1. Extreme Value Prediction in Sloshing Response Analysis
CeSOS WORKSHOP
ON RESEARCH CHALLENGES
IN PROBABILISTIC LOAD
AND RESPONSE MODELLING
Extreme Value Prediction
in Sloshing Response Analysis
Mateusz Graczyk
Trondheim,

2. Scope Characteristics of sloshing Stochastic methods
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. Sloshing phenomenon 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
...

4. Sloshing parameters Fluid motion pattern Tank motion
Location in the tank
Fluid spatial / temporal pattern
...
Tank motion
Filling level
Wave heading angle
...

5. long-term description
Procedure of determining structural response
long-term description
of random
sea
ship
motion
fluid motion
in the tank
pressure
time history
structural response
Hs, m
Tz, s

6. long-term description
Procedure of determining structural response
long-term description
of random
sea
ship
motion
fluid motion
in the tank
pressure
time history
structural response
Scope
critical conditions
experiments
statistics

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% Irregular ship motion 4 DOF
Rigid walls
Sensors’ location
Scaling

9. Probabilistic methods
Distribution of individual maxima
Gaussian process:
initial sample normally distributed
sample of maxima Rayleigh/Rice distributed
Arbitrary process:
no general relation established
Probabilistic methods
maxima distribution of interest rather than initial process distribution
maxima distribution sought

10. Probabilistic methods
Distribution of largest maximum
Order statistics:
individual maxima distribution FX(x)
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
threshold level ?
Asymptotic extreme value theory:
dividing the sample into a number of even epochs
new sample: the largest element from all epochs
epochs’ size ?
“new sample” size in experiments ?

11. Probabilistic methods
Characteristic extreme values α
xp E[fXmax(x)] xα
fXmax
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 ?

12. Choice and fit of the model
3-parameter Weibull model:
Generalized Pareto model (peak-over-threshold method) :
where FX(x) asymptotically follows the generalized Pareto distribution and can be expressed by:
Parameters’ estimation: method of moments
Models’ evaluation: by plotting in the corresponding probability paper
Characteristic 3-hours extreme value x :
for  = 0.1

13. Fit of the models

14. Interesting feature:“clusters” of results
various physical phenomena ?
spatial/temporal pattern ?
...

15. Fit of the models (Pareto: 87% threshold)
Threshold selection in POT method
Pickands’ theorem implies a high threshold level
too high threshold level reduces the accuracy

16. Fit of the models (Pareto: 87% threshold)

17. Threshold selection in POT method

18. Threshold selection in POT method
Estimates of characteristic extreme value (generalized Pareto model) with the threshold level as parameter

19. the same order of magnitude !
Variability
10 runs with identical motion time histories
5 runs with different motion time histories
HIGH
10 runs with identical motion time histories
HIGH
10 runs with different motion time histories
LOW
LOW
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

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 ?