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FM-BIEM with regularised integral equation

In this section we use the regularised integral equation (3.16) for the formulation. We now compute the integral on the right-hand side of (3.16) over a subset of S denoted by Sy for $\mbox{\boldmath$\space x $ }$which is away from Sy. Using (3.17) we obtain

 \begin{displaymath}- n_j(\mbox{\boldmath$ x $ }) e_{jpl}\int_{S_y} \frac{\partia...
...ghtarrow{O\mbox{\boldmath$ x $ }})
\widetilde{M}_{j,n,m}(O),
\end{displaymath} (3.30)

where $\widetilde{M}_{j,n,m}(O)$ is the multipole moment centred at O, defined by
 
$\displaystyle \widetilde{M}_{j,n,m}(O) = - e_{jpl}\int_{S_y} \frac{\partial R_{...
...$ y $ }) e_{iql}\frac{\partial\phi}{\partial y_q}(\mbox{\boldmath$ y $ }) dS_y.$     (3.31)

In (3.30) we have assumed that the inequality $\vert\overrightarrow{O\mbox{\boldmath$\space x $ }}\vert > \vert\overrightarrow{O\mbox{\boldmath$\space y $ }}\vert$ ( $\mbox{\boldmath$\space y $ }\in S_y$) holds. The multipole moment is transformed when the centre of the multipole moment is translated from O to O'
 
$\displaystyle \widetilde{M}_{j,n,m}(O') = \sum_{n'=0}^{n}\sum_{m'=-n'}^{n'} R_{n',m'}(\overrightarrow{O'O})
\widetilde{M}_{j,n-n',m-m'}(O),$     (3.32)

where we have used (3.21) and (3.31). $\widetilde{M}_{j,n,m}(O)$ has the same property as that for Mn,m(O) given in (3.25):
 
$\displaystyle \widetilde{M}_{j,n,-m}(O) = (-1)^m \overline{\widetilde{M}_{j,n,m}}(O) \quad (m \ge 0).$     (3.33)

The integral in (3.30) can be evaluated with the coefficient of the local expansion $\widetilde{L}_{j,n,m}(\mbox{\boldmath$\space x $ }_0)$ as follows:
 
$\displaystyle - n_j(\mbox{\boldmath$ x $ }) e_{jpl}\int_{S_y} \frac{\partial G(...
...x $ }_0\mbox{\boldmath$ x $ }})\widetilde{L}_{j,n,m}(\mbox{\boldmath$ x $ }_0),$     (3.34)

where $\widetilde{L}_{j,n,m}(\mbox{\boldmath$\space x $ }_0)$ is expressed in terms of $\widetilde{M}_{j,n,m}(O)$ by
 
$\displaystyle \widetilde{L}_{j,n,m}(\mbox{\boldmath$ x $ }_0) = \sum_{n'=0}^{\i...
...',m+m'}}(\overrightarrow{O\mbox{\boldmath$ x $ }_0})\widetilde{M}_{j,n',m'}(O).$     (3.35)

In the derivation of this formula we have used (3.20) and have assumed that the inequality $\vert\overrightarrow{O\mbox{\boldmath$\space x $ }_0}\vert > \vert\overrightarrow{\mbox{\boldmath$\space x $ }_0\mbox{\boldmath$ x $ }}\vert$holds. Also, $\widetilde{L}_{j,n,m}(\mbox{\boldmath$\space x $ }_0)$ has the same property as that for $L_{n,m}(\mbox{\boldmath$\space x $ }_0)$ given in (3.28):
 
$\displaystyle \widetilde{L}_{j,n,-m}(\mbox{\boldmath$ x $ }_0) = (-1)^m \overline{\widetilde{L}_{j,n,m}}(\mbox{\boldmath$ x $ }_0)
\quad (m \ge 0).$     (3.36)

The coefficient of the local expansion is translated according to the following formula when the centre of the local expansion is shifted from $\mbox{\boldmath$\space x $ }_0$ to $\mbox{\boldmath$\space x $ }_1$

 
$\displaystyle \widetilde{L}_{j,n,m}(\mbox{\boldmath$ x $ }_1) = \sum_{n'=n}^{\i...
...}_0\mbox{\boldmath$ x $ }_1})\widetilde{L}_{j,n',m'}(\mbox{\boldmath$ x $ }_0),$     (3.37)

where we have used (3.21) and (3.34).

Notice that in FM-BIEM with the regularised integral equation the multipole moments and the coefficients of the local expansion have three components for a given pair of n and m. This means that this formulation requires three times as much computational cost for M2M, M2L and L2L translation as FM-BIEM with the hypersingular integral equation obtained in the previous section and moreover the memory requirement of the regularised formulation is also triple that of the hypersingular one. However, because in the regularised formulation we can compute the multipole moments analytically (See the following section), FM-BIEM with regularisation can be expected to yield more accurate results than that without regularisation. Hence, we investigate the difference between numerical results obtained with two formulation through numerical experiments.

We have thus prepared all the formulae needed for FM-BIEM. Using formulae presented above and the algorithm described in chapter 2, one can implement FM-BIEM.


next up previous contents
Next: Numerical procedure Up: Crack problems for three-dimensional Previous: FM-BIEM with hypersingular integral
Ken-ichi Yoshida
2001-07-28