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Discretisation of the regularised integral equation with piecewise constant shape functions

Discretising the regularised integral equation (3.47) with piecewise constant shape functions, one obtains the following equation:
 
$\displaystyle t_{a}^{\infty}(\mbox{\boldmath$ x $ })$ = $\displaystyle \int_{S} n_{b}(\mbox{\boldmath$ x $ }) C_{abik} e_{rck} C_{cdjl}
...
...i_{d}(\mbox{\boldmath$ y $ })}{\partial y_q}
n_{s}(\mbox{\boldmath$ y $ }) dy_r$  
  = $\displaystyle \sum_J \oint_{\partial S_J} n_{b}(\mbox{\boldmath$ x $ }) C_{abik...
... y_l}
\Gamma_{ij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }) dy_r \phi_d^J,$ (3.80)

where SJ is a plane element in S and $\phi^J_d$ represents $\phi$ on SJ. In (3.80) the right-hand screw convention is applied to the direction of the integration along $\partial S_J$. We use (3.80) for the direct computation instead of (3.47). In the similar way $\widetilde{M}^{1}_{kj,n,m}(O)$ and $\widetilde{M}^{2}_{k,n,m}(O)$ are rewritten as
  
$\displaystyle \widetilde{M}^1_{kj,n,m}(O)$ = $\displaystyle \sum_J \oint_{\partial S_J}
e_{rck} C_{cdjl}\frac{\partial}{\partial y_l}
R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }})dy_r \phi_d^J,$ (3.81)
$\displaystyle \widetilde{M}^2_{k,n,m}(O)$ = $\displaystyle \sum_J \oint_{\partial S_J} e_{rck} C_{cdjl}
\frac{\partial}{\par...
...th$ y $ }})_j R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }}))
dy_r \phi_d^J.$ (3.82)



Ken-ichi Yoshida
2001-07-28