Magnetic and Combined Field Integral Equations Based on the Quasi-Helmholtz Projectors

Abstract

Boundary integral equation methods for analyzing electromagnetic scattering phenomena typically suffer from several of the following problems: (i) ill-conditioning when the frequency is low; (ii) ill-conditioning when the discretization density is high; (iii) ill-conditioning when the structure contains global loops (which are computationally expensive to detect); (iv) incorrect solution at low frequencies due to current cancellations; (v) presence of spurious resonances. In this paper, quasi-Helmholtz projectors are leveraged to obtain a magnetic field integral equation (MFIE) formulation that is immune to drawbacks (i)-(iv). Moreover, when this new MFIE is combined with a regularized electric field integral equation (EFIE), a new quasi-Helmholtz projector combined field integral equation (CFIE) is obtained that also is immune to (v). Numerical results corroborate the theory and show the practical impact of the newly proposed formulations.

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

“…In addition, the augmented EFIE [19] has successfully been extended to lossy conductors [20] and inhomogeneous media [21] but these extensions require additional matrices to be computed and stored, thus increasing their computational cost. Among the different approaches, the use of quasi-Helmholtz projectors [22]- [24] offers many benefits over previous techniques, including an improved stability, an implicit handling of multiply connected geometries, and a compatibility with fast solvers operating with quasi-linear computational complexity [25], [26]. However, the approaches developed in [22], [24] are only available for perfectly conducting objects.…”

“…Among the different approaches, the use of quasi-Helmholtz projectors [22]- [24] offers many benefits over previous techniques, including an improved stability, an implicit handling of multiply connected geometries, and a compatibility with fast solvers operating with quasi-linear computational complexity [25], [26]. However, the approaches developed in [22], [24] are only available for perfectly conducting objects. Similarly, the formulation introduced in [23] has been derived for purely dielectric (lossless) materials but the proposed low frequency regularizer is not applicable to the lossy case because it cannot rescale separately the upper and lower diagonal blocks of the PMCHWT which do not follow the same frequency behavior under eddy current conditions.…”

“…Performing a Taylor series expansion of the Green function in (51) and (52) when ω → 0 and keeping the dominant real and dominant imaginary terms will yield the low frequency behavior of the near and far field scattering operators. To obtain the correct behavior, however, the aforementioned cancellations should be explicitly enforced in the expansion when considering the solenoidal components (as is customarily done at very low frequency [24]). Once the six scalings per operator have been obtained, they can be multiplied with those of the physical solutions (Table I (b)) to derive the far and near fields asymptotic scalings that are presented in Table I (c) and (d).…”

“…The scheme however is fully compatible with high frequency simulations for frequencies where a loop-star decomposition should not be used. In this case, it is sufficient to set to 1 all coefficients multiplying the projectors as is customary done in similar frameworks (see [24] and references therein). A smooth and automatic transition between the two regimes is possible and will be the topic of a future communication.…”

A Fast Converging Resonance-Free Global Multi-Trace Method for Scattering by Partially Coated Composite Structures

“…Because of this, a novel methodology is developed in this contribution. Our new method is inspired by the role that boundary layer operators at imaginary wavenumbers play in the construction of resonance free CFIE type integral equations [36]- [38].…”