1

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

11

12

13

14

15

16

17

18

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

0.14

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

8

Experiments

Filling: 92.5%, 30%Filling: 92.5%, 30%

Irregular ship motionIrregular ship motion

4 DOF4 DOF

Rigid wallsRigid walls

Sensors’ locationSensors’ location

ScalingScaling

9

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

10

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 ?

11

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

12

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

13

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

14

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 ?

• ...

15

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

16

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

17

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

18

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

19

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

20

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 ?