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1 Copyright © 2015 by ASME

Proceedings of the ASME 2015 34th International Conference on Ocean, Offshore and Arctic Engineering OMAE2015

May 31-June 5, 2015, St. John’s, Newfoundland, Canada

OMAE2015-42421

REDUCED ORDER MODEL FOR MOTION FORECASTS OF ONE OR MORE VESSELS

William M. Milewski Applied Physical Sciences Corp.

Groton, CT USA

Benjamin S. H. Connell Applied Physical Sciences Corp.

Groton, CT USA

Valerie J. Vinciullo Applied Physical Sciences Corp.

San Diego, CA USA

Ivan N. Kirschner Applied Physical Sciences Corp.

Groton, CT USA

ABSTRACT The reduced order model at the heart of the APS

Environmental and Ship Motion Forecasting system must retain

the accuracy of a higher fidelity seakeeping code while

simultaneously meeting computational speed required to provide

motion forecasts minutes into the future for two or more ships

operating in close proximity. We describe the mathematical

formulation of the reduced order model and efficient modeling

techniques to construct databases of wave force Response

Amplitude Operators and Impulse Response Functions before

presenting comparisons between the reduced order model, a

higher fidelity seakeeping code called AEGIR, and experimental

data for the R/V Melville and two multi-ship configurations.

INTRODUCTION The U.S. Office of Naval Research is sponsoring the

development of an Environmental and Ship Motion Forecasting

(ESMF) System to provide operator guidance for cargo transfer

operations between vessels at sea. ESMF is one of several

emerging technologies which will enable the sea basing concept

of operation, providing the capability to use the sea in the same

way U.S. forces currently use land bases to pre-position troops

and materiel. ESMF is envisioned to extend safe at-sea

operations between the Mobile Landing Platform and other

Military Sealift Command vessels through Sea State 4. Other

potential applications for ESMF include launch and recovery

operations for air, surface and underwater vehicles and at-sea

transfer between vessels in offshore applications.

Applied Physical Sciences (APS) has developed and

conducted preliminary demonstrations of an ESMF system on

the R/V Melville. At the heart of the APS ESMF system is a

reduced order model (ROM) for ship motions which is based on

the generalized Cummins equations of motion for one or more

vessels. The ROM solves for motions in the time domain, using

databases of wave force Response Amplitude Operators (RAO)

and Impulse Response Functions (IRF) to evaluate the forcing

terms in the equations of motion. These databases are assembled

via a suite of diffraction and impulsive motion simulations using

AEGIR, a B-Spline-based high-order three-dimensional

boundary integral equation time-domain seakeeping code, for a

range of operational and environmental conditions. The RAO

and IRF are interpolated and combined with the measured phase-

resolved wave field and ship motion history to forecast time-

accurate ship motions. Recent enhancements to the ROM

include the addition of models for mooring line and fender forces

which are required to forecast the motions of vessels moored

together in the skin-to-skin configuration.

Extensive verification and validation of the ROM was

conducted using model-scale motion data for the R/V Melville.

These comparisons show that the ROM retains the accuracy of

the higher fidelity numerical model while meeting the required

computational efficiency to forecast motions 30 seconds into the

future. A preliminary validation of the ROM is also presented

using model-scale data measured at MARIN in 2006 for a two-

ship configuration composed of the T-AKR 300 and T-AK 3008

vessels operating in a skin-to-skin configuration.

NOMENCLATURE B beam [m]

F external force in the k-th mode [N, Nm]

g gravitational constant [m2/s]

k wavenumber [m-1]

L length [m]

N number of vessels [-]

S water plane area/transverse separation [m2]

s transverse separation [m]


T draft [m]

t time [s]

U ship speed [m/s]

a, A∞ high-frequency limit added inertia

b linear damping coefficient

c linear restoring coefficient

m body inertia [kg, kgm2]

i,j hydrodynamic response in the i-mode

due to motion in the j-mode

[-]

x translational/rotational displacement [m, rad]

x translational/rotational velocity [m/s,

rad/s]

bQ quadratic damping

bU speed-dependent linear damping

cQ quadratic restoring

K impulse response function

density [kg/m3]

angle of U-tube water level relative to

ship

[rad]

frequency [rad/s]

relative wave heading [deg]

displacement [m3]

wave length [m]

wave height [m]

D complex diffraction force RAO [N/m,

Nm/m]

η complex wave amplitude [m]

TIME-DOMAIN MODEL The Reduced Order Model is a lumped parameter time-

domain model for the ship system forced by ambient ocean

surface waves. One equation is solved for each degree of

freedom using the Cummins (1962) formulation.

