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FMBIEM with regularised integral equation
In this section we use the regularised integral equation
(3.47) for the formulation. We now compute the integral on the
righthand side of (3.47) over a subset of S denoted by
S_{y} for
which is away from S_{y}. Using
(3.49), we obtain




(3.66) 
where
and
are the
multipole moments centred at O, expressed as
Notice that
has nine components and
has three components for a given pair of n and
m, and therefore the number of multipole moments in this formulation is
12. Also, the multipole moments have the following properties:



(3.69) 



(3.70) 
The multipole moments are translated according to the following
formulae as the centre of multipole expansion is shifted from O to O':
where we have used (3.21), (3.67) and
(3.68). In the evaluation of the integral on the
righthand side of (3.47) one can use not only the
multipole moments but also the coefficients of local expansion in the
following manner:




(3.73) 
where
and
are
the coefficients of the local expansion centred at
and are
expressed with
and
by
In these formulae we have used (3.20) and have assumed
that the inequality
holds. Notice
that
has nine components and
has three component for a given pair of n and m, and therefore the number of coefficients of the local
expansion in this formulation is 12. Also, the coefficients of the
local expansion have the following properties:



(3.76) 



(3.77) 
The coefficients of the local expansion are translated according to the
following formulae when the centre of the local expansion is shifted
from
to
where we have used (3.21) and (3.73).
As we shall see in FMBIEM with the regularised integral equation we can
compute the multipole moments analytically (See the following
section), but the multipole moments and the coefficients of the local
expansion have twelve components for a given pair of n and
m. Namely, the number of the multipole moments in the regularised
formulation is triple that in the hypersingular one. This tradeoff
issue will be settled through a numerical experiment in section
3.3.5.
Next: Numerical procedures
Up: Crack problems for threedimensional
Previous: FMBIEM with hypersingluar integral
Kenichi Yoshida
20010728