AOR-SC69

Email address: clauss@ism.tu-berlin.de

(G.F. Clauss)

- 1 -

Dramas of the Sea – Episodic Waves and their Impact on

Offshore Structures

G.F. CLAUSS

Technical University of Berlin, Dept. of Naval Architecture and Ocean Engineering, SG 17, Salzufer 17-19, 10587

Berlin, Germany)

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 non-linear 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 - 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 and - by introducing the electro-hydraulic and hydrodynamic transfer functions of the wave

generator - the 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 3.2m 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 the appropriate 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 wave – the Draupner “New Year Wave” is simulated and

generated in a physical wave tank. Also a “Three Sisters” wave sequence with succeeding wave heights

H

s

…2H

s

…H

s

, embedded in an extreme sea, is synthesised.

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

semisubmersible in the Draupner New Year Wave, embedded in extreme irregular seas is numerically and

experimentally evaluated.

Keywords: Rogue Waves, Wave/Structure interaction, Deterministic Seakeeping Tests

AOR-SC69

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INTRODUCTION

The impossible will happen one day. Considering

rogue wave events as rare phenomena - according to

Murphy’s law - beyond our present modelling

abilities, Haver (2000) 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

45

1010

−−

−

as the Rayleigh

model underpredicts the highest crest heights

indicating that real processes may be strictly affected

by higher order coefficients. In addition to the ULS

(ultimate limit state) based on a 100-year design wave

an ALS (accidental limit state) with a return period of

10000 years is suggested. Based on observations

Faulkner (2000) suggests the freak or abnormal wave

height for survival design

sd

HH 5.2≥

. 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 (Kjeldsen an Myrhaug,

1979).

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

s

HH 3.2

max

>

(Wolfram et al, 2000). 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

significant consequences on green water loading on the

fore deck and hatch covers of a bulk carrier (Drake,

1997). Heavy weather damages caused by giant waves are

presented by Kjeldsen (1996), including the capsizing of

the semisubmersible Ocean Ranger. Faulkner and

Buckley (1997) 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,

(2000) report on substantial damage of the jacket

platform Draupner when a giant wave (

mH 63.25

max

=

)

with the crest height

m

c

5.18

=

η

hit the structure in

m70

water depth on January 1, 1995 (Fig. 1-top). Related to

the significant wave height

mH

s

92.11

=

, the maximum

wave rises to

s

HH 15.2

max

=

with a crest of

max

72.0 H

c

=

η

.

Not as spectacular but still exceptional are wave data

from the Norwegian Frigg field – water depth

m4.99

(

mH

s

49.8

=

,

mH 98.19

max

=

m

c

24.12

=

η

) (Kjeldsen,

1990) and the Danish Gorm field – water depth

m40

(

mH

s

9.6

=

,

mH 8.17

max

=

,

m

c

13

≈

η

) (Sand et al.,

2000). Also remarkable are wave records of the Japanese

National Maritime Institute measured off Yura harbour at

a water depth of

m43

(

mH

s

09.5

=

,

mH 6.13

max

=

,

m

c

2.8

≈

η

) (Mori et al., 2000) (Fig. 1-bottom).

All these wave data – with

15.2/

max

>

s

HH

and

6.0/

max

>

H

c

η

– prove, 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 heights can develop in a certain sea-state

characterized by

s

H

and

p

T

. 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 parameters

s

H

and

p

T

the wave effects on a

structure depend on superposition and the interaction of

wave components, i.e. on local wave characteristics.

Phase relations and nonlinear interactions are key

parameters to specify the relevant surface profile at the

structure. If wave kinematics and dynamics are known,

cause-effect relationships can be detected.

AOR-SC69

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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 sea state data (

s

H

,

p

T

) the

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 are

complementing 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.

LINEAR TRANSIENT WAVE DESCRIPTION

The method for generating linear wave groups is

based on the wave focussing technique of Davis and

Zarnick (1964), and its significant development of

Takezawa and Hirayama (1976). Clauss and Bergmann

(1986) recommended a special type of transient waves,

i.e. Gaussian wave packets, which have the advantage

that their propagation behaviour can be predicted

analytically. With increasing efficiency and capacity of

computer the restriction to a Gaussian distribution of

wave amplitudes has been abandoned, and the entire

process is now performed numerically (Clauss and

Kühnlein, 1995). 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 high accuracy the technique

is capable of generating special purpose transient waves.