𝑚𝑖�� + ∑ 𝑎𝑖𝑗 ��𝑗

6𝑁

𝑗=1

+ ∑ 𝑏𝑖𝑗��𝑗

6𝑁

𝑗=1

+ ∑ 𝑐𝑖𝑗𝑥𝑗

6𝑁

𝑗=1

=

∑ ∫ 𝐾𝑖𝑗(𝑡 − 𝜏, 𝑈)��𝑗

𝑡

−∞

(𝜏)𝑑𝜏

6𝑁

𝑗=1

+

∑ ∑ 𝐷��(𝜔. 𝛽, 𝑈)𝑒𝑖𝜔𝑡��(𝜔, 𝐵) + ∑ 𝐹𝑖𝑘𝑘𝛽𝜔 (1)

This expression describes the response of the ith mode of

motion for a system of N vessels with up to 6DOF per vessel.

The dynamic responses, xi, are solved about a steady state

coordinate system translating at the forward speed U.

This formulation is well-suited to the ESMF application

because: the equations of motion are integrated in the time

domain; the ROM can retain the accuracy of the three-

dimensional potential flow code; the numerical implementation

runs significantly faster than real-time, providing a forecasting

capability; current and prior state observations can be integrated

directly into the forecasts as initial conditions and system

memory; nonlinear and other external forces can be integrated

into the numerical model; and, it is readily scalable to the multi-

body problem. MARIN’s workhorse code, aNySIM, uses a

similar Cummins approach, finding utility in this simple,

efficient, and robust formulation (Serraris, J.J., 2009).

The coefficients and functions that make up Eqn. (1) can be

computed via numerical simulations or obtained from model

tests. We use the AEGIR numerical seakeeping code to provide

initial estimates of these values, and utilize model- or full-scale

ship data to fine tune them. AEGIR is a time-domain Rankine

code that uses a high-order B-spline discretization of the hull(s)

and free surface to solve the three-dimensional potential-flow

around a ship (Kring et al, 2004). It has been used extensively by

APS and others to simulate the response of a variety of mono-

hull and multi-hull vessels operating alone or in close proximity

to one another.

The added inertia and hydrostatic restoring terms are

calculated by solving a double body problem in AEGIR with a

NURBS representation of the ship geometry and the assumed

mass characteristics. The IRF and Force RAO functions are

calculated with AEGIR for discrete values of ship speeds and

wave frequencies and directions, yielding a database which

characterizes the hydrodynamic forcing to the ship for all

relevant operating conditions. The forcing function databases are

interpolated for the particular operational and environmental

conditions. This approach assumes linear seakeeping theory

where the hydrodynamic forcing can be decomposed into the

incident wave, diffraction and radiation forces (Newman, 1977).

The forces are assumed to scale linearly with wave amplitude

and allow superposition over a collection of wave components.

External Models The coefficients of Equation (1) that are calculated with

AEGIR provide a linear potential-flow reduced-order model for

ship seakeeping. This is a very good model for predicting the

ship response in heave and pitch, modes that are dominated by

potential-flow effects. However, viscous effects not captured in

AEGIR play a significant role in the dynamics in roll, surge,

sway, and yaw responses. To obtain an accurate ROM for all

modes, appropriate external models for these other important

effects, with particular emphasis on roll, must be included. Many

of these are approximate models derived from the ship geometry.

