Application of CFD in Long-Term Extreme Value Analyses of Wave Loads

Transcription

1 Application of CFD in Long-Term Extreme Value Analyses of Wave Loads By Jan Oberhagemann 1,*, Vladimir Shigunov 2 & Ould el Moctar 1 ABSTRACT This paper discusses ways to embed time-domain field methods in extreme value predictions. Approaches are suggested that appear to give most reliable results. They rely on Monte-Carlo simulations, a reduction of parameter variations and extrapolation of exceedance rates over significant wave height. The computational effort is large, yet it can be handled with modern cluster computers. Key words: CFD, Seakeeping, Reliability 1 Abbreviations CDF CFD CRRW ERDW MCS MLRW PDF VBM cumulative distribution function computational fluid dynamics conditional random response wave equivalent regular design wave Monte-Carlo simulations most likely response wave probability density function vertical bending moment 2 Introduction Computational Fluid Dynamics (CFD) methods have their merits in applications where nonlinearities are of major concern, e.g. in predictions of extreme value distributions of wave loads acting on ships and other marine structures. In the literature, corresponding applications of field methods have mostly been restricted to relatively short simulations. Regular waves corresponding to target 1 University Duisburg-Essen, Duisburg, Germany 2 Germanischer Lloyd SE, Hamburg, Germany * Corresponding author. Present address: jan.oberhagemann@uni-due.de 4

2 design values are often employed in such simulations. Irrespectively of the nonlinearities included in such simulations, the assumptions made during establishing the wave load scenarios affect the results to a high degree, often putting their informational value for design and dimensioning of the structure in question. The major reason for the strongly simplified analyses is the large computational effort related to such simulations. Procedures involving simulations with field methods could be advantageous because of the higher degree of nonlinearities implicitly included, but suffer from the high computational costs they impose. Nevertheless, attempts to apply more sophisticated techniques are presented in the recent literature, e.g. Seng and Jensen (2012). We discuss approaches available for the generation of simulation scenarios, and suggest a computational procedure for long-term extreme analyses based on transient simulations. Still relying on linear theory in the first step, the final extreme value distribution depends rather weakly on the probability distributions of linear responses. We use two different codes: a linear boundary-element method (BEM) of Papanikolaou and Schellin (1991), based on zero-speed Green functions with encounter-frequency correction to account for the effect of non-zero forward speed, is applied for efficient preparatory frequency-domain analyses. Sectional loads can be nonlinearly corrected in a post-processing step. The correction is based on hydrostatic extrapolation of the total pressure to the instantaneous wave elevation including dynamic swell-up. For the time-domain simulations, we couple the transient Navier-Stokes (NS) equations solver COMET with a non-linear solver of six degrees of freedom of ship motions and a linear structure solver, based on a Finite-Element Timoshenko beam approach. COMET implements an Eulerian multiphase approach for free-surface flows and the SIMPLE (Semi-Implicit Pressure-Linked Equations) pressure correction scheme for incompressible fluids, see Ferziger and Perić (2002). A description of the coupled method is presented in Brunswig and el Moctar (2004) and Oberhagemann et al. (2008), and comprehensive validation examples are given in Moctar et al. (2011). Oberhagemann et al. (2012a) discuss issues related to computing short-term statistics of ship responses with the coupled method. The coupled time-domain method is suitable to account for effects of elastic hull girder vibration in response to resonant or impulsive wave excitation. Hull girder vibration is an important issue for modern large containerships due to their flared bow and overhanging stern regions and low hull girder natural frequencies. Hence, application examples given herein focus on vertical bending moment of large containerships. 3 Ship Response Statistics During its service life at sea, a ship sails in continuously varying seaway conditions with varying operational parameters. Loads on the ship structure depend on many variables; as a minimum, we consider significant wave height H S, zero-upcrossing period T z, average forward speed of the ship v, angle between the ship course and the mean wave direction µ, and the loading condition LC. For short time intervals, these variables can be assumed statistically stationary. The wave process in a stationary sea state can be discretised as a superposition of a finite number n of harmonic components. The amplitude of each component is defined by the spectral energy distribution of the sea state, S ζ (ω). For a coordinate system moving with the average ship forward speed v, the wave elevation process ζ(x, y, t) at a position x = [0, 0] T and time t is represented by ζ(t) = n a ζ,i [U i cos(ω e i=1 i t) + V i sin(ωi e t)] where the amplitudes of components are a ζ,i = S ζ (ω i, µ i ) ω µ are derived from the spectral energy density distribution S ζ (ω, µ). Coefficients U i and V i are independent standard normal random variables, and ω e i = ω i k i v cos(µ i ) Ship Technology Research Schiffstechnik VOL. 59 / NO. 3 August

