Helmholtz's partial differential equation is approximated by a system of linear algebraic equations by minimising an energy norm. The
finite elements expression for central vertices is:
φ
xx+φ
yy+ε
dk²φ=0
(1+h²ε
dk²/8)[φ
x-1,y+φ
x+1,y+φ
x,y-1+φ
x,y+1]+(h²ε
dk²/2-4)φ
x,y=0
where ε
d is the dielectric constant.
The
Dirichlet boundary condition φ=0 is pertinent to electric waves, while the
Von Neumann boundary condition, ∂φ/∂n=0, applies to magnetic fields.
Finite elements impose the equivalent Eüler condition
everywhere along the element, while the Helmholtz equation is only enforced at the nodes for
finite differences. Therefore improved convergence is expected. A further benefit, if general triangular elements are used, is that they provide a better fit to curved parts of the boundary. Strict application of finite element theory yields a
symmetric matrix which has certain computational advantages.
Symmetry will be exploited at all times in field problems. Convergence will be verified by increasing the number of elements and checking that the results are much the same.
The same problems introduced in the finite differences section will be reexamined: The solution to the 1-d problem is only useful as a measure of the accuracy obtained. This
program provided the results shown in the following table, corresponding to a simple rectangular waveguide of cross section length
a in the
TE10 mode:
| Elements\ (2x/a) |
1 | .8 | .6 | .4 | .2 | 0 |
| 6 | 1 | .950 | .808 | .587 | .308 |
0 |
| 18 | 1 | .951 | .809 | .588 | .309 | 0 |
| sin(πx/a) | 1 | .951 | .809 | .588 | .309 | 0 |
The
cutoff wavelength for this waveguide is plain to see, but for more sophisticated waveguides, it will have to be calculated: The smallest (non zero) eigenvalue yields the
cutoff wave number.
The dielectric loaded, rectangular waveguide results will not be repeated here; there is no difference between the formulae when the terms in k² are omitted.
The
ridge
waveguide program provided the data in the
following table (ridge dimensions
d x
d/2):
| Elements | kd |
| 35 | 2.33 |
| 59 | 2.30 |
| 89 | 2.28 |
| Extrapolation | 2.24 |
| Reference | 2.25 |
The results have been improved by using
Aitken's extrapolation.
The field values for the ridge waveguide dominant H-mode have actually been calculated on a 69x69 grid, but are shown on a 9x9 grid (for 1/2 of the cross section). Otherwise, there is no difference between the
program shown and the program used.
| 797 | 784 | 745 | 680 | 592 | 481 | 355 | 217 | 73 |
| 809 | 796 | 758 | 693 | 603 | 489 | 360 | 219 | 73 |
| 832 | 820 | 783 | 720 | 627 | 506 | 368 | 223 | 74 |
| 864 | 854 | 822 | 765 | 670 | 528 | 377 | 227 | 75 |
| 901 | 893 | 870 | 830 | 777 | 573 | 392 | 231 | 76 |
| 938 | 933 | 918 | 898 | 882 | | | | |
| 969 | 966 | 958 | 947 | 941 | | | | |
| 990 | 988 | 983 | 977 | 974 | | | | |
| 1000 | 995 | 991 | 987 | 984 | | | | |
The cutoff wavelength and field amplitude for the
circular waveguide (of diameter
d) have been obtained in a similar way: The
eigenCirc program confirmed the value for the fundamental frequency in the literature, while the
circular program generated the figures in the next table: Again, the program calculations for the principal TE mode were performed on a 67x67 mesh (for the entire circle), but one quarter of a cross section is shown on a 9x9 mesh. In every other respect, the program used is exactly like the program supplied:
| Elements | kd |
| 76 | 3.65 |
| 126 | 3.70 |
| 190 | 3.68 |
| Extrapolation | 3.69 |
| Reference | 3.682 |
And the wave amplitude:
| | | | | | | 490 | 998 | 1000 |
| | | | | 880 | 925 | 959 | 981 | 987 |
| | | | 790 | 841 | 881 | 912 | 930 | 937 |
| | | 685 | 730 | 771 | 805 | 830 | 846 | 852 |
| | 562 | 599 | 636 | 669 | 697 | 718 | 731 | 735 |
| | 452 | 484 | 514 | 540 | 562 | 578 | 588 | 591 |
| | 326 | 349 | 370 | 389 | 404 | 415 | 422 | 424 |
| 164 | 185 | 199 | 211 | 222 | 230 | 236 | 240 | 241 |
| 35 | 38 | 41 | 43 | 45 | 47 | 48 | 49 | 49 |
I practise what I preach! The finite differences section mentioned exploiting symmetry whenever possible. Only one quarter of the
circular waveguide needs to be simulated (not one half), which doubles the resolution for the same amount of work. Also, the value of the cutoff wavelength was not entirely satisfactory. The second version of the
circular eigenvalue program yields a result within 0.1% of the reference.
In the first reference below, the derivation of finite element formulae and the corresponding geometric aspects are discussed. It is stressed that the Euler equation shown includes a Dirichlet boundary condition. The second paper explicitly includes final formulae for inhomogeneous waveguides.
References:
P. L. Arlett, A. K. Bahrani and O. C. Zienkiewicz: 'Application of finite elements to the solution of Helmholtz's equation', Proc IEE, Vol. 115, No.12, pp. 1762-1766
Ahmed, S. and Daly, P.: 'Finite-element methods for inhomogeneous waveguides', Proc. IEE, Vol. 116, No. 10, pp. 1661-1664
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