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]
2 Copyright © 2015 by ASME
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 +
3 Copyright © 2015 by ASME
∑ ∑ 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
4 Copyright © 2015 by ASME
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
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5 Copyright © 2015 by ASME
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
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6 Copyright © 2015 by ASME
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%
7 Copyright © 2015 by ASME
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
8 Copyright © 2015 by ASME
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.
9 Copyright © 2015 by ASME
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
10 Copyright © 2015 by ASME
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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