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Numerical simulations and model experiments can enable accurate assessment of the damage survivability of passenger ships
By Dracos Vassalos, Andrzej Jasionowski, Jakub Cichowicz, and Nikolaos Tsakalakis
The recent and nearly catastrophic accident of the Costa Concordia demonstrated clearly that survivability of passenger vessels is more than simply risk management. It is in fact a business instrument that has a profound impact on performance—not only of a single company, but indeed the whole industry.
Collision and grounding incidents are not artifacts of the past. They happen more often than common perception might suggest. In fact, recent studies show that, statistically, we can expect to witness one catastrophic accident (those claiming more than 1,000 lives) every 20 years. There is human error behind many accidents that have occurred, but even the most advanced operational procedures will not prevent these kinds of incidents. There are countless possibilities of system malfunctions, human negligence, or carelessness leading to disasters, and the risk of a ship sinking or capsizing cannot be eradicated. Instead, it should be understood, measured, and accounted for. The risk is manageable; therefore, it should be managed.
However, treating safety as a business instrument means that there has to be a rationale behind risk management. Safety must serve industry and not prevent its growth. For obvious reasons, this cannot be achieved by lowering standards—these are already debatable—and therefore the only way forward is better understanding of the physics of the problem. The issues of capsizing and sinking cannot be solved, at least in a rational manner, by increasing the number of bulkheads or imposing minimal height of residual freeboard. Nature is far more inventive than scientists and engineers. On the contrary, the legislative instruments regulating safety must be robust enough to ensure that the rightful safety levels that might have prevented past accidents have been set, which also will minimize the risk of future disasters.
The currently in-force probabilistic framework for damage stability, known as SOLAS 2009, is a good example of such thinking. It is based on knowledge derived from the past, but by abandoning deterministic artifacts, it aims at allowing innovative designs. However, shortly after being introduced, the new regulations provoked serious concerns for passenger ships with respect to accuracy in predicting the probability of surviving damage. Specifically, it was postulated that the new SOLAS overestimated the probability of survival in the case of RoPax vessels, while survivability of large cruise vessels was said to be underestimated. In order to address these issues the European Commission founded in 2009 a research project entitled GOALDS (GOAL-Based Design for Damage Stability). The results presented here are based on studies performed within this project.
Modeling damage survivability
A major difficulty in modeling damage survivability stems from the complexity of the underlying physical phenomena. More specifically, the problem lies in predicting ship behavior in waves while accounting for floodwater dynamics and the flooding process in complex geometries typical of passenger ships. For individual cases, the prediction can be based on computational fluid dynamics CFD solutions. However, due to high demand in computational power and long computation time, such an approach cannot be followed on an industrial scale where large number of cases and long runs are envisioned (physical experiments share similar limitations). Therefore, studies on diversified samples, necessary for statistical generalization of the results, require simplified mathematical models, while physical tests and CFD experiments provide data for benchmarking of numerical results.
Usually, numerical simulations are carried out by computer codes based on linear or linearized theories of ship motions. This enables employing the superposition principle, which simplifies the process of solving the equations of ship motions to a great extent. A similar effect occurs with the assumptions made with respect to flow-through damage (external) and internal openings; for example, that the flow through an opening commences without delay whenever there is a pressure difference on either sides of the opening, or that water spreads instantaneously in a damaged compartment. Furthermore, while accounting for ship-floodwater interaction, it is often assumed that free surface of the floodwater remains horizontal regardless of the orientation of the ship (hydrostatic modeling) or that the whole floodwater is reduced to its centroid and such defined lump-mass moves within the compartment over predefined surfaces (approximate hydrodynamic modeling).
Inevitably, the computational efficiency of the numerical techniques comes at the expense of accuracy of the results. Moreover, performing formal quantification of intrinsic uncertainties is difficult and consequently the models are usually described as being of “satisfactory engineering accuracy.” In practice, it is virtually impossible to provide accurate assessment for epistemic (systematic) uncertainties for such compound models. Furthermore, quantification of aleatory (statistical) errors is very difficult. This derives mainly from the fact that, unlike in the case of an intact ship, the ergodicity assumption does not hold—in principle—for dynamical processes associated with the motions of a damaged ship. As a result, the quantification of these processes in the time domain as an ensemble domain assessment (i.e., analysis of large collection of short runs) becomes inapplicable. The time domain quantification requires very long runs, making CFD and EFD experiments impractical.
