Evaluation of the critical wave groups method for calculating the probability of extreme ship responses in beam seas

Transcription

1 Proceedings of the 6 th International Shi Stability Worsho, 5-7 June 207, Belgrade, Serbia Evaluation of the critical wave grous method for calculating the robability of extreme shi resonses in beam seas Panayiotis A. Anastooulos, Deartment of Naval Architecture and Marine Engineering, National Technical University of Athens, Greece, anasto@central.ntua.gr Kostas J. Syrou, Deartment of Naval Architecture and Marine Engineering, National Technical University of Athens, Greece,.syrou@central.ntua.gr ABSTRACT The aer investigates the accuracy of the current formulation of the critical wave grous method for calculating the robability of extreme resonses of vessels rolling in beam seas. The method emloys short duration regular excitations to identify critical for shi stability wave events that cause slight exceedance of a given roll angle threshold. The robability of any exceedance of the roll angle threshold is then estimated by the robability of encountering any wave sequence higher than the determined critical, based on wave height and eriod distributions derived from sectral methods. In this study the critical wave grous method is extended by incororating realistic wave grou forms, characterized by high robability of occurence. Both the regular and the irregular wave grou schemes are alied to evaluate the robability of exceedance for several roll angle thresholds for two shi models. To increase the accuracy of the aroach, wave grou statistics are obtained from direct simulations of the wave field rather than from sectral methods. The results are tested against Monte Carlo simulations of shi roll motion. Keywords: wave grou, robability, instability, roll, dynamics, resonance, rare events.. INTRODUCTION The study of large amlitude shi roll motions in stochastic beam seas is a non-trivial tas exanding in both the fields of non-linear dynamics and robability. As nown, roll statistics deviate from Gaussianity with increasing level of nonlinearity, leading to robability distributions with heavy-tailed structure (Beleny et al., 206b). However, calculating the robability of extreme roll events by emloying brute force methods suffers from a number of deficiencies. First, the accuracy of a direct counting definition of robability becomes questionable when dealing with rare events. At the same time, the fact that shi resonse is not essentially an ergodic random rocess in the case of a non-linear system further increases the comutational burden for tracing the comlex shae of the tails (Beleny et al., 998). Several methods have been roosed to treat the so called roblem of rarity, described in the above. Extraolation methods emloy statistics based on a limited number of realizations to redict the robability of an event that is too rare to be observed. The concet derives from Extreme Value Theory which rovides asymtotic exressions for the distribution of the maximum of a samle of indeendent and identically distributed random variables. Thus, the objective is the estimation of the arameters of an extreme value distribution by fitting the latter to a set of exerimental or simulation data. The method has been demonstrated in several studies and much effort has been ut into addressing ractical issues regarding its alication for shi stability assessment (e.g., Beleny et al., 206a; Cambell et al., 206). On the other hand, wave grou methods offer an alternative solution to the roblem by focusing on secific time intervals when dangerous wave events occur. One of them is the critical wave grous method which quantifies instability tendency through the robability of encountering any wave grou that could have rovoed the instability (Themelis and Syrou, 2007). In the deterministic art of the method, regular wave trains are emloyed to identify critical, in terms of shi stability, height thresholds. Then, in the robabilistic art, the robability of encountering any wave sequence higher than the secified thresholds is calculated using distributions of wave

