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Formulation for original FMM

In the application of FMM to BIEM our starting point is to expand the fundamental solution $\Gamma_{ij}(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$ into a series of products of functions of $\mbox{\boldmath$\space x $ }$ and those of $\mbox{\boldmath$\space y $ }$. From the expression of (4) one finds that it is necessary to expand $\vert\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ }\vert$ into a series. Yoshida et al.[7] obtained an expansion for $\vert\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ }\vert$ as follows:
 
$\displaystyle {\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert=}$
    $\displaystyle \sum_{n=0}^{\infty}\sum_{m=-n}^{n}
\left(\frac{S_{n,m}(\overright...
...t^2
\overline{R_{n,m}}(\overrightarrow{O\mbox{\boldmath$ y $ }})}{2n+3}
\right.$  
    $\displaystyle \left. \qquad -
\frac{\vert\overrightarrow{O\mbox{\boldmath$ x $ ...
...}})
\overline{R_{n,m}}(\overrightarrow{O\mbox{\boldmath$ y $ }})}{2n-1}
\right)$  
    $\displaystyle \qquad(\vert\overrightarrow{O\mbox{\boldmath$ x $ }}\vert > \vert\overrightarrow{O\mbox{\boldmath$ y $ }}\vert)$ (7)

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

\begin{eqnarray*}R_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})&=&\frac{1}{(n...
...h$ x $ }})&=&(n-m)!P_n^m(\cos\theta)e^{im\phi}\frac{1}{r^{n+1}},
\end{eqnarray*}


$(r,\theta,\phi)$ are the polar coordinates of the point $\mbox{\boldmath$\space x $ }$, 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
 
$\displaystyle {S_{n,m}(\overrightarrow{\mbox{\boldmath$ y $ }\mbox{\boldmath$ x $ }})=}$
    $\displaystyle \sum_{n'=0}^{\infty}\sum_{m'=-n'}^{n'}
\overline{R_{n',m'}}(\over...
...\mbox{\boldmath$ y $ }})S_{n+n',m+m'}(\overrightarrow{O\mbox{\boldmath$ x $ }})$  
    $\displaystyle (\vert\overrightarrow{O\mbox{\boldmath$ y $ }}\vert<\vert\overrightarrow{O\mbox{\boldmath$ x $ }}\vert) \qquad$ (8)


 
$\displaystyle {R_{n,m}(\overrightarrow{\mbox{\boldmath$ y $ }\mbox{\boldmath$ x $ }})=}$
    $\displaystyle \sum_{n'=0}^{n}\sum_{m'=-n'}^{n'}
R_{n',m'}(\overrightarrow{\mbox{\boldmath$ y $ }O})R_{n-n',m-m'}(\overrightarrow{O\mbox{\boldmath$ x $ }})$  
    $\displaystyle (\mbox{This holds for arbitrary} \ \overrightarrow{O\mbox{\boldmath$ x $ }} \ \mbox{and} \ \overrightarrow{O\mbox{\boldmath$ y $ }})
\qquad$ (9)

Using (7), one rewrites (4) as[7]
 
$\displaystyle {\Gamma_{ij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ })}$
  = $\displaystyle \frac{1}{8\pi\mu}\sum_{n=0}^{\infty}
\sum_{m=-n}^{n}\left(F^S_{ij...
...h$ x $ }})
\overline{R_{n,m}}(\overrightarrow{O\mbox{\boldmath$ y $ }}) \right.$  
    $\displaystyle \quad + \left. G^S_{i,n,m}(\overrightarrow{O\mbox{\boldmath$ x $ ...
... $ }})_{j}
\overline{R_{n,m}}(\overrightarrow{O\mbox{\boldmath$ y $ }}) \right)$ (10)

where FSij,n,m and GSi,n,m are functions defined as[7]
  
$\displaystyle {F^S_{ij,n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})=}$
    $\displaystyle \frac{\lambda+3\mu}{\lambda+2\mu}\delta_{ij}S_{n,m}
(\overrightar...
...
\frac{\partial}{\partial x_i}S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})$  
      (11)
$\displaystyle {
G^S_{i,n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})=
\frac{\l...
...frac{\partial}{\partial x_i}
S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})}$

We now compute the integral on the right hand side of (5) over a subset of S denoted by Sy for $\mbox{\boldmath$\space x $ }$ which is away from Sy. Using (10) we obtain

