Partial Differential Equations - Modelling and ... - ResearchGate
Partial Differential Equations - Modelling and ... - ResearchGate Partial Differential Equations - Modelling and ... - ResearchGate
Electromagnetic Scattering 103 Table 1. Propagation of a plane harmonic TE wave in a waveguide. δ ≈ λ/15 Scheme spc time, s C A Yee 21 7 0.99613 1.00 Co-volume 46 29 0.99850 1.00 FETD 44 3151 1.0015 0.723 δ ≈ λ/30 Scheme spc time, s C A Yee 43 61 0.99896 1.00 Co-volume 106 263 0.99964 1.00 FETD 89 23040 1.0008 0.96 h , he Fig. 4. Propagation of a plane harmonic TE wave in a waveguide showing variation of the computed phase velocity with ∆t/〈h〉 (∆t/〈h e〉 for FETD). Solid symbols and solid line: δ ≈ λ/15; open symbols and dotted line: δ ≈ λ/30. Here 〈h〉 is the averaged Voronoï edge length, 〈h e〉 is the averaged minimal triangle height. 6.2 Scattering by a Circular PEC Cylinder The second example is the simulation of scattering of a plane single frequency TE wave by a perfectly conducting circular cylinder of diameter λ. The objective is to use this example to illustrate the order of accuracy that can be achieved with the co-volume solution technique and the FETD technique on unstructured meshes. The problem is solved on a series of unstructured meshes, with mesh spacings ranging from λ/8 toλ/128. The minimum distance from the rectangular PML to the cylinder is λ. When the spacing is
104 I. Sazonov et al. (a) (b) Fig. 5. Scattering of a plane TE wave by a circular PEC cylinder of diameter λ showing (a) an unstructured mesh with δ ≈ λ/16, (b) the corresponding computed total magnetic field. Scattering Width, dB Viewing Angle, degrees Fig. 6. Scattering of a plane TE wave by a circular PEC cylinder of diameter λ showing a comparison between the computed and analytical scattering width distributions. λ/16, the mesh employed, excluding the PML region, and the corresponding distribution of the computed total magnetic field is shown in Figure 5. The computed scattering width distributions are compared to the exact distribution in Figure 6. For each simulation undertaken, the error, E SW , in the solution is determined as the maximum difference, in absolute value, between the computed and analytical scattering width distributions. The variation of this computed error, with the number of elements per wavelength, λ/δ, forboth the FETD and co-volume schemes, is shown in Figure 7. It can be observed that a convergence rate of around O(δ 2 ) is obtained with both methods on these unstructured meshes, indicating that second-order accuracy is achieved. It is likely that the error in the FETD results is slightly less because the approach adopted for the evaluation of the scattering width integral requires an interpolation, in the co-volume scheme, to obtain all the field components at
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104 I. Sazonov et al.<br />
(a)<br />
(b)<br />
Fig. 5. Scattering of a plane TE wave by a circular PEC cylinder of diameter λ<br />
showing (a) an unstructured mesh with δ ≈ λ/16, (b) the corresponding computed<br />
total magnetic field.<br />
Scattering Width,<br />
dB<br />
Viewing Angle, degrees<br />
Fig. 6. Scattering of a plane TE wave by a circular PEC cylinder of diameter<br />
λ showing a comparison between the computed <strong>and</strong> analytical scattering width<br />
distributions.<br />
λ/16, the mesh employed, excluding the PML region, <strong>and</strong> the corresponding<br />
distribution of the computed total magnetic field is shown in Figure 5. The<br />
computed scattering width distributions are compared to the exact distribution<br />
in Figure 6. For each simulation undertaken, the error, E SW , in the solution<br />
is determined as the maximum difference, in absolute value, between the<br />
computed <strong>and</strong> analytical scattering width distributions. The variation of this<br />
computed error, with the number of elements per wavelength, λ/δ, forboth<br />
the FETD <strong>and</strong> co-volume schemes, is shown in Figure 7. It can be observed<br />
that a convergence rate of around O(δ 2 ) is obtained with both methods on<br />
these unstructured meshes, indicating that second-order accuracy is achieved.<br />
It is likely that the error in the FETD results is slightly less because the approach<br />
adopted for the evaluation of the scattering width integral requires an<br />
interpolation, in the co-volume scheme, to obtain all the field components at
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