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Ocean Engineering 35 (2008) 589597
Comparison of hydroelastic computer codes based
on the ISSC VLFS benchmark
H.R. Riggsa,, Hideyuki Suzukib, R.C. Ertekinc, Jang Whan Kimd, K. Iijimae
aDepartment of Civil and Environmental Engineering, University of Hawaii at Manoa, Honolulu, HI 96822, USAbDepartment of Environmental and Ocean Engineering, University of Tokyo, Tokyo 108-8639, Japan
cDepartment of Ocean and Resources Engineering, University of Hawaii at Manoa, Honolulu, HI 96822, USAdTechnip USA, Houston, TX 77079, USAeOsaka University, Osaka 565-0871, Japan
Received 3 August 2007; accepted 10 January 2008
Available online 25 January 2008
Abstract
There has been substantial development in computer codes for linear hydroelasticity in recent years, driven in part by the motivation to
investigate the wave-induced response of very large floating structures (VLFSs). A recent International Ship and Offshore Structures
Congress (ISSC) state-of-the-art report on VLFS design and analysis [ISSC, 2006. Report of Specialist Task Committee VI.2, very large
floating structures. In: Frieze, P.A., Shenoi, R.A. (eds.), Proceedings of the 16th International Ship and Offshore Structures Congress,
Elsevier, Southampton, UK, pp. 397451] included a brief comparative study of the simulation results from different computer codes
for a pontoon (mat-like) VLFS. The codes covered a mix of both fluid models (potential and linear GreenNaghdi) and structural models
(3-D grillage, 2-D plate, 3-D shell). A more detailed comparison of the results from a select group of models from that study is provided
and discussed herein. The similarities in the results increase the confidence level of the state-of-the-art in predicting the hydroelastic
response of such structures, and the differences, including in computational efficiency, lead to an understanding of the significance of
specific modeling assumptions and their impact on the predicted response.r 2008 Elsevier Ltd. All rights reserved.
Keywords: Very large floating structure; VLFS; Hydroelasticity; ISSC; Potential theory; GreenNaghdi theory
1. Introduction
There has been substantial development in hydroelastic
computer codes over the last two decades. Much of the
development of 3-D formulations has followed the foun-
dational work of Wu (Price and Wu, 1985;Wu, 1984). The
driving force, especially in the 1990s, for the developmenthas been the Japanese Megafloat project (Suzuki, 2005)
and the US Mobile Offshore Base project (Palo, 2005). The
former project was concerned principally with floating
civilian airports in protected, shallow water, and focused
on pontoon-type very large floating structure (VLFS). The
latter project was concerned principally with a floating
military airbase in the deep open ocean, and hence focused
on multi-module semi-submersible-type VLFS. This latter
project included a comparison of the predictive response of
different computer codes for the latter application (BNI,
1999; NFESC, 2000; Wung et al., 1999). However, the
computer codes were based on very similar formulations,
for the most part, and the system consisted of flexibly
connected rigid modules. The results are not directlytransferable to the pontoon-type VLFS and the more
different structural and fluid models that are used for this
class of structure. It should be noted that one of the
computer codes (HYDRAN) used in that comparative
study is also used here.
The interest in VLFS prompted the International Ship
and Offshore Structures Congress (ISSC) to establish a
special task committee on VLFS, whose task was to
consider the state-of-the-art in VLFS analysis and design
procedures, including the application of hydroelasticity
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www.elsevier.com/locate/oceaneng
0029-8018/$ - see front matterr 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.oceaneng.2008.01.012
Corresponding author. Tel.: +1 808956 6566; fax: +1 808956 5014.
E-mail address: [email protected] (H.R. Riggs).
http://www.elsevier.com/locate/oceanenghttp://localhost/var/www/apps/conversion/tmp/scratch_5/dx.doi.org/10.1016/j.oceaneng.2008.01.012mailto:[email protected]:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_5/dx.doi.org/10.1016/j.oceaneng.2008.01.012http://www.elsevier.com/locate/oceaneng -
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(ISSC, 2006). One section of that report included a
comparative study on the wave-induced response deter-
mined by several computer programs, each based on
significantly different models. Space limitations prevented
a detailed comparative study of the methodology and the
results (ISSC, 2006). Therefore, a more detailed presenta-
tion is made herein, including additional and some up-dated results). Such a comparison is important because:
(1) similarities in the results increase the confidence level of
the state-of-the-art in predicting the hydroelastic response
of such structures, and (2) differences lead to an under-
standing of the significance of specific modeling assump-
tions and their impact on the predicted response. A similar
comparative study (Jiao et al., 2006) has been published
recently, but it was restricted to a shallow draft plate. In
addition, it considered only displacements. For VLFS
design, prediction of stresses is very important, which the
current study considers.
