Capsizing Probability of Dead Ship Stability in Beam Wind and Wave for Damaged Ship

The International Maritime Organization has developed the second-generation intact stability criteria. Thus, damage stability criteria can be established in the future. In order to identity the capsizing probability of damaged ship under dead ship condition, this paper investigates two methods that can be used to research the capsizing probability in time domain, which mainly focus on the nonlinear righting lever GZ curve solution. One method subjects the influence of damaged tanks on the hull shape down to the wind and wave, and the other method is consistent with the real-time calculation of the GZ curve. On the basis of one degree of freedom rolling equation, the solution is Monte Carlo method, and a damaged fishery bureau vessel is taken as a sample ship. In addition, the results of the time-domain capsizing probability under different loading conditions are compared and analyzed. The relation of GM and heeling angle with the capsizing probability is investigated, and its possible reason is analyzed. On the basis of combining the time-domain flooding process with the capsizing probability calculation, this research aims to lay the foundation for the study of capsizing probability in time domain under dead ship condition, as well as provide technical support for capsizing mechanism of dead ship stability and damage stability criteria establishment in waves.


Introduction
The International Maritime Organization (IMO) is currently working on formulating and revising the stability standards, which are applicable to satisfy the standards in still water but still capsize in waves (IMO SDC 1/INF, 2014b). Ships in waves pitch, roll, and sway, and the amplitude of motion can become unfavorable under bad conditions. If a large-scale rolling motion occurs and the restoring moment is insufficient, then ships will capsize. Ship capsizing in wind and wave is a strongly nonlinear problem, which involves nonlinear rolling motion, nonlinear restoring moment, nonlinear wave moment, and its interaction with the ship hull. Dead ship behavior is the resonance roll when ships lose power without a forward speed; this behavior is one of the five major capsizing modes that the IMO must cover through new-generation intact stability criteria (Ogawa, 2009). Once ships are damaged under dead ship condition, their motion in waves and the strong nonlinearity of the system complicate ship capsizing in wind and waves. Therefore, investigating the dead ship stability and capsizing mechanism of damaged ships under severe sea conditions is necessary.
Researchers have investigated dead ship stability through experimental and theoretical analyses. A series of studies have been conducted to understand dead ship stability. A piecewise linear approximation of the restoring moment is proposed in accordance with one degree-of-freedom (1-DOF) rolling equation and the nonlinear roll damping coefficient (Belenky, 1993), then a systematic study of the piecewise linear method is performed (Umeda et al., 2002;Paroka et al., 2006). The capsizing probability of a tumblehome ship is investigated (Zeng et al., 2014(Zeng et al., , 2015, and a solution has been determined through an optimization method (Lyu et al., 2015); the first-order reliability method is adopted (Choi et al., 2017) to reduce the computational effort. These methods mainly address the capsizing probability of intact ships. On the other hand, the model test and numerical simulation for progressive flooding of damaged barges or passenger ships is performed (Ruponen, 2007;Paroka et al., 2006;Ogawa and Ohashi, 2011), but the capsizing probability in time domain under dead ship condition is unclear; that is, the problem of combining the investigation of the time-domain dead ship stability with the capsizing probability for damaged ships must be solved. There-fore, the present research aims to discuss two methods that can be used to investigate the capsizing probability of damaged ships in time domain. One method subjects the influence of damaged tanks on the hull shape to the wind and wave, and the other method is consistent with the real-time calculation of the GZ curve for damaged ships. Also, the 1-DOF rolling equation of ships in irregular beam wind and wave is adopted in this paper, combining nonlinear damping and nonlinear restoring moment with random initial conditions. The equation is solved by fourth-order Runge-Kutta algorithm in time domain to estimate the capsizing probability under different wave parameters. The nonlinear damping coefficients and other variables included in the rolling motion equation are obtained through theoretical calculation. This paper combines the time-domain calculation of flooding process with the capsizing probability research, the calculation results within the two methods at different stages are compared and analyzed. In addition, this research has selected a fishery bureau vessel as a typical high-speed ship, which is easy to capsize given the dead ship behavior. This investigation is expected to be useful in the steps toward the development of the damage stability criteria in waves.

