Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design

ABS TECHNICAL PAPERS 23 Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design Y.S. Shin, Associate Member, American Bureau of Shipping, V.L. Belenky, Member, American Bureau of Shipping, W.M. Lin, Member, Science Applications International Corporation, K.M. Weems, Member, Science Applications International Corporation, A.H. Engle, Associate Member, Carderock Division, Naval Surface Warfare Center. ABSTRACT This paper describes recent developments and new applications in the field of numerical simulation for nonlinear ship motions using the Large Amplitude Motion Program (LAMP), following up on our 1997 paper (Shin, et al. 1997). The objective of LAMP s development is to develop an analysis tool for highly realistic prediction of wave loads and behavior for a ship in severe seas. This approach is based on physics and does not rely on statistical information gathered from model tests or existing ships, so it is expected to be especially useful for new ship types. The kernel of LAMP is panel-based potential flow solution of the ship-wave hydrodynamic problem with many numerical options, including a recently-added Rankine singularity model with a damping beach and an option to rapidly compute the perturbation potential using pre-computed Impulse Response Function (IRF) potentials. The paper describes important implementation details for these and other computational options added since 1997. The LAMP System is structured so that numerical or empirical models of other systems or effects can be directly incorporated into the time domain ship motions and load calculations using a series of optional features. One such feature is a multi-level green-water-on-deck model including a finite-volume solution of 3-D shallow water flow over the deck. This model can be used for analyzing ship behavior with water on deck and loads caused by green-water-on-deck. Numerical results illustrate the effect of green-water-on-deck on the pitch motion and vertical bending moment of a cruiser in head seas and on the roll behavior of a fishing vessel. Another feature involves anti-roll fin and tank systems, including an integrated anti-roll tank model that solves for fluid motion in a U-tube tank concurrently with the wave-body hydrodynamics. The paper evaluates the use of passive anti-roll tanks to mitigate parametric roll resonance. Recent development has focused on applying LAMP to unconventional ships, multi-hull high-speed displacement ships, and non-ship-like configurations. Results are presented from studies for an advanced Naval hull form, a trimaran, and a semi-submersible platform. INTRODUCTION New ship designs could be characterized by large variation of lines and speed range. New container carriers have not only become larger, but their hull shapes have also experienced a significant transformation as large bow flares and stern overhangs become standard features. Bulk carriers have also become larger and wider, but relatively more shallow (Shin, 21). Increasing demand for high speed generates more multi- and mono-hull high-speed designs both for commercial and naval applications. Such a variety of new and unusual hull forms poses a new challenge for the maritime industry as design methods based on semi-empirical processes become suspect. Even direct calculation methods with proven track records for accuracy and reliability may lose their applicability, as their assumptions are no longer met. A typical example for such a situation is a recent case of parametric rolling in head seas (France et al., 23). The physical phenomenon of parametrically excited roll motions in following seas has been known since the 195s (Paulling and Rosenberg, 1959), but the appearance of parametric roll in head seas conditions was unexpected. The cause was the large bow flare and stern overhang that created strong coupling between longitudinal and transverse ship motions, a phenomena that is not observed for more traditional hull shapes. This resulted in unusually large roll motions for which the cargo securing system was not designed. Another example of new problems generated by modern hull shapes is bow flare slamming in oblique seas (Finn, et al., 21). Problems associated with the influence of green-water-on-deck on ship motions and loads are also becoming serious issues for new hull shapes where operational experience has not yet been gathered. Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design 257

ABS TECHNICAL PAPERS 23 The natural reply to this challenge is to develop calculation methods suitable for these new designs. Such new methods were thoroughly reviewed by Beck and Reed (21), who pointed out that time domain simulations of ship motions based on potential flow formulations is becoming a more practical way to achieve engineering results for new ship designs. BRIEF DESCRIPTION OF LAMP The LAMP System is a time-domain simulation model specifically developed for computing the motions and loads of a ship operating in extreme sea conditions. LAMP System development began with a 1988 DARPA project for advanced nonlinear ship motion simulation, and has continued under the sponsorship of the U.S. Navy, U. S. Coast Guard, the American Bureau of Shipping (ABS), and Science Applications International Corporation s (SAIC) IR&D program. LAMP uses a time stepping approach in which all forces and moments acting on the ship, including those due to the wave-body interaction, appendages, control systems, and green-water-on-deck, are computed at each time step and the 6-DOF equations of motions are integrated in the time-domain using a 4 th -order Runge- Kutta algorithm. In addition to motions, LAMP also computes main hull-girder loads using a rigid or elastic beam model and includes an interface for developing Finite-Element load data sets from the 3-D pressure distribution (Shin et al., 1997; Weems et al., 1998). The central part of the LAMP System is the 3-D solution of wave-body interaction problem in the timedomain (Lin and Yue, 199, 1993). A 3-D perturbation velocity potential is computed by solving an initial boundary value problem using a potential flow boundary element or panel method. A combined body boundary condition is imposed that incorporates the effects of forward speed, the ship motion (radiation), and the scattering of the incident wave (diffraction). The potential is computed using a hybrid singularity model that uses both transient Green functions and Rankine sources (Lin et al., 1999). Once the velocity potential is computed, Bernoulli s equation is then used to compute the hull pressure distribution including the second-order velocity terms. The perturbation velocity potential can be solved over either the mean wetted surface (the body linear solution) or over the instantaneously wetted portion of the hull surface beneath the incident wave (the body nonlinear approach). In either case, it is assumed that both the radiation and diffraction waves are small compared to the incident wave and the incident wave slope is small so that the free-surface boundary conditions can be linearized with respect to the incident-wave surface. Similarly, the incident wave forcing (Froude- Krylov) and hydrostatic restoring force can also be computed either on the mean wetted surface or on the wetted hull up to the incident wave. The combinations of the body linear and body nonlinear solutions of the perturbation potential and the hydrostatic/froude-krylov forces provide multiple solution "levels" for the ship-wave interaction problem. These levels are: LAMP-1 (Body linear solution): Both perturbation potential and hydrostatic/froude-krylov forces solved over the mean wetted hull surface LAMP-2 (Approximate body nonlinear solution): The perturbation potential is solved over mean wetted hull surface while the hydrostatic/froude-krylov forces are solved over the instantaneous wetted hull surface LAMP-4 (Body nonlinear solution): Both the perturbation potential and the hydrostatic/froude- Krylov forces are solved over the instantaneous wetted hull surface For most problems, the most practical level is the approximate body-nonlinear (LAMP-2) solution, which combines the body-linear solution of the perturbation potential with body-nonlinear hydrostatic-restoring and Froude-Krylov wave forces. This latter approach captures a significant portion of nonlinear effects in most shipwave problems at a fraction of the computation effort for the general body-nonlinear formulation. However, bodynonlinear hydrodynamics and nonlinear incident wave effects can be important depending on ship geometry and operating conditions. Mixed Source Formulation In the context of time-domain potential-flow boundary-element methods, the most commonly used approaches fall in two categories: (1) methods that use transient Green functions and (2) methods that use Rankine sources (Weems, et al., 2). For the methods in the first category (e.g. Lin et al., 1994), the transient Green function satisfies the linearized free surface boundary conditions and radiation conditions in the far field, so that the singularities need to be distributed only on the wetted portion of the body surface. For ships with non-wall-sided geometry, numerical difficulties may arise in the area where the intersection angles between the body surface and the free surface become small, as with wave-piercer ships. This is mainly due to the highly oscillatory nature of the transient Green function adjacent to the free surface. For the methods in the second category (e.g. Nakos et al., 1993), the Rankine source is used as a kernel in the boundary integral equation. Rankine sources are fairly robust for modeling either wall-sided or non-wall-sided geometries. To satisfy the free surface boundary condition, Rankine sources must be distributed not only on the body surface but also on the free surface. In order to limit the size of the computation domain, the free surface region is typically truncated at several ship lengths away from the ship, and a numerical damping zone is employed to absorb wave energy. 258 Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design