Therefore, the associated coefficients must be tuned using model

test data and/or full scale results to fully capture the properties of

a particular ship. The extended form of the ROM equation is:

𝑚𝑖𝑗�� + ∑ 𝑎𝑖𝑗��𝑗

6𝑁

𝑗=1

+ ∑ 𝑏𝑖𝑗 ��𝑗

6𝑁

𝑗=1

+ ∑ 𝑏𝑈𝑖𝑗𝑈��𝑗

6𝑁

𝑗=1

+

∑ 𝑏𝑄𝑖𝑗|��𝑗|��𝑗

6𝑁

𝑗=1

+ ∑ 𝑐𝑖𝑗𝑥𝑗

6𝑁

𝑗=1

+ ∑ 𝑐𝑈𝑄𝑖𝑗𝑈2𝑥𝑗

6𝑁

𝑗=1

=

∑ ∫ 𝐾𝑖𝑗(𝑡 − 𝜏, 𝑈)��𝑗

𝑡

−∞

6𝑁

𝑗=1

(𝜏)𝑑𝜏 +


∑ ∑ 𝐷��(𝜔. 𝛽, 𝑈)𝑒𝑖𝜔𝑡��(𝜔, 𝐵) + ∑ 𝐹𝑖𝑘𝑘𝛽𝜔 (2)

This general form allows quadratic damping, linear-speed-

dependent damping, and quadratic-speed-dependent restoring.

Instead of forming the equations with equivalent-linear

coefficients, our time-domain application allows direct inclusion

of the nonlinear form. This offers an advantage over a frequency-

domain approach which would require use of ever-changing

equivalent-linear coefficients.

The new terms associated with viscous forcing in the roll,

surge, sway, and yaw modes are associated with appendages,

bilge keels, hull circulation, and propulsor effects. The

functional form and initial evaluation of the associated

coefficients were determined by drawing upon previous

analytical and experimental studies (Schmitke, 1978), (Himeno,

1981) (McTaggart, 2004, 2005), (ITTC, 2011). While many of

these previous works derive equivalent linear forms, our

coefficient calculation was for the nonlinear form that is used

directly in our model.

Anti-Roll Tank The ESMF Phase 1B demo was performed on the R/V

Melville, a ship which includes an anti-roll tank system to

improve the seakindliness of the vessel. The scale model used in

the tank testing did not include a roll tank, so all validation was

performed for the ship without one. The ROM used at sea in the

Phase 1B demo was enhanced to include an anti-roll rank system.

Preliminary numerical studies showed that the roll tank, as

expected, significantly reduced the roll response. The ship

equations of motion were supplemented with an additional

equation of motion to represent the passive operation of an anti-

roll tank system (Gawad et al, 2001; Youssef et al, 2002). This

equation, solved for the angle of the U-tube water level relative

to the ship,, is

𝑎𝜏𝜏�� + 𝑏𝜏𝜏�� + 𝑐4𝜏𝜏 + (𝑎𝜏4��4 + 𝑐𝜏4𝑥4) = 0 (3)

where x4 is the roll mode. The anti-roll tank dynamics are

coupled to the roll equation of motion (Eqn. 1) by adding the

term

(𝑎4𝜏�� + 𝑐4𝜏𝜏) (4)

on the left-hand side of the roll equation of motion. The

additional coefficients were derived from the specifications

provided for the R/V Melville installation of the Intering anti-

roll tank system. Simulated testing of the effect of the system

predicted a substantial reduction in the roll response associated

with inclusion of the roll tank, as can be seen in the example

simulation of Figure 1. Roll amplitude is reduced by

approximately 50%, indicating the importance of the anti-roll

tank model to the overall ship roll dynamics. The implemented

roll tank model was only a partial representation of the Melville

system because it did not include an active control model for the

valves designed to alter the dynamics of the system under certain

conditions. This is but one example of the model uncertainty

which suggests the need for tuning parameters associated with

the roll response in any full-scale ship motion forecasting model.

Figure 1 – Impact of the roll tank on RV Melville roll

response in Sea State 4 beam seas.

WIGLEY-BARGE SIMULATIONS The Wigley-Barge two-ship configuration was original

studied experimentally in a wave basin (Kashiwagi et al 2005)

and has subsequently been the subject of several computational

studies, including Xiang and Faltinsen (2009) who validated a

3D Rankine source panel method against the measured added

mass and damping forces. The characteristics of each model are

summarized in Table 1. Both vessels have the same length, beam

and draft, but the displaced volume of the Wigley, with its finer

ends and smaller waterplane, is only about half of the

displacement of the barge.

Table 1. Principal characteristics of Wigley-Barge ship models.

