*4.1. PEC Scatterer*

To test the efficiency of the method, a PEC scatterer is initially defined by radius ρ = 3λ, arc displacement *d* = 0.5ρ = 1.5λ, incidence angle ϕ*inc* = 0, auxiliary surface radius ρ*aux* = ρ − 0.075λ, and originally 136 CPs and therefore 136 inner ASs. The solution to the problem without any deformation yields the results of Figure 5a, which depicts the quantified error in the generic boundary condition (BC) of the *E* field along the boundary stretch, i.e.,:

$$
\Delta E\_{\rm hc} = \frac{\left| \hbar \times (E\_{\rm in} - E\_{\rm out}) \right|}{|E\_{\rm inc}|\_{\rm max}},
\tag{9}
$$

where in (9) *n*ˆ is the normal unit vector on the boundary, pointing outwards, and *Ein*, *Eout* are the electric fields just inside and just outside the scatterer, respectively, the former obviously vanishing for the PEC case.

**Figure 5.** Plots of *E* field boundary condition error for a perfectly electric conducting (PEC) scatterer with initial ρ*aux* = ρ − 0.075λ using: (**a**) the standard, conformal auxiliary surface; and (**b**) the deformed auxiliary surface.

To improve the satisfaction of the BC, the deformation scheme proposed above was implemented. After several trials, the following parameters were finally invoked: All ASs were displaced as well as densified and the proximity factor was set equal to *s* = 0.88, whereas *Dstart* = 0.9. No extra ASs were added since their presence proved to achieve only incremental improvement. The results are displayed in Figure 5b, which shows that the overall *E* field error at MPs was substantially reduced.

It is worth noting that the proper placement of the auxiliary surface is of great importance in order to avoid caustics exclusion by the auxiliary surface or any potential resonance effects, as presented in [23]. To illustrate the possibly detrimental influence on the BC error, specifically, in this case, keeping the multitude of CPs consistent (and thus the multitude of ASs consistent) but moving the auxiliary surface further away from the scattering boundary (ρ*aux* = ρ − 0.15λ) produced the results in Figure 6a,b, before and after applying the deformation scheme, respectively. Severe complications are revealed in Figure 6a, while it may be deduced from Figure 6b that utilizing an improved auxiliary surface layout, compared to a conformal one, may be much less prone to accuracy degradation, presumably because caustics are still retained within the auxiliary surface.

**Figure 6.** Plots of *E* field boundary condition error for a PEC scatterer with initial ρ*aux* = ρ − 0.15λ using: (**a**) the standard, conformal auxiliary surface; and (**b**) the deformed auxiliary surface.

## *4.2. Lossless dielectric scatterer*

Next, a lossless dielectric scatterer was considered, with radius ρ = 3λ, arc displacement *d* = 0.5ρ = 1.5λ, relative permittivity ε*<sup>r</sup>* = 2.56, incidence angle ϕ*inc* = 0, inner and outer auxiliary surface radius ρ*aux in* = ρ − 0.3λ and ρ*aux out* = ρ + 0.45λ, and, originally, 160 CPs (hence 160 inner and 160 outer ASs). Like in the PEC case, the efficiency of MAS was initially tested using a conformal auxiliary surface and the extracted results of the BC error are presented in Figure 7a, where the upper subplot depicts the quantified error for the *E* field (Δ*E* %) and the lower one for the *H* field (Δ*H* %), both being relatively significant near the wedge tips.

**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.

Afterwards, the improvement of the BC was examined by employing the aforementioned deformation scheme with the following parameters, obtained through multiple trials: the portion of ASs to be displaced was 1/5 for the inner and 1/8 for the outer ones, with a proximity factor of *s* = 0.75 and *s* = 0.65, respectively, while all of them were densified with *Dstart* = 0.80. The addition of extra ASs offered no further refinement and was skipped once more. Figure 7b depicts the considerably decreased quantified BC error, and the corresponding radar cross section (RCS) is displayed in Figure 7c.

#### *4.3. Lossy Dielectric Scatterer*

Finally, a lossy dielectric scatterer of radius ρ = 5λ, arc displacement *d* = 0.5ρ = 2.5λ, relative permittivity ε*<sup>r</sup>* = 2.56 − 0.102 *j*, incidence angle ϕ*inc* = 0, inner and outer auxiliary surface radius ρ*aux in* = ρ − 0.5λ and ρ*aux out* = ρ + 0.75λ, 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 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 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 *Dstart* = 0.80. Adding extra ASs did not prove to be beneficial and, therefore, was not employed. The maximum *<sup>E</sup>* field error at MPs was reduced from 5.13 <sup>∗</sup> 10−<sup>2</sup> to 7.27 <sup>∗</sup> 10−<sup>3</sup> and the maximum *<sup>H</sup>* field error from 5.11 <sup>∗</sup> <sup>10</sup>−<sup>1</sup> to 4.14 <sup>∗</sup> <sup>10</sup><sup>−</sup>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◦ to 360◦ in Figure 8e, in the azimuthal direction where Δ*E* and Δ*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 <sup>s</sup> = 0.99 for both inner and outer ASs lead to inadequate error reduction, namely 3.16 <sup>∗</sup> <sup>10</sup>−<sup>2</sup> for the *<sup>E</sup>* field and 4.43 <sup>∗</sup> <sup>10</sup>−<sup>1</sup> for the *<sup>H</sup>* 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.

**Figure 9.** (**a**) Symmetric geometry with non-symmetric auxiliary points allocation, (**b**) RCS, and (**c**) ΔE and ΔH error.

