Eddy Current Modeling in Multiply Connected Regions via a Full-Wave Solver Based on the Quasi-Helmholtz Projectors

Abstract

Eddy currents are central to several industrial applications and there is a strong need for their efficient modeling. Existing eddy current solution strategies are based on a quasi-static approximation of Maxwell's equations for lossy conducting objects and thus their applicability is restricted to low frequencies. On the other hand, available full-wave solvers such as the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) equation become highly ill-conditioned and inaccurate in eddy current settings. This work presents a new well-conditioned and stable full-wave formulation which encompasses the simulation of eddy currents. Our method is built upon the PMCHWT equation and thus remains valid over the entire frequency range. Moreover, our scheme is also compatible with structures containing holes and handles (multiply connected geometries). The effectiveness of quasi-Helmholtz projectors is leveraged to obtain a versatile solver, which is computationally efficient and allows for a seamless transition between low and high frequencies. The stability and accuracy of the new method are demonstrated both theoretically and through numerical experiments on canonical and realistic structures.

An efficient full wave solver for eddy currents

“…Between these two extremes it is necessary to solve the MTP without approximations. The full-wave solvers for the MTP in the eddy current regime that we have found in the literature are [24,29,4]. Rucker et al [24] give an overview of full-wave solvers, which give at best a relative error of 1% in scattering situations comparable to our Figure 3.…”

“…Zhu et al [29] describe a full-wave solver used for an open source program FastImp. Chhim et al [4] present a full-wave solver based on the PMCHWT BIE. We lack data to judge the accuracy of the BIEs in [29] and [4].…”

“…Chhim et al [4] present a full-wave solver based on the PMCHWT BIE. We lack data to judge the accuracy of the BIEs in [29] and [4]. The analysis in [4] appears to be limited to ω → 0 for fixed σ, leaving a possible gap to the green PEC regime in Figure 1.…”

“…We lack data to judge the accuracy of the BIEs in [29] and [4]. The analysis in [4] appears to be limited to ω → 0 for fixed σ, leaving a possible gap to the green PEC regime in Figure 1. Also, numerical evaluation of the field representations is missing in [4], cf.…”

“…For our numerical method to be efficient when Im (k + ) is large, we always assume that |k + |L 50, so our standing assumption in numerical evaluations is that (4) 0 < k − L |k + |L 50.…”