ABS TECHNICAL PAPERS 23 In view of the pros and cons of these two approaches, a hybrid numerical approach was developed to use both transient Green functions and Rankine sources (Lin et al., 1999). This approach is implemented in the LAMP System as the mixed source formulation. In the mixed source formulation, the fluid domain is split into two regions as shown in Figure 1. The outer domain is solved with transient Green functions distributed over an arbitrarily shaped matching surface, while the inner domain is solved using Rankine sources. The advantage of this formulation is that Rankine sources behave much better than the transient Green function near the body and free surface juncture, and the matching surface can be selected to guarantee good numerical behavior of the transient Green functions. The transient Green functions satisfy both the linearized free surface boundary condition and the radiation condition, allowing the matching surface to be placed fairly close to the body. This numerical scheme has resulted in robust motion and load predictions for hull forms with non-wall-sided geometries. Another advantage of the mixed formulation is that the local free surface elevation is part of the solution, and no additional evaluation is needed as in the case of the transient Green function approach. In addition, a nonlinear free surface boundary condition can be implemented at modest computational cost. Figure 1. Mixed source formulation In the LAMP System, a 2nd-order free surface boundary condition can be applied on the local portion of the free surface; see more details in Weems et al. (2). However, in the case of nonlinear free-surface boundary condition in the local portion of the free surface, the matching surface has to be placed further away from the body to minimize errors caused by a mismatch of the free surface condition. Rankine Singularity with Damping Beach While the mixed-source singularity model works very well for low to modest speeds (Fr.5), it can be difficult to obtain a stable solution at higher speeds. For this reason, an alternative singularity model has been implemented that replaces the external domain and the matching surface with a numerical damping region on the outer edge of the inner region s free surface. The body and free surface boundary conditions are otherwise identical to the mixed-source singularity model. While this singularity model typically requires a considerably larger free surface grid then the mixedsource model, it has been successfully applied at Fr=.85. The model also allows shallow water to be modeled in the hydrodynamic problem by panelizing the bottom or using image sources. IRF-Based Formulation A drawback to time domain hydrodynamics is the computational cost. To mitigate this, an Impulse Response Function (IRF) based hydrodynamic formulation (Liapis 1986, King et al., 1988, Bingham et al. 1993) was integrated into the LAMP System to complement the mixed source formulation. In the IRF formulation, velocity potentials are pre-computed for steady forward speed, impulsive motion in up to six modes, and impulsive incident waves for each speed and heading angle. The hydrodynamic problem is reduced to a convolution of the IRF potentials with the actual ship motions and incident wave elevation, thereby allowing numerical simulations to be performed faster than real time using modest computational resources and without compromising the accuracy of the hydrodynamic calculation. The IRF potentials are convoluted and summed on a panel-by-panel basis, so that the complete potential distribution of the hull can be computed in the time domain. This allows the panel pressure to be computed directly, including the nonlinear terms in Bernoulli s equation, in the same fashion as in the mixed source formulation. The only restriction is that the IRF formulation can only be used with body linear (LAMP-1) and approximate body nonlinear (LAMP-2) hydrodynamics. In the IRF-based formulation, the perturbation velocity potential on each body panel is decomposed as 6 r r r r Φ x, t) = Φ ( x, t) + Φ ( x, t) +Φ ( x, ) k ( k 7 8 t k= 1 where the Φ k, k=1..6 are the radiation potentials for the six rigid-body motions, Φ 7 is the diffraction potential related to the incident wave potential Φ, and Φ 8 is the steady state potential related to the constant forward speed U. To solve for the six radiation potentials Φ k, six corresponding impulse response functions φ k are introduced via the convolution integral t r r Φ ( x, t) = ϕ ( x, t τ) X& ( τ) dτ k k where X k is the ship motion in mode k and the dot signifies the derivative with respect to time. Each of the radiation IRF potentials is the solution for an impulsive velocity in that mode and is obtained by solving the initial boundary value problem: 2 ϕ ϕ k n k 2 ϕk 2 = = n δ( t) + m h( t) k ϕk + g t z ϕk ϕk = = t k = at t = in on S on S V b on S f f Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design 259

