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New FM-BIEM

Now we consider a source point located at (y1,y2,y3) and a target point at (x1,x2,x3) and denote a cell including a source point by Cs and a cell including a target point by Ct. Suppose that each of the cells Cs and Ct has a volume of d3. Here we use the integral representation (4.1) again. The integral in (4.1) converges under an assumption that the inequality x3 > y3 is valid. This is why we restrict the discussion to the case where Ct is located in +x3 direction of Cs 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 X1j(p,q;O) and X2(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)

 = (4.30) = (4.31)

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):

 = (4.32) = (4.33)

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 three-dimensional Previous: Crack problems for three-dimensional
Ken-ichi Yoshida
2001-07-28