Transfer of Non-Linear Seakeeping Loads to FEM Model Using Quasi Static Approach

Date June 2009

Author

Tuitman, J.T., FiX. Sireta, S. Malenica and T.N. Bosman

Mdress

Delft University of Technology

Ship Hydromechanics Laboratory

Mekelweg 2, 2628 CD Deift

Transfer of Non-Linear Seakeeping Loads to

FEM Model Using Quasi Static Approach

by

3.T. Tuitman, F.X. Sireta, S. Malenica and

T.N. Bosman

Report No. 1633-P

2009

Proceedings of the 1gth International Offshore and Polar

Engineering Conference, Osaka, )apan, June 21-26, 2009,

ISBN 978-1-880653-53-1, ISOPE2009

TU Delit

Deift University of Technology

ISBN 978-1-880653-53-I

_{(Vols. 1-4 Full Proceedings Set)}

ISSN 1098-6189

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The Proceedings of

The Nineteenth

_{(2009) international}

OFFSHORE AND POLAR

_ENGINEERING

CONFERENCE

Osaka, Japan, June 2 1-26, 2009

Proceedings of the Nineteenth (2009) International Offshore and Polar Engineering Conference

Osaka, Japan, June 21-26, 2009

Copyright © 2009 by The International Society ofOffshore and Polar EAgineers (ISOPE)

ISBN 9 78-1-880653-53-1 (Set); ISSN 1 098-618

'fransfer of Non-Linear Seakeeping Loads to FEM Model Using Quasi Static Approach

J. T Tuitnian', FX, Sireia2,.S Malenica2 and TN. Bosnian' Delft University of Technology, The Netherlands

2Bureau Veritas, France

ABSTRACT

This paper presents a methodology to transfer the non-linear_time do-main seakeeping loads to Ihe 3Dstructural finite element_{method (FEM)} model. The presented approach ensures that the loading and accelera-tions of the seakeeping and FEM calculation is almost indentical. The applied accelerations and forces at the FEM model ensures a well bal-anced structural model.

KEY WORDS: non-linear, seakeeping, structure loading and_response

INTRODUCTION

To be able to investigate the local response of the shipstructurein waves

it is necessary to transfer theseakeeping loadingto_{a 3D structural model}

of the structure. The usual approach 1510 transfer the linearfrequency

do-main seakeeping loads calculatedby a hydrodynamic boundaryelement

method (BEM) to the structural finite element method (FEM)model. The transfer of the hydrodynamic loading to the structural model is not trivial. The meshes used for the BEM and FEM models are normally quite different. A common approach is to interpolate the total hydrody-namic pressure to the FEM mesh. This interpolation will cause some inaccuracies in the loading. An other problem is the accelerations of the FEM model. The accelerations should fully balance the_applied pres-sure loadingto avoid additional stress by the supportreactions, Supports or constrained nodes arenecessary forsolving the quasi static structural problem. Reaction forces can easily be avoided by using the "relieve

acceleration" in the FEM package. The FE-packages will calculate the accelerationsneededto fully balance the loading. But theremay be a

dif-ference between these "relieve" accelerations and the original

seakeep-ing motions. If so, there will bealsoa'discrepancy between _{loading and} accelerationsbecause the hydrodynamicforcesare basedon the motions.

The approach presented in (Malenica et al., 2008) ensures that the lin-ear hydrodynamic loading is consistently transfered tothe FEM model. In this approach the hydrodynamic boundary value problem (B VP)is

601

solved using thehydrodynamic mesh, but interpolation isavoided by

in-tegrating the potentials over the FEM mesh. The equation of motions

are solved using these coefficients and the mass distribution of theFEM model. The resulting loading can directly be transfered to the FEM model without any interpolation. The. calculated accelerations_balance the hydrodynamic loads fully which ensures that the support reactionare negligible. The forces and accelerations at the structural model _{are very} close to the results of the normal seakeeping calculation.

Linear seakeeping analyse can be utilised as long the wave height re-mains small. For calculation of the structural response in extreme_sea states it is necessary to utilize non-linear time domain seakeeping theory andtransfer these non-linear loads to the structural model.

In this paper we present an approach to transfer the loadingfrom a time domain seakeeping program with non-linear load components to the FEM model. The approach as presented in(Malenica et al, 2008) is extended for the time domain withnon-linear load_{components. The} pre-sented method ensures that both theseakeeping forcesand_{accelerations} are consistently transfered to the FEM model.

