Time-domain numericalanalysis of hydroelasticoating structure edgedsubmerged horizontal p

i, JiUniv

odel is built basede in hydrodynamicnd accurate evalu-tions and its spatialerpolationetabula-ools and the modelnse-reduction ef-ically assessed forters, such as plates a result of theted as the optimalporosity and the

porous parameter is developed by using the least-squares ttingscheme. After simulation and verication, the dual anti-motionplates which are the perforated-impermeable-plate combina-tion attached to the fore-end and back-end of the VLFS, are

* Corresponding author. Tel.: 86 84706757.E-mail address: deep_1@dlut.edu.cn (Y. Cheng).

Contents lists available at ScienceDirect

Marine Structuresjournal homepage: www.elsevier .com/locate/

Marine Structures 39 (2014) 198e224Anti-motion deviceHydroelastic responsesPerforate plateSubmerged plate

interaction problem. A quarter of numerical mon the symmetry of ow eld and structurforces, and special care is paid to the rapid aation of time domain free-surface Green funcderivatives in nite water depth by using inttion method. Using the developed numerical ttests conducted in a wave basin, the respociency of the perforated plates is systematvarious wave and anti-motion plate paramewidth, porosity and submergence depth. Aparametric study, the porosity 0.11 is selecporosity, and the relationship between theVLFSTime domain

which the uid ows across the perforated anti-motion plate byapplying the Darcy's law, is applied to the uidestructureYong Cheng*, Gang-jun ZhaDeepwater Engineering Research Center, Dalian

a r t i c l e i n f o

Article history:Received 29 August 2013Received in revised form 19 May 2014Accepted 18 July 2014Available online

Keywords:http://dx.doi.org/10.1016/j.marstruc.2014.07.0070951-8339/ 2014 Elsevier Ltd. All rights reserved

Downloaded from http://www.eleand experimentalresponse of a very largewith a pair oflates

nping Ouersity of Technology, Dalian 116024, China

a b s t r a c t

This paper is concerned with the hydroelastic problem of apontoon-type, very large oating structure (VLFS) edged withthe perforated plates, non-perforated plates or their combinationanti-motion device both numerically and experimentally. Adirect time domain modal expansion method, taking amount ofthe time domain Kelvin sources in hydrodynamic forces, in

marstruc.

arnica.ir

pliedd a

and numerical methods. Based on Mindli[26] presented the dynamical behavior of(periodic load). Karmakar and Soares [2

r ofbe c

Y. Cheng et al. / Marine Structures 39 (2014) 198e224 199mooring lines at its corners in watelines and lateral pressure loads can

deection of the VLFS.oating exible plate of a nite small draft using the analyticaln thick plates theory andWeinereHopf technique, Zhao et al.oating elastic plates acted upon by a localized external load7] studied the wave scattering of a VLFS connected withnite depth and shallow water approximation. The mooringonsidered as wave absorber, which reduces signicantly thetype, circular VLFS. Kashiwagi [24] apAndrianov and Hermans [25] analyzethe B-spline panels for the analysis of a zero draft VLFS. Thendesigned for more wave energy dissipation and added damping.Considering variation of the water depths in offshore, discussionon the effectiveness of these anti-motion devices at differentwater depths is highlighted.

2014 Elsevier Ltd. All rights reserved.

1. Introduction

Very large oating structure (VLFS) can offer an alternative option of birthing land from sea,however, they may sometimes be placed in a location where the sea state is rather harsh, and thussomeways [1] for mitigating themotion responses of a VLFS have been proposed. The traditional way isby building the breakwater around the VLFS to reduce the height of the incident wave, such as two oilstorage facilities, which are located in Shirashima and Kmigoto respectively, Japan, are surrounded bythe breakwater. However, many cases such as the environmental protection, construction time andeconomics restrict the development of the upright breakwater. If we adopt oating breakwater [2,3],dynamic characteristics of the mooring system have complex effects on the oating body. Anotherinnovative approach is semi-rigid connections into the VLFS instead of rigid connections. The pro-ponents of this approach are Paulling [4], Riyansyah et al. [5] and Gao et al. [6].

Another interestingmethod is to the introduction of anti-motion devices attached to the fore-end ofa VLFS. This submerged vertical or horizontal plate can mitigate the motion of a VLFS by dissipatingwave energy and increasing added hydrodynamic coefcients. Masanobu et al. [7] proposed to use thesubmerged vertical plate for reducing the hydroelastic response, however it demands a larger space toachieve desirable performance, and the thin plate may undergo the elastic deections under thetrapped waves. Takagi et al. [8] performed the hydroelastic responses of a VLFS with a simple verticalanti-motion box but the anti-motion device result in an increase of steady drift force. Ohta et al. [9] andWatanabe et al. [10] also investigated the wave response of a VLFS, of which a submerged horizontalplate is attached at the fore-end. Pham et al. [11] further extended method to develop a horizontalsubmerged annular plate attached around perimeter of a pontoon-type, circular VLFS.

All above response-reduction performances of VLFS aim at the effectiveness of non-perforatedvertical/horizontal anti-motion plates. In order to further reducing the transmitted and reectedwave, the perforated horizontal anti-motion plate with viscous dissipation in this paper has manyattractive merits in reducing the motion of VLFS, such as low transmission, low reection and smallwave force acting on it. The wave energy dissipation of perforated plates is based on the boundarycondition for uid across perforated plate by using Darcy's law [12e17], which suggested that thevelocity for uid across the plate with nite pores is linearly dependent on the pressure difference bycomplex-valued frequency relation parameter. Following this idea, in the present paper, the perforatedanti-motion plates are attached to the edges of VLFS for dissipating wave energy from open sea, andthen we propose a perforated-impermeable-plate combination device.