These constitute clear disadvantages of simplified techniques, but it should be stressed here that they have less significant impact on predicting the behavior of a damaged ship than they do in the case of an intact vessel. This follows from the fact that damaged vessels do not, generally, experience large motions. Putting it colloquially, the ship becomes heavier: The natural frequency of roll shifts towards the lower range of the spectrum while the presence of floodwater causes considerable increase of damping. As a result, the dynamic processes become quasi-static. Consequently, the higher-order effects cease to play a significant role in the dynamics of a damaged ship, which in turn compensates for simplifications in the mathematical models. Roll angles, for instance, rarely exceed a few degrees; therefore, ship response to wave excitation seldom departs from linearity. Hence, the models can be inaccurate on their own, but it can be expected that, in many applications, they will be “accurate enough,” making numerical codes a perfect tool for damage survivability assessment.
Figure 1: Numerical simulations of the sequence of flooding stages leading to capsize of large passenger vessels.
One of the easiest to obtain, yet important characteristics of a damaged ship is the so-called capsize band: a range of sea states within which a transition from unlikely to certain capsize is observed. The width of the band varies with damage characteristics and ship loading conditions; in fact, the band is broader and covers higher sea states in case of ships characterized by larger residual stability. For this reason, the capsize band is often plotted as a family of curves (sigmoids) obtained by varying the vertical position of the center of gravity (KG) with the ordinate corresponding to the rate of observed capsizes.
Figure 2: CFD computations showing visualization of sloshing in a flooded compartment.
An important notion originating from the concept of capsize band is critical significant wave height (HScrit) becoming the foundation of the SOLAS s-factor. The s-factor—a measure of the probability of surviving collision damages—is one of the core elements introduced in SOLAS 2009. Originally, during Project HARDER, the s-factor was linked to the critical significant wave height, understood as a sea state at which a ship exposed for half an hour to the action of waves would have a 50% chance of survival (abscissa of the inflection point of the sigmoid curve). The HScrit was correlated with residual stability parameters (range of positive stability, or range, and maximum righting lever within the range, GZmax) and the s-factor obtained by mapping of the HScrit through the statistical distribution of sea states encountered during collisions.
In the process of formulating amendments, an explicit reference to the HScrit was removed from SOLAS and the current formulation of the s-factor is based only on the aforementioned stability parameters. Given its importance, the concept of the capsize band was revisited during the GOALDS Project when an attempt was made to improve accuracy of the survivability prediction. One of the most important findings was that the capsize band contracts towards its lower boundary as the time of simulation increases. In the limiting case of infinite time of observation the sigmoid distribution of capsize rate could be approximated by a step function. The observation led to the conclusion that it is more appropriate to define HScrit as the highest sea state at which no capsizes are observed within half-hour runs (full scale). For practical reasons, it has been decided to assign HScrit to sea state of capsize rate (probability) of 5%. Consequently, the critical significant wave height constitutes a boundary below which a damaged vessel is unlikely to capsize, while exposure to sea states higher than HScrit will eventually lead to capsize.
Figure 3: Response amplitude operator (left) and non-dimensional roll damping coefficient (right) in calm-water forced-roll oscillations. Physical experiments carried out at the Ship Stability Research Centre. The dry hull condition corresponds to an intact ship. In damaged condition, part of the solid ballast was removed from the flooded compartment to maintain the same mass and center of gravity of the system (mass + floodwater) as in the case of an intact vessel. The second peak on the RAO of the flooded ship—at approximate frequency 1.125—is practically invisible due to ten-fold increase in roll damping.
Here is where another important notion emerges from the concept of capsize band–time to capsize. The time to capsize is a random quantity; it cannot be calculated precisely but it can be derived by means of a probability distribution. Without going into unnecessary details, one can conclude, based on the concept of critical significant wave height, that because the HScrit is a boundary below which it is unlikely to observe capsizes, that this boundary should, in theory, constitute an asymptote of the time to capsize distribution. Indeed, this intuitive observation has been confirmed by numerical simulations. The question therefore is, what happens if a damaged ship is exposed to sea states exceeding the HScrit? Here the simulations provided a rather disturbing answer: It can be expected that time to capsize decreases rapidly with sea state. More precisely, the time to capsize is inversely proportional to the difference between the actual sea state and HScrit.