ABS TECHNICAL PAPERS 23 where δ(t) and h(t) are the Dirac delta and Heavyside step functions with respect to time t, respectively, and k runs from 1 to 6. To solve for the diffraction potential Φ 7, the diffraction IRF φ 7 is introduced via the convolution integral Φ 7 = ϕ 7 ζ ( τ) dτ where ζ is the incident wave elevation at the origin of the ship-fixed frame. The diffraction IRF φ 7 is the diffraction potential due to an impulsive unit incident wave and is obtained by solving the initial boundary value problem: 2 ϕ 7 = ϕ ϕ 7 7 = t in = V ϕ 7 ϕ = on S b n n 2 ϕ 7 ϕ + g k7 = on 2 t z S f at t = on S f where φ is the potential due to an impulsive unit incident wave at the origin of the ship-fixed frame: ϕ g = Re 2π i e ω kz ik ( x cosβ+ y sin β) + iωt dω where β is the wave heading angle relative to the ship, ω is the wave frequency, and ω e is the frequency of encounter. For the head seas (π/2 β 3π/2) and zero speed cases, there is single valued relationship between ω and ω e so there is a single impulsive wave, and hence a single diffraction IRF, for each wave heading angle. However, for the following seas cases, there are up to three wave frequencies for each frequency of encounter, so there are three impulsive incident waves, and hence three diffraction IRFs, for each wave heading angle. The final component of the decomposed wave potential, the steady forward potential, can be determined by simply solving a forced steady forward speed problem with no incident waves until a converged solution is reached. All of the IRF potentials can be computed using the mixed-source formulation s numerical scheme and saved in a series of files. Each of the IRFs depends only upon the ship geometry, speed, and incident wave heading angle. In principle, the convolution of the radiation IRFs must start at the beginning of the calculation and the convolution of the diffraction IRFs must run from τ=- to τ=. In practice, however, the radiation convolutions need only include the most recent 4 or so time steps (for a typical time step of.15 seconds) while the diffraction convolutions need only include ±4 steps for head seas and ±8 steps for following seas. As a result, the IRFs e are not excessively expensive to compute a full set of IRFs for a long crested head sea case requires the same calculation effort as about 2 seconds of simulation using the mixed-source formulation. The computational cost of IRFs for short-crested and/or following sea cases will be higher as additional diffraction IRFs will be required. Once computed, however, the resulting IRF-based LAMP calculation can be run 4-1 times faster than the mixed source formulation. This allows faster-than-realtime calculations for most configurations using very modest computational resources. Furthermore, the IRFs are independent of the incident waves frequency and amplitude, so an entire series of regular or irregular wave conditions can be run with the same IRFs (Weems et al., 2). As mentioned earlier, a restriction of the IRF-based formulation is that it can only be used with the bodylinear solution for the perturbation potential where the perturbation potential is solved over the mean wetted surface of the hull. However, this does not imply that the IRF formulation is restricted to a classical linear seakeeping solution. Classical linear seakeeping linearizes Bernoulli s equation, hydrostatic restoring forces, and the equations of motion and assumes a direct linear relation between the incident wave and the response. In contrast, the IRF-based velocity potentials due to ship motion (radiation) are linear in the instantaneous ship velocity as computed by the general solution of the equations of motion. Also, the IRFs are used to compute a distribution of velocity potential, so that the second order terms in velocity in Bernoulli s equation are preserved. In the approximate body-nonlinear calculation (LAMP-2), the incident wave forcing and hydrostatic restoring forces are computed using a nonlinear approach. Additional forces such as those due to viscosity, control surfaces, green-water-on-deck, etc. can also be treated using this nonlinear approach. As a result, LAMP s approximate body-nonlinear solution is capable of capturing significant large amplitude wave phenomena that are missed in classical linear seakeeping, such as parametric rolling and the dependence of motions and loads on wave slope. Using the IRF-formulation with the approximate body-nonlinear calculation can provide faster-than-real-time calculation of large amplitude responses, thereby facilitating the large number of relatively lengthy simulations required to properly characterize a ship s response in large amplitude seas. LARGE AMPLITUDE WAVE LOADS Comparisons Between the Mixed and Rankine Source Formulations In an effort to assess possible differences in results between the Mixed and Rankine Source formulations in LAMP, several comparisons were made against available model test data. The seakeeping and loads model data used for these comparisons were compiled during an advanced hull form development study at Carderock Division, Naval Surface Warfare Center (NSWC/CD). 26 Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design

