Results of the 3-d Finite Element Software (Hexahedral Edge Elements)

  Problem #1
  We solved the 3-d Wave Equation for a rectangular waveguide (Fig. 1.) driven in TE10 mode. The analytic solution of this problem is known. The boundary values of the E field vectors for the all the edges on x=0, x=4, y=0 and y=3 surfaces are set to (0+j0), for z=0 and z=5 surfaces the boundary values of the E field vectors defined for the edges which are parallel to x & z axis are set to (0+j0), and the boundary values of the E field vectors for the edges parallel to y-axis are calculated from the formula below :
  We run the program for three different meshes, the detailed results are given in Table 1. The results were very close to the analytical results, the table shows the deviation from the analytical results (both in magnitude and phase).
  Fig. 1. A rectangular waveguide

Table 1. Performance data of 3-d FEA software for the geometry in Fig 1.
RUN | 1 | 2 | 3
Number of elements used | 1 | 1 | 1
[x,y,z] resolution for each element | 4x2x5 | 8x3x10 | 20x5x25
Number of hexahedra | 40 | 240 | 2500
Number of nodes | 90 | 396 | 3276
Number of edges | 207 | 1009 | 9000
Number of unknowns | 55 | 473 | 6100
Total execution time (sec.) | 0.2 | 1.4 | 58.76
Percent Error in magnitude | 6.7 - 10.6 % | 0.9 - 3.3 % | 0.15 - 0.28 %
Percent Error in phase | 1.8 - 2.1 % | 0.3 - 1.1 % | 0.005 - 0.05 %

  The results obtained with the fine mesh (3rd run) were approximately equal to the analytical values.

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