Wigley Barge

L 2.0 2.0

B 0.3 0.3

T 0.125 0.125

S 0.416 0.60

0.04205 0.075

The models were separated by 1.097 m for the experimental

diffraction study and by 1.797 m for the experimental added

mass/damping study, corresponding to non-dimensional

separations, s/L, of 0.5485 and 0.8985, respectively. While this

is larger than the anticipated separation between two vessels in a

skin-to-skin sea base configuration (s/L ~ 0.02-0.05), it is

representative of vessel separation in the approach phase, and the

physics captured in the Wigley-Barge simulations remain

relevant even though the separation is greater.

In Figure 1 we show the basic two-vessel configuration from

the model tests, with a rectangular barge and modified Wigley

hull arranged side-by-side, separated by a distance S between the

centerlines of the two models. In the diffraction study, waves

were incident from the beam of the Wigley, which was located

on the weather side of the barge. The motions of both vessels

were constrained, and forces and moments were measured. In


the added mass and damping study, the Wigley hull was forced

in either the heave (vertical) or sway (side-to-side) directions

while the barge was constrained. No incident waves were used.

The heave and sway forces on both vessels were measured. The

models were not translated with a mean forward speed in either

type of measurement.

Figure 2. Wigley-Barge Diffraction Set-up.

Linear Diffraction Forces As a first validation case we examine the non-dimensional

heave and sway forces due to linear diffraction, using small

amplitude incident waves approaching from the beam of the

Wigley hull. Separate AEGIR simulations were performed for

six wave frequencies that were selected for the study. The period

of the incident wave ranged from 0.715 to 1.454 seconds, which

corresponds to normalized wavelengths, /L, ranging from 0.4

to 1.65. The size of the computational domain and other

numerical parameters in the computations were set to capture the

longest waves within the central part of the free surface

computational domain without artificial damping. The free

surface grid density was selected to resolve the smallest waves

of interest.

In Figure 3 we compare the computed and measured non-

dimensional heave force acting on the Wigley hull. The top row

presents the magnitude of the force, and the bottom row presents

the response phase relative to the phase of incident wave. Both

quantities are plotted as a function of the normalized wavelength.

Each image shows measurements (red circles), the Aegir-

computed quantities (black squares), and results from

Kashiwagi’s panel code (blue lines). The force amplitudes have

been normalized by 𝜌𝑔𝜁𝑆𝑖. The correlation between the AEGIR-

computed results, the experimental results, and other

computations is very good in general. AEGIR is able to capture

the sheltering effect of the Wigley, which is manifested as

smaller forces on the barge compared to the Wigley. The code

also captures the resonance conditions associated with waves

sloshing between the two models which appear as force peaks

and rapid changes in phase like at /L of 0.6. We chose not to

simulate the longest waves because Kashiwagi et al (2005)

believe these data may be affected by reflections off of the tank

walls.

Figure 3. Magnitude (top) and phase (bottom) of

normalized heave force – Wigley

We compare the computed and measured non-dimensional

sway force acting on the Wigley in Figure 4. Again, the

correlation between the AEGIR predictions and other results is

very good.

Added Mass and Damping Forces The Kashiwagi data set also includes frequency-dependent

added mass and damping for the configuration with the Wigley

hull forced harmonically in heave or sway. We replicated these

experiments in AEGIR using two different simulation

approaches. First, we directly simulated the experiment, forcing

the Wigley hull harmonically in heave or sway. With this

approach, a very extensive suite of simulations is required to

compute added mass and damping over the entire frequency

range of interest because each frequency uses a separate AEGIR

run. Added mass force is the component of the hydrodynamic

force that is 180o out of phase with the acceleration; damping

force is component that is 180o out of phase with the velocity. A

second, and more efficient approach, uses impulse response (IR)

