numerical study of two vessels seakeeping in waves

Copyright © 2014 by ASME

NUMERICAL STUDY OF TWO VESSELS SEAKEEPING IN WAVES

Dexin Zhan Oceanic Consulting Corporation

St. John’s, NL, Canada

Don Bass Don Bass Marine Consulting


David Molyneux Oceanic Consulting Corporation


ABSTRACT This paper presents a numerical study of seakeeping in regular

waves for two vessels in close proximity using commercial

seakeeping software HydroStar and an in-house code

MOTSIM. The objective was to study the possible sheltering

effect of the larger vessel (FPSO) on the smaller one (OSV)

during personnel transfer between the two vessels, where one

vessel was at some angle relative to the other vessel and there

was no connection line between them. The study mainly

focused on the OSV motion resulting from the interaction of the

FPSO when the OSV was at different headings and wave

directions. Initially the OSV motions close to the FPSO (and

parallel) were compared with those for the OSV alone. For an

un-parallel position of the two vessels, an objective function

based on the OSV RAOs motion in roll, pitch and heave

directions was used to optimize the OSV position. Finally

comparisons between HydroStar and MOTSIM results are

provided.

The main conclusions are:

1) When the FPSO and OSV are located in parallel, the OSV

motions in sway, roll and yaw are larger than the single OSV

motions in head waves while surge, heave and pitch are almost

the same. The OSV motions in most of the six degrees of

freedom are smaller than the single OSV motions when the

waves are from other directions (always on the port side of the

FPSO), which means that there is a sheltering effect.

2) The simulation results from different OSV rotation angles

show that the hydrodynamic interaction between the FPSO and

OSV e.g. the sheltering effect is related to the OSV angle and

the wave heading. The objective function in roll, pitch and

heave RAOs indicates that the OSV should maintain a close to

parallel position with the FPSO to minimize motion when the

waves come from the port side of the FPSO from 180 to 240

degrees. When the wave direction is around 240 degrees the

OSV should have relatively small motion in waves for any OSV

rotation angle.

3) A comparison of HydroStar and MOTSIM results shows

that the MOTSIM results of a single vessel seakeeping

simulation is in a good agreement with HydroStar. In two

vessels situation more validation work needs to be done.

1. INTRODUCTION

There are many hydrodynamic interactions between bodies in

waves in oceanic engineering. For example, a shuttle tanker

may be alongside a FPSO while it takes on oil, or an offshore

supply vessel may be alongside a larger vessel which would be

taking on supplies. Another example is an underway

replenishment of long-term naval operations where two ships

travel in close proximity. One of the activities specific to

offshore operation is the transfer of personnel between support

vessels and other offshore structures. In the process of crew

transfer the two vessels/structures could be in different positions

or angles depending on the specific environment (such as wave,

wind or current). In some cases, the larger vessel is used to

provide a sheltering effect which can improve the likelihood of

a successful transfer. Therefore a study of the hydrodynamic

interaction between the two vessels is necessary and important.

In comparison to the single ship case, numerical seakeeping

simulation of the two-ship case is more complex because it has

12 degrees of freedom. Furthermore, hydrodynamic terms such

as added mass, damping, and wave diffraction force must

account for the presence of two ships in the flow field.

McTaggart et al. (2003) developed a frequency domain code

SHIPINT to predict motions in waves of two ships in close

proximity. A seakeeping experiment was conducted for

validation of this method using two semi-captive models in a

towing tank. In their research the two vessels were restricted to

a parallel position to model the replenishment process. Chen

and Fang (2001) also investigated hydrodynamic interactions

between two ships using a three-dimensional potential-flow

theory based on the source distribution technique. Islam (2013)

carried out a numerical study of dynamic interaction of parallel

moving ships in close proximity using speed dependent Green

function. A significant effect of the sway, roll and yaw for two

parallel moving ships was revealed. At present, research reports

of two un-parallel ships are few. In this paper the sheltering

1

Proceedings of the ASME 2014 33rd International Conference on Ocean, Offshore and Arctic Engineering OMAE2014

June 8-13, 2014, San Francisco, California, USA

OMAE2014-23269

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effect between vessels in the process of personal transfer was

studied where the one vessel could be at some angle relative

another vessel. An optimization function based on the roll, pitch

and heave motion RAOs is presented to evaluate the supply

vessel movement at different rotation angles and wave

directions. Commercial software HydroStar and an in-house

code MOTSIM were used and their results were compared.

