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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