1 cesos workshop on research challenges in probabilistic load and response modelling extreme value...
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CeSOS WORKSHOPON RESEARCH CHALLENGES
IN PROBABILISTIC LOAD AND RESPONSE MODELLING
Extreme Value PredictionExtreme Value Prediction in Sloshing Response Analysisin Sloshing Response Analysis
Mateusz Graczyk
Trondheim, 24.03.2006
2
Scope
• Characteristics of sloshing – Problem definition
– Procedure of determining structural response
– Methods of analysis
– Sloshing experiments
• Stochastic methods– Classification
– Choice and fit of models
– Threshold selection for POT method
– Variability of results
3
• Violent resonant fluid motion
in a moving tank with free surface
• Complex motion patterns, coexistence of phenomena:– breaking and overturning waves
– run-up of fluid
– slamming
– two-phase flow
– gas cushion
– turbulent wake
– flow separation
– ...
Sloshing phenomenon
4
• Tank motion
• Filling level
• Wave heading angle
• ...
Sloshing parameters
• Fluid motion pattern
• Location in the tank
• Fluid spatial / temporal pattern
• ...
5
shipmotion
long-term descriptionof random
sea
fluid motionin the tank
pressuretime history
structural response
2 5 10 1
00
Tz (sec)
Hs
(m)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1
2
3
4
5
6
7
8
9
10
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Frame 001 05 Jan 2003 iacs34
Tz, s
Hs, m
-1000
0
1000
2000
3000
7626.150 7626.151 7626.152 7626.153 7626.154 7626.155 7626.156 7626.157
Pressure [kPa]
Time [s]
Run number: 1016
Ch17-Loc751 Ch18-Loc752 Ch21-Loc757 Ch22-Loc759Ch25-Loc761 Ch26-Loc760 Ch29-Loc765
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
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0.15
0.16
0.17
0.18
5 6 7 8 9 10 11 12 13 14 15 16 17 18
RA
O P
ITC
H A
CC
. η(5
)/kA
[(d
eg/s
²)/d
eg
]
WAVE PERIOD [sec]
ACCELERATIONS
Project: Untitled
Untitled ; 5.00kn 0.0° Untitled ; 5.00kn 15.0°Untitled ; 5.00kn 22.5° Untitled ; 5.00kn 30.0°
Procedure of determining structural response
6
shipmotion
long-term descriptionof random
sea
fluid motionin the tank
pressuretime history
structural response
statistics
Scope
experiments
critical conditions
Procedure of determining structural response
7
Methods for analyzing the sloshing
• analytical solutions, applicability limited to “regular” cases
• numerical concepts – more versatile application
– mesh-based methods (boundary element method, finite element method, finite volume method, finite difference method) and meshless methods (Smoothed Particle Hydrodynamics)
– not full knowledge about interacting phenomena
– computational expenses (temporal and spatial accuracy, simulation time)
• experiments despite cost and uncertainties → most reliable, thus ultimate method in determining pressures
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Experiments
Filling: 92.5%, 30%Filling: 92.5%, 30%
Irregular ship motionIrregular ship motion
4 DOF4 DOF
Rigid wallsRigid walls
Sensors’ locationSensors’ location
ScalingScaling
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Probabilistic methods
Gaussian process: initial sample normally distributed
sample of maxima Rayleigh/Rice distributed
Arbitrary process: no general relation established
maxima distribution sought
maxima distribution of interest rather than initial process distribution
Distribution of individual maxima
Probabilistic methods
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Probabilistic methods
Order statistics:rearranging the sample in ascending order
largest maximum distribution FXmax(x) = FX(x)n
- combined with a Peak-over-Threshold method:
only peaks over a certain, high threshold considered
Pickand’s theorem: generalized Pareto distribution
Distribution of largest maximum
individual maxima distribution FX(x)
Asymptotic extreme value theory:
dividing the sample into a number of even epochs
new sample: the largest element from all epochs
threshold level ?
epochs’ size ?
“new sample” size in experiments ?
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Probabilistic methods Characteristic extreme values
the most probable largest maximum, xp
expected value of the largest maximum, E[fXmax(x)]
value exceeded by the certain probability level α, xα
choice of probability level α ?
