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

spherical scatterers. Other analysis is carried out by Shirron and BabusÏ ka, 58;59 who
reveal a somewhat paradoxical result. The usual in®nite elements give better results
in the ®nite element mesh, but worse results in the in®nite elements themselves. But
the wave envelope elements give worse results in the ®nite elements, and better results
in the far ®eld. This result, which is ascribed to ill-conditioning, does seem to be
counterintuitive. Astley 41 and Gerdes 42 have also surveyed current formulations and
accuracies.
8.15 Transient problems
Recently Astley 60ÿ63 has extended his wave envelope in®nite elements, using the prolate
and oblate spheroidal coordinates adopted by Burnett and Holford, 51ÿ53 and has
shown that they give accurate solutions to a range of periodic wave problems. With
the geometric factor of Astley, which reduces the weighting function and eliminates
the surface integrals at in®nity, the sti€ness, K, damping, C, and mass M matrices
of the wave envelope in®nite element become well de®ned and frequency independent,
although unsymmetric. This makes it possible to apply such elements to unbounded
transient wave problems. Figure 8.10 shows the transient response of a dipole.
More results from the application of in®nite elements to transient problems are
given by Cipolla and Butler, 64 who created a transient version of the Burnett in®nite
element. There appear to be more di culties with such elements than with the wave
envelope elements, and a consensus that the latter are better for transient problems
seems to be emerging. Dampers and boundary integrals can also be used for transient
problems. Space is not available to survey these ®elds, but the reader is directed,
again, to Givoli 29 and Geers. 30 One set of interesting results was obtained using
transient dampers by Thompson and Pinsky. 65
Dimensionless acoustic pressure
1.5
1.0
0.5
0
–0.5
–1.0
P(A)
P(B)
P(C)
–1.5 0 0.5 1.0 1.5 2.0 2.5 3.0
Fig. 8.10 Transient response of a dipole, Astley. 63
Dimensionless time T (= tD/c)
Transient problems 265