Combined Semi-analytical and Finite Element Approach for Hydro StructureInteractions dur ing Sloshing Impacts - " Sloshel Project"

Malenica S.(1)

, Korobkin A.A.(2)

, Ten I.(1)

, Gazzola T.(1)

, Mravak Z.(1)

, De-Lauzon J.(1)

& Scolan Y.M(3)

(1)Bureau Veritas, Paris, France

(2)University of East Anglia, Norwich, UK

(3)Ecole Centrale de Marseille, Marseille, France

ABSTRACT

The paper discusses the development of the combined semi-analytical (fluid flow) and numerical FEM (structure) models forhydro structure interactions during the sloshing impacts in thetanks of the membrane type LNG carriers. This is a very challeng-ing problem and lot of work, both experimental and numerical,has been done in the past. However, it is fair to say that no fullyconsistent solution exists up to now. Indeed, the small modeltests, which are usually conducted in this context, suffer fromscale effects and numerical CFD calculations suffer from numer-ous numerical problems and prohibitive CPU time requirements.There is a clear necessity for full scale measurements with the realLNG and the real containment system. Unfortunately this is tech-nically extremely difficult and, up to now, there is no documentedwork on this, at least not in open literature. In the absence ofthe real full scale measurements, some ”quasi” full scale measure-ments were performed. These measurements consist in impact-ing the real containment system structure through the drop teststechnique Kim et all (2008), or through the more sophisticatedwave generated impacts (Sloshel project - Malenica et al (2009),Brosset et al (2009)). The developments which are discussed inthis paper were conducted within the Sloshel project which con-centrated on the structural response of the real (full scale) con-tainment system under the breaking wave impacts. Very rich andambitious project objectives have been defined, with the finalgoal of improving the current methodologies used in the struc-tural assessment of the containment system and the supportingship structures. At the beginning of the project, it was decidedthat the effort in the numerical part would be concentrated onthe development of semi-analytical methods for the local fluidflow, which will be combined with the complex structural FEMmodels for containment systems. The use of the classical CFDmethods for the local analysis of extremely complex sloshing phe-nomena was avoided at the beginning of the project, but was in-cluded later. The simplified semi-analytical models for fluid flow,keeping the main physical parameters allow for the assessment of

the phenomena within the reasonable computational effort. Thedrawback is that the impact situations should be highly simplified.In that respect, the sloshing impacts were preliminary classifiedinto 3 main groups:

• impact without inclusion of air (steep wave impact),

• impact with entrapped air pocket (Bagnold type impact),

• impact with aerated fluid.

Some intermediate impact types were also identified but theyreceived less attention so far. For each impact type, the dedicatedsemi-analytical model is developed and the computational resultsare compared with the full scale Sloshel tests.During the project some additional modeling difficulties aroseand one of them is the modeling of the wave propagation whichis necessary for proper definition of the main flow parametersjust before the impact happens. Different approaches, basedon the potential flow modeling of the wave propagation, wereproposed by ECM and were shown to be very efficient in thepresent context. These wave propagation models are also brieflydiscussed.

INTRODUCTION

Transportation of Liquified Natural Gas by membrane type LNGcarriers is rather safe and cost efficient at present. However,the increased size of the recent LNG carriers (up to 240 000m3) and new operation requirements (partial fillings) introducethe important challenges with respect to safety and structuralintegrity of LNG tanks. Structural integrity is of concern, in par-ticular, in terms of the so called sloshing-slamming loads whichare induced by violent liquid motion inside the LNG tanks. Dueto the fact that these impact loads are most often characterizedby extremely high pressure peaks localized in space and in time,the evaluation of the structural response can not be decoupledfrom the evaluation of the fluid flow and coupled hydro-structurenumerical models are necessary. Usually we talk about the

Proceedings of the Nineteenth (2009) International Offshore and Polar Engineering ConferenceOsaka, Japan, June 21-26, 2009Copyright © 2009 by The International Society of Offshore and Polar Engineers (ISOPE)ISBN 978-1-880653-53-1 (Set); ISSN 1098-618

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fully coupled or hydroelastic analysis, in contrast to the mostcommon quasi static approach where the hydrodynamic loadingis calculated independently by assuming the rigid structure, andthe resulting pressure field is simply applied to structural modelas a static loading at each time step. Unfortunately most often,it is not possible to avoid the complex hydroelastic calculationsin the present context where the duration of impact load is veryshort so that the dynamic amplification factor can significantlyvary. This can be clearly seen from Figure 1, where the wellknown dependence of the maximum dynamic response of anelastic system on the impulse duration, is presented. The mainparameter here is the ratio of the impulse duration to the naturalperiod of the elastic system. In sloshing impact situations thisparameter can take any particular value. This Figure alsoshows that the hydroelastic effects can be both beneficial ordangerous, depending on t1/T0 ratio. In order to address these

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7

X_m

ax/X

_0

t1/T_0

Figure 1: Dynamic amplification for triangular impulsive load-ing (t1 impulse duration, T0 - natural period). The loading issymmetrical around t = t1/2 where it takes the maximum valueF0 = kx0.

challenges and resolve the difficulties associated with the hy-droelastic analysis, combined approach based on semi-analyticalhydrodynamics and FEM structural analysis, is proposed. Thisapproach accounts for physical effects such as hydroelasticity,mixing processes between the LNG and the gas, compressibilityof the fluid and other effects only then and there when andwhere they are of primary importance. The overall methodologyof the combined approach was developed for a general case ofviolent sloshing in LNG tanks (see Korobkin A.A. & MalenicaS., 2006 for more details). In this paper the methodology forthe sloshing assessment at low fillings is presented. Low fillingsituation is characterized by possible formation of steep andbreaking waves, which hits the tank walls and produce very highimpact pressures. This was of main concern in the Sloshel Project.