3 are the wave encounter frequencies, with the wave numbers for deep-water waves k i = ωi 2 /g and the gravity acceleration g. Wave encounter angles µ = 0 and 90 correspond to following waves and waves from starboard, respectively. The spectral energy distribution is usually obtained from standard spectra; we use the Pierson-Moskowitz spectrum recommended by the International Association of Classification Societies (IACS) where S ζ (ω) is a function of H S and T z, IACS (2001). Assuming a linear dependency of ship responses on the wave amplitude, the associated linear (index l) response process is r l (t) = n Y ω i=1 i a ζ,i [U i cos(ωi e t + θ ωi ) + V i sin(ωi e t + θ ωi )] (1) with the absolute value of the response amplitude operator (RAO) Y ω and phase θ ω. In a linear case, Y ω and θ ω are independent of a ζ (ω) and can be obtained from the frequency-domain seakeeping analysis. Calculation of life-cycle distributions of ship responses requires establishing conditional probability distributions of all variables mentioned above. Wave statistics provide information about the probability distributions of sea state parameters for the intended areas of operation. Probability distributions of loading conditions and operational parameters follow real life observations and are chosen in a conservative manner. The expected number N of exceedances of a target response level R, denoted N(r > R), during the total service life at sea T D becomes N(r > R) = T D χ(r > R T z, H S, v, µ, LC)p(T z, H S )p(v, µ LC, T z, H S )p LC T z H S v µ LC where p(t z, H S ) is the probability of encountering, at an arbitrary time instant, a seaway with the zero-upcrossing period and significant wave height within small ranges around their characteristic values T z and H S, respectively. p(v, µ LC, T z, H S ) is the probability of sailing with the forward speed and wave encounter angle within small ranges around values v and µ, respectively, conditional on the loading condition, wave zero-upcrossing period and significant wave height. For brevity, the combination [T z, H S, v, µ] will be referred to as service parameters. χ(r > R T z, H S, v, µ, LC) is the short-term exceedance rate of R, i.e. the expected number of exceedances of the level R per time conditional on T z, H S, v, µ and LC. χ(r > R) is the inverse of the return period T R (r > R) and has the unit [s 1 ]. The definition of the design value R of a wave response may differ in detail between different authors and classification societies; without loss of generality, we assume in this paper that the design value R of a wave response r is the response amplitude which is exceeded on average by one response peak during service life at sea T D : N(r > R ) = 1 RAOs obtained from frequency-domain seakeeping analyses allow computing the short-term exceedance rates with relatively small effort, assuming ship responses as narrow-banded zero-mean Gaussian processes with spectrum S r (ω, µ), S r (ω, µ) = [Y ω (ω, µ)] 2 S ζ (ω, µ) We will use the zero up-crossing period of the response process T zr instead of its mean period between positive peaks; the error of this approximation is compensated largely by the error of the assumption of a narrow-banded process. The rate of amplitudes exceeding a level R l thus becomes χ(r l > R l ) = 1 m2r exp ( R 2 ) l (2) 2π m 0r 2m 0r where m jr is the j-th spectral moment of S r, m jr = 0 (ω e ) j S r (ω) dω 6

4 Nonlinearities of the response process make the relation between excitation and response statistics more complex. Structural vibration response amplitudes depend nonlinearly on the excitation amplitudes, and the narrow-band assumption may also not be valid for certain responses. For example, the total response including vibrations has typically a spectrum with more than one peak. 4 Deterministic Design Scenarios 4.1 General Idea Several approaches exist based on numerical methods that include higher-order effects, e.g. Jensen and Dogliani (1996) and Naess (1996), but they are often limited to frequency-domain methods. Time-domain simulations are computationally more expensive, but allow, in principle, to include all nonlinearities of interest. Field methods solving the NS equations have a high accuracy of physical modelling and allow to include additional physical effects. Thus they are best suited for simulation of extreme events, although the computational expense prohibits long-term statistical analyses with direct simulations. For example, even using reasonably coarse discretisations, a single second simulation time with our coupled code takes roughly half an hour of computational time when executed in parallel on a dual core processor. Therefore, approaches need to be developed that allow a reduction of the required simulation time by several orders of magnitude. Most approaches to minimise the required simulation time reduce to one common idea: Use a fast but less accurate code (predictor method) for a long-term or short-term statistical analysis, identify one or several critical conditions representative of the target probability level, and carry out a limited number of simulations with a more complex method (corrector method) in the selected critical conditions. For simplicity, we will use the term linear in the following to refer to predictor methods and corresponding results, although they may already account for certain nonlinear effects. The approaches differ not only in the underlying assumptions of the predictor method, but also in the number and length of the time-domain simulations required. A common feature of these approaches is the fact that the result of the corrector method is conditional on the input of the predictor method. While nonlinear response is sought to correct the linear response, the associated exceedance probability is assumed equal to that defined by the predictor method. Some approaches attempt to overcome this shortcoming, albeit at the price of significant computational effort. This section discusses two approaches that establish one or several wave sequences of short duration designed to cause a response peak of pre-defined probability of exceedance according to the predictor method: the well established equivalent design wave method and a theoretically more consistent technique to generate transient wave sequences. 4.2 Equivalent Regular Design Waves The Equivalent Regular Design Wave (ERDW) concept is a simple way of adding nonlinear effects to a linear extreme value. The design wave is constructed so that the ship response amplitude to this wave corresponds to a target design value R l, obtained by the predictor method. R l is chosen to match a target safety level which could be defined as, e.g., N(r l > R l ) = 0.01 or T R (r l > R l ) = 100 years. The result of the simulation in this wave using the corrector method is regarded as a correction of the linear response. For a given wave frequency ω d, the amplitude of the equivalent regular design wave is a ζ (ω d, R l ) = R l /Y(ω d ) (3) In principle, a regular wave of any frequency can be used as ERDW by choosing the wave amplitude a ζ according to eq. (3). The wave frequency of the ERDW is usually selected as the frequency corresponding to the peak of the transfer function over all µ and v for selected loading conditions. Experience-based corrections are used sometimes, or responses to multiple ERDWs are computed and compared. Still, the resulting nonlinear response is strongly related to the selection criteria and hence arbitrary. In addition, the obtained nonlinear response is assumed to have the Ship Technology Research Schiffstechnik VOL. 59 / NO. 3 August