2. Problem description
The model is a rectangular, pontoon structure 500 m
long and 100 m wide. It is 2 m high, the draft is 1 m, and it
is in 20m of seawater. Transverse and longitudinal
bulkheads are spaced at 50 and 20 m, respectively. The
top deck, bottom hull, bulkheads, and sides are steel plates
with a nominal thickness of 20mm. To include the
additional bending stiffness associated with stiffeners that
are not modeled, the plate bending thickness is 150 mm for
all plates. The mass density of steel is 7850 kg/m3, and the
structural mass is based on the nominal steel volume given
the 20 mm plate thickness. The modulus of elasticity, E,and Poissons ratio, n, for steel were taken to be 200 GPa
and 0.3, respectively. To model the non-structural mass,
the top and bottom plates are assigned an additional mass
density of 17,131 kg/m3. As a result, the CG is located at
the geometric center of the cross-section. Note that all
dimensions are midplane dimensions consistent with a shell
finite element model of the structure. The density of
seawater, r, is 1025 kg/m3, and gravitational acceleration,
g, is 9.81 m/s2. Structural damping is not included in the
dynamic analysis. The seawater is assumed unbounded
horizontally and the seabed is flat.
The model has been designed to have significant flexible
response under waves. Of interest herein is the global
response, and in particular displacements and stresses.
Results are reported for wave periods between 5 and 30 s,
and for wave angles of 01 (head seas), 301, and 451.
3. Computer codes
Three computer programs were used to obtain the
dynamic response: HYDRAN (OCI, 2005), VODAC
(Iijima et al., 1997) and LGN (Kim and Ertekin, 1998).
In all programs, the fluid is assumed to be incompressible
and inviscid and the flow is assumed to be irrotational.
In the first two programs, the fluid model is based on
linear, 3-D potential theory. In the last program, the
fluid model is based on the linear GreenNaghdi equations
for long waves. HYDRAN uses a traditional constant
panel Green function formulation for the fluid and a
3-D shell finite element model for the structure. VODAC
uses a traditional constant panel Green function formula-
tion, together with interaction theory, for the fluid and a 3-D grillage model for the structure. LGN uses the Green
Naghdi equations in the fluid domain, as mentioned, and a
linear Kirchhoff plate model; the governing equations are
matched at the juncture boundaries and they are solved by
a combination of eigenfunction expansion method and the
boundary-integral equation method. All of the codes
determine the response in the frequency domain.
4. Fluid modelsHYDRAN, VODAC
The fluid models in these programs are based on the
linear potential theory. The fluid motion is assumed small,and hence the total velocity potential F for a regular wave
at frequency o can be written as a function of position
r [x1,x2,x3]T and structural displacements u as
F fI fD io/Rueiot (1)
in which fI, fD, and /Rand the incident, diffraction, and
radiation potentials, respectively. The diffraction and
radiation potentials must satisfy the same governing
equation in the fluid domain, O, and the same boundary
conditions on the free surface and at the seabed
r2f 0; r2 O, (2)
o2f gqf
qx3 0 on x3 0, (3)
limx3!1
qf
qx30. (4)
They must also satisfy Sommerfelds radiation condition
on the still water surface at infinitely large radial distances
from the body. On the wetted body surface, S, the
diffraction potential must satisfy
qfDq
n
qfIq
n
on S, (5)
where n is the outward-directed unit normal vector to the
wetted surface, while the radiation potentials must satisfy
qfRqn
nn on S. (6)
n* is the generalized normal, i.e., the normal displacement
of the body. Once the velocity potential is determined, the
relationship between velocity and pressure, p, from Eulers
integral is used to determine the linear, dynamic pressure
on the structure
p rqF
qt . (7)
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5. Fluid modelLGN
In the LGN model, a simple kinematics of fluid motion
is used under the assumption that the wave length of the
free-surface motion is much longer than water depth, h.
The horizontal and vertical velocity profiles along the
depth are assumed to be constant and to vary linearly,respectively. In the case of small amplitude wave motion,
the horizontal velocity field ~Vx;y; t and the verticalvelocity component w(x,y,z,t) are given in terms of the
complex mean velocity potential c(x,y) (see Ertekin and
Kim, 1999)
Vx;y
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7. Structural modelVODAC
VODAC was originally developed for the analysis of
semi-submersible type VLFS supported by multiple floa-
ters. Although the structural modeling available in the code
is 3-D beamcolumn grillage, the structural model for this
study is a simpler plane grillage, as shown in Fig. 3.