HYDROELASTIC MODELS

Basic principles of coupling

In the developed hydroelastic models it is assumed that the globalkinematics of the sloshing is independent of the tank flexibilitybut the local impact pressures and their distributions are affectedby the elastic properties of the containment system and the as-sociated ship structure. The wall elasticity is accounted for onlyduring the impacts and only near the places were such impactshappen.

It is assumed that the analysis of global fluid motion in LNGtank by using CFD simulations has been performed, and both

the places and times of the different impacts were identified. Al-ternatively this can be also done by proper analysis of the smallscale model tests which are believed to be representative of theoverall sloshing motion inside the tank in spite of the possiblescale effects.

The local hydroelastic models are valuable also on their ownand can be used independently. Indeed, the local analysis pro-vides reference cases, which distinguish dangerous impact typesand impact conditions with respect to the failure modes of thecontainment system. Moreover, the results of the local analysismay help to distinguish the most important failure modes. Thelocal hydroelastic models can be validated independently by com-paring their predictions with the laboratory experimental data.

The local hydroelastic models are applicable only during theimpact stages, when the hydrodynamic loads are high and theelastic response of the structure is maximal. By definition theimpact stage is of short duration. This makes it possible to disre-gard many effects, which are of main concern in the CFD analysis,such as large dimensions of the tank and its real shape, real pro-file of the free surface far from the impact region, viscosity of thefluid, its surface tension and gravity effects. However, some ef-fects, which are believed to be of minor importance in the CFDanalysis, should be taken into account in the local analysis. Theseeffects are compressibility of the fluid, presence of the gas abovethe fluid surface and in the impact region, aeration of the fluid inthe impact region, jetting and fine details of the flow in the jetroot region, rapid increase of the wetted surface of the tank walland the flexibility of the wall.

Short duration of the impact stage allows us to simplify thelocal analysis and to use a combination of analytical and numer-ical methods instead of direct numerical calculations as in theglobal sloshing analysis by CFD. By introducing small time scaleand appropriate scales of other variables and unknown functionswhich describe coupled hydrodynamics and structural response,we arrive at the complex boundary-value problem of hydroelas-ticity with a small parameter (see Korobkin A.A. & Malenica S.,2006, section 2.3). The small parameter has the order of the ratiobetween the liquid displacement and the length scale of the orig-inal problem. This ratio is small for impact situations. This factallows us to use perturbations methods of asymptotic analysisto distinguish effects providing minor contributions and formu-late the coupled problem of hydroelasticity with proper accountfor physical effects providing the most important contributions tothe local loads and structure response.

The asymptotic analysis of local fluid-structure interaction inthe impact region is performed only for the hydrodynamic partof the problem. The structural part is presented by the originalFEM model. Computations for simple structures were done byFEM and by analytical models with the aim to verify the localmodels of hydroelasticity (Korobkin A.A. et al. 2004, MalenicaS. et al. 2006, Korobkin A.A.et al. 2006, Ten I. et al. 2008).

It is well accepted that fine details of the fluid flow andpressure distribution are not very important for the stressdistribution caused by the impact. Even the measured orcalculated pressures vary significantly, the stress evolution isbelieved to be much less sensitive to such variations. Unfortu-nately, the proper sensitivity analysis has not been performedso far for the hydroelastic problems of fluid impact onto thecontainment system or even for hydroelastic problems with

144

simplified structural characteristics in 2D. There is a necessity toperform this analysis in order to verify if the stress distributionsin a structure are weakly dependent on the impact details butstrongly dependent on the impact type, because this might allowfor further simplification of the different models by using onlyfew parameters specifying, approximately, the initial conditionsof the impact. This idea is rather attractive because it allowsus to perform the local analysis in semi-analytical way withfull control of the results. Reasons for such simplificationsare discussed by Korobkin and Iafrati (2005) with respect tonumerical solutions of water impact problems (see also otherpapers by Iafrati on this subject). This idea is also attractivebecause the numerical results for sloshing in LNG tanks byclassical CFD codes are available with the spatial accuracy ofthe order of 0.5m. This means that actually the fine details ofboth the free surface shape and velocity field are not available,and the initial conditions for the impact stage are known veryapproximately. This implies that, in some sense, attemptsto reproduce all details of the flow and pressure distributionduring the impact have no meaning in practice, even if they leadto very interesting mathematical problems (Malenica et al. 2006).

