# Application of Machine Learning for Bulbous Bow Optimization Design and Ship Resistance Prediction
## 2 Materials and Methods
### Free-Form Deformation Method  
The free-form deformation (FFD) method in ship structure modeling allows for the visualization of shape alterations on a computer, thereby enhancing control over the hull configuration. Proposed by [PERSON] and [PERSON] [19], it has become a key tool in modern ship design, after years of development. This approach does not directly alter the object. Instead, it incorporates the object into the spatial geometry and achieves deformation  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Name** & **Parameter Symbol** & **Parameter Values** \\ \hline Length between perpendiculars (m) & \(L_{pp}\) & 142.0 \\ Ship’s width (m) & \(B\) & 19.06 \\ Type depth (m) & \(D\) & 10.0 \\ Drainage volume (m\({}^{3}\)) & \(D_{t}\) & 8459.8 \\ Wet surface area (m\({}^{2}\)) & \(S_{W}\) & 2972.6 \\ Design speed (kn) & \(V\) & 30 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The DTMB5415 boat model’s main parameters.  
Figure 1: DTMB5415 hull model.  
by adjusting the moving geometry control points. A local coordinate system \(\overset{\rightarrow}{S},\overset{\rightarrow}{T},\overset{\rightarrow}{U}\) is formed on the geometry, designating \(X_{0}\) as the coordinate origin, with any grid point on the geometry expressible as follows:  
\[X=X_{0}+s\overset{\rightarrow}{S}+t\overset{\rightarrow}{T}+u\overset{ \rightarrow}{U},0\leq s,t,u\leq 1 \tag{3}\]  
where \(s,t,u\) is the coordinate value of any grid point in the local coordinate system.  
Upon establishment of the local coordinate system, the mapping relationship between the control points of the bulbous bow surface and the control points of the geometry are defined using polynomial basis functions:  
\[X=\sum_{i=0}^{l}\sum_{j=0}^{m}\sum_{k=0}^{n}R_{i}(\,s\,)R_{j}(\,t\,)R_{k}(\,u\,) P_{i,j,k} \tag{4}\]  
where \(R_{i}(\,s\,)\), \(R_{j}(\,t\,)\), \(R_{k}(\,u\,)\) are polynomial basis functions along the directions \(\overset{\rightarrow}{S},\overset{\rightarrow}{T},\overset{\rightarrow}{U}\); \(P_{i,j,k}\) denotes the coordinate value of the control point; and \(l,m,n\) denote the number of control points for the three polynomial basis functions.  
In the choice of polynomial basis functions, the NURBS basis function greatly increases the flexibility of the deformation by introducing a weighting factor that defines the P-times NURBS basis function \(N\) in the \(\overset{\rightarrow}{S}\) direction:  
\[N_{i,p}(\,s\,)=\frac{\omega_{i}B_{i,p}(\,s\,)}{\sum_{i=0}^{l}\omega_{i}B_{i,p} (\,s\,)} \tag{5}\]  
where \(\omega_{i}\) is the weight coefficient along the \(\overset{\rightarrow}{S}\) direction, and \(B_{i,p}(\,s\,)\) is the B-spline basis function along the \(\overset{\rightarrow}{S}\) direction.  
Substituting the other NURBS basis functions into the above definition, the mapping relation becomes as follows:  
\[X_{NURBS}=\sum_{i=0}^{l}\sum_{j=0}^{m}\sum_{k=0}^{n}\frac{\omega_{i}B_{i,p}(\,s \,)}{\sum_{i=0}^{l}\omega_{i}B_{i,p}(\,s\,)}\frac{\omega_{i}B_{j,q}(\,t\,)}{ \sum_{i=0}^{l}\omega_{i}B_{j,q}(\,t\,)}\frac{\omega_{i}B_{k,r}(\,u\,)}{\sum_{i =0}^{l}\omega_{i}B_{k,r}(\,u\,)}p_{i,j,k} \tag{6}\]