dramas of the sea_ episodic waves and their impact on offshore structures.pdf

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Dramas of the sea: episodic waves and their impact on offshore structures

Gunther F. Clauss

Division of Naval Archaeology and Ocean Engineering, Technical University of Berlin,

SG 17, Salzufer 17-19, Berlin 10587, Germany

Received 10 July 2002; accepted 11 August 2002

Abstract

For the design of safe and economic offshore structures and ships the knowledge of the extreme wave environment and related

wave/structure interactions is required. A stochastic analysis of these phenomena is insufficient as local characteristics in the wave pattern are

of great importance for deriving appropriate design criteria.

This paper describes techniques to synthesize deterministic task-related rogue waves or critical wave groups for engineering

applications. These extreme events, represented by local characteristics like tailored design wave sequences, are integrated in a random or

deterministic seaway with a defined energy density spectrum. If a strictly deterministic process is established, cause and effect are clearly

related: at any position the nonlinear surface elevation and the associated pressure field as well as the velocity and acceleration fields can be

determined. Also the point of wave/structure interaction can be selected arbitrarily, and any test can be repeated deliberately. Wave

structure interaction is decomposable into subsequent steps: surface elevation, wave kinematics and dynamics, forces on structure

components and the entire structure and structure motions.Firstly, the generation of linear wave groups is presented. The method is based on the wave focussing technique. In our approach the

synthesis and up-stream transformation of arbitrary wave packets is developed from its so-called concentration point where all component

waves are superimposed without phase-shift. For a target Fourier wave spectrum a tailored wave sequence can be assigned to a selected

position. This wave train is linearly transformed back to the wave maker andby introducing the electro-hydraulic and hydrodynamic

transfer functions of the wave generatorthe associated control signal is calculated.

The generation of steeper and higher wave groups requires a more sophisticated approach as propagation velocity increases with wave

height. With a semi-empirical procedure the control signal of extremely high wave groups is determined, and the propagation of the

associated wave train is calculated by iterative integration of coupled equations of particle tracks. With this deterministic technique freak

waves up to heights of 3.2 m have been generated in a wave tank.

For many applications the detailed knowledge of the nonlinear characteristics of the flow field is required, i.e. wave elevation, pressure

field as well as velocity and acceleration fields. Using a finite element method the velocity potential is determined, which satisfies the Laplace

equation for Neumann and Dirichlet boundary conditions.

In general, extremely high rogue waves or critical wave groups are rare events embedded in a random seaway. The most efficient and

economical procedure to simulate and generate such a specified wave scenario for a given design variance spectrum is based on theappropriate superposition of component waves or wavelets. As the method is linear, the wave train can be transformed down-stream and up-

stream between wave board and target position. The desired characteristics like wave height and period as well as crest height and steepness

are defined by an appropriate objective function. The subsequent optimization of the initially random phase spectrum is solved by a

sequential quadratic programming method (SQP). The linear synthetization of critical wave events is expanded to a fully nonlinear

simulation by applying the subplex method. Improving the linear SQP-solution by the nonlinear subplex expansion results in realistic rogue-

waves embedded in random seas.

As an illustration of this technique a reported rogue wavethe Draupner New Year Wave is simulated and generated in a physical wave

tank. Also a Three Sisters wave sequence with succeeding wave heights Hs,,2Hs,,Hs, embedded in an extreme sea, is synthesized.

For investigating the consequences of specific extreme sea conditions this paper analyses extreme roll motions and the capsizing of a RO

RO vessel in a severe storm wave group. In addition, the seakeeping behaviour of a semi-submersible in the Draupner New Year Wave,

embedded in extreme irregular seas is numerically and experimentally evaluated. q 2002 Elsevier Science Ltd. All rights reserved.

Keywords:Rogue waves; Wave/structure interaction; Deterministic seakeeping tests

0141-1187/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.

PII: S 0 1 4 1 - 1 1 8 7 ( 0 2 ) 0 0 0 2 6 - 3

Applied Ocean Research 24 (2002) 147161

www.elsevier.com/locate/apor

E-mail address:[email protected] (G.F. Clauss).