STRUCTURAL ANDHYDROMODELS

The staflingpoint for the calculation is theFEM model. The FEMmodel

should have the correct mass distribution, this is normally achieved_by adding additional masses andlor increased density to account for equip-ment and cargo weights. The mass distribution of the structural model will be used for the seakeeping calculations.

The elements that represents the ship's outer hullare extracted_{from the} FEM model and will be used for all remaining steps. The BEM_{mesh is} created based on the FEM model. The first step is creating, in longitu-dinal directions, many transversal sections, see figure I. These sections are created by making acut trough the FEM mesh. Knucklepoints, like the skeg, are preserved and the remaining section is smoothed. Thenext

step is to balancethe structural model in still water. The.draft and trim are obtained using an iterative procedure. The BEM mesh iscreated

us-ing a meshus-ing program, optimised for hydrodynamic_{meshes, with the}

panel. The translations of the location for pressure evaluation are usually very small because both meshes describes the same geometry. sections and calculated draft and trim as input.

ILIW ..I5 UI

iI II iIi'l

't\L%

II

'!ifij

/1

)

Figure I: Ship lines extracted from FEM model.

The potential around the ship is the basic solution of the hydrodynamic

BVP. Usually the pressures caused by the potentials are integrated over

the hydrodynamic mesh to obtain the hydrodynamic coefficients for the rigid body modes. The panel pressures can be obtained by evaluating the potentials at the panel centers. For a consistent transfer of hydro-dynamic pressure to the FEM model, the FEM-mesh will be usedto

integrate the potentials to the rigid body hydrodynamic coefficients. As will be explained later, the hydro dynamic loading will be transfered by nodal forces instead of pressures. Therefore the potentials have alsoto be integrated over the FEM mesh to nodal force coefficients.

The hydrodynamic pressure are integrated over the FEM mesh by using Gauss points at the FEM elements. Note that these Gauss points are only

used for integration and have nothing to do with the Gauss points used in the theory of the structural FEM program. The accuracy of the inte-gration can be controlled by changing the number of Gauss points per

element. The pressure evaluating at the Gauss points at the FEM mesh should be done carefully. Although the BEM and FEM mesh will model

the same geometry, the meshing itself can be very different. Especiallyat

curved parts of the geometry the Gauss points at the FEM mesh will not

be exactly at the HEM mesh. A part of the Gauss points will fall inside the BEM mesh. This will cause problems with forward speed. The

hy-drodynamic pressure will also depend on the gradient of the potential for

forward speed problems. The gradient of the potential is discontinuous across the hydrodynamic mesh. Due to this discontinuity the pressure

can only be evaluated at or outside the hydrodynamic mesh.

For the integration of the hydrodynamic pressures over the FEM mesh the pressure is evaluated for every Gauss point. It is first checked if the point is outside the hydrodynamic mesh before evaluating thepressure

at a Gauss point. If this is not the case, the pressure will be evaluated at a point near the Gauss point which is just outside the hydrodynamic mesh to ensure a valid evaluation of the hydrodynamic pressure. This is illustrated in figure 2. The figure is a zoom of the mesh of the at

body seen from inside. The dots are the Gauss points and the small lines

indicate the translation towards the point of pressure evaluation, If a

Gauss point is inside the BEM mesh the pressure is evaluated at a point

moved to the projection of the original Gauss point at the closest BEM

SEEKEEPING CALCULATIONS

BEM mesh

- FEM mesh _j- Gauss point

- Translation

Figure 2: Translation from Gauss points for pressure evaluation

The theory used in the non-linear seakeeping calculation is described in

(Tuitman et al., 2009). The theory accounts for the non-linear Froude-Kryloff and hydrostatic forces. The diffraction and radiation is kept lin-ear in this theory. The coefficients for the diffraction and radiation are obtained by solving a boundary value problem in frequency domain

us-ing the potential flow assumption.

