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Analytical computation of the multipole moment $\widetilde{M}^1_{kj,n,m}$ and $\widetilde{M}^2_{k,n,m}$

Noting that Rn,m is a homogeneous polynomial of degree n, one can compute the right-hand side in (3.81) and (3.82) analytically as follows when SJ is a polygon:
 
$\displaystyle {\widetilde{M}^1_{kj,n,m}(O)=\sum_{J}\oint_{\partial S_J}
e_{rck}...
...}{\partial y_l}
R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }})dy_r \phi_d^J}$
    $\displaystyle =\sum_{J}\sum_{I=1}^K\sum_{n'=1}^{n}\sum_{m'=-n'}^{n'}
R_{n-n',m-...
...rightarrow{P^J_{I}P^J_{I+1}}) (\overrightarrow{P^J_{I}P^J_{I+1}})_{r} \phi_d^J,$ (3.83)


 
$\displaystyle {\widetilde{M}^2_{k,n,m}(O)=\sum_{J}\oint_{\partial S_J} e_{rck} ...
...$ y $ }})_{j} R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }}))
dy_r \phi_d^J}$
    $\displaystyle =\sum_{J}\sum_{I=1}^K\sum_{n'=0}^{n}\sum_{m'=-n'}^{n'}
R_{n-n',m-...
...(\frac{1}{n'+1}\delta_{lj}
R_{n',m'}(\overrightarrow{P^J_{I}P^J_{I+1}}) \right.$  
    $\displaystyle \left.+\frac{1}{n'+1}(\overrightarrow{P^J_{I}P^J_{I+1}})_{j}
\fra...
...tial}{\partial y_l}R_{n',m'}(\overrightarrow{P^J_{I}P^J_{I+1}})
\right)\phi_d^J$  
    $\displaystyle +\sum_{J}\sum_{I=1}^K\sum_{n'=1}^{n}\sum_{m'=-n'}^{n'}
R_{n-n',m-...
...c{\partial}{\partial y_l}
R_{n',m'}(\overrightarrow{P^J_{I}P^J_{I+1}})\phi_d^J,$ (3.84)

where we have used (3.21). In (3.83) and (3.84) $P^J_I (I=1,2,\ldots ,K)$ are the vertices of SJ and PJK+1=PJ1. In FM-BIEM with the regularised integral equation we use (3.83) and (3.84) for the computation of the multipole moments in (3.67) and (3.68).


  
Figure 3.16: Line integration for analytical computation of $\widetilde{M}^1_{kj,n,m}$ and $\widetilde{M}^2_{k,n,m}$
\begin{figure}
\begin{center}
\epsfile{file=FIG/polygon2.eps,scale=0.8} \end{center} \end{figure}


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