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

250 Waves
agreement with the analytical series solution. In this problem ka ˆ 8 , ˆ 0:25a,
radius of cylinder, a ˆ 1, and the mesh extends to r ˆ 7a. For a conventional
radial ®nite element mesh, the requirement of 10 nodes per wavelength would lead
to a mesh with 424 160 degrees of freedom. But in the results shown, with 36
directions per node and 252 nodes there are only 9072 degrees of freedom. The
dramatic reduction in the number of variables merits further investigation and
development of the method.
The method still has a number of uncertainties regarding the conditioning of the
system matrix and the stability of the technique and a signi®cant problem remains
in the numerical cost of integrating the element matrix.
8.6 Waves in unbounded domains (exterior surface wave
problems)
Problems in this category include the di€raction and refraction of waves close to
®xed and ¯oating structures, the determination of wave forces and wave response
for o€shore structures and vessels, and the determination of wave patterns adjacent
to coastlines, open harbours and breakwaters. In electromagnetics there are
scattering problems of the type already described, and in acoustics we have various
noise problems. In the interior or ®nite part of the domain, ®nite elements, exactly
as described in Sec. 8.2 can be used, but special procedures must be adopted for
the part of the domain extending to in®nity. The main di culty is that the problem
has no outer boundary. This necessitates the use of a radiation condition. Such a
condition was introduced in Chapter 19 of Volume 1, as Eq. (19.18), for the
case of a one-dimensional wave, or a normally incident plane wave in two or
more dimensions. Work by Bayliss et al. 26;27 has developed a suitable radiation
condition, in the form of an in®nite series of operators. The starting point is the
representation of the outgoing wave in the form of an in®nite series. Each term
in the series is then annihilated by using a boundary operator. The sequence of
boundary operators thus constitutes the radiation condition. In addition there is
a classical form of the boundary condition for periodic problems, given by
Sommerfeld. A summary of all available radiation conditions is given in Table 8.1.
8.6.1 Background to wave problems
The simplest type of exterior, or unbounded wave problem is that of some exciting
device which sends out waves which do not return. This is termed the radiation
problem. The next type of exterior wave problem is where we have a known incoming
wave which encounters an object, is modi®ed and then again radiates away to in®nity.
This case is known as the scattering problem, and is more complicated, in as much as we
have to deal with both incident and radiated waves. Even when both waves are linear,
this can lead to complications. Both the above cases can be complicated by wave refraction,
where the wave speeds change, because of changes in the medium, for example
changes in water depth. Usually this phenomenon leads to changes in the wave direction.
Waves can also re¯ect from boundaries, both physical and computational.