Stability Analysis of the EFIE-IBC Formulation and Regularization via Spatial Filtering

The Impedance Boundary Condition (IBC) is a homogenization approximation of great importance, especially in the design of metasurfaces. However, the standard Electric-Field Integral-Equation formulation of the IBC boundary-value problem (EFIE-IBC) has been shown to lead to numerical instabilities for some impedance ranges of practical interest, in particular inductive reactances. This contribution shows that the numerical instabilities are due to an intrinsic ill-conditioning of the EFIE-IBC operator for the concerned surface impedance values, that can degenerate into an ill-posedness that does not allow for definite solution. Hence, the stable discretization of the EFIE-IBC operator requires a regularization. The analysis leads to a proposed regularization by systematically limiting the wavenumber spectrum of the basis functions, which amounts to a spatial filtering. This is implemented using entire-domain basis functions. Given the possible ill-posedness, we devise two “ground truth” test examples starting from a physical metasurface, then approximated via IBC. Comparison to ground truth results shows that the standard EFIE-IBC may lead to significant errors, and that these may be challenging to detect. Conversely, the regularized system yields stable results that well match the ground truth of the physical structure of which the IBC is an approximation.