In performing the hydroelastic response of the VLFS, there are two computing approaches referredto as frequency domain method and time domain method. Watanabe et al. [18], Eatock [19] and Chenet al. [20] presented a review of these methods. The earlier solution to the uidestructure interactionproblem was given by Wu et al. [21] in which they applied eigenfunction method to the two dimen-sional hydroelastic problem. Jin et al. [22] have extended the analytical method to three dimensionalstudies. Watanabe et al. [23] presented benchmark solutions for hydroelastic problems of a pontoon-

hydrodynamic effects. Our approach, instead, is to analyze directly the transient responses by using adirect time domainmodal expansionmethod and themethod that solves hydrodynamic diffraction and

radiationproblemsbyapplying the time-domain free-surfaceGreen functionmethod. In order to reducethe memory and CPU time of computer, the interpolationetabulation method based on bilinear/tri-linear interpolation is applied to the rapid and accurate evaluation of Green functions and its spatialderivatives in nite water depth. Taking account of symmetry of ow eld and structure, we onlyestablish boundary integral equations of a quarter VLFS model.

Though several model experiments of a VLFS have been conducted [37e39], there are peculiarproblems in these model experiment because of the denition very large. i.e. how to strictly satisfythe geometrical scale of model and generate short waves, while simultaneouslymake the small verticalbending rigidity similar to the rigidity of the actual VLFS. In this paper, a sandwich-type model [36] ofVLFS attached to submerged horizontal anti-motion plates, is conducted by overcoming these dif-culties, and then the authors study the effectiveness of perforated anti-motion plates and the com-bination of an upper horizontal perforated and a lower horizontal non-perforated plate in reducing thehydroelastic responses at the edge and the interior of the VLFS.

2. Numerical analysis

2.1. Fluid part

We consider a pontoon-type VLFS edged with a pair of submerged horizontal plates, and incidentwave propagates along the positive x-direction. The uidestructure system and Cartesian coordinatesystem are shown in Fig. 1. The z-axis is pointing upwards, and the xey plane is on themean position ofthe free surface, where h is thewater depth, A is the amplitude of the incident wave, d is the draft of theVLFS in z direction, L and B denote the length and width of the structure, respectively. The perforatedand non-perforated horizontal plates of the width 2a are attached at both the fore-end and the back-end of the VLFS, and their submergence depths are d1 and d1 d2, respectively. The problem at hand isto determine the modal deections under the wave action. Below, the basic assumptions are usuallymade:

(1) The uid is incompressible, inviscid and irrotational.(2) The horizontal motion of VLFS is restricted, and the vertical motionwhich is consists of the rigid

and elastic displacement is considered in this paper.(3) The amplitude of the incident wave and the motions of the VLFS are both small and only verticalAll the above analytical or numerical solutions are the frequency domain approaches when deter-mining the hydroelastic response amplitude operator (RAO) of theoating bodyand pertinent responseparameters in a steady state condition. However, in real situation, we would often like use some timeseries of the responses [28,29]. Wantanabe et al. [30], Qiu and Liu [31], and Shin et al. [32] applied thenite element methods (FEM) to the transient response analysis of a VLFS due to impulsive landing/takeoff of an airplane, however, the response analysis is required for a few seconds and this simulationmodel is very much simplied in the treatment of structure or uid. Kashiwagi [33,34] developed aindirect time-domainmethod inwhich the hydrodynamic effect is evaluated fromgood performance inthe computation of the memory-effect function. Lee et al. [35] proposed a boundary element-niteelement (BE-FE) hybrid method to solve the transient responses indirectly by using transient equa-tions, which are derived from the Fourier inverse transform of harmonic equations of motion and thecausality condition. Based on the BE-FE combinedmethod, Endo [36] simulated the takeoff/landing runof an airplane in regular wave conditions. These transient responses are solved indirectly by usingtransient equations which are derived from the Fourier inverse transform of harmonic equations ofmotion and the causality condition, and some difculties in carrying out their time-domain simulationgive restriction themathematical model. For example, the integration of memory-effect function is stilltime-consuming for evaluating at some high-frequency range, and the evaluation of hydrodynamiccoefcients such as added mass at innite frequency commonly neglect some cross-coupling terms in

Y. Cheng et al. / Marine Structures 39 (2014) 198e224200deformation is considered.

Y. Cheng et al. / Marine Structures 39 (2014) 198e224 201Under the rst assumption given above, a velocity potential F(x,y,z,t) exists, and would be given by,

Fx; y; z; t FIx; y; z; t FSx; y; z; t (1)where FI(x,y,z,t) and FS(x,y,z,t) are the incident and scattering potential, respectively.

The velocity potential must satisfy the following Laplace's equation (2) and the following boundaryconditions (3)e(9):

V2FSx; y; z; t 0 (2)the linearization of boundary conditions on the free surface Sf:

v2FSx; y; z; t2 g

vFSx; y; z; t 0 on Sf ; t >0 (3)

Fig. 1. Geometry of a VFLS edged with the perforated-impermeable plate combination and corresponding coordinate system.vt vz

with the boundary conditions on the sea-bed Sd, the innity S, on the wetted surface of the oatingbody Sb (the bottom surface) and Ss (the side surface), and on the surface of the submerged horizontalplates Sau, Sab:

vFIx; y; z; tvn

vFsx; y; z; tvn

Vn on Sb Ss; t >0 (4)

vFIx; y; z; tvn

vFsx; y; z; tvn

Vn s1u

vF

vt vF

vt

!on Sau1 Sau2 (5)

vFIx; y; z; tvn

vFsx; y; z; tvn

Vn s2u

vF

vt vF

vt

!on Sab1 Sab2 (6)

vFsx; y; z; tvn

0 on Sd; t >0 (7)

