Progressive Flooding Assessment of the Intermediate Damage Cases as an Extension of a Monte-Carlo based Damage Stability Method

Transcription

1 Proceedings of the PRADS October, 213 CECO, Changwon City, Korea Progressive Flooding Assessment of the Intermediate Damage Cases as an Extension of a Monte-Carlo based Damage Stability Method Hendrik Dankowski, Stefan Krüger Institute of Ship Design and Ship Safety Hamburg University of Technology, Hamburg, Germany Abstract The computation of intermediate flooding stages for current damage stability rules is a very complex and time consuming task for the design engineer and usually only performed manually for certain critical cases and stages. A progressive flooding method is presented in this paper, which computes the flux between the compartments based on the Bernoulli equation. Large and partly flooded openings are taken into account as well as optional air compression and flooding through completely filled rooms. The method is validated with the standard benchmark model test recommended by the ITTC. Especially for multi-compartment damage combinations, the correct treatment of possible critical intermediate stages of flooding is unclear and only briefly described in the current SOLAS 29 regulations. The method presented here uses a typical damage opening based on the generated damage cubes by a Monte Carlo simulation to perform a direct progressive flooding assessment for each critical intermediate case. This method has the advantage to remove the limitation of typical three to four discrete intermediate stages of flooding, that are usually assumed. Instead, it allows to investigate each intermediate damage case in the time domain to gain a more detailed view on the severity of the flooding process and the vulnerability of the ship with regard to flooding of the watertight integrity after damage. The generation of the damage cases and the required openings is completely automated by chopping the section based hydrostatic model of the hull and the compartments with an appropriate damage cube. A robust and fast algorithm taking into account complex compartment geometries including negative sub-spaces will be presented. The combination of flooding calculations with a Monte Carlo method extends the classical damage stability calculations to the time domain, which allows a more accurate estimation of the overall safety level of a ship to withstand damage. Keywords Damage Stability; Flooding Simulation; Intermediate Damage Cases; Computational Geometry Introduction The introduction of the probabilistic damage stability concept in the SOLAS 29 provides great advantages for the ship designer. However, the new concept also introduces some problems with regard to the correct application of the regulations during the early design stage. The first problem to mention is the generation of the damage cases and the determination of the correct probabilities for each case. This problem can be solved by applying a Monte Carlo simulation (short: MC) based approach to the damage stability calculations. This concept has for example been described in Dankowski (21) or Koelman (26). Details of this concept are revised in the first part of this paper. Another main problem is the correct treatment of intermediate cases of flooding. For some damage cases, the flooding process to reach the final equilibrium stage of flooding is more hazardous to the stability of the ship than the final stage. Especially for multi-compartment damage cases, the correct identification of critical intermediate flooding stages is very challenging and error-prone. This problem can be solved by simulating the flooding process of each critical intermediate stage in the time domain by applying an appropriate time domain flooding simulation. The missing link to the MC damage stability calculation is the correct and automatic generation of the damage opening to the watertight integrity of the ship hull based on the current damage case. The determination of the actual opening(s) is only possible with an MC approach, because only here the damage cuboid, which is assumed to penetrate the hull is known. The generated openings serve as an input to the flooding calculations in the time domain. The geometric algorithm to determine the cor-

2 rect hole automatically from the hydrostatic model and the damage cuboid will be the core topic of this paper. The Monte Carlo Approach for Damage Stability Calculations The probabilistic concept to assess damage stability problems has first been introduced by Wendel (196). The SOLAS 29 regulations harmonises the damage stability requirements for ships and applies this probabilistic concept to both cargo and passenger ships. Even though this probabilistic approach has many advantages, it also requires more complex computations, which can essentially be simplified with a Monte Carlo approach for this type of problem. A short introduction to the Monte Carlo method applied to probabilistic damage stability calculations will be given in the following, while further details of this method can for example be found in Dankowski (21). Basic Concept The damaging of ships can be in general considered as a statistical process. The size and location of damages occurred to