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Analytical computation of the multipole moment $\widetilde{M}_{j,n,m}$

Noting that Rn,m is a homogeneous polynomial of degree n, one can compute the right-hand side in (3.39) analytically when SL is a polygon as follows:
 
$\displaystyle \widetilde{M}_{j,n,m}(O)$ = $\displaystyle \sum_{L=1}^{N_L} \oint_{\partial S_L} e_{jpl}
\frac{\partial R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }})}{\partial y_p} dy_l \phi_L$  
  = $\displaystyle \sum_{L=1}^{N_L} \sum_{n'=0}^{n}\sum_{m'=-n'}^{n'} \sum_{I=1}^{K}...
...,m-m'}(\overrightarrow{P^L_I\mbox{\boldmath$ y $ }})}{\partial y_p} dy_l \phi_L$  
  = $\displaystyle \sum_{L=1}^{N_L} \sum_{I=1}^{K}\sum_{n'=0}^{n-1}\sum_{m'=-n'}^{n'...
...c{\partial R_{n-n',m-m'}(\overrightarrow{P^L_IP^L_{I+1}})}{\partial y_p}\phi_L,$ (3.40)

where we have used (3.21) in (3.40), NL is the number of the plane elements in Sy, $P^L_I (I=1,2,\ldots ,K)$ are the vertices of SL and PLK+1=PL1 (See Fig.3.4). In FM-BIEM with the regularised integral equation we can use (3.40) for the computation of the multipole moment in (3.31). The computational cost for $\widetilde{M}_{j,n,m}(O)$ for a given pair of n and m is O(3NLKn2).
  
Figure 3.4: Line integration for analytical computation of $\widetilde{M}_{j,n,m}$
\begin{figure}
\begin{center}
\epsfile{file=FIG/polygon.eps,scale=0.7} \end{center}\end{figure}

On the other hand, we compute the integral in (3.23) numerically with Gaussian quadrature as follows:
$\displaystyle M_{n,m}(O) = \int_{S_y} \frac{\partial R_{n,m}(\overrightarrow{O\...
...\boldmath$ y $ }^I})}
{\partial n_y}\phi(\mbox{\boldmath$ y $ }^I) \Delta(S_L),$     (3.41)

where wI and $\mbox{\boldmath$\space y $ }_I^L$ are the Gaussian weight and Gaussian point on SL, Ng is the number of the Gaussian points and $\Delta(S_L)$ is the area of SL. Therefore, the computational cost for Mn,m(O)for a given pair of n and m is O(Ng NL ). As n increases, the computational cost for $\widetilde{M}_{j,n,m}(O)$ for a given pair of n and m becomes large whereas, that for Mn,m(O) does not depend on n. Since the computational cost for the multipole moments is not so dominant in FMM, this is not so serious as we shall see. As we have mentioned above, however, the difference in the cost for M2L translations is considerably more serious than that in the cost for the multipole moments.


next up previous contents
Next: Algorithm Up: Numerical procedure Previous: Discretisation of the regularised
Ken-ichi Yoshida
2001-07-28