Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

246 Waves
of 10 nodes or thereabouts per wavelength, makes the necessary ®nite element meshes
prohibitively ®ne. To take one example, radar waves of wavelength 1 mm might
impinge on an aircraft of 10 m wing span. It is easy to see that the computing requirements
are truly astronomical.
8.5.1 Transient solution of electromagnetic scattering problems
The penalty in using a ®ne mesh of conventional ®nite elements in solving wave
problems, referred to above, is the storage and solution of the system matrix. The
approach of Morgan et al. 16;17 is to treat the problem as transient and not to assemble
and solve the system matrix. The Maxwell equations are
@E
" 0
@t ˆ curl H and @H
0 ˆÿcurl E …8:12†
@t
where E and H are the electric and magnetic ®eld intensity vectors respectively. The
equations are combined and expressed in the conservation form
@U
@t
ˆ X3
j ˆ 1
@F j
ˆ 0 where U ˆ
@xj E
H
and the ¯ux vectors, F, are derived from the curl operators. That is
F 1 ˆ ‰ 0 H3 ÿH2 0 ÿE3 E2 Š T
F 2 ˆ ‰ ÿH3 0 H1 E3 0 ÿE1 Š T
F 3 ˆ ‰ H2 ÿH1 0 ÿE2 E1 0 Š T
…8:13†
…8:14†
The algorithm used is the characteristic±Galerkin (or Lax±Wendro€) method as
described in Chapter 2. Details of the algorithm as applied to the electromagnetic problem
are given by Morgan et al. Improved CPU e ciency and reduced storage requirements
are obtained by the use of a representation in which each edge of the tetrahedral
mesh is numbered and the data structure employed provides the numbers of the two
nodes which are associated with each edge. Because of the massive computations
needed for problems of scattering by short waves, parallel processing has also been
used. The problem of radar scattering by an aircraft is shown in Fig. 8.1(a), and Fig.
8.1(b) (also in colour plate included in Volume 1) shows the radar cross-section
(RCS) obtained for the aircraft using a mesh with about 20 million degrees of freedom.
It would be desirable to simulate radar scattering in the millimetre wavelength range,
however even the above described scheme is computationally too intense at this time.
8.5.2 Finite elements incorporating wave shapes
Another approach is to tailor the shape functions within the elements to the known
nature of the wave solution. The ®rst attempt to do this was the in®nite elements of
Bettess and Zienkiewicz. 11;18 The ®rst attempt on ®nite elements was that of Astley, 19;20
using his wave envelope, or complex conjugate weighting method. See Sec. 8.13.