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Electromagnetic Scattering 101
6 Numerical Examples
A number of examples will be presented which enable a comparison to be
made between the accuracy and the performance of the FETD approach and
the co-volume algorithm on unstructured meshes.
6.1 Narrow Waveguide
The first example involves the simulation of the propagation, in the positive
x-direction, of a plane harmonic TE wave, of wavelength λ, in a narrow rectangular
waveguide. The waveguide occupies the region 0 ≤ x ≤ 200λ and its
width, 0.4λ, is small enough to avoid the generation of any wave normal to the
direction of propagation. Two unstructured meshes, with spacing δ ≈ λ/15
and δ ≈ λ/30, are generated using the stitching method. The majority of the
elements are almost equilateral triangles which exhibit all the desired mesh
quality properties [ZM06]. To enable a comparison with the results produced
by the traditional Yee scheme, two structured triangular grids are generated,
using the vertex spacings δ = λ/15 and δ = λ/30. On these meshes, the covolume
scheme of the equations (11) and (12) reduces to the classical Yee
scheme. Figure 2 shows the structured mesh with δ = λ/15 and the unstructured
mesh with δ ≈ λ/15. The solution is advanced for 170 cycles, using the
maximum allowable time step. For each case considered, the computed distribution
of the magnetic field, between x = 139λ and x = 141λ, is compared
with the exact distribution in Figure 3. It can be seen that the Yee scheme on
the structured grid and the co-volume scheme on the unstructured grid maintain
the amplitude of the propagating wave, while the FETD scheme fails
to maintain the amplitude. It can also be observed that the phase velocity
is under-predicted by both the Yee and the co-volume schemes and is overpredicted
by the FETD scheme. However, the phase velocity obtained on the
unstructured meshes with the co-volume scheme is more accurate than the
phase velocity obtained using the traditional Yee scheme on the structured
(a)
(b)
Fig. 2. Details of the meshes employed for the propagation of a plane harmonic TE
wave in a waveguide: (a) the structured mesh with δ = λ/15; (b) the unstructured
mesh with δ ≈ λ/15.