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Contribution à la modélisation numérique en électromagnétisme statique stochastique

Abstract : In electromagnetism, for most numerical models (so-called deterministic models) solving Maxwell Equations, all input data are supposed to be perfectly known. Unfortunately, geometry and material characteristics would rather present uncertainties (aging...). The problem is then how to broadcast the uncertainties of the entry data to the output data. In those conditions, probabilistic models would be more appropriated to take into account those uncertainties than a deterministic model. Methods have been already proposed mainly in mechanics, but few in electromagnetism. First, the Monte Carlo Simulation Method (MCSM) is simple and robust, but is very time consuming. Besides, other methods have been proposed. The “perturbation method” consists in expanding the unknown field around its mean. This method is very useful to determine the moment of the first and second orders (mean and variance) of the unknown field. But the extension to moments of higher orders is very difficult and time consuming. In Neumann Expansion Method operators are expanding into Neumann series but the convergence rate seems to be weak. Finally, methods based upon chaos polynomial expansions such as the so-called Spectral Stochastic Finite Element Method (SSFEM) has been proposed by Ghanem or non intrusive projection methods have been proposed. It is used to solve spatial differential equation's problem, by using a discretization simultaneous onto spatial domain and random domain. This method has been already applied mainly in mechanics. In this thesis, we present the Monte Carlo Simulations, Non intrusive projection method, and SSFEM to solve a electrokinetics problem where the conductivies are supposed to be piece wise random variable. Those two lasts, are based upon Hermite chaos polynomials expansion, this family of polynomial will be presented. The conductivities can be expanded in Hermite Polynomials; we will study the effect of the truncate. Finally, we will propose method to compute global values as the current, and we will use this method to calculate the current in an industrial case: a line joint. Keywords: numerical electromagnetism, random, uncertainty quantification, polynomial chaos, stochastic finite element method, projection non intrusive method, global quantities computation.
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Submitted on : Tuesday, June 23, 2009 - 8:00:00 AM
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  • HAL Id : pastel-00005115, version 1

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Roman Gaignaire. Contribution à la modélisation numérique en électromagnétisme statique stochastique. Sciences de l'ingénieur [physics]. Arts et Métiers ParisTech, 2008. Français. ⟨NNT : 2008ENAM0005⟩. ⟨pastel-00005115⟩

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