Electromagnetics - Modelling, Simulation, Control and Industrial Applications - Abstract
Schmidt, Kersten
Shielding sheets are commonly used in the protection of electronic devices. With their large aspect ratios they become a serious issue for the direct application of the finite element method, as many small cells are required to resolve the sheets, as well as the direct application of the boundary element method (BEM) due to the occuring almost singular integrals. Impedance transmission conditions (ITCs), posed on the sheet mid-line $Gamma$, allow for finite element formulations in the exterior of the sheet mid-line or boundary element formulations on this mid-line only. We introduce the ideas behind the commonly used ITCs for the time-harmonic eddy current problem in two dimensions. Some of the ITCS, like those by Levi-Cevita (1902) or the shielding element by Nakata et. al (1990) assume the sheet thickness $d$ small compared to the skin depth, whereas the thin sheet conditions by Mayergoyz et. al. (1974) assume larger field variation in thickness direction due to the skin effect. All the impedance transmission conditions can be written in the general form with (possibly differential) operators depending on frequency, conductivity, sheet thickness and sheet geometry (e.g. with the curvature). We show by an asymptotic analysis that all the previously models are robust with respect to the skin depth or frequency, i.e., the accuracy depends on the sheet thickness, but not on the frequency, and that the commonly used perfect electric conductor (PEC) boundary condition is not robust. By asymptotic expansions we derive two families of impedance boundary conditions ITC-1-N and ITC-2-N where N denotes the order. The family ITC-1-N is derived for a conductivity or frequency scaled like $1/d$ and for ITC-2-0 it is scaled like $1/d^2$. We find ITC-1-0 the natural limit for $d to 0$ which coincides with Levi-Cevita's conditions, and derived with ITC-2-1 conditions of at least $O(d^2)$ in all the asymptotic regimes $sigmaomega sim 1$, $sigmaomega sim 1/d$ and $sigmaomega sim 1/d^2$. In the second part of the talk we will propose boundary integral equations (BIE) and boundary element methods of the classical and recent impedance transmissions conditions. Here, ITC-1-0, ITC-1-1 and thin sheet conditions lead to BEMs of second kind where we introduce mixed BEMs of first kind for the shielding elements, ITC-1-2 and ITC-2-1. For the latter stable pairs of finite element spaces are required to obtain stability even in the limit $d to 0$. For large thickness to skin depth ratios or in presence of differential operators the BIE are singularly perturbed. We analyse the dependence of the regularity of the solution on the model parameters and the sheet regularity. The research is in collaboration with Sébastien Tordeux (INRIA Bordeaux South-Ouest), Alexey Chernov (University of Bonn) and Ralf Hiptmair (ETH Zurich).