calculations to first compute time series of the forces associated

Incident wave direction

s

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1Heave Wigley

/L

Ma

gn

itu

de

Kashiwagi Calculation

Experiment

Aegir

0 0.5 1 1.5 2 2.5 3-150

-100

-50

0

50

100

150

200

/L

Ph

as

e


Experiment

Aegir

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7Heave Barge

/L

Ma

gn

itu

de


Experiment

Aegir

0 0.5 1 1.5 2 2.5 3-200

-150

-100

-50

0

50

100

150

200

/L

Ph

as

e


Experiment

Aegir

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1Heave Wigley

/L

Ma

gn

itu

de


Experiment

Aegir

0 0.5 1 1.5 2 2.5 3-150

-100

-50

0

50

100

150

200

/L

Ph

as

e


Experiment

Aegir

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7Heave Barge

/L

Ma

gn

itu

de


Experiment

Aegir

0 0.5 1 1.5 2 2.5 3-200

-150

-100

-50

0

50

100

150

200

/L

Ph

as

e


Experiment

Aegir


with wave radiation. In our implementation of the IR

calculation, we apply an impulsive velocity to the body in a

particular mode at t=0. The free surface boundary conditions and

hull forces are advanced in time using AEGIR. Estimates of the

impulse response functions are found by numerically

differentiating the computed force and moment time series with

respect to time. Added mass and damping forces are related to

the impulse response functions by the Kronig-Kramers relations

(Perez and Fossen, 2008):

A(ω) = A∞-1

ω∫ K(t) sin(ωt) dt

∞

0 (5)

𝐵(𝜔) = ∫ 𝐾(𝑡) cos(𝜔𝑡) 𝑑𝑡∞

0 (6)

Figure 4. Magnitude (top) and phase (bottom) of

normalized sway force - Wigley One impulse response simulation is performed for each free

mode of motion. For the Wigley-Barge validation study

presented here, only a single IR for heave motion was required.

K(t) is more commonly found from an inverse transform of

the damping and added mass coefficient. One advantage of

transforming the directly-computed K(t) to estimate the

frequency-dependent added mass and damping is that the low

and high frequency limits are directly captured without requiring

an asymptotic form. Resolution of the low-frequency A() and

B() depends on the size of the free surface computational

domain, whereas resolution at high frequencies depends on the

density of the free surface grid.

In Figure 5 we present the non-dimensional, frequency-

dependent heave added mass for the Wigley hull as a function of

the non-dimensional wave length, kL, where k is the wave

number of a surface gravity wave of frequency and L is the

ship length. The added mass is normalized by the ship mass 𝜌∇

after 𝐴∞ was subtracted. Experimental data are shown as red

circles, results from the AEGIR forced harmonic motion

simulations are shown as black squares and the transformed

results from the AEGIR impulse response simulations are shown

by the blue lines with crosses. The correlation between the

computed and experimental values is very good over the entire

frequency range.

In Figure 6 we present the non-dimensional, frequency-

dependent heave damping coefficient for the Wigley hull as a

function of the non-dimensional wave length, kL. The damping

coefficient is normalized by the product of the ship mass and

forcing frequency 𝜌∇𝜔. The experimental data are shown as red

circles, the results from the AEGIR forced harmonic motion

simulations are shown as black squares and the transformed

results from the AEGIR impulse response simulations are shown

by the blue lines with crosses. The correlation between the

computed and experimental values is again very good over the

entire frequency range. The resonances in the damping curve at

kL ~ 3π, 5π, 7π,…(2n-1)π, (n = 1,2,3…) corresponding to

longitudinal sloshing modes along the ship length are nicely

captured in the computations and match the experimental

observations.

The correlation between the computed and observed added

mass and damping coefficients is also very good. By running

both impulse response and forced harmonic motion simulations

we were able to verify the equivalence between the two

approaches for computing and representing the radiation (added

mass and damping) forces. The impulse response approach is

preferred because it allows for much more efficient calculation

of the databases for radiation loading. A single simulation is

required for each free mode of motion (12 modes total for each

two-ship configuration with a specific orientation) instead of

many simulations for each mode to capture a wide range of

frequencies.

