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No credit card required COMBINATION OF REGIONS 201
not give an accurate indication of the behaviour of the system towards
infinity but the effect of the far region on the domain of interest is
introduced (see Bettess
2
).
Example 7.4
10
Diffraction and refraction of waves by a
parabolic shoal, surmounted by a cylindrical island
Figure 7.18 shows the geometry of the problem and the element mesh
used. The finite element mesh is enclosed by a number of infinite
elements, the shape function in the radial direction is given by
P(r)e-^
L
Q-
iKr
(a)
(for time dependence e
lwi
). Here P(r) is a polynomial in r and L is the
so-called decay length, and κ is the wavenumber corresponding to the
frequency ω. This shape function satisfies the Sommerfeld radiation
condition.
L was chosen so that near the island the decay e~
3/L
roughly
matched the decay of the first term of the general series solution,
H
(
0
2)
(/cr), where H
(
0
2)
is the Hankel function of the zeroth order of the
second kind. See Bettess and Zienkiewicz
10
for a fuller explanation of
the above theory.
Figure 7.19, from Bettess and Zienkiewicz
10
, shows a comparison
of the elevations at the island itself compared with the analytic
solution given in Homas
1
*
and Vastano and Reid.
12
The agreement is
very good especially at moderate wavelengths.
The weakness in this method is that to determine the parameter L
some knowledge of the exact solution is required before it can be used.
However, the method is very efficient if this information is available.
7.5 COMBINATION OF FINITE AND BOUNDARY ELEMENTS
The idea of combining both techniques can be attributed to Wexler
13
,
who started to use integral equation solutions to represent the
unbounded field problem early in the 1970s, the advantage being that
this allowed for the use of appropriate conditions to represent the
infinite domain. The integral equation technique is also of interest
when regions of high stress or potential gradients exist, but finite
elements are adequate for other parts of a body and may be simpler to 202 COMBINATION OF REGIONS
60 80 100 120
Angle around island,
Θ
(deg )
Figure 7.19 Relative amplitudes on cylinder
use in cases such as layered continua, anisotropic and non-linear
materials. Hence it is important for the analyst to be able to represent
a body using finite or boundary element techniques, depending on the
particular geometry, boundary conditions, etc.
The first combination of the two methods for elastostatics appears
to be by Osias
14
, although for wave propagation problems, the
method was used by Mei
7
in 1975 who explained the method of
combining both solutions using variational techniques.
Lachat
3
, following the work of the Southampton University group,
developed a technique which allowed for variations of displacements
and tractions along parts of the boundary surface or elements. The
idea of using interpolation functions to define the variables along
these elements allows for the combination of finite and boundary
elements without any loss of continuity. In addition the work at
Southampton
15
concentrated on the common basis of the differerit
methods
16
and the equivalence of the direct and indirect boundary
element techniques.

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