5 same probability of exceedance as the response obtained by the predictor method. The impossibility to improve this probability estimation is another drawback of the ERDW method. 4.3 Response Conditioned Wave Sequences Instead of using a single RAO, the whole information included in the transfer function for a given combination of [µ,v,lc] and a given sea state [H S,T z ] could be used to establish a wave excitation corresponding to R l. Based on the work of Adegeest et al. (1998) and Friis-Hansen and Nielsen (1995), Dietz (2004) developed the Most Likely Response Wave (MLRW) method. The idea is to determine the coefficients V = [U i, V i ] in eq. (1) in such a way that the response process has a peak of amplitude R l exactly at time t 0, i.e., r l (t 0 ) = R l and ṙ l (t 0 ) = 0. For this purpose, he used the Hilbert transform ˆr l (t) of the response process and described the response as an envelope process. Following Dietz (2004), introduce a vector of coefficients A = [A 1, A 2, A 3, A 4 ], A 1 = n a r,i [U i cos(θ ωi ) + V i sin(θ ωi )] r l (t 0 ) i=1 A 2 = n a r,i ω e i=1 i [U i sin(θ ωi ) V i cos(θ ωi )] ṙ l (t 0 ) A 3 = n a r,i [ U i sin(θ ωi ) + V i cos(θ ωi )] ˆr l (t 0 ) i=1 A 4 = n a r,i ω e i=1 i [U i cos(θ ωi ) + V i sin(θ ωi )] ˆṙ l (t 0 ) with a r,i = Y ω,i a ζ,i, r l (t 0 ) = R l, ṙ l (t 0 ) = 0, ˆr l (t 0 ) = 0 and ˆṙ l (t 0 ) = ω r R l. Here ω r is the arbitrary instantaneous frequency of the response at time instant t 0. Now vector V of random variables U i and V i is replaced by random constrained coefficients [U c,i, V c,i ] = {V A = [0, 0, 0, 0]} = V Cov [ V, A T ] Cov [ V, A T ] 1 A Substituting the coefficients U c,i and V c,i in eq. (1) yields a response process with a peak of amplitude R l at time t 0. Dietz (2004) introduced the notation Conditional Random Response Wave (CRRW); the mean of all such processes is called the Most Likely Response Wave (MLRW). The conditional mean constrained coefficients of the MLRW are found by solving equations Ū c,i = E {U i A = [0, 0, 0, 0]} V c,i = E {V i A = [0, 0, 0, 0]} If the instantaneous frequency ω r of the response at t 0 is set equal to the mean frequency of the response process, ω r = m 1r /m 0r, the mean constrained coefficients can efficiently be calculated using spectral moments of the response process, see Dietz (2004) for further details. The MLRW corresponds to the mean response process causing a response peak of magnitude R l according to the predictor method, but it produces only one wave train and thus, only one maximum nonlinear response, which is assumed to have the same exceedance probability as the linear response R l. On the other hand, multiple simulations in CRRW trains allow to obtain the probability density function (PDF) of nonlinear response peaks conditional on the linear response, i.e. f(r r l = R l, T z, H S, v, µ, LC). Moreover, it is possible to obtain the nonlinear cumulative distribution function (CDF) using multiple CRRW simulations for a broad range of R l. Unconditioning with respect to the linear PDF, f(r l T z, H S, v, µ, LC), then yields the unconditional nonlinear CDF: 8 F(r < R T z, H S ) = 0 F(r < R r l = R l, T z, H S )f(r l T z, H S )dr l (4) Dietz (2004) and Drummen et al. (2009) demonstrated the feasibility of this approach and found good agreement with the nonlinear response CDF obtained from Monte-Carlo simulations using nonlinear potential theory codes. Seng and Jensen (2012) recently presented application examples of the MLRW approach using a NS solver and a rigid hull representation. They calculated the most probable wave based on a nonlinear strip method, while Oberhagemann et al. (2012b) carried out

6 a simplified CRRW study for a flexible ship based on a linear 3D panel predictor method. Both studies are not conclusive yet and further research is required, as several open questions could not be addressed yet, for example, when it is more feasible to use NS simulations directly to calculate response distributions, without relying on a predictor method. Further considerations will also be inevitable when attempting to use conditioned wave sequences for low exceedance rates in such steep sea states where nonlinearities of the waves become important, e.g. due to wave breaking. Predictor methods with nonlinear wave models may help here. 5 Approaches Based on Monte-Carlo Simulations Calculation of the long-term exceedance probability of some extreme load level R requires averaging over all possible short-term seastates, as well as all ship speeds, wave encounter angles and loading conditions. A straightforward approach would be to carry out Monte-Carlo simulations for sufficiently many service parameter combinations, discretising the entire service parameter space with a sufficient resolution. In addition to the large number of service parameter combinations to be considered, many long random realisations are required for each combination of parameters to obtain reliable statistical estimates; the overall duration of such simulations would be many durations of the ship s service life. Hence, direct Monte-Carlo simulations (MCS) can be used to obtain long-term extreme value distributions only with codes that are orders of magnitude faster than real time. This is not the case with the methods discussed here. The expected probability of exceedance is calculated as a sum of conditional probabilities for each parameter combination times the frequency of occurrence of the parameters within the corresponding cell. This sum is usually dominated by few terms, which allows efficient reduction of the parameter combinations to be considered. A statistical analysis of linear seakeeping results can be used to find out the combinations of the service parameters which contribute most to the exceedance rate of a given linear long-term reaction Rl. Baarholm and Moan (2000) proposed a quantitative measure for the selection of such dominating parameter combinations, denoted coefficient of contribution (CoC). It is equal to the number of exceedance events for a particular combination of short-term parameters, divided by the total number of exceedance events: CoC(R l ; T z, H S, v, µ) = T D χ {r l > R l T z, H S, v, µ} p TzH S p µv T D T zh S vµ χ {r l > R l T z, H S, v, µ} p TzH S p µv (5) where χ(r > R) is the exceedance rate of a ship response level R, T D is the service time at sea, and p TzHS p µv T D is the expected sojourn time of a ship in short-term service conditions in a small domain around their characteristic values T z, H S, v and µ. The denominator in eq. (5) becomes 1 when CoC is calculated the for the response Rl, which is exceeded on average by one linear response peak during service life at sea. Baarholm and Moan (2000) demonstrated that only few parameter combinations significantly contribute to the extreme value distribution, and these typically correspond to sea states with low probabilities of occurrence. The parameter combination with largest CoC can be identified and used as a design sea state to carry out nonlinear Monte-Carlo simulations. When doing so, two issues need to be addressed: First, the service parameter combination most relevant for the linear extreme reaction Rl may not necessarily lead to the nonlinear reaction with the same probability of exceedance, i.e. nonlinearities not accounted for by the predictor method may change the significance of the parameter combinations with respect to R. The second issue concerns the duration of the simulations for the selected service parameter combination. If it is simply taken equal to the expected duration of this combination over the operational life, then the expected number of occurrences of a linear response would become N(r l > Rl ) = CoC < 1, which means an underprediction of the linear response R l and, as a consequence, also the nonlinear R. Ship Technology Research Schiffstechnik VOL. 59 / NO. 3 August