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1. Introduction

The impossible will happen one day. Considering rogue

wave events as rare phenomenaaccording to Murphys

lawbeyond our present modelling abilities, Haver [1]

suggests that freak waves (unexpected large crest height/

wave height, unexpected severe combination of wave height

and wave steepness, or unexpected group pattern) should be

defined as wave events which do not belong to the

population defined by a Rayleigh model. To yield a

sufficiently small contribution to the overall risk of

structural collapse, the structure should withstand extreme

waves corresponding to an annual probability of exceedance

of say 1025 2 1024 as the Rayleigh model underpredicts

the highest crest heights indicating that real processes maybe strictly affected by higher order coefficients. In addition

to the ultimate limit state based on a 100-year design wave

an accidental limit state with a return period of 10 000 years

is suggested. Based on observations, Faulkner [2] suggests

the freak or abnormal wave height for survival design Hd $

2:5Hs:It is also recommended to characterize wave impact

loads so they can be quantified for potentially critical

seaways and operating conditions. Present design methods

should be complemented by survival design procedures, i.e.

two levels of design wave climates are proposed:

The operability envelope which corresponds to the best

present design practice The survivability envelope based on extreme wave

spectra parameters which may lead to episodic waves

or wave sequences (e.g. the Three Sisters) with

extremely high and steep crests.

Wave steepness, characterized by front and rear

steepness as well as by horizontal and vertical wave

asymmetries seems to be a parameter at least as

important as wave height [3].

A probability analysis of rogue wave data recorded at

North Alwyn from 1994 to 1998 reveals that these waves are

generally 50% steeper than the significant steepness, with

wave heights Hmax . 2:3Hs [4]. The preceding and

succeeding waves have steepness values around half the

significant values while their heights are around the

significant height.Steep-fronted wave surface profiles with significant

asymmetry in the horizontal direction excite extreme

relative motions at the bow of a cruising ship with signi-

ficant consequences on green water loading on the fore deck

and hatch covers of a bulk carrier [5]. Heavy weather

damages caused by giant waves are presented by Kjeldsen

[6], including the capsizing of the semi-submersible Ocean

Ranger. Faulkner and Buckley [7] describe a number of

episodes of massive damage to ships due to rogue waves,

e.g. with the liners Queen Elisabeth and Queen Mary. Haver

and Anderson[8]report on substantial damage of the jacket

platform Draupner when a giant wave Hmax 25:63 m

with the crest height hc 18:5 m hit the structure in 70 mwater depth on January 1, 1995 (Fig. 1, top). Related to the

Fig. 1. Rogue wave registrations.

G.F. Clauss / Applied Ocean Research 24 (2002) 147161148


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significant wave height Hs 11:92 m; the maximum wave

rises to Hmax 2:15Hs with a crest ofhc 0:72Hmax:

Not as spectacular but still exceptional are wave data

from the Norwegian Frigg fieldwater depth 99.4 m Hs

8:49 m;Hmax 19:98 m; hc 12:24 m[9]and the Danish

Gorm fieldwater depth 40 m (Hs 6:9 m; Hmax 17:8

m; hc < 13 m)[10]. Also remarkable are wave records of

the Japanese National Maritime Institute measured off Yura

harbour at a water depth of 43 m Hs 5:09 m; Hmax

13:6 m;hc < 8:2 m [11](Fig. 1, bottom).

All these wave datawith Hmax=Hs . 2:15 and

hc=Hmax . 0:6prove, that rogue waves are serious events

which should be considered in the design process. Although

their probability is very low they are physically possible.It is a

challenging question which maximum wave and crest heightscan develop in a certain seastate characterized byHsand Tp.

Concerning wave/structure interactions, with respect to

response based design loads and motions or reliability based

design: Is the highest wave with the steepest crest the most

relevant design condition or should we identify critical wave

sequences embedded in an irregular wave train? In addition to

the global parametersHs and Tp the wave effects on a structure

depend on superposition and the interaction of wave com-

ponents, i.e. on local wave characteristics. Phase relations and

nonlinear interactions are key parameters to specify the

relevant surface profileat thestructure.If wavekinematicsand

dynamics are known, cause effect relationships can be

detected.This paper presents a numerical as well as an

experimental technique for the generation of rogue waves

and design wave sequences in extreme seas. Based on

selected global seastate data Hs; Tpthe wave field is fitted

to predetermined characteristics at a target location, such as

wave heights, crest heights and periods of a single or a

sequence of extreme individual waves. Starting with a linear

approximation of the desired wave train by optimizing an

initially random phase spectrum for a given variance

spectrum we obtain an initial guess for the wave board

motion. This control signal is systematically improved to fit

the wave train to the predetermined wave characteristics at

target location. Numerical and experimental methods arecomplementing each other. If the fitting process is

conducted in a wave tank all nonlinear free surface effects

and even wave breaking are automatically considered.

