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

254 Waves
Relative errors (%)
10
8
6
4
2
0
–2
–4
–6
–8
–10
–16
–34
–35
Cylinder
Havelock. The higher-order dampers are clearly a big improvement over the plane
and cylindrical dampers, for little or no extra computational cost. Engquist and
Majda have also earlier proposed dampers for these problems, 32 but instead of
using a hierarchy of operators like Bayliss et al., they use a di€erent method. The
e€ect is the same, in that a hierarchy of boundary operators is de®ned, but the
terms are di€erent to those of Bayliss et al.
8.8.1 Other damper-related approaches
A great variety of methods have appeared recently based on dampers, and variants of
the concept. There is not enough space to review them all in detail here and the reader
is referred to the book by Givoli 29 and the volume edited by Geers, 30 which gives
access to recent developments. The papers in the Geers volume by Bielak, Givoli,
Hagstrom, Hariharan, Higdon, Pinsky and Kallivokas should be consulted. An
interesting development is the method of the perfectly matched layer (PML), or
`sponge layer', the idea being that the outgoing wave is not absorbed on a boundary,
but in a domain which extends beyond the boundary. In this domain the wave is
1
Incident
waves
2 3 4 5 6 7
Outer radius
of finite element
domain
Plane damper
Cylindrical damper
Second-order damper
Fig. 8.4 Damper solutions for waves diffracted by circular cylinder. Comparison of relative errors for various
outer radii, …ka ˆ 1†. Relative error ˆ…abs… n†ÿabs… a††=abs… a†.