Frequency domain

The total velocity potential is decomposed into the incident, diffracted

and 6 radiated components:

{

= 0 in the fluid

z=0

on Sb

ZIW)]=0

R.cx

(I) (2)

Where Sb is the ship surface. V denotes the body boundary conditions

which depends on the considered potential:

ClçQD )co' UWR

aii

au'

au where: P1 -incident potential - diffraction potential WR, -j-th radiation potential

- rigid body motion

The so called encounter frequency approach is used to include the

for-ward speed. This is a common approach as it is still very difficult to solve

the BVP with the forward speed included correctly. Using the encounter

frequency approach the BVP reduces to the zero forward speedcase:

This BVP issolved numerically using constantsource distribution over

the hydro mesh panels. After solving the BVP, the diffracted _and radi-ation pressure components for every Gauss point_{at the structural mesh}

are calculated:

PD= ulJpcoD ₍₄₎

PR, =U)PcOR, ₍₅₎

The hydrodynamic coefficients for the total body areobtained by inte-grate these pressures over the hull using the Gauss points of the FEM

mesh: F1.D

=

IL PDIt,dS

(6) wA + iwB

=

IL1 PRjit,

dS (7) Time domain

The non-linear seakeeping problem is solved in the time domain using thecummings equation (Cummins, 1962):

(A()+m)+B(oo).+f K(tr).r)dr=

JD(E,t)+Jk((t)+fh()+f5(()

(8) Where:

m genuine mass matrix

A added mass matrix

B damping matrix K retardation functions ID diffraction force

f/k

Froude-Kryloff force 15 hydrostatic force Is gravity force displacements

It has been shown in (Ogilvie, 1964) that the retardation functionscan

becalculatediusing the frequency domain damping:

K(t)

=

f (B(w) -

B(oo)) .cos(w ₎dü (9) The location of the ship (ã = (x, y)) is equal to:

+ Ui.

(10) The total displacement in the direction of the waves is:

Xw=XCO5/L+YSIflp

_(II)

Where p isihe heading of the ship.

The wave elevationof asingle wave component at location x,, is equal

to:

((t,xv,w)

(12) with: 603 (,. wave amplitude E phase angle k wave number

This results in a total wave elevation of:

Nfrq

((t,x)

=

The Froude-Kryloff force by thewave is:

pa = _{z < 0 and z <((t, c,,,)}

= pg((t, a,,)

(i

<(t

))

z > 0 and z <((t, a1,)

Pa=O

z((t,x)

The hydrostatic pressure isequal to:

fPh = p92 z <Oandz <((,x,,,)

= 0 z

0orz

((t,x,,)

Both the Froude-Kryloff and hydro Static pressure areintegrated over the structural mesh to obtain the force vector.

The diffraction force is equal to:

Nlreq

j'D(t, x) =

(o(wI) (R(iD())cos(wt + e((w) + xk(w,)) +

sin (wf + Ei,) +

xk(w1)))

(16) The gravity force is the total mass times gravity acceleration vector

be-cause the center of gravity isusedas origin for all forces

The time domain seakeeping theory accounts for large amplitude mo-tions. The rotations cause a difference between the earth and body

system. Euler angels are used to define the transformation matrix. From the body system to the earth system the transformation matrix is (Lewandowski, 2004):

s indica es the sin function using of the next variable and e is the cosine

function. i, 0 and i' are the roll, pitch and yaw. The transformation matrix from the earth system to the bodysystein is:

Tb = T (18)

It is assumed that the linear hydrodynamic coefficients act in the earth

reference frame. Equation (8) is solved in the body system, see (Thitman

et al., 2009). Therefore, all vectors and matrix's which represents the

linear hydrodynamic components are transfered to the body system using (18).

The result of equation (8) is the acceleration vector, . This vector is

transfered to the earth system and integrated with a fourth-order Runge-Kutta method to velocities and displacements of the ngidbody modes in the earth reference system.

T,

= C4,bCO .qzJ'c9

sO

cibs9Ø - 3/)C) s/s9st + ebr, c9s CT,bSOS4, +8J)CJ1

s,cti -

cij's4, cOc (17)

TRANSFER LOADS TO STRUCTURAL MODEL

The program to transfer the time domain seakeeping _{loads to the}

struc-tural model is a modified version of the time domainseakeeping program

presented in the previous section. The program is now used as post-processor of the results of the time domain results. In theory it would

be possible to generated the structural load case directly during the

sea-keeping calculations, but there are two major disadvantages_{using that}

approach:

The retardation function and convolution integral has_{to be}

evalu-ated for every point-at-the structural mesh This willLbeasignificant

overhead.

Usually only a smallipart of the time domain calcUlationis selected

to perform a structural analysis. This is-Usually the interval where

the most extreme loading occurs. Using-apost-processor allows to select these events.