Firstly, the linear procedure is presented, and illustrated by

the generation of deterministic wave packets as well as the

synthesis of the above target wave train into an irregular sea.

Next, the nonlinear approach with its experimental validation

is presented. Finally, the nonlinear fitting process of the target

wave sequence embedded in irregular seas is developed.

2. Linear transient wave description

The method for generating linear wave groups is

based on the wave focussing technique of Davis and

Zarnick[12], and its significant development by Takezawa

and Hirayama [13]. Clauss and Bergmann [14] recom-

mended a special type of transient waves, i.e. Gaussian

wave packets, which have the advantage that their

propagation behaviour can be predicted analytically. Withincreasing efficiency and capacity of computer the restric-

tion to a Gaussian distribution of wave amplitudes has been

abandoned, and the entire process is now performed

numerically[15]. The shape and width of the wave spectrum

can be selected individually for providing sufficient energy

in the relevant frequency range. As a result the wave train is

predictable at any instant and at any stationary or moving

location. In addition, the wave orbital motions as well as the

pressure distribution and the vector fields of velocity and

acceleration can be calculated. According to its highaccuracy the technique is capable of generating special

purpose transient waves.

A continuous real-valued wave record zt may be

represented in frequency domain by its complex Fourier

transform Fv which is calculated by Eq. (1). Applying

the inverse Fourier transformation, Eq. (2), gives the

original record zt

Fv 121

zte2ivt dt 1

zt1

2p 1

21

Fveivt dv 2

where t represents the time and v 2pf the angular

frequency. In polar notation, the complex Fourier trans-

form can be expressed by its amplitude and phase

spectrum:

Fv lFvlei argFv 3

In practice, it is necessary to adopt a discrete and finite

form of the Fourier transform pair described by Eqs. (1)

and (2)

FrDv DtXN21k0

zkDte2i2prk=N r0; 1; 2; ;N=2 4

zkDt Dv

2p

XN=2r0

FrDvei2prk=N

k0; 1; 2; ; N2 1;

5

where the values zkDt represent the available data

points of the discrete finite wave record, with Dt

denoting the sampling rate and Dv2p=NDt the fre-

quency resolution. The summation in Eqs. (4) and (5)

can be efficiently completed by the fast Fourier transform

(FFT) and its inverse algorithm (IFFT).

Extreme wave conditions in a 100-year design stormarise from the most unfavourable superposition of com-

ponent waves of the related severe sea spectrum. Freakwaves have been registered in standard irregular seas

when component waves accidentally superimpose in

G.F. Clauss / Applied Ocean Research 24 (2002) 147161 149


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phase. Extensiverandomtime-domain simulationof theoceansurface for obtaining statistics of the extremes, however, is

very time consuming. In generating irregular seas in a wave

tank the phase shift is supposed to be random, however, it is

fixed by the control program on the basis of a pseudo-random

process: consequently, it is also a deterministic parameter.

Why should we wait for these rare events if we can achieve

these conditions by intentionally selecting a suitable phase

shift, and generate a deterministic sequence of waves, which

converge at a preset concentration point? Assuming linear

wave theory, the synthesis and up-stream transformation of

wave packets is developed from this concentration point. At

this position all waves are superimposed without phase shift

resulting in a single high wave peak. From its concentrationpoint, the Fouriertransform of the wavetrain is transformed to

the upstream position at the wave board[16].

The Fourier transform is characterized by theamplitude spectrum and the related phase distribution.

During propagation the amplitude spectrum remains

invariant, however, the phase distribution and the related

shape of the wave train varies with its position. At the

concentration point all wave components are super-

imposed in phase, and a single high wave is observed

[17]. As the process is strictly linear and deterministic,

wave groups can be analysed back and forth in time and

space. They also can be integrated into a specified

irregular sea.

3. Nonlinear transient wave description

The generation of higher and steeper wave sequences,

Fig. 2. Genesis of a 3.2 m rogue wave by deterministic superposition of component waves (water depth d 4 m).