Y. Cheng et al. / Marine Structures 39 (2014) 198e224202Fsx; y; z; t;Fstx; y; z; t;VFsx; y; z; t/0 on S; t >0 (8)

Fsx; y; z; t Fstx; y; z; t 0 on Sf ; t 0 (9)where g is the gravitational acceleration, Vn is the normal velocity of the structure, n is a unit normalvector (the positive direction points out of the uid domain),Fst(x,y,z,t) represent the time derivative ofscattering potential, Superscript (Eqs. (5) and (6)) means ed1 or e(d1 d2), the complex-valuedfrequency dependent parameters s1 and s2 denote both viscous and inertial effects of the upper andlower anti-motion plate, respectively.

Referring to Cho and Kim [16], the imaginary part of s1 or s2 in Darcy's law represents the inertiaeffect and thus it has nothing towith energy dissipation. It can be neglected when the submerged plateis thin and the hole sizes are small. However, the positive real part is called the porous-effectparameter, which can be obtained from experiment and is related to the porosity coefcient and dy-namic viscosity. According to Cho and Kim [16], the porosity parameter b used in the present method isexpressed as:

b 2psk

(10)

when b 0, the plate corresponds to a impermeable plate such as the lower plate, while for b , itmeans that the plate is transparent.

The boundary value problems given by Eqs. (2)e(9) can be solved by using the Green's functionmethod. If the free-surface Green's function satisfying the boundary conditions given by Eqs. (3) and(7)e(9) is considered, the boundary integral equation for each of the scattering potential can bederived as follows:

aFSQ ; t SbtSstSsutSabt

FSP; tvG0P;Q

vnpdSp

SbtSstSsutSabt

G0P;Q vFSP; tvnp

dSp Zt0

dt SbtSstSsutSabt

"FSP; t

vGftPt;Qt; t tvnp

GftPt;Qt; t tvFSP; tvnp

#dSp 1g

Zt0

dt Cbt

FSP; tGfttPt;Qt; t t

GftPt;Qt; t tvFSP; tvnp

$VnP; tdSp

(11)

where a represents the solid angle, Q(x0,y0,z0) and P(x,y,z) represent the source and eld point,respectively, G0 and Gf represents the instantaneous and memory term of the free-surface Greenfunction, respectively, Cb(t) represents the instantaneous waterline of the intersection between thebody and the free-surface, and others are the same as above.

Considering a nite water depth h, the Green function may be formulated:

GP; t;Q ; t G0P;Q Gf P; t;Q ; t 1r 1r2 2

Z0

ekhcosh kz0 h

cosh khcosh kz hJ0kRdk

2Z0

cosh kz hcosh kh sinh kh

cosh kz0 hn1 cos

ht t

gk tanh kh

p ioJ0kRdk

(12)

Y. Cheng et al. / Marine Structures 39 (2014) 198e224 203where J0 denotes the Bessel function of the rst kind, order zero, R denotes the horizontaldistance between eld and source point, r denotes the distance between eld and source point,and r2 denotes the distance between eld and the mirror image of the source eld about watersurface.

The instantaneous term G0 and its spatial derivatives have been systematically investigated [40],and thus we start by considering the non-dimensional space and time parameters for rst order timederivative Gft as follows:

X Rh; Y z0

h; Z z

h; T t tg=h1=2 (13)

then

Gft g1=2h3=2FX;Y Z; T FX;2 Y Z; T (14)

FX;V ; T Z0

k tanh k

p

cosh k sinh ksinTk tanh k

p coshkVJ0kXdk

2Z0

k tanh k

psinTk tanh k

p ekV2 ekV21 e4k J0kXdk (15)

The rate of convergence of the integral in Eq. (15) can be accelerated by adding and subtracting theappropriate function F which may be given in the form

FX;V ; T limk/

FX;V ; T 2Z0

k

psinTk

p ekV2J0kXdk (16)

where the Vertical coordinate V is restricted to the uid domain (1, 2).If the spherical coordinate is adopt, the non-dimensional parameters on r are dened by.

r hX2 V 22

i1=2; t T

.r1=2; k kr; V 2 r cos q; X r sin q (17)

where t and q are lies in the interval (0, ) and (0, 2p). Thus, we will reduce three arguments to twoarguments by substituting Eq. (17) into Eq. (16). The function F and its spatial derivatives are obtained

F 2r32Im8

The function F is solved by referring to the paper by Newman [40] and Huang [41], then thefunction FeF may be solved directly by Gaussian integration. Finally, F (X,V,T) is obtained by.

FX;V ; T F F F (22)

2.2. Structural part

It is nowwidely accepted that VLFS response in terms of the vertical deection can be captured wellby modeling the whole VLFS as an elastic plate. In this formulation, assuming the VLFS as an elastic,isotropic, thin plate, the motion of the oating body is governed by the equation of a thin plate restingon a uniform elastic foundation [33]:

DV4Wx; y; z; t ms W Px; y; z; t rgW (23)whereD EI is the bending rigidity, E is modulus of elasticity, I is the cross sectional moment of inertia,ms denotes themass per unit area, rdenotes density ofuid, and thedynamic pressure P(x,y,d,t) relatesto the velocity potential on the bottom surface of the VLFS from the linearized Bernoulli's equation

Px; y;d; t r vFx; y;d; tvt

(24)

