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, 24.03.2006

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

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 ...

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

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

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

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

Experiments Filling: 92.5%, 30% Irregular ship motion 4 DOF Rigid walls Sensors’ location Scaling

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

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 ?

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 ?

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

Fit of the models

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

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

Fit of the models (Pareto: 87% threshold)

Threshold selection in POT method

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

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

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 ?