7 Combination

of Regions

7.1 INTRODUCTION

Boundary solutions offered important advantages over the 'domain'

type techniques. In particular, we can use a smaller number of

unknowns to analyse the same problems, as unknowns are not needed

inside the domain. This characteristic renders boundary solutions

specially attractive for three-dimensional problems for which the

ratio of the external surface to the volume is small, such as concrete

nuclear pressure vessels, gravity dams, etc.

The idea of having the unknowns varying over part of the surface

with known interpolation functions gives origin to the boundary

element technique. The technique is similar to taking finite elements

only on the surface of the body. Boundary elements allow us to

describe better the geometry of the body under consideration and in

addition, are easy to combine with finite elements.

One of the main disadvantages of the finite element method is its

obvious inability to model domains extending to infinity. Boundary

elements on the contrary can use fundamental solutions which

naturally obey the conditions at infinity. It may be difficult however to

find fundamental solutions for all cases but some approximation may

equally well be applied. This idea has not yet been fully exploited in

boundary solutions, although it has been used with a certain degree of

success for domain techniques, i.e. the so-called 'infinite elements'.

1,2

Its inherent ability to model domains extending to infinity has

made the boundary element method popular for modelling problems

such as those occurring with the Helmholtz type equation, i.e.

problems such as diffraction, harbour resonance, etc. There are also

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