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Research Challenges in Probabilistic Load and Response Modeling for Extreme Value Prediction in Sloshing Response Analys

This workshop explores sloshing phenomenon characteristics, methods of analysis, sloshing parameters, and probabilistic methods for extreme value prediction in sloshing response analysis. It covers experimental procedures, statistical methods, and conclusions for accurate pressure estimation.

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Research Challenges in Probabilistic Load and Response Modeling for Extreme Value Prediction in Sloshing Response Analys

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  1. CeSOS WORKSHOP ON RESEARCH CHALLENGES IN PROBABILISTIC LOAD AND RESPONSE MODELLING Extreme Value Prediction in 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. 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 • Location in the tank • Fluid spatial / temporal pattern • ... • Tank motion • Filling level • Wave heading angle • ...

  5. 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. 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 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. fXmax α xpE[fXmax(x)] xα 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 ?

  12. Choice and fit of the model 3-parameter Weibull model: Generalized Paretomodel (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. Variability 10 runs with identical motion time histories 5 runs with different motion time histories HIGH HIGH 10 runs with identical motion time histories 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 ?

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