Impacts classification

The detailed classification of the impact types can be found inKorobkin and Malenica (2006) and here below we briefly summa-rize the main ideas. Three basic types of the impact (steep wave,

y

H

Uy

x

y

VH

O x

������������������������������������

������������������������������������

y

H U

x

Figure 2: Different types of wave impact (steep wave impact,breaking wave impact and impact by aerated wave.

breaking wave and aerated wave) are defined in Figure 2.It was shown that the fully coupled hydroelastic equation for in-compressible liquid impact can be written in the following form:

∂2

∂t2

n[M + S(b)]W

o+KW = Qr(b), (1)

where b(t) defines the dimension of the contact region, M andK are structural mass matrix and stiffness matrix of the struc-ture, S(b) is the added mass matrix, which is calculated at eachtime instant during the impact stage, Qr(b) represents the gen-eralized hydrodynamic loads with respect to the natural modesof the structure calculated without account for the structure de-flections. There are also additional equations for the dimensionsof the contact region between the fluid and the elastic structure.These equations are not shown here. Once the added mass ma-trix S(b) is known, the problem of hydroelasticity is reduced toa system of ordinary differential equations with respect to thedimensions of the contact region and the generalized coordinatesW(t) of the structure response.

In the models with compressible fluid one needs to solve asystem of integro-differential equations of the form

∂2

∂t2

nMW +

Z t

0

S(t− τ )W(τ )dτo

+ KW = Qr(t). (2)

It is important to notice that equations (1) and (2) do notrequire calculations of the hydrodynamic pressures but the timeintegrals of the pressure distribution is used instead. This highlyimproves accuracy and robustness of hydroelastic computations.Moreover, for several configurations the added mass matrix Swas calculated analytically which allows the use of very largenumber of modes.

Steep wave impact

Within the model of steep wave impact two cases were distin-guished with: (1) almost vertical front of the wave approachingthe vertical elastic wall and (2) the wave front inclined from thewall (Wagner type of steep wave impact). Wagner type of steepwave impact was covered in Korobkin (2008). Second order loadsacting on a rigid surface during wave impact were investigatedin Korobkin (2007). The generalized Wagner approach which ac-count for actual geometry of the impact, was studied by Malleronet al. (2007). Three-dimensional Wagner problem was analyzedby Gazzola et al. (2005) for rigid bodies and by Gazzola et al.(2006) for elastic structures.

Here we restrict ourselves to the first case where the wavefront comes into the contact with an elastic vertical wall almostinstantly. In this case the fluid compressibility has to be includedinto the hydroelastic model. However, the corresponding modelis the simplest hydroelastic model of wave impact. The hydro-dynamic parameters of the problem are the fluid depth H , thewave height Hw, and the sound speed of the fluid c0 (see Figure2). The wave speed V is actually not a parameter in this linearproblem because all unknown quantities are proportional to thewave speed. In this model the incident wave is two-dimensionalbut the structural response and the distribution of the hydrody-namic loads can be three-dimensional. Two-dimensional prob-lem of step wave impact was studied by Korobkin and Malenica(2007). Ten et al. (2008) considered 3D problem of wave impactonto a complex elastic structure. In Korobkin, Malenica (2007)it was also shown that the wave impact problem can be treatedwithin the incompressible fluid model but only for very flexiblestructures. Note that the actual panels of LNG tanks are ratherstiff with the period of the lowest mode being comparable withthe acoustic time scale. Within the acoustic approximation theflow is described by the velocity potential ϕ(x, y, z, t). The prob-

145

U

H

Hw

L

z1

z2

y1 y2

Figure 3: 3D configuration of the problem (left) and the wallwith mounted elastic panes (right).

lem is formulated in non-dimensional variables which are chosenin such a way that the fluid depth, impact velocity, sound speedand fluid density are equal to unity. The boundary-value problemwith respect to the velocity potential has the form8>>>>>>>><

>>>>>>>>:

ϕtt = ∇2ϕ in flow domain,ϕy = 0 y = 0 and y = l,ϕ = 0 z = 1,ϕz = 0 z = 0,ϕx = −χ1(y, z) + wtχ2(y, z) x = 0,ϕ = ϕt = 0 t = 0,q(y, z, t) = −ϕt(0, y, z, t) ,

where w(y, z, t) is the deflection of the elastic panel, q(y, z, t) is thehydrodynamic pressure distribution over the wall, and χ1(y, z),χ2(y, z) are characteristic functions of the impact region and theregion of the panel, respectively. The panel deflection is repre-sented as

w(y, z, t) =∞X

n=1

an(t)Wn(y, z),

where the principal coordinates an(t) satisfy the differential equa-tions

αnd2an

dt2+ dn

„an + γ

dan

dt

«=

ZS

q(y, z, t)Wn(y, z)dS,

an(0) = an(0) = 0.

Here the normal modes Wn(y, z) of the panel vibration in air andcoefficients αn, dn (n = 1, 2, ...) are computed by 3D FEM codeABAQUS, γ is the coefficient of structural damping. The integralterm is calculated analytically which makes it possible to reducethe coupled problem to the following system of integro-differentialequations

d2bn

dt2+ dn

„an + γ

dan

dt

«= Pn(t),

αnan +

∞Xm=1

Z t

0

am(τ )Knm(t− τ )dτ = bn(t),

where the functions Knm(t) and Pn(t) are given by analyticalformulae. This system is solved numerically.The developed numerical codes (2D, 3D plate, 3D complex elasticstructure) were validated with respect to:a) convergence with respect to the time step of integration andnumber of modes retained;b) comparison between predictions by 2D and 3D models;c) comparison of the results obtained within the compressible andincompressible fluid models;d) comparison of the 3D results obtained for uniform elastic plateby normal mode method and by coupling with 3D-FEM code.