Standard, 2-node 3-D beamcolumn elements are used, six
degrees of freedom per node (three translations and three
rotations). Member forces are axial force, vertical and
horizontal shears, torsion, and vertical and horizontal
bending moments. VODAC uses a substructure method to
enhance computational efficiency. This approach is effec-
tive especially for VLFS, which often have a repetitive
structure.For the structural model used herein, the nodal spacing
is 12.5 m in the longitudinal direction (40 elements per
section line) and 10m in the transverse direction (10
elements per section line), resulting in a total of 400
elements. The elements are located so as to determine the
deflections along the longitudinal and transverse bulk-
heads. The axial, shear, torsional, and bending rigidities
are determined so as to be equivalent to the structural
properties of the pontoon-type VLFS, including the
Poisson effect. For example, the bending rigidity EI for
the transverse cross-section is approximately 880 GN-m2
according to the following formula:
EI E
1 n21
2BH2t
, (12)
where B is breadth of the structure, H is height of the
structure, and t is the actual plate bending thickness; this
considers contributions from the top and bottom plates
only. The bending rigidity is equally distributed to the
beam elements that represent the cross-section; i.e., all
beams in a given direction have the same properties.
A similar approach is used for the beams that model the
longitudinal cross-section. The following equivalent sec-
tion properties are used: for longitudinal elements,
A 0.8m2, Iyy 0.88 m4, Izz 26.7 m4, J 1.6 m4; and
for transverse elements, A 1.0m2, Ixx 1.1 m4, Izz
52.1 m4, J 2.0 m4. The moments of inertia include the
term 1/(1n)2 to account for the Poisson effect.
8. Structural modelLGN
The rectangular pontoon is modeled as a homogeneous,
isotropic plate with uniform mass distribution (per unit
area), m, and flexural rigidity, D Et3eq=121 n2, where
teq 0.792 m is the effective plate thickness. This thickness
is defined so as to result in the same moment of inertia
as the actual built-up, yz section of the pontoon. The
moment of inertia of thexzsection is very similar, and so
the isotropic assumption is reasonable. The verticaldisplacement, z(x,y,t) of the plate is assumed to be
governed by the thin-plate theory (see e.g., Rayleigh, 1894):
mq
2z
qt2 DD2z pf (13)
in whichpfis the spatial part of the time-harmonic pressure
on the bottom of the plate and D q2/qx2+q2/qy2 is the
two-dimensional Laplacian on the horizontal plane. Since
the plate is freely floating, the bending moment and shear
force should vanish at the edges of the plate
q2zqn2
n q2z
qs20; q
3zqs3
2 n q3z
qs2qn0 on J, (14)
where nand s denote the normal and tangential directions
on the boundary J, andn is Poissons ratio. Eq. (14) can be
written alternatively as
Dz 1 nq
2z
qs20;
q
qn Dz 1 n
q2z
qs2
0 on J.
(15)
At the corners of the plate, there can be concentrated
shear forces to compensate for the torsional moment
along the edges of the plate. The vanishing of this shear
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Fig. 3. VODAC structural model.
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force leads to
q2z
qxqy 0. (16)
Combining Eqs. (9), (10), and (13), the following coupled
equation can be obtained in Region II:
DD3cII rgDcII m rh
3
o2DcII
ro2
h cII 0.
(17)
In Region I, the atmospheric pressure is neglected, i.e.,
pf=0, and therefore, we have
DcI k2cI 0. (18)
Along the juncture boundary, J, the continuity of mass
flux and depth-mean pressure leads to the following match-
ing conditions:
qcI
qn
qcII
qn ; cI cII on J. (19)
9. Solution procedureHYDRAN
HYDRAN uses the 3-D source distribution, or the
Green function, method to determine the velocity poten-
tials. The Green function satisfies all but the body
boundary conditions (Wehausen and Laitone, 1960), and
so only the wetted surface must be discretized. It uses the
constant-strength panel formulation. This well-known
method has been used for rigid body hydrodynamics by
many authors (Faltinsen and Michelsen, 1974; Garrison,1974, 1975) and more recently for hydroelasticity (Price
and Wu, 1985;Wu, 1984).
To reduce the number of radiation potentials that must
be solved, a reduced-basis approach is used. In particular,
the structural displacements are represented by a combina-
tion of a limited number of the dry (i.e., in-air) normal
modes of vibration (Price and Wu, 1985; Wu, 1984).
Specifically,u Wp, where W is the matrix of normal mode
shapes and p is the vector of generalized displacements.
