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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