## 5 Hull Form Optimization Process
### Practical Considerations for Optimization  
Table 6 lists the relevant parameters of hull form optimization such as the initial value, the range, and the step size for calculation of gradient for nine design variables introduced in Section 3. For instance, the initial value of \(X_{\text{C}}^{\text{SAC}}\) for the SAC variation was set at 16.297 ST, where the prismatic coefficient \(C_{P}\) becomes equal to 0.5. Here, all the quantities related to the design variable defined in the \(X\) direction were expressed in terms of station (ST), which is \(1/20\) of \(LBP\). Similarly, the quantities defined in the \(Y\) and \(Z\) direction are normalized with respect to the breadth \(B\) and the draft \(T\), respectively. Regarding the section shape variation, the reference \(Z\) positions were placed at \(Z_{D}=T\) (at design draft) and \(Z_{L}=T/3\). The initial values of the design variables \(X_{\text{C}}\_Z_{D}^{Section}\) and \(X_{\text{C}}\_Z_{L}^{Section}\) were set at \(X=17.215\) ST and \(X=15.852\) ST, respectively. These positions were obtained at the \(X\) coordinates where the waterlines at \(Z_{D}\) and \(Z_{L}\) cross the quarter-breadth position \(Y=B/4\). Figure 6 illustrates how these initial values were determined in terms of the SAC and the waterline shapes of the original KCS hull form. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Modification** & **Design** & **Initial** & **Range** & **Step Size, \(\epsilon\)** \\  & **Function** & **Variable** & **Value** & & \\ \hline \multirow{2}{*}{SAC Shape} & \multirow{2}{*}{\(f_{1}(X)\)} & \(X_{\text{C}}^{\text{SAC}}\) & 16.297 ST & \(-2.0\) ST, \(+2.0\) ST & 0.05 ST \\ \cline{3-6}  & & \(\Delta X_{\text{C}}^{\text{SAC}}\) & 0.0 ST & \(-1.0\) ST, \(+1.0\) ST & 0.02 ST \\ \hline \multirow{4}{*}{Section Shape} & \multirow{4}{*}{\(f_{2}(X,Z)\)} & \(X_{\text{C}}\_Z_{D}^{Section}\) & 17.215 ST & \(-2.0\) ST, \(+2.0\) ST & 0.05 ST \\ \cline{3-6}  & & \(\Delta Y_{\text{0C}}\_Z_{D}^{Section}\) & 0.0 B & \(-0.03\) B, \(+0.03\) B & \(3.11\times 10^{-4}\) B \\ \cline{3-6}  & & \(\Delta Y_{\text{C}}\_Z_{D}^{Section}\) & 0.0 B & \(-0.03\) B, \(+0.03\) B & \(3.11\times 10^{-4}\) B \\ \cline{3-6}  & & \(X_{\text{C}}\_Z_{D}^{Section}\) & 15.852 ST & \(-2.0\) ST, \(+2.0\) ST & 0.05 ST \\ \cline{3-6}  & & \(\Delta Y_{\text{C}}\_Z_{L}^{Section}\) & 0.0 B & \(-0.03\) B, \(+0.03\) B & \(3.11\times 10^{-4}\) B \\ \hline \multirow{2}{*}{Bulb Shape} & \(f_{3}(X,Z)\) & \(\Delta Z^{Bulb}\) & 0.0 T & \(-0.09\) T, \(+0.09\) T & \(1.85\times 10^{-3}\) T \\ \cline{2-6}  & \(f_{4}(X,Z)\) & \(\Delta Y^{Bulb}\) & 0.0 B & \(-0.03\) B, \(+0.03\) B & \(6.21\times 10^{-4}\) B \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Initial value and range of design variables for SQP optimization. As addressed above, the most significant concern in the implementation of SBD hull form optimization is the computational time required for the repeated RANS simulations. Here, a practical consideration on this is given. Since the SQP is a gradient-based optimization algorithm, the finite differencing to calculate gradient demands \((2N+1)\) times RANS simulations for one iteration with \(N\) design variables. If the iteration number \(K\) is needed to obtain the converged solution to the SQP optimization, then at least \((2N+1)\times K\) simulation times will be required. If we assume the computational time for each simulation to be \(H\) hours, then the total computation time will become \((2N+1)\times K\times H\) hours. As discussed in Section 4, the value of \(H\) can be decreased from 5 to 2 by adopting \"Fixed with static sinkage & trim\" running attitude. The total computational time could be further reduced by means of parallel computing. If two simulations required for the calculation of the gradient can be performed simultaneously, then the computational time will be reduced to \((1+1)\times K\times H\) hours. However, the number of CPU cores allocated should be increased by \(2N\)\((N=9)\) times in return for speeding up the process.