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requires a more sophisticated approach as propagation

velocity increases with height. Consequently, it is not

possible anymore to calculate the wave train linearlyupstream back to the wave generator to determine the

(nonlinear) control signal of the wave board. To solve this

problem, Kuhnlein [16] developed a semi-empirical pro-

cedure for the evolution of extremely high wave groups

which is based on linear wave theory: the propagation of

high and steep wave trains is calculated by iterative

integration of coupled equations of particle positions.

With this deterministic technique freak waves up to

3.2 m high have been generated in a wave tank[18].Fig. 2

shows the genesis of this wave packet and presents

registrations which have been measured at various locations

including the concentration point at 84 m.

The associated wave board motion which has been

determined by the above semi-empirical procedure is the

key input for the nonlinear analysis of wave propagation. As

has been generally observedat wave groups as well as at

irregular seas with embedded rogue wave sequenceswe

register substantial differences between the measured time-

series and the specified design wave trainat target location if a

linearlysynthesized control signal is used for the generation of

higher and steeper waves. As shown in Fig. 3, however, the

main deviation is localized within a small range [19]. This

promising observation proves that it is sufficient for only a

short part of the control signal in the time-domain to be fitted.

As a prerequisite, however,the computercontrolled loop in the

experimentalgeneration process should imply nonlinear wave

theory and develop the wave evolution by using a numericaltime-stepping method. The two-dimensional fully nonlinear

free surface flow problem is analysed in time-domain using

potential flow theory.Fig. 4summarizes the basic equations

and boundary conditions.

A finite element method developed by Wu and Eatock

Taylor[20,21]is used to determine the velocity potential,

which satisfies the Laplace equation for Neumann and

Dirichlet boundary conditions. The Neumann boundary

condition at the wave generator is introduced in form of

the first time derivative of the measured wave board

motion. To develop the solution in time-domain the

fourth order Runge Kutta method is applied. Starting

from a finite element mesh with 8000 triangular elements

Fig. 3. Comparison between target wave and measured time-series at target

location.

Fig. 4. Numerical wave tank[29].

Fig. 5. Finite element mesh for nonlinear analysis.

Fig. 6. Nonlinear numerical simulation of transient waves.



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(401 nodes in x-direction, 11 nodes in z-direction, i.e.

4411 nodes) (Fig. 5) a new boundary-fitted mesh is

created at each time-step. Lagrangian particles concen-

trate in regions of high velocity gradients, leading to a

high resolution at the concentration point. This mixedEulerian Lagrangian approach has proved its capability

to handle the singularities at intersection points of the

free surface and the wave board. Fig. 6 shows wave

profiles with associated velocity potential as well as

registrations at different positions. Note that the pressure

distribution as well as velocity and acceleration fields

including particle tracks at arbitrary locations are

deduced from the velocity potential.

Fig. 7 presents numerical results as well as experi-

mental data to validate this nonlinear approach. Excellent

agreement of numerical and experimental results is

observed. Note that all kinematic and dynamic charac-

teristics during wave packet propagation are deducedfrom the velocity potential, i.e. registrations at any

position (top, left) with associated Fourier spectra, wave

Fig. 7. Wave packet registrations at different positions as well as instantaneous wave profiles at selected instantsnumerical calculations validated by

experimental results[33].

Fig. 8. Maximum (crest) and minimum (trough) surface elevations (zmax,

zmin) as well as wave height zmax 2 zmin:



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profiles at arbitrary instants (top, right) as well as

velocity, acceleration and pressure fields.Fig. 8shows the maximum (crest) and minimum (trough)

surface elevation in the wavetank, zmaxand zmin,aswellasthe

difference, i.e. the wave height zmax 2 zmin:Note the sudden

rise of water level (crest andtrough) at theconcentration point.

Fig. 9shows numerically calculated orbital tracks of particles

with starting locations at 126 m, which is very close to the

concentration point. Generally, the orbital tracks are not

closed. Particles with starting locationsz . 21 m are shifted

in thex-direction, and due to mass conservation particles with

lowerz-coordinates are shifted in the opposite direction.