2. METHODOLGY FOR TWO VESSELS SEAKEEPING

2.1 HydroStar Software HydroStar is the commercial hydrodynamic software which has

been developed by Bureau Veritas since 1991. It provides a

complete solution of the first order problem of wave diffraction

and radiation and also the QTF for second order low-frequency

wave loads on a floating body with or without forward speed in

deep water and in finite water depth. Even though the classical

potential theory is applied, addition damping terms are included

in HydroStar to take into account energy dissipation that does

not exist when the inviscid and irrotational flow are assumed.

The HydroStar modules are shown in Figure 1, where the

commonly used modules are hslec (read geometry mesh), hsrdf

(compute radiation and diffraction), hsmec (compute ship

motions), and hsrao (output RAO files) etc. In version 7.02 of

HydroStar any number of bodies can be modeled [4]

. In multi-

body simulation the number of bodies and the first and last

identification numbers of the panels associated to each body are

defined in the input file while the coordinates and panels of

each body can be defined separately. The bodies’ relative

position are defined by two keywords TRANS and ROTA,

where the TRANS means translation in x- and y- direction, and

ROTA means rotation of the body mesh in the horizontal plane.

More details are given in Reference [4]. An example of two

vessel’s meshes and position for the FPSO and OSV in a

parallel position is displayed in Figure 2.

It should be noted that the CG positions of bodies also need to

be modified in the body motion input file when these vessels

change their relative position. For each translation and rotation

the new CG position of x and y coordinates of the vessels

should be recalculated by user. The matrix of inertia and

moments also needs be updated to respond to the global

coordinate system if you make a body rotation.

In HydroStar a RAOs file of the multi bodies is output when

running the hsrao module. The RAOs in directions of surge,

sway, heave, roll, pitch and yaw are relative to the global

coordinate system in the HydroStar software.

In this paper the main interest is focused on the OSV motion

under the potential sheltering effect of the FPSO. To compare

the OSV RAO motion at different rotation angles it is better to

express these RAOs in the OSV local coordinate system. In

order to solve the problem, instead of rotating the OSV we keep

the OSV fixed and translate/rotate the FPSO in the geometry

input file. In the process of translating and rotating the FPSO

the CG distance between the FPSO and OSV is kept the same as

the CG distance when translating/rotating the OSV. Meanwhile

the wave heading was also rotated with the FPSO to correspond

the wave from correct side of the FPSO. According to a

principle of relative motion the simulation results for these

movements are the same. But keeping in mind all the OSV

RAO results are relative to the OSV local coordinate system

when rotating the FPSO. For convenience, we still display the

position figure of the FPSO and OSV using rotating OSV in the

following section of this paper.

Figure 1: HydroStar Modules

Figure 2: Example of FPSO and OSV Bodies Meshes in

Parallel for HydroStar

2.2 MOTSIM Code

MOTSIM is a nonlinear time domain panel method based on a

weak scatterer hypothesis developed by Professor Don Bass in

association with NRC/IOT [5]

. Recently an extended version of

MOTSIM was developed in order to determine motions for two

vessels in close proximity.

The new version, MOTSIM12, basically doubles everything

that the older (six degrees of freedom) version does. For

example the ‘single’ version solves 21 equations, while the

‘double’ version solves 42 equations. The same ODE solver is

2



used and practically all the routines in the six degrees of

freedom version are modified in a straight forward manner for

the 12 degrees of freedom case. The most significant changes

occur in the routines for memory function and scattering forces

calculation. They are changed significantly because the

hydrodynamics of two vessels oscillating side by side may

differ significantly from those for the individual oscillating

vessels. The hydrodynamics of two vessels operating side by

side are represented (in linear theory) by four sets of

hydrodynamic coefficients. If the vessels are named A and B,

there are four coefficients as follows:

(1) Added mass and damping coefficients for A due to the

oscillations of A in the presence of the vessel B at rest;

(2) Added mass and damping coefficients for A at rest with B

moving;

(3) Added mass and damping coefficients for B for B at rest

and A moving and;

(4) Added mass and damping coefficients for B moving in the

presence of A that is at rest.

