Partial Differential Equations - Modelling and ... - ResearchGate
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Electromagnetic Scattering 107 Table 3. Simulation of scattering of a plane TE wave by a square PEC cylinder of side length λ. Mesh FETD Co-volume Speed up resolution spc time, s E SW spc time, s E SW ratio a 61 18. 2.64 45 0.4 0.21 45 b 90 27. 1.66 88 0.8 0.25 34 c 182 58. 0.38 164 1.3 0.14 44 (a) (b) (c) Fig. 9. Details of the meshes employed for the simulation of scattering of a plane TE wave by a square PEC cylinder of side length λ showing (a) mesh a, (b) mesh b, (c) mesh c. E SW,dB Near-Corner Resolution Fig. 10. Simulation of scattering of a plane TE wave by a square PEC cylinder of side length λ showing the variation in the computed error with the near corner mesh resolution. error in the FETD results on mesh c is similar to the error in the co-volume results obtained on mesh a. The constant error in the co-volume results confirm the belief that no special modification of the scheme is required in the vicinity of geometrical singularities. Table 3 also displays information about the calculations performed on meshes b and c. For this example, the co-volume scheme is faster than FETD by a factor that ranges between 34 and 45. This level of variation in the speed-up factor is probably due to the difficulty in determining exactly the small times required for the co-volume solution.
108 I. Sazonov et al. 6.4 Scattering by a PEC NACA0012 Aerofoil The next example involves the simulation of scattering of a plane single frequency wave, directed along the x-axis, by a perfectly conducting NACA0012 aerofoil of length λ. The aim of this example is to analyse the performance of the numerical schemes when the geometry exhibits high curvature. A benchmark solution is computed using an unstructured mesh with spacing λ/120. The unstructured mesh is generated, outside the aerofoil, in the region −λ ≤ x, y ≤ λ. The scattering width distributions computed on this mesh with the co-volume scheme and the FETD scheme proved to be identical. An unstructured mesh was generated to meet the spacing requirement of λ/15. Another unstructured mesh, providing better representation of the leading edge curvature, is generated by locally reducing the mesh spacing in the vicinity of the leading edge of the airfoil by a factor of 2. A view of both these meshes is shown in Figure 11. The computed scattering width distributions are compared with the benchmark distribution in Figure 12. It can be observed that the co-volume results are better on the uniform mesh and that the accuracy of the FETD results improve with the local refinement in the leading edge region. For this example, Table 4 shows the values of spc, time and E SW . The co-volume method is approximately 30 times faster than FETD for this example. 6.5 Scattering by a PEC Cavity The final example considers the simulation of scattering of a plane single frequency wave by a U-shaped PEC cavity. The thickness of the cavity walls is equal to 0.4λ, the internal cavity width is 2λ and the internal cavity length is 8λ. In the simulation, the wave is incident upon the open end of the cavity and propagates in a direction which lies at an angle θ =30 ◦ to the main axis of the cavity. An unstructured mesh is employed, with typical spacing λ/15, in the region that lies within a distance of λ from the scatterer, as (a) (b) Fig. 11. Details of the unstructured meshes employed for the simulation of scattering of a plane TE wave by a PEC NACA0012 aerofoil of length λ showing (a) the uniform mesh, (b) the locally refined mesh.
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Electromagnetic Scattering 107<br />
Table 3. Simulation of scattering of a plane TE wave by a square PEC cylinder of<br />
side length λ.<br />
Mesh FETD Co-volume Speed up<br />
resolution spc time, s E SW spc time, s E SW ratio<br />
a 61 18. 2.64 45 0.4 0.21 45<br />
b 90 27. 1.66 88 0.8 0.25 34<br />
c 182 58. 0.38 164 1.3 0.14 44<br />
(a) (b) (c)<br />
Fig. 9. Details of the meshes employed for the simulation of scattering of a plane<br />
TE wave by a square PEC cylinder of side length λ showing (a) mesh a, (b) mesh<br />
b, (c) mesh c.<br />
E SW,dB<br />
Near-Corner Resolution<br />
Fig. 10. Simulation of scattering of a plane TE wave by a square PEC cylinder<br />
of side length λ showing the variation in the computed error with the near corner<br />
mesh resolution.<br />
error in the FETD results on mesh c is similar to the error in the co-volume<br />
results obtained on mesh a. The constant error in the co-volume results confirm<br />
the belief that no special modification of the scheme is required in the<br />
vicinity of geometrical singularities. Table 3 also displays information about<br />
the calculations performed on meshes b <strong>and</strong> c. For this example, the co-volume<br />
scheme is faster than FETD by a factor that ranges between 34 <strong>and</strong> 45. This<br />
level of variation in the speed-up factor is probably due to the difficulty in<br />
determining exactly the small times required for the co-volume solution.
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