 
$\displaystyle {- \mbox{p.f.}\int_{S_y} \frac{\partial}{\partial x_k}
\frac{\par...
... $ }) C_{cdjl} n_c(\mbox{\boldmath$ y $ })\phi_d(\mbox{\boldmath$ y $ }) dS_y }$
  = $\displaystyle -\frac{1}{8\pi\mu} \sum_{n=0}^{\infty} \sum_{m=-n}^{n}
\left( \fr...
...,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})
\overline{M^1_{j,n,m}}(O) \right.$  
    $\displaystyle \left. + \frac{\partial}{\partial x_k}G^S_{i,n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})
\overline{M^2_{n,m}}(O)\right)$ (13)

where M1j,n,m and M2n,m are the multipole moments centred at O, expressed as
  
M1j,n,m(O)=
    $\displaystyle \int_{S_y} C_{cdjl} \frac{\partial}{\partial y_l}
R_{n,m}(\overri...
...th$ y $ }}) \phi_d(\mbox{\boldmath$ y $ }) n_c(\mbox{\boldmath$ y $ }) dS_{y} ,$ (14)
$\displaystyle { M^2_{n,m}(O)=}$
    $\displaystyle \int_{S_y} C_{cdjl} \frac{\partial}{\partial y_l}
((\overrightarr...
...ath$ y $ }}))\phi_d(\mbox{\boldmath$ y $ }) n_c(\mbox{\boldmath$ y $ }) dS_{y}.$  

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')=
    $\displaystyle \sum_{n'=0}^{n}\sum_{m'=-n'}^{n'}
R_{n',m'}(\overrightarrow{O'O})M^{1}_{j,n-n',m-m'}(O)$  
      (15)
M2n,m(O')=
    $\displaystyle \sum_{n'=0}^{n}\sum_{m'=-n'}^{n'} R_{n',m'}(\overrightarrow{O'O})
\bigl(M^2_{n-n',m-m'}(O)$  
    $\displaystyle - (\overrightarrow{OO'})_{j} M^1_{j,n-n',m-m'}(O) \bigr)$ (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:
 
$\displaystyle {- \mbox{p.f.}\int_{S_y} \frac{\partial}{\partial x_k}
\frac{\par...
... $ }) C_{cdjl} n_c(\mbox{\boldmath$ y $ })\phi_d(\mbox{\boldmath$ y $ }) dS_y }$
  = $\displaystyle -\frac{1}{8\pi\mu} \sum_{n=0}^{\infty} \sum_{m=-n}^{n}
\left( \fr...
...$ x $ }_0\mbox{\boldmath$ x $ }})
L^1_{j,n,m}(\mbox{\boldmath$ x $ }_0) \right.$  
    $\displaystyle \left. + \frac{\partial}{\partial x_k}G^R_{i,n,m}(\overrightarrow...
...ath$ x $ }_0\mbox{\boldmath$ x $ }})
L^2_{n,m}(\mbox{\boldmath$ x $ }_0)\right)$ (17)

where L1j,n,m and L2n,m are expressed with M1j,n,m and M2n,m by
  
$\displaystyle {L^1_{j,n',m'}(\mbox{\boldmath$ x $ }_{0})=}$
    $\displaystyle \sum_{n=0}^{\infty}\sum_{m=-n}^{n}(-1)^{n'}\overline{S_{n+n',m+m'}}
(\overrightarrow{O\mbox{\boldmath$ x $ }_{0}})M^1_{j,n,m}(O)$  
      (18)
$\displaystyle { L^2_{n',m'}(\mbox{\boldmath$ x $ }_{0})=}$
    $\displaystyle \sum_{n=0}^{\infty}\sum_{m=-n}^{n}(-1)^{n'}\overline{S_{n+n',m+m'}}
(\overrightarrow{O\mbox{\boldmath$ x $ }_{0}})$  
    $\displaystyle \times
(M^2_{n,m}(O)-(\overrightarrow{O\mbox{\boldmath$ x $ }_{0}})_{j} M^1_{j,n,m}(O))$ (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 $\vert\overrightarrow{O\mbox{\boldmath$\space x $ }_0}\vert > \vert\overrightarrow{\mbox{\boldmath$\space x $ }_0\mbox{\boldmath$ x $ }}\vert$ 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 $\mbox{\boldmath$\space x $ }_0$ to $\mbox{\boldmath$\space x $ }_1$

  
$\displaystyle { L^1_{j,n'',m''}(\mbox{\boldmath$ x $ }_{1})=}$
    $\displaystyle \sum_{n'=n''}^{\infty}\sum_{m'=-n'}^{n'}R_{n'-n'',m'-m''}(\overrightarrow{x_{0}x_{1}})
L^1_{j,n',m'}(\mbox{\boldmath$ x $ }_{0})$  
      (20)
$\displaystyle {L^2_{n'',m''}(\mbox{\boldmath$ x $ }_{1})=\sum_{n'=n''}^{\infty}...
...n'',m'-m''}(\overrightarrow{\mbox{\boldmath$ x $ }_0\mbox{\boldmath$ x $ }_1})}$
    $\displaystyle \times
\left(L^2_{n',m'}(\mbox{\boldmath$ x $ }_{0})-(\overrighta...
...\mbox{\boldmath$ x $ }_1})_{p}L^1_{p,n',m'}(\mbox{\boldmath$ x $ }_{0})
\right)$ (21)

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


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