Low frequency scaling of the mixed MFIE for scatterers with a non-simply connected surface

Abstract

-The Magnetic Field Integral Equation (MFIE) is a widely used integral equation for the solution of electromagnetic scattering problems involving perfectly conducting objects. It is usually discretized by means of RWG functions as both basis and test functions. This discretization of the MFIE is well-known for its good condition number. However, it is equally well-known for the inferior accuracy of its solution when compared to the Electric Field Integral Equation (EFIE). What is less-known is that this accuracy problem becomes even more serious when the frequency is lowered. Recently it has been proved that the so-called mixed discretization of the MFIE, also called 'mixed MFIE', eliminates this low-frequency accuracy problem on simply connected scatterers. The mixed MFIE utilizes the so-called Buffa-Christiansen or Chen-Wilton functions for testing. In this contribution, the low frequency behavior of the mixed MFIE is investigated for scatterers with a non-simply connected surface. An analysis shows the presence of an approximate nullspace in the mixed MFIE at low frequencies. This nullspace becomes exact when the frequency is zero. This behavior matches known results for the continuous MFIE. Numerical results are presented that confirm this analysis. Despite the approximate nullspace at low frequencies, numerical results indicate that the mixed MFIE still delivers accurate results for toroidal scatterers.

Low-frequency stability of the Razor blade tested MFIE

“…For each wave number, the smallest singular value was plotted in Figure 3. Again, the result is as expected from [7].…”

“…Indeed, it was shown that a mixed discretization scheme, in which the MFIE is tested using rotated Buffa-Christiansen (BC, [2]) or Chen-Wilton (CW, [3]) functions, delivers a much better accuracy [4,5] than the standard discretization scheme. In addition, the mixed discretization yields physical solutions at low frequencies [6,7], which is in stark contrast to the standard scheme. A drawback of the mixed scheme is the need for a barycentrically refined mesh, which increases the number of triangle-triangle interactions that needs to be evaluated for each impedance matrix element.…”

“…As a consequence, there are F − 1 loops, hence the requirement for low-frequency stability is met for simply connected structures. In the same spirit as [7], it can also be concluded that this testing scheme for the MFIE on non-simply connected scatterers will lead to an exact null-space in the DC limit, which is physical because it is also found in the continuous MFIE [11].…”

A well-conditioned combined field integral equation based on quasi-helmholtz projectors

“…The new operator will be discretized by adopting a mixed-discretization strategy [7,8] where the magnetic operators are tested with Buffa-Christiansen (BC) basis functions. For the sake of brevity we consider a discretization for the case of simply connected structures; minor modifications are required for the non-simply connected case.…”

On a Well-Conditioned Electric Field Integral Operator for Multiply Connected Geometries

“…If these two components are not separated during the solution process, numerical cancelations that deteriorate the accuracy of the far field computation ensue. This phenomenon has been first pointed out in [24], and further studied in [25], [26] and [27] and [28]. The reader should be aware of the meaning of "very low frequency" in this context.…”