7 Baarholm and Moan (2000) addressed the first issue using contour lines of CoC in the space of service parameter combinations. Nonlinear simulations used the combination with the largest CoC as a starting point; iteratively, more parameter combinations were included, until the estimate of the nonlinear long-term extreme converged. Oberhagemann et al. (2012b) demonstrated the feasibility of this contour line approach using a field method exemplarily for two container ships in long-crested head waves. The second issue was not addressed. When the set of selected parameter combinations growth, the iterative contour line procedure converges towards a certain nonlinear response R. However, the procedure is non-conservative in that the nonlinear long-term extreme reaction R remains always above the upper bound of the calculated extreme. If O is the total number of service parameter combinations in the discretised space of service parameters and O s is the subset of parameter combinations, selected for nonlinear simulations, then O s CoC < O CoC = 1 This issue can be addressed, consistently with the CoC approach, as follows: the total duration of the states, corresponding to the selected service parameter combinations, should be increased in such a way that the expected linear extreme is the same as the original linear extreme, computed over all possible states. This means that the duration time of each individual parameter combination should be scaled by the ratio of the CoC sum over all possible parameter combinations (1 for the case of value exceeded on average once per service time at sea Rl ) to the CoC sum over the reduced set of parameter combinations selected for nonlinear simulations. 6 Extrapolation of Exceedance Rates in Stationary Sea States The First Order Reliability Method (FORM), e.g. der Kiureghian (2000), uses the basic concept of a reliability index, β, as a measure of the probability of exceeding a certain response level. Using linear theory, the reliability index is a function of the response spectrum, β l = R l / m 0r, and hence eq. (2) becomes χ(r l ) = 1 T r0 exp( β 2 l /2) Here, T r 0 = 2π m 0r /m 2r is the mean period between consecutive zero-upcrossings of the response. Jensen (2010) and Jensen (2011) suggested to assume the inverse proportionality between β and the square root of the response spectrum irrespectively of the nonlinearity of the response. When the response characteristics do not change significantly with H S, it should be possible to derive exceedance rates for one significant wave height H S,1 from exceedance rates obtained for a different significant wave height H S,2. In linear theory, m 0r is directly proportional to HS 2, therefore a constant B exists such that β = B /H S. Jensen (2010) proposed a more general formulation for the reliability index, β = A + B /H S (6) This expression proves to fit well to MCS results, obtained with a nonlinear time-domain strip method, including a momentum approach for slamming and hull girder elasticity effects. Since the return period of a response peak of level R decreases with increasing significant wave height, the required simulation time can be significantly reduced by using a large H S and exploiting the assumed dependency of the reliability index on the significant wave height. For T z = 11.5 s, v = 10.0 knots, head seas (µ = 180 ) and a typical loading condition, Fig. 1 shows exceedance rates of vertical bending moment (VBM) sagging peaks of a large container vessel in sea states with significant wave heights H S = 8.0 to 14.0 m. These data were determined from Monte-Carlo simulations using COMET with simulation lengths T S = to s, Table 1. 10

8 Fig. 1: Exceedance rates of sagging VBM peaks. Tab. 1: Monte-Carlo simulation durations for different H S. H S [m] T 10 4 [s] Peaks were identified as maxima between consecutive response zero-upcrossings. Small cycles related purely to vibration should therefore not affect the peak period. Using these MCS data, we will try to establish a linear relationship between the logarithm of the outcrossing rate and the reciprocal of the significant wave height squared. Instead of eq. (6), we test a formulation that has been proposed by Söding and Tonguć (1986), see also application in Shigunov et al. (2010) to excessive roll motions of containerships: ln [χ(r)] = A + B/H 2 S (7) Selecting a characteristic exceedance period T r0 of the response in such a way that A = ln(t r0 ), we can rewrite eq. (7) as ln [χ(r) T r0 ] = B/H 2 S (8) This formulation is asymptotically equivalent to eq. (6), i.e. both give the same dependency of the exceedance rate on H S for small exceedance rates. Figure 2 shows logarithm of the non-dimensionalised exceedance rates ln [χ(r)t r0 ] as a function of H 2 S for six wave heights and a range of response levels from 3000 to 7500 MNm. Least squares regression determined B for each response level according to eq. (8), while T r0 was determined beforehand using eq. (7) by averaging over all response levels. Although using eq. (7) with coefficient A, individually adjusted to each response level, produces better fits, results in Fig. 2 indicate fair overall agreement of the MCS data with the approximation eq. (8). At least for the Ship Technology Research Schiffstechnik VOL. 59 / NO. 3 August

