Dramas of the Sea – Episodic Waves and their Impact on ...

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Oct 30, 2013 (3 years and 5 months ago)

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AOR-SC69
Email address: clauss@ism.tu-berlin.de
(G.F. Clauss)
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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
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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.
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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
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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.
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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).
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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).
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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

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

. 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)(
,


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.

REFERENCES

Bouws, E., Günther, H., Rosenthal, W. and Vincent,
C. (1985): Similarity of the wind wave spectrum
in finite depth water – 1. Spectral form, Journal of
Geophysical Research, 90(C1)
Clauss, G. and Bergmann J. (1986): Gaussian wave
packets – a new approach to seakeeping tests of
ocean structures, Applied Ocean Research, 8(4)
Clauss, G. (1999): Task-Related wave groups for
seakeeping tests or simulation of design storm
waves, Applied Ocean Research, 21, pp 219-234
Clauss, G. and Habel, R. (2000): Artificial reefs for
coastal protection – transient viscous computation
and experimental evaluation, 27
th
International
Conference on Coastal Engineering (ICCE), Sydney,
Australia
Clauss, G. and Hennig, J. (2001) Tailored transient wave
packet sequences for computer controlled seakeeping
tests, 20
th
International Conference on Offshore
Mechanics and Arctic Engineering (OMAE), Rio de
Janeiro, Brazil
Clauss, G. and Hennig, J. (2002) Computer controlled
capsizing tests using tailored wave sequences, 21
st

International Conference on Offshore Mechanics and
Arctic Engineering (OMAE), Oslo, Norway
Clauss, G. and Kühnlein, W. (1995): Transient wave
packets – an efficient technique for seakeeping tests of
self-propelled models in oblique waves, Third
International Conference on Fast Sea Transportation,
Lübeck-Travemünde, Germany
Clauss, G. and Kühnlein, W. (1997): Simulation of
Design Storm Wave Conditions with Tailored Wave
Groups, 7
th
International Offshore and Polar
Engineering Conference (ISOPE), pp. 228-237.
Honolulu, Hawaii, USA.
Clauss, G., Pakozdi, C. and Steinhagen, U. (2001):
Experimental Simulation of Tailored Design Wave
Sequences in Extreme Seas, 11
th
International
Offshore and Polar Engineering Conference (ISOPE),
Stavanger, Norway.
Clauss, G. and Steinhagen, U. (1999): Numerical
Simulation of Nonlinear Transient Waves and its
Validation by Laboratory Data, 9
th
International
Offshore and Polar Engineering Conference (ISOPE),
Brest, France.
Clauss, G. and Steinhagen, U. (2000): Optimization of
Transient Design Waves in Random Sea, 10
th

International Offshore and Polar Engineering
Conference (ISOPE), Seatle, USA.
Clauss, G. and Steinhagen, U. (2001): Generation and
Numerical Simulation of Predetermined Nonlinear
Wave Sequences in Random Seaways, 20
th
OMAE
Symposium, Rio de Janeiro, Brazil
Clauss, G., Schmittner, C. and Stutz, K. (2002): Time-
domain investigations of a semisubmersible in rogue
waves, 21
st
International Conference on Offshore
Mechanics and Arctic Engineering (OMAE), Oslo,
Norway
Davis, M. and Zarnick, E. (1964): Testing Ship Models in
Transient Waves, 5
th
Symposium on Naval
Hydrodynamics.
Drake K. (1997): Wave profiles associated with extreme
loading in random waves, RINA International
AOR-SC69

- 17 -
Conference: Design and Operation for Abnormal
Conditions, Glasgow, Scotland
Faulkner (2000): Rogue Waves – Defining Their
Characteristics for Marine Design, Rogue Waves
2000, Brest, France.
Faulkner, D. and Buckley, W. (1997): Critical
survival conditions for ship design, RINA
International Conference: Design and Operation
for Abnormal Conditions, Glasgow, Scotland
Haver, S. (2000): Some evidence of the existence of
socalled freak waves, Rogue Waves 2000, Brest,
France
Haver, S. and Anderson, O.J. (2000): Freak Waves:
Rare Realizations of a Typical Population or
Typical Realization of a Rare Population?, 10
th

International Offshore and Polar Engineering
Conference (ISOPE), Seattle, USA.
Kjeldsen, S.P. (1996): Example of heavy weather
damage caused by giant waves, Techno Marine,
Bul. Of the society of Naval Architects of Japan,
No. 820
Kjeldsen, S.P. (1990): Breaking Waves. Water Wave
Kinematic, Kluwer Academic Publisher, NATO
ASI Series, ISBN 0-7923-0638-4, pp. 453-473.
Kjeldsen, S.P. and Myrhaug, D. (1997): Breaking
waves in deep water and resulting wave forces,
Offshore Technology Conference, OTC 3646
Korsmeyer,F., Bingham, H. and Newman, J.
(1999):TiMIT- A panel-method program for
transient wave-body interactions, Research
Laboratory of Electronics, Massachusetts Institute
of Technology
Kühnlein, W. (1997): Seegangsversuchstechnik mit
transienter Systemanregung, PhD Thesis,
Technische Universität Berlin, D83.
Mori, N., Yasuda, T. and Nakayama, S. (2000):
Statistical Properties of Freak Waves Observed in
the Sea of Japan, 10
th
International Offshore and
Polar Engineering Conference (ISOPE), Seattle,
USA.
Nelder, J. and Mead, R. (1965): A Simplex Method
for Function Minimization. Computer Journal, 7,
pp. 308-313.
Pakozdi, C. (2002): Numerische Simulation
nichtlinearer transienter Wellengruppen – Bericht
zum DFG-Vorhaben Cl 35/30-1, Selbstverlag
Rowan, T. (1990): Functional Stability Analysis of
Numerical Algorithms, PhD Thesis, University of
Texas at Austin.
Sand, S.E., Ottesen, H.N.E., Klinting, P., Gudmestad,
O.T.

and

Sterndorff, M.J. (1990): Freak Wave
Kinematics. Water Wave Kinematics, Kluwer
Academic Publisher, NATO ASI Series, ISBN 0-
7923-0638-4, pp. 535-549.
Steinhagen, U. (2001): Synthesizing Nonlinear Transient
Gravity Waves in Random Seas, PhD Thesis,
Technische Universität Berlin, D83.
Takezawa, S. and Hirayama, T. (1976): Advanced
Experiment Techniques for Testing Ship Models in
Transient Water Waves, 11
th
Symposium on Naval
Hydrodynamics
Wolfram, J., Linfoot, B. and Stansell, P. (2000): Long-
and Short-Term Extreme Wave Statistics in the North
Sea: 1994-1998, Rogue Waves 2000, Brest, France.
Wu, G-X., Eatock Taylor, R. (1994): Finite Element
Analysis of Two-Dimensional Non-Linear Transient
Water Waves, Applied Ocean Research, 16(6), pp.
363-372.
Wu, G-X., Eatock Taylor, R. (1995): Time Stepping
Solutions of Two-Dimensional Non-Linear Wave
Radiation Problem, Ocean Engineering, 22(8), pp.
785-798.