2 Proceedings of the 6 th International Shi Stability Worsho, 5-7 June 207, Belgrade, Serbia 2 heights and eriods derived from sectral methods. A first attemt to validate the concet was resented by Shigunov et al. (202) who selected a modern 8000 TEU containershi to calculate the robability of exceedance for a 40degrees roll angle threshold. The results were tested against Monte Carlo simulations and fair coincidence was noted in the case of beam seas excitation. As a next ste, in this aer we emloy the critical wave grous method to redict the robability of exceedance for a number of roll angle thresholds for two different shi models. At the same time, our recent wor towards imroving the determistic art of the aroach is continued, by incororating more realistic wave grou forms. The idea is to identify critical wave events in terms of the most exected wave grous of a given sea state using the method develoed by Anastooulos et al. (206). To eliminate the imact of sectral methods on the accuracy of the robabilistic art, desired height and eriod distributions are obtained from direct simulations of the wave field. Finally, the conditions under which the critical wave grous method roduces comarable results with those obtained from Monte Carlo simulations of roll motion are investigated and the focus is set on the region of extreme resonses where the accuracy of the latter is disutable. 2. MATHEMATICAL FORMULATION In the field of ocean and coastal engineering, wave grous are traditionally considered as sequences of waves with heights exceeding a certain reset level and slightly varying eriods (Masson and Chandler, 993; Ochi, 998). Desite that several threshold-based definitions have been utilized in the ast to study wave grouiness measures, one would argue that, from shi dynamics ersective, wave grous are sequences of waves which are sufficiently high to rovoe instabilities. Now, let us assume that we are interested in estimating the robability that a vessel exceeds a roll angle threshold ϕ crit. The ey idea of the critical wave grous method is to first identify the wave events that cause the exceedance and then, calculate the robability of encountering them. The essence of the aroach is resented below: ϕ > ϕcrit wg, i, ic wg, i, ic i i = () where wg i, is a wave grou event with characteristics i, determined for the th set of initial conditions { ϕ0, ϕ 0} of the vessel at the moment of the encounter. From a reliminary investigation, Themelis and Syrou (2008) concluded that for sea states of moderate severity the influence of initial conditions may not be very significant and thus, examining only the uright ϕ, ϕ = 0,0, denoted by osition of the vessel { } { } 0 0 = 0, can be somehow accetable: (2) Eventually, the method is imlemented in two arts: a urely deterministic one, focused on the identification of the so called critical wave grous, i.e., those wave successions leading to only slight exceedance of ϕ crit ; and a robabilistic art to calculate the robability of encountering any wave grou higher than the determined critical. As realized, the accuracy of the method deends exlicitly on the shae of the critical wave grous which are in fact height thresholds for the wave events that result in ϕ > ϕ crit. By assuming that individual wave grou occurrences are indeendent events, eq. (2) is reformulated as (Themelis and Syrou, 2007): [ ϕ ϕ ] > = crit [ ϕ ϕ ] > = crit wg ic ic wg ic [ ] i, 0, i 0 i i [ ϕ ϕ ] > = crit N N N wg0, i wg0, i, wg0, r+ i= i= r= i+ N ( ) wg0,, wg0,2,..., wg0, N + wg, wg,..., wg = wg. where 0, 0,2 0, N 0, i i= N (3) A significant challenge in eq. (3) is to ensure that wave grous that rovoe exceedance of ϕcrit form

3 Proceedings of the 6 th International Shi Stability Worsho, 5-7 June 207, Belgrade, Serbia 3 a set of mutually exclusive and collectively exhaustive events. To avoid ossible overlas in the calculations, it is convenient to identify wave grous with resect to their run length j, which is the number of consecutive heights exceeding a critical threshold: wg = j= i > Τ where H = { H H } and = { T T} (4) i,..., i T i,..., i are vectors of random variables referring resectively to the heights H n and eriods T n of an individual n i, wave grou event with run length i ( ) cr, i = { hcr,,..., hcr, i} h is a deterministic vector for the heights of a critical wave grou with run length i and Τ is the m th range within which the cr,m critical eriods are considered to vary. In the case of regular wave grous the width T w of all critical ranges Τ cr, m ( m =,2,..., M ) is fixed. Modelling