9 investigated H S, coefficient B as the slope parameter of the nondimensional exceedance rate can be regarded a function of R, but independent from H S. Fig. 2: Exceedance rates of sagging VBM peaks according to MCS: ln [χ(r)t r0 ] as a function of 1/H 2 S and regression lines with fixed T r0 for selected load levels. Another least-squares fitting was used to establish a functional relation B(R); a second-order polynomial fitting was found best for the available MCS data: B(R) = b 1 R + b 2 R 2 (9) The exceedance rate χ in a stationary service condition [T z,v,µ,lc] can now be expressed as a function of the response level R and the significant wave height H S : 12 χ(r, H S ) = 1 exp [ ( b 1 R + b 2 R 2 ) /H 2 ] S T r0 We used regression analysis of MCS data to determine the parameters T r0, b 1, b 2 from MCS results separately for each H S to find parameters A, b 1 and b 2. Figure 3 shows the resulting coefficients as a function of H S. Corresponding T r0 and b 2 computed with the linearly and nonlinearly corrected BEM are added for comparison. Despite large scatter, especially for H S = 12.0 and 13.0 m, trends can be observed for the coefficients derived from MCS. We tested three different approaches to obtain functional dependencies of T r0, b 1 and b 2 on H S. These are referred to as type I, type II and type III, Table 2. Here, constant and linear denote constant and linear relations, respectively. Least squares fitting (LSQ) was applied where more data were used than required. The approaches use MCS data for H S = 14 m (type I), H S = 8 and 14 m (type II) and all H S (type III). The use of the linear BEM value b l = H 2 S /(2m 0r) for b 2 of type I is equivalent to assuming a linear response at H S = 0 m. The introduction of the artificial data points [0, 0] to establish b 1 (H S ) for types I and II is justified as follows: The response process including structural vibration due to impulsive excitation can be considered as a combination of two random processes, a basically Gaussian process in response to Gaussian wave excitation and transient vibrations. Amplitudes of the former process are well (10)

10 Fig. 3: T 0r, b 1 and b 2 as functions of H S ; comparison of approaches I, II and III. Tab. 2: Data used to establish T r0, b 1, b 2 as functions of H S. T r0 b 1 b 2 Type I T r0 (14 m) 0, b 1 (14 m) b l, b 2 (14 m) constant linear linear Type II T r0 (8 m), T r0 (14 m) 0, b 1 (8 m), b 1 (14 m) b 2 (8 m), b 2 (14 m) constant (LSQ) linear (LSQ) linear Type III all MCS all MCS all MCS constant (LSQ) linear (LSQ) linear (LSQ) approximated by a Rayleigh distribution (which corresponds to b 2 ) and the vibration amplitudes are exponentially distributed (which corresponds to b 1 ). The peak distribution of a combination of both processes may therefore be well approximated by eq. (10). Since the excitation of transient vibrations will tend to zero for H S 0, we can assume b 1 = 0 at H S = 0.0 m. With the type III regression, b 1 turned out to be very close to zero at H S = 0.0 m, which was not forced and seems to be a coincidence. Figure 4 exemplifies resulting curves of B(R) for H S = 14 m compared to the original MCS data. Type I regression fits best to the MCS results in this case and requires the least computational effort. On the other hand, this approach increasingly deviates from the MCS data with decreasing H S, because it is a linear blending between the linear results at H S = 0 m and the nonlinear MCS data at H S = 14 m. Type III has the best overall agreement with the MCS data, but requires most computational effort. Thus it is only regarded as a reference here. Type II, as a compromise, shows a fair agreement with all MCS data. Figure 5 compares the original MCS data at all investigated wave heights to the exceedance rates from the regression model type II. Fair overall agreement is found except for the tail distributions of the MCS data, where the statistical uncertainty becomes large anyway. Ship Technology Research Schiffstechnik VOL. 59 / NO. 3 August

11 Fig. 4: B as a function of R for H S = 14.0 m: original regression (filled squares), regression types I (dashed line), II (solid line) and III (dotted line), and MCS results (small dots). Fig. 5: Exceedance rates of sagging VBM peaks resulting from type II regression and original MCS data, T z = 11.5 s, v = 10 knots, µ =

12 At least for the present example with a limited range of wave heights tested, extrapolation of exceedance rates towards sea states with lower significant wave height appears practicable and not related to significant errors. In turn, the resulting reduction in required simulation times is noteworthy: Instead of simulating for a number of different significant wave heights, a single set of Monte-Carlo simulations for each combination of T z, v, µ and LC can be used to cover the whole relevant range of significant wave heights. Further on, exceedance rates of large response levels are more efficiently obtained from Monte-Carlo simulations in higher significant wave heights due to the shorter return periods. Taking for instance Fig. 5, the exceedance rate of R = 8000 MNm is two orders of magnitude higher at H S = 14 m than at H S = 8 m. It appears therefore meaningful to test the model based on eq. 9 on other nonlinear problems. 7 Extrapolation Towards Grid-Independent Solutions Numerical results of field methods are sensitive to both spatial and temporal discretisation. Insufficient grid resolution or bad quality grids may lead to inaccurate results or give wrong tendencies. Even with increasingly fine discretisations, one may still not obtain fully converged results, mainly due to the asymptotic accuracy when discretising differential equations. Keeping this in mind, one may arrive at two contrary conclusions: a) Since the solution significantly depends on the discretisation in space and time, field methods are often regarded as inaccurate, unreliable or random. Additionally, these methods may be regarded excessively computational time-consuming since very fine grids are required to minimise the numerical error. b) For many responses, coarse discretisations give results of sufficient accuracy, or at least indicate a trend towards accurate results which could be obtained on finer grids. Grid studies then allow to estimate grid-independent solutions. Provided that governing parameters are identified, one may try to transfer the error estimates to similar problems. Doing so indeed affects the accuracy of the results to a certain extent, but the savings in computation time can often outweigh this loss of accuracy. For instance, in 3D time domain simulations, the number of unknowns is a function of the grid spacing to the power of four, while the error is typically a function of the grid spacing to the power of one or two, depending on the discretisation scheme. Adopting the more optimistic point of view (b), the asymptotic accuracy can be understood as a chance to calculate the uncertainty of the numerical result. This is a potential advantage when assessing the reliability of a marine structure against failure due to hydrodynamic loading. In the context of this study, wave dispersion of high-frequency wave components, local pressure distributions due to slamming, and radiated waves due to ship motions are examples of flow features that require a fine grid resolution in space and time. Numerical damping and underestimation of local peaks are typical consequences of insufficient refinement. Moctar et al. (2011) discuss discretisation requirements in the context of ship responses to wave excitation. Whereas Richardson extrapolation is widely accepted for uncertainty and error estimation of steady, quasi-steady or periodic solutions, there is little experience with irregular transient problems, especially for coupled fluid and structure dynamics. We will, however, attempt to quantify errors in response statistics introduced by coarse discretisations. Global responses of rigid ships, such as vertical bending moment midships, can be calculated with sufficient accuracy even on coarse grids in the fluid domain, of the order 300, 000 control volumes, but vibratory responses require finer resolution in space and time, Oberhagemann et al. (2012a). In the former case, low frequency and large amplitude wave components dominate the reponse process. In contrast, ship vibration responses occur at smaller space and time scales. Radiated waves due to vibration are comparatively high in frequency and small in amplitude, and vibrations may also be excited by short waves. Furthermore, pressure impulses (slamming) as a main source of transient vibrations are very sensitive to the local flow and require high resolution in space and time. Ship Technology Research Schiffstechnik VOL. 59 / NO. 3 August

