## 4 Results
### Lossy Dielectric Scatterer  
Finally, a lossy dielectric scatterer of radius \(\rho=5\lambda\), arc displacement \(d=0.5\rho=2.5\lambda\), relative permittivity \(\varepsilon_{r}=2.56-0.102\ j\), incidence angle \(\varphi_{inc}=0\), inner and outer auxiliary surface radius \(\rho_{aux\ in}=\rho-0.5\lambda\) and \(\rho_{aux\ out}=\rho+0.75\lambda\), and originally 180 CPs (hence 180 inner and 180 outer ASs) was analyzed. Similar BC tests were conducted for conformal auxiliary surface, once again resulting in a relatively sizable BC error in the neighborhood of the wedge tips (Figure 8a).  
Figure 7: Results for lossless dielectric scatterer: (**a**) \(E\) and \(H\) field boundary the condition error by employing a conformal auxiliary surface; (**b**) \(E\) and \(H\) field boundary the condition error by employing a deformed auxiliary surface; (**c**) the Radar Cross Section (RCS) acquired by utilizing the auxiliary surface deformation scheme.  
The deformation scheme proposed above was also implemented in order to enhance the satisfaction of the boundary condition. The parameters found to produce the lowest BC error, shown in Figure 8b, are as follows: 1/5 of the inner ASs and 1/8 of the outer ones were displaced, with a proximity factor set equal to \(s=0.75\) and \(s=0.65\), respectively, whereas all of them were included in the densification with \(D_{start}=0.80\). Adding extra ASs did not prove to be beneficial and, therefore, was not employed.  
Figure 8: Results for lossy dielectric scatterer: (**a**) error in the generic boundary condition (BC) of the \(E\) and \(H\) field via conformal auxiliary surface placement; (**b**) error in the generic BC of the \(E\) and \(H\) field via improved auxiliary surface placement; (**c**) computed RCS for conformal auxiliary surfaces; (**d**) computed RCS for improved auxiliary surfaces; (**e**) RCS compared for both auxiliary surface placements (conformal shown in red, improved shown in blue), focused in angles near the backscattering direction.  
The maximum \(E\) field error at MPs was reduced from \(5.13*10^{-2}\) to \(7.27*10^{-3}\) and the maximum \(H\) field error from \(5.11*10^{-1}\) to \(4.14*10^{-2}\).  
The computed RCSs, using conformal and deformed auxiliary surface layouts, are depicted in Figure 8c,d, respectively. Although not visually discernible in these plots due to their wide dynamic range, comparing their values at angles close to the scatterer's wedges exhibits an observable improvement, e.g., from \(300^{\circ}\) to \(360^{\circ}\) in Figure 8e, in the azimuthal direction where \(\Delta E\) and \(\Delta H\) obtain their greatest values.  
It is worth mentioning that placing ASs too close to the wedge tip is not advisable. Indeed, an AS right on the wedge tip causes severe ill-conditioning of the linear system matrix, whereas proximity factor \(\mathrm{s}=0.99\) for both inner and outer ASs lead to inadequate error reduction, namely \(3.16*10^{-2}\) for the \(E\) field and \(4.43*10^{-1}\) for the \(H\) field.  
Furthermore, the symmetry of the geometry should be accompanied by symmetry in the ASs/CPs. As a counterexample, the same lossy dielectric scatterer was experimentally analyzed via unequal CP densities in the two arcs, namely 45 for the upper and 90 for the lower one, leading to excessively high errors and completely wrong RCS plots, as demonstrated in Figure 9.