Mathematical model under dead ship condition
In the calculation of the capsizing probability, ships are assumed to be under dead ship condition in irregular waves and gusty winds for a specified exposure time. The wind state is characterized by a mean wind speed and gustiness spectrum. The rolling motion of ships is also modeled as a 1-DOF system as follows: where is the roll angle (rad), is the linear roll damping coefficient (1/s), is the nonlinear roll damping coefficient (1/rad), and here, the damping coefficients are obtained based on Ikeda's method (Ikeda and Katayama, 2000). W is the ship displacement (N), I xx is the roll moment of inertia (tf·m·s 2 ), J xx is the added moment of mass inertia (tf·m·s 2 ), GZ is the righting lever that corresponds to different heeling angles (m), M wind (t) is the heeling moment (N·m), and M wave (t) is the wave-exciting moment based on Froude-Krylov assumption (N·m). The roll diffraction and radiation moments due to a sway can contradict when the wavelength is longer than the ship breath.

Wind-exciting moment
The excited moment caused by wind M wind (t) is calculated as follows: ρ air where is the air density (kg/m 3 ), C m is the aerodynamic drag coefficient, U w is the mean wind velocity (m/s), U(t) is the time-varying wind velocity (m/s), A L is the lateral windage area (m 2 ), and H C is the height of the wind force center from the hydrodynamic reaction force center (m). The fluctuating wind velocity is calculated by using the following equation: where .

Wave-exciting moment
The Froude-Krylov wave exciting moment M wave (t) is calculated as follows: where is the effective wave slope coefficient, which is calculated according to the ship's rolling frequency (IMO SDC 1/INF, 2014a), and is the wave slope (Francescutto et al., 2004); the calculation equation can be as follows: where N w is the number of regular waves, H s is the significant wave height (m), and T 01 is the mean wave period (s).

Flooding calculation
The flooding calculation is performed through the buoyancy loss method, which enables unchanging displacement and compensates the lost buoyancy with an increased draft. The flooding water is assumed as quasi-static at each time step in time domain. In addition, the time domain model of the flooding process is established on the basis of Bernoulli equation. The influence of damaged position and hull motion is corrected by flow coefficients; moreover, the flooding volume and time are determined through the trapezoidal integral method (Hu et al., 2017). The present research proposes the following two main ideas to handle the impact of damaged compartments on the GZ curve.
(1) Based on the GZ curve in still water, the influence of damaged tanks on the floating position of a ship is considered a gust. The GZ value at different heeling angles is adopted to correct the righting lever.
(2) The GZ curve is calculated in real time after ships are damaged. On the basis of this real-time calculation during the flooding process, the real-time GZ value is obtained. Such a value is the input for solving Eq. (1), thus combining the time-domain calculation of damaged ships with the capsizing probability research. This combination provides the solution for the time-domain calculation of the capsizing probability under dead ship condition.
On the basis of the two methods, the short-term capsizing probability is calculated under various sea conditions at different times. Moreover, Fig. 1 illustrates that the longterm capsizing probability of damaged ships is obtained based on the long-term characterization of the standard environmental conditions given by a wave scatter diagram (IMO SDC 3/WP, 2016). The capsizing probability in time domain is also gained. Here, the significant wave heights H s and mean wave period T 01 are calculated for all sea states appearing in the North Atlantic. Fig. 2 depicts the flowchart of calculation.

Capsizing probability model of dead ships
The Monte Carlo simulation (Maki, 2017) is adopted to solve the 1 DOF roll motion equation in the time domain through the fourth-order Runge-Kutta algorithm solution (Wandji and Veritas, 2018). The assumption that the calculation time is 1 h (3600 s) denotes that the capsizing probability of a ship is considered p (0<p<1) per time; thus, the calculation is repeated n times under the assumption of the Bernoulli process, and the number of capsizing is n c . Moreover, the capsizing probability during n times is p c =n c /n, and the equation is as follows: The process satisfies normal distribution N[p, p(1−p)] when n is sufficiently large to ensure that the deviation of probability p c from probability p does not exceed 5% of the total standard deviation p(1−p). The confidence interval reaches 95%, in which each wind and wave condition simulation runs at least 1600 times. A remarkable advantage of this method is its accuracy.