In the present case, the VLFS is not constrained in the vertical elastic displacement along its edges,the following boundary conditions for a free edge must be satised:

v2Wvn2

v v2Wvs2

0; v3Wvn3

2 v v2W

vnvs2 0 (25)

where v is Poisson's ratio, n and s denote the normal and tangential directions, respectively.The vertical elasticity displacement W(x,y,z,t) of the VLFS is the sum of various modes as follows:

Wx; y; z; t XNj1

zjtfjx; y (26)

where zj(t) is the vibration amplitude of the j mode, fj(x,y) is modal function of the j mode. The modalfunctions can be expressed by the product of modal function at x direction and modal function at ydirection, thus a case study of a beam model with free end is discussed as follows. In this paper, weemploy the following modal functions (dry-modal functions) combining the rigid body motions andthe sinusoidal functions because they satisfy the free-end condition of the beam and their convergencehas already been proved by Eatock [42] and Etaock and Ohkusu [43] :

fmcx; y fmxfcy (27)

f1x 1; f2x 2x=L fm1x sinj 2pxL 12

; m 2 (28)

f1y 1; f2y 2y=B fc1y sinj 2pyB 12

c 2 (29)

By substituting Eqs. (24) and (26) into Eq. (23), we have

XM XNmsfmxfcyzmct

DV4fmxfcy rgfmxfcy

zmct r

vFI FSvt

(30)

Y. Cheng et al. / Marine Structures 39 (2014) 198e224204m1 c1

wherem, n are themodal numbers in x and y direction, respectively. Multiplying the above equation byfi(x) fj(y) and integrating along the bottom of the VLFS, we can obtain a conventional set of equationsgiven by

XMm1

XNc1

Mmc;ijzmct XMm1

XNc1

Kmc;ijzmct Fij i 1;;M; j 1;;N (31)

where

Mmc;ij Sb

msfmxfcyfixfjyds (32)

Kmc;ij DSb

V2fmxfcyV2fixfjyds D1 v

Sb

(v2fmxfcy

vx2v2fixfjy

vy2 v

2fmxfcyvy2

v2fixfjyvx2

2 v2fmxfcyvxvy

v2fixfjyvxvy

)ds

Sb

rgfmxfcyfixfjyds

(33)

Fij SbSab

r vFI FSvt

fixfjyds Sau

r vFI FSvt

fixfjyds (34)

It shouldbenoted thatMmc, ij,Kmc, ijand Fijare thegeneralizedmass, generalized stiffness andgeneralizedwave force, respectively. And the generalized stiffness Kmc,ij shown by Eq. (33) has been obtained bytaking account of the free-edge boundary conditions, Eq. (25), referring to the paper by Kashiwagi [33].

In order to solve the deection of the VLFS, Eq. (31) are solved by using the fourth order Run-geeKutta method.

2.3. Interpolationetabulation method

Fast evaluation of the time-domain free-surface Green function with desired accuracy is the key tothe solution of the velocity potential from the boundary integral Eq. (11) on the body surface. Theinterpolationetabulation method is applied to the solutions of the auxiliary function F and FeF inEq. (22) as follows.

Referring to Eqs. (16)e(21), the three argumentX,V, Tof the function F are changed to twoargumentst and cos q, which are divided 800 and200 parts, respectively. The solution for cos q 0.7which has beendescribedbyBeck [44] is obtainedas above, elsewhen cos q>0.7, the Filon integral scheme is determinedto calculate the function F directly. For improving the calculating efciency, the bilinear interpolationwas applied to the effective approximation of F and its' derivation in hydrodynamic forces.

However, the FeF function has three arguments V, X and T. Here, the space non-dimensionalparameters V and X are restricted to the region (1, 2) and (0, 20), and the time non-dimensionalparameters T lies in the interval (0, 20). First seven XeT planes are adopted in V direction. And thenevery XeT plane is divided 40 parts in X and T direction, respectively. The slowly-varying function FeFand its' derivations, which are determined by Gaussian integration, can be approximated effectively bythe tri-linear interpolation method.

2.4. Symmetry of structure

Considering symmetry of the VLFS about xez plane and yez plane shown in Fig. 2, with the dis-cretization of the constant boundary elements, Eq. (11) may be expressed as form of the linear

Y. Cheng et al. / Marine Structures 39 (2014) 198e224 205equations

Y. Cheng et al. / Marine Structures 39 (2014) 198e2242062664A11 A12 A13 A14A21 A22 A23 A24A31 A32 A33 A34A41 A42 A43 A44

37758>>>>>:F1SF2SF3SF4S

9>>>=>>>; 8>>>:

B1B2B3B4

9>>=>>; (35)

The symmetric relationships for the matrix [A] may be formulated:

A11 A22 A33 A44 (36)

A12 A21 A34 A43 (37)

A13 A31 A24 A42 (38)

A14 A41 A23 A32 (39)In order to reduce the dimensions of the matrix, the conversions are obtained by taking

Fh

S

EfFSg;

B_

EfBg; I 1

bE2 (40)

Fig. 2. Sketch of the district of symmetry.where thematrix [E] is a transitionmatrix, [I] is identity matrix. After substituting Eq. (40) into Eq. (35),the linear equations may be obtained for each region from 1 to 4 dened in gure as follows:

In the region 1, we haveA_1

Fh 1

S

B_1

(41)

A_1

A11 A12 A13 A14 (42)

Fh 1

S

nF1S F2S F3S F4S

o(43)

Y. Cheng et al. / Marine Structures 39 (2014) 198e224 207B_1 fB1 B2 B3 B4g (44)

In the region 2, we haveA_2

Fh 2

S

B_2

(45)