The detailed validation procedure allowed us to select opti-mal parameters for numerical integration of the derived system

of integro-differential equations. Once the principal coordinates ofthe normal modes have been computed, we return to the 3D-FEMcode and compute the stress distribution inside the structure anddisplacements of the structure elements. Examples of the struc-ture responses to steep wave impacts are shown in Figures 4.

t = 0.0043 sec

t = 0.0099 sec

t = 0.0014 sec

t = 0.0157 sec

Figure 4: Deformations of NO96 box during steep wave impact.The box (1.1920 m × 0.99 m) is fitted into a larger cover box(1.655 m × 1.655 m). The centers of the boxes coincide. Waveheight Hw = 1.5 m, water depth H = 2 m, wall width L = 2 m,center of the cover box yb

c = 1 m zbc = 1.1725 m.

Breaking wave impact

Breaking wave impact occurs when the approaching wave isoverturning in front of the wall and the cavity filled with air orgas is closed at the impact instant. The presence of the cavity isa main feature of this type of the impact. After the time instant,when the cavity is closed, it starts to be compressed and oscillateslater on. These oscillations are of relatively large periods, whichare defined by the volume of the trapped cavity and the propertiesof the gas in the cavity.

In general case this problem is very complicated and requiresa solver with extremely high resolution. However, during thestage of high hydrodynamic loads the problem can be simplifiedassuming that the shape of the cavity does not change signifi-cantly during this initial stage. This implies, in particular, thatthe projection of the cavity onto the wall does not change duringthe stage of interest.

We consider a wave in a channel with vertical wall at x =0. The wave breaks in front of the wall and traps a cavity. Inthe projections of the wave and the trapped cavity onto the wallshown in Figure 5, all regions are assumed rectangular. In thehydroelastic analysis it is suggested to use the Bagnold model ofbreaking wave impact. This model was generalized to 3D case ofwave impact onto elastic structures. Within the Bagnold modelthe cavity projection onto the wall does not change with time andthickness of the cavity is assumed to be a function of time only.The cavity is modelled as a chamber covered with a piston of smallmass m. Motion of the piston is governed by the hydrodynamicpressure in the fluid and the aerodynamic pressure in the cavity.The pressure in the cavity pc(t) is calculated as

pc(t) = patm[(s0/s(t))γ − 1],

146

Impact region

Cavity

Elastic structure

xy

z

zc1

zc2

zp1

zp2

1

hw

yc1 yc

2yp1 yp

2 l0

Figure 5: Projections of the wave and the trapped cavity ontothe wall. The cyan stripped area corresponds to the impact re-gion, the white rectangular corresponds to the cavity, the grayrectangular is the elastic panel mounted into the wall.

where s(t) is the thickness of the cavity, s0 = s(0) and γ is theadiabatic index of the gas in the cavity. The hydrodynamic pres-sure in the fluid is calculated with the help of the acoustic modelfrom previous section but now the boundary condition on the wallhas the form

ϕx = −χ1(y, z) + wtχ2(y, z) + [v(t)− v(0)]χ3(y, z) x = 0,

where v(t) is the time derivative of the cavity thickness andχ3(y, z) is the characteristic function of the cavity projection ontothe wall.

Balance of forces acting on the free surface of the cavity pro-vides differential equation for the cavity thickness s(t), which canbe integrated once the hydrodynamic pressure over the wall isknown. Therefore, we calculate the hydrodynamic pressure, thecavity thickness and the deflections of the elastic panel mountedinto the wall at the same time. The problem is solved by usingthe method of normal modes. The normal modes of a structureare computed by 3D-FEM code. The obtained nonlinear systemof integro-differential equations with respect to the principal co-ordinates of the structural vibration and the cavity thickness isintegrated in time numerically. The numerical procedure and themethod were properly validated. Examples of structure responsesto breaking wave impact computed with the help of this hydroe-lastic model (with account for the gas cavity) and with the help ofthe model from the previous section (without cavity) are shownin Figure 6.

Aerated wave impact

The fluid of depth H , see Figure 3 (left), occupies the domainx < 0, 0 < y < L and 0 < z < H , where z = 0 corresponds tothe flat rigid bottom, and the vertical z−axis is directed upward.Before the impact a part of the wall 0 < z < H−Hw is in contactwith the fluid. A wave with vertical front (hydraulic jump) ofheight Hw approaches the wall in the region 0 < y < L andH−Hw < z < H at the velocity U and comes in contact with theelastic wall at the initial time instant t = 0. The wall is assumedpartly elastic. The fluid is assumed non-uniformly aerated closeto the wall, −D < x < 0. The aerated fluid is modelled as afictitious weakly compressible medium with reduced sound speed.Concentration of the gas in the fluid α(x, y, z) is given. In the

Without cavity With cavity

t = 0.0613 sec

t = 0.0867 sec

Figure 6: Deformations of a steel box during breaking wave im-pact. Wave height Hw = 0.65 m, water depth H = 2 m, wallwidth L = 2 m, center of the box is at yb

c = 1 m zbc = 1.25 m, box

dimensions 1 m×1m, the thickness of the each surface except thefront one is 2 cm, the thickness of the front plate is 2mm. Thecavity dimensions are L0.5 m×H0.3 m×W0.1 m. The center ofthe rectangular cavity is at yc

c = 1 m zcc = 1.5 m. Impact velocity

is 1m/s and the initial velocity of the cavity surface is 50cm/s.