North Sea – Draupner jacket platform (Haver and Anderson, 2000) – New Year wave 01-01-95

H

s

= 11.92m,

H

max

= 25.63m = 2.15

H

s

;

c

η

= 18.5m = 0.72

H

max

(water depth 70m)

Japanese Sea – off Yura harbour – Japanese National Maritime Institute (Mori et al., 2000)

H

s

= 5.09m,

H

max

= 13.6m = 2.67 H

s

;

c

η

= 8.2m = 0.6

H

max

(water depth 43m)

Fig. 1

Rogue wave registrations

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A continuous real-valued wave record

)(t

ζ

may

be represented in frequency domain by its complex

Fourier transform

)(

ω

F

which is calculated by

Eq.(1). Applying the inverse Fourier transformation,

Eq.(2), gives the original record

)(t

ζ

:

dtetF

ti

∫

+∞

∞−

−

=

ω

ζω )()(

(1)

ωω

π

ζ

ω

deFt

ti

∫

+∞

∞−

= )(

2

1

)(

(2)

where t represents the time and

f

π

ω

2=

the angular

frequency. In polar notation, the complex Fourier

transform can be expressed by its amplitude and

phase spectrum:

)(

)()(

ω

ωω

Fargi

eFF =

(3)

In practice, it is necessary to adopt a discrete and

finite form of the Fourier transform pair described by

Eqs. (1) and (2):

∑

−

=

−

∆∆=∆

1

0

/2

)()(

N

k

Nrki

etktrF

π

ζω

(4)

2/,,2,1,0 Nr l=

∑

=

∆

∆

=∆

2/

0

/2

)(

2

)(

N

r

Nkri

erFtk

π

ω

π

ω

ζ

(5)

)1(,,2,1,0 −= Nk

l

,

where the values

)( tk∆

ζ

represent the available data

points of the discrete finite wave record, with

t

∆

denoting the sampling rate and

)/(2 tN∆=∆

π

ω

the

frequency 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

storm arise from the most unfavourable superposition

of component waves of the related severe sea

spectrum. presents the simulation of an irregular sea

state by random phase superposition. As a rare – but

possible – event, a very high freak wave is observed.

Freak waves have been registered in standard

irregular seas when component waves accidentally

superimpose in phase. Extensive random time domain

simulation of the ocean surface 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

concentration point, the Fourier transform of the wave

train is transformed to the upstream position at the wave

board (Kühnlein, 1997).

The Fourier transform is characterized by the

amplitude 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

superimposed in phase, and a single high wave is

observed (Clauss, 1999). 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.

NONLINEAR TRANSIENT WAVE DESCRIPTION

The generation of higher and steeper wave

sequences, requires a more sophisticated approach as

propagation velocity increases with height. Consequently,

it is not possible anymore to calculate the wave train

linearly upstream back to the wave generator to determine

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

this problem, Kühnlein (1997) developed a semi-

empirical procedure 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

(Clauss and Kühnlein, 1997). 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 observed – at wave groups as well

as at irregular seas with embedded rogue wave sequences

– we register substantial differences between the

measured time series and the specified design wave train

at target location if a linearly synthesized control signal is

used for the generation of higher and steeper waves. As

illustrated in Fig. 3, however, the main deviation is

localized within a small range (Clauss et al., 2001). This

promising observation proves that it is sufficient for only

AOR-SC69

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a short part of the control signal in the time domain to

be fitted. As a prerequisite, however, the computer

controlled loop in the experimental generation

process should imply nonlinear wave theory and

develop the wave evolution by using a numerical

time-stepping method. The two dimensional fully

nonlinear free surface flow problem is analysed in

time domain using potential flow theory. Fig. 4

summarises the basic equations and boundary

conditions.

