# Statutory and Operational Damage Stability by a Monte Carlo Based Approach

## Abstract

**:**

## 1. Introduction

## 2. Background

_{i}that this compartment or group of compartments is damaged needs to be calculated according to the SOLAS 2020 B1 regulations as follows:

_{i}(x

_{1},x

_{2},y,z) = p

_{i}(x

_{1},x

_{2}) r

_{i}(y,x

_{1,}x

_{2}) v

_{i}(z)

_{i}denotes the total probability that the compartment or group of compartments is damaged. The individual probabilities p

_{i}, r

_{i,}and v

_{i}are related to the damage cuboid which extends from x

_{1}to x

_{2}in longitudinal direction. The transversal penetration depth is y, measured from the damage waterline, and the vertical extension is from the base line to the damage height is z. p

_{i}(x

_{1},x

_{2}) is the probability that the damage extends from the aft position located at x

_{1}to the forward position x

_{2}. r

_{i}(y) is the probability that the penetration depth extends to the local breadth y and v

_{i}(z) is the probability that the damage extends to the height z. Formula (1) is applicable for so called one-compartment damages (see Figure 1), and it should be noted that the probability r

_{i}is depending not only on the penetration depth, but also on the length of the damage as given by x

_{1}and x

_{2}. Further, the SOLAS requires also to investigate damages where the lower boundary in z is above the baseline (so called lesser extent cases), but this investigation is performed during the calculation of the individual S

_{i}-factors (survivability of the damage) according to the SOLAS requirements.

_{1}, x

_{2,}and x

_{3}, the probability p

_{i}

_{,2}for a damage which opens only the two compartments (and not a single compartment) is to be calculated as follows (see Figure 1):

_{i}

_{,2}(x

_{1},x

_{3}) = p

_{i}(x

_{1},x

_{3}) − p

_{i}(x

_{1},x

_{2}) − p

_{i}(x

_{2},x

_{3})

_{i,}the penetration depth y and damage height z need to be considered to obtain the probabilities r

_{i}and v

_{i}. When the total probability P

_{i}of the damage case has been determined from the combination of Formulae (1) and (2), the probability of survival S

_{i}needs to be computed. This requires a hydrostatical analysis of the damage case. The product P

_{i}S

_{i}then contributes to the total attained index A. If that index A is larger than the required index R, the calculation is finished. The calculation has to be made for three drafts, assuming damages from both port and starboard side. It has to be repeated many times during the design of a ship, as many variations of possible compartmentations in combination with possible values of GM have to be investigated. As a matter of fact, this process is quite inefficient with respect to modeling effort and also computational time. At first, the manual input of longitudinal damage zones, penetration depths and damage heights results in the fact that the ship’s compartmentation is actually modelled twice, which boosts the modeling effort. If during the design phase of the ship the compartmentation is modified, each modification must be reprocessed in the modeling of the damage cases. Besides possible consistency errors, this requires a lot of time. For this reason, the damage stability calculation process has been reengineered in our ship design software by inverting the problem, see Krüger and Dankowski [1]. Comparable developments have been made by Koelmann [2], Bulian [3,4,5,6], Ruponen [7], and Mauro [8]. This inverted method is based on the concept that instead of assuming a damage case and calculating the related probabilities, it is much more efficient to assume a damage extent and to compute the outcome of that particular damage extent, which can be most efficiently solved by a classical Monte Carlo Simulation technique. This method has been proven as a very efficient tool during the design phase of a ship, as the A-index and the limiting stability curves can be calculated extremely fast. The problem now exists that despite these improvements, still a significant amount of man hours is required to generate the approval documents, even if the compartmentation and the limiting stability curves are already fixed. Therefore, it would be extremely useful if this efficient simulation principle could be extended for the generation of approval documents. This requires the generation of additional data which cannot directly be obtained from the Monte Carlo simulation method. Most challenging here is to determine the required individual probabilities p, r, and v, to sort damage cases into damage zones and to split damage cases afterwards according to their individual penetration depths. The following sections give an overview about this development, which is a kind of reverse engineering of the conventional calculation principle based on the Monte Carlo simulation. First, the Monte Carlo simulation principle is briefly introduced.

## 3. Monte Carlo Realization

_{i}computations according to the principles stated in the SOLAS. Bulian et al. [6] have shown an interesting alternative method to include other lower damage boundaries than base line in the Monte Carlo simulation, but this would deviate from the SOLAS and would then require alternative approval procedures.

- Draw the damage cuboid from the damage distributions.
- Find the corresponding damage case.
- Integrate the hits for each individual damage case.

