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FM-BIEM with regularised integral equation

In this section we use the regularised integral equation (3.47) for the formulation. We now compute the integral on the right-hand side of (3.47) over a subset of S denoted by Sy for $\mbox{\boldmath$\space x $ }$ which is away from Sy. Using (3.49), we obtain
 
$\displaystyle { \int_{S_y} e_{rck} C_{cdjl}\frac{\partial}{\partial y_l}
\Gamma...
...{d}(\mbox{\boldmath$ y $ })}{\partial y_q}
n_{s}(\mbox{\boldmath$ y $ }) dS_y }$
    $\displaystyle =\frac{1}{8\pi\mu}\sum_{n=0}^{\infty}\sum_{m=-n}^{n}(
\overline{F...
..._{i,n,m}}(\overrightarrow{O\mbox{\boldmath$ x $ }})\widetilde{M}^2_{k,n,m}(O)),$ (3.66)

where $\widetilde{M}^{1}_{kj,n,m}(O)$ and $\widetilde{M}^{2}_{k,n,m}(O)$ are the multipole moments centred at O, expressed as
  
$\displaystyle \widetilde{M}^1_{kj,n,m}(O)$ = $\displaystyle \int_{S_y}
e_{rck} C_{cdjl}\frac{\partial}{\partial y_l}
R_{n,m}(...
..._{d}(\mbox{\boldmath$ y $ })}{\partial y_q}
n_{s}(\mbox{\boldmath$ y $ }) dS_y,$ (3.67)
$\displaystyle \widetilde{M}^2_{k,N,M}(O)$ = $\displaystyle \int_{S_y}e_{rck} C_{cdjl}
\frac{\partial}{\partial y_l}((\overri...
..._{d}(\mbox{\boldmath$ y $ })}{\partial y_q} n_{s}(\mbox{\boldmath$ y $ }) dS_y.$ (3.68)

Notice that $\widetilde{M}^{1}_{kj,n,m}(O)$ has nine components and $\widetilde{M}^{2}_{k,n,m}(O)$ has three components for a given pair of n and m, and therefore the number of multipole moments in this formulation is 12. Also, the multipole moments have the following properties:
  
$\displaystyle \widetilde{M}^1_{kj,n,-m}(O) = (-1)^m \overline{\widetilde{M}^1_{kj,n,m}}(O)
\quad (m \ge 0),$     (3.69)
$\displaystyle \widetilde{M}^2_{k,n,-m}(O) = (-1)^m \overline{\widetilde{M}^2_{k,n,m}}(O)
\quad (m \ge 0).$     (3.70)

The multipole moments are translated according to the following formulae as the centre of multipole expansion is shifted from O to O':

  
$\displaystyle \widetilde{M}^1_{kj,n,m}(O')$ = $\displaystyle \sum_{n'=0}^{n}\sum_{m'=-n'}^{n'}
R_{n',m'}(\overrightarrow{O'O})\widetilde{M}^{1}_{kj,n-n',m-m'}(O),$ (3.71)
$\displaystyle \widetilde{M}^2_{k,n,m}(O')$ = $\displaystyle \sum_{n'=0}^{n}\sum_{m'=-n'}^{n'}
R_{n',m'}(\overrightarrow{O'O})...
...,m-m'}(O)
-(\overrightarrow{OO'})_{j} \widetilde{M}^1_{kj,n-n',m-m'}(O) \bigr),$ (3.72)

where we have used (3.21), (3.67) and (3.68). In the evaluation of the integral on the right-hand side of (3.47) one can use not only the multipole moments but also the coefficients of local expansion in the following manner:
 