From these coefficients, four added mass and damping files are

derived. The coefficients are calculated in a modified version of

the code Green3D using panel data from vessels A and B.

Memory functions for each of these four cases are derived using

Fourier series from the damping coefficients taken over a range

of 30 or 40 frequencies. The memory functions are integrated

with the appropriate accelerations using convolution integrals to

determine the inertial and scattering forces and moments acting

on the two vessels. Sectional accelerations and ‘average’

accelerations in the various modes and cross modes are

determined for these four cases according to their memory

functions. Similar convolution integrals are determined for

forward speed conditions using sectional or ‘averaged’

velocities instead of accelerations.

In order to overcome the drifting motion of the vessels in

MOTSIM, tethering lines are used to restrict vessels’ drifting. It

should also be noted that the OSV surge, sway and heave

motion in MOTSIM were transformed from global coordinate

to the OSV local coordinate in a post-processor in order to

compare with HydroStar result when the OSV had a rotation

angle.

3. SIMULATION PARAMETERS AND DEFINITIONS 3.1 Vessel particulars

Principal dimensions for the two vessels, Floating Production,

Storage and Offloading (FPSO) vessel and Offshore Supply

Vessel (OSV), are listed in Table 1. Only the loaded condition

was considered for the FPSO.

3.2 Definition of FPSO and OSV Positions

A global coordinate system OXYZ is established in Figure 3

which is the initial position of the FPSO, where origin O is

located at the water line and Z axis points upward. A local

coordinate system O’X’Y’Z’ for the OSV is also shown in the

figure. The translations in x and y directions are XT and YT, and

the rotation angle of OSV is α. As we mentioned in Section 2,

the RAO results of the OSV motion in this paper are relative to

the local coordinate system O’X’Y’Z’ for HydroStar and

MOTSIM.

Table 1: FPSO and OSV Principal Characteristics

Draft Units FPSO OSV

Lpp m 258 76.08

Breadth m 46 18

Draft m 18.35 6.47

Depth m 26.6 8.56

Displacement t 182409 6337

CG forward of

midship m 1.53 1.63

CG above keel m 15.33 6.7

GMt m 3.64 1.6

Kxx m 17.37 6.3

Kyy m 57.89 17.33

Kzz m 58.89 17.33

Figure 3: Definition of FPSO and OSV Positions

3.3 Wave heading definition

The wave heading angle definition is shown in Figure 4, where

the 1800 means head wave and 270

0 means beam wave from the

port. In this report four wave headings (180, 210, 240 and 270

degrees) are relative to the FPSO position, e.g. the global

coordinate system. And only regular waves were considered in

the simulation.

3



Figure 4: Wave Heading Definition

4. SIMULATION RESULTS AND DISCUSSION 4.1 FPSO and OSV in Parallel Position

When the FPSO and OSV are parallel, the OSV angle α in

Figure 3 is 0 degree. The translation values of XT and YT are

138.2m and -62m, respectively, which means the OSV midship

is 47.2m ahead of the FPSO midship and the lateral gap

between the two vessels is 30m. Four wave headings (head

wave to beam wave) were used in the simulation to study the

sheltering effect. This might be the most common operational

scenario in crew transfer. The RAOs for the OSV next to the

FPSO are compared with the results of a single OSV as follows.

In this paper, the RAO is defined by motion amplitude divided

by the wave amplitude (roll, pitch and yaw are in degrees).

