The discussion in the previous section has been restricted to the case where C_{t} is in +x_{3} direction of C_{s}. In this section we shall remove this assumption to generalise the discussion in the same way as in 4.1. If the target cell is included in lists except the uplist of C_{s} (See Fig.4.2) we rotate the coordinate system so that the target cell is in the positive direction viewed from the source cell, where denotes the new axis. In general the multipole moments in the new coordinate system are obtained as follows:
We next describe the generalised M2L translation process in the new FMM.
First we rotate the multipole moments via (4.34) and (4.35) so as to make the procedure presented in 4.2.1 applicable. The specific forms of (4.34) and (4.35) depend on the location of C_{t} and are described as follows:
= | (4.38) | ||
= | (4.39) |
= | (4.40) | ||
= | (4.41) |
= | (4.42) | ||
= | (4.43) |
= | (4.44) | ||
= | (4.45) |
Compute the coefficients of the exponential expansion via
(4.28) and (4.29) as follows:
As the centre of the exponential expansion is shifted from
the centroid of C_{s} (O) to the centroid of C_{t} (
), the coefficients of the exponential expansion are
translated according to (4.30) and (4.31) as follows:
Compute the coefficients of the local expansion from the
exponential expansion according to (4.32) and (4.33)
as follows:
= | (4.56) | ||
= | (4.57) |
= | (4.58) | ||
= | (4.59) |
= | (4.60) | ||
= | (4.61) |
= | (4.62) | ||
= | (4.63) |