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## Formulation for original FMM

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 (4) one finds that it is necessary to expand into a series. Yoshida et al.[7] obtained an expansion for as follows:

 (7)

where Rn,m and Sn,m are solid harmonic functions defined as

are the polar coordinates of the point , Pnm is the associated Legendre function and a superposed bar indicates the complex conjugate. The use of solid harmonics in FMM has been suggested by Perez-Jorda and Yang[24]. The functions Rn,m and Sn,m satisfy the following relations given by

 (8)

 (9)

Using (7), one rewrites (4) as[7]

 = (10)

where FSij,n,m and GSi,n,m are functions defined as[7]

 (11)

We now compute the integral on the right hand side of (5) over a subset of S denoted by Sy for which is away from Sy. Using (10) we obtain

 = (13)

where M1j,n,m and M2n,m are the multipole moments centred at O, expressed as

 M1j,n,m(O)= (14)

The multipole moments are translated according to the following formulae as the centre of multipole expansion is shifted from Oto O':

 M1j,n,m(O')= (15) M2n,m(O')= (16)

where we have used (9), (14) and (15). In the evaluation of the integral on the right hand side of (5) one can use not only the multipole moments but also the coefficients of local expansion in the following manner:

 = (17)

where L1j,n,m and L2n,m are expressed with M1j,n,m and M2n,m by

 (18) (19)

and FRij,n,m and GRi,n,m are functions obtained by replacing Sn,m by Rn,m in (11) and (12). In these formulae we have used (8) and have assumed that the inequality holds. The procedures given by (19) and (20) are called M2L translation.

The coefficients of the local expansion are translated according to the following formulae when the centre of the local expansion is shifted from to

 (20) (21)

where we have used (9) and (18).

Next: Algorithm for original FMM Up: Original FMM Previous: Original FMM
Ken-ichi Yoshida
2001-03-26