Figure 5 shows the comparison of RAOs between double

vessels and single vessel at 180 degrees waves (head wave)

using HydroStar software. It can be seen that when the FPSO

was present, the OSV sway, roll and yaw motion had some

changes compared to a single OSV, while the surge, heave and

pitch motions were close to the single OSV results (the values

of sway, roll and yaw of single OSV are all close to 0). This is

due to hydrodynamic interaction between the two ships when

they are in parallel and close proximity. These effects were

predicted by other researchers in numerical modeling and

physical model experiments for two parallel moving vessels[1][3]

.

For other wave directions (from the port side of the FPSO),

simulation results show that the OSV RAOs in most of 6

directions are smaller than results of a single OSV, which

indicates that the sheltering effect is more obvious when the

waves come from the port side of the FPSO. In these wave

conditions the OSV is located in a wave shadow of the FPSO

and the FPSO creates an obstruction to the propagation of

waves. Figure 6 shows an example of OSV RAOs comparison

of wave heading 240 degrees for double vessels and single

vessel (other wave angles are not shown in this paper). The

solid line represents the OSV result for the double vessels and

the dashed line with delta symbol is the ones of single OSV.

Frequency (r/s)

Surg

eR

AO

s(m

/m)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Double, Wave 180 deg

Single, Wave 180 deg

OSV Surge RAOs, Wave 180

(a) Surge RAO

Frequency (r/s)

Sway

RA

Os

(m/m

)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1



OSV Sway RAOs, Wave 180

(b) Sway RAO

Frequency (r/s)

Hea

veR

AO

s(m

/m)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6



OSV Heave RAOs, Wave 180

(c) Heave RAO

4



Frequency (r/s)

Rol

lRA

Os

(deg

/m)

0 0.5 1 1.5 20

2

4

6

8

10



OSV Roll RAOs, Wave 180

(d) Roll RAO

Frequency (r/s)

Pitc

hR

AO

s(d

eg/m

)

0 0.5 1 1.5 20

1

2

3

4



OSV Pitch RAOs, Wave 180

(e) Pitch RAO

Frequency (r/s)

Yaw

RA

Os

(deg

/m)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1



OSV Yaw RAOs, Wave 180

(f) Yaw RAO

Figure 5: OSV RAO Comparisons for Double Vessels and

Single Vessel in Wave Angle 1800 (HydroStar)

Frequency (r/s)

Surg

eR

AO

s(m

/m)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1



OSV Surge RAOs, Wave 240

(a) Surge RAO

Frequency (r/s)

Sway

RA

Os

(m/m

)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1



OSV Sway RAOs, Wave 240

(b) Sway RAO

Frequency (r/s)

Hea

veR

AO

s(m

/m)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6



OSV Heave RAOs, Wave 240

(c) Heave RAO

5



Frequency (r/s)

Rol

lRA

Os

(deg

/m)

0 0.5 1 1.5 20

5

10

15



OSV Roll RAOs, Wave 240

(d) Roll RAO

Frequency (r/s)

Pitc

hR

AO

s(d

eg/m

)

0 0.5 1 1.5 20

1

2

3

4



OSV Pitch RAOs, Wave 240

(e) Pitch RAO

Frequency (r/s)

Yaw

RA

Os

(deg

/m)

0 0.5 1 1.5 20

0.5

1

1.5



OSV Yaw RAOs, Wave 240

(f) Yaw RAO

Figure 6: OSV RAO Comparisons for Double Vessels and

Single Vessel in Wave Angle 2400 (HydroStar)

4.2 FPSO and OSV with Relative Heading Angles to Waves

In this scenario the FPSO and OSV are in a position with a

translation and a rotation. The OSV rotation is relative to the

point O’, and the rotation angle α is defined as positive in

clockwise direction which could change from 0 to 180 degrees.

The XT and YT are still fixed at 138.3m and -62m, respectively.

In this situation a concern is focused on an analysis of sheltering

effect based on the all OSV motion RAOs in different OSV

rotating angles and wave directions.

