Implementation of High-Order Impedance Boundary Conditions in Some Integral Equation Formulations

Abstract

The scattering problem from an object coated with materials is solved in the frequency domain. The coating is modeled by an impedance boundary condition (IBC) implemented in an integral equation (IE) prescribed on its outer surface. The Leontovich IBC (IBC0) is local and constitutes a poor approximation for low index coatings. A possible remedy is to employ higher order IBCs. These, as well as the IBC0, are implemented in two IE formulations: 1) CFIE with the electric current as sole unknown and 2) EFIE+MFIE where the unknowns are both the electric and magnetic currents. The problems raised by the discretization of the surface div and surface curl operators involved in the IE and IBC formulations are solved by employing a simple and computationally cheap technique, the accuracy of which is numerically assessed. The performances of the various IBC and IE formulations are evaluated by calculating the radar cross section (RCS) of some axisymmetric objects.

A Well-Posed and Effective High-Order Impedance Boundary Condition for the Time-Harmonic Scattering Problem From a Multilayer Coated 3-D Object

“…For a thin monolayer of thickness d, effective and well-posed high order IBCs (HOIBCs) of order k have been obtained via a multiscale asymptotic expansion of the exact solution with respect to d [2][3][4], and it has been mathematically and numerically demonstrated that, if Γ is smooth enough, the error between this expansion and the exact solution scales as O(d k+1 ). Regarding a low index and not necessarily thin multilayer, essentially heuristic HOIBCs involving tangential derivatives of the tangent electric and magnetic fields on Γ are available in the literature [5][6][7][8][9][10][11][12][13]. A generalized IBC was first proposed in the space domain for 2-D problems [5] where the coefficients in the IBC are derived from the exact reflection coefficient calculated for a plane wave incident on a planar layer.…”

“…In the spectral domain (infinite plane or 2D circular cylinder) an effective HOIBC with five coefficients has been proposed in [7] where the IBC coefficients are calculated in such a way as to yield an impedance as close as possible to the exact one (local planar or cylindrical approximation: LPA or LCA) in a large angular range including evanescent incident waves, and this HOIBC has been successfully applied to axisymmetric bodies [7]. A corresponding space domain formulation, termed here IBC3, has been proposed in [8] that can be applied to arbitrary 3D objects, and has been numerically implemented in MoM formulations [11][12][13].…”

“…The important issue of uniqueness of the solution of the Maxwell's problem (i.e., Maxwell's equations in free-space, radiation condition at infinity and HOIBC on Γ) is discussed in [9,11] where sufficient uniqueness conditions (SUCs) are proposed for some HOIBCs. New SUCs specific to the IBC3 with five coefficients are presented here.…”

“…This problem has been circumvented in [16] for the IBC0 and in [12] for the IBC3 by using Buffa-Christianssen functions that, however, increase the computational cost. Instead, the computationally very cheap div2curl and curl2div transformations [10,11,17] are employed here, and the costly inversion of a full impedance matrix involved in the formulation proposed in [11] is suppressed.…”

Approximate impedance for time-harmonic Maxwell's equations in a non planar domain with contrasted multi-thin layers

“…Our numerical examples indicate that the third order IBC consistently gives the most accurate results up to a value of δ = 0.09. Important numerical evaluations concerning high order impedance boundary conditions are performed also in many references, we cite, e.g., [17,28,29].…”