13 Low resolution suffices to detect the occurrence of slamming events and to resolve global effects, but the impulses are typically diffused in space and time and less pronounced, Moctar et al. (2011). Summing up the above, care must be taken when calculating responses of a flexible ship in irregular waves. Comparison with model tests is the best means to not only validate a code, but also to develop strategies for grid generation and simulation tuning. Still, it is potentially not possible to work towards a perfect matching of experimentally and numerically obtained time series data. For example, Fig. 6 (top) shows VBM time series of a large containership in severe head seas. Low-frequency responses agree well, but the vibratory modulations differ more. Still the time series agree with respect to the level of vibration observed. Comparing single responses may be very misleading in this case, because of the large scatter of individual peaks. Having a look at the response cycle spectra obtained from rainflow counting, Fig. 6 (bottom), however, suggests a fair agreement for a large part of response levels. Note that the computation was performed on a relatively coarse grid with approximately 500, 000 cells. Deviations are mainly observed for small and large numbers of response cycles, and are most significant for the largest responses. Here, slamming events and corresponding vibrations dominate the response process. Insufficient grid resolution causes less pronounced impulses and, as a consequence, underpredicted responses. Small response cycles are, on the other hand, dominated by pure vibration cycles of small magnitude. These typically occur after the initial vibration excitation as a consequence of vibration decay. Here, coarse discretisation typically causes an overprediction of damping, i.e., underprediction of cycles at a certain response level. The example given in Fig. 6 is rather unusual in that damping at small response levels is in the numerical results slightly lower than in experiments. Therefore, computations yield overpredicted numbers of load cycles at small response levels. The structural damping was difficult to determine from model tests, therefore a large uncertainty is related to structural damping in this case. It follows from these considerations, that the number of exceedances, N, of a response level R is grid dependent, and that the ratio of N i (computed on a grid of refinement level i) to N 1 (computed on the initial coarsest grid) can be expressed as a function of the refinement ratio i / 1, which can be written as i = x i t i (11) 1 x 1 t 1 where x and t represent grid node spacing and time step size, respectively. According to this approach, the initial grid is always the reference grid with unity refinement ratio, and = 0 corresponds to the grid-independent solution. Eq. (11) suggests equal effects of space and time step refinements, which is probably not correct. Refinements should reasonably be made in all space and time dimensions. We recommend to use the same refinements for all dimensions, x i / x 1 = t i / t 1, to yield similar Courant numbers. Moreover, estimates of discretisation error are possible only if all grids in a grid-dependency study use the same refinement strategy. We suggest the following shape function to account for the assumed decrease of discretisation error with grid refinement: Ξ(R, i ) = N i /N 1 = 1 + a 0 ( i ) exp [ a 1 ( i )R] + a 2 ( i )R 3 (12) The exponential term accounts for the discretisation error in vibration damping, whereas the cubic term is the best fit to account for the underprediction of large response magnitudes in a number of grid studies. Coefficients a 0, a 1 and a 2 are found from linear regression over all response levels and available grids. Figure 7 gives an example of fitting eq. (12) to VBM cycles of a large containership, calculated on three different grids. Simulations comprised 2700 s simulation time on each grid using the same wave process realisations. Although there is scatter, especially for large R, the fit agrees fairly well with the data. Figure 8 shows the corresponding fitted functions for a 0, a 1 and a 2. In order to compare grid-dependency studies at different significant wave heights, VBM responses were non-dimensionalised with respect to ρgbl 2 PP H S, where ρ is water density, g gravity acceleration, B ship breadth and L PP length between perpendiculars. 16

14 Fig. 6: Time series of VBM (top) and rainflow evaluation of VBM cycles (bottom); comparison of model test with numerical results. Ship Technology Research Schiffstechnik VOL. 59 / NO. 3 August

15 Fig. 7: Increase in number of exceedance events on refined grids, N i /N 1, as a function of response level R; H S = 14 m, T z = 11.5 s, v = 10 knots, µ = 180 ; polynomial fit denotes the extrapolated grid-independent solution. Fig. 8: Coefficients a 0, a 1, a 2 of extrapolation function Ξ. 18

