## 3 Numerical Results and Discussion
### 3D-FR-NWT with Radiation Boundary Conditions  
Based on the optimal mesh system, in this study, we applied three types of radiation boundary conditions (1: Sommerfeld radiation condition (SRC), 2: artificial damping zone (ADZ), 3: combined radiation condition between the SRC and ADZ) and compared their performance. First, the SRC was applied only at the sidewall boundary surface. Figure 4 compares the hydrodynamic coefficients (added mass and radiation damping coefficient) and wave excitation vertical forces on the hemispheric heaving buoy. The results were in agreement with the WAMIT results, except for under low-frequency conditions. This is because low-frequency radiated waves (long waves) are reflected at the end boundary of the computational domain (see Figure 5). Hence, in the case of the rectangular calculation domain, if only the Sommerfeld radiation condition is applied, the low-frequency wave near the apex cannot be properly radiated on the end boundary. Therefore, when numerical modeling is performed in a rectangular wave tank, an artificial damping zone must be adopted to solve the wave radiation problem. An artificial damping scheme was used on the free surface to prevent the reflected or re-reflected waves in the fluid domain. Figure 6 presents a plan view image of the free surface with artificial damping zones installed at all boundaries of 3D-FR-NWT. Two artificial damping terms were added to the linearized free surface boundary condition (see Table 2). The efficiency of the artificial damping technique can be increased by applying two damping terms instead of one to the free surface boundary condition [14]. The \(h\)\(f\)-type damping method substitutes the damping terms proportional to the wave elevation and velocity potential to the linearized free surface boundary condition. The \(F_{n}\)\(h\)-type uses \(f_{n}\) and \(h\) damping terms for the free surface boundary condition based on a previous study [13]. The relationship between the two coefficients can be expressed as \(m_{1}^{2}=4m_{2}\) \begin{table}
\begin{tabular}{c c c} \hline \hline Characteristics & Value & Unit \\ \hline Wave frequency (\(w\)) & 1.5–3.5 & rad/s \\ Wave height (H) & 1.0 & m \\ Radius of a floating body (R) & 1.0 & m \\ Draft of a floating body (d) & 1.0 & m \\ Water depth (h) & 20.0 & m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculation conditions for Sommerfeld radiation condition. Figure 3: Convergence test for the number of nodes per wavelength with Sommerfeld radiation condition (\(w\) = 3.2 rad/s): (**a**) heave added mass; (**b**) heave radiation damping coefficient. Based on the optimal case of [14], Figure 7 compares three shape functions of the artificial damping coefficient for various angles (\(q\)). All shapes have a small initial slope when entering the damping zone so that the occurrence of the reflected waves at the initial point of ADZ was reduced. In the case of'shape 1', the slope of damping coefficients increased rapidly compared to the cases of'shape 2' and'shape 3'. Figure 8 shows the radiated wave amplitude to which the three ramp shapes were applied to compare the wave damping effect. The wave frequency was 1.5 rad/s. In the results using'shape 1' and'shape 2', some fluctuations were observed at the starting point of ADZ. This is the effect of the reflected wave at the front of ADZ because the initial slope of the shape function was relatively large. In the case of'shape 3', however, fluctuation of the wave amplitude did not occur. This means that the waves were attenuated properly at the start point of the artificial damping zone (\(x/l=1.0\)). Therefore, in this study, we adopted the'shape 3' ramp function for stable numerical analysis. The ramp function to which the damping coefficient was applied is presented for each artificial damping zone, as expressed in Equations (20)-(22).