The Wigley-Barge impulse response simulations also

provide insight into the computational requirements for the

AEGIR simulations. In particular, the size of the free surface

grid controls the low frequency end of the radiation forces. The

free surface domain must be large enough to capture the longest

waves that will excite large ship. The AEGIR free surface grid

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2Sway Wigley

/L

Ma

gn

itu

de


Experiment

Aegir

0 0.5 1 1.5 2 2.5 3-150

-100

-50

0

50

100

150

200

/L

Ph

as

e


Experiment

Aegir

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4Sway Barge

/L

Ma

gn

itu

de


Experiment

Aegir

0 0.5 1 1.5 2 2.5 3-200

-150

-100

-50

0

50

100

150

200

/L

Ph

as

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Experiment

Aegir

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2Sway Wigley

/L

Ma

gn

itu

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Experiment

Aegir

0 0.5 1 1.5 2 2.5 3-150

-100

-50

0

50

100

150

200

/L

Ph

as

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Experiment

Aegir

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4Sway Barge

/L

Ma

gn

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Experiment

Aegir

0 0.5 1 1.5 2 2.5 3-200

-150

-100

-50

0

50

100

150

200

/L

Ph

as

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Experiment

Aegir


spacing, equivalently the number of free degrees of freedom on

the free surface, controls the high frequency behavior of the

radiation forces. A grid must provide enough degrees of freedom

to resolve multiple longitudinal wave modes and the first

transverse mode in the gap between ships. Higher order

transverse gap modes are not likely important in skin-to-skin

configuration because they corresponds to very high frequencies.

They may, however, be important during the approach and break-

away phases. Finally, the length of the impulse response

simulation implicitly sets the frequency resolution of the

radiation forces. The duration must be sufficient to capture the

resonances associated with the longitudinal and transverse wave

modes in the gap between the two vessels. The temporal and

spatial requirements for AEGIR impulse response simulations

are similar to the requirements for AEGIR free motion

simulations, further confirming that the same physics are being

captured in the ROM databases.

Figure 5. Added mass coefficient for the Wigley.

Figure 6. Damping coefficient for the Wigley

RV MELVILLE SIMULATIONS A series of model tests were performed in the Maneuvering

and Seakeeping Basin (MASK) at NSWCCD with a 1/23rd scale

model of the R/V Melville to provide validation data for ESMF

performers. The model was self-propelled with remote control.

Free-decay tests were performed for roll at zero speed, 8 knots,

and 12 knots (full scale). Seakeeping response of the model to

incident waves at a range of frequencies and from a range of

relative directions, both at 8 knots and 12 knots. Tests were

performed in regular and irregular waves over a range of

frequencies, headings and speeds to provide time series of roll

angle, pitch angle and wave elevation. Time series of the

accelerations were also recorded for the other modes. The

filtered heave acceleration signal was integrated twice in time to

calculate time series of the vertical (heave) displacement.

Three different views of the model are shown in Figure 7.

Figure 7. Model 5720 tethered to the MASK carriage.

Reproduced from Minnick et al, (2012).

Validation Against Tank Data We first consider the validation of the ROM motion

predictions against the model test data for heave and pitch

response. Motion RAOs for heave and pitch are shown for head

seas at 8 and 12 knots in Figure 8 and Figure 9, respectively. The

differences between the ROM prediction and model test data are

shown in Table 2.

Table 2. Differences between ROM prediction and model test data

Heave

Avg. Diff

Heave

Peak Diff

Pitch

Avg. Diff

Pitch

Peak Diff

8 kts 9% 4% 6% 8%

12 kts 4% 6% 9% 21%


Figure 8. Computed and measured Heave RAO.

Figure 9. Computed and measured pitch RAO.

The average difference of the predicted values to those

found in the test is less than 10%. A comparison of the pitch

response in the time domain is shown for an irregular-wave head

seas case in Figure 10, again showing good comparison between

the ROM predictions and model tests. To run the irregular-wave

case in the ROM we used a Fourier Transform on the measured

time history of wave elevation from the model test, and used the

complex amplitudes of these Fourier wave components as the

input wave field in the ROM simulation. The performance of the

ROM model without tuning is consistent with expectations for

heave and pitch, as these modes should be largely dominated by

the potential flow effects that are accounted for in the wave and

impulse response databases.

Figure 10. Time domain pitch comparison for

irregular head seas case at 8 knots.

T-AKR 300/T-AK 3008 SKIN-TO-SKIN SIMULATIONS Numerical simulations using AEGIR and a two-ship ROM

were performed for the USNS Bob Hope (T-AKR 300), a Large

Medium Speed Roll on/Roll off (LMSR) vessel, and the USNS

Lt. John P. Bobo (T-AK 3008), a Marine Prepositioning vessel.

Both vessels are operated by the Military Sealift Command

(MSC). The objectives of this study were to verify the ROM

against AEGIR for a realistic two-ship sea base configuration

and to validate the simulations against model scale data.