The hydrodynamic loads will be appliedto the structural_{model as nodal} forcesinstead of pressures. The main _{reason for using this approach is} that the seakeepingtheory accounts for large amplitude motions. Asthe

linearhydrodynamic components-are assumed to-act in the earth system,

the force vector should be transfered to the body system to be used in

the structural calculation. This can not be done whenpressures

load-ing is used. Another reason is that the pressure integration routines_of the different FEM packages are-avoided by using forces These differ-ent routines might result in a slightly differdiffer-ent total loading which will disturb theperfect balance-of the-structural model.

Load case

The balance between the hydrodynamic loading and acceleration- forces at the structural model is ensured by solving the-dynamic equation again.

The nodal contribution of all hydro dynamic forcecomponents are cal-culated. These forces are based on thedisplacements and velocity calcu-lated by the seakeeping calculation. The acceleration is-solved using:

nnod

(A(oo) +

_{m) . ((t) =}

+

(j(t,

+

iI

+ j(t,))

(19)

This equation is- similar to (8). The superscript ' indicates force at a nodes of the-structural model. The infinite added mass is still at the left hand-side to ensure that the hydrodynamic loading at the structural model is consistent with the accelerations. Note that the summationof the nodal

forces in equation (19) should give exactly the total force vector as the

total force vector in equation (8) as the same mesh is usedto integrate the pressures. However, there will be a small differences in the radiation force as a different approach is used for calculation of the nodal forces for

this force component as will be explained in the last part of this section.

The total:hydrodynamic loading atthe structUral-model is calculated after

solving the.motion equation:

= j(t,) + f((t)) + fA(0).() + If k,(t,)

+ ft',1(t,))

(20) The hydrodynamic-force is balanced by nodal acceleration:forces:

=in (aj, + g) (21)

Where:

m'

nodal mass

tZ, accelerationsin body system at node

7 gravity acceleration vector in body system

The hydrodynamic and gravity forces are combined and usedas load case for a- quasi-static FEM calculations.

Nodal forces components

The remainder-of this section explains the-calculation of the nodal forces needed in-equation (19) and (20).

Both the diffraction and radiation pressure (5) areintegrated to nodal val-ues using the Gauss points at which the pressures are-calculated. These

nodal values are indicated as F" and F' and are still in frequency

domain.

The radiation force isthe most difficult force cOmponent to compute An

approach couldibeto create and evaluate the retardation function for

ev-ery structural node This approach should result in exact the same total

radiation force as computed by the seakeeping program. But the-calcu-lation effort to do this is huge. An other approach could be_{to use just} the frequency domain RAO values. This will not be correct if itis based on the motion calculated in the frequency domain. Therefore _{a Fourier}

transform-on the seakeeping velocity is used to obtain the amplitudeand

phases-of thedifferent frequencies. These should result ina

total-radia-tion force-that is very close to-the radiatotal-radia-tion force calculated by the-sea-keeping program There will be some differences because the_creation

of-the retardation function (9) assumes that the linear hydrodynamic

co-efficients fully comply to the Kramers-Kronig relations, which is not always the case when forward speed- is included. Also the length and discretization of the retardation function may cause some differences.

The discretized Fourier transform is the last source of possible

differ-ences between results of the-radiation force calculated using retardation functions or a frequency reconstruction.

The frequency content of the velocity in the earth reference frame is

equal to:

(w) = ₍₂₂₎

The nodal force in the body reference frame due to radiation, withthe infinite frequency added mass-subtracted is:

nfreq

j;(i) =

Tb

-

{i (f(i)

-

f'(oo))

- ((1(i)

+

(J'(i)) -

(1(i)

-

e) }

(23)

The nodal force by infinite frequency added mass is:

= Tb -

_(1(oo))

- t) (24)

The nodal diffraction is equal to:

Nfrcq-J1(t, x) = Tb

(w1) ((P') cos(wt+e<(w)-i-xk(w))

NUMERICAL RESULTS

A typical frigate model is used to illustrate the presented theory. The used structural (FEM) and hydro (BEM) meshes are_{presented in figure}

3. The hydro model is obtained by creating lines from the _structural

model. Unfortunately, the used mesher to create the BEM mesh is un-able to make a mesh with the correct deck height. This can give some difference when non-linear calculations are performed using the hydro mesh. The linear calculations uses only the mesh under the still water line which is very close to the original shape.

