## 2 Numerical Methods
### Potential Flow Theory  
[PERSON] et al. (2017, 2019, 2020) reviewed two mathematical theories for calculating floating body hydrodynamic forces (i.e., [PERSON]'s equation and potential flow theory). Potential flow theory can counter the effects of wave radiation and diffraction, thus can more accurately predict floating body hydrodynamic forces compared with [PERSON]'s equation. Thus, this study adopts potential flow theory to calculate wind turbine hydrodynamic forces. Specifically, a potential function \(\phi(x,\ y,\ z)\) is utilized, whose governing equation can be expressed as  
\[\
abla^{2}\phi=0, \tag{9}\]  
where \(\partial\phi/\partial x=u,\partial\phi/\partial y=v\), and \(\partial\phi/\partial z=w\) with \(u,v\), and \(w\) as water particle velocities in \(x,y,\) and \(z\) directions, respectively. After determining free water surface, structural surface, and seabed boundary, the entire water particle velocity distribution in the numerical wave basin can be computed by solving the potential function. Subsequently, the floating body hydrodynamic force (\(\overset{\rightarrow}{F}_{WAP}\)) can be calculated as  
\[\overset{\rightarrow}{F}_{WAP}=\overset{\rightarrow}{F}_{W}+\overset{ \rightarrow}{F}_{HS}+\overset{\rightarrow}{F}_{RD}+\overset{\rightarrow}{F} _{AM}, \tag{10}\]  
in which, \(\overset{\rightarrow}{F}_{W}\) denotes wave load; \(\overset{\rightarrow}{F}_{HS}\) means static water pressure; \(\overset{\rightarrow}{F}_{RD}\) represents the hydrodynamic force resulted from wave radiation; and \(\overset{\rightarrow}{F}_{AM}\) is the hydrodynamic force  
Figure 1: (**a**) Schematic of blade element momentum method and (**b**) blade element forces.  
caused by added water mass. These four parts of total hydrodynamic force can be respectively computed as  
\[\overset{\rightarrow}{F}_{W}=\frac{1}{N}\sum_{k=-\frac{N}{2}-1}^{\frac{N}{2}} \mathrm{W}[k]\sqrt{\frac{2\pi}{\Delta t}S_{\xi}^{2-sided}(\omega)}X(\omega,\ \beta)\Bigg{|}_{\omega=k\Delta\omega^{\prime}} \overset{\rightarrow}{N}, \tag{11}\]  
\[\overset{\rightarrow}{F}_{HS}=\rho gV_{0}\delta_{3}-C^{Hydrostatic}x, \tag{12}\]  
\[\overset{\rightarrow}{F}_{RD}=-\int_{0}^{t}K_{1}(t-\tau)\dot{x}(\tau)d\tau, \tag{13}\]  
\[\overset{\rightarrow}{F}_{AM}=-AM_{RP}\overset{\rightarrow}{a}_{p}, \tag{14}\]  
In Equation (11), \(\mathrm{W}[k]\) means the fast Fourier transform result of white noise time series; \(S_{\xi}^{2-sided}(\omega)\) denotes wave spectrum (JONSWAP spectrum is utilized in this study); \(X(\omega,\ \beta)\) is the wave force matrix normalized by unit wave amplitude; \(\omega\) means wave frequency; and \(\beta\) indicates inflow wave direction. In Equation (12), \(\rho gV_{0}\delta_{3}\) represents floating body buoyancy with \(V_{0}\) as under-water floating body volume; \(C^{Hydrostatic}\) means hydrostatic recovery matrix; and \(x\) is floating body rigid displacement. In Equation (13), \(K_{1}\) denotes radiation kernel in potential flow theory; and \(\dot{x}(\tau)\) means floating body rigid displacement velocity. In Equation (14), \(AM_{RP}\) denotes added mass force caused by wave radiation; and \(\overset{\rightarrow}{a}_{p}\) means floating body rigid displacement acceleration.