Fig. 10finally proofs that the technique for generating

nonlinear wave packets is adaptable to different wave

machines. The diagrams present results for a two-flap wave

generator, i.e. the angular motions (and speed) of the lower

and upper flaps as well as the resulting wave group

registration. Excellent agreement between numerical and

experimental results is observed. Note that the short leading

waves are generated by the upper flap. As the lower flap

starts working, the motion of the upper flap is reduced, and

finally oscillates anti-phase with the lower flap[22,23].

4. Integration of design wave groups in irregular

seaslinear approach

In general, extremely high rogue waves or critical wave

groups are rare events embedded in a random seaway.

As long as linear wave theory is applied, the seastate can

be regarded as superposition of independent harmonic

waves, each having a particular direction, amplitude,

frequency and phase. For a given design variance spectrum

of an unidirectional wave train, the phase spectrum is

responsible for all local characteristics, e.g. the wave height

and period distribution as well as the location of the highest

wave crest in time and space. For this reason, an initially

random phase spectrum arg Fv is optimized to generate

the desired design wave train with specified local proper-

ties. The phase values b b1;b2; bnT are bounded

by 2p # b# p and are initially determined from bi 2pRj 2 0:5 whereRj are random numbers in the interval

01[24].

The set-up of the optimization problem is illustrated for a

high transient design wave within a tailored group of three

successive waves in random sea. The crest front steepness of

the design wave in time-domain 1i as defined by Kjeldsen

[9]:

1t 2pzcrest

gTriseTzd6

is maximized during the optimization process. zcrestdenotesthe crest height, Trise the time between the zero-upcrossing

and crest elevation, and Tzd the zero-downcrossing periodwhich includes the design wave.

The target zero-upcrossing wave heights of the leading,

Fig. 9. Particle tracks with starting location at x 126 m.

Fig. 10. Motions of a two-flap wave generator and related wave group

registration comparing numerical and experimental results.



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the design and the trailing wave are defined by Hl, HdandHt. The target locations in space and time of the design

wave crest height zd are xtarget and ttarget. These data

define equality constraints. The maximum values of

stroke xmax, velocity umax, and acceleration amax of the

wave board motion xbt define inequality constraints to be

taken into account. Hence the optimization problem is

stated as

minimizeb[Rn

fb 21t 7

subject to g1 Hi21 2Hl 0;

g2 Hi 2Hd 0;

g3 Hi1 2Ht 0;

g4 zxtarget; ttarget 2 zd 0;

g5 max{lxbtl} 2xmax # 0;

g6 max{l_xbtl} 2 umax # 0;

g7 max{lxbtl} 2 amax # 0;

g7j 2p2 bj # 0; j 1; ; n

g7nj 2p bj # 0; j 1; ; n

where fb is the objective function to be minimized.

The general aim in constrained optimization is totransform the problem into an easier subproblem that

can be solved, and is used as the basis of an iterative

process. A sequential quadratic programming (SQP)

method is used which allows to closely imitate Newtons

method for constrained optimization just as is done for

unconstrained optimization [29].

For evaluating the objective function and constraints, thecomplex Fourier transform is generated from the amplitude

and phase spectrum. Application of the IFFT algorithm

yields the associated time-dependent wave train at target

location. Zero-upcrossing wave and crest heights as well as

the crest front steepness 1tof the design wave are calculated.

The motion of the wave board xbt is determined by

transforming the wave train at x xtarget in terms of the

complex Fourier transform Ftargetv to the location of the

wave generator at x 0 and applying the complex

hydrodynamic transfer function Fhydrov which relateswave board motion to surface elevation close to the wave

generator

xbt IFFTFtargetvFtransvFhydrov 8

with Ftransvj expikjxtarget: The maximum stroke of

the wave board is set to xmax 2 m; maximum velocity

to umax 1:3 m=s and maximum acceleration to amax

1:7 m=s2: The optimization terminates if the magnitude

of the directional derivative in search direction is

less than 1023 and the constraint violation is less than

1022.

In our example the design variance spectrum is chosen tobe the finite depth variant of the Jonswap spectrum known

as TMA spectrum[25]

v4pEq

g2 aq2

5 tanh2kd

12kd=sinh2kde21:25q

24

ge2r

2=2

9

whereq v=vp f=fprepresents the normalized frequency

with respect to the peak frequency fp 1=Tp: The Jonswap

peak enhancement factor g is set to 3.3 and the spectral

width parameter sp

to 0.07 for q # 1 and 0.09 for q . 1

with r q 2 1=sp: The frequency-dependent wave

number k is calculated from the dispersion relationship

v2 gktanhkdwhereg is the acceleration due to gravityandd the water depth.

