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Local expansion

We can efficiently evaluate the integral in (2.1) using the multipole moments alone. However, in order to enhance this efficiency we use the local expansion in FMM.

Suppose that one can expand $\phi_i$ as follows:

 
$\displaystyle \phi_i(\overrightarrow{O\mbox{\boldmath$ x $ }})= \sum_{l} \xi_l(...
...\mbox{\boldmath$ x $ }})
\beta^i_l(\overrightarrow{O\mbox{\boldmath$ x $ }_0}),$     (2.8)

where $\xi_l$ is a function, $\beta^i_l$ is a certain coefficient of the expansion and $\mbox{\boldmath$\space x $ }_0$ is a certain point such that the inequality $\vert\overrightarrow{\mbox{\boldmath$\space x $ }_{0}\mbox{\boldmath$ x $ }}\vert < \vert\overrightarrow{O\mbox{\boldmath$\space x $ }_0}\vert$ is valid (See Fig.2.5).

Substituting (2.8) into (2.3), one can evaluate the integral in (2.1) as follows:

 
$\displaystyle \int_{S_y} K(\mbox{\boldmath$ x $ },\mbox{\boldmath$ y $ }) \varphi(\mbox{\boldmath$ y $ }) dS$ = $\displaystyle \sum_{i} \phi_i(\overrightarrow{O\mbox{\boldmath$ x $ }}) M_i(O)$  
  = $\displaystyle \sum_{i} \sum_{l} \xi_l(\overrightarrow{\mbox{\boldmath$ x $ }_{0...
...\boldmath$ x $ }})
\beta^i_l(\overrightarrow{O\mbox{\boldmath$ x $ }_0}) M_i(O)$  
  = $\displaystyle \sum_{l} \xi_l(\overrightarrow{\mbox{\boldmath$ x $ }_{0}\mbox{\b...
...h$ x $ }})
\sum_{i}\beta^i_l(\overrightarrow{O\mbox{\boldmath$ x $ }_0}) M_i(O)$  
  = $\displaystyle \sum_{l}\xi_l(\overrightarrow{\mbox{\boldmath$ x $ }_{0}\mbox{\boldmath$ x $ }}) L_l(\mbox{\boldmath$ x $ }_0),$ (2.9)

where $L_l(\mbox{\boldmath$\space x $ }_0)$ is the coefficient of the local expansion centred at $\mbox{\boldmath$\space x $ }_0$ given by
 
$\displaystyle L_l(\mbox{\boldmath$ x $ }_0) = \sum_{i} \beta_l^i(\overrightarrow{O\mbox{\boldmath$ x $ }_0}) M_i(O).$     (2.10)

This formula (2.10) is used to translate the multipole moments centred at O to the coefficients of the local expansion centred at $\mbox{\boldmath$\space x $ }_0$ (See Fig.2.5) and this translation is called ``M2L (Multipole moment to(2) Local expansion) translation''.
next up previous contents
Next: L2L translation Up: Formulation for FM-BIEM Previous: M2M translation
Ken-ichi Yoshida
2001-07-28