A finite element method developed by Wu and

Eatock Taylor (1994, 1995) is used to determine the

velocity potential, which satisfies the Laplace

Fig. 2

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

Fig. 3

Comparison between target wave and

measured time series at target location.

AOR-SC69

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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 forth order

Runge-Kutta method is applied. Starting from a finite

element mesh with 8000 triangular elements (401

nodes in x-direction, 11 nodes in z-direction, i.e. 4411

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

created at each time step. Lagrangian particles

concentrate in regions of high velocity gradients,

leading to a high resolution at the concentration point.

This mixed Eulerian-Langrangian 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

experimental data to validate this nonlinear approach.

Excellent agreement of numerical and experimental

results is observed. Note that all kinematic and dynamic

characteristics during wave packet propagation are

deduced from the velocity potential, i.e. registrations at

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

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

velocity, acceleration and pressure fields.

Fig. 8 shows the maximum (crest) and minimum

(trough) surface elevation in the wave tank

max

ζ

and

min

ζ

as well as the difference, i.e. the wave height

minmax

ζζ

−

. Note the sudden rise of water level (crest and

trough) at the concentration point. Fig. 9 illustrates

numerically calculated orbital tracks of particles with

starting locations at

mx 87

=

and at

mx 126

=

, which is

very close to the concentration point. Generally, the

orbital tracks are not closed. Particles with starting

locations

mz 1−>

are shifted in the x-direction, and due

to mass conservation particles with lower z-coordinates

are shifted in the opposite direction.

Surface elevation and associated velocity potential

Φ

Fi

g

. 6

Nonlinear numerical simulation of transient waves.

Fig. 5

Finite element mesh for nonlinear analysis.

Fig. 4

Numerical wave tank (Steinhagen, 2001).

AOR-SC69

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Fig. 9

Particle tracks with starting location at

x=126m.

Fig. 8

Maximum (crest) and minimum

(trough) surface elevations (

max

ζ

,

min

ζ

) as

well as wave height

minmax

ζζ −

.

Registrations wave profiles (“photos”)

Fourier spectra

Fig. 7

Wave packet registrations at different positions as well as instantaneous wave

profiles at selected instants – numerical calculations validated by experimental results

(Clauss and Steinhagen, 1999).

AOR-SC69

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Fig. 10 finally 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 (Pakozdi, 2002, Hennig, 2001).

INTEGRATION OF DESIGN WAVE GROUPS

IN IRREGULAR SEAS – LINEAR 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 sea

state 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

ω

F

is optimized to

generate the desired design wave train with specified

local properties. The phase values

T

n

),,(

21

ββββ =

are bounded by

π

β

π

≤≤−

and are initially determined

from

)5.0(2 −=

ji

R

πβ

where

j

R

are random numbers in

the interval 0 to 1 (Clauss and Steinhagen, 2000).

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

t

ε

as

defined by Kjeldsen (1990):

zdrise

crest

t

TTg

ζ

π

ε

2

=

(6)

is maximized during the optimization process.

crest

ζ

denotes the crest height,

rise

T

the time between the zero-

upcrossing and crest elevation, and

zd

T

the zero-

downcrossing period which includes the design wave.

The target zero-upcrossing wave heights of the

leading, the design and the trailing wave are defined by

l

H

,

d

H

and

t

H

. The target locations in space and time

of the design wave crest height

d

ζ

are

target

x

and

target

t

.

These data define equality constraints. The maximum

values of stroke

max

x

, velocity

max

u

, and acceleration

max

a

of the wave board motion

)(tx

b

define inequality

constraints to be taken into account. Hence the

optimization problem is stated as

t

Rβ

fminimize

n

εβ

−=

∈

)(

tosubject

{ }

{ }

{ }

njg

njg

atxg

utxg

xtxg

txg

HHg

HHg

HHg

jjn

jj

b

b

b

dtargettarget

ti

di

li

,,1,0

,,1,0

,0)(max

,0)(max

,0)(max

,0),(

,0

,0

,0

7

7

max7

max6

max5

4

13

2

11

l

l

DD

D

=≤+−=

=≤−−=

≤−=

≤−=

≤−=

=−=

=−=

=−=

=−=

++

+

+

−

βπ

βπ

ζζ

(7)

where

)(

β

f

is the objective function to be minimized.