For the selected spectrumsignificant wave height,

Hs 0:7 m; peak period, Tp 4:43 s; water depth,

d 5:5 ma high transient design wave within a

tailored group of three successive waves in random

sea is optimized. The target zero-upcrossing wave

height of the design wave is Hd 2Hs with a maximum

crest height zdxtarget; ttarget 0:6Hd 1:2Hs: Target

location is at a distance of xtarget 100 m from the

wave generator, and target time is ttarget 80 s: The

heights of the leading and the trailing waves adjoiningthe design wave are set to be Hl Ht Hs: Note that

this wave sequence is quite representative for roguewave groups as has been proved by Wolfram et al. [4]

who classified 114 extremely high waves with their

Fig. 11. Optimized phase spectra and associated wave trains resulting from

different initial phase distributions.



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immediate neighbours out of 345 245 waves collected

between 1994 and 1998 at North Alwyn.

As shown in Fig. 11, the optimization process finds

local minima, i.e. a number of different wave trains,

which depend on the initial phase values. Hence the

random character of the optimized seastate is not

completely lost.

From this linear approach we obtain an initial guess of

the wave board motion which yields the design wave

sequence at target location.

5. Integration of a nonlinear rogue wave sequence into

extreme seas

In Section 4 it is shown how a tailored group of three

successive waves is integrated into a random sea using a

SQP method. As shown in Fig. 12 (which is one of the

realizations of the wave trains in Fig. 11) all target

features regarding global and local wave characteristics,

including the rogue wave specification Hmax 2Hs and

hc 0:6Hmax are met.

Of course, this result is only a first initial guess as

linear wave theory used is not appropriate for describing

extreme waves since nonlinear free surface effects

significantly influence the wave evolution. However, the

linear description of the wave train is a good starting

Fig. 12. Linear wave train with predetermined wave sequence.

Fig. 13. Nonlinear wave train simulation with predetermined wave

sequence. Wave board motion optimized with the linear SQP method.

Fig. 14. Comparison of optimized wave board motions.

Fig. 15. Nonlinear wave train simulation with predetermined wave

sequence. Wave board motion optimized with subplex method.



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point to further improve the wave board motion (i.e. time-

dependent boundary conditions) required in the fully

nonlinear numerical simulation. If the control signal from

thelinear approach and therelated wave board motionis used

as an input for the nonlinear evolution of the wave train,Fig.

13shows that the nonlinear wave train significantly deviates

from the target values if this first guess of the wave board

motion is used in the numerical simulation.

As a consequence, the nonlinear wave train at target

location that originates from the first optimization

process must be further improved. This is achieved by

applying the subplex method developed by Rowan [26]

for unconstrained minimization of noisy objective

functions. The domain space of the optimization

problem is decomposed into smaller subdomains whichare minimized by the popular Nelder and Mead

simplex method [27]. The subplex method is introduced

because SQP cannot handle wave instability and break-

ing since the gradient of the objective function is

difficult to determine in this case. Nonlinear free surface

effects are included in the fitting procedure since the

values of objective function and constraints are

determined from the nonlinear simulation in the

numerical wave tank.

The target wave characteristics define equality con-

straints. The maximum values of stroke xmax 2 m;

velocity umax

1:7 m=s; and acceleration amax

2:2 m=s2

of the wave board motion xBt define inequality

constraints to be taken into account.

The subplex minimization problem is formulated as

minimizec[Rn

fc Hi212H1;target

H1;target

!2

Ti212T1;target

T1;target

!2

Hi2H2;target

H2;target

!2

Ti2T2;target

T2;target

!2

zc;i2zc;target

zc;target

!2

tzc;i2 tzc;target

tzc;target

!2

Hi12H3;target

H3;target

!2

Ti12T3;target

T3;target

!2

sxBt2sxBtinitial

sxB

tinitial

2

10

subject to g1 max{lxBtl}2xmax # 0;

g2 max{l_xBtl}2umax # 0;

g3 max{lxBtl}2amax # 0;

11

wheresxBt is the standard deviation of the wave board

motion. As a resultFig. 14shows the improved wave board

motion. The zero-downcrossing characteristics of the wave

train are presented in Fig. 15. The target values of the

transient wave are significantly improved. Note that the

rogue wave sequence is exactly fitted, with Hmax 2Hs andh

c 0:6H

max: As a result we obtain a control signal of the

wave generator which yields a specified rogue wave

sequence embedded in an extreme irregular seaway

Fig. 16. Evolution of rogue wave sequenceregistrations atx 5 m;50 m and 100 m (left) as well as wave profiles at t 75 s;81 s and 87 s (right hand side)

(water depth h 5 m, Tp 3:13 s).



