# Numerical Study on Hydrodynamic Performance of a New Semi-Submersible Aquaculture Platform
## 3 Numerical Model
### Main Structure  
#### 3.1.1 Potential Flow Model  
For the main structure of the platform, the linear wave theory is adopted. The fluid is assumed to be inviscid and the motion is irrotational. The flow field's total velocity potential \(\varphi\) consists of the incident wave velocity potential \(\varphi_{i}\), the diffraction potential \(\varphi_{d}\), and the radiation potential \(\varphi_{r}\). \[\varphi=\varphi_{i}+\varphi_{d}+\varphi_{r} \tag{2}\]  
The governing equations and boundary conditions are given as follows [21; 28]:  
Laplace equation:  
\[\
abla^{2}\varphi=0 \tag{3}\]  
Linearized free surface condition:  
\[\frac{\partial\varphi}{\partial n}-\frac{\omega^{2}}{g}\varphi=0 \tag{4}\]  
Structure surface condition:  
\[\frac{\partial\varphi}{\partial n}=n_{j} \tag{5}\]  
Sea bed condition:  
\[\
abla\varphi\to 0 \tag{6}\]  
Infinity condition:  
\[\lim_{R\rightarrow\infty}\sqrt{R}(\frac{\partial\varphi}{\partial R}-ik\varphi )=0 \tag{7}\]  
where \(n\) is the unit normal vector; \(\omega\) is the wave frequency; \(g\) is the gravitational acceleration; and \(k\) is the wave number. The boundary element method is used to calculate wave loads on the main structure. Based on the linearized Bernoulli equation, the distribution of hydrodynamic pressure \(p\) is calculated by:  
\[p=-\rho\frac{\partial\varphi}{\partial t} \tag{8}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Operation Condition** & **Survival Condition** \\ \hline Significant wave height \(Hs\) (m) & 1 & 3.5 \\ Peak wave period \(Tp\) (s) & 4.3 & 6.5 \\ Wave spectrum factor \(\gamma\) & 1 & 1 \\
1-h wind speed (m/s) & 25.8 & 25.8 \\ Current speed (m/s) & 0.5 & 0.88 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Environment conditions. where \(\rho\) is the density of water with \(\rho=1025\,\mathrm{kg/m^{3}}\). The forces and moments of the six-degree-of-freedom can be calculated by integrating the wave pressure acting on the outer surface of the structure:  
\[f_{i}=\iint_{S}-\rho\frac{\partial\varphi}{\partial t}n_{i}dS \tag{9}\]  
The first-order equation of motion for the structure in the frequency domain can be established as follows:  
\[(m_{ij}+\mu_{ij})\ddot{x}_{j}+(\lambda_{ij}+b_{ij})\dot{x}_{j}+c_{ij}x_{j}=f_{ i}^{fk}+f_{i}^{d},\;\;\;i,j=1,2,\ldots,6 \tag{10}\]  
where \(m_{ij}\) is the generalized mass matrix; \(c_{ij}\) is the generalized hydrostatic restoring stiffness; \(\mu_{ij}\) is the frequency-dependent added mass; \(\lambda_{ij}\) is the frequency-dependent radiation damping; \(b_{ij}\) is the additional damping; \(f_{i}^{fk}\) is the Froude-Krylov force; and \(f_{i}^{d}\) is the diffraction force. This study considers only the linear additional damping resulting from the fluid viscosity. Due to the lack of relevant model test data for the platform, the additional linear damping in heave, roll, and pitch directions are assumed to be 5%, 8%, and 8% of the critical damping, respectively. The added mass matrix, radiation damping matrix, and the first-order wave force are obtained using the three-dimensional Green function method, which is based on the wave diffraction and radiation theory. The motion response characteristics can be quantified by normalizing the response amplitude to the input amplitude. This normalization process yields the response amplitude operator (RAO). The RAOs for the motion of the structure are determined by solving the frequency domain motion Equation (10). In addition to the first-order wave force \(f_{i}^{fk}+f_{i}^{d}\), the structure is also subjected to the second-order wave force, which includes the mean wave drift force, difference-frequency wave force, and sum-frequency wave force. In this study, the second-order wave force is calculated by the middle-field method proposed by [PERSON] [41].