The scale-model data set was obtained from seakeeping

measurements that were conducted at MARIN in late 2005 for

the U.S. Office of Naval Research using 1/45th scale models of

the T-AKR 300 and T-AK 3008. This test sequence included

measurements in regular and irregular waves for a range of

separations (longitudinal and transverse) between vessels,

speeds, and relative wave headings. A photograph of the two

ship models is shown in Figure 11. All of the results presented

herein are for a full-scale separation of 3 m, which is typical of a

skin-to-skin arrangement. These tests focused on the

hydrodynamic interactions between vessels due to wave

diffraction and the local hydrodynamic pressure field because

mooring lines and fenders were not used in these tests.

Principal particulars for the two vessels are summarized in

Table 3.

Testing was done using a specialized carriage system that

permitted measurements of time series of the heave and pitch

displacements of each ship model. Time series of the

hydrodynamic forces and moments were measured on both ship

models for the blocked degrees of freedom.

AEGIR was previously validated against data from these

measurement as part of a code evaluation study conducted by


NSWCCD (Silver et al, 2008). In general, the magnitude and

phase of the AEGIR-computed motion and force RAO compared

favorably to the experimental data and performed as well or

better than the other codes in the evaluation. However, the grid

resolution used in the study was limited because a 32-bit AEGIR

executable was used by the authors. This limitation is overcome

in the present study, allowing us to more fully explore

convergence with respect to the resolution of the free surface.

Figure 11. USNS Bob Hope (right) and USNS Lt. John P. Bobo (left) models in the MARIN tank. Reproduced

from Silver et al (2008)

Table 3. Principal Particulars of the USNS Hope and USNS Bobo.

T-AKR 300 T-AK 3008

LWL 279.16 181.69

B 32.26 32.16

TF 7.83 5.19

TA 9.76 8.18

49,168 30,034

LCG aft of FP 125.35 89.88

VCG 11.55 11.46

KYY 66.55 45.9

AEGIR Simulations Numerical Set-up A suite of AEGIR simulations were

conducted with the two ships sailing side-by-side at 5 knots with

3m separation (full scale) as part of our verification and

validation study. The two vessels were aligned midship-to-

midships, a configuration that is representative of two ships

moored together skin-to-skin in a sea base. However, the tests

did not include any mechanical connections (e.g. mooring lines

and fenders) between vessels so the ship-to-ship coupling is

purely hydrodynamic in nature. The run suite included free

motion simulations, where the potential flow and ship equations

of motions were integrated in time using AEGIR, as well as

diffraction and impulse response simulations to construct a

partial database for the ROM. This approach allows for direct

comparison of the two numerical methods, offering the chance

to directly demonstrate that the ROM retains the numerical

accuracy of the more computationally-expensive AEGIR

simulations.

The two vessels were centered within a 1 km x 1 km square

patch of free surface with the USNS Bob Hope to starboard of

the USNS Bobo as shown in Figure 12. The grid dimensions

were selected to support the longest wave of interest (~360m)

while also accommodating a numerical absorbing layer to

enforce a radiation condition along the outer boundary of the free

surface computational grid. For headings less than 180 deg, the

USNS Hope was upwave of the USNS Bobo, while for headings

greater than 180 deg, the USNS Bobo was the upwave vessel.

We explored convergence with free surface discretization as

part of our initial numerical studies, finding improved correlation

between the predicted and measured RAO for a few frequencies

and headings with the finer free surface grid (~3 m resolution)

compared to the coarser free surface (~5 m resolution). All of

the results presented here are based on computations on the finer

grid. Both of these grids were larger and better resolved than the

grid used in the prior NSWCCD study.

Figure 12. Computational Grid for TAKR-300/T-AK

3008 simulation.

Comparison against Experimental Data We compare

the AEGIR-computed and measured heave, pitch and roll

moment RAO for both vessels in this section. Free motion

simulations were run for following (0 deg), quartering (45 and

315 deg), beam (90 deg), bow (135 deg) and head (180 deg) seas

using a six-component spectrum of long-crested waves. The

wave amplitudes and phases were selected to correspond to the

regular wave measurements at each heading. The magnitude and

phase of the RAO were computed from discrete Fourier

transforms of the motion (force) and wave elevation time series

at the encounter frequency. In Figure 13 and Figure 14 we show

the magnitude of the head seas pitch and heave RAO for the two

vessels. Heave RAO is expressed in units of m/m; pitch RAO is

expressed in units of deg/m. The relative errors, defined by Eqn.