$\displaystyle { \int_{S_y} e_{rck} C_{cdjl}\frac{\partial}{\partial y_l}
\Gamma...
...{d}(\mbox{\boldmath$ y $ })}{\partial y_q}
n_{s}(\mbox{\boldmath$ y $ }) dS_y }$
    $\displaystyle =\frac{1}{8\pi\mu}\sum_{n=0}^{\infty}\sum_{m=-n}^{n}(
F^R_{ij,n,m...
... }_0\mbox{\boldmath$ x $ }})\widetilde{L}^2_{k,n,m}(\mbox{\boldmath$ x $ }_0)),$ (3.73)

where $\widetilde{L}^1_{kj,n,m}(\mbox{\boldmath$\space x $ }_0)$ and $\widetilde{L}^2_{k,n,m}(\mbox{\boldmath$\space x $ }_0)$ are the coefficients of the local expansion centred at $\mbox{\boldmath$\space x $ }_0$ and are expressed with $\widetilde{M}^{1}_{kj,n,m}(O)$ and $\widetilde{M}^{2}_{k,n,m}(O)$ by
  
$\displaystyle \widetilde{L}^1_{kj,n',m'}(\mbox{\boldmath$ x $ }_{0})$ = $\displaystyle \sum_{n=0}^{\infty}\sum_{m=-n}^{n}(-1)^{n'}
\overline{S_{n+n',m+m'}}(\overrightarrow{O\mbox{\boldmath$ x $ }_{0}})\widetilde{M}^1_{kj,n,m}(O),$ (3.74)
$\displaystyle \widetilde{L}^2_{k,n',m'}(\mbox{\boldmath$ x $ }_{0})$ = $\displaystyle \sum_{n=0}^{\infty}\sum_{m=-n}^{n}(-1)^{n'}
\overline{S_{n+n',m+m...
...)-(\overrightarrow{O\mbox{\boldmath$ x $ }_0})_{j}\widetilde{M}^1_{kj,n,m}(O)).$ (3.75)

In these formulae 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. Notice that $\widetilde{L}^1_{kj,n,m}(\mbox{\boldmath$\space x $ }_0)$ has nine components and $\widetilde{L}^2_{k,n,m}(\mbox{\boldmath$\space x $ }_0)$ has three component for a given pair of n and m, and therefore the number of coefficients of the local expansion in this formulation is 12. Also, the coefficients of the local expansion have the following properties:
  
$\displaystyle \widetilde{L}^1_{kj,n,-m}(\mbox{\boldmath$ x $ }_0) = (-1)^m \overline{\widetilde{L}^1_{kj,n,m}}(\mbox{\boldmath$ x $ }_0)
\quad (m \ge 0),$     (3.76)
$\displaystyle \widetilde{L}^2_{k,n,-m}(\mbox{\boldmath$ x $ }_0) = (-1)^m \overline{\widetilde{L}^2_{k,n,m}}(\mbox{\boldmath$ x $ }_0)
\quad (m \ge 0).$     (3.77)

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

  
$\displaystyle \widetilde{L}^1_{kj,n'',m''}(\mbox{\boldmath$ x $ }_{1})$ = $\displaystyle \sum_{n'=n''}^{\infty}\sum_{m'=-n'}^{n'}
R_{n'-n'',m'-m''}(\overr...
...mbox{\boldmath$ x $ }_1})
\widetilde{L}^1_{kj,n',m'}(\mbox{\boldmath$ x $ }_0),$ (3.78)
$\displaystyle \widetilde{L}^2_{k,n'',m''}(\mbox{\boldmath$ x $ }_1)$ = $\displaystyle \sum_{n'=n''}^{\infty}\sum_{m'=-n'}^{n'}
R_{n'-n'',m'-m''}(\overr...
...math$ x $ }_1})_{j}\widetilde{L}^1_{kj,n',m'}(\mbox{\boldmath$ x $ }_0)\right),$ (3.79)

where we have used (3.21) and (3.73).

As we shall see in FM-BIEM with the regularised integral equation we can compute the multipole moments analytically (See the following section), but the multipole moments and the coefficients of the local expansion have twelve components for a given pair of n and m. Namely, the number of the multipole moments in the regularised formulation is triple that in the hypersingular one. This trade-off issue will be settled through a numerical experiment in section 3.3.5.


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