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New FMBIEM
Suppose that we evaluate the contribution from a cell C_{s} to the
coefficients of the local expansion in a cell C_{t} contained in the
interaction list of C_{s}. The cells C_{s} and C_{t} are called a
source cell and a target cell, respectively. We divide the
interaction list of C_{s} into 6 lists called the uplist, downlist,
northlist, southlist, eastlist and westlist of C_{s}. The uplist and
downlist contain target cells located in +x_{3} and x_{3}directions of C_{s}, respectively. The northlist and southlist contain
target cells located in +x_{2} and x_{2} directions of C_{s}except those in the uplist or downlist, respectively. The eastlist and
westlist contain target cells located in +x_{1} and x_{1}directions of C_{s} except those in the uplist, downlist, northlist or
southlist, respectively. In this section we discuss the case where
target cells are included in the uplist of C_{s} and each of the cells is
a cube having a volume of d^{3}.
Figure 4.2:
Uplist of source cell

Suppose that a source point
is located at
(y_{1},y_{2},y_{3}) in C_{s} and a target point
at
(x_{1},x_{2},x_{3}) in C_{t}. It is easy to see that the following
integral representation[61] holds:




(4.1) 
where we have assumed that the inequality x_{3} > y_{3} holds. The inner
integral with respect to
in (4.1) is computed with
the trapezoidal rule and the outer integral with generalised Gaussian
quadrature rules[83], yielding
where
is given by
is the error term and the numbers
,
M(k), Gaussian
weights
and nodes
are given in Yarvin and
Rokhlin[83]. These numbers are available from Netlib
(http://www.netlib.org/pdes/multipole/). One may determine these
parameters considering the required accuracy.
Noting (4.2) and the following formulae: (See
(3.19) and Appendix B)
one can evaluate the integral in (3.22) in the
following manner: (See Appendix J.1)
where X(k,j;O) is the coefficient of the exponential expansion at O,
defined by



(4.6) 
This formula converts the multipole moments into the exponential
expansion coefficients. We call the procedure given by (4.6)
``M2X (Multipole moment to(2) eXponential expansion)
translation''. The coefficients of the exponential expansion is
translated according to the following formula when the centre of the
exponential expansion is shifted from O to
:
(See Appendix J.2)



(4.7) 
where
stand for the components of the vector
.
We call
the procedure given by (4.7) ``X2X (eXponential expansion
to(2) eXponential expansion) translation''.
We next need to convert the coefficients of the exponential expansion
into the coefficients of the local expansion (See
(3.27)). Noting that the integral in (3.22) is
expressed with
as follows:


= 

(4.8) 
and using (3.26) and the following formula (See
(3.18) and Hobson[43]):



(4.9) 
we obtain the following formula which converts
the coefficients of the exponential expansion into the coefficient of the local
expansion: (See Appendix J.3)



(4.10) 
We call the procedure given by (4.10) ``X2L (eXponential
expansion to(2) Local expansion) translation''. In the new
FMBIEM we replace M2L translation given by (3.27) with M2X, X2X
and X2L translation given by (4.6), (4.7) and
(4.10).
Next: Rotation of coefficients
Up: Crack problems for threedimensional
Previous: Crack problems for threedimensional
Kenichi Yoshida
20010728