7, for the heave and pitch RAO are shown in Figure 15 and

Figure 16. In general, the correlation between the computed and

measured response is very good across the entire frequency

range, although it is slightly better for heave than pitch, and

slightly better for the USNS Hope than for the USNS Bobo.


𝜀 =𝑋𝑐−𝑋𝑀𝑒𝑎𝑠

max𝜔,𝛽

(𝑋𝑀𝑒𝑎𝑠) (7)

Figure 13. Head seas pitch RAO for the USNS Hope

(top) and USNS Bobo (bottom).

Figure 14. Head seas heave RAO for the USNS Hope

(top) and USNS Bobo (bottom).

Comparison between AEGIR and the ROM The ROM is

designed to be a forecasting model for the ESMF system, and

therefore must retain the accuracy of established high fidelity

codes like AEGIR, but run at a small fraction of real time. We

have found that a 300-second forecast with the ROM takes about

0.4 sec for a single ship and about 0.8 sec for the USNS

Hope/USNS Bobo two-ship configuration on a desktop

workstation. The equivalent highly-resolved AEGIR two-ship

calculation takes approximately 105 times longer.

The accuracy of the ROM is related to the quality of the

wave force RAO and impulse response functions, which in turn

can be tied back to the accuracy of the AEGIR diffraction and

impulse response computations used to generate the databases.

In Figure 17 we demonstrate the accuracy of the ROM by

comparing time series of the vertical displacement at the deck

edge near the side port ramp from the two codes. Heave, roll and

pitch all contribute to the displacement of this point which is

forward and outboard of the ship center of gravity. In this

particular example, the ships are translated at 5 knots in fully-

developed bow seas with a significant wave height of 2.4m and

cosine-squared spreading. Heave, roll and pitch of each ship

were active; the other degrees of freedom were held fixed. The

ROM reproduces AEGIR very well. The phase of the two

signals are well correlated over the run duration, and the RMS of

the amplitude is within about 10%.

Figure 15. Relative Error of Heave RAO.

Figure 16. Relative Error of Pitch RAO.

SUMMARY AND RECOMMENDATIONS The ESMF system must include an accurate and

computationally-efficient ship motion model to forecast time-

accurate motions of one or more ships operating in a seaway.

These requirements are achieved with a reduced order model

based on the generalization and extension of the Cummins

(1962) formulation. Our numerical implementation


accommodates ingestion of the measured motion history via the

convolution integral so that each new forecast uses the recently

observed motion history in the estimation of the radiation forces.

The forces associated with the seaway, both incident and

diffracted, are estimated by combining observations of the waves

with a force Response Amplitude Operators. For the validations

presented herein, these measurements were made using wave

probes. In the system deployed at sea, waves are observed using

a specialized Doppler radar. Details on the radar, overall system

and its performance in a full-scale trial are presented in a

companion paper by Connell et al (2015).

Figure 17. AEGIR and ROM comparison for USNS

Hope and USNS Bobo Operating Side-by-Side.

Our time-domain formulation accommodates the

introduction of nonlinear and external dynamics models. These

terms have the biggest impact on the forecasts of modes of

motion that are dominated by viscous effects and not especially

good at creating waves. They are also included to capture the

dynamics of other ship subsystems which influence the overall

motions of the ship. For example, an anti-roll tank model is used

in the ROM for the full-scale R/V Melville. We are currently

testing a mooring system model to capture the mechanical loads

associated with mooring lines and fenders that are deployed in a

skin-to-skin two-ship configuration. The mooring model is

nonlinear, and like other parts of the ROM, includes coefficients

that must be tuned in situ because of uncertainty in mooring line

material properties.

Our numerical studies with the ROM show that it can

reproduce the predictions of a higher fidelity time-domain

seakeeping code at a very small fraction of the computational

cost. The accuracy of these forecasts depends on the quality of

the underlying force databases which have been generated with

AEGIR. Solutions obtained with both AEGIR and the ROM

correlate well with the model-test data for the single and two-

ship configurations presented herein.

ACKNOWLEDGMENTS This material is based upon work supported by the Office of

Naval Research (Dr. Paul Hess III, Program Manager) under

Contract No. N00014-11-D-0341. Any opinions, findings and

conclusions or recommendations expressed in this material are

those of the author(s) and do not necessarily reflect the views of

the Office of Naval Research.

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