ships are collected in databases as described for example in Lützen (22). Based on these statistics, the probability of damaging a certain part of a ship can be computed by integrating the probability functions for the location and extent of the damage. The determination of the damage probabilities for one damage case is indirectly given in the harmonised SOLAS 29 damage stability regulations. However, the correct calculation of this probability for arbitrary damage cases becomes very difficult, especially for multi-compartment cases with complex inner subdivision arrangements. The Monte Carlo simulation approach simplifies this calculation. The idea is to perform the integration by an numerical experiment: A sufficient number of random damage cubes are drawn from the marginal cumulative distribution functions. A uniform distributed random number between zero and one is for example converted to the non-dimensional damage length by computing the inverse of the cumulative distribution (see Fig. 1). These cubes are then used to virtually penetrate the ships compartmentation and simply counting the number of hits for each generated case. In the limit of infinite drawings, this gives the exact integration value, while in practice a number of about 1, drawings is sufficient. Generation of Damage Cases The damage generation process is sketched as follows: - Draw damage cube from the statistics - Identify the penetrated compartments - Count number of hits for each case All four degrees of freedom of the damage cube (location, length, depth and height) are computed from the marginal distributions of these values. Any geometric or statistical dependencies of the values are taken into account by either truncating the values or redrawing the damage cumulative distribution function (CDF) random number resulting length damage length/subdivision length Figure 1: Drawing of the damage length for one random number cube until a valid one has been found (see also IMO SLF 55/INF.7, 212). The overlap volume between the damage cube and the compartments in question determines precisely if the compartment is penetrated by the cube or not. Combined with a pre-check of the bounding boxes, this algorithm works very robust and fast resulting in a damage generation time of less than a minute for 1, drawings. Depending on the complexity of the inner subdivision and the number of compartments, about 1,-3, damage cases are generated for each damage group, i.e. different draughts and sides. The probabilities are the number of hits of each damage case divided by the total number of hits. The Monte Carlo approach has the advantage, that this generation of cases is completely automatic resulting in more damage cases compared to a manual method. This means that the reached attained index is typically also slightly larger, because each additional case may contribute to the index. Furthermore, the generation of the cases is independent from the inner subdivision, since the perspective is from the damage side, which allows to handle arbitrary complex geometries of the inner subdivision of the ship design. The main advantage for the ship design engineer is the direct access to the most important damage cases. These can be identified by sorting the cases according to their overall probabilities, which are not directly given by the SOLAS 29 formulas. In addition, the MC method can not only be applied to any probabilistic damage rule, but also to deterministic formulations, if for example a uniform distribution is assumed within the given limits of the deterministic damage extent. The Numerical Flooding Simulation The first step towards a time dependent assessment of each damage case and its intermediate flooding stages is provided by a quasi-static treatment of the flooding process. The floodwater ingress and the spreading of the floodwater inside the vessel is computed by an hydraulic model for the water fluxes. For each time step, the new distribution of the floodwater inside the complex inner subdivision of the ship is computed and a new floating equilibrium position is determined based on the

3 new resulting hydrostatic moments caused by the floodwater. Details of the methods can be found in Dankowski (212a) and will roughly be sketched in the following. Determination of Water Fluxes The pressure head differences at the openings lead to an water in- or egress to the watertight integrity of the ship or between two inner compartments. The flow velocity is determined by the incompressible, stationary and viscous and rotational free Bernoulli equation for a streamline connecting two points a and b 1 Outside 3 5 R21S 2 R R21 6 R R21P 7 R22 p a p b ρ g + u2 a u 2 b 2 g + z a z b ϕ ab g which results in the following flow velocity = dh, (1) u = 2 g dh. (2) By integrating the velocity over the area of the opening, the volume flux is determined assuming a perpendicular flow direction to the opening. Any dissipative losses are taken into account by an semi-empirical discharge coefficient C d : V = V t = Q = C d u n da = C d u da. (3) A A The integration becomes more complicated for a free discharge into air through a larger opening, since the cross section and the fluid velocity may vary over the extent of the opening. In this