Case study
The Monte Carlo method is combined with the loss buoyancy method to calculate the capsizing probability in time domain in this research using Visual Basic 6.0 language. This research uses the fishery bureau ship as the sample. The loading conditions (including full-, light, and half-load displacement) are used to calculate the capsizing probability under intact and damaged conditions through the abovementioned methods. Table 1 summarizes the main data of this ship.
The damage conditions are indicated as shadow parts, which lies in the aft of the starboard side, in the schematic demonstrated in Fig. 3. Damaged tanks 1 and 2 are connected to a medium hole (30 mm×30 mm). Thus, water flows to Tank 2 after filling Tank 1. The damaged opening is assumed at the side. Furthermore, a circle is considered, and the diameter is 0.4 m. However, air compression is disregarded during the flooding process because the rooms are modeled as fully vented. The location of the side damage opening only affects the flooding time, not the final floating position. Therefore, this research adopts the location of the damaged opening with z = 1.5 m as an initial condition. The floating position under different loading conditions in time domain is presented as follows: Figs. 4 and 5 exhibit the          ships, "C" represents the long-term capsizing probability, Method 1 is based on the GZ curve of intact ships, and Method 2 relies on the real-time calculation of damaged ships' GZ curve, as mentioned in Section 2.3. The results show that various loading conditions can still lead to different flooding times even with the same damage condition.
For each loading condition, the capsizing probability increases with the flooding time. Thus, the capsizing probability is the maximum until the final equilibrium. The comparison of results indicates a difference between the two methods, and the maximal difference is within 8%. The possible reason may be explained in Fig. 13, in which GM is not the same considering the difference in GZ curves. Therefore, the GM with a small value can lead to a significant rolling period. The small change in the rolling period may cause a significant change in the capsizing probability. As the stability condition in which Method 1 is better than that of Method 2, the final capsizing probability of Method 1 is smaller than Method 2. Fig. 11 illustrates certain evidence of this conclusion at the final equilibrium status.
Figs. 14-16 depict the capsizing probability change with a heel under different loading conditions. These figures are consistent with the tendency demonstrated in Figs. 10-12. For each loading condition, the capsizing probability increases with the heeling angle, thus reaching the maximum until the final equilibrium. The tendency indicates that the research method proposed herein is feasible. Such a method can also substitute the GZ curve of intact ships for the realtime GZ curve of damaged ships when necessary. Fig. 17 exhibits the capsizing probability change under different loading conditions of intact ships. The tendency is that the capsizing probability decreases with the increase in GM, thereby complying with the relevant conclusion of hydrostatics. For damaged ships, the capsizing probability change with GM initially increases and then decreases (Fig. 18). This condition differs from the tendency of intact ships. This phenomenon maybe due to the flooding water can affect GM, which is correlated with the rolling period, thus     HU Li-fen et al. China Ocean Eng., 2019, Vol. 33, No. 2, P. 245-251 249 deviating or approaching the natural period and obtaining different capsizing probabilities.

Discussion
In order to investigate the difference within the two proposed approaches, the calculated short-term capsizing times in 100 occurrences under full load condition are shown in Fig. 19-20. Since the natural rolling frequency of the damaged ship is 6.7, the short-term capsizing times by two methods concentrate around T z =6.5, which is close to the natural rolling frequency and marked with a red circle in Figs. 19-20. The short-term capsizing times distribution under other loading conditions are similar. As the GZ curves are different, the capsizing times distributions are not the same but the final capsizing probabilities are consistent with each other, which can be easily understood in Fig. 12 (the capsizing probability at the final equilibrium). The deficiency of the method is that the reason analysis is only based on three conditions of the sample ship, so the research on the same or similar ship type is needed to verify the proposed methods for the capsizing mechanism investigation. Also, by means of the Monte-Carlo method, the cal-     culation with the proposed two approaches requires large amount of simulation times, which is the main disadvantage to be overcome in the next step.

Conclusion and further research
This research discusses two approaches to consider the GZ curve effect on the capsizing probability study on a damaged fishery ship. The 1-DOF rolling equation is formulated through the Monte Carlo method, which combines the flooding process in the time domain of damaged ships with the capsizing probability research. Two approaches have been proposed to manage the effect of damaged tanks, which rely on the GZ curve of damaged and intact ships, and then study the capsizing probability. The calculation results are also compared and analyzed. The comparison of results indicates that it can substitute the GZ curve of intact ships for the real-time GZ curve of damaged ships, if necessary. However, only three conditions of a fishery ship are analyzed in this research. Therefore, further verifications of the proposed approaches are necessary. In the future, the feasibility analysis of other ship types will be conducted, and how to reduce the calculation time is being carried out. The capsizing probability based on the 4-DOF rolling equation will also be initiated. This research aims to provide technical support for the capsizing mechanism of the stability under dead ship condition and lay the foundation for the capsizing probability study of damaged ships.