A_2 A11 A12 A13 A14 (46)Fh 2

S

nF1S F2S F3S F4S

o(47)

B_2 fB1 B2 B3 B4g (48)

In the region3, we haveA_3

Fh 3

S

B_3

(49)

A_3 A11 A12 A13 A14 (50)Fh 3

S

nF1S F2S F3S F4S

o(51)

B_3 fB1 B2 B3 B4g (52)

In the region 4, we haveA_4

Fh 4

S

B_4

(53)

A_4 A11 A12 A13 A14 (54)Fh 4

S

nF1S F2S F3S F4S

o(55)

B_4 fB1 B2 B3 B4g: (56)

3. Experiment

3.1. Model and the experimental set-up

In order to obtain experimental data for comparison with the results of the numerical calculationbased on the theory described so far, the model test was carried out in Ocean Engineering Basin (40 mlong, 8 mwide, and 1.1 m deep) at Dalian University of Technology. Table 1 lists the principal details ofthe VLFS model, which is a sandwich-type with the combination of aluminum plate, polyethylenefoam, aluminum plate and polyethylene foam. The exible deck is two aluminum plates with a suf-cient weight and rigidity. The middle polyethylene foam between two-layer plates can adjust therigidity by changing the height of the polyethylene foam without changing the weight. The poly-ethylene foam attached to the lower aluminum plate is used as a buoyancy material, which hasnegligible weight and rigidity. The anti-motion plates are attached to the fore-end and back-end of theVLFS, respectively as shown in Fig. 3, and the pertinent information for these anti-motion devices issummarized in Table 2. Here, M1 is a basic sandwich-type VLFS model without anti-motion device. M2is a sandwich-type VLFS model edged with the anti-motion device which consists of two non-perforated horizontal plates (32 cm length, 1.2 m width and 5 mm thick) attached to the edges ofthe VLFS model, respectively. The anti-motion device of the model M3_1, M3_2 and M3_3 is twoperforated horizontal plates (32 cm length, 1.2 mwidth and 5mm thick) with three different porositiesp 0.038, 0.11 and 0.18, respectively. Based on the characters of M2 andM3, M4's anti-motion device isthe perforated-impermeable-plate combination. The upper perforated plate has the same porosity asthe anti-motion device of the M3_2 model, while the lower plate is impermeable b 0.

In order to prevent drifting inwaves, two models of mooring device were made from standing steelframes in thewater basin as shown in Fig. 4: the one is chainmodel (four 200-cm-longwire rope-springswith universal joints at both ends are attached to themiddle location of the VLFS and the standing steelframes, which represent the mooring dolphins), the other is fender model (ten 25-cm-long wire rope-springs are arranged at the side of VLFS model). A set of monochromatic waves with the period be-tween 0.599 s and 3.703 s is generated by a user-dened timeevoltage input to the wave maker to theVLFS model at incident angle q 00. The wave amplitude was maintained around 20 mm. The verticaldisplacements can be measured accurately at various points with the displacement sensor having anaccuracy of 0.1mm, inwhich the instantaneous positions ofmeasured points of upper deck are detectedby a data acquisition system. Twenty-one sites were selected for elastic response measurements, asindicated in Fig. 5, where the data for fore-end, mid-position, and the back-end of the VLFS are from theaveraged data (#1#12#17)/3, (#6#14#19)/3 and (#11#16#21)/3, respectively.

3.2. Law of similarity

Once the scale ratio of the model is decided upon, the following problems that satisfy the law ofsimilarity are discussed in the present experiment.

1. In order to satisfy the geometrical similarity of the structure and uid, since the scale is 1:50, thewave periods for the real structure are from 4.24 s to 26.18 s, respectively. The wave amplitude usedin the experiment (2 cm) corresponds to 1m for the real structure. On the other hand, the periods ofwaves that are observed in near shore areas, are 5e15 s, and the wave amplitudes are 0e3 m, thusthe waves used in the experiment are reasonable.

2. Based on the dimensions in the experiment, the inertial modulus I (0.013630.013)/121.2 m4 andthe Young's modulus E 0.702 1011 N/m2, therefore, the bending rigidity EI 1.0653 104 Nm2.Considering the design dimensions of the VLFS, the target bending rigidity EI of the model is

Y. Cheng et al. / Marine Structures 39 (2014) 198e2242081.061 104 Nm2. Thus the rigidity almost satises the targeted gures.

3. The weight W L B t g 8 1.2 0.0018 2700 2 93.312 kg. Considering the designdimensions of the VLFS, the target weightWof themodel is 98.4 kg. Thus theweight almost satisesthe targeted gures.

4. Results and discussions

We consider the box-type VLFS model and sea state condition as described in Section 3. First, theeffectiveness of the submerged horizontal plates attached to both ends of the VLFS is compared withthe solutions without anti-motion plate and single plate only at fore-end of the VLFS in Figs. 6 and 7 for

Table 1Principal details of the VLFS model.

Prototype 1/50 model

Length 400 m 8 mBreadth 60 m 1.2 mHeight 3.7 m 0.0736 mDraft 0.5 m 0.00948 mEI 3.3129 1012 Nm2 1.0653 104 Nm2Water depth 20 m, 25 m, 30 m 0.4 m, 0.5 m, 0.6 mWavelength 28 me400 m 0.56 me8 m

Y. Cheng et al. / Marine Structures 39 (2014) 198e224 209Fig. 3. The VLFS model used for the experiment (unit is millimeter in scale, and scale ratio is 1/50).

Table 2Main parameters of the perforated and non-perforated plates.