case, the opening is discretised in several smaller z-stripes. s s 1 y 1 y(s) y Figure 2: Stripe in z-direction An analytic solution for the free outflow into air for such a stripe shown in Fig. 2 is given by: Q = 2 2 g 3 s z [ y 1 h y h (y ] 1 y ) 5 (z 1 z ) (h h 5 2 ). (4) This approach allows to accurately compute the water flux through any kind of opening shape, which can be discretised by such z-stripes. The Flooding System The connection of all compartments by openings can be modelled by directed graphs. Each compartment is represented by a node and the openings are the corresponding edges. A simple example for such a flooding graph is shown in Fig. 3. This flooding graph is derived from the geometric setup of the model test case A as described in Ruponen (27). s z 1 z Figure 3: Simple flooding graph The flooding graph is used to obtain the mass balance for each compartment flooded through the connected openings. Time Domain Integration The resulting ordinary differential equation to be solved for the flooding simulation based on a hydraulic model is of first order and nonlinear. The flux to one compartment must be equal to the flux through all connected openings to this compartment. The non-linearity is introduced by the square root relation of the flow velocity to the water height differences and the typically nonlinear relation of the water height to the fluid volume in one compartment. Due to the character of the equation system, the integration in the time domain is performed by an explicit predictor-corrector scheme. This efficiently stabilises the numerical scheme and avoids oscillations in the water heights. Further Aspects The numerical simulation of the flooding process of real ships, for example to reconstruct an accident, requires further aspects to be considered: - Conditional openings - Air compression effects - Propagation through completely flooded compartments - Grounding effects - Additional heeling moments The conditional openings are for example used to simulate breaking windows or doors. If a certain water height column above a window is exceeded it breaks and stays open for the rest of the flooding process. For certain cases it may be required to take into account entrapped air. The occurrence of air pockets are determined and the pressure in this compartment is increased according to Boyle s law. It has been shown in other test cases that the air flow itself has usually not to be modelled. If a compartment is completely filled with water, the simple hydraulic approach has to be extended, because the

4 boundary condition, that the overall flux to this compartment over one time step is zero has to be fulfilled. In this case, a typically small nonlinear system of equations has to be solved, which determines the required pressure in this compartment leading to a propagation of the water through this compartment without any total water ingress. Most of the severe accidents involving large scale floodings of ships occur in shallow water. In this case, it may be useful to account for the forces and moments resulting from the grounding of the ship, whose influence can be modelled by a spring model for the interaction with the seabed, see Dankowski and Dilger (213) (not yet published) for details. This also allows to estimate the correct flooding sequence by comparing the final position of the vessel on the seabed with the results from the numerical simulation. For accident investigations, as for example performed in Dankowski (212b), additional heeling moments caused for example by a cargo shift, a turning of the vessel or wind influences can be taken into account by adding these moments when the new equilibrium floating position is determined. Simulation Overview Each time step of the numerical simulation consists of the following parts: 1. Check for conditional openings 2. Iterate pressure for full compartments 3. Determination of the opening fluxes 4. Mass balance determination 5. Update of new filling levels 6. Predictor-corrector step repeating the last three steps 7. Optional air compression 8. Update optional external moments 9. Computation of a new floating equilibrium position 1. Convergence check The simulation stops either if convergence is reached, i.e. the water levels in the compartments and the floating position does not change anymore, or if the requested simulation time is exceeded. Validation The presented numerical flooding method has been validated with the standard ITTC benchmark test for these kind of problems (van Walree and Papanikolaou, 27). Details of the test cases with a box-shaped barge can be found in Ruponen (27). As an example, some results of the validation process for test case C are presented in the following. This test case is an example for a slow up-flooding from a damage in the bottom to the upper decks. The setup is shown in Fig. 4 R12 R11 8 Side View A 6 R22 R21 z x DB1 2 1 DB2 A 7 3 Front Section View A-A ps sb R22 R21P 4 5 DB2 7 y z CL R21 R21S Figure 4: Model test case C - Side and section view of openings and compartments This test case involves a complicated progressive flooding of the model including air compression effects in the double bottom compartment DB1. For the numerical simulation, a time step of.1 seconds is