- The number of hits even for a very complex combination of compartments can directly be computed. There is no need to look at any subcases and their probabilities.
- As counting of hits is simply a binary event (yes/no), also very complicated geometries can easily be handled (in contrast to the procedure described in the SOLAS Explanatory Notes, which is in parts reflected by Figure 1).
- Sorting the damage cases according to the frequency gives direct access to the important cases for the subdivision design. This shows the designer immediately which compartment combinations have the largest impact on the subdivision index A.
- Additionally, substantially more damage cases will be found compared to the conventional method. For sufficiently large samples, all possible damages cases are found by the simulation. This is very important for validation purposes.

_{i}can be computed according to the regulations in the SOLAS. The only requirements for this method are the damage distribution functions, a random number generator (e.g., Matsumoto [14]), and a reliable method to obtain the combination of damaged compartments from the geometry of the damage cube. It was shown by Kehren [9] that the obtained probabilities do clearly converge for large numbers of samples. For practical damage stability assessments, a sufficiently large sample is assumed to be 1.0 × 10

^{6}drawings. As shown by Dankowski and Krüger [1], this simulation method shortens the damage stability computation time drastically and the method has successfully been applied during the initial design phase of complex ship subdivisions by the industry.

- It can only deliver the total probability p
_{i}(x_{1},x_{2}) r_{i}(y,x_{1},x_{2}) v_{i}(z) and not the individual probabilities p_{i}, r_{i,}and v_{i}. However, the calculation of the individual probabilities is required for the approval by the classification societies. - The output of the method consists of individual damage cases with their contribution to the total index. For the generation of approval documents, it is necessary to group the damage cases into so called damage zones, which can be defined either by the user or (as a default) by a program. All damage cases located in a damage zone must then be sorted according to their damage heights and their penetrations.

_{i}can in some cases take values larger than 1 which requires further measures to solve this problem.

## 4. Determination of the Probabilities P_{i}, R_{i} and V_{i}

#### 4.1. Principle Approach

- The determination of the probability p
_{i}by an additional drawing. - The adding of damage zones.
- The splitting of damage cases to cope with probabilities r
_{i}that are larger than 1.

_{i}requires a modification of the Monte Carlo simulation method. This modification is described in the following: From Equation (1) it becomes obvious, that r

_{i}(y,x

_{1},x

_{2}) and v

_{i}(z) become exactly 1 if the damage cuboid extends to the maximum possible damage height z

_{max}and maximum possible damage penetration y

_{max}. If now r

_{i}(y

_{max},x

_{1,}x

_{2}) = 1 and v

_{i}(z

_{max}) = 1 hold, then it becomes immediately obvious that P

_{i}(x

_{1},x

_{2},y

_{max},z

_{max}) = p

_{i}(x

_{1},x

_{2}), which means that our simulation principle will determine the probability p

_{i}(x

_{1},x

_{2}) instead of the total probability P

_{i}(x

_{1},x

_{2},y,z). This does in practice mean that we have to modify the simulation principle by adding so called “fully extent damage cases” which are obtained from a second drawing with a modified cuboid as follows: before we determine the damage case from the cuboid extensions as obtained from the underlying CDF data, we modify each damage cuboid by setting z = z

_{max}and y = y

_{max}. From this additional cuboid we can obtain the combination of compartments breached by this “fully extent” cuboid and we obtain the so called “fully extent damage case”. If we do now integrate the hits for each full extent damage case, we can obtain the probability in the same way as before. However, we do know that this probability must be equal to p

_{i}(x

_{1},x

_{2}), as r

_{i}and v

_{i}are 1 by definition. As we do perform both drawings simultaneously, we know for each individual damage case to which full extent damage case it belongs, and consequently, we know the probability p

_{i}(x

_{1},x

_{2}) for all these damage sub cases.

^{6}samples has most probably detected all possible full extent configurations, but there should be remaining combinations of penetration depths and damage heights which have not yet been detected. However, from a practical viewpoint, this is not a problem because the index contribution of these remaining cases (even if they would be survived) is in the order of magnitude of about 1.0 × 10

^{−5}. The total number of damages cases with a sample of 1.0 × 10

^{6}is 1776, which means that each full extend damage case has in average 7.78 sub damage cases. In fact, the number of sub cases varies from 1 to 34 sub cases for each full extent damage case. By this second drawing, we have now not only established a relationship between a damage case and its full extent damage case, but we can now at the same time compute the required probability p

_{i}. All sub cases of all full extent damage cases must then be damage cases where either r

_{i}or v

_{i}or both do not equal 1. It should be mentioned in this context that the computed index A

_{i}is astonishingly robust: with only 10,000 drawings, one obtains an index of already 0.7222, where the computed index for 1.0 × 10

^{6}drawings amounts to 0.7212. The difference is about 1‰.