main flow domain, x < −D, the fluid is not aerated and is treatedas an incompressible medium. The fluid flow is assumed 3D andpotential in the whole flow domain. The mathematical problem

H

-D

Impact area

Incompressible

Compressiblewith c(x,z)

U

1.00

1.0

Non

dim

ensi

onal

dep

th

Impa

ct a

rea

Are

a un

der t

he im

pact

Sound speed, c(z)/c0

0.75

1

0.50

Elas

tic

beam

Figure 7: Configuration of aerated fluid impact (left) and sketchof the sound speed dependence on the depth (right) used in nu-merical calculations.

is formulated in the non-dimensional variables in the same way asin the model of steep wave impact but for aerated fluid the waveequation is

c−2(x, y, z)ϕtt = ∇2ϕ,

where the sound speed c(x, y, z) is given function of spatial co-ordinates only in the region of fluid/air mixture and is set to be+∞ where x < −D/H = d. Note that the change of the fluiddensity due to aeration is not taken into account in this model.The velocity potential ϕ(x, y, z, t) and the structure deflection aresought in the forms

ϕ(x, y, z, t) =

∞Xk=1

φk(t)un(k)(x)Vm(k)(y, z),

147

t = 0.0064 s t = 0.0256 s

t = 0.0832 s t = 0.1408 s

Figure 8: Deformations of a steel box during aerated fluid im-pact. Wave height Hw = 0.65 m, water depth H = 2 m, wallwidth L = 2 m, center of the box is at yb

c = 1 m zbc = 1.25 m, box

dimensions 1 m×1m, the thickness of the each surface except thefront one is 2 cm, the thickness of the front plate is 2mm. Impactvelocity is 1m/s. The thickness of aerated part of the fluid is50cm.

w(y, z, t) =∞X

n=1

an(t)Wn(y, z)

where the functions Vm(y, z) and Wn(y, z) are orthonormal, re-spectively, in the cross section of the flow region, 0 < y < 1,0 < z < 1, and the time-dependent coefficients φk(t) and an(t) areto be determined from the body boundary condition and the waveequation with non-constant sound speed. These time-dependentcoefficients are computed as solution to the system of ordinarydifferential equations

8><>:

d2φ

dt2= S

−1 ◦ Tda

dt− S

−1 v − S−1 ◦ D

2φ

αd2a

dt2+ Ka + γdK

da

dt= −T

∗ dφ

dt,

where φ = {φ1, φ2, ...}, a = {a1, a2, ...}, v = {v1, v2, ...}, D =diag{d1, d2, ...}, K = diag{βμ4

1, βμ42, ...}, T = {Tkn}, S = {Skl},

dk and μn are eigenvalues which correspond to un(k)(x)Vm(k)(y, z)and Wn(y, z), respectively. In particular, the elements of thematrix S are given by the integral

Skl =

0Z−d

lZ0

1Z0

c−2un(k)un(l)Vm(k)Vm(l)dzdydx.

The derived system of ordinary differential equations is writtenin matrix form

dX

dt= MX + C, X(0) = 0

for extended unknown vector X(t) = {φ, φ,a, a}∗. The matrix M

and vector C are given by

M =

0BB@

0 I 0 0−S

−1 ◦ D2 0 0 S

−1 ◦ T

0 0 0 I

0 −α−1T∗ −α−1

K −γd α−1K

1CCA ,

C =

0BB@

0−S

−1v00

1CCA .

Note that the matrix M and the vector C are independent oftime. This makes it possible to derive the exact solution to thisinitial value problem. To do this, the matrix M is represented asa product M = R ◦ L ◦ R

−1, where R is complex matrix made byeigenvectors and L is the diagonal matrix formed by the corre-sponding eigenvalues of matrix M. Then, the solution is

X(t) =`R ◦ EL(t) ◦ R

−1 − I´ “

D−2v,0,0,0

”∗, EL(t) = eLt.

Convergence of the method was investigated and validated interms of the number of the modes and time step of integration.The method was also validated by performing several tests com-paring the results obtained: a) by using 3D and 2D numericalcodes; b) by using analytical and numerical modes for the elas-tic panel; c) by 3D code for aerated fluid and 3D code for fluidwithout gas bubbles. In all calculations the sound speed dependson the vertical coordinate z only. The sketch of sound speed asfunction of the vertical coordinate is shown in Figure 7 (right).

An example of structure response to aerated fluid impactcomputed with the help of the developed hydroelastic model isshown in Figure 8.

Steep wave impact using variational inequality

Parallel to the semi analytical models of steep wave impact,the equivalent numerical model based on variational inequalitymethod was also undertaken. The developments are based on thework by Gazzola (2007) where the basic principles and all the de-tails of the method can be found. The method was developed inthe context of the 3D hydroelastic Wagner impact which is muchmore complicated than the 2D one, because of the difficulties re-lated to the determination of the wetted part of the structure ateach time step. This method is extremely strong in applicationsto truly 3D cases and to complex geometrical configurations.Below we present basic principles of the method and some vali-dation results by comparing the predictions by this method withthe previously obtained semi analytical results.