Figure 3: Structural and hydro model of frigate

The loading and response is calculated in a extreme_{sea state with a} sig-nificant wave height of 10 meters. The frigate has no velocity and is heading towards the incoming waves. The frequency domainsurge, sway

and yaw are imposed by adding Lagrange multipliers (Thitman et al., 2009) to equation (8). This avoids drift during the time domain

calcu-lations and is easier than adding springs/dampers or full propulsion and manoeuvring system to keep the ship at the correct speed and heading. Seakeeping calculation

The seakeeping loading and response is calculated for the same wave train using three different methods:

Hydro seakeeping This is the normal or classical approach_{to calculate} the time domain non-linear seakeeping response of the ship. The hydro mesh is used in all steps of the seakeeping calculation. The structural mesh is not used in this case.

Struct seakeeping This is the seakeeping calculation as described in this paper. The hydro mesh is used to solve the BVP and the struc-tural mesh is used for integrating the hydrodynaniic coefficients and calculating the non-linear terms in the time domain

calcula-tion.

Struct FEM This is the calculation or reconstruction of the nodal forces at the structural mesh. The summation of these nodal forces_are presented as result of this calculation. There are no motion results for this calculation because it is a post-processing of the motions of the structural seakeeping calculation. The rigid body forces should be almost indentical to the forces found in structural seakeeping

calculation.

605

The mass distribution of the structural (FEM) model is used for all

cal-culation methods. This ensures that difference in response can only be caused by difference in the calculated loading.

The different force components in heave direction can be found in figure 4. Figure 5 shows the total loading for heave and pitch. The _resulting motion is shown in figure 6. The accelerations are not shown. As the used mass distribution is the same for all methods the difference in

ac-celeration is identical to the differences in total force.

The hydrodynamic loading and resulting motion calculated usingthe dif-ferent methods are very comparable. The diffraction force (16) is

identi-cal for all method. This force depends only on the location of theship,

which is the same for all methods as the frequency domain_{surge, sway} and yaw are imposed, Obtaining the same diffraction force_{proves that} integration of the hydrodynamic coefficients over differentmeshes gives

consistent results.

There are small differences in the calculated motion between the sea-keeping calculation with the hydro and structural mesh. These

differ-ences are caused by differdiffer-ences in one or more force components. But as soon there is a difference in the resulting motion, there will bea

dif-ference in Froude-Kryloff, hydrostatic, gravity and radiation loading _as these depends on the motion. The gravity loading depends only on the mass distnbution and the orientation of the ship so this component will not cause the differences in motion. The linear hydrodynamic coeffi-cients are almost identical for all methods, as already shown forthe

diffraction force. The radiation force is also based on the linear hydro-dynamic coefficients and will result in almost equal force if the motions

would be equal. The most likely cause of the difference in motion are the

Froude-Kryloff and hydrostatic loading. Due to the difference in geome-try of the aft body, see figure 3. the calculated load will be a bit different.

Note that in this severe sea state the aft deck is sometimes wetted. A Fourier transform (22) together with a frequency reconstruction (23) is used to calculate the nodal retardation force. This approach avoids_to calculate and evaluate retardation functions for every node, whichsaves

a lot of calculation effort. The last approach is used for the rigid body motions in the time domain seakeeping calculations. Figure 4 shows that there is a large difference in the result of the nodal structuraland

seakeeping radiation force at the start of the calculation. The seakeep-ing radiation force is incorrect at the start of the calculation because the evaluation of the retardation function needs the velocity history which is lacking at the start of the calculation. As soon_{as there is enough history,} the diffraction force is almost the same for the structural seakeeping and the nodal forces. There are some minor differences in the remaining sig-nal, but they are hardly visible in the graph. This results_{proves that the} approach used to calculate the nodal diffraction forces, is consistent. After 10 seconds there is almost no differences between the total load-ing calculated by the structural seakeepload-ing and nodal loadsat the FEM

model. This indicates that the hydrodynamic and acceleration loading at the structural model represent the seakeeping loading and accelera-tions very well. The good agreement between the seakeeping calcula-tions with the hydrodynamic mesh and structural mesh shows that the structural mesh can be used to perform seakeeping calculations.

