# Numerical Modelling of a Floating Wind Turbine Semi-Submersible Platform
## 2 Materials and Methods
### Mesh
#### 2.2.2 Mesh Independence Study  
A mesh convergence study is needed to understand how the mesh definition affects the response of the platform in the different tests, and to achieve the best possible ratio between definition and computational cost. To achieve this aim, 3 grids were designed to compare their results in one of the cases. A decrease in all the dimensions of 50% was presented to know how the response was sensitive to the cell size change. The free surface cells, the smallest ones of the tank mesh, are 50, 100 and 200 times smaller in relation to the height of the device. The overset cells have a height between 50% and 70% of the free surface cell, depending on each of the grids. Thus, 3 grids of clearly different computational weights were obtained, which can be seen in Table 1.  
All the grids have been simulated with the same time-step to focus just on the dependency of the cell size. In Figure 2, it can be seen that there is almost no difference between the 3 grids. However, some tendencies can be seen when analysing the damping ratios of the signals. \begin{table}
\begin{tabular}{c c} \hline \hline Grid & Number of Cells \\ \hline Grid 1 & 2976.088 \\ Grid 2 & 11,898.568 \\ Grid 3 & 26,111.669 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Number of total cells for each of the grids considered in the mesh convergence study. The damping ratios can be observed by comparing the mean amplitudes of each of the periods and the decrease in amplitude that can be seen in the period itself. This method is the same as that which [PERSON] and [PERSON] have used [34]. To do that, different equations are needed. The decrease in amplitudes is compared logarithmically:  
\[\delta=\log_{10}\left(\frac{\eta_{i+2}}{\eta_{i}}\right) \tag{12}\]  
where \(\eta_{i}\) is the peak or drought of the decay signal. It is always compared in intervals of 2 to have a comparison between continuous peaks or droughts. The value used as a decrease in the peak or drought value is then used to calculate the damping ratio in that period [31]. \[DR=\sqrt{\frac{1}{1+\frac{\delta}{2\pi}}} \tag{13}\]  
Then, to compare the results obtained from this mesh dependency study, the tendency of the points has been defined. A comparison of linear, quadratic and cubic tendencies has been done to see if the highest complexity of the curve has a relationship with the degrees of freedom unlocked. The damping ratios calculated for each of the meshes in the size dependency study can be seen in Figure 3. In it, the different fittings are set to see how each mesh compare to the other two. The different fittings show that similarities between the two finest meshes increase when the grade of the fitting increases. The most similar fittings are when the grade of the fitting is the same as the degrees of freedom unlock in the simulations. Besides the mesh sensitivity study, a time-step dependency study was run to see which time-step in relation to the period was needed. The first step of this process was to calculate the Courant number to ensure that all time-steps considered were good enough theoretically. It arises in the numerical analysis of explicit time integration schemes, when these are used for the numerical solution. As a consequence, the time-step must be less than a certain time in many explicit time-marching computer simulations; otherwise the simulation produces incorrect results. Figure 2: Direct comparison of the signals obtained in the mesh sensitivity study. This can be seen in Table 2 in which for the three time-steps considered, and the cell size, all the possible Courant numbers in each of the meshes were reasonable. These 3 values are \(\mathrm{T_{N}}/1000\), \(\mathrm{T_{N}}/1800\) and \(\mathrm{T_{N}}/3200\), being \(\mathrm{T_{N}}\) the natural period of the degree of freedom analysed. All the values are low, reinforcing the idea of the use of these time-steps,  
Figure 3: Comparison of the relation of the tendencies used for the case with linear (**up**), quadratic (**mid**) and cubic (**down**) fitting.