On the asymptotic expansion of the magnetic potential in eddy current problem: a practical use of asymptotics for numerical purposes

Abstract : Asymptotics consist in formal series of the solution to a problem which involves a small parameter. When truncated at a certain order, the finite serie provides an approximation of the exact solution with a given accuracy, and the coefficients of this sum are solution to elementary problem that do not depend on the small parameter, which can be for instance the thickness of the domain or a small or high conductivity coefficient. This a useful tool to obtain approximate expressions of the solution to the so-called Eddy Current problem, which describes the magnetic potential in a material composed by a dielectric material surrounding a conductor. However such expansions are derivatives consuming, in the sense that to go further in the expansion, it is necessary to compute the higher derivatives of the first orders terms, and it also requires a precise knowledge of the geometry, since derivatives of the parameterization of the interface dielectric/conductor are involved. From the numerical point of view, this leads to instability which may restrict or prevent a direct use of the asymptotic expansion. The aim of this report is to present a numerical way to tackle such drawbacks by using the property that the coefficients of the expansion are real of the source term is real, making it possible to identify numerically the first two terms of the expansion.
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[Research Report] RR-8749, INRIA Bordeaux; INRIA. 2015
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Laurent Krähenbühl, Victor Péron, Ronan Perrussel, Clair Poignard. On the asymptotic expansion of the magnetic potential in eddy current problem: a practical use of asymptotics for numerical purposes. [Research Report] RR-8749, INRIA Bordeaux; INRIA. 2015. 〈hal-01174009〉

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