We deal with the displacement potential in this method. Theboundary value problem for the displacement potential φ, (whichis the integral with respect to time of the velocity potential) isthen 8>>>><

>>>>:

∇2φ = 0 in Ωφ = 0 on the free surfaceφn = 0 on the bottom wall

φn = f + w − h− s on the wet surfaceφ → 0 in the far field,

(1)

where the functions f , w, h and s describe respectively the shapeof the wall before the impact, deflection of the wall for the hydro-elastic case, the wave displacement and the wave shape.We introduce the quadratic form:

148

a(u, v) =

ZZZΩ

∇u · ∇v dΩ. (2)

and the linear form l :

l(v) =

ZZImpacted wall

(f + w − h− s) · v dΣ. (3)

The problem 1 can be reduced to the following variational in-equality:

a(φ, v) ≥ l(v),∀v ∈ K, (4)

where K is a convex set of functions. The problem is well posed,and has a unique solution.

The structure is assumed elastic. Its deformation W is gov-erned by the classical structural dynamic equation which can beformally written in the following form:

M∂2W

∂t2+ C

∂W

∂t+ KW = P, (5)

where M , C and K are the mass, damping and stiffness matrices.P is the pressure linearised term from the fluid onto the structure.We denote by exponent the time step number n (with t = n·Δt).For example, W n−1 is the structural displacement at time instantt = (n− 1)·Δt. The non linear coupled system of equations takesthe following form:

8>>><>>>:

MW n+1 − 2W n + W n−1

Δt2+ C

W n+1 −W n−1

2Δt

+KW n+1 + W n−1

2 −R

Φn+1 − 2Φn + Φn−1

Δt2,

Φn+1 = H ◦ tR(W n+1),

(6)

where R is an operator coming from the fluid-structure non-conforming interface, and H is the fluid operator from the vari-ational inequalities. At time step tn+1 the unknowns are W n+1

and Φn+1. The coupled system (6)is solved at each time step byusing the Newton algorithm. The method is validated by compar-

-3

-2

-1

0

1

2

3

4

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

xi_1

, 10*

xi_3

t [s]

Mode_1 VIMode_3 VI

Mode_1 SWMode_3 SW

Figure 9: Generalised coordinates of the first and third structuralmode (VI - variational inequality, SA - semi-analytical).

ing the results with the semi analytical predictions for the steepwave impact on steel plate from Figure 3 (y1 = 0.5m, y2 = 1.5m,z1 = 1.0m and z2 = 2.0m at the instant of impact). The flexibleplate is 1 m × 1 m, and its thickness is 2 cm. The plate is 2.0cmthick and is simply supported on its upper and lower boundarieswhile it is free on the left and right boundaries. The structureis at rest at the initial time instant, Hw = 1.5m and the impactvelocity is 1.0m/s. The generalised coordinates of the first andthird mode are given in Figure 9. Good agreement is observed.

MODELING OF THE CONTAINMENT SYSTEM ANDTHE SHIP STRUCTURE

In this work we concentrate on the NO96 containment systemonly. This containment system is shown in Figure 10. It consists

Figure 10: NO96 containment system.

U, Magnitude

+0.000e+00+8.333e−02+1.667e−01+2.500e−01+3.333e−01+4.167e−01+5.000e−01+5.833e−01+6.667e−01+7.500e−01+8.333e−01+9.167e−01+1.000e+00+1.526e+00

Step: test74Increment 8507: Step Time = 8.7004E−03Primary Var: U, MagnitudeDeformed Var: U Deformation Scale Factor: +5.000e+01

ODB: test74_dp.odb ABAQUS/EXPLICIT Version 6.5−1 Tue Sep 23 17:05:29 Paris, Madrid 2008

Step: test74 Frame: 174

X

Y

Z

Figure 11: FE model of NO96 box.

of two insulation layers: primary and secondary boxes. Each ofthem is an assembly of plywood plates attached together by sta-ples. The two boxes are separated by an Invar membrane, whichrigidity, being negligible compared to plywoods, is not modeledhere. Fixation of the box onto the rigid wall is represented bya clamping boundary condition (all degrees of freedom equal tozero) on the bottom plate of the secondary box. The primarybox is the part in contact with the incoming wave. The four sidepanels are plywood 9mm thick. The box is filled with 7 plywoodbulkheads; 2 are 24mm thick and 5 are 12mm thick. Cover plateis composed of two plywood plates 12mm thick assembled withstaples; translational degrees of freedom are constrained betweenthese two plates to model the contact interaction. The secondarybox is the one located closer to the rigid wall. The four sidepanels are 9mm thick. The box is filled with 6 transverse bulk-heads 12mm thick, oriented 90◦ with respect to the primary boxbulkheads. The plate in contact with the rigid wall is a plywood6.5mm thick, and the top plate in contact with the primary boxis 12mm thick; translational degrees of freedom are constrainedbetween top plate of secondary box and primary box to model thecontact interaction. The bulkheads are assembled to the rest ofthe box with staples, which allow relative rotations between them.A simple and efficient way to represent this effect is to constrainthe rotations between these parts URpart2

⊥ = χ · URpart1⊥ ; χ can

149

be determined through comparison between calculated bucklingeigen-modes and buckling test results.

