*3.4. Parametric Method*

The parametric generation of the damage cases was divided into two phases. In the first, the box-shaped damages were generated considering every single room laying on the hull shell. In the latter, the single-room damage cases were parametrically combined to define additional damage cases involving neighboring rooms.

In a previous study [17], it was observed that the longitudinal position of the damage center has only a limited impact on progressive flooding, whereas its vertical position has a relevant effect. Hence, all the single-room damages were applied at half of the room longitudinal extension assuming an *Ld* equal to the room longitudinal extension. On the other hand, at least three different vertical positions of damage center *Zd* were considered for each room: the room's bottom, half-height, and top. If a room extends over multiple decks, additional intermediate positions were considered corresponding to the main decks' heights from the baseline. The damage center height was corrected for top and bottom damages, considering the applied damage area as shown in Figure 4, to avoid damages extending outside the room boundaries. For each possible damage location (*Xd*, *Zd*), multiple damage sizes were considered. The area of the *i*-th damage in the *j*-th room, considering the *k*-th center, is evaluated as:

$$\frac{1}{A\_{ijk}} = \frac{1}{A\_{\max\_{jk}}} + \frac{k}{n\_d} \left( \frac{1}{A\_{\max\_{jk}}} - \frac{1}{A\_{\min\_{jk}}} \right) \tag{12}$$

with *k* = [1, 2, ..., *nd*], where *nd* is the so-called number of divisions, which is the main parameter governing the database size. Besides, *Amin* and *Amax* were defined for each room and each position of the damage center. The minimum area was defined as the one corresponding to an initial inflow equal to the bilge pumps' capacity. To define the maximum area, note that very large damages result in the almost instantaneous filling of the damaged room. Hence, the maximum damage area was defined as the area that causes the room filling in 15 s. Furthermore, every single room was assumed as lost at the beginning of progressive flooding (instantaneous flooding), defining an additional single-room damage case. After the definition of the single-room damage cases, they were combined with the ones related to the neighboring rooms. Namely, all the possible combinations of the damage areas of damages having the same center height (bottom, half-height, top) and sharing a boundary (watertight bulkhead, deck) were considered. For instance, at the intersection of a deck and a transverse bulkhead, the combinations were defined considering up to four rooms. Here, only one or two compartments' damages were generated. Nevertheless, the parametric generation technique can be easily extended to a higher number of contiguous compartments if required.

**Figure 4.** Correction of the vertical position of the damage center according to the damage area for a top damage.