of wave successions as Marov chains has been one of the most successful aroaches in wave grou theory. Kimura (980) was the first to elaborate on wave grou statistics assuming that wave heights and related eriods are Marov rocesses. Ever since the concet has been tested several times against numerical simulations and real wave field measurements with remarable success (e.g., Stansell et al., 2002). In this context, the robability of encountering dangerous wave grous with certain secifications, as in eq. (4), is exressed as: (5) where m =,2,..., M denotes different cases of critical eriod segments and: In the above, {, H h, T } 0, i i cri, i cr, m m wg0, i = i + + (, ) f h t dtdh = 0 H, T hcr, Τcr, m f H, T H, T n n n n (,, ) f h t h t dt dh 0 Hn, Tn Hn, Tn n n n n n n n= 2 hcr, n Τcr, m (6) is the conditional robability density function (PDF) of two consecutive wave heights and related eriods and f H, T is the joint PDF of the height and eriod of a single wave. Equation of roll motion In this study shi motion is modelled under the Froude-Krylov assumtion using the following simle uncouled equation, written in terms of the relative roll angle ϕ : ( I + A ) + D( ) + gδ GZ( ) = M ( t) ϕ ϕ ϕ (7) with I 44 and A 44 being the roll moment of inertia and the added moment of inertia, resectively, Δ is the shi dislacement, g is the gravitational acceleration and D is the daming moment: ( ) = + 2 D ϕ B ϕ B ϕ ϕ The restoring arm in still water is given as: ( ϕ) = ϕ, =,3,5,... GZ C (8) (9) When information about the roll resonse amlitude oerator (RAO) is available, the wave induced moment is estimated from: MM ( ω) = ( ω) 2 ( ω) S RAO S (0) where S ηη is the energy sectrum of the water surface elevation which is a stationary ergodic Gaussian rocess. Alternatively, in the resence of long incident waves, the concet of instantaneous wave sloe at the middle of the shi α can be emloyed (Wright and Marshfield, 980): () = () M t I t 44α Dividing eq. (7) by I44 + A 44 we finally obtain: 2 () ϕ + b ϕ + b ϕ ϕ + c ϕ = m t ηη () (2) Construction of realistic wave grous Anastooulos et al. (206) extended the Marovian model of Kimura (980) to develo a method for the systematic construction of irregular wave grou rofiles, characterized by high robability of occurrence. The ey is to select the

4 Proceedings of the 6 th International Shi Stability Worsho, 5-7 June 207, Belgrade, Serbia 4 height H c and eriod T c of the highest wave of the grou to initiate the following iterative scheme: (, ) h = h f h h t dh i i i i i H H, T i i i i 0 t = t f ( t h, h, t ) dt i i i i i i T H, H, T i i i i i 0 (3) (4) Now, let us assume that we are interested in generating a sequence of j wave grou heights and related eriods with H c and T c occuying the i th osition ( i j ). Forward alication of eqs. (3) and (4) will rovide the heights and eriods of the waves succeeding the initial (highest) one. Then, the most exected ast outcomes are identified by alying the same rocedure bacwards in time. The calculation of the conditional exectation in eq. (3) recedes that of eq. (4) so as to tae into account the correlation between the height and eriod of a redicted wave. The transition PDFs can be obtained either from sectral methods (Anastooulos et al., 206) or by analyzing data collected from Monte Carlo simulations of the wave field (Anastooulos and Syrou, 206). The next ste is to construct the continuoustime counterarts of the generated sequences. To this end, we ot for a reresentation of water surface elevation η of the form: η ( 5j 3 )/ 2 x, t = anfn x, t ( ) n= 0 ( ) (5) In our earlier studies the f n basis functions were derived from the alication of the Karhunen- Loève theorem (Sclavounos, 202). Here, aiming at reducing the comutational cost related to the solution of the Karhunen-Loève eigen-roblem, we emloy the widely used Fourier basis functions. The number of terms et in eq. (5) is selected so as to satisfy a set of geometrical constraints which ensure that the shae of the roduced waveform is comatible with the redictions of eqs. (3) and (4). More details can be found elsewhere (e.g., Anastooulos and Syrou, 206). It is noted however that the truncation order in eq. (5) is lower than the originally recommended ( 6 j ) since it was recently observed that fewer terms were enough to generate desired waveforms. 