# Numerical Study on Hydrodynamic Performance of a New Semi-Submersible Aquaculture Platform
## 4 Results and Discussion  
### Verification of Numerical Results  
Hydrodynamic coefficients, including the added mass, damping coefficient, the first-order wave force, and the second-order wave force, are obtained using the software package HydroStar v8.00 [47]. This software is based on the three-dimensional Green function potential flow theory in a finite water depth. The project site has a water depth of only 20 m, which classifies it as a shallow-water area. Therefore, the effects of the second-order differential frequency forces become important. It is necessary to calculate the full quadratic transfer function (QTF). In this study, the wave frequency ranges from 0.05 rad/s to 2.0 rad/s, with an interval of 0.05 rad/s, resulting in a total of 40 wave frequencies. The wave frequency range covers the most common wave frequencies observed in our study area, and the interval ensures a sufficient resolution for the analysis. The wave direction ranges from 0\({}^{\circ}\) to 345\({}^{\circ}\), with an interval of 15\({}^{\circ}\), giving a total of 24 wave directions. The wave direction range represents the full circle of possible wave directions, and the interval is consistent with the wind direction data we collected from the meteorological station. Basically, the same selection of the range and interval of the wave frequency and wave direction for the common frequency domain analysis can be found in [38]. The mesh convergence analysis is conducted to examine the effects of mesh size on the numerical results of hydrodynamic coefficients. Three different mesh sizes of 1.0 m, 1.5 m, and 2.0 m are selected for the analysis, with respective total mesh numbers of 4846, 2221, and 1022, respectively. A quadrilateral mesh is utilized, and the mesh size is decreased in areas with complex surface shapes. Figures 10-12 present the RAO results of surge, heave, and pitch under three different sizes. Based on these results, a mesh size of 1.0 m is adopted to balance the accuracy and efficiency of the numerical results in this study. The mesh model of the platform is illustrated in Figure 13. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Loading** & & & & **Wind** & **Current** & **Wave** & **Wind** & **Current** & **Line** \\
**Case** & & & & **Speed** & **Speed** & **Direction** & **Direction** & **Direction** & **Condition** \\  & & & & **(m/s)** & **(m/s)** & **(\({}^{\circ}\))** & **(\({}^{\circ}\))** & **(\({}^{\circ}\))** & **(\({}^{\circ}\))** \\ \hline LC1 & 3.5 & 6.5 & 1 & 25.8 & 0.88 & 180 & 180 & 180 & Intact \\ LC2 & 3.5 & 6.5 & 1 & 25.8 & 0.88 & 157.5 & 157.5 & 157.5 & Intact \\ LC3 & 3.5 & 6.5 & 1 & 25.8 & 0.88 & 135 & 135 & 135 & Intact \\ LC4 & 3.5 & 6.5 & 1 & 25.8 & 0.88 & 112.5 & 112.5 & 112.5 & Intact \\ LC5 & 3.5 & 6.5 & 1 & 25.8 & 0.88 & 90 & 90 & 90 & Intact \\ LC6 & 3.5 & 6.5 & 1 & 25.8 & 0.88 & 180 & 180 & 180 & Line 2 damaged \\ LC7 & 3.5 & 6.5 & 1 & 25.8 & 0.88 & 157.5 & 157.5 & 157.5 & Line 2 damaged \\ LC8 & 3.5 & 6.5 & 1 & 25.8 & 0.88 & 135 & 135 & 135 & Line 1 damaged \\ LC9 & 3.5 & 6.5 & 1 & 25.8 & 0.88 & 112.5 & 112.5 & 112.5 & Line 1 damaged \\ LC10 & 3.5 & 6.5 & 1 & 25.8 & 0.88 & 90 & 90 & 90 & Line 1 damaged \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Loading cases of survival conditions. Figure 11: RAO results of heavy for different mesh sizes at the 180\({}^{\circ}\) heading angle. Figure 12: RAO results of pitch for different mesh sizes at the 180\({}^{\circ}\) heading angle. Figure 10: RAO results of surge for different mesh sizes at the 180\({}^{\circ}\) heading angle. The middle-field method, proposed by [PERSON] [41], has been implemented in the software HydroStar v8.00 [47]. The mean drift forces are calculated using both the far-field method and the middle-field method. Figures 14-16 compare the mean drift forces of surge, sway, and yaw under different wave incident angles obtained using these two methods.