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Direct computation

Discretising the regularised integral equation (3.119) with piecewise constant shape functions, one obtains
 
$\displaystyle -\frac{\partial u_I}{\partial n_x}(\mbox{\boldmath$ x $ })
= \sum...
...\boldmath$ y $ }) G(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ })
dS_y \phi_J,$     (3.133)

where SJ is a plane element in Sy and $\phi_J$ represents $\phi$on SJ. In (3.134) the right-hand screw convention is applied to the direction of the integration along $\partial S_J$. In the evaluation of the integral in (3.134) we divide (3.117) into the static part (the fundamental solution of three-dimensional Laplace's equation) and the residual part as follows:

\begin{eqnarray*}G(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }) &=& G^{S}(\mbo...
...uad R = \vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert.
\end{eqnarray*}


When kR is small we use the following series for the computation of $G^R(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$

\begin{eqnarray*}G^R(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }) = \sum_{n=1}^{\infty} \frac{(ik)^n R^{n-1}}{n!}
\end{eqnarray*}


We compute the integrals related to the static part $G^S(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$analytically and the rest numerically with Gaussian quadrature.

Ken-ichi Yoshida
2001-07-28