3. RESULTS AND DISCUSSION In this section, the critical wave grous method is alied to two different shi models in order to redict the robability of exceedance for several roll angle thresholds. Both regular and irregular wave grou excitations are emloyed and the results are tested against Monte Carlo simulations of roll motion. To imrove the overall accuracy of the aroach, the PDFs of successive wave heights and eriods aearing in eqs. (5) and (6) are comuted from direct simulations of the water surface dislacement instead of sectral methods. Regarding the construction of irregular wave grou shaes, the transition robabilities in eqs. (3) and (4) were calculated according to the method described in Anastooulos et al. (206) with the only difference that the necessary correlation arameters were estimated from the generated wave data. In this way, the efficiency of the Marov model for determining the most exected wave height and eriod sequences is enhanced. Shi model An ocean surveillance shi, referred in the study of Su (202), was selected as the first shi model. Main arameters of the vessel are given in Table. Table : Main arameters of the ocean surveillance vessel. Parameter Dimensional value I44 + A g m Δ g b s b s s The shi is assumed to oerate in a sea state described by the modified Pierson-Mosowitz (PM) sectrum with significant wave height H s = 4m and ea eriod T = 6s : S ηη ( ω) gH 5 ω s = ex 5 4 ω T 4 ω (6)

5 Proceedings of the 6 th International Shi Stability Worsho, 5-7 June 207, Belgrade, Serbia 5 were ω is the ea frequency. The wave induced moment is modelled using eq. (0) and the roll resonse amlitude oerator RAO ( ω) of the vessel is resented in Figure. deends on the distance of the highest wave from the root. Figure : Roll resonse amlitude oerator RAO( ω ) for shi model. For the simulations of the wave field, the model of Longuet-Higgins (952) was adoted: (7) were ε n are random variables uniformly distributed over [ 0,2π ), ω n are the frequencies of the wave comonents and d ω is the frequency resolution. In total, 8853 waves were analyzed from a set of 24 records of hour. Finally, statistics of roll motion were estimated without assuming the ergodic roerty for the resonse (Beleny et al., 998). As a corollary, the analysis was erformed on a 5 collection of aroximately 5 0 short-duration realizations, samled at a fixed time instant t = 50s. s ηη n n n n n () t = 2S ( ) d cos( t+ ) η ω ω ω ε In Figure 2 the iterative scheme of eqs. (3) and (4) is alied in order to redict the characteristics of the most exected wave grous of the examined sea state for various cases of { Hc, T c}, values, here denoted by red nodes. The vertical axis shows the heights that derive from successive iterations and the horizontal axis shows the corresonding eriods. The evolution of the rocedure for a given set of { Hc, T c} arameters is indicated by blac crosses along the dashed lines. The root of this tree-shaed diagram is the stationary state of the Marovian system and the structure of the most exected wave grous Figure 2: Characteristics of the most exected height and eriod sequences generated for the PM sectrum. In Figure 3 the results of the Monte Carlo simulations (MC sim.) are resented in the same lot with the estimates of the critical wave grous method using regular wave grous with j 6. For the latter two different cases of critical eriod range widths T w were studied. As illustrated, for roll angles below 40degrees the method consistently underestimates the robability of exceedance. This demonstrates that for intermediate roll angle thresholds it is rather unliely that the exceedance has been rovoed by wave grouing henomena. For larger angles, however, the accuracy of the method is imroved but it is sensitive to the selection of T w. The reason is that T w is actually a measure of tolerance for the detection of resonant henomena and as realized, the condition that T = s is ossibly too strict. w On the other hand, the method erforms better for roll angle thresholds before the tail region in the case of irregular wave grous, as shown in Figure 3. In this imlementation, however, the method is sensitive to the maximum eriod of the highest wave T c,max. The reason is that, for irregular wave grous, the critical eriod ranges Τ cr,m are defined as the difference of the shortest from the longest eriod encountered within a generated sequence. As shown in Figure 2, for increasing T c the highest wave rogressively deviates from the mean eriod of the wave grou and the critical eriod ranges Τ cr,m become larger. Therefore, the tolerance for the detection of resonant henomena is relaxed and the method overestimates the robability of