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characterized by the selected global parameters Hs and Tp[28].

Fig. 16 shows the evolution of this design wave

sequence, with registrations at 5, 50 and 100 m (target

position) behind the wave board (left side) as well as wave

profiles (photos of surface elevation) at t 75 s, 81 s

(target) and 87 s (water depth h 5 m,Tp 3:13 s).

The associated energy flux at the locations x 5;50 and

100 m is shown inFig. 17. As has been expected the energy

flux focuses at the target position.

From the velocity potential which has been determined

as a function of time and space all kinematic and dynamic

characteristics of the wave sequence are evaluated. Fig. 18

presents the associated velocity, acceleration and pressurefields[29]. Note that the effects of the three extremely high

waves are reaching down to the bottom.

The above optimization method has also been applied to

generate the Yura wave and the New Year Wave (Fig. 1) in

the wave tank (Fig. 19). Firstly, for the specified design

variance spectrum, the SQP-method yields an optimized

phase spectrum which corresponds to the desired wave

characteristics at target position. The wave generator

control signal is determined by transforming this wave

train in terms of the complex Fourier transform to the

location of the wave generator. The measured wave train at

target position is then iteratively improved by systematic

variation of the wave board control signal. To synthesize thecontrol signal wavelet coefficients are used. The number of

free variables is significantly reduced if this signal is

compressed by low-pass discrete wavelet decomposition,

concentrating on the high energy band. Based on deviations

between the measured wave sequence and the design wave

group at target location the control signal for generating the

seaway is iteratively optimized in a fully automatic

computer-controlled model test procedure (Fig. 19).

Fig. 20presents the evolution of the Yura wave at a scale

of 1:112. The registrations show how the extremely high

wave develops on its way to the target position at x 7 m.

As compared to full-scale data the experimental simulation

is quite satisfactory.The evolution of the Draupner New Year Wave is shown

inFig. 21. Again the full-scale data correlate quite well with

Fig. 17. Energy flux of nonlinear wave at x 5;50 and 100 m (target).

Fig. 18. Kinematic and dynamic characteristics of rogue wave sequence

Hs,,2Hs,,Hsat target time t 81 s.

Fig. 19. Computer controlled experimental simulation of tailored design

wave sequences.



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numerical (time-domain) approach and model test results

[31].

For the numerical simulations the program TiMIT (Time-

domain investigations, developed at the Massachusetts

Institute of Technology) is used, a panel-method program

for transient wave-body interactions [32] to evaluate

the motions of the semi-submersible. TiMIT performs

linear seakeeping analysis for bodies with or without

forward speed. In a first module the transient radiation and

diffraction problem is solved. The second module provides

results like the steady force and moment, frequency-domain

coefficients, response amplitude operators, time-histories of

body response in a prescribed sea of arbitrary frequency

content on the basis of impulse-response functions.

The drilling semi-submersible GVA 4000 has been

selected as a typical harsh weather offshore structure to

investigate the seakeeping behaviour in rogue waves in

time-domain. The wetted surface of the body is discretized

into 760 panels (Fig. 24). The number of panels is sufficient

to simulate accurate results.

For validating TiMIT results of wave/structure inter-

actions in extreme seas the Draupner New Year Wave

(Fig. 1) has been synthesized in a wave tank at a scale of

1:81. Using the proposed wave generation technique, the

wave board signal is calculated from the target wave

sequence at the selected wave tank location.

Fig. 25 presents the modelled wave train at target

location. For comparison the exact New Year Wave is also

shown to illustrate that we have not reached an accurateagreement so far. However, this is not detrimental since the

associated numerical analysis is based on the modelled

wave train, registered at target position.

