## 2 Overall Analysis Approach and Implemented Methods
### Parametric Definition of Tanks Geometry
#### 2.3.2 Part-Section Layout  
Part-section layouts are defined with algorithms similar to the full-section one by adding upper and/or lower constraints along the \(y\) direction to the available space and removing the symmetry conditions with respect to the horizontal axis.  
Figure 11: Part-section layouts: (**a**) available space in the upper part of the cross-section; (**b**) available space in the lower part of the cross-section.  
Figure 12: Examples of full-section layout for \(t_{0}=40^{\circ}\): (**a**) without catwalk (\(e_{a}=83\%\); \(e_{p}=31\%\)), (**b**) with catwalk (\(e_{a}=67\%\); \(e_{p}=45\%\)), (**c**) with catwalk and minimum radius constraint (\(e_{a}=56\%\); \(e_{p}=100\%\)).  
Figure 11 shows the two possible options in which tanks are installed in the upper part of the fuselage (a), for cases in which the space above the cabin might be sufficient for tank installation, or in the lower part of the fuselage (b), which represents the case in which the lower deck is used for LH\({}_{2}\) storage with a partial or total reduction in the volume assigned to containers. It is worth mentioning that these layouts have the possible advantage of avoiding the reduction in cabin space, hence number of transported passengers, but at the same time can embark smaller LH\({}_{2}\) quantities, hence shorter flyable ranges, if compared to full-section solutions.  
In the case depicted in Figure 11a, where the available space is designated through the constraint \(y\geq y_{U}\), up to 5 circles can be constructed starting from the circle \(\mathbb{C}0_{U}\) which depends on the tangent point belonging to the ellipse \(T0_{U}\), whose position can be defined using the polar coordinate \(t_{0U}\) which is analogous the full-section parameter \(t_{0}\). Then, \(\mathbb{C}0_{U}\) radius (\(r_{0U}\)) and centre \(C_{0U}\equiv(x_{0U},\ y_{0U})\) can be defined using Equations (13) and (14).  
\[C_{0U}:\ \left\{\begin{array}{c}x_{0U}=cos\ t_{0U}+f\text{-}cot\ t_{0U}\text{-} \left(y_{0U}-f\text{-}sin\ t_{0U}\right)\\ y_{0U}=\frac{y_{U}+f\text{-}sin\ t_{0U}\sqrt{\left(f\text{-}cot\ t_{0U}\right) ^{2}+1}}{1+\sqrt{\left(f\text{-}cot\ t_{0U}\right)^{2}+1}}\end{array}\right. \tag{13}\]  
\[r_{0U}=\sqrt{\left(x_{0U}-cos\ t_{0U}\right)^{2}+\left(y_{0U}-f\text{-}sin\ t _{0U}\right)^{2}} \tag{14}\]  
Once \(\mathbb{C}0_{U}\) is known, \(\mathbb{C}1_{U}\) and \(\mathbb{C}2_{U}\) are the defined setting of the tangent conditions to the ellipse, the line \(y=y_{U}\) and to \(\mathbb{C}0_{U}\). The same procedure is then perfomed to define \(\mathbb{C}3_{U}\) and \(\mathbb{C}3_{U}\), using the tangent condition to \(\mathbb{C}1_{U}\) and \(\mathbb{C}2_{U}\), respectively. An as example, Equation (15) shows the system used for the construction of \(\mathbb{C}1_{U}\), whose solution was found numerically using MATLAB _prasolve_ function. As in Equations (13) and (14), the first two lines set the tangency to the ellipse and to the line \(y=y_{U}\), the third equation sets the tangency to \(\mathbb{C}0_{U}\), and the fourth line specifies on which side of \(\mathbb{C}0_{U}\) this condition has to be obtained. Similar systems of equations can be written to define the remaining circles.  
\[\left\{\begin{array}{c}x_{1U}=cos\ t_{1U}+f\text{-}cot\ t_{1U}\text{-} \left(y_{1U}-f\text{-}sin\ t_{1U}\right)\\ y_{1U}=\frac{y_{U}+f\text{-}sin\ t_{1U}\sqrt{\left(f\text{-}cot\ t_{1U}\right) ^{2}+1}}{1+\sqrt{\left(f\text{-}cot\ t_{1U}\right)^{2}+1}}\\ \sqrt{\left(x_{1U}-x_{0U}\right)^{2}+\left(y_{1U}-y_{0U}\right)^{2}}-r_{0U}= \sqrt{\left(x_{1U}-cos\ t_{1U}\right)^{2}+\left(y_{1U}-f\text{-}sin\ t_{1U} \right)^{2}}\\ t_{1U}>t_{0U}\end{array}\right. \tag{15}\]  
For the case given in Figure 11b, i.e., considering as constraint \(y\leq y_{L}\), the procedure for defining the circles is analogous to the one here presented.