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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

ARTICLE IN PRESS

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: riggs@hawaii.edu (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:riggs@hawaii.edumailto:riggs@hawaii.eduhttp://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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0

10

20

30

40

50

60

70

5

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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