A Quasi-Helmholtz Projector Stabilized Full Wave Solver Encompassing the Eddy Current Regime

Abstract: Despite its several qualities, the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation for simulating scattering by dielectric media suffers from numerical instabilities and severe ill-conditioning at low frequencies. While this drawback has been the object of numerous solution attempts in the standard low-frequency breakdown regime for scattering problems, the eddy-current regime requires a specific treatment. In this contribution, we present an extension of the recently introduced quasi-Helmholtz projectors based preconditioning of the PMCHWT to obtain an equation stable at low frequencies and specifically in three regimes relevant for eddy currents analyses: (i) when the frequency decreases with a constant conductivity, (ii) when the conductivity increases at fixed frequency and (iii) when the frequency decreases while keeping the product of the frequency and the conductivity constant. Being based on quasi-Helmholtz projectors our new strategy does not further degrade the conditioning of the original equation and is compatible with existing fast solvers. The resulting full-wave formulation is capable of smoothly transitioning from simulations at high frequencies to the different low-frequency regimes which encompass, in particular, eddy currents applications. Numerical results demonstrate the validity of our approach in all the regimes with a special emphasis given to eddy currents.