ABS TECHNICAL PAPERS 23 To support this analysis, a free running 1/2-scale segmented model was built, with a continuous longitudinal backspline beam and instrumented to provide structural loads data (see Figure 2). The test program consisted of calm water, regular wave, and irregular wave measurements over a range of ship speeds and wave headings. Wave At CG, ft 2 1-1 -2 a Model test LAMP4 Time, s 1 2 3 4 5 6 7 1 b 5 Figure 2. Advanced monohull structural loads model Subsequent to testing, the LAMP System was used to compare motions and loads predictions with model test results. The visualization of the LAMP simulation for an extreme wave case is shown in Figure 3. Pitch, deg. Heave, ft -5-1 2 1-1 -2 c Model test LAMP: Rankine LAMP: mixed Time, s 1 2 3 4 5 6 7 Model test LAMP: Rankine LAMP: mixed Time, s 1 2 3 4 5 6 7 Figure 3. LAMP simulation for advanced Monohull form A sample of the body nonlinear formulation (LAMP- 4) results obtained using both the Rankine and the mixed formulations is shown in Figures 4 and 5. The first series of figures compare the two formulations for a regular wave run with a wave steepness of 1/3 and a ship speed of 15 knots. Although there are some difference in phasing between the model test results and the predictions, both formulations show fairly good agreement. The next set of comparisons shows the model in a Hurricane Camille seaway. As can be seen from Figure 5, predicted motions and loads using either formulation are quite good, the main differences between the predicted and measured loads being that slam-induced whipping predictions were not performed. Hence the high frequency component to the load time history is not accounted for. Bow accelerations. g' s VBM, ft lb 1-6 1..5. -.5-1. -1-2 -3-4 d e Model test LAMP: Rankine LAMP: mixed 1 2 3 Model test LAMP: Rankine LAMP: mixed 1 2 3 Figure 4. Run 727: λ/l=1., H/λ=1/3, 15kts. (a) Wave at CG. (b) pitch, (c) heave, (d) bow acceleration, and (e) vertical bending moment 4 4 5 5 Time, s 6 6 7 7 Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design 261

ABS TECHNICAL PAPERS 23 Heave, ft (full-scale) 2 1-1 a Model test LAMP: mixed LAMP: Rankine Application of LAMP-FEM Interface for Computation of Wave Loads for Large Container Carrier Shin et al. (1997) contains detailed information on the interface between LAMP and finite element method (FEM) analysis software. Another sample application is shown in Figure 6. This time, a large container ship was the subject of the case study. Since no experience existed in designing a container ship of such a size, the designer recognized the simulation results as an important source of information for making technical decisions. -2 25 5 75 1 125 15 175 Time, s (full-scale) a 1 b Model test LAMP: mixed LAMP: Rankine Pitch (deg) 5-5 b 25 5 75 1 125 15 175 c Time (sec, full-scale) VBM, ft-lb 1 (full-scale) -1-2 -3 Model test c LAMP: mixed LAMP: Rankine 25 5 75 1 125 15 175 Time (s) (full-scale) d e Figure 6. Structure responses of large container ship for different loading cases (a) hogging; (b) sagging; (c-e) extreme torsion for different location of wave crest along the ship Figure 5. Run 127: (a) heave, (b) pitch, (c) vertical bending moment 262 Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design

ABS TECHNICAL PAPERS 23 LARGE MOTION PHENOMENA AND ASSOCIATED LOADS Parametric Roll and its Stabilization Parametric excitation in head seas is known to be a factor in the very high amplitude roll motion that caused lost cargo (France et al., 23). One possible way to mitigate this kind of motion is a roll stabilization system. A method for approximating a U-tube type passive anti-roll tank system has been implemented as an optional plug-in module in LAMP. The system calculates fluid motion in the U-tube tank and determines the coupled nonlinear forces acting on a floating body in full six degrees of freedom. These forces are added to LAMP s calculation of motion and loads as an external (nonpressure) force. A schematic of the anti-roll tank is shown in Figure 7. An initial investigation into the implementation and effectiveness of the anti-roll tank module was performed for a regular synchronous roll case. In this example, the sample container ship shown in Figure 8 was simulated at 15 knots in regular beam sea waves. Figure 8. Panel representation of a sample large modern container ship The wave period was equal to the ship s natural roll period of 25 seconds, and the wave height was 12 meters. Figure 9 compares roll motions for a case with no anti-roll tanks and a case with two identical anti-roll tanks. The case with no anti-roll tanks has a steady-state roll angle of approximately 22 degrees while the case with the anti-roll tanks has a maximum roll angle of 3.5 degrees. This investigation clearly shows the ability of an appropriately tuned anti-roll tank system to reduce resonant roll motions. Figure 7. Schematic of anti-roll tank system The LAMP anti-roll tank formulation is based on the improved passive anti-roll tank model developed by Youssef, et al. (22). It has been successfully applied in the nonlinear motion range (including parametric roll suppression) as shown in the above reference. Several assumptions are used in the anti-roll tank formulation including: Vertical tank columns and connecting tubes are assumed to be circular. Shear stress at the wall is related to a friction factor that is dependant on Reynolds number and based on either the Colebrook formula or the Swamee and Jain (1976) formula. Energy loss in the elbows connecting the vertical columns to the cross pipe is expressed in terms of a loss coefficient. Tank free surface correction does not account for sloshing effects and assumes a flat calm free surface that remains parallel to the global horizontal plane. Figure 9. Roll motion in regular beam seas (wave period = ship s natural roll period) Several calculations were performed on a sample container ship to determine the effectiveness of the passive anti-roll tank system in reducing the ship s susceptibility to parametric roll. Figure 1 shows the predicted maximum roll angles for this container ship in regular head seas as a function of encounter frequency. A range of encounter frequencies in which this particular ship might be susceptible to parametric roll can be clearly seen in the curve with no anti-roll tanks. The roll natural frequency for this ship is.251 rad/sec, which corresponds to a 25 second period, so parametric rolling would be expected near an encounter frequency of.52 rad/sec, which is clearly shown in the results. The nonlinear dependence of parametric roll on the incident wave amplitude is also clearly illustrated. This ship had a Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design 263

