## 3 Hull Form Variation and Design Parameters  
Hull form variation is important in the entire hull form SBD process because it affects many significant aspects such as the variety of hull forms, the correlation between design variables and resulting hull form, and the fairness of hull form. In this study, the hull form variation was carried out by varying the SAC, the sections shape, and the bulb shape of the original hull form of KCS. In order to maintain the length overall (LOA), the bulb length was kept unchanged. The hull form variation can be expressed in terms of parametric modification function \(f_{n}\) as follows:  
\[H_{new}(X,Y,Z)=H_{old}(X,Y,Z)+\sum_{n=1}^{4}f_{n}(X,Y,Z). \tag{1}\]  
Here, the subscript \(n\) denotes variation of different kind of design variable; \(n=1\) for the SAC variation, \(n=2\) for the section shape variation, \(n=3\) for the bulb height variation, \(n=4\) for the bulb breadth variation. As pointed out earlier, the fairness of the new hull form \(H_{new}\) is automatically guaranteed as long as the fairness of old hull form \(H_{old}\) and the modification function \(f_{n}\) are maintained. The initial hull form of KCS and the coordinate system are illustrated in Figure 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Symbol** & **Full Scale** & **Model Scale** \\ \hline Scale ratio & \(\lambda\) & & 31.6 \\ \hline Length between & & & \\ perpendiculars [m] & \(LBP\) & 230.0 & 7.2786 \\ \hline Breadth [m] & \(B\) & 32.2 & 1.0190 \\ \hline Draft [m] & \(T\) & 10.8 & 0.3418 \\ \hline Block coefficient & \(C_{B}\) & & 0.6505 \\ \hline Wetted surface area [m\({}^{2}\)] & \(WSA\) & 9530 & 9.5541 \\ \hline Design speed (m/s) & & 12.3466 (24 knots) & 2.197 \\ \hline Froude number & \(Fn\) & & 0.26 \\ \hline Reynolds number & \(Rn\) & \(2.58\times 10^{9}\) & \(1.40\times 10^{7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Principal particulars of the initial hull form at the sea trial condition. Figure 1: Body plan and side view of the initial hull form of the objective ship. Figure 2: Hull surface mesh of KCS and the coordinate system for hull form variation. Each parametric modification function affects hull surface coordinates differently; in other words, the SAC variation changes only \(X\) coordinates, the section shape variation involves only \(Y\) coordinates, and the bulb height and breadth variation correspond to the \(Z\) and \(Y\) coordinate translation of bulb part. These can be expressed as follows:  
\[X_{new}=X_{old}+f_{1}(X), \tag{2}\]  
\[Y_{new}=Y_{old}+f_{2}(X,Z), \tag{3}\]  
\[Z_{new}=Z_{old}+f_{3}(X,Z), \tag{4}\]  
\[Y_{new}=Y_{old}+f_{4}(X,Z). \tag{5}\]  
Each coordinate translation defines the incremental change of coordinate, which can then be regarded as the design parameters to be optimized. The changes in hull form are illustrated in connection with the corresponding design parameters in Figure 3. Table 2 describes the design parameters in detail. The SAC modification function \(f_{1}(X)\) was consisted of two B-splines defined by three design parameters: \(X_{\text{C}}^{SAC}\), \(\Delta X_{\text{OC}}^{SAC}\), and \(\Delta X_{\text{C1}}^{SAC}\). The section shape modification function \(f_{2}(X)\) was a calculated by combining two waterline modification curves and section shape modification function which are defined by six design variables \(X_{\text{C}}\_Z_{\text{D}}^{Sectin}\)  
Figure 3: Relation between design variables and hull form variation.