As described above the crew transfer was conducted between

the FPSO and OSV. For a safe transfer in waves the smaller the

movement of the OSV, the lower the risk. According to the

transfer process, comparing with the OSV surge, sway and yaw

motions the roll, pitch and heave motions of the OSV play an

important role in the risk assessment. Normalized values of the

roll, pitch and heave motions were calculated by selecting the

maximum RAO motion in the frequency ranges divided by a

maximum value of all waves and OSV angles in the

corresponding motion. Then an objective function Y for an

optimization calculation based on the normalized data is created

as follows:

heavepitchroll YYY=Y

Where Yroll, Ypitch and Yheave are the normalized RAOs in roll,

pitch and heave, respectively. The Y is a function of the wave

direction and the OSV angle. Figure 7 shows the function Y

changes with wave headings while Figure 8 shows the function

Y changes with the OSV rotation angles (0~90 degrees).

From Figure 7 it can be seen that when the wave heading

changes from 180 to 240 degrees, the smallest value of Y is

appeared in OSV angle 0 degree. It means that the OSV should

keep in around 0 degree in order to keep a lower motion in roll,

pitch and heave directions when the waves come from 180 to

240 degrees. From Figure 8 it also can be seen that the lower

parts of function Y appears when the wave heading is 240

degrees, which means that comparing with the other wave

headings the sheltering effect is stronger when the waves come

from the 240 degrees whatever the OSV rotation angle is.

4.3 Comparison between HydroStar and MOTSIM Results

HydroStar and MOTSIM were developed by different

organizations. One is commercial software and another is an in-

house code. Their mathematic models applied to seakeeping

simulation are different. For example, a frequency domain

solver is used in HydroStar, a time domain is used in MOTSIM.

The most important is that the wave mean force (second order

wave force) was considered in MOTSIM, but HydroStar didn’t

calculate it in current RAO simulation in regular waves. The

wave drifting force affects ship’s motion in a considerable

range, especially for smaller frequency and higher amplitude

waves. This was observed in MOTSIM simulation when the

frequency is less than 0.3 rad/s and the simulation was

corrupted.

6



Figure 9 shows a validation comparison between MOTSIM and

HydroStar results for a single OSV simulation in wave 210

degrees. A good agreement was achieved for these RAO results.

Figure 10 shows a comparison for double vessels simulation

while both vessels are in the parallel position (OSV 0 rotation

angle). Results of four wave directions are provided in the

figure. The dashed line with delta symbol represents MOTSIM

result while the solid line is HydroStar result. From the figure it

can be seen that in most of six directions MOTSIM data are

larger than the ones of HydroStar, and this phenomenon is also

appeared in other OSV angles (not provided in this paper). It is

considered that the current version of MOTSIM12 does not take

into account the sheltering type diffraction effects. A new

modification is on the way.

Wave Angle (deg)

Nor

mal

ized

Obe

jctiv

eFu

nctio

nY

160 180 200 220 240 260 280 300

1

1.5

2

2.5 OSV Angle 0 degOSV Angle 30 degOSV Angle 45 degOSV Angle 60 degOSV Angle 90 deg

Figure 7: OSV Normalized Objective Function Y Changes

with Wave Angles

OSV Angle (deg)

Nor

mal

ized

Obj

ectiv

eFu

nctio

nY

0 20 40 60 80 1001

1.5

2

2.5

3

3.5

4

Wave Angle 180 degWave Angle 210 degWave Angle 240 degWave Angle 270 deg

Figure 8: OSV Normalized Objective Function Y Changes

with OSV Angles

Frequency (r/s)

Hea

veR

AO

s(m

/m)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

HydroStar, Wave 210 deg

MOTSIM, Wave 210 deg

Single OSV Heave RAOs Comparison

(a) Surge RAO

Frequency (r/s)

Sway

RA

Os

(m/m

)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2



Single OSV Sway RAOs Comparison

(b) Sway RAO

Frequency (r/s)

Hea

veR

AO

s(m

/m)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2



Single OSV Heave RAOs Comparison

(c) Heave RAO

7



Frequency (r/s)

Rol

lRA

Os

(deg

/m)

0 0.5 1 1.5 20

5

10

15



Single OSV Roll RAOs Comparison

(d) Roll RAO

Frequency (r/s)

Pitc

hR

AO

s(d

eg/m

)

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4



Single OSV Pitch RAOs Comparison

(e) Pitch RAO

Frequency (r/s)