6 Proceedings of the 6 th International Shi Stability Worsho, 5-7 June 207, Belgrade, Serbia 6 exceedance. However, it is not clear at the moment if such cases should be included in the robability calculations since the eriod of the highest wave distorts the grouing character of the rest eriod sequence. maximum wave grou steeness), for the case of irregular wave grous, shading has been alied between the two lines in order to enhance the contrast against the regular-wave-grous line. For j = 2, height thresholds defined by regular and irregular wave grous are, in the mean sense, relatively close. The shift of instability region towards the area of long waves has already been reorted in Anastooulos and Syrou (206). However, for j = 3 the dangerous zone is enlarged for the case of irregular wave grous. Figure 3: Probability of exceedance for shi model using regular wave grous. Figure 4: Probability of exceedance for shi model using irregular wave grous. In the deterministic art of the method, critical 0 wave grou arameters, identified for ϕ crit = 45, are summarized in Figure 5 in the form Transient casize diagrams. These are lots of the wave steeness of a critical wave grou against its eriod, here normalized with the natural eriod of the vessel T o = 5.9s (Rainey and Thomson, 99). Regular wave grous are given by long dashed curves while irregular wave grous are reresented both by their mean steeness (short dashed line) and by the steeness of the highest wave (solid line), always against the normalized eriod of the latter. As one obtains two boundary lines (deending on whether he emloys the mean or the Figure 5: Transient casize diagrams for shi model for different run lengths j and φ crit =45deg. Shi model 2 A modern 4800 TEU Panamax containershi with arameters listed in Table 2 and natural eriod T o = 5.2s is the second shi model that was studied. The restoring arm coefficients in eq. (9) were rovided directly from the loading manual of the vessel. Since no information was available about the RAO function, wave excitation was aroximated by eq. (). In this alication the JONSWAP sectrum, given in eq. (8), with arameters H s = 0m, T = 4s and γ =.932 was selected to describe the sea state of oeration. In the same sirit, 24 records of hour length were generated according

7 Proceedings of the 6 th International Shi Stability Worsho, 5-7 June 207, Belgrade, Serbia 7 to eq. (7), corresonding to a total oulation of 7875 waves. Monte Carlo simulations of roll motion were erformed with the same setu as for shi model, however samled at t = 200s. S ηη ( ω) Table 2: Main arameters of the Panamax containershi. Parameter Dimensional value I44 + A g m Δ g b s b ωω 2 4 ex ω 20.08ω g ex γ = ω 4 ω.667 s s c s s c s s (8) The results obtained from the imlementation of the critical wave grous method when shi model 2 is excited by regular and irregular wave grous is shown in Figures 6 and 7, resectively. Again, for intermediate angle thresholds, better redictions are achieved by irregular waveforms. In the tail region, direct simulations of roll motion (MC sim.) fail to redict exceedances due to the roblem of rarity while, in the same range, both schemes of the critical wave grous method yield reliable estimates. Finally, Figure 8 comares regular and irregular critical wave grous with run lengths j = 2 and j = 3 in terms of their individual robability of exceedance P j. The calculations were made for the critical eriod arameters that rovided the best agreement with the simulation results according to Figures 6 and 7. Thus, T w = 2s and T c,max = 5s were selected for the regular and the irregular case, resectively. The contribution of run lengths with j > 6 to the total robability of exceedance was found negligible. Figure 7: Probability of exceedance for shi model 2 using irregular wave grous. Figure 6: Probability of exceedance for shi model 2 using regular wave grous. Figure 8: Contribution of individual run lengths j to the robability of exceedance for shi model 2.