_{max}-value of all the spaces of all compartments breached each cuboid. As a consequence, we obtain a reliable value for the maximum damage height z of each individual damage case. Applying the same procedure also for the y-coordinates of each cuboid, we can obtain the maximum penetration depth for each individual damage case. However, this task is now much easier, as we have already pre-sorted the damages according to their individual damage heights. We have decided to start with the damage heights first due to the fact that this direction is geometrically less complex compared to the y-penetration.

_{i}(z) for each group of penetration depths can directly be computed for each of these individual damage cases from the basic CDF Formulae. As already mentioned, the probability p

_{i}is obtained from the full extent case. The remaining probability r

_{i}(which is the most challenging one, because it depends on both penetration and damage length) can then simply obtained by inverting Equation (1):

_{i}(x

_{1},x

_{2},y) = P

_{i}(x

_{1},x

_{2},y,z)/p

_{i}(x

_{1},x

_{2})/v

_{i}(z)

#### 4.2. Procedural Problems during the Determination of P_{i}, R_{i,} and V_{i}

_{i}(z = 16.00) = 1, and at the same time v

_{i}(z = 9.20) = 0.2769. The problem now exists that a damage case which does breach only the steering gear compartment (and no other) is possible by two basically different damage types:

- Any damage which does not breach the space “Emergency Exit” but only the space “Steer Gear CL” must have a maximum damage height of less or equal to 9.20 m. The damage penetration is the maximum penetration possible.
- Or the damage breaches only the space “Emergency Exit”, and then the maximum damage height is 16 m with a limited penetration.

_{i}(x

_{1},x

_{2}) is obtained from the full extent case, and our methodology would result in z

_{max}= 16.20 m for this particular case, resulting in v

_{i}= 1 and r

_{i}according to Equation (3). This result is obviously a possible and a reasonable result at the same time, but it differs from a conventional damage stability calculation: From the viewpoint of a manual damage case generation, the damage case “steering gear compartment” must be divided into two separate damage zones to correctly obtain the three individual probabilities. The Monte Carlo simulation is due to its principle able to deliver the total probability P

_{i}correctly, but the splitting of the total probability into the three individual probabilities does obviously depend on the splitting of damage cases into longitudinal damage zones. We will later show that a comparable problem exists for the transversal penetration, too. The following subsections show how these difficulties can be overcome in the framework of the Monte Carlo simulation.

#### 4.3. The Necessity of Adding Damage Zones

#### 4.4. The Necessity of Splitting Damage Cases in a Damage Zone

_{i}obtained from Equation (3) may become larger than 1. This is due to the following reason: The void space (and only the void space) can be damaged by principally two different cuboids. One group of cuboids has the full damage height, but the penetration depth is limited to the inner longitudinal bulkhead. If the penetration depth would be larger, then also the lower hold would be damaged, which leads to a different damage case. This set of cuboids is denoted by 1 in Figure 5. Alternatively, the cuboid has a damage height lower than z

_{min}of the lower hold, but a penetration depth which extends to the pipe duct in the center. This set of damage cuboids is denoted by 2 in Figure 5.

_{i}and p

_{i}. After the simulation is finished, these probabilities P

_{i}and p

_{i}are correctly determined. When now the product r

_{i}v

_{i}is analyzed according to the aforementioned procedure, it is found that the maximum damage height of this damage case is surely z

_{max}, which comes from the cuboid group 1. The maximum penetration depth is now the transversal position of the pipe duct, which comes from damage cuboid group 2. However, both extremes at the same time are not possible, because that would result in a different damage case. If now the probability v

_{i}is obtained from z

_{max}, this results in the fact that the obtained r

_{i}from Equation (3) is now larger than 1, which is due to the fact that r includes contributions from both damage scenarios at the same time. Although the product r

_{i}and v

_{i}is obtained correctly from the simulation, it is formally difficult to explain that one individual probability shall take a value larger than 1.

_{i}and v

_{i}can be obtained. As p

_{i}must be the same for both sub cases, the two values of the total probabilities P

_{i}can then be obtained from Equation (1). This splitting of the damage cases can now automatically and iteratively be processed for all situations where r

_{i}> 1 is detected.

## 5. Numerical Studies

_{i}is to be computed as

_{i}= k (h/0.2 r/20)

^{1/4}

_{i}= k (h/0.12 r/16)

^{1/4}

_{i}. E.g., a maximum range of one degree and a maximum righting lever of 1 cm result in an S

_{i}value of 0.223. As a consequence, the numerical accuracy of the calculation model is of importance for the computed index, as the index is the sum of many small numbers. To illustrate this, we have created two models of the EMSA2 RoPax. Besides the buoyant hull, the ship has 72 compartments which consist in total of 241 spaces. Figure 6 shows the coarse model of the RoPax and a finer model.