chosen. The results for the motion of the barge are shown in Fig. 5. The curves labelled with meas. are the measured val- Trim angle (deg) Time (s) meas. trim R. calc. trim R. calc. trim D. meas. heave R. calc. heave R. calc. heave D. Figure 5: Model test case C - Trim and heave motion ues from the model tests, the ones with calc. R. are the computed values from Ruponen (27) and the calc. D. curves are the results from the presented flooding code. The results match quite well. However, the trim and heave motions from the measurements and the calculated values by Ruponen are slightly larger compared to our method. These small differences can be explained by the fact, that only air compression but no air flow is taken into account by our method. Small amounts of entrapped air in DB1 can escape through the opening to DB2 before the air is compressed, which allows more water to enter through the bottom opening. The comparison of the obtained water levels for the double bottom compartments are given in Fig. 6. Again, the comparison of the water levels shows a quite good agreement. The computed values are only slightly smaller than the measured ones. Therefore, the influence of the air flow on the overall flooding process can be neglected Heave (mm)

5 Water height (mm) meas. DB1 R. calc. DB1 R. calc. DB1 D. 2 meas. DB2 R. calc. DB2 R. calc. DB2 D Time (s) Figure 6: Model test case C Water Levels double bottom compartments Geometric Algorithm for the Generation of the Damage Openings 29), is composed of three spaces divided in vertical direction. A stripe of triangles is added between two section polygons by finding the shortest diagonal connecting two points between the sections for each generated triangle. The first and last section polygons are triangulated by an appropriate algorithm like the ear clipping method (O Rourke, 1998). The obtained surface model is shown in Fig. 7b. For a better visibility, the triangulated first and last section are not shown. Due to the fact, that each compartment may consist of several spaces, parts of the obtained surface model may overlap. These duplicate elements have to be removed first. This can be accomplished by clipping each surface element with all other elements by applying a polygon clipping algorithm (Murta, 1997). The cleaned surface model without duplicates is shown in Fig. 7c. It can be seen, that the decks between the spaces are now removed. The combination of the MC method and the numerical flooding simulation is only useful, if no manual interaction is required to compute any intermediate stage of flooding in the time domain. Only if the process is completely automated, sufficiently fast and reliable for any possible damage case, a benefit is obtained from this coupling. This means, that the generation of the openings from the damage cube is the essential part of the missing link between both methods. The following substeps are required to obtain the correct openings from a given damage cube: 1. Convert section into surface model a) Fill elements between sections b) Close first and last section c) Remove duplicate elements (a) Hydrostatic sections (b) Surface model (c) Removed duplicates Figure 7: Conversion from hydrostatic sections to surfaces 2. Chop hydrostatic sections with damage cube 3. Identify openings and connections 4. Remove openings located on the damage cube The substeps of the conversion and the geometric algorithms are illustrated in the following. Generation of the Surface Model A hydrostatic data model typically consist of several hydrostatic sections represented by planar polygons with a constant longitudinal x-position. Each compartment may consists of several sub-parts called spaces. A surface model is usually not required, since only the integrated values of the volume and moments are needed for hydrostatic evaluations. However, to obtain the correct geometry of the openings, it is mandatory to have a correct surface representation of the boundaries of the compartments. The conversion process is illustrated in Fig. 7. First, the hydrostatic sections of the compartment are shown in Fig. 7a. This wing compartment, a store of a typical RoPax ferry (for details of the design see Valanto et al., Chop Damage Cube with Compartment Model The next step is to chop the sections with the damage cube to keep only the part of the compartment which is penetrated by the damage. The section polygons are chopped by the six side planes of the cube resulting in a new section model of the part of the compartment inside the damage cube. The integrated volume of these remaining sections is also used as very robust check, if a compartment is really penetrated. If the resulting value of the volume is greater than zero, the associated compartments are hit by the damage in question. This approach allows also to reliable handle even negative sub-spaces. These sections are again converted to a surface model. This time, the duplicate surface elements between two compartments are not removed but saved for later. Only duplicate internal elements between two spaces of the same compartment are removed. This process is applied to a more complicated damage case involving four different compartments. The surface model of the compartments is shown in Fig. 8.