It is noteworthy that the rate at which survival time diminishes at higher sea states depends not only on residual stability, but also on the complexity of the internal architecture of the ship. It is much higher in the case of RoPax ships than in the case of cruise liners. The difference derives from the distinct watertight architecture of the vessels and specifically from the presence of large undivided cargo decks/holds typical for RoPax ships. For this reason, RoPax vessels are more vulnerable than cruise ships to rapid capsizes with catastrophic consequences. However, suffering collision or grounding damage in sea states higher than HScrit specific for the damage will inevitably lead to capsize regardless of the ship type. Depending on the severity of the damage, time to capsize can be shorter or longer, but, as the results suggest, it may often be too short for orderly evacuation and abandonment.
It is a well known fact, nearly a truism, that capsize is caused by accumulation of floodwater in the open-to-sea internal spaces of the ship. Hence, the HScrit must constitute a limiting sea state above which the flooding process becomes progressive, to the point at which there is either stability or floatability failure. The ship either capsizes or sinks. In the same way, one can reason that, if the damaged ship is exposed to sea states lower than critical, the flooding is confined; the amount of floodwater may undergo instantaneous changes but, on average, it should remain constant. This is an important finding, because if the average amount of floodwater does not change there is no formal distinction between intact and damaged condition; floodwater could be treated in a way similar to liquid cargo.
In order to validate the assumption it was necessary to investigate the process of accumulation of floodwater in damaged spaces. The task, however, is not trivial for a number of reasons. Most importantly, the analysis should be performed for surviving cases because only in the absence of progressive flooding can the process can be stationary; that is, its statistical characteristics do not change with time or position. This makes the analysis simpler and enables reasoning based on collection of short-time realizations instead of individual long runs.
Figure 4: The graph on the left shows capsize bands recorded during model tests on RoPax vessel. Graph on the right depicts contraction of the capsize band with increased observation time.
On the other hand, it is usually very difficult to detect capsize on the basis of time histories. With RoPax, in particular, there is seldom a hint that the ship is about to capsize until the process has already begun. Therefore, the first question to be asked is: How to decide whether the particular run survives or not? Indirectly, the answer is already known as, during a surviving run, an average amount of floodwater must remain constant. In GOALDS, however, a slightly different approach was followed. Namely, instead of looking at the average amount of floodwater, the maximum (95th percentile) values were recorded at times corresponding to multiples of wave peak periods, Tp. The characteristic obtained for surviving cases at a sea state corresponding to critical significant wave height was taken along with its upper confidence limit as a reference, with all other characteristics plotted against it.
The results revealed interesting features of the process of floodwater accumulation. First, if the reference curve did not have a horizontal asymptote (limit), this would indicate that some of the surviving runs will result in capsize with longer simulation times. Second, the characteristics obtained for surviving runs in sea states exceeding HScrit would fall within the confidence limits of the reference curve; that is, despite a gradual increase with wave height, statistically all the curves would be identical. Correspondingly, if the floodwater characteristic exceeded the maximum reference value at any time, given by the upper confidence limit, the corresponding run would eventually result in capsize.
Interestingly, particularly in the case of RoPax ships, it was observed that capsize would not occur immediately after the limit had been exceeded; in many cases, it would take considerable time before the ship lost stability. Furthermore, even if the ship returned temporarily to the safe band after exceeding the floodwater limit, capsize was inevitable. These conclusions are important for two reasons. One, they provide additional evidence supporting the revisited concept of critical significant wave height, HScrit; and two, they show that an indication as to whether a ship will capsize or not is not to be found at the time immediately preceding capsize, as the capsize process may originate at earlier stages of the flooding. Moreover, the results demonstrate clearly that the margin separating a safe from an unsafe amount of floodwater accumulated within damaged compartments can be very narrow.
