave elevation (\(w=1.5\) rad/s): (**b**) nondimensional heave-added mass; (**c**) nondimensional heave-damping coefficient. Figure 10 presents the heave-induced radiated wave amplitude with an artificial damping zone. The radiation problem was performed by forcibly oscillating a heaving buoy. No nonphysical waves around corners of the computational fluid domain were observed in either frequency case, unlike Figure 5. Therefore, the artificial damping zone works well in the high-frequency and low-frequency regions. Therefore, FR-NWT adopted the \(\hat{F}_{n}\)\(h\)-type damping terms with the shape 3 ramp function. A parametric study of the damping coefficients was conducted. For this analysis, the radiation problem was performed with various damping coefficients in \(w=1.5\) and \(3.0\) rad/s, as shown in Figure 11. The results were compared with those of WAMIT. When the artificial damping coefficient is more than 2.0, the heave-added mass and damping coefficient converged and agreed well with the WAMIT results. Another parametric study on the length of the ADZ was performed (Figure 12). Figure 3:0 Convergence test on the artificial damping coefficients (\(w=1.5\) rad/s and \(w=3.0\) rad/s): (**a**) nondimensional heave-added mass: (**b**) nondimensional heave radiation damping coefficient. Figure 3:1 Radiated wave amplitude distribution on the free surface using ADZ: (**a**) low wave frequency (\(w=1.5\) rad/s): (**b**) high wave frequency (\(w=3.0\) rad/s).