The general aim in constrained optimization is to

transform the problem into an easier subproblem that can

Fig 10

Motions of a two-flap wave generator and

related wave group registration comparing numerical

and experimental results.

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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

Newton's method for constrained optimization just as

is done for unconstrained optimization.

For evaluating the objective function and

constraints, the complex 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

t

ε

of the design wave

are calculated. The motion of the wave board

)(tx

b

is

determined by transforming the wave train at

target

xx =

in terms of the complex Fourier transform

)(ω

target

F

to the location of the wave generator at

0=x

and applying the complex hydrodynamic

transfer function

)(ω

hydro

F

which relates wave board

motion to surface elevation close to the wave

generator:

]

[

)()()()( ωωω

hydrotranstargetb

FFFIFFTtx ⋅⋅=

(8)

with

)exp()(

targetjjtrans

xikF

=ω

. The maximum stroke

of the wave board is set to

mx

2

max

=

, maximum

velocity to

smu

/3.1

max

=

and maximum acceleration

to

2

max

/7.1 sma =

. The optimization terminates if the

magnitude of the directional derivative in search

direction is less than

3

10

−

and the constraint violation

is less than

2

10

−

.

In our example the design variance spectrum is

chosen to be the finite depth variant of the Jonswap

spectrum known as TMA spectrum (Bouws et al,

1985):

2

2

4

25.1

2

5

2

4

)2(sinh/21

)(tanh

)(

r

e

q

p

e

kdkd

kd

q

g

qE

−

−

−−

+

= γα

ω

(9)

where

pp

ffq//== ωω

represents the normalized

frequency with respect to the peak frequency

pp

Tf/1=

. The Jonswap peak enhancement factor

γ

is set to 3.3 and the spectral width parameter

*

σ

to

0.07 for

1≤q

and 0.09 for

1>q

with

*

/)1(

σ

−= qr

.

The frequency-dependent wave number k is

calculated from the dispersion relationship

)tanh(

2

kdgk=

ω

where g is the acceleration due to

gravity and d the water depth.

For the selected spectrum – significant wave height

mH

s

7.0=

, peak period

sT

p

43.4=

, water depth

md 5.5=

– a 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

sd

HH 2=

with a maximum crest height

dtargettargetd

Htx 6.0)(

,

=ζ

s

H2.1=

. Target location is at a

distance of

mx

target

100=

from the wave generator, and

target time is

st 80

target

=

. The heights of the leading and

the trailing waves adjoining the design wave are set to be

stl

HHH ==

. Note that this wave sequence is quite

representative for rogue wave groups as has been proved

by Wolfram et al. (2000) who classified 114 extremely

high waves with their immediate neighbours out of

345245 waves collected between 1994 and 1998 of North

Alwyn.

As illustrated in Fig. 12, 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 sea state 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.

Fig. 12

Optimized phase spectra and associated wave

trains resulting from different initial phase

distributions.

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INTEGRATION OF A NONLINEAR ROGUE

WAVE SEQUENCE INTO EXTREME SEAS

In the previous section it is shown how a tailored

group of three successive waves is integrated into a

random sea using a Sequential Quadratic

Programming (SQP) method. As illustrated in Fig. 13

(which is one of the realisations of the wave trains in

Fig. 12) all target features regarding global and local

wave characteristics, including the rogue wave

specification

s

HH 2

max

=

and

max

6.0 H

c

=

η

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 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 the linear approach and the

related wave board motion is used as an input for the

non-linear evolution of the wave train, Fig. 14

illustrates 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 (1990) for unconstrained

minimization of noisy objective functions. The

domain space of the optimization problem is

decomposed into smaller subdomains which are

minimized by the popular Nelder and Mead simplex

method (Nelder and Mead, 1965). The subplex

method is introduced because SQP cannot handle wave

instability and breaking 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

constraints. The maximum values of stroke

mx 2

max

=

,

velocity

smu/7.1

max

=

, and acceleration

2

max

/2.2 sma =

of the wave board motion

)(tx

B

define inequality

constraints to be taken into account.