6 Further Aspects The mentioned internal openings between two spaces in one compartment could also be used to further subdivide the compartment model for the purpose of the flooding simulation. This allows, for example to automatically generate openings for cross flooding arrangements and to test if this device is capable of removing the asymmetric flooding setup within an appropriate time duration. In addition, the influence of non-watertight A-class boundaries could be further investigated with this concept. Required Extensions Figure 8: Compartment surface model of the example damage case The resulting openings together with the hydrostatic sections of the four compartments and the damage cube in red are shown in Fig. 9. The red surfaces are the external openings to the sea, the green coloured surfaces are the internal openings connecting two compartments. Figure 9: Resulting openings of the damage case At this point, all information for the start of the flooding simulation is available. The loading condition like the draught and the vertical centre of gravity is taken from the damage calculations. The openings obtained from the algorithm sketched above allow the floodwater exchange between the compartments and the outside sea. In addition, any other openings which allow further progressive flooding can be defined in advance and included in the flooding simulation. Extensions of the Monte Carlo Damage Calculations For the MC damage calculations a characteristic damage cube for one damage case must be found. Each damage case consists of hits by several different random damage cubes taken from the damage statistics. Possible choices for a cube belonging to a damage case are to take the largest, the smallest or the mean damage cube of all cubes leading to this specific damage case. The only other extension required, is to start the flooding simulation from within the MC damage calculations for the selected cases of interest. Extensions of the Flooding Simulation The obtained surface elements from the geometric algorithm sketched above describing the openings, have to be mapped to the description of the openings within the flooding simulation module. In addition, the current loading condition is derived from the damage case in question. The safety of a certain damage case regarding the intermediate stages of flooding can be tested by adding external heeling moments to the flooding simulation. If the ship keeps a stable floating equilibrium during the flooding simulation, i.e. providing a positive metacentric height together with a positive righting lever arm without capsizing, the damage case is judged to be sufficiently safe. This simple and obvious approach does not require to evaluate a lever arm curve at any intermediate flooding stage, which would actually be problematic. The reason for this is, that the definition of a lever arm curve during flooding is unclear, because the behaviour of the floodwater during the inclination of the heeling angle is unknown. Results General Setup For one specific damage case, the combination of the classical damage calculation with a time domain flooding simulation is presented. The selected damage case, as already mentioned, involves four compartments in the aft part of a typical RoPax ferry. A time step of.1 seconds and a default discharge coefficient of.6 for the generated openings is selected. The lever arm curves for the intact loading condition and the damage case is shown in Fig. 1.