Yaw

RA

Os

(deg

/m)

0 0.5 1 1.5 20

0.5

1

1.5

2



Single OSV Yaw RAOs Comparison

(f) Yaw RAO

Figure 9: OSV RAO Comparisons between HydroStar and

MOTSIM for Single Vessel in Wave 2100

Frequency (r/s)

Surg

eR

AO

s(m

/m)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

HydroStar, Wave 180 degHydroStar, Wave 210 degHydroStar, Wave 240 degHydroStar, Wave 270 degMOTSIM, Wave 180 degMOTSIM, Wave 210 degMOTSIM, Wave 240 degMOTSIM, Wave 270 deg

OSV Surge RAOs Comparison(Double Vessels, OSV 0)

(a) Surge RAO

Frequency (r/s)

Sway

RA

Os

(m/m

)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


OSV Sway RAOs Comparison(Double Vessels, OSV 0)

(b) Sway RAO

Frequency (r/s)

Hea

veR

AO

s(m

/m)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


OSV Heave RAOs Comparison(Double Vessels, OSV 0)

(c) Heave RAO

8



Frequency (r/s)

Rol

lRA

Os

(deg

/m)

0 0.5 1 1.5 20

5

10


OSV Roll RAOs Comparison(Double Vessels, OSV 0)

(d) Roll RAO

Frequency (r/s)

Pitc

hR

AO

s(d

eg/m

)

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4


OSV Pitch RAOs Comparison(Double Vessels, OSV 0)

(e) Pitch RAO

Frequency (r/s)

Yaw

RA

Os

(deg

/m)

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4


OSV Yaw RAOs Comparison(Double Vessels, OSV 0)

(f) Yaw RAO

Figure 10: OSV RAO Comparisons between HydroStar and

MOTSIM for Double Vessels in OSV Angle 00 (Parallel)

5. CONCLUSIONS This paper presents a numerical seakeeping simulation of two

vessels in close proximity with parallel or unparallel headings

using HydroStar and an in-house code MOTSIM. A normalized

optimization function based on vessel’s roll, pitch and heave

motion shows that the sheltering effect is stronger when the

supply vessel is located in a wave shadow of the FPSO which

means smaller movement of OSV and less riskiness for crew

transfer. Comparing with result of HydroStar, MOTSIM needs

more validation and modification in considering the sheltering

effect for two vessels seakeeping. It should be noted that the

viscous dissipation in the gap of two ships was not considered

in either software. The dissipation may largely damp the

vessel’s motion in reality[4]

although this usually applies to

much smaller gaps than the ones considered in this paper.

One observation made during the course of this research was

that there is very little data that can be used for validating

numerical simulations other than the head sea case, with

forward speed. This type of data would be very useful for

developing an understanding of the behavior of two ships in

close proximity for offshore engineering applications.

ACKNOWLEDGMENTS This work was carried out at Oceanic as part of a project which

was supported by the Atlantic Innovation Fund administered by

Atlantic Canada Opportunities Agency and additional funding

from Husky Energy. This support is gratefully acknowledged.

REFERENCES 1. Kevin McTaggart, David Cumming, C. C. Hsiung, Lin Li,

“Seakeeping of Two Ships in Close Proximity”, Ocean

Engineering 30 (2003), pp1051-1063.

2. Gung-rong Chen, Ming-chung Fang, “Hydrodynamic

interactions between Two Ships Advancing in Waves”,

Ocean Engineering 28 (2001), pp1053-1078.

3. M. Rafiqul Islam, Motohiko Murai, “Dynamic Interaction of

Parallel Moving Ships in Close Proximity”, J. Marine Sci.

Appl. 12 (2013), pp261-271.

4. HydroStar for Experts User Manual, Research Department

BUREAU VERITAS, March 2011.

5. Jacek S. Pawlowski, Don W. Bass, "A Theoretical and

Numerical Model of Ship Motion in Heavy Seas", SNAME

Transaction, Vol.9, 1991, pp. 319-352.

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numerical study of two vessels seakeeping in waves

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