## 2 The Quick Preliminary Design Model
### Hull Forms  
The present study is focused on torpedo-shaped vehicles (see Figure 2). This is the shape of approximately 55% of AUV models [5]. The profile definition of the bow and stern cones utilizes the curves proposed by [PERSON] [17]. These equations describe the contour with the lowest drag coefficient given a body length to maximum diameter ratio (slenderness ratio). The variables that define the hull forms are total length, \(L\); length to diameter ratio, \(L/\phi\); length of the bow cone to total length ratio, \(L_{\textit{low}}/L\); length of the stern cone to total length ratio, \(L_{\textit{stem}}/L\); length of the cylindrical body, \(L_{cc}\); exit angle at the stern cone, \(\alpha\); and nose flattening index, \(n\). Figure 2 shows the layout of the hull form of the vehicles under study. As indicated above, this work uses the AUV models Hugin 1000, Hugin 3000, and Hugin 4500 for purposes of comparison. As a consequence, the variables that define the hull forms are the same as those of the model AUVs. This fact is reflected in Table 2, where the hull form variables adopt fixed values, and they are treated as constraints. \begin{table}
\begin{tabular}{r c c c c c c c c c} \hline \hline  & & & \multicolumn{3}{c}{**Model 1000**} & \multicolumn{3}{c}{**Model 3000**} & \multicolumn{3}{c}{**Model 4500**} \\ \multicolumn{1}{c}{**Symbol**} & \multicolumn{1}{c}{**Description**} & \multicolumn{1}{c}{**N. bits**} & \multicolumn{1}{c}{**Min**} & \multicolumn{1}{c}{**Max**} & \multicolumn{1}{c}{**N. bits**} & \multicolumn{1}{c}{**Min**} & \multicolumn{1}{c}{**Max**} & \multicolumn{1}{c}{**N. bits**} & \multicolumn{1}{c}{**Min**} & \multicolumn{1}{c}{**Max**} \\ \hline \(e\) (h) & Endurance & 10 & 10 & 150 & 10 & 10 & 150 & 10 & 10 & 150 \\ \(Dph2 DC\) (mm) & Cph diameter to outer diameter ratio & 8 & 0.5 & 1 & 8 & 0.5 & 1 & 8 & 0.5 & 1 \\ \(tphB\) (mm) & Bow pressure hull shell thickness & 3 & 10 & 14 & 3 & 10 & 14 & 3 & 10 & 14 \\ \(tphC\) (mm) & Central pressure hull shell thickness & 4 & 25 & 40 & 4 & 25 & 40 & 4 & 25 & 40 \\ \(tphS\) (mm) & Stern pressure hull shell thickness & 3 & 23 & 30 & 3 & 23 & 30 & 3 & 23 & 30 \\ \(tphHB\) (mm) & Bow pressure hull head and thickness & 2 & 8 & 15 & 2 & 8 & 15 & 2 & 8 & 15 \\ \(VARIABLES\) & \(tphHC\) (mm) & Central pressure hull head end thickness & 4 & 25 & 40 & 4 & 25 & 40 & 4 & 25 & 40 \\ \(tphHS\) (mm) & Stern pressure hull head end thickness & 3 & 15 & 22 & 3 & 15 & 22 & 3 & 15 & 22 \\ \(phDiv\) & Number of frame spans & 3 & 2 & 9 & 3 & 2 & 9 & 3 & 2 & 9 \\ \(lsw\) (mm) & Frame web height & 6 & 40 & 103 & 6 & 40 & 103 & 6 & 40 & 103 \\ \(asw\) (mm) & Frame web width & 5 & 5 & 15 & 5 & 5 & 15 & 5 & 5 & 15 \\ \(hsf\) (mm) & Frame flange height & 4 & 15 & 30 & 4 & 15 & 30 & 4 & 15 & 30 \\ \(asf\) (mm) & Frame flange width & 5 & 44 & 75 & 5 & 44 & 75 & 5 & 44 & 75 \\ \hline \(L\) (m) & Length overall & - & 4.5 & - & 5.5 & - & 6 \\ \(LBc2L\) & Bow cone length to total length ratio & - & 0.1 & - & 0.1 & - & 0.1 \\ \(LphB\) (mm) & Bow pressure hull length & - & 400 & - & 400 & - & 400 \\ \(LphS\) (mm) & Stern pressure hull length & - & 750 & - & 750 & - & 750 \\ \(LSc2L\) & Stern cone length to total length ratio & - & 0.2 & - & 0.2 & - & 0.2 \\ \(LD\) & Length to outer diameter ratio & - & 6 & - & 5.5 & - & 6 \\ \(DphB\) (mm) & Bow pressure hull outer diameter & - & 280 & - & 280 & - & 280 \\ \(DphS\) (mm) & Stern pressure hull outer diameter & - & 520 & - & 520 & - & 520 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: DMFC-AUV\(\backslash\)QpDM variables and constraints. Upper section: definition of the number of bits for each variable and their variation range. Lower section: definition of the values of each constraint. Cph, central pressure hull. Figure 2: AUV profile. \(L\), total length. \(L_{\textit{low}}\), bow cone length. \(D\), outer diameter. \(DphC\), central pressure hull diameter. \(\alpha\), exit angle. BCn, bow cone. Bph, bow pressure hull. Cph, central pressure hull. Sph, stern pressure hull. SCN, stern cone.