## 2 Model Statement
### Platforms' Design and Advanced Computations  
Using MultiSurf software, the geometry of the platforms can be designed. Three varied platforms have been compared. Considering the first platform, It is a square-shaped barge called standard barge platform, shown in Figure 2a. On the other hand, the second model is an OWC-based barge platform with four OWCs housed at the platform's corners when the OWCs' valves are closed as illustrated in Figure 2b. Figure 2c shows the third model which is a platform with four OWCs at the platform's corners when the OWCs' valves are open.  
Through MultiSurf tool, the body's wetted portion, only, is meshed in its undisplaced position [26]. In this research, the standard barge platform has been meshed with 8960 rectangular panels within the whole body while the second model is modeled with 9940 rectangular panels and the third model is meshed with 9840 total rectangular panels. In the second and third models, the geometry of the OWCs has a distance of one meter from the sides of the platform with the measurements 5 m \(\times\) 5 m \(\times\) 10 m.  
Figure 2: Platforms’ geometry for (**a**) Standard barge platform. (**b**) 4 OWC-based barge platform with closed valves. (**c**) 4 OWC-based barge platform with open valves.  
After designing the platforms in MultiSurf, the advanced panel method WAMIT tool has been employed to introduce the matrices \(A_{Hydro}(\omega)\), \(B_{Hydro}(\omega)\), \(C_{Hydro}\) and \(f_{Hydro}(\omega)\) expressed in Section 2.2. This advanced computational software performs the analysis of the interaction between wave inputs and floating offshore platforms. WAMIT calculates the hydrodynamics loads caused by water pressure on the wetted areas, which may be connected to MultiSurf to employ the geometric floating model.  
The added mass and damping matrices may be described as:  
\[A_{ij}-\frac{i}{\omega}B_{ij}=\rho\iint_{S\mathrm{b}}n_{i}\varphi_{j}dS \tag{33}\]  
Therefore, the normalized added mass and damping matrices can be defined by:  
\[\bar{A}_{ij} = A_{ij}/\rho L^{k} \tag{34}\] \[\bar{B}_{ij} = B_{ij}/\rho L^{k}\omega \tag{35}\]  
here \(L\) is the length scale, \(k=3\) for \((i,j=1,2,3)\), \(k=4\) for \((i=1,2,3)\), \((j=4,5,6)\) or \((i=4,5,6)\), \((j=1,2,3)\) and \(k=5\) for \((i,j=4,5,6)\).  
All hydrostatic data may be described in the form of surface integrals over the mean body wetted surface \(S_{b}\), by virtue of [PERSON] divergence theorem. Volume can be given by:  
\[\forall=-\iint_{S_{b}}n_{1}x\ dS=-\iint_{S_{b}}n_{2}y\ dS=-\iint_{S_{b}}n_{3}z\ dS \tag{36}\]  
Matrix of hydrostatic and gravitational restoring components may be described by:  
\[C_{Hydro}=\left[\begin{array}{ccccc}0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&\rho g\iint_{S_{b}}n_{3}\ dS&\rho g\iint_{S_{b}}yn_{3}\ dS&-\rho g\iint_{S_{b}}xn_{3}\ dS&0\\ 0&0&0&\rho g\iint_{S_{b}}y^{2}n_{3}\ dS+\zeta&-\rho g\iint_{S_{b}}xyn_{3}\ dS& \rho g\forall x_{b}+mgx_{g}\\ 0&0&0&\rho g\iint_{S_{b}}x^{2}n_{3}\ dS+\zeta&-\rho g\forall y_{b}+mgy_{g}\\ 0&0&0&0&0\end{array}\right] \tag{37}\]  
where \((x_{g},y_{g},z_{g})\) are the coordinates of the center of gravity, \(\zeta=\rho g\forall z_{b}-mgz_{g}\) and thr coordinates of center of buoyancy are given by:  
\[x_{b} = \frac{-1}{2\forall}\iint_{S_{b}}n_{1}x^{2}\ dS \tag{38}\] \[y_{b} = \frac{-1}{2\forall}\iint_{S_{b}}n_{2}y^{2}\ dS\] (39) \[z_{b} = \frac{-1}{2\forall}\iint_{S_{b}}n_{3}z^{2}\ dS \tag{40}\]  
The advanced computations performed using the meshed platform for the aforementioned matrices may be included in the equations of motions to carry out numerous simulations in FAST and MATLAB.