Probability of surviving collision damage
One of the principal goals of the GOALDS Project was to derive new, more robust, and accurate formulation for estimating the probability of surviving collision and grounding damages in waves—the s-factor. Similar to the earlier approach pursued in the Project HARDER, it was decided to base the new formulation on the concept of HScrit and to derive the probability in two steps—first, to evaluate the HScrit and then map the critical significant wave height through the distribution of probability of suffering collision or grounding damages in a sea state equal to HScrit. The latter step involves elementary calculation and therefore does not require elaboration. The former step, evaluation of HScrit for a specific damage scenario, is more difficult. It requires relatively large and diverse samples and for this reason it has to be based on numerical simulations, as neither CFD nor physical experiments can provide sufficient data in a short time frame.
Numerical simulations, however, are based on simplified mathematical models and, as discussed earlier, the prediction can be significantly affected by uncertainties. Therefore, regression models should be based on unbiased and readily verifiable parameters; for example, instead of referring to the amount of accumulated floodwater, it is more appropriate to account for damage extent, and so on. However, going further in this direction, it soon became apparent that the only reliable parameter that could be derived from numerical simulations is the HScrit. All other parameters can be based only on geometrical characteristics of the vessel and damage, and on static stability data.
As indicated previously, the SOLAS formulation for calculating the s-factor employs only two parameters: maximum righting lever, GZmax, and range of positive stability, called range. These parameters, however, are insufficient to account for dynamic effects. Indeed, this was proved by exploiting a design of experiments (DoE) technique, which demonstrated clearly that these parameters cannot accurately represent the collected data. The parameter sensitivity studies carried out in GOALDS revealed that, apart from the static stability descriptors, the formulation should involve some measure of ship size. In order to account for dynamic effects, the studies commenced with examining possibilities of correlating ship roll period with wave peak period corresponding to HScrit. This led to a very simple and accurate formulation involving only three parameters (without any other coefficients) – maximum residual righting lever, GZmax, metacentric height in flooded condition, GMf, and residual volume, VR (volume of watertight spaces not involved in damage definition).
The first two parameters were taken as a ratio, GZmax / GMf, which may seem counterintuitive, as GMf is naturally (and rightly) perceived as a parameter positively affecting survivability. However, without going into unnecessary details, it can be said that the ratio of these two quantities is closely related to the angle of maximum righting lever within the range of positive stability. This angle, in turn, can be used as a conservative estimate of the angle of unstable equilibrium—the dynamic equivalent of the angle of vanishing stability. Therefore, following dimensional analysis, the initial GOALDS proposal allowed estimating of the HScrit by means of using the product of the ratio GZmax / GMf multiplied by the cubic root of the residual volume. Applying the formula to numerical data resulted in moderate improvement compared to SOLAS; the GOALDS prediction was characterized by comparable to SOLAS scatter but with a smaller mean deviation from the numerical data.
However, when applied to experimental data (model tests) it proved to be generally conservative. Bearing in mind that, as a rule of thumb, the numerical prediction underestimates survivability, it was decided to base further refinements on results of physical tests instead of numerical data. The final expression was obtained by very simple manipulation, which resulted in replacing the ratio GZmax / GMf with AGZ/( 0.5 . Range . GMf), where AGZ corresponds to area under the GZ curve. In fact, in the hypothetical case of a triangular GZ curve, both ratios are equivalent as AGZ can be expressed as 0.5 . Range . GZmax.
The modified formula allowed very high correlation with experimental data and high accuracy of prediction. Furthermore, validation with a DoE technique demonstrated that the choice of parameters is completely sufficient to represent the sample of measured critical significant wave heights.
Predicting survivability of damaged ships in a seaway is a complex task, involving analysis of complex physical phenomena. This, however, can be done only if the available experimental data is of high quality and derived from large and diverse samples. These two requirements are conflicting as, at present, it is practically impossible to perform sufficiently large number of CFD or physical experiments within reasonable time, even if the work can be carried out by a number of research institutions. Numerical simulations, on the other hand, are much more time-efficient, but the efficiency comes at the cost of lower accuracy. However, this can be overcome if the numerical prediction is used primarily in order to derive dominant parameters, while tasks requiring higher precision and validation are performed with the use of more accurate, but less effective, techniques.
The authors are with the Ship Stability Research Centre in the Department of Naval Architecture and Marine Engineering at the University of Strathclyde, in Glasgow, United Kingdom.