ABS TECHNICAL PAPERS 23 large range of encounter frequencies where parametric roll is predicted, which indicates that the ship is very susceptible to parametric roll. subsequently updated, after which the green-water-ondeck pressure and forces are passed back to the ship motions calculation to be used in solving the equations of motions and computing sectional loads. The structure of this computational approach is illustrated in Figure 12. Figure 1. LAMP computed parametric roll domain for sample container ship The other cases plotted in Figure 1 show the same container ship with a single passive anti-roll tank where the tank mass is equivalent to.27,.35,.71, and 1.4 percent of the ship s displacement, with a tank system natural period equal to the ship s natural roll period of 25 seconds. The.27 percent case shows a reduction in the bandwidth where parametric roll occurs, but there is still a region of large roll angles. Both the.35 percent and.71 percent cases show virtually no roll past the initial 5- degree roll angle at the start of the simulation. The 1.4 percent case shows a region of slightly elevated roll motion (<1 degrees) for low encounter frequencies. The increased roll motion is due to the large mass in the tank system, which is not in equilibrium at the beginning of the simulation. The tank fluid continues to oscillate for a short period of time until it eventually damps out as the simulation progresses. Analogous principles and procedures have been used to couple a numerical solution of sloshing in cargo and anti-rolling tanks with time-domain ship motions (Kim, 22). Influence of Green-Water-on-Deck on Motions and Loads Recent efforts have developed a sophisticated greenwater-on-deck model that can be integrated directly into LAMP, allowing water-on-deck effects to be included in time domain calculations of nonlinear ship motions and loads. In the approach that was selected, the ship motions and green-water-on-deck calculations run concurrently in the time domain. As illustrated in Figure 11, this approach subdivides the computational domain into an outer problem where the ship-wave interaction problem is solved and an inner problem where the green-water-ondeck problem is solved (Belenky, et al., 22). At each time step of the LAMP calculation, the ship motion and wave definition are used to compute the relative motion at the deck edge (often called the deck edge exceedance), the deck tilt, and the deck acceleration. These values are in turn passed to the green-water-ondeck calculation. The green-water-on-deck calculation is Figure 11. Decomposition of the computation domain Figure 12. Structure of the LAMP System with greenwater-on-deck The green-water-on-deck calculation in the LAMP System has been integrated in a modular fashion so that different levels of calculations are available. Hydrostatic and Froude-Krylov pressure: Deck pressure is computed from the submergence of each deck element and the linear incident wave pressure ( Φ/ t). This is a straightforward modeling of the deck as a bodynonlinear hydrostatic/froude-krylov surface and represents the basic deck-in-water calculation. Semi-empirical model: Water-on-deck is calculated directly from deck edge exceedance using an empirical expression derived from Zhou et al. (1999). The resulting deck pressure, including the effects of deck motion, can then be evaluated following the approach proposed by Buchner (1995). This model is intended to provide a quick estimate of the effect of foredeck green-water-on-deck on ship motions and main girder loads. Shallow water flow calculation: Computes the longitudinal and transverse flow over the deck with the assumption of shallow water using a finite-volume technique. This is the principal green-water-on-deck 264 Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design

ABS TECHNICAL PAPERS 23 calculation used in the present study and is described in detail below. Fully 3-D flow calculation: Computes the flow over the deck, including vertical gradients, using a 3-D volume grid and a finite-volume technique. This new technique is under development and is a likely future path for greenwater-on-deck flow calculation methods. The most sophisticated of LAMP s current greenwater-on-deck calculation methods is the one based on the solution of the shallow water flow equations using a novel finite volume strategy. In this method, the equations for conservation of mass and momentum are solved in the time domain, shallow-water flow is assumed, and viscous effects are ignored except for a relatively simple deck friction term. The shallow water assumptions are that the fluid acceleration normal to the deck can be ignored while the tangential fluid velocity and pressure are constant across the depth of water-on-deck (Stoker, 1957). These assumptions reduce the 3-D fluid domain to a 2-D computational domain. This method can handle a variety of boundary and initial conditions, and it is capable of supporting arbitrary motions and general ship-deck geometries, including partial height walls (e.g. bulwarks), infinite walls (e.g. deckhouses), and stepped or raised sections (e.g. hatch covers). This approach has been validated with available experimental data and has been successfully integrated with the LAMP System (Liut, et al., 22). For this approach to be useful as an integrated part of the LAMP simulation, the green-water-on-deck calculation must be reasonably fast, robust, and capable of calculating the flow on a deck that is moving with largeamplitude six-degrees-of-freedom motions and also exchanging water with the environment. Because of these requirements, a considerable portion of the effort in developing and implementing the numerical method has been spent in ensuring a stable and reasonable solution even when the assumptions of shallow water flow are stretched to their limits. In order to test and validate the numerical solution of the shallow-water calculation, computational studies have been made for several configurations for which detailed theoretical and experimental data are available. One such study examined the dam-breaking problem, studying the level of water behind a dam after the dam is suddenly removed. A second study examined the outflow from a box whose end is suddenly removed. Both studies show good agreement between the present method and published data (Liut et al., 22). In order to test the green-water-on-deck implementation in the LAMP System, a series of calculations were made for a US Navy CG-47 Class cruiser, a U.S. Coast Guard Cutter, and a large modern container ship. The qualitative validation continues with the fishing boat study described below. The fishing vessel analyzed in the present study is the 21-meter stern dragger Italian Gold, which sank in heavy seas off Massachusetts in 1994. All calculations were made at a heavily loaded condition with a displacement of 194 long tons and the center of gravity.8 meters above the mean waterline. The resulting transverse metacentric height (GM t ) is.22 meters. The basic geometry model used in the hydrodynamic and hydrostatic calculations is shown in Figure 13. Figure 13. Fishing boat geometry In the simulations, the skeg and rudder were modeled using non-pressure force models, so their effect was included in the equations of motions but not in the wavebody interaction potential. Green-water-on-deck was calculated over the afterdeck, with inflow/outflow allowed over both sides and the stern of the ship. The first and, to this point, most extensive portion of the study involved a series of roll decay calculations using different computational models and options. For clarity, these calculations were made in calm water at zero speed with the ship free only to roll. The initial roll angle was 3 degrees, at which the deck edge and bulwark are reasonably deeply submerged. For the case involving the shallow water green-water-on-deck calculation, the deck elevations were initialized with stationary water-on-deck up to the calm waterplane Z =. An initial calculation was made for the boat with no water-on-deck calculation of any kind on the afterdeck. A time history of the roll angle, in degrees, is shown in red in Figure 14. The blue square-tooth at the bottom of the graph indicates the relative motion of the minimum freeboard point of the deck edge; a high value indicates that the deck edge is submerged while a low value indicates that it is above the wave surface. This point, which is located just aft of the deckhouse, submerges at about 1 roll angle in calm water. To some extent, this indicates the beginning of the transition from water-ondeck to deck-in-water. 3 φ, deg Deck enters water 2 1 1 2 2 4 t, s Deck enters water 3 Figure 14. Roll decay angle and deck edge submergence without effect of water-on-deck The initial roll period is 5.17 seconds and increases slightly to 5.47 seconds as the roll angle decreases. This change in roll period is due primarily to the change in Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design 265

