## 3 Results  
### Case Study 1: Scattering from a Perfect Conductor Cylinder  
The first case study is a benchmark case that is a detailed analysis of an example provided in [26]: A perfectly conducting cylinder, infinitely long along \(z\), is illuminated by a TM\({}_{\text{z}}\) plane wave, with wave-vector, \(\mathbf{k}\), oriented along the \(x\) axis (see Figure 2):  
\[E^{i}(r,\theta)=E_{0}e^{-jkross\theta}. \tag{20}\]  
For this problem, the scattering field may be given in closed form [26] as:  
\[E^{s}(r,\theta)=\sum_{n=-\infty}^{\infty}w_{n}H_{n}^{(2)}(kr)e^{jn\theta}, \tag{21}\]  
where coefficients \(w_{n}\) are given by:  
\[w_{n}=-j^{-n}\frac{J_{n}(ka)}{H_{n}^{(2)}(ka)}E_{0}. \tag{22}\]  
For this case study, a frequency of 3 GHz was chosen for the exciting wave, corresponding to a free-space wavelength \(\lambda=10\) cm. A cylinder of radius \(a=\lambda\) was assumed. The distribution of the scattered field, evaluated as in Expansion (21), is shown in Figure 3 and will be taken in the following as the reference solution.  
In implementing the DtN solution, a circular boundary, \(\Sigma\), of radius \(R=1.2\lambda\) was chosen, therefore locating the boundary of the computational domain in close proximity to the scatterer, in the near-field zone. The finite element mesh was made by triangular elements. The number of the degrees of freedom corresponds to the number, \(N\), of mesh nodes.  
The PML approach is instead implemented by using a commercial simulator (COMSOL Multiphysics(r), [32]), which requires a layer of thickness, \(\delta\), to be added externally, as pointed out. Such a thickness is here chosen in an optimal way, as discussed in the following.  
The DtN and PML numerical solutions are compared in terms of the relative error with respect to the reference solution, defined by using norm-2:  
\[\varepsilon\equiv\frac{\left\|E_{n}^{\mathrm{s}}-E_{r}^{\mathrm{s}}\right\|} {\left\|E_{r}^{\mathrm{s}}\right\|}, \tag{23}\]  
where the subscripts \"\(n\)\" and \"\(r\)\" indicate the numerical and the reference solutions, respectively. The numerical solutions can be obtained by using the DtN or the PML approaches. In the latter case, the layer thickness, \(\delta\), is set to a value that optimizes the curve of the performances in terms of the error (23). In this case, this value was found to be \(\delta=1.4\lambda\).  
Figure 3: Case study 1 (perfectly conducting cylinder). Relative error as a function of the number of mesh nodes, for the DtN and perfect matching layer (PML) approaches. The PML solution refers to the optimized value of the layer thickness, \(\delta=1.4\lambda\).  
Figure 2: Case study 1. (**a**) Geometry and references for evaluating the scattering from an infinitely long conducting cylinder, illuminated by a TM\({}_{2}\) plane wave. (**b**) Map of the amplitude of the scattered field, normalized to that of the incident field. The axes \(x\) and \(y\) are normalized to the wavelength value [26].  
The errors are plotted in Figure 3, versus the number, \(N\), of mesh nodes, which is related to the computational cost, as pointed out before. The result validates the proposed DtN-based numerical scheme and clearly indicates its advantage with respect to the PML approach. Indeed, for a given value of \(N\) the relative error for DtN is about one order of magnitude smaller. Conversely, for a fixed accuracy, the DtN approach requires much less elements: For instance, a relative error \(\varepsilon=0.01\) would be obtained with \(N=501\) for DtN and \(N=11305\) for PML, hence with a reduction of the DoFs of about \(23\times\). Similar performances may be observed in the case of a single penetrable cylinder, or for arbitrarily shaped scatterers, as shown in [27].