15 10 8 6 -2 -4 -8 10 Hydroseakeeping --Struct seakeeping Siruct FEM-Hydro seakeeping SIruct seakeeping StructF M

Figure 4: Force components

Hydr seakeepin'g Struct seakeepIn Struct FE E 2 0 a-12 to 8

-4--6 a 0 0.3 0.2 -0.1

0'

-0.1 -0.2 0.3 --0.4 -0.5 -0.6 0 3 2.5 2 1.5 0.5 0 -0.5 -1.5 -2 -2.5 0 10 8 6 4 a, 20 40 60 Time (sJ - _{Hr,oseakeeping} - seakeeping Struct FEM 20 40 60 80 100 Time (SI

Figure 5: Total loading

I-Iydroseakeepidg

-J---Stnict seakee in---StwCtI?E

80 100

20 40 60 80 100

Time [sJ

Figure 6: Motions using structural and hydro models

20 40 60 80 100 Time 181 -33 -33.1 -33.2 -33.3 a,

I

-33.4 -33.5 0 20 40 60 Time Isi 80 100 80 100 0 20 40 60 Time 181 0 20 40 60 80 100 Time (sJ

FEM results

The structural response is calculated using a quasi static FEM solution. Some nodes are constrained to be able to solve the quasi static_problem. These nodal constraints are placed such that theysuppress the rigid body motions but will not cause any addition loading due to lateral contraction.

The structural response is calculated for all time steps of the seakeep-ing calculation. The resultseakeep-ing forces at the constrained nodes indicates how well the hydrodynamic and acceleration loads are balanced. The maximum force at a constrained nodes is below the 40N for all time

steps. _{This force is neglectable compared to the total loading at the}

model which shows that the model is well balanced. The found reac-tion force is in the order of i0 _{of the total loading at the ship which} can be explained by the numerical accuracy of the FEM package. The stress in the deck between the superstructure and hangar_is evalu-ated in figure 7. The effect of the non-linear loading is investigevalu-ated_by doing an other seakeeping and structural calculation where the Froude-Kryloff and hydrostatic loading is linearised. The stress due to the still water bending moment is subtracted from the non-linear results_{to make} a comparison possible. 60 40 20 0 -20 -40 -60 -80 -100 120 0 Nonlinear -Linear 20 40 60 Time_(sI

Figure 7: Stress in deck

Taking into account the non-linear Froude-Kryloff and hydrostatic forces

decreases the stress due to hogging and increase the stress due to sag-ging. These differences in structural response is very interesting when the ultimate limit state of the ship structure is investigated.

607

CONCLUSIONS

A methodology to calculate the structural loading based on a time do-main seakeeping calculation with non-linear force components has been presented. The numerical comparison shows that the presented method ensures that the calculated loading and acceleration at the structural model are consistent with the original seakeeping loads and

accelera-tions. The negligible support reactions shows that the hydrodynamic forces are fully balanced by the accelerations forces at the structural

model.

This approach can be used to included the effect of non-linear

Froude-Kryloff and hydrostatic forces when the structural_{response is} investi-gated in heavy seas.

The next step us to add flexible modes and slamming loading_{to the} pre-sented approach. Then it will possible to investigate the effects of whip-ping and springing on the structural response. Slamming_{can be included} by combining (Malenica et al., 2009) and (Tuitman and Malenica, 2009) in the presented theory.

ACKNOWLEDGEMENT

A part of this work was done within the MARSTRUCT European project.

REFERENCES

Cummins, W. (1962). The impulse response function and ship motions.

sch:ffstechnjk 47,101-109.

Lewandowski, E, M. (2004). The dynamics of marine craft. World Sci-entific.

Malenica, ., E. Stumpf, F. Sireta, and X. Chen (2008). Consistent

hydra-structural interface for evaluation of global structuralresponses

in linear seakeeping. In 26th OMAE Conf

Malenica, _{., S. Tomasevic, F. Sireta, J. Thitman, and I. Schipperen} (2009). Local hydro-structure interactions due to slamming. In2nd

ml. Conference on Marine Structures.

Ogilvie, T. F. (1964). _{Recent progress toward the understanding and} prediction of ship motions. InProc. 5th Symposium on Naval Hydro-dynamics.

Thitman, J. and . Malenica (2009). Direct coupling between seakeeping and slamming calculations. _{Journal of Engineering for the Maritime}

Environment submjgfed:JEME-S-09-000J4

Tuitman, .J., . Malenica, and R. van 't veer (2009). Generalized

modes for time domain seakeeping calculations. _{Applied ocean}

re-search submitted:APOR-D-08.00038