ABS TECHNICAL PAPERS 23 hydrostatic restoring moment of the body-nonlinear formulation. A second calculation computed water-on-deck pressure from hydrostatics. The results for this calculation are shown in Figure 15. In addition to the roll angle and deck edge submergence shown in the bottom graph, the top graph shows a red line indicating the volume of wateron-deck in m 3 and a blue line shows the resulting heeling moment in Newton-meters. Since the model contains no lag for water entering or leaving the deck, the volume of water and the heeling moment are exactly in77 phase with the roll angle. In other words, only the deck-in-water situation is modeled. The initial roll period increases to 5.96 seconds, an effect caused by the water-on-deck heeling moment, which in this case effectively decreases the restoring moment when the deck is submerged. As the roll angle decreases, the roll period decreases to 5.47 seconds, matching the period from the previous calculation. A small but noticable decrease in the roll damping can be observed over the early portion of the run, where the longer roll period has decreased the roll velocity and reduced the damping due to skin friction and appendages. 4 2 M GW, 1-4, Nm V GW, m 3 V GW as the water flows off the deck, although there are small periods of inflow during subsequent deck edge submergences. The initial roll period is similar to the hydrostatic pressure case, which is reasonable as the water-on-deck heeling moment is similar, while subsequent roll periods match both of the previous calculations at similar roll angles. The influence of wateron-deck and deck-in-water is realized mainly through a significant increase of roll damping (see Figure 16). Since the problem is otherwise nearly identical to that of the previous calculation, the damping influence must be credited to the water-on-deck. The most likely explanation is that the lag of the water flowing off the deck or to the other side of the boat when the boat rights itself results in a significant volume of water on the rising side of the deck, where the resulting deck pressure produces a significant moment in phase with but opposed to the roll velocity. Evidence for this interpretation is offered in Figure 17, which shows a snapshot of the boat, which was initially healed to starboard, with the computed water-ondeck surface as it rolls through -degree heel in the first cycle of the calculation. 6 4 2 M GW, 1-4, Nm V GW, m 3 M GW 2 4 t, s V GW 2 4 t, s 2 4 4 φ, deg 2 4 2 φ, deg 2 2 4 t, s 2 4 t, s 2 2 Figure 16. Roll decay results including finite-volume water-on-deck effect without bulwarks 4 Figure 15. Roll decay results including hydrostatic forces on deck: water-on-deck volume (V GW, top) and roll moment (M GW, top), roll angle (φ, bottom), and deck edge submergence (bottom) The next calculation uses the finite-volume solution of the shallow water flow over a deck with bare deck edge (i.e. no bulwark). The volume of water-on-deck starts out large (as noted above, the finite-volume model is initially full to mean Z = ) and generally decreases in volume Figure 17. Computed water-on-deck surface for the fishing boat as it rolls though º heel 266 Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design

ABS TECHNICAL PAPERS 23 The next calculation, whose results are shown in Figure 18, introduces a solid (no freeing ports) bulwark with height.853 meters to the finite-volume calculation. The bulwark, which traps the water-on-deck and raises the angle at which water can enter the deck in calm water to over 2 degrees, changes the whole picture of motions. Damping is such that the second peak of roll angle is too small to immerse the bulwark top or to allow significant inflow. The amount of green-water-on-deck stays constant and the ship quickly assumes a static heel angle of about 7 degrees. 4 2 2 4 6 M GW 2 4 φ, deg 2 2 M GW, 1-4, Nm V GW, m 3 V GW 2 4 6 t, s t, s Figure 18. Roll decay results including finite-volume water-on-deck calculation with.853 m bulwarks The second portion of the fishing boat study involved calculations for the boat at zero speed in regular (single frequency) beam waves. As in the roll decay calculation, the boat was only free to roll. While not a realistic situation, it simplifies the comparison of the different calculations and reduces the risk of capsize, especially during the initial transient period. The wave height was 3.5 meters with the period of 6.28 sec, which makes the wave forcing close to the roll resonance regime. We consider only steady state rolling in the present discussion, so the results shown in Figures 19-22 are plotted beginning 52 seconds into the simulations. The initial transient behavior, which is both interesting and very important, is simply beyond the scope of the present discussion. Figure 19 shows the results of the regular wave calculation for the fishing boat with no water-on-deck model at all. As for the roll decay results, the blue square-tooth indicates the submergence of the deck s minimum point, which now is a function of the roll angle and the incident wave elevation. 4 Figure 19. Regular wave roll without the effect of wateron-deck 1 1 8 4 4 8 4 2 2 52 54 56 58 6 M GW, 1-4, Nm V GW, m 3 V GW 52 54 56 58 6 φ, deg t, s M GW 52 54 56 58 6 t, s t, s Figure 2. Regular wave results including hydrostatic forces on deck: water-on-deck (v gw, top, solid) and roll moment (m gw, top, dashed), roll angle ( bottom), and deck edge submergence (bottom) Figure 2 shows the results of the regular wave calculation with the deck pressure calculated as the hydrostatic and linear incident wave (Froude-Krylov) pressure only. The response is very regular and primarily periodic at the forcing frequency. An increase in roll amplitude, attributed mostly to the decrease in restoring moment and increase in wave forcing when the deck is submerged, leads to an increasing amount of time deckin-water. Figure 21. Regular wave results including finite-volume water-on-deck calculation but with no bulwark The next regular wave calculation used the finitevolume calculation of the shallow water flow over the deck with bare deck edges (no bulwark). The volume of water-on-deck, plotted in red on the top graph, changes by approximately 8 percent over the wave cycle. The roll response is smaller than the hydrostatic and linear incident wave pressure calculation, probably because of the wateron-deck damping seen in the roll decay study, and shows Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design 267