16 The exponential term of Ξ(R, i ) in eq. (12) accounts for the small amplitude cycles due to vibration, which lead to an S-shape of the load cycle spectrum, Fig. 6, Oberhagemann et al. (2012b). When looking instead at maximum response peaks between consecutive zero-upcrossings, the exponential term should be omitted and eq. (12) reduces to Ξ(R, i ) = N i /N 1 = 1 + a 2 ( i )R 3 (13) For given T z, v, µ and LC, grid-dependency studies should be carried out for more than one significant wave height in order to investigate the influence of H S. This allows approximating the coefficients a 0, a 1,a 2 as functions of H S. Function Ξ(R, i, H S ) can then be applied to the extrapolated exceedance rate distributions in sea states with lower H S. Figure 9 shows VBM exceedance rates as a result of extrapolation type II and a grid study according to eq. (13) using MCS data for H S = 8 and 14 m. The MCS data are still the same as used in Section 6. Function Ξ(R,, H S ) increases with H S, which is in line with the expected increase of discretisation error along with increasing importance of slamming and transient vibrations. Function Ξ also depends on service parameters, therefore, grid-dependency studies are recommended for at least a representative number of parameter combinations. Fig. 9: Comparison of final processed MCS data (solid lines), linear BEM (dotted lines) and nonlinearly corrected BEM (dashed lines). 8 Outline of a Procedure for Extreme Value Prediction For a comprehensive long-term extreme value analysis we suggest a combination of the contour line approach according to Baarholm and Moan (2000) with modifications, Monte-Carlo simulations in irregular waves, extrapolation over wave height according to Section 6, and simulations in random conditioned wave sequences. The proposed procedure is: I) Start with a linear long-term statistical analysis to determine the contribution of service parameters in a small domain around particular significant wave height H S, zero-upcrossing Ship Technology Research Schiffstechnik VOL. 59 / NO. 3 August

17 period T z, ship forward speed v, wave heading µ and loading condition LC to the linear long-term extreme distribution of wave loads. II) Calculate short-term distributions of exceedance rates of selected global wave loads for T z, v, µ and LC combinations of interest using nonlinear simulations. Use artificially increased H S to reduce required Monte-Carlo simulation times for small wave heights. III) Perform grid refinement studies to estimate the discretisation error of computed exceedance rates. IV) Prolongate the resulting exceedance rate distributions towards lower probability levels, if necessary, using nonlinear simulations in conditioned wave sequences. V) Extrapolate the resulting exceedance rates to yield the short-term distributions for all H S of interest. VI) Determine the long-term distribution by summing up all short-term distributions. VII) Increase the number of selected service parameter combinations T z, v, µ and LC using CoC as indicator and repeat the procedure. Continue to add additional parameter combinations until the extreme wave loads of interest are converged with the required tolerance. Monte-Carlo simulations become very time consuming for low exceedance rates. On the other hand, the calculation of exceedance rates with conditioned wave sequences according to eq. (4) only becomes efficient for small probabilities of exceedance, since a large number of approximately 50 to 100 nonlinear simulations is required for each response level. Combining both techniques allows to reduce the required simulation time for a short-term distribution of exceedance rates: For small response levels with large exceedance rates, MCS should be used, and conditioned wave sequences are used for the rare events with small exceedance rates. The need for random conditioned wave sequences to prolongate the available short-term exceedance rate distributions (step IV) is not clear yet and requires further study. Figure 10 shows return periods of two response levels as a function of H 2 S and gives an alternative presentation of Fig. 9. The methods displayed predict significantly different return periods of R = 6000 MNm for H S = 4 m, which vary about an order of magnitude. On the other hand, all of these return periods are some orders of magnitude longer than the relevant periods for the exceedance probability of the reaction R = 6000 MNm. Thus, extreme value distributions will be negligibly affected by the errors of values, corresponding to very long return periods, which agrees with the suggested reduction of parameter combinations following the CoC method. 9 Estimation of Required Computer Resources Estimations of the computational effort required for extreme wave load predictions according to the outlined procedure, given in this section, are intended to give an idea of required computer power, which may vary depending on the investigated ship. The preliminary investigation consists of a linear seakeeping analysis and its long-term statistical evaluation, and involves negligible computational costs of less than 50 hours on a desktop PC. Investigations of this type are quite standard in the maritime industry today. The estimates of computation time for the consecutive CFD simulations given here are based on the results presented above and on studies presented in Moctar et al. (2011), Oberhagemann et al. (2012a) and Oberhagemann et al. (2012b). The estimation is based on type II extrapolation; using type I will halve the required time. Using a computer cluster, the wall clock time required for MCS can be significantly reduced. It turns out to be most efficient to run several simulations of relatively short duration in parallel, in addition to the common spatial parallelization. Computation times for nonlinear simulations are lowest for long-crested head waves due to the midship symmetry and short encounter periods. Here, 10 4 s simulation time can be calculated in 5 days on a computer cluster with 80 CPU cores. Efficiency drops for short-crested and oblique waves, where the number of grid cells approximately doubles. In 20

18 Fig. 10: Extrapolated return periods of two different response levels (type II and comparison with linearly and nonlinearly corrected BEM results for H S = 4.0 m. the latter case, 6 to 7 days on a computer with 160 cores are required. For the further estimates, we will assume short-crested waves. Assume that v and LC are constant for all H S and T z combinations, and only a limited number of wave headings, say 4, are of interest; further assume that only sea states with 5 different T z need to be investigated. This results in 2 20 short-term probability distributions. A Monte-Carlo simulation time of 10 4 s corresponds to 50 outcrossings of a response level with an exceedance rate χ = s 1. This is probably the limit where Monte-Carlo simulations start to be less efficient than the application of CRRW wave sequences. For these, 50 simulations of 50 s duration should be sufficient to cover one response level. Assuming that it is sufficient to calculate the conditional nonlinear CDF for each order of magnitude (χ = 10 3, 10 4,..., s 1 ), additional 15, 000 s simulation time is required to obtain a short-term distribution of outcrossing rates up to χ = 10 8 s 1. From the above follows that 100, 000 to 120, 000 days CPU time are required for a nonlinear long-term extreme value analysis, even when applying the above reductions in parameter variations. Further reductions of T z and µ variations can probably be made based on experience, and response levels with outcrossing rates of χ = 10 5 s 1 in severe to extreme sea states with large H S are most probably sufficient for long-term analysis. Then it is possible to reduce the required CPU time to 50, 000 days. Hence, such an analysis can be managed in approximately 50 days computing time on a cluster with 1000 CPU cores. Grid studies will take about 3 to 5 days in addition. 10 Conclusions Extreme value predictions of marine structures like ships require appropriate hydrodynamic codes capable of modelling all relevant nonlinearities. For reliability assessments, these have to be embedded in computational procedures defining load scenarios. Disregarding important load effects or using simplistic procedures will result in over- or underestimated design loads, which eliminates the usefullness of directly assessed design loads. Ship Technology Research Schiffstechnik VOL. 59 / NO. 3 August

