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FMBIEM with hypersingluar integral equation
In this section we use the hypersingular integral equation
(3.46) for the formulation. In the application of FMM to BIEM
our starting point is to expand the fundamental solution
into a series of products of functions of
and those of
.
From the expression of (3.45) one
finds that it is necessary to expand
into a
series. Thus, we expand
as (See Appendix F)



(3.48) 
Using (3.48), we rewrite (3.45) as
(See Appendix G)



(3.49) 
where
F^{S}_{ij,n,m} and
G^{S}_{i,n,m} are functions defined as
We now compute the integral on the righthand side of
(3.46) over a subset of S denoted by S_{y} for
which is away from S_{y}. Using (3.49), we obtain




(3.52) 
where
M^{1}_{j,n,m}(O) and
M^{2}_{n,m}(O) are the multipole
moments centred at O, expressed as
M^{1}_{j,n,m}(O) 
= 

(3.53) 
M^{2}_{n,m}(O) 
= 

(3.54) 
Notice that
M^{1}_{j,n,m}(O) has three components and
M^{2}_{n,m}(O) has one component for a given pair of n and m, and therefore the
number of multipole moments in this formulation is 4. Fu et
al.[18] applied FMBIEM to threedimensional elastostatics and
used 4 and 12 multipole moments for the single and doublelayer
potentials in their formulation, whereas in our formulation we use 4
multipole moments for ordinary problems where the single and
doublelayer potentials are used as well as for crack problems
(details of formulations for the single and doublelayer potentials
are given later). It is seen that the number multipole moments
in our formulation is related to the fact that in the NeuberPapkovich
representation the displacement field is expressed in terms of four
harmonics functions. Also, Fukui and Kutsumi[27] use 4
moment formulation for ordinary problems.
Noting (A.5) and the fact that
is realvalued one can
find that the multipole moments have the following properties:



(3.55) 



(3.56) 
The multipole moments are translated according to the following
formulae as the centre of multipole expansion is shifted from O to O':
M^{1}_{j,n,m}(O') 
= 

(3.57) 
M^{2}_{n,m}(O') 
= 

(3.58) 
where we have used (3.21), (3.53) and
(3.54) (See Appendix H.1). The procedures given by
(3.57) and (3.58) are called ``M2M
translation''.
In the evaluation of the integral on the righthand side of
(3.46) one can use not only the multipole moments
but also the coefficients of local expansion in the following manner:




(3.59) 
where
and
are the
coefficients of the local expansion centred at
and are
expressed with
M^{1}_{j,n,m}(O) and
M^{2}_{n,m}(O) by
Also,
F^{R}_{ij,n,m} and
G^{R}_{i,n,m} are functions obtained by
replacing S_{n,m} with R_{n,m} in (3.50) and
(3.51). In these formulae we have used (3.20) and
have assumed that the inequality
holds (See Appendix H.2). The procedures given by
(3.60) and (3.61) are called ``M2L
translation''. Notice that
has three components
and
has one component for a given pair of n and m,
and therefore the number of the coefficients of the local expansion
in this formulation is 4. Also,
and
have the following properties:



(3.62) 



(3.63) 
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
(3.59) (See Appendix H.3). The procedures given by
(3.64) and (3.65) are called ``L2L
translation''.
Next: FMBIEM with regularised integral
Up: Crack problems for threedimensional
Previous: Integral equation
Kenichi Yoshida
20010728