MODELING OF WAVE PROPAGATION

As mentioned in the abstract, during the SlosHel project, propermodelling of the wave propagation was identified as necessary fordefinition of impact conditions. This work was carried out inECM and more details about the different methods can be foundin Kimmoun and Scolan (2009).

A desingularized technique combined with conformal map-pings of the inner tank lead to an optimized algorithm for freesurface tracking method. Basically we use the theoretical develop-ments by Krasny (1985) and Tuck (1997). The tank is rectangularbut may have a bathymetry. Conformal mappings consist af suc-cessive changes of variable, so that the inner domain of the tank issimply turned into a half plane or even a quarter of a plan. Greenfunctions are hence calculated in this transformed plan with theiradditional images with respect to the solid boundaries. The cor-responding Rankine sources (log function) are distributed alonga line which is at a small distance of the actual free surface butoutside the fluid domain. This small distance is determined toget stable numerical results. More details can be found in Scolanet al. (2007) and quite similar works are shown in Bredmose etal. (2007). A time differential system follows from kinematic anddynamic free surface conditions, they read

dφ

dt=

1

2(U2 + V 2)− g(Y − h),

dX

dt= U,

dY

dt= V (7)

where (U, V ) are the cartesian coordinates of the velocity in

0

5

10

15

20

0 5 10 15 20

y (m

)

x (m)

2

3

4

5

6

7

0 1 2 3 4 5

y (m

)

x (m)

Figure 12: Bathymetry and corresponding wave profile beforeimpact.

the physical plane, (X, Y ) are the cartesian coordinates of themarkers along the free surface, g is the gravity and h the liq-uid depth. Elaborated algorithms are very stable and robust.Few computational resources are required: a typical time stepfor 200 markers on the free surface costs about 0.19s on a stan-dard processor. This technique allows parametric studies even ifwavemaker cannot be modeled. In fact, we use an artifact whichconsists in starting from an initial deformed free surface. Thisinitial condition depends on few parameters. Depending on themean free surface level and bathymetry a large range of waves canbe reproduced. As exposed in Scolan et al. (2007), kinematicsbeneath the free surface can be accurately reproduced by per-forming some identification with free surface profile only. Thatmakes it possible to yield kinematics in the overturning crest asinitial conditions to impact model. As an example the followinginitial profile is used with a bathymetry as drawn below (left)

The presence of the bathymetry is necessary to inhibit the strongrun-up which unavoidably occurs for this kind of wave generation.The bathymetry is thus an artifact. In order to avoid this arti-fact, further development by coupling of Boussinesq model withthe present code is under progress. More details are exposed inKimmoun and Scolan (2009).

COMPARISONS WITH SLOSHEL EXPERIMENTS

Validation of the FEM model

First we consider the validation of the finite element modeling byapplying pressures measured during full scale testing as a loadin a dynamic finite elements analysis. For each time step of theAbaqus/Explicit calculation procedure, the pressures (measuredat 20 points of the cover plate) are interpolated at each integrationpoint of the cover plate mesh.The results of this calculation are compared with values measuredduring full scale tests: reaction force at the back of the NO96 boxand strains in the NO96 box. Examples of these comparisons areplotted in Figure 13, showing the reaction force at the back of thebox and the strains evolution at one point in the cover plate. The

-50

0

50

100

150

200

250

300

0 0.005 0.01 0.015 0.02

forc

e (k

N)

time (s)

reaction force on the back of the NO96 box

dampingno damping

measure

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0 0.005 0.01 0.015 0.02

stra

in

time (s)

strains - cover plate #6

dampingno damping

measure

Figure 13: Comparison of reaction force (left) and strain in the coverplate.

comparison of reaction forces shows relatively good agreement.The strains however show an underestimation of the measuredvalues. The reasons for that lies, both in the inaccuracies of thepressure measurements and their numerical interpolation over thewetted part of the FE model. On the other hand, we should alsokeep in mind the complexity of the real N096 box so that the FEmodel can only be approximate.

Direct comparisons with Sloshel experiments

In the following section we present few preliminary comparisonsof the numerical results and the Sloshel experiments.During the experiments, it was difficult to distinguish which casecorresponds to Wagner, steep, breaking or aerated wave impacttype. For each experiment on the same graph we plotted mea-sured data by all pressure gauges located on one line. It wasfound three typical behaviors of the pressures shown in Fig. 14 Inthe different figures each line corresponds to the pressure evolu-tion measured by the gauge PN . The gauges are numbered fromhighest to the lowest position. From the figure 14 left, we can seethat there is a shift between two neighbor gauges. It is due to theangle between the wall and the wave front. On the other hand, ifthe wave front is almost vertical, all gauges record the change ofthe pressure almost at the same time, so the figure in the middlecorresponds to the steep wave impact. Finally, because of theoscillation of the cavity in the system, there are oscillations afterthe impact. This is shown in figure 14 right. From the theory,the maximum pressure during the impact is estimated as ρUc,

150

Concrete P03Concrete P04Concrete P05Concrete P06Concrete P07Concrete P08

150

100

50

0

0.00 0.05 0.10 0.15 0.20

Pres

sure

, kPa

time, sec

Concrete P03Concrete P04Concrete P05Concrete P06Concrete P07Concrete P08

50

0.0 0.1 0.2 0.3 0.4 0.5

0

10

20

30

40

Pres

sure

, kPa

time, sec

Concrete P03Concrete P04Concrete P05Concrete P06Concrete P07Concrete P08

350

300

250

200

150

100

-50

0

50

0.0 0.40.30.20.1

Pres

sure

, kPa

time, sec

Figure 14: Three typical behaviors of the pressures in the ex-periment: Wagner, steep and breaking wave impacts. Pres-sure gauges are counted from the top.

where ρ is the fluid density, U is the impact velocity, and c isthe sound velocity in the fluid. If during the steep wave impact,the maximums of peaks happen almost in the same time, butthe values are different, then we can suppose that it is becauseof variation of the sound speed and, thus, aerated wave impactshould be considered. Following the described method above, weclassified the experimental data to the Wagner, steep, breakingand aerated fluid impacts.