8 Proceedings of the 6 th International Shi Stability Worsho, 5-7 June 207, Belgrade, Serbia 8 4. CONCLUDING REMARKS In this study the critical wave grous method was alied to redict the robability of largeamlitude shi motions in beam seas. The method was extended by incororating realistic wave excitations reresenting the most exected wave grous of a sea state. Both the regular and the irregular wave grou schemes were alied to two different shi models to estimate the robability of exceedance for several roll angle thresholds and comarisons with Monte Carlo simulations of roll motion were resented. The results indicate good coincidence in the tail region where the efficiency of direct simulations is generally low. For intermediate roll angle thresholds the critical wave grous method erforms better when irregular wave grous are emloyed due to realistic modelling of wave eriod successions. However, the robability calculations are sensitive to the degree of variability that is allowed in the wave eriod grouings. The extent u to which wave grou eriod variations are resonsible for resonant henomena is a toic of future research. ACKNOWLEDGEMENTS The final year thesis wor of Mr. Georgios Paageorgiou, who graduated earlier this year from our School, has motivated artly the current study. REFERENCES Anastooulos P.A., Syrou K.J., Bassler C.C. and Beleny V.L., 206, Towards an Imroved Critical Wave Grous Method for the Probabilistic Assessment of Large Shi Motions in Irregular Seas, Probabilistic Engineering Mechanics 44, Anastooulos P.A. and Syrou K.J., 206, Shi Dynamic Stability Assessment Based on Realistic Wave Grou Excitations, Ocean Engineering 20, Beleny V.L., Degtyarev A.B. and Bouhanovsy A.V., 998, Probabilistic Qualities of Nonlinear Stochastic Rolling, Ocean Engineering 25(), Beleny V.L., Weems K. and Lin W.M., 206a, Slit-Time Method for Estimation of Probability of Casizing Caused by Pure Loss of Stability, Ocean Engineering 22, Beleny V.L., Glozter D., Piiras V. and Sasis T.P., 206b, On the Tail of Nonlinear Roll Motions, Proceedings of the 5 th International Shi Stability Worsho, Stocholm, Sweden, Cambell B., Beleny V.L. and Piiras V., 206, Alication of the Enveloe Peas Over Threshold (EPOT) Method for Probabilistic Assessment of Dynamic Stability, Ocean Engineering 20, Kimura A., 980, Statistical Proerties of Random Wave Grous, Proceedings of the 7 th International Coastal Engineering Conference, Sydney, Australia, Longuet-Higgins M.S., 952, On the Statistical Distribution of the Heights of Sea Waves, Journal of Marine Research (3), Masson D. and Chandler P., 993, Wave Grous: A Closer Loo at Sectral Methods, Coastal Engineering 20, Ochi M., 998, Ocean Waves: The Stochastic Aroach, Cambridge University Press, Cambridge, England, ISBN: Rainey R.C.T. and Thomson J.M.T., 99, The Transient Casize Diagram A New Method of Quantifying Stability in Waves, Journal of Shi Research 35(), Sclavounos P.D., 202, Karhunen Loève Reresentation of Stochastic Ocean Waves, Proceedings of the Royal Society A 468(245), Shigunov V., Themelis N. and Syrou K.J., 202, Critical Wave Grous vs. Direct Monte-Carlo Simulations for Tyical Stability Failure Modes of a Container Shi, Proceedings of the th International Conference on Stability of Shis and Ocean Vehicles, Athens, Greece, Stansell P., Wolfram J. and Linfoot B., 2002, Statistics of Wave Grous Measured in the Northern North Sea: Comarisons between Time Series and Sectral Predictions, Alied Ocean Research 24, Su Z., 202, Nonlinear Resonse and Stability Analysis of Vessel Rolling Motion in Random Waves Using Stochastic Dynamical Systems, Texas A&M University. Themelis N. and Syrou K.J., 2007, Probabilistic Assessment of Shi Stability, Transactions of SNAME 5, Themelis N. and Syrou K.J., 2008, Probabilistic Assessment of Shi Stability Based on the Concet of Critical Wave Grous, Proceedings of the 0 th International Shi Stability Worsho, Daejeon, Korea, Wright J.H.G. and Marshfield W.B., 980, Shi Roll Resonse and Casize Behavior in Beam Seas, RINA Transactions 22,