ABS TECHNICAL PAPERS 23 a bias of about 5 toward the up-wave direction. A weak subharmonic roll response can be observed. 5 5 1 4 2 2 52 54 56 58 6 52 54 56 58 6 4 Figure 21. Regular wave results including finite-volume water-on-deck calculation but with no bulwark 1 5 5 1 8 4 4 M GW, 1-4, Nm V GW, m 3 φ, deg M GW, 1-4, Nm V GW, m 3 M GW 52 54 56 58 6 φ, deg M GW V GW 52 54 56 58 6 V GW t, s t, s t, s t, s 8 Figure 22. Regular wave results including finitevolume water-on-deck calculation with.853 m bulwarks Figure 22 shows the regular wave results with a.843 m bulwark added to the finite-volume calculation. The maximum volume of water-on-deck is slightly higher than in the no-bulwark calculation, but the mean value of water-on-deck has increased substantially. Subharmonic roll response and increased roll amplitudes are observed. Comparing Figures 21 and 22, it can be concluded that accumulation of water-on-deck has a major impact on ship dynamics: the subharmonic character of the response might be attributed to water-on-deck influence. Garkavy (1991) observed subharmonic rolling of a ship with small freeboard (and as a result, one with water-on-deck) during model tests. Observations of subharmonic roll could be a result of period-doubling bifurcation, which is known to happen for nonlinear rolling in beam seas; see, for example, Nayfeh and Sanchez (199). Accumulation of water-ondeck decreases the instantaneous restoring moment; as a result, ship roll could be shifted close to the nonlinear region in the vicinity of the GZ curve s peak, which might facilitate the bifurcation of the periodic solution. However, in order to state that the observed behavior is the manifestation of period-doubling bifurcation, it is necessary to analyze the stability of steady state motion, which is beyond of the scope of this study. At the same time, the period-doubling bifurcation could be seen as a possible physical explanation for the observed results and as an indicator of validity for the numerical model of water-on-deck. To check influence of the green-water-on-deck on longitudinal motions and global loads another set of calculations have been made for the US Navy CG-47 Class cruiser in two sea conditions: Head seas, steep regular waves, λ = 81 feet and H = 48 feet (ship length is 528 feet) Head seas, storm sea condition, H 1/3 = 37 feet The latter case matches an experimental recreation of the waves measured during a hurricane and would be classified as Sea State 8. For the green-water-on-deck calculations, a CG-47 input geometry was prepared with an additional component representing the deck forward of the superstructure. This input geometry is presented in Figure 23. Figure 23. LAMP geometry for CG-47 including forward deck The first of the two test cases is somewhat unrealistic but was intended to test the robustness of the calculation and exaggerate the green-water-on-deck effects for evaluation purposes. The heave and pitch motion histories are shown in Figure 24. Note that the heave is normalized by ship length, the pitch is in radians, and a positive pitch angle is bow down in this and all subsequent motion plots. The horizontal axis is nondimensional time, in which unit time would correspond to about 4. seconds for the fullscale ship. As seen in these plots, water-on-deck does not significantly change the motion response. 268 Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design

ABS TECHNICAL PAPERS 23 Heave, (-) Pitch, (-).4.2.2 No.4 greenwater 2 effect 4 6 8 1 12 14 16 Hydrostatics / FK Time, (-) Semi-empirical Finite volume model.2.1.1.2 2 4 6 8 1 12 14 16 Time, (-) Figure 24. Water-on-deck effects on heave (top) and pitch (bottom) motion for CG-47 in large regular waves Figure 25 shows a detail of the pitch response for two cycles of the calculation. All four of the calculations agree at the maximum bow down position (+ pitch) but three calculations that include water-on-deck effects all predict a lag in the bow s recovery and some differences in the bow up ( pitch) peak. Pitch, (-).15.1.5.5.1.15No 1 greenwater 11 effect 12 13 14 15 16 Hydrostatics / FK Semi-empirical Time, (-) Finite volume model Figure 25. Detail of water-on-deck effects on pitch motion of CG-47 in large regular waves The second of the two test cases duplicates a model test that recreated the sea spectrum derived from wave probe data taken during a major hurricane. This case is thought to represent physically realistic extreme sea conditions that a ship might encounter. As for the regular wave case, the various green-water-on-deck models do not have a significant apparent effect on the motions, shown in Figure 27, although the bow emerges from the water somewhat more slowly when green-water-on-deck effects are present. VBM, (-) 4 1 5 No greenwater 2 effect 4 6 8 1 12 14 16 Hydrostatics / FK Time, (-) 1 1 4 Semi-empirical Finite volume model VBM, (-) 6 1 5 4 1 5 2 1 5 2 1 5 5 1 5 5 1 5 1 1 4 2 4 6 8 1 12 14 16 Time, (-) Figure 26. Water-on-deck effects on vertical bending moment at X/L=.26 (top) and X/L=.49 (bottom) in large regular waves (VBM is normalized by ρgl 4 ) Figure 28 shows the total green-water-on-deck force and moment for the semi-empirical and 3-D shallow water models. The results for the two models are similar, although the forces from the 3-D model appear to be larger and slightly delayed. The force results also show how the 3-D model allows water on deck to linger somewhat after the free surface falls below the deck edge The green-water-on-deck effect is far more noticeable in the load calculations shown in Figure 26. In all of the calculations, the water-on-deck reduces the peak sagging (+ VBM) moment. The magnitude of this reduction is similar for the three methods but the details of the peak differ. As might be expected, the water-on-deck effect is greatest for the loads near the bow and smallest for those near the stern. Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design 269