In Figure 15 comparison between numerical and experimentalresults for breaking wave impact is presented. Both the pressurein the cavity and the strain in the cover plate are shown. Impactparameters are derived from Sloshel iCAMs measurements, whichallow us to determine the shape of the wave just before it impactsthe wall.

In the present case, the pressure from the experiments is de-caying due to some physical effects, while in the theory it os-cillates. In order to include the decaying effects, an artificialdamping was introduced in the numerical code. The strains areobtained by multiplying the general modal coordinates an by themodal contributions which are obtained by Abaqus. As we can seethe agreement is reasonably good. The most violent impacts ob-

CalculationsExperiment S042

Pres

sure

in th

e ca

vity

, kPa

200

0.40.30.20.10.0

0

50

100

150

time, sec

CalculationsExperiment S042

400

0.200.150.100.050-300

-200

-100

0

100

200

300

μStr

ains

time, sec

Figure 15: Pressure in the cavity (left) and strain (right) inthe cover plate during the breaking wave impact.

served during Sloshel measurement campaign were of flip-throughtype. The code developed using Wagner approximation is used forthe simulation of these impacts: actual shape of the wave frontjust before the impact was used with polynomial interpolationof the iCAMs measurements. In Figure 16 structural responsecalculated for one impact at two locations on the cover plate ofthe NO96 box is presented. The numerical results obtained withand without hydroelastic coupling are presented. A fairly goodagreement can be observed.

-0.0005

0

0.0005

0.001

0.0015

0.002

0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08

strain gauge #05

sloshel measurecoupled simulation

non coupled simulation

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08

strain gauge #07

sloshel measurecoupled simulation

non coupled simulation

Figure 16: Comparison at strain gauges #5 (left) and #7 (right).

CONCLUSIONS

Sloshing induced impacts are very important in the design of aship tank. Many physical effects may have to be considered suchas gas cushion, liquid compressibility, boiling of liquid cargoesand hydroelasticity. When analyzing sloshing impacts, one mustalways have the structural response in mind. An important con-sideration is the time scale of a particular hydrodynamic effectrelative to wet natural periods for structural modes contributingsignificantly to large structural stresses. More structural modesmay be included for membrane structures analyses than for steelstructures. Some of the important structural modes for mem-brane structures may have relatively lower natural periods thanfor steel structures. If the time scale of a hydrodynamic effectas for instance acoustic effects is very small relative to importantstructural natural periods, the structure has a negligible reactionand therefore the particular hydrodynamic/hydroelastic effect canbe neglected. When the hydrodynamic loads occur on the timescale of important structural modes, hydroelasticity must be con-sidered. This implies that the fluid (liquid, gas) flow must besolved simultaneously with the dynamic elastic structural reac-tion.

In spite of all the efforts which were made in order to properlysolve the sloshing impact problems, it is fair to say that still lotof uncertainties persist and it is not fully clear how they could beproperly solved. The full scale monitoring of the real LNG shipsduring their normal operation would certainly be very helpfulbut, for the time being, it seems to be very difficult to performthese kinds of measurements. Recently reported damages on largeLNG carriers clearly show that the important improvements of themethodology for structural assessment of the LNG containmentsystems are necessary.

In this paper we proposed rational methodology which mightrepresent a step forward toward the final solution of this diffi-cult problem. Indeed, only relatively simple models can afford fora quick sensitivity analysis of the structural responses with re-spect to the different physical phenomena which occur during thesloshing impacts. In spite of rather important simplifications webelieve that these models are representative of what happens inreality. It is hopeless to try to look for quantitative assessment ofthe structural responses during the lifetime ship operations, be-cause of an infinite number of cases which need to be considered.Only approximate approaches, as those presented in this paper oreven simpler approaches based on estimates of loads and stressesbut not on their evaluations, seem to be of practical meaning. Webelieve that the combination of either small scale model tests, orglobal CFD sloshing analysis, for determination of the impacttypes, with the simplified local hydroelastic analysis, is the onlyfeasible and practical approach today. Having say that, it is clear

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that more validations of the present models must be done, and theSloshel project seems to be a good place to achieve that because,in this project, the complex structure of containment system isrepresented at full scale.

The opinions expressed in this paper are those of authors aloneand do not necessarily represent the views of Sloshel JIP consor-tium.

ACKNOWLEDGEMENTSBureau Veritas would like to acknowledge the support provided bythe other Sloshel consortium members that have made the Sloshelproject possible: ABS, LR, Ecole Centrale Marseille, Chevron,ClassNK, DnV, GTT, MARIN and Shell.

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