Case 1: Damage probability prediction in ship design

Ship design regulations prescribe that a ship must be subdivided into watertight compartments to ensure its safety. One measure for safety is the probabilistic damage stability. The measure was introduced in the 1960s and 1970s as a more flexible and more realistic alternative to its

deterministic counterpart. Ship designer were supposed to be able to create a safe ship without being bothered by regulations limiting their creativity. The deterministic damage stability calculation implies that there is a maximum volume for each compartment, whereas the probabilistic method looks only at the effects of the selected subdivision and does not prescribe any geometric boundaries. The probabilistic calculation is used to determine the safety of a ship by assessing the effect of damages in case of a collision. The presumed effect of such an accident is that water enters the hull, which changes the distribution of weight inside the ship and thus its heeling angle and stability characteristics. To examine these effects, it is crucial to know the position of watertight bulkheads and decks, as they determine the boundaries of potential damages. Assuming the ship hull form has already been chosen, one can say that these positions are the main design parameters in the computation of the damage stability.

The regulations we are referring to can be found in IMO [1991] and IMO [1992]. In these two references, the probabilistic damage stability calculation is explained step by step and a requirement is formulated, being the minimum stability that a ship is allowed to possess. It can be argued that since a minimum stability value is defined, the optimum ship subdivision is the one with exactly that value. It is nevertheless interesting to find the subdivision that provides the best stability. The difference between the theoretic maximum and the imposed minimum can then be regarded as additional freedom for the designer, who is enabled to decide which ship form and subdivision, along with cargo capacity and distribution give the best ship. Hence, we will call the ship with the highest damage stability the optimum and any compromise that the designer makes to improve other ship characteristics are considered sub-optimal.

The principle of the probabilistic damage stability is to make an inventory of all damages that may occur in a collision and assign a probability of occurrence to each damage type, based on statistics concerning accidents in the past. Next, the effects of leakages are quantified in terms of the ship survivability, which can be calculated using a static stability analysis. The Attained Subdivision Index (A) is a measure that quantifies the damage stability. It equals the sum of the product of the aforementioned probabilities for all damage cases denoted by the set C, so we can define

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than to the probability of sinking. The attained subdivision index should exceed the required damage stability, which depends on the ship length only, albeit that the number of passengers is taken into account too if a passenger ship is concerned.

The probability of an accident is a given constant. The stability of the ship on the other hand is difficult to compute and has a dynamic character. In case of a collision, the damaged compartments will be flooded and this continuous process changes the ship’s stability from second to second. However, it is practically impossible and probably not even useful to consider the stability at every moment between the collision and the final equilibrium stage (if there is one), but somehow this flooding process has to be accounted for. The solution found is to compare the survivability numbers corresponding to different filling percentages of the damaged compartments. Since the weight distribution plays an important role in this calculation, it is necessary to repeat this investigation for sufficiently many different loading conditions of the ship.

Calculation of the attained subdivision index is complicated, since the number of damage cases is always very high. Usually the simplest cases (consisting of only one flooded compartment) have the largest contributions to the attained subdivision index, whereas the least likely cases (consisting of more than a dozen flooded compartments) lead to unnoticeable increments of the index, but one cannot say in advance after how many damage cases the computation can be aborted. Note that for the checking of compliance with regulations, there is no objection to ending the calculation prematurely. If a sufficient safety index has been reached after a number of damage cases has been handled, it can only profit from the remaining cases.

However, the attained subdivision index is only an approximation, because it is partly based on statistics from previous accidents. Realistic scenarios in which the base of the damage does not coincide with the bottom of the ship, or in which multiple holes are observed, are not covered by the regulations (see Jensen [1995]).

In order to utilize computer power for such complex design problems, many researchers have proposed optimization methods in the quest for the best design. The first steps were taken with conventional gradient methods (for example Broyden’s method, Broyden [1965]) and with non- gradient methods (e.g., the Simplex method of Nelder and Mead [1965] and Hooke and Jeeves [1961]). These methods were applied to design subjects such as the optimization of the local hull shape and the minimization of the construction weight (see Nowacki [2003] and Bertram [2003]). These conventional methods have proven to be very efficient for continuous functions, as opposed to functions of a discontinuous nature like the attained subdivision index as a function of bulkhead

positions. As a consequence, genetic algorithms have been introduced in ship design applications including the field of the probabilistic damage stability in recent years (see Sommersel [1997], Gammon and Alkan [2003], Zaraphonitis et al. [2003], Ölçer et al. [2003], Guner and Gammon [2003] and Chen et al. [2003]). It appeared that even though a good solution can always be found, a large number of iteration cycles are wasted with the (time-consuming) calculation of impossible or irrelevant bulkhead configurations.

Figure 24. Kriging models of the attained subdivision index as a function of the position of the bulkheads.

The MDDO is specifically suited to these situations. Not only will it need fewer iterations than genetic algorithms, it will also give the designer insight into the safety as a function of the bulkhead positions. To prove this, we have applied the MDDO method to an actual ship design, namely a general cargo ship of 100 meters containing only two transverse bulkhead positions as design parameters. The selected bulkheads are named trans01 and trans02 and their original locations are

at 66.00 and 83.50 meters, respectively. We have allowed them to move 6 meters back or forward. Since the behaviour of the Attained Subdivision Index is expected to be non-linear in terms of the bulkhead positions, Kriging models are used in the meta-modeling step. It is up to the designer to determine how detailed the Kriging model should be. A rough estimate of the best subdivision can be obtained with a Maximin LHD consisting of only 10 experiments. However, a Kriging model based on a Maximin LHD consisting of 100 experiments is much more accurate. See Figure 24.

Figure 25. A rough meta-model of the attained subdivision index (left) and the rough meta-model that is made more accurate in the interesting part

(right). The design sites that are used for the construction of the right meta-model are shown in the middle.

Alternatively, first a rough approximation of the design space can be made, e.g. using a Kriging model based on 20 experiments. This meta-model can then be used for the identification of interesting regions in the design space. The meta-model can be made more accurate in the interesting region by adding experiments in this part of the design space. See Figure 25. See also Kleijnen and Van Beers [2004].

We conclude that the MDDO method is an efficient methodology for the ship design problem, because the number of simulations that is needed to estimate the optimal design is limited compared to the amount usually needed by Genetic Algorithms. Further, the designer has control over the number of simulations and the designer gains more insight compared to other optimization methods.