M2 M3_1 M3_2 M3_3 M4

The perforatedplate

Rows in longitudinaldirection

1 3 5 3

Rows in transversedirection

20 20 20 20

Width 2a 32 cm 32 cm 32 cm 32 cmLength B 1.2 m 1.2 m 1.2 m 1.2 mHole radius R 1.5 cm 1.5 cm 1.5 cm 1.5 cmPorosity (p) 0.038 0.11 0.18 0.11Depth d1 5 cm, 10 cm,

15 cm, 20 cm5 cm, 10 cm,15 cm, 20 cm

5 cm, 10 cm,15 cm, 20 cm

10 cm

The non-perforatedplate

Width 2a 20 cm, 24 cm,28 cm, 32 cm

32 cm

Y. Cheng et al. / Marine Structures 39 (2014) 198e224210the case plate width 2a/L 0.03, submergence depth 2d1/L 0.05, water depth 2 h/L 0.125, wavenumbers kL/2 31.416 and 15.708. Figs. 6 and 7 show that the deections at the fore-end, mid-positionand back-end of the VLFS as functions of dimensionless time parameters t/T. The numerical solutionsand experimental results are in good agreement, and the effect of the anti-motion plate in reducing thehydroelastic response is clearly observed. In this case, considering the deections at fore-end and mid-position, the multiple plates are attached to both edges of the VLFS are almost as efcient as the singleplate attached only to the fore-end of the VLFS. However, for the back-end of the VLFS, the multipleplates are more effective in reducing the deections compared with the single plate. Further toconsider the real wave condition such as the multi-directional incident waves and the reections fromoffshore, the design with multiple plates is selected in this paper.

Next, the sensitivity of the width of the submerged horizontal plate is shown in Fig. 8 as a functiondimensionless wave numbers kL/2. The y-axis is the vertical displacement amplitude obtained after thetime step of four times the wave period. Five different plate widths, 2a/L 0, 0.025, 0.03, 0.035 and0.04, are considered with the submergence depth of 2d1/L 0.025 and water depth 2h/L 0.125. In allthe cases, the numerical solutions agree well with the experimental tests. It can be found that

Length B 1.2 m 1.2 mPorosity (p) 0 0Depth d1/d2 d1: 5 cm, 10 cm,

15 cm, 20 cmd2: 5 cmextremely long incident waves (kL/2/ 0), the submerged plates in reducing the deections of the VLFSare ineffective (the deection amplitudes of the VLFS tend to the same values). However, when thewave numbers in a range of 18 < kL/2 < 35, the anti-motion plate with width 2a/L 0.04 moreeffectively interact with surface waves to lessen the deections of the VLFS. As the wavelength de-creases to a moderate-length (kL/2 > 35), all anti-motion plates are very effective. In the aspect of

Fig. 4. Mooring conguration of the VLFS model.

Y. Cheng et al. / Marine Structures 39 (2014) 198e224 211design for submerged horizontal plate, the anti-motion plate with width 2a/L 0.04 are chosen for allcalculations in this section owing to the signicant incident wave energy at least when L/l > 6 in realsea areas [10].

In Figs. 9 and 10, the empirical relationship between the porous parameter b and plate porosity pcan be determined from the experimental results with plates of various porosities. These perforatedplate width and submergence depth are xed at 2a/L 0.04 and 2d1/L 0.025, respectively. In thesegures, the lines represent the calculated results applying the BEM while the symbols stand for theexperimental data. The x-axis is the non-dimensional wavelength and y-axis is the RAO of thedeection for the VLFS model with submerged horizontal plates with different porosity. Referring toCho and Kim [18], the porous parameter b depend mainly on the porosity if the hole sizes are suf-ciently small and arrayed uniformly. Then, the empirical relationship between the porous parameter band plate porosity p was tted by a line function as follows:

b yp BX (57)

Fig. 5. The locations of the displacement sensors (unit is millimeter in scale, and scale ratio is 1/50).where y is the slope of the linear function, and BX is the b intercept. Then, the least-squares ttingscheme is applied byS P

ib bb2 P

ib yp BX2, and the line function is determined by

minimizing the squared error S:

vSvBX

0 (58)

vSvy

0 (59)

The tted curve (see Fig. 11) can be obtained in this paper as soon as the slope and intercept aregiven by solving above equations:

b 54:015p 0:379 (60)

All the curves of Figs. 9 and 10 are given at the fore-end and mid-position of the VLFS as thefunctions of the dimensionless wavenumber kL/2 for various porous parameters (p 0, 0.038, 0.11 and0.18). In these gures, the deections of the VLFS is not sensitive to the use of the submerged plates

Fig. 6. Comparison of the deection amplitude between numerical and experimental results for the VLFS without and with single plate/multiple plates as a function of t/T at kL/2 31.416. (a)Fore-end, (b) mid-position, (c) back-end.

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Fig. 7. Comparison of the deection amplitude between numerical and experimental results for the VLFS without and with single plate/multiple plates as a function of t/T at kL/2 15.708. (a)Fore-end, (b) mid-position, (c) back-end.

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Y. Cheng et al. / Marine Structures 39 (2014) 198e224214with various porous parameters in low frequency region kL/2 < 10 but becomes obviously discrepantwhen kL/2 > 10. In general, the cases of the perforated anti-motion plates perform better than that ofthe non-perforated plates (Figs. 9(a) and 10(a)). The effectiveness of the perforated plates becomesclearer when the porosity p increases near to 0.11 (Figs. 9(b) and 10(b)), whereas the overall perfor-mance (p 0.18) is interestingly worsened when the porosity further increase more than 0.11. Thenumerical calculated curves correlate well with the experimental results for such range of plateporosities.

