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

as the line integral, of Eq. (8.22), we must transform the nodal values, either to T or
to R. This can be done simply by enforcing the change of variable, which leads to a
contribution to the `right-hand side' or `forcing' term. 28
8.7 Unbounded problems
There are several methods of dealing with exterior problems using ®nite elements in
combination with other methods. Some of these methods are also applicable to ®nite
di€erences. The literature in this ®eld has grown enormously in the past 10 years, and
this section will therefore be selective. The monograph by Givoli 29 is devoted
exclusively to this ®eld and gives much more detail on the competing algorithms. It
is a very useful source and gives many more algorithms than can be covered here.
The book edited by Geers, 30 from an IUTAM symposium, gives a very useful and
up-to-date overview of the ®eld.
The main methods include:
. boundary dampers, both plane and cylindrical (also called non-re¯ecting boundary
conditions);
. linking to exterior solutions, both series and boundary integral (also called Dirichlet
to Neumann mapping);
. in®nite elements.
8.8 Boundary dampers
The nomenclature of boundary dampers comes from engineering applications. Such
boundary conditions are also called local non-re¯ecting boundary conditions by
mathematicians. As was seen in Chapter 19 of Volume 1, we can simply apply the
plane damper at the boundary of the mesh. This was ®rst done in ¯uid problems
by Zienkiewicz and Newton. 6 However the more sophisticated dampers proposed
by Baylisss et al. 26;27 can be used at little extra computational cost and a big increase
in accuracy. The dampers are developed from the series given in Table 8.1. Full details
are given in reference 31. For the case of two-dimensional waves the line integral
which should be applied on the circular boundary of radius r is
… " #
2
2 @
A ˆ ‡ ds …8:23†
ÿ 2 2 @s
where ds is an element of distance along the boundary and
ˆ 3=4r2 ÿ 2k 2 ‡ 3ik=r
2=r ‡ 2ik
and ˆ
1
2=r ‡ 2ik
Boundary dampers 253
…8:24†
For the plane damper, ˆ 0 and ˆ ik. For the cylindrical damper ˆ 0and
ˆ ik ÿ 1=2r. The corresponding expressions for three-dimensional waves are
di€erent. Non-circular boundaries can be handled but the expressions become
much more complicated. Some results are given by Bando et al. 31 Figure 8.4 shows
the waves di€racted by a cylinder problem for which there is a solution, due to