Fig. 26presents the modelled wave train as well as the

heave and pitch motions of the semi-submersible comparing

numerical results and experimental data (scale 1:81). The

Fig. 22. Roll motion of the RORO vessel in a severe storm wave train

Tp 14:6 s, Hs 15:3 mat GM 1:36 m v 15 kn,Z-manoeuvre with

m ^108.

Fig. 23. RORO vessel in a severe storm.

Fig. 24. Semi-submersible GVA 4000: main dimensions and discretization

of the wetted surface using 760 panels.

Fig. 25. Comparison of model wave (scale 1:81) as compared to the

registered New Year Wave[8] presented as full-scale data.



14/15

airgap as a function of time is also shown. Note that this

airgap is quite sufficient, even if the rogue wave passes

the structure. However, wave run-up at the columns

(observed in model tests) is quite dramatic, with the

consequence that green water will splash up to the platform

deck.

As a general observation, the rogue wave is not

dramatically boosting the motion response. The semi-

submersible is rather oscillating at a period of about 14 s

with moderate amplitudes.

Related to the (modelled) maximum wave height of

Hmax 23 m we observe a maximum measured double

heave amplitude of 7 m. The corresponding peak value fromnumerical simulation is 8.6 m. As a consequence, the

measured airgap is slightly smaller than the one from

numerical simulation. The associated maximum double

pitch amplitudes compare quite well. Note that the impact

results in a sudden inclination of about 38. Considering the

complete registration it can be stated that the numerical

approach gives reliable results. At rogue events the

associated response is overestimated due to the disregard

of viscous effects in TiMIT calculations.

8. Conclusions

For the evaluation of wave structure interactions the

relation of cause and effects is investigated deterministically

to reveal the relevant physical mechanism. Based on the

wave focussing technique for the generation of task-related

wave packets a new technique is proposed for the

synthetization of tailored design wave sequences in extreme

seas.

The physical wave field is fitted to predetermined global

and local target characteristics designed in terms of

significant wave height, peak period as well as wave height,

crest height and period of individual waves. The generation

procedure is based on two steps: firstly, a linear approxi-

mation of the desired wave train is computed by a SQP

method which optimizes an initially random phase spectrum

for a given variance spectrum. The wave board motion

derived from this initial guess serves as starting point for

directly fitting the physical wave train to the targetparameters. The subplex method is applied to improve

systematically a certain time-frame of the wave board

motion which is responsible for the evolution of the design

wave sequence. The discrete wavelet transform is intro-

duced to reduce significantly the number of free variables to

be considered in the fitting problem. Wavelet analysis

allows one to localize efficiently the relevant information of

the electrical control signal of the wave maker in time and

frequency domain.

As the presented technique permits the deterministic

generation of design rogue wave sequences in extreme seas

it is well suited for investigating the mechanism of arbitrary

wave/structure interactions, including capsizing, slammingand green water as well as other survivability design

aspects. Even worst case wave sequences like the Draupner

New Year Wave can be modelled in the wave tank to

analyse the evolution of these events and evaluate the

response of offshore structures under abnormal conditions.

Acknowledgements

The fundamentals of transient wave generation and

optimization have been achieved in a research project

funded by the German Science Foundation (DFG). Appli-cations of this technique, i.e. the significant improvement of

seakeeping tests and the analysis of wave breakers and

artificial reefs in deterministic wave packets have been

funded by the Federal Ministry of Education, Research and

Development (BMBF). Results are published in outstanding

PhD theses (J. Bergmann, W. Kuhnlein, R. Habel,

U. Steinhagen). The technique is further developed to

synthesize abnormal rogue waves in extreme seas within the

MAXWAVE project funded by the European Union

(contract number EVK-CT-2000-00026) and to evaluate

the mechanism of large roll motions and capsizing of

cruising ships (BMBF funded research project ROLL-S).

The author wishes to thank the above research agencies fortheir generous support. He is also grateful for the invaluable

contributions of Dr Steinhagen, Dipl.-Ing. C. Pakozdi,

Fig. 26. Results of numerical simulation and experimental tests for semi-

submersible GVA 4000: heave, pitch and airgap (measured at a scale 1:81,

presented as full-scale data).



15/15

Dipl.-Math. Techn. Janou Hennig and Dipl.-Ing. C.

Schmittner.

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