The subplex minimization problem is formulated as

2

2

,3

,31

2

,3

,31

2

,

,

,

2

,

,,

2

,2

,2

2

,2

,2

2

,1

,11

2

,1

,11

))((

))(())((

)(

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

=

+

+

−

−

∈

initial

initial

target

targeti

target

targeti

targetc

targetc

i

targetc

targetcic

target

targeti

target

targeti

target

targeti

target

targeti

Rc

txσ

txσtxσ

T

TT

H

HH

t

tt

T

TT

H

HH

T

TT

H

HH

cfminimize

B

BB

c

n

ζ

ζζ

ζ

ζζ

(10)

tosubject

{

}

{ }

{ }

,0)(max

,0)(max

,0)(max

max3

max2

max1

≤−=

≤−=

≤−=

atxg

utxg

xtxg

B

B

B

(11)

Fig. 13

Linear wave train with predetermined wave

sequence.

Fig. 14

Nonlinear wave train simulation with

predetermined wave sequence. Wave board motion

optimized with the linear SQP method.

AOR-SC69

- 11 -

where

))((

tx

B

σ

is the standard deviation of the wave

board motion. As a result Fig. 15 shows the improved

wave board motion. The zero-downcrossing

characteristics of the wave train are presented in Fig.

16. The target values of the transient wave are

significantly improved. Note that the rogue wave

sequence is exactly fitted, with

s

HH

2

max

=

and

max

6.0

H

c

=

η

. 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 characterized by the selected global parameters

s

H

and

p

T

(Clauss and Steinhagen, 2001).

Fig. 17 illustrates the evolution of this design wave

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

Fig 16

Nonlinear wave train simulation with

predetermined wave sequence. Wave board motion

optimized with Subplex method.

Fig. 15

Comparison of optimized wave board

motions.

Fig. 17

Evolution of rogue wave sequence – registrations at

x

= 5m, 50m and 100m (left) as well as wave

profiles at

t

= 75s, 81s and 87 s (right hand side) (water depth h=5m, T

p

=3.13s).

AOR-SC69

- 12 -

(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,

T

p

=3.13s).

The associated energy flux at the locations

x

=5m,

50m and 100m is shown in Fig. 18. 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. 19 presents the

associated velocity, acceleration and pressure fields

(Steinhagen, 2001). 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 (see Fig. 1) in the wave tank (Fig. 20). Firstly,

for the specified design variance spectrum, the SQP-

method yields an optimised 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 synthesise

the control 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

optimised in a fully automatic computer-controlled model

test procedure (Fig. 20).

Fig. 18

Energy flux of nonlinear wave at

x

= 5 m, 50 m

and 100 m (target).

Fig. 20

Computer controlled experimental simulation

of tailored design wave sequences.

Fig. 19

Kinematic and dynamic characteristics of

rogue wave sequence

H

s

... 2

H

s

...

H

s

at target time

t

= 81 s.

AOR-SC69

- 13 -

Fig. 21 presents 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

=7m. As compared to full scale data the

experimental simulation is quite satisfactory.

The evolution of the Draupner New Year Wave

is shown in Fig. 22. Again the full scale data correlate

quite well with model test results at target position

x

=7.9m. The wave tank tests illustrate how these

extremely high waves are developing from rather

inconspicuous wave trains and disperse shortly later.

If wave/structure interactions are investigated the tank

tests allow for considering memory effects. In

addition, the mechanism of nonlinear structure

dynamics is evaluated, and cause-effect relationship

can be analysed.

COMPUTER CONTROLLED CAPSIZING TESTS

USING TAILORED WAVE SEQUENCES

The technique of generating deterministic wave

sequences embedded in irregular seas is used to analyse

the mechanism of large roll motions with subsequent

capsizing of cruising ships (Clauss and Hennig, 2002).

The parameters of the model seas – transient wave

sequences consisting of random seas or regular wave

trains with an embedded deterministic high transient

wave – are systematically varied to investigate the ship

model response with regard to metacentric height, model

velocity, and course angle for each of both ship types.

