## 2 Numerical Theories
### Potential Flow Theory  
The research by [PERSON] [38; 42] suggested two theories for calculating the hydrodynamic loads. One is the Morison's equation and the other is the potential flow theory. [PERSON]'s equation is applicable for calculating the hydrodynamic loads on slender cylindrical structures when the effects of diffraction and radiation damping are negligible. In order to model the interaction between the waves and the structures in a more accurate way, the potential flow theory should be adopted, which is capable of considering the forces induced by both the diffraction and the radiation. Therefore, in the present study the potential flow theory is applied. By defining a potential function \(\phi(x,\ y)\), the governing equation for \(\phi\) is:  
\[\
abla^{2}\phi=0, \tag{9}\]  
where \(\partial\phi/\partial x=u,\partial\phi/\partial y=v\), and \(\partial\phi/\partial z=w\). \(u\), \(v\), and \(w\) are the velocity components of the flow fields in the Cartesian coordinates. After determining the boundary conditions at the free surface, the surface of the structures, and the sea bed, the potential function can be solved and the velocity distribution can be obtained. A more detailed introduction to the potential flow theory can be found in the study by [PERSON] [43]. Then the forces acting on the floating structures can be determined as:  
\[\overset{\rightarrow}{F}_{WAP}=\overset{\rightarrow}{F}_{W}+\overset{ \rightarrow}{F}_{HS}+\overset{\rightarrow}{F}_{RD}+\overset{\rightarrow}{F}_ {AM}, \tag{10}\]  
\[\overset{\rightarrow}{F}_{AM}=-AM_{RP}\overset{\rightarrow}{a}_{p}, \tag{11}\]\[\overset{\rightarrow}{F}_{W}=\frac{1}{N}\sum_{k=-\frac{N}{2}-1}^{\frac{N}{2}}W[k] \sqrt{\frac{2\pi}{\Delta t}}S_{\xi}^{2-sided}(\omega)X(\omega,\beta)|_{\omega=k \Delta\omega}\omega^{\frac{2\pi\mathrm{bit}}{N}}\,, \tag{12}\]  
\[\overset{\rightarrow}{F}_{HS}=\rho gV_{0}\delta_{3}-C^{Hydrostatic}x, \tag{13}\]  
\[\overset{\rightarrow}{F}_{RD}=-\int\limits_{0}^{t}K_{1}(t-\tau)\dot{x}(\tau) d\tau, \tag{14}\]  
where \(\overset{\rightarrow}{F}_{WAP}\) is the total loads at the reference point; \(\overset{\rightarrow}{F}_{W}\) is the incident-wave excitation force at reference point; \(\overset{\rightarrow}{F}_{HS}\) is the hydrostatic forces at the reference node; \(\overset{\rightarrow}{F}_{RD}\) is the radiation memory-effect force at the reference point; \(\overset{\rightarrow}{F}_{AM}\) is the total added mass forces from all contributions; \(AM_{RP}\) is the added mass due to radiation at the reference point; \(\overset{\rightarrow}{a}_{p}\) is the linear acceleration of the structure; \(W[k]\) is Fourier transform of a white noise time series with unit variance; \(S_{\xi}^{2-sided}(\omega)\) is the wave spectrum (P-M spectrum in this paper); \(X(\omega,\beta)\) is the wave-induced force array normalized per unit wave amplitude; \(\omega\) is the frequency; \(\beta\) is the incident wave direction angle; \(\rho gV_{0}\delta_{3}\) is the buoyancy and equals to the platform weight; \(V_{0}\) is the volume of the immersed part of the platform; \(\delta_{3}\) is the component of the Kronecker-Delta function; \(C^{Hydrostatic}\) is the hydrostatic-restoring matrix; \(x\) is the platform motions; \(t\) is time; \(K_{1}\) is the radiation kernel from potential flow theory; and \(\dot{x}(\tau)\) is the platform velocity. In the in-house code, the code will use the given wave parameters and the selected wave numerical theory to perform wave kinematics calculations at the first step; secondly, the total added mass forces \(\overset{\rightarrow}{F}_{AM}\) and the hydrostatic forces \(\overset{\rightarrow}{F}_{HS}\) can be calculated by the initial motion state of the platform and the given related matrix; the incident-wave excitation force \(\overset{\rightarrow}{F}_{W}\) can be calculated by the P-M wave spectrum and the radiation memory-effect force \(\overset{\rightarrow}{F}_{RD}\) will be obtained by the platform initial velocity.