ABS TECHNICAL PAPERS 23.6 regular wave case, the green-water-on-deck models all predict a reduction in the peak sagging bending moment. Heave, (-).4.2 1 1 4 No greenwater effect Hydrostatics / FK Semi-empirical Finite volume model.2 5 1 5 Pitch, (-).4 No greenwater effect 5 1 15 2 Hydrostatics / FK Time, (-) Semi-empirical Finite volume model.2.1.1.2 5 1 15 2 Time, (-) Figure 27. Water-on-deck effects on heave (top) and pitch (bottom) motions for CG-47 in irregular head seas, H 1/3 = 37 feet. VBM, (-) 5 1 5 1 1 4 5 1 15 2 Time, (-) Figure. 29 Green-water-on-deck effects on vertical bending moment at X/L=.49 for CG-47 in head storm seas For the storm sea run, pressure output files were written that include the deck geometry and computed pressure for any green-water-on-deck calculations. LAMP s pressure data post-processor can use the computed deck pressure, along with the computed hull pressure distribution, to calculate nodal pressure and FE load data sets. Figure 3 shows a plot of the deck and hull pressure distributions at T=1.4 of the storm sea run. 1 1 4 Pitch moment Time 1 1 4 Surge force 2 1 4 Vertical force 3 1 4 5 1 15 2 1 1 4 1 1 4 2 1 4 3 1 4 Pitch moment Time Surge force Vertical force 5 1 15 2 Figure 28. Water-on-deck force (normalized by ρgl 3 ) and moment (normalized by ρgl 4 ) for CG-47 in irregular head seas by semi-empirical (upper) and 3-D shallow water (lower) models for water-on-deck. Figure 29 shows the predicted vertical bending moment near midships for the storm sea case. As with the Figure 3. Plot of hull and deck pressure (normalized by ρgl) for CG-47 in storm seas at T=1.5 Multi-Hull Applications With its 3-D formulation and general geometry model, the LAMP System is well suited for analyzing multi-hull ships such as catamarans and trimarans. Figure 31 shows the geometry of a notional trimaran design for which some LAMP validation has been performed. 27 Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design

ABS TECHNICAL PAPERS 23 used both the linear and 2 nd -order free surface boundary conditions. Figure 34 shows the predicted local wave field at one time instant of a calculation for a regular wave propagating down the long axis of the platform. Figure 31. Geometry of sample trimaran This trimaran was designed by a team of graduate students and tested at the U.S. Naval Academy (Harris, 1999). The tests analyzed several longitudinal and transverse locations of the outer hulls and included seakeeping tests in regular and irregular head seas. Figure 31 compares the predicted and measured heave and pitch motions for the regular wave runs at a Froude number of.15. Like the experiments, the LAMP predictions were computed as a series of regular wave runs that used the approximate body-nonlinear (LAMP-2) model and matched the wave slope of the experimental runs. Figure 32. RAO for heave (m/m) and pitch (deg/m) motion in head seas for trimaran at Fr=.15 Figure 33 offers a similar comparison at a Froude number of.3. The comparison for both cases is good. LAMP calculations have been made for this configuration for speeds up to a Froude number of.74, but no experimental data is available at that higher speed. Figure 33. RAO for heave (m/m) and pitch (deg/m) motion in head seas for trimaran at Fr=.15 The LAMP System has also been used to analyze large offshore structures, including a detailed evaluation of the disturbed wave field beneath a large semisubmersible platform (Weems et al., 1999). The study, which was part of an effort to examine the risk of wave impact on the underside of the main platform structure, Figure 34. Predicted local wave elevations for semi-submersible platform LARGE AMPLITUDE MOTIONS IN FREQUENCY DOMAIN One of the main advantages of the time-domain simulation approach is its more precise description of the forces on a ship, compared with frequency domain calculations. Once the amplitude of response grows large, its nonlinear nature becomes evident. Figure 35 illustrates this through motion and load predictions for a 195 m container ship. The heave, pitch and VBM response is shown in dimensionless form in Figure 35, so it is very clear how the responses lose their sinusoidal shape with increasing wave amplitude. It is noticeable that the VBM reacts more to increasing wave amplitude than do heave and pitch. However, the heave and pitch responses are also quite nonlinear for the 8.12 m wave amplitude. Increased fidelity of ship motion and load predictions comes with a price. Results are nonlinear and they no longer can be presented in a form of an RAO, because the amplitude of response (and shape, too) now depends on the amplitude of wave excitation. Moreover, the amplitude of a nonlinear response may depend on initial conditions, so several responses may be possible at the same frequency. At the same time, amplitude vs. frequency is a conventional way to present a design s ship motion. This makes it important to find the appropriate method of frequency domain presentation for nonlinear simulation results. The natural way to do it is to apply methods of nonlinear dynamics. However, these methods were developed for objects, behavior of which can be described by a system of ordinary differential equations. This is not the case for a ship in waves. Radiated and diffracted waves interfere with incident waves, which causes excitation to depend on motion history beyond the previous time step. As a result, initial conditions (including the excitation phase) are no longer enough to determine the status of a dynamical system. There are additional issues that make the application of nonlinear dynamics non-trivial. Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design 271