19 The above described methods have already been applied in combination with field methods and gave reasonable results. The estimated computation times are based on experience and are expected to give a realistic picture of the required computational effort. At first glance, the computing time appears prohibitively long; it actually is for standard application in design or classification. However, one should keep in mind that comparable model test campaigns take a similar time at significantly higher daily costs. Experience may also allow for reductions of the overall computation time by further reducing the number of computed short-term probability distributions a priori or reducing the simulation times for Monte-Carlo simulations. Thus, it makes sense to perform such analysis for some selected cases to derive less demanding procedures or cross-check existing procedures. Further studies are recommended to gain more experience on the applicability of the extrapolation over the significant wave height, as well as on the uncertainties related to this extrapolation. The proposed distribution function of response peaks according to eq. (10) fit well the data presented here. Its motivation was to account for a combined response process including vibrations. The applicability of this approach to other ships or response types, e.g. accelerations, should be investigated. Discretisation errors are a critical issue. As illustrated above, these will affect the response statistics, and careful grid-dependency studies are required. The estimates of computation times are based on the assumption of using very coarse grids that may especially in case of calculations for elastic ships lead to underestimated responses. Experience with grid studies so far indicates, however, the feasibility of using coarse grids combined with extrapolation. The extrapolation approach presented in section 7 appears reasonable and has been successfully used in comparisons with short sequences from model tests, but proper validation is still pending. References Adegeest, L., Braathen, A. and Løseth, R. (1998). Use of nonlinear sea loads simulations in design of ships. In Proc. 7-th Int. Symp. on Practical Design of Ships and Other Floating Offshore Structures, The Hague Baarholm, G. and Moan, T. (2000). Estimation of nonlinear long-term extremes of hull girder loads in ships. Marine Structures, 13(6): Brunswig, J. and el Moctar, O. (2004). Prediction of ship motions in waves using RANSE. In Proc. 7-th Numerical Towing Tank Symposium, Hamburg der Kiureghian, A. (2000). The geometry of random vibrations and solutions by FORM and SORM. Probabilistic Engineering Mechanics, 15(1):81 90 Dietz, J. (2004). Application of Conditional Waves as Critical Wave Episodes for Extreme Loads on Marine Structures. Ph.D. thesis, Technical University of Denmark, Lyngby Drummen, I., Wu, M. and Moan, T. (2009). Numerical and experimental investigations into the application of response-conditioned waves for long-term nonlinear analyses. Marine Structures. Doi: /j.marstruc Ferziger, J. H. and Perić, M. (2002). Computational Methods for Fluid Dynamics. Springer Verlag, Berlin Heidelberg New York, 3rd edn. Friis-Hansen, P. and Nielsen, L. P. (1995). On the New Wave model for the kinematics of large ocean waves. In Proc. 14-th Int. Conf. on Offshore Mechanics and Arctic Engineering, Copenhagen IACS (2001). Recommendation No. 34. Standard wave data. URL: http : // Guidelines_and_recommendations/PDF/REC_34_pdf186.pdf Jensen, J. J. (2010). Extreme value predictions using Monte Carlo simulations with artificially increased wave height. In Proc. 11-th PRADS Conference, Rio de Janeiro Jensen, J. J. (2011). Extreme value predictions using Monte Carlo simulations with artificially increased load spectrum. Probabilistic Engineering Mechanics, 26: Jensen, J. J. and Dogliani, M. (1996). Wave-induced ship hull vibrations in stochastic seaways. Marine Structures, 9: Moctar, O. E., Oberhagemann, J. and Schellin, T. E. (2011). Free-surface RANS method for hull girder springing and whipping. In Proc. SNAME 2011 Annual Meeting, Houston Naess, A. (1996). A second-order theory for the response statistics of wave-induced ship hull vibrations in random seas. Marine Structures, 9: Oberhagemann, J., Moctar, O. E. and Schellin, T. E. (2008). Fluid-structure coupling to assess whipping effects on global loads of a large container vessel. In Proc. 27-th Symp. on Naval Hydrodynamics, Seoul Oberhagemann, J., Ley, J. and Moctar, O. E. (2012a). Prediction of ship response statistics in severe sea conditions using RANS. In Proc. 30-th Int. Conf. on Ocean, Offshore and Arctic Engineering. Rio de Janeiro. Paper No. OMAE Oberhagemann, J., Ley, J., Shigunov, V. and Moctar, O. E. (2012b). Efficient Approaches for Ship Response Statistics using RANS. In Proc. 22-nd ISOPE Conf., Rhodes Papanikolaou, A. D. and Schellin, T. E. (1991). A three-dimensional panel method for motions and loads of ships with forward speed. Ship Technology Research, 39 Seng, S. and Jensen, J. J. (2012). Slamming simulations in a conditional wave. In Proc. 31-st Int. Conf. on Ocean, Offshore and Arctic Engineering. Rio de Janeiro. Paper Nr. OMAE Shigunov, V., Moctar, O. E. and Rathje, H. (2010). Operational guidance for prevention of cargo loss and damage on container ships. Ship Technology Research, 57(1):6 23 Söding, H. and Tonguć, E. (1986). Computing capsizing frequencies of ships in a seaway. In Proc. 3-rd Int. Conf. on Stability of Ships and Ocean Vehicles, Gdansk 22