This modal-superposition-type strategy reduces the radia-
tion potentials from the order of the number of structural
degrees of freedom to a relatively small number. It alsomeans that the radiation potentials correspond to global
displacement shapes rather than local finite element shape
functions. The equation of motion becomes
o2 Mns Mn
f
i Cns oC
n
f
KnS K
n
f
h ip FnW
(20)
in which
Mns WTMsW; K
n
S WTKSW; K
n
f WTKfW,
whereMs,KS, andKfare the structural mass, stiffness, and
hydrostatic stiffness matrices, respectively, and Cns is the
diagonal matrix of modal hysteretic damping coefficients
(not used herein). Mns and Kn
S are diagonal matrices
because the dry normal modes are used. The terms in the
vector of wave exciting forces are obtained from
FnWj iro
Z fI fDn
n
j dS, (21)
while the terms in Mn
f and Kn
fare given respectively by
mijr
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condition required for Kagemotos interaction theory.
However, it has been shown from experiments and
simulations that VODAC is applicable to pontoon-type
VLFS with good accuracy (Iijima et al., 1999). The present
problem is formulated as a hydrodynamic interaction
problem between 100 (20 in longitudinal direction 5 in
transverse direction) floating bodies. Each body has 220panels, based on a mesh discretization of 10 10 3 in the
x, y, and z directions, respectively. Diffraction character-
istics (Goo and Yoshida, 1990), which represent the re-
lations between incident wave and scattered wave from a
unit body, are calculated in advance by using the normal
singularity distribution method. Once the diffraction
characteristics are obtained, we have only to calculate the
diffraction coefficients representing the hydrodynamic
interactions by solving a system of equations that describe
the relations between the diffraction coefficients. The
source distribution is hierarchically calculated using the
diffraction coefficients and the diffraction characteristics.
As mentioned, the structural modeling is also hierarch-
ical. The substructure method is applied to the dynamic
analysis case, and the whole structure is expressed as a
repetition or an assembly of the substructures. The stiffness
matrix of a substructure is condensed into a smaller matrix
with all the degrees of freedom eliminated except those of
the boundary nodes. In this process, dynamic condensation
is performed instead of static condensation. The final
matrix to be solved is obtained by summing the condensed
matrices regarding the boundary nodes. The displacements
of the boundary nodes are solved first, together with the
diffraction coefficients. Then the displacements of the inner
nodes are calculated through backward calculations usingthe relations obtained in the condensation stage.
Additional details of the solution procedure can be
found inIijima et al. (1999).
The CPU time for a single frequency and wave heading
was 18.86 s on a Pentium 4 PC, 2.16 GHz with 2 GB RAM,
running Windows XP.
11. Solution procedureLGN
The numerical solution of Eq. (18) defined in Region I,
where there is no floating plate, can be given by
distribution of the Green function along the boundary J
cIIx;y cwx;y p
2i
ZJ
H10 kjx isjsis ds, (24)
wherecw(x,y) is the velocity potential for the incident wave,
which is known. H10 is the Hankel function and s(s) is the
unknown source strength defined along the juncture
boundary, J. To obtain the numerical solution, the jun-
cture boundary is discretized into a finite number of line
elements, where the source strengths are assumed to be
piecewise constant. The unknown source strengths are
determined by matching the solution with the solution in
Region II through the matching conditions given by Eq. (19).
For this problem, 480 line elements (200 and 40 along each
edge of the plate in x and y directions, respectively) were
used. No other discretization is necessary because the model
uses semi-analytical models for both the fluid and the plate.
In Region II, the governing equation, Eq. (17), can be
solved by the eigenfunction-expansion method because the
domain is a simple rectangular one. The coefficients of the
eigenfunctions are obtained by enforcing the matchingcondition, Eq. (19), and the free-edge conditions given by
Eqs. (15) and (16). The details of this approach are given in
Kim and Ertekin (1998).
The CPU time for each frequency and each wave
heading was 1.85s on a Pentium 4 PC, 2.0 GHz, with
384 MB RAM, running Windows 2000.
12. Results
12.1. Natural periods and modes
As mentioned previously, the HYDRAN solution used
the first 30 dry normal modes for the reduced basis. The
first dry deformation mode of the shell model corre-
sponded to vertical bending in the longitudinal direction
and had a natural period of 24.6s, while the second
bending mode had a period of 8.96 s and the third bending
mode had a period of 4.62 s. The first bending mode in the
transverse direction had a natural period of 1.32 s. The
30th mode had a natural period of 0.81 s and a spatial
variation unlikely to attract significant forces for the wave
periods considered.