The wave elevation at the position of the ship model at

any position in time and space is calculated (and

controlled by registrations during model tests) in order to

relate wave excitation to the resulting roll motion.

Fig. 23 presents a model test with a RO-RO vessel

(GM=1.36 m, natural roll period

T

R

= 19.2 s,

v

= 15 kn) in

extremely high seas from astern (ITTC spectrum with

H

s

=15.3 m,

T

p

= 14.6 s, z-manoeuvre: target course

µ

=±10°). The vessel broaches and finally capsizes as the

Fig. 21

Evolution of the Yura wave (scale 1:112)

(full scale wave data collected by National Maritime Research Institute, Japan (Mori et al., 2000)).

Fig. 22

Evolution of the New Year Wave (scale 1:175)

(full scale wave data collected by Statoil (Haver and Anderson, 2000)).

AOR-SC69

- 14 -

vessel roll exceeds 40 degrees and the course

becomes uncontrollable (Fig. 24). Note that the wave

elevation refers to the ship center (moving frame),

and has been calculated from the registration at a

stationary wave probe - 10 meters in front of the wave

board. Thus, relevant wave elevation is directly

related to the associated ship motions.

DYNAMICS OF SEMISUBMERSIBLES IN

ROGUE WAVES

The method of synthesizing extremely high

waves in severe irregular seas is also applied to

analyse the impact of reported rogue waves on

semisubmersibles. As the procedure is strictly

deterministic we can compare the numerical (time-

domain) approach and model test results (Clauss et

al., 2002).

For the numerical simulations the program TiMIT

(T

ime-domain investigations, developed at the

M

assachusetts I

nstitute of T

echnology) is used, a panel-

method program for transient wave-body interactions

(Korsmeyer et al, 1999) to evaluate the motions of the

semisubmersible. 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 semisubmersible GVA 4000 has been

selected as a typical harsh weather offshore structure to

investigate the seakeeping behaviour in rogue waves in

Fig. 23

Roll motion of the RO-RO vessel in a severe

storm wave train (

T

p

=14.6s,

H

s

=15.3m) at GM=1.36m

v

=15kn, Z-manoeuvre with

µ

= ±

10°.

Fig. 24

RO-RO vessel in a severe storm.

Fig. 25

Semisubmersible GVA 4000- main dimensions

and discretization of the wetted surface using 760

panels.

AOR-SC69

- 15 -

time-domain. The wetted surface of the body is

discretized into 760 panels (Fig. 25). The number of

panels is sufficient to simulate accurate results.

For validating TiMIT results of wave/structure

interactions in extreme seas the Draupner New Year

Wave (see 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. 26 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

accurate agreement so far. However, this is not

detrimental since the associated numerical analysis is

based on the modelled wave train, registered at target

position.

Fig. 27 presents the modelled wave train as well

as the heave and pitch motions of the

semisubmersible comparing numerical results and

experimental data (scale 1:81). The airgap as 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

semisubmersible is rather oscillating at a period of

about 14s with moderate amplitudes.

Related to the (modelled) maximum wave height

of

H

max

= 23m we observe a maximum measured

double heave amplitude of 7m. The corresponding

peak value from numerical simulation is 8.6m. 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 3°. 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.

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 approximation of the desired wave train is

computed by a sequential quadratic programming 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 target parameters.

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 introduced 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

Fig. 26

Comparison of model wave (scale 1:81) as

compared to the registered New Year Wave (Haver and

Anderson, 2000) presented as full scale data.

Fig. 27

Results of numerical simulation and

experimental tests for semisubmersible GVA 4000:

Heave, pitch and airgap (measured at a scale 1:81,

presented as full scale data).

AOR-SC69

- 16 -

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,

slamming and 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). Applications 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. Kühnlein,

R. Habel, U. Steinhagen). The technique is further

developed to synthesise 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 for their

generous support. He is also grateful for the

invaluable contributions of Dr. Steinhagen, Dipl.-Ing.

C. Pakozdi, Dipl.-Math. techn. Janou Hennig and

Dipl.-Ing. C. Schmittner.

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