Comparison of all numerical results with measured values in Figs. 12 and 13 are shown for a givenplate submergence depth. In order to further examine the sensitivity of the position of the submergedplates, the developed theory and corresponding results with the VLFS model M2 (p 0), M3_1(p 0.038), M3_2 (p 0.11) and M3_3 (p 0.18) are compared against the experiments, respectively.Figs. 12 and 13 show the deection magnitude at the fore-end, mid-position and back-end of the VLFSedged with non-perforated plates for dimensionless wavelength l/L 0.1 and l/L 0.2. Four differentsubmergence depth of plates, 2d1/L 0.125, 0.025, 0.0375 and 0.05 are selected with the platewidth of2a/L 0.04. In all these case, the experimental results can agree well with the numerical simulation. Itis seen that as the non-perforated plates move away from the VLFS, the hydroelastic responsesdecrease. However when the depth of the plates increase to 2d1/L 0.05, the motion of the VLFS ismagnied as compared with the results of depth 2d1/L 0.0375. For the VLFS model edged with theperforated anti-motion plates, such as M3_1, M3_2 and M3_3, the effects of submergence depth 2d1/Lon maximum values of the deections are listed in Tables 3 and 4 for kL/2 31.416 and 15.708 whileother parameters are xed by the given values. The position is considered at fore-end, mid-position

Fig. 8. Nondimensionalized deection amplitude of the VLFS with the non-perforated plates as the non-dimensional wavenumberKL/2 and plate width for 2d1/L 0.025, h/L 0.125, b 0: lines are for BEM solutions and marks are for experimental values. (a)Fore-end, (b) mid-position, (c) back-end.

Fig. 9. Deection amplitude of the VLFS with the perforated horizontal plates at fore-end as a function of non-dimensionalwavenumber kL/2 for 2d1/L 0.025, 2a/L 0.04: Lines are for the BEM solutions and marks are for experimental results.

Y. Cheng et al. / Marine Structures 39 (2014) 198e224 215and back-end of the VLFS, respectively. Note that, the best level of the perforated plates in reducing themotion of the VLFS is 2d1/L 0.025. As the perforated plates move near to the depth, the effectivenessof these plates becomes clearer and then the deections of the VLFS at the edges and the interiorincrease with the increase of 2d1/L. Accordingly, both the model test results and the numerical resultsindicate that here exists an optimal submergence depth near 2d1/L 0.0375 for the non-perforatedplates, however, for the perforated plates, the optimal submergence depth is near 2d1/L 0.025.

Considering the best position and merits of the perforated and non-perforated plates in reducingthe deections of the VLFSmodel under wave action, we propose a simple anti-motion device, which isthe perforated-impermeable-plate combination attached to the fore-end and back-end of the VLFSmodel M4. Figs. 14 and 15 show the effect of different anti-motion plates on the hydroelastic responsesof the VLFS model at kL/2 31.416 and 15.708, respectively. In the design regards of the dual anti-motion plates for VLFS model M4, the upper perforated plate (p1 0.11) is placed at the depths2d1/L 0.025, whereas the location of the lower non-perforated plate (p2 0.) is xed at 2(d1 d2)/L 0.0375 with plate width 2a/L 0.04. The details of the VLFS model M1, M2, M3_2 are described inthe previous section. To be clear, the displacement amplitudes of model M4 are compared with theresults of model M3_2 in Table 5. In this particular, the combination of the perforated and non-Fig. 10. Deection amplitude of the VLFS with the perforated horizontal plates at mid-position as a function of non-dimensionalwavenumber kL/2 for 2d1/L 0.025, 2a/L 0.04: Lines are for the BEM solutions and marks are for experimental results.

perforated plates could achieve more reductions in the motion of the VLFS for the short wavelength kL/2 31.416 but the deections are magnied near the edges of the VLFS when the wavelength increasesto kL/2 15.078. In other words, the installation of the lower plate does not contribute much to theoverall the response-reducing effectiveness when the wave numbers kL/2 < 15.708 i.e. the upper plateplays a major role for the given case (see Tables 3 and 4). However, the dual perforate- impermeableplates still are advantageous when the incident waves are short waves.

Finally, we would like to investigate in Table 6 performance of the VLFS model M4 as the waterdepth changes. The porosities of the two horizontal perforated and non-perforated anti-motion plateswith a constant gap 2d2/L 0.0125 are p10.11 and p2 0, respectively. Four water depths 2h/L 0.1,0.125 and 0.15 with respect to the rst column are considered and the vertical deection amplitudes ofthe VLFS model are listed in Table 6 for kL/2 31.416 and 15.708, respectively. As it can be observedfrom this table, our calculated results are in good agreement with the experimental data, and the

Fig. 11. Linear Fit between the porous parameter and the porosity.

Y. Cheng et al. / Marine Structures 39 (2014) 198e224216hydroelastic responses at the edges and the interior of the VLFS are magnied with increasing waterdepth.

5. Conclusions

The response-reduction efciency of a VLFS by adopting submerged horizontal non-perforated,perforated or their combination plates has been investigated in the context of numerical theory andexperimental test. A direct time domain modal expansion method, in which the time-domain-Kelvin-source-based BEM solutions are carried out for hydrodynamic forces, is applied to obtain numericalresults. Special care is paid to the fast and accurate evaluation of the time-domain free-surface Greenfunction and its spatial derivatives in nite water depth by using the bilinear or tri-linear inter-polationetabulation method. The numerical solutions with different porous parameter of anti-motionplates are then compared with corresponding experimental results, from which the relationship be-tween the submerged horizontal plate porosity and the porous parameter is determined by a cure-tting technique. The measured data and simulation results correlate well each other.