Although not needed for the analysis, the program also
estimates the wet natural periods and modes, which include
the added mass and the hydrostatic stiffness. The programfound 6 wet natural periods between 8 and 30 s: 8.2, 8.4,
8.6, 10.5, 14.50, and 15 s.
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0
0.2
0.4
0.6
0.8
1
-250
HYDRAN
VODAC
LGN
Maximumverticaldisplacem
entin10secwave(m/m)
Position (m)
Wave angle = 0
-200 -150 -100 -50 50 100 150 200 2500
Fig. 4. Maximum vertical displacement in 10 s head sea.
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The other two programs do not require natural periods
and mode shapes, so no comparison can be made. The
results are reported principally to reveal a little bit more the
fundamental characteristics of the structure.
12.2. Wave-induced response
The maximum displacements along the centerline
induced by a wave with a period of 10 s and an incidence
angle of 01(propagating in the xdirection) are shown in
Fig. 4. The shape is strongly dependent on the wave angle,
as can be seen by comparing Figs. 4 and 5. These results
show generally good comparisons between the programs,
especially considering the different numerical models that
were used to obtain the results. However, in the oblique sea
case,Fig. 5, the results for LGN differ significantly in the
interior of the plate. The reason for this is unknown, but itmay be that the twisting response induced by the oblique
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0
10
20
30
40
50
60
70
80
90
5
HYDRAN
VODAC
LGN
Longitudinalstressatcenter(MPa/m)
Wave Period (s)
Wave angle = 0
Wave angle = 45
Wave angle = 30
10 15 20 25 30
Fig. 8. Longitudinal stress at center.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
5
HYDRAN
VODAC
LGN
Verticaldisplacementatfore(m/m)
Wave Period (s)
Wave angle = 0
Wave angle = 45
Wave angle = 30
10 15 20 25 30
Fig. 7. RAO of vertical displacement at bow.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
5
HYDRAN
VODAC
LGN
Verticaldisplaceme
ntatcenter(m/m)
Wave Period (s)
Wave angle = 0
Wave angle = 45
Wave angle = 30
10 15 20 25 30
Fig. 6. RAO of vertical displacement at center.
0
0.2
0.4
0.6
0.8
1
1.2
-250
HYDRAN
VODAC
LGN
Maximumverticaldisplacementin10secwave(m/m)
Position (m)
Wave angle = 45
-200 -150 -100 -50 0 50 100 150 200 250
Fig. 5. Maximum vertical displacements in 10 s oblique sea.
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sea is not captured as well by the plate model. All three
models agree quite well in the end displacements, however.
The programs agree well for the RAOs for the vertical
displacement at the center and bow of the structure (again
along the centerline) for the 3 wave angles of 01, 301, and
451; seeFigs. 6 and 7.
RAOs of the longitudinal stresses at the center and 50 m
back from the bow, also along the centerline, obtainedfrom the programs show strikingly good agreement; see
Figs. 8 and 9. The difference in peak stresses is less than
10% for 01 and 301, while it is about 15% for 451.
Considering the strikingly different structural models, the
agreement is surprising, especially in oblique seas.
13. Conclusions
The comparative study considered three programs for
the linear hydroelastic response of a pontoon-type VLFS.
Two of the programs were based on potential theory, while
one used linear GreenNaghdi theory for long waves. All
three programs used very different structural models. The
results for this pontoon-type VLFS, considering a wide
range of wave periods and wave angles, showed good
agreement between the programs. Even the stresses agreed
quite well. This demonstrates that all three models can be
used with confidence, especially for preliminary studies.
The computer times show that LGN is the most com-
putationally efficient, although there is little practical
difference between time requirements for it and VODAC.
HYDRAN is the most computationally demanding. LGN,
however, is limited to a structure that can be modeled
as a uniform plate and for relatively shallow water. The
LGN theory is established for shallow-water waves, and
although the agreement of the LGN results to the linear
potential theory results for smaller wave periods is
remarkable, it is likely that for deeper water the results
will show a larger difference. (For more discussion of this
issue, the reader is referred toKim and Ertekin (2001), who
compared the LGN predictions with the linear potential
theory predictions for the dispersion relation for waterwaves and for hydroelastic waves on a mat-like structure.)
The present results also confirm the conclusion that
VODAC, which uses interaction theory of separate bodies,
can be applied to a pontoon VLFS that consists of a single
body (Iijima et al., 1999).
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50
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HYDRAN
VODAC
LGN
Longitudinalstressfore(MPa/m)
Wave Period (s)
Wave angle = 45
Wave angle = 0
Wave angle = 30
10 15 20 25 30
Fig. 9. Longitudinal stress 50m back from bow.
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