#### *4.4. Investigation of the Solution Behavior for Various Geometry Modifications*

To check the robustness of the method described above, several modifications of the geometry were carried out and the respective results were extracted. In all cases below, lossy dielectric material with relative permittivity ε*<sup>r</sup>* = 2.56 − 0.102 *j* is considered, whereas the incidence angle is always ϕ*inc* = 0 with an arc radius of ρ = 3λ.

First, a scatterer that deviates only slightly from a circle was studied, i.e., a layout with small arc displacement *d* = 0.1ρ = 0.3λ, followed by a scatterer with a moderate deviation from a circle, i.e., a layout with mediocre arc displacement *d* = 0.5ρ = 1.5λ, and, finally, a scatterer with a significant distortion with respect to a circle, which, therefore, encompasses sharp wedges, i.e., a layout with large arc displacement *d* = 0.7ρ = 2.1λ. All three geometries and their computed results are depicted in Figures 10–12, respectively, while their MAS parameters and the quantified error in the generic boundary condition of the electromagnetic field are presented in Table 1.

**Figure 10.** Small arc displacement *d* = 0.1ρ = 0.3λ : (**a**) geometry and conformal auxiliary surfaces; (**b**) geometry and improved auxiliary surfaces; (**c**) conformal Δ*E* and Δ*H* error; (**d**) improved Δ*E* and Δ*H* error; (**e**) conformal RCS; and (**f**) improved RCS.

**Figure 11.** Mediocre arc displacement *d* = 0.5ρ = 1.5λ: (**a**) geometry and conformal auxiliary surfaces; (**b**) geometry and improved auxiliary surfaces; (**c**) conformal Δ*E* and Δ*H* error; (**d**) improved Δ*E* and Δ*H* error; (**e**) conformal RCS; and (**f**) improved RCS.

As a verification of the broad applicability of the technique, an asymmetric scatterer was also analyzed. Namely, two unequal circular arcs were connected, forming a "sorrowful" eye. The geometry parameters were chosen as follows: radius ρ = 4λ and arc displacement *d* = 0.5ρ = 2λ for the upper arc, radius ρ = 5λ and arc displacement *d* = 0.5ρ = 2.5λ for the lower arc, originally 160 CPs (hence 160 inner and 160 outer ASs), 1/2.5 of the inner ASs and 1/4 of the outer ones were displaced, with proximity factor set equal to *s* = 0.6 and *s* = 0.6, respectively. No densification was implemented and no extra ASs were added. The maximum *<sup>E</sup>* field error at MPs was reduced from 4.19 <sup>∗</sup> <sup>10</sup>−<sup>2</sup> to 1.57 <sup>∗</sup> <sup>10</sup>−<sup>2</sup> and the maximum *<sup>H</sup>* field error from 4.36 <sup>∗</sup> <sup>10</sup>−<sup>1</sup> to 1.38 <sup>∗</sup> <sup>10</sup><sup>−</sup>1. The results are depicted in Figure 13.

**Figure 12.** Large arc displacement *d* = 0.7ρ = 2.1λ : (**a**) geometry and conformal auxiliary surfaces; (**b**) geometry and improved auxiliary surfaces; (**c**) conformal Δ*E* and Δ*H* error; (**d**) improved Δ*E* and Δ*H* error; (**e**) conformal RCS; and (**f**) improved RCS.

Judging from the outcome of the aforementioned tests, the method proposed herein is very robust, producing reliable results for blunt, acute, or non-symmetric wedges.

**Figure 13.** Non-symmetric scatterer: (**a**) geometry and conformal auxiliary surfaces; (**b**) geometry and improved auxiliary surfaces; (**c**) conformal Δ*E* and Δ*H* error; (**d**) improved Δ*E* and Δ*H* error; (**e**) conformal RCS; and (**f**) improved RCS.

**Table 1.** Geometry definition, method of auxiliary sources (MAS) parameters, and computed BC error of the electromagnetic field.


**Table 1.** *Cont*.


<sup>1</sup> Boundary condition error at midpoints (MPs).

#### **5. Discussion**

The applicability range of the approach presented above may span various aspects of computational electromagnetics. MAS has already been used in the literature to simulate practical problems, such as scattering by a raindrop [24], although that particular geometry is smooth, without wedges. Wedge treatment, as discussed in this paper, enhances MAS accuracy for applications such as jet engine inlet modeling [25–27], where interior blades contain sharp edges, or even further military aircraft scattering simulation, such a stealth design. Moreover, additional possible applications may include automotive modeling, for example, the functionality of a telecommunications antenna in the presence of the vehicle surface. Furthermore, the algorithm may be used in the design of absorbers and cloaking materials [28–30], or generally wave manipulation [31] when the geometry layout contains wedges.

Although the numerical results presented were very good, they are limited to wedges formed by intersecting circular arcs only. It is anticipated that analogous techniques would be equally efficient in more general cases (i.e., arbitrarily shaped curved wedges); however, this is yet to be proven in future research. Likewise, other issues to be investigated include TE (Transverse Electric) incidence, the three-dimensional counterpart of this problem, unconventional materials, as well as a robust methodology to choose the optimal deformation parameters.

Furthermore, by studying various geometries, proposing schemes for wedge treatment and obtaining solutions by choosing alternative configurations for the incident wave, material properties, etc., an extensive data set can be acquired. The latter, a combination of input variables and their respective exported results may be used as training data for building a mathematical model via Machine Learning algorithms. Machine Learning, the usage of computational methods in order to produce/"learn" information directly from data without relying on explicit instructions, improves its accuracy and performance proportionally to the amount of available training data. A larger data set is, therefore, required to find hidden patterns or intrinsic properties that can be utilized in the procedure of determining the optimal settings for the scattering problem and, hopefully, automating it.