Through the parametric study of various wave and anti-motion device conditions, such as the wavelength, submerged plate width, submergence depth and the porosity, it can be found that there existsan optimal design with the anti-motion plate width 2a/L 0.04, the plate porosity p 0.11, and thesubmergence depth 2d1/L 0.025 in reducing the deections of the VLFS for given water depth. Tofurther reduce the motion of the VLFS, a simple perforated-impermeable-plate combination anti-motion device is employed in the developed program, and the numerical response reduction has

Fig. 12. Defection amplitudes of the VLFS model without and with the non-perforated plates for kL/2 31.416 (a) fore-end, (b) mid-position, (c) back-end.

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Fig. 13. Defection amplitudes of the VLFS model without and with the non-perforated plates for kL/2 15.708 (a) fore-end, (b) mid-position, (c) back-end.

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Table 3Deection amplitudes of the VLFS models edged with the perforated horizontal plates for kL/2 31.416.Modelposition2d1/L

M3_1 (p 0.038) M3_2 (p 0.11) M3_3 (p 0.18)Fore-end Mid-position Back-end Fore-end Mid-position Back-end Fore-end Mid-position Back-end

EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL.

0 0.313 0.315 0.164 0.164 0.158 0.158 0.312 0.315 0.163 0.164 0.158 0.158 0.312 0.315 0.163 0.164 0.158 0.1580.0125 0.276 0.271 0.118 0.115 0.162 0.163 0.167 0.168 0.087 0.087 0.139 0.134 0.197 0.191 0.132 0.129 0.149 0.1470.025 0.176 0.178 0.092 0.093 0.110 0.112 0.115 0.106 0.068 0.069 0.101 0.102 0.173 0.174 0.094 0.095 0.123 0.1200.0375 0.221 0.222 0.103 0.102 0.124 0.126 0.153 0.155 0.087 0.084 0.131 0.131 0.180 0.176 0.120 0.116 0.121 0.1210.05 0.240 0.243 0.105 0.110 0.131 0.134 0.187 0.181 0.094 0.096 0.158 0.158 0.184 0.184 0.126 0.125 0.131 0.130

Notes: The dimensionless depth 0 indicates the VLFS model without anti-motion device, which is the model M1.

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Table 4Deection amplitudes of the VLFS models edged with the perforated horizontal plates for kL/2 15.708.Modelposition2d1/L

M3_1 (p 0.038) M3_2 (p 0.11) M3_3 (p 0.18)Fore-end Mid-position Back-end Fore-end Mid-position Back-end Fore-end Mid-position Back-end

EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL.

0 0.338 0.346 0.137 0.136 0.233 0.245 0.338 0.346 0.137 0.136 0.233 0.245 0.338 0.346 0.137 0.136 0.233 0.2450.0125 0.262 0.264 0.121 0.125 0.242 0.244 0.181 0.175 0.090 0.092 0.221 0.221 0.280 0.270 0.082 0.083 0.244 0.2380.025 0.132 0.136 0.079 0.083 0.071 0.070 0.112 0.113 0.072 0.069 0.166 0.166 0.117 0.122 0.085 0.085 0.174 0.1740.0375 0.229 0.232 0.089 0.092 0.127 0.122 0.121 0.124 0.092 0.091 0.196 0.190 0.125 0.125 0.096 0.096 0.187 0.1820.05 0.277 0.269 0.098 0.099 0.124 0.120 0.143 0.143 0.096 0.096 0.212 0.215 0.152 0.152 0.115 0.114 0.214 0.219

Notes: The dimensionless depth 0 indicates the VLFS model without anti-motion device, which is the model M1.

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Fig. 14. Comparison of deection of the VLFS M1, M2, M3_2 and M4 as a function of t/T at kL/2 31.416. (a) Fore-end, (b) mid-position, (c) back-end.

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Fig. 15. Comparison of deection of the VLFS M1, M2, M3_2 and M4 as a function of t/T at kL/2 15.708. (a) Fore-end, (b) mid-position, (c) back-end.

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Y. Cheng et al. / Marine Structures 39 (2014) 198e224 223been conrmed by experimental analysis. Finally, the hydroelastic responses of the VLFS edged withdual anti-motion plates are assessed for various water depths. Both the numerical analysis andexperimental results indicate that as the water depths increase, the motion of the VLFS model at edgesand interior is magnied.

Acknowledgment

The authors are grateful to the National Science Foundation for Creative Re-Search Groups of China(Grant No. 50921001) for supporting this work.

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Table 5Comparison of the displacement amplitudes of model M4 with the results of model M3_2 for kL/2 31.416 and 15.709.kL/2 model 31.416 15.709

Fore-end Mid-position Back-end Fore-end Mid-position Back-end

EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL.

M3_2 0.115 0.106 0.068 0.069 0.101 0.102 0.112 0.113 0.072 0.069 0.166 0.166M4 0.083 0.086 0.054 0.053 0.067 0.065 0.114 0.115 0.065 0.065 0.195 0.183

Table 6Deection amplitudes of the VLFS model M4 at various water depth for kL/2 31.416 and 15.709.kL/2 position 31.416 15.709

Fore-end Mid-position Back-end Fore-end Mid-position Back-end

EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL. EXP. CAL.

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Time-domain numerical and experimental analysis of hydroelastic response of a very large floating structure edged with a pa ...1 Introduction2 Numerical analysis2.1 Fluid part2.2 Structural part2.3 Interpolationtabulation method2.4 Symmetry of structure

3 Experiment3.1 Model and the experimental set-up3.2 Law of similarity

4 Results and discussions5 ConclusionsAcknowledgmentReferences