Eddy Currents and Corner Singularities

Abstract

Eddy current problems are addressed in a bidimensional setting where the conducting medium is non-magnetic and has a corner singularity. For any fixed parameter δ linked to the skin depth for a plane interface, we show that the flux density |∇A δ | is bounded near the corner unlike the perfect conducting case. Then as δ goes to zero, the first two terms of a multiscale expansion of the magnetic potential are introduced to tackle the magneto-harmonic problem. The heuristics of the method are given and numerical computations illustrate the obtained accuracy.

Perfect Conductor and Impedance Boundary Condition Corrections via a Finite Element Subproblem Method

“…A modified IBC can be defined in the vicinity of corners in 2-D or edges in 3-D, as given by a reference problem [2]. If Γ c,p,i has one corner singularity located at the origin X = 0, of angle β in Ω c,p,i , then the scaling X = x/δ p gives a "profile" term V α that is independent of δ p and satisfies a reference problem described in details in [2], with a reference corner of the same opening β as in Ω c,p,i .…”

“…If Γ c,p,i has one corner singularity located at the origin X = 0, of angle β in Ω c,p,i , then the scaling X = x/δ p gives a "profile" term V α that is independent of δ p and satisfies a reference problem described in details in [2], with a reference corner of the same opening β as in Ω c,p,i . The surface impedance close to the corner can be then approximated by…”

Corner asymptotics of the magnetic potential in the eddy‐current model

“…Few authors have proposed heuristic impedance modifications close to the corners [5,6], but these modifications are neither satisfactory nor proved. In particular in [6], the modified impedance appears to blow up near the corner, which does not seem valid for nonmagnetic materials with a finite conductivity as presented in [7].…”

“…In particular, when the conductor has a convex corner, its surrounding domain C has a nonconvex corner; thus, the Dirichlet problem has non-C 1 singularities, in opposition to the problem for any finite Ä 0 . This apparent paradox can be solved by a delicate multiscale analysis, whose heuristics are exposed in [7]. Roughly speaking, there exist profiles in R 2 that have the singular behavior of the operator C 2i1 near the corner and that connect at infinity with the singular functions of the Dirichlet problem in C , as described by (13) in [7].…”

“…This apparent paradox can be solved by a delicate multiscale analysis, whose heuristics are exposed in [7]. Roughly speaking, there exist profiles in R 2 that have the singular behavior of the operator C 2i1 near the corner and that connect at infinity with the singular functions of the Dirichlet problem in C , as described by (13) in [7]. This is why the knowledge of the singularities of C iÄ 0 1 at given Ä and is a milestone in the full multiscale analysis.…”