7 1.8 GZ (m) intact GM: 3.94m GZ (m) damage GM: 4.33m -1-2 dkg=. m dkg=1. m dkg=1.8 m dkg=1.9 m Righting lever (m) Heeling angle (deg) Heel (deg) Time (s) Figure 12: Heeling lever over time for different KG Figure 1: Lever arm curves of the selected loading condition Both, the intact and the damage case, show a sufficient initial stability and a relative large range of the lever arm curves. To illustrate the geometric setup, one frame of the flooding sequence is shown in Fig. 11. Figure 11: Detailed view from inside showing the openings and the compartments as cubes The cubes are the reduced bounding boxes of the compartments coloured according to their filling level. Red means completely full and blue means empty. The opening elements are coloured in red, if a positive absolute water flux is present. It can depicted from the figure, that the elements above the waterline (shown as a blue surface) are not red, i.e. no water in- or egress is present here. Sensitivity Analysis Having all required data automatically generated, it is very easy to perform case studies and sensitivity analysis for a certain damage case. As a simple example, the vertical centre of gravity KG is increased for this damage case until the ship capsizes. The development of the heeling angle over time for different KG values is shown in Fig. 12. The influence of the KG value is very interesting to observe. After only a few seconds, a stable stage of a relative small heeling angle of less than six degrees is reached for a maximum increase of 1.8 m. However, if a certain threshold value is exceeded, in this case an increase of 1.9 m, the heeling angle rapidly increases and the ship capsizes, finding a new floating equilibrium upside down at a heeling angle of around 18 degrees. Conclusion and Outlook A robust and fast geometric algorithm has been presented, which generates the holes from an damage cuboid based on the hydrostatic sectional model of the ship. This serves as an input for a flooding simulation in the time domain to gain an in-depth view of each intermediate stage of flooding for any arbitrary damage case. In addition, this concept also simplifies the investigation of ship accidents, since the damage openings can be generated automatically. The direct and automatic coupling of the MC simulation with a progressive flooding simulation allows to perform case studies to identify the main contributing factors leading to hazardous situations during intermediate stages of the flooding process. Furthermore, it is very useful at the early design stage to identify critical intermediate stages of flooding. For the shown example, a sudden capsize during the flooding process is observed after the vertical centre of gravity is increased by a specific threshold. The influence of other contributing factors like the inner subdivision arrangement and the statistical evaluation of all critical intermediate damage cases for a specific ship design could be part of further research. Acknowledgement This work is part of the research project LESSEO, which is funded by the German Federal Ministry of Economics and Technology (BMWi). The authors gratefully thank the BMWi for supporting this research work. References Dankowski, H. (21). On the safety level of the SOLAS 29 damage stability rules for RoPax vessels. In 11th International Symposium on Practical Design of Ships and Other Floating Structures. Dankowski, H. (212a). A Fast, Explicit Method for the Simulation of Flooding and Sinkage Scenarios on Ships. Ph.D. Thesis, Hamburg University of Technol-

8 ogy, Institute of Ship Design and Ship Safety. Dankowski, H. (212b). An Explicit Progressive Flooding Simulation Method. In Spyrou, K. J., Themelis, N., and Papanikolaou, A. D., editors, 11th International Conference on the Stability of Ships and Ocean Vehicles, Athens, Greece. Dankowski, H. and Dilger, H. (213). Investigation of the Mighty Servant 3 Accident by a Progressive Flooding Method. In Proceedings of the ASME st International Conference on Ocean, Offshore and Arctic Engineering, number ISBN IMO SLF 55/INF.7 (212). The GOAL based Damage Stability project (GOALDS) - Derivation of updated probability distributions of collision and grounding damage characteristics for passenger ships. 55th session. Koelman, H. J. (26). A new method and program for probabilistic damage stability. In Grimmelius, H., editor, COMPIT 6, pages , Oegstgeest, Netherlands. Lützen, M. (22). Harder: Damage Distributions. (2-22-D ). O Rourke, J. (1998). Computational Geometry in C. Cambridge University Press. Ruponen, P. (27). Progressive Flooding of a Damaged Passenger Ship. PhD thesis, Helsinki University of Technology. Valanto, P., Krüger, S., and Dankowski, H. (29). EMSA Study on Damage Stability of ROPAX Vessels. Technical report, European Maritime Safety Agency. Murta, A. (1997). GPC General Polygon Clipper library. toby/alan/software/. last access: van Walree, F. and Papanikolaou, A. (27). Benchmark study of numerical codes for the prediction of time to flood of ships: Phase I. In Proceedings of the 9th International Ship Stability Workshop, pages 45 52, Hamburg, Germany. Maritime Research Institute Netherlands (MARIN), National Technical University of Athens (NTUA). Wendel, K. (196). Die Wahrscheinlichkeit des Überstehens von Verletzungen. Schiffstechnik, 4.