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New FMBIEM
Now we consider a source point
located at
(y_{1},y_{2},y_{3}) and a target point
at
(x_{1},x_{2},x_{3}) and denote a cell including a source point by
C_{s} and a cell including a target point by C_{t}. Suppose that each of
the cells C_{s} and C_{t} has a volume of d^{3}. Here we use the
integral representation (4.1) again. The integral in
(4.1) converges under an assumption that the
inequality x_{3} > y_{3} is valid. This is why we restrict the discussion
to the case where C_{t} is located in +x_{3} direction of C_{s} for the
present.
Noting (4.2) and (4.3)(4.5), one can
evaluate the integral in (3.14) in the following manner:
(See Appendix K.1)
where
X^{1}_{j}(p,q;O) and
X^{2}(p,q;O) are the coefficients of the
exponential expansion at O, defined in terms of the multipole moments
(3.53) and (3.54) as



(4.28) 



(4.29) 
and
are operators defined as
These formulae (4.28) and (4.29) convert the multipole
moments into the coefficients of the exponential expansion. We call the
procedures given by (4.28) and (4.29) ``M2X translation''.
The coefficients of the exponential expansion is translated according to
the following formulae when the centre of the exponential expansion is
shifted from O to
:
(See Appendix K.2)
where
and
are the coefficients of
the exponential expansion at
and
stand for the
components of the vector
.
We call the procedures given by
(4.30) and (4.31) ``X2X translation''.
We next need to convert the coefficients of the exponential expansion
into the coefficients of the local expansion. Noting that the integral
in (2.3) is evaluated with
and
as follows: (See Appendix K.3)
and using (2.9) and (4.9),
we obtain the following formulae which convert
the coefficients of the exponential expansion into the coefficients of
the local expansion (3.60) and (3.61):
We call the procedures given by (4.32) and (4.33) ``X2L
translation''.
The procedures given by (4.28)(4.33) correspond to the
M2L translation given by (3.60) and (3.61) in
the original FMM described in 3.3.
Next: Rotation of coefficients
Up: Crack problems for threedimensional
Previous: Crack problems for threedimensional
Kenichi Yoshida
20010728