Low-Frequency Stabilizations of the PMCHWT Equation for Dielectric and Conductive Media: On a Full-Wave Alternative to Eddy-Current Solvers

TL;DR

The paper tackles instability of the PMCHWT boundary integral equation at low frequencies and in conductive media, which hinders accurate full-wave simulations for penetrable bodies. It introduces a quasi-Helmholtz projector-based preconditioner that left-right preconditions the PMCHWT operator, stabilizing conditioning across quasi-static, eddy-current-free, and skin-effect-dominated regimes while preserving the dominant current components needed for accurate field reconstructions. A dominant-component analysis informs which quasi-Helmholtz currents must be retained, and regime-specific scalar coefficients are chosen to avoid loss of accuracy and to eliminate artificial nullspaces, even on multiply connected geometries. Numerical results on simply and multiply connected structures demonstrate frequency- and conductivity-independent conditioning and high fidelity impedance and field predictions, with compatibility to fast solvers and standard acceleration strategies. The proposed approach thus enables robust, full-wave simulations that seamlessly span low to high frequencies and dielectric to highly conductive materials, including complex multi-loop geometries.

Abstract

We propose here a novel stabilization strategy for the PMCHWT equation that cures its frequency and conductivity related instabilities and is obtained by leveraging quasi-Helmholtz projectors. The resulting formulation is well-conditioned in the entire low-frequency regime, including the eddy current one, and can be applied to arbitrarily penetrable materials, ranging from dielectric to conductive ones. In addition, by choosing the rescaling coefficients of the quasi-Helmholtz components appropriately, we prevent the typical loss of accuracy occurring at low frequency in the presence of inductive and capacitive type magnetic frill excitations, commonly used in circuit modeling to impose a potential difference. Finally, leveraging on quasi-Helmholtz projectors instead than on the standard Loop-Star decomposition, our formulation is also compatible with most fast solvers and is amenable to multiply connected geometries, without any computational overhead for the search for the global loops of the structure. The efficacy of the proposed preconditioning scheme when applied to both simply and multiply connected geometries is corroborated by numerical examples.

Figures (8)

• Figure 1: Graphical representation of the low-frequency limit in the three regimes: the blue arrow represents the limit direction in the QSR, the black arrow in the ECFR, and the red arrow in the SEDR.
• Figure 2: Condition number of the formulations applied to torus with major and minor radii of 1m and 0.3m discretized at $h\simeq 0.44m$. The preconditioning applied is tailored to inductive (c) and capacitive (d) excitations.
• Figure 3: Condition number of the formulations applied to a sphere of radius 1m of conductivity 10S/m excited at 1e5Hz at different mesh refinements: comparison between the original PMCHWT formulation ($\mathsfbfit{Z}$), the Loop-Star preconditioned formulation ($\mathsfbfit{\Lambda} \mathsfbfit{H} \mathsfbfit{\Sigma}$ precond.), and this work ($\mathsfbfit{L}\mathsfbfit{Z}\mathsfbfit{R}$).
• Figure 4: Full-wave evaluation of the impedance of a toroidal structure with major and minor radii of 1m and 0.2m of different conductivities: relative difference with respect to circuit theory expectations $R_{ct}$ and $L_{ct}$.
• Figure 5: Torus of major and minor radii of 1m and 0.2m of conductivity 1S/m excited by a magnetic frill imposing 1V at 10mHz: induced magnetic field.