is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible.This is an author-deposited version published in: http://sam.ensam.eu Handle
is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible.This is an author-deposited version published in: https://sam.ensam.eu Handle ID: .http://hdl.handle.net/10985/7274 To cite this version :Duy Hung MAC, Stéphane CLENET, Jean-Claude MIPO -Transformation Methods for Static Field Problems With Random Domains -IEEE Transactions on Magnetics -Vol. 47, n°5, p.1446-1449 -2011 Any correspondence concerning this service should be sent to the repository Administrator : archiveouverte@ensam.euThe numerical resolution of the Maxwell equations enables the development of accurate models of electromagnetic systems. To solve numerically these partial differential equations, the Finite Elements Method (FEM) has been widely used. In several cases, the available input data are known with a finite level of confidence. These uncertainties can arise for instance from the aging of the materials or from imperfections of the manufacturing processes. Since the numerical models are more and more accurate due to the improvement of numerical methods (in 3D for example) and also due to the increase in computer performances, some of these uncertainties cannot be considered negligible any more. In several works, a probabilistic approach using random variables is used in order to take into account these uncertainties [1]. Methods have been presented in the literature to take into account the uncertainties on the behavior law [2], [3]. However, the case of uncertainties on the geometry is much less studied. In [4] one method which transforms the problem with uncertainties on the geometry into a problem with uncertainties on the behavior law is proposed. The challenge of this method is how to determine an efficient one to one random mapping that transforms the random domain into a deterministic domain.In this paper, a comparison between two methods to calculate the random mapping is proposed. One is based on the resolution of the Laplace equations. The second is based on a geometric transformation. First, we present briefly the transformation method and we will show how the problem with uncertainties on the geometry can be transformed into a problem with uncertainties on the behavior law using a random mapping. Second, we will detail the two methods proposed to determine the random mapping addressed above. Finally, these two methods are compared on a stochastic magnetostatic example. II. TRANSFORMATION METHODIn this part, we will recall briefly the transformation method [5] used to solve electromagnetic problems with random domains.The stochastic magnetostatic problem on a domain D(θ) with random inner interfaces or boundaries can be written:where θ, the outcome, refers to randomness of the problem. The uncertainties on the geometry can be taken into account by random interfaces Γ k (θ) between two sub-domains D i and D j . The permeability µ depends on the position x and also on the outcome θ. Actually, for a point x located close to a random inte...
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