α
xp E[fXmax(x)] xα
fXmax
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Choice and fit of the model
3-parameter Weibull model:
Generalized Pareto model (peak-over-threshold method) :
Characteristic 3-hours extreme value x : nxF1
1)( for = 0.1
cxexF
)/)((1)(
Parameters’ estimation: method of moments
)()(1)()( uFuFxXFxF
0,
/)(1
0,1
/)(11)(c
uxe
ccuxcuxG
where FX(x) asymptotically follows the generalized Pareto distribution and can be expressed by:
Models’ evaluation: by plotting in the corresponding probability paper
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Fit of the models
0 1000 2000 3000 4000 5000
10-4
10-3
10-2
10-1
100 1-F(x)
x, kPa128 128 276 22154
0.921
0.873
0.8
0.692
0.545
0.368
0.192
0.066
0.011
0.0011-F(x) Weibull probability paper
x, kPa265 1026 2281 4351
0.082
0.05
0.03
0.018
0.011
0.007
0.004
0.002
0.002
0.0011-F(x) Pareto probability paper
x, kPa
0 1000 2000 3000 4000 5000
10-4
10-3
10-2
10-1
100 1-F(x)
x, kPa99 146 495 3073 22118
0.999
0.998
0.993
0.982
0.951
0.873
0.692
0.368
0.066
0.0011-F(x) Weibull probability paper
x, kPa1396 2322 3345 4476 5725
0.082
0.05
0.03
0.018
0.011
0.007
0.004
0.002
0.002
0.0011-F(x) Pareto probability paper
x, kPa
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Interesting feature:“clusters” of results
0 1000 2000 3000 4000 50000.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
p, kPa
F(p
)
0 500 1000 1500 2000 2500 3000 35000.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
p, kPa
F(p
)
0 500 1000 1500 2000 2500 30000.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
p, kPa
F(p
)
• various physical phenomena ?
• spatial/temporal pattern ?
• ...
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Fit of the models (Pareto: 87% threshold)
0 1000 2000 3000 4000 5000 6000 7000 80000.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
p, kPa
F(p
)
• Pickands’ theorem implies a high threshold level
• too high threshold level reduces the accuracy
Threshold selection in POT method
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Fit of the models (Pareto: 87% threshold)
0 1000 2000 3000 4000 5000 6000 7000 80000.98
0.982
0.984
0.986
0.988
0.99
0.992
0.994
0.996
0.998
1
p, kPa
F(p
)
WeibullGen.Pareto
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Threshold selection in POT method
0 1000 2000 3000 4000 5000 6000 7000 80000.98
0.982
0.984
0.986
0.988
0.99
0.992
0.994
0.996
0.998
1
p, kPa
F(p
)
0.870.950.99
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Threshold selection in POT method
0.8 0.85 0.9 0.95 15000
5050
5100
5150
5200
5250
5300
5350
5400
5450
5500
5550x
, kP
a
0.8 0.85 0.9 0.95 1
4700
4750
4800
4850
4900
4950
5000
5050
5100
5150
5200
x ,
kPa
0.8 0.85 0.9 0.95 1
4000
4050
4100
4150
4200
4250
4300
4350
4400
x ,
kPa
10211025
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 12500
3000
3500
4000
4500
5000
x ,
kPa
10211025
Estimates of characteristic extreme value (generalized Pareto model) with the threshold level as parameter
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Variability
10 runs with identical motion
time histories
5 runs with different motion
time histories
sloshing “inherent” variation
of estimates
variation of estimates due to
randomness in ship motion as well as sloshing response
the same order of
magnitude !
Higher variability of results by the generalized Pareto distribution
HIG
H
HIG
H
10 runs with identical motion
time histories
10 runs with different motion
time historiesLO
W
LO
W
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Conclusions
• order statistics approach for the distribution of largest maximum• good fit of the models to sloshing pressure data samples• underestimation of the highest data points• more conservative estimates by GPD• value for α / long-term estimates ?• threshold level for POT method / length of experimental runs ?• variability of results: number of experimental runs ?