_{i}= 0.7212, and the coarse model delivers a

_{i}= 0.7191. The difference is small, but still remarkable. The calculations are based on 1.0 × 10

^{6}drawings of damage cuboids (plus 1.0 × 10

^{6}cuboids with full height and full penetration). On a standard LapTop (Intel core i7 processor, 9th generation), the computational times are as follows (valid for one single sub index A

_{i}

_{)}:

- Course model: 9 s for the generation of the damages, 10 s for the hydrostatic calculations to obtain the individual S
_{i}-values. In total, this means the 19 s for the calculation of one A_{i}-index, and then 114 s to calculate the total A-index. - Fine model: 12 s for the damage generation, 32 s for the S
_{i}-computations which means 44 s for one A_{i}-index and 264 s for the total A-index.

_{i}-computations include the investigation of all possible lesser extent damages (where the lower z-boundary is above the base line) and the computation of any intermediate stages of flooding, e.g., due to cross flooding devices and/or A-class bulkheads, if applicable.

## 6. Damage Stability Output

_{i,av}for all damage cases of that group. The limit values can be set by the user, and the defaults which are also used in the following figures are S

_{i,av}> 0.85 for a green triangle and S

_{i,av}< 0.3 for a red triangle.

_{i}into p

_{i}and the related r

_{i}and v

_{i}probabilities.

## 7. Operational Damage Stability

_{i}as a function of the ship’s draft (see also Table 1). In the stability booklet of the ship, an equivalent figure is plotted showing the required GM-values. The stability booklet also contains a loading condition “Max Pax Departure” where the ship has a (mean) draft of 5.565 m (or 17,260 t displacement), which is shown in Figure 9. Following the damage stability requirements of the stability booklet, the ship must have a GM of 4.10 m to cope with the damage stability requirements. From Figure 8, the fact becomes obvious that for the 5.565 m draft, the index A

_{i}can be interpolated as 0.717. This means that if we perform an operational damage stability calculation, this calculation must result in an index A

_{i}for that particular loading condition of no less than 0.717 (means of port side and starboard side). For RoRo-passenger ships, it must also be verified that any A

_{i}must be larger than 0.9 R. However, this must hold for all indices presented in Figure 8, and then it holds automatically for any interpolated index, too. In the framework of the research project DIGILECK (see acknowledgements) it was investigated under which boundary conditions this proposed procedure can be covered by the existing SOLAS explanatory notes.

_{i}of 0.8067 on port side and 0.7935 on starboard side (see Figure 10), which is substantially more than 0.717. The difference is due to the effect of the fluid exchange of the tanks, as we did for the moment not alter the permeabilities of the RoRo-cargo holds. If it is sufficient to reach an index of 0.717, the GM can be lowered to 3.70 m, which is a substantial gain and significantly increases the flexibility of the ship, especially at the lower drafts. Additionally, it must again be pointed out that the calculation does not take any substantial effort.

## 8. Conclusions

_{i}. Secondly, the grouping of the resulting sub damage cases according to the penetration depths and damage heights allows the determination of v

_{i}and r

_{i}. This enables the simulation method to produce exactly the same output as the conventional calculation method, and the results can be approved by the classification society during the standard approval process. The extended method is in use by our industrial partners, and two damage stability calculations by the extended method have already passed the class approval.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Principle of the Monte Carlo Method for damage stability problems. Here: Penetration depth for side damages.

**Figure 3.**Simulation results for the EMSA2-RoPax. Black curve: Number of damage cases. Red curve: 5× Number of “fully” damaged cases.

**Figure 10.**Output of the operational damage stability calculation for ps (left) and stb (right). GM = 3.70 m.

**Table 1.**Damage stability results for the EMSA 2 RoPax [15].

Denomination | Draft A.P. | Draft F.P. | Displacement | GM | Index | Index | Index |
---|---|---|---|---|---|---|---|

[m] | [m] | [t] | [m] | PS | STB | Mean | |

Light | 5.74 | 4.75 | 16,081 | 4.50 | 0.724 | 0.680 | 0.702 |

Partial | 6.17 | 6.17 | 19,933 | 4.10 | 0.761 | 0.725 | 0.743 |

Deepest | 6.80 | 6.80 | 22,875 | 4.10 | 0.735 | 0.700 | 0.718 |

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**MDPI and ACS Style**

Krüger, S. Statutory and Operational Damage Stability by a Monte Carlo Based Approach. *J. Mar. Sci. Eng.* **2023**, *11*, 16.
https://doi.org/10.3390/jmse11010016

**AMA Style**

Krüger S. Statutory and Operational Damage Stability by a Monte Carlo Based Approach. *Journal of Marine Science and Engineering*. 2023; 11(1):16.
https://doi.org/10.3390/jmse11010016

**Chicago/Turabian Style**

Krüger, Stefan. 2023. "Statutory and Operational Damage Stability by a Monte Carlo Based Approach" *Journal of Marine Science and Engineering* 11, no. 1: 16.
https://doi.org/10.3390/jmse11010016