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FM-BIEM with hypersingular integral equation

In this section we use the hypersingular integral equation (3.15) for the formulation. We first expand the fundamental solution $G(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$ into a series of products of functions of $\mbox{\boldmath$\space x $ }$ and those of $\mbox{\boldmath$\space y $ }$. To this end we introduce the following well-known identity: (See Appendix A)
 
$\displaystyle \frac{1}{\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert}...
...mbox{\boldmath$ x $ }}\vert>\vert\overrightarrow{O\mbox{\boldmath$ y $ }}\vert,$     (3.17)

where Rn,m and Sn,m are the solid harmonics defined as
  
$\displaystyle R_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})$ = $\displaystyle \frac{1}{(n+m)!}P_n^m(\cos\theta)e^{im\phi}r^n ,$ (3.18)
$\displaystyle S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})$ = $\displaystyle (n-m)!P_n^m(\cos\theta)e^{im\phi}\frac{1}{r^{n+1}},$ (3.19)

$(r,\theta,\phi)$ are the polar coordinates of the point $\mbox{\boldmath$\space x $ }$, Pnm is the associated Legendre function and a superposed bar indicates the complex conjugate, respectively. The functions Rn,m and Sn,m satisfy the relations given by (See Appendix B)
  
$\displaystyle S_{n,m}(\overrightarrow{\mbox{\boldmath$ y $ }\mbox{\boldmath$ x $ }})$ = $\displaystyle \sum_{n'=0}^{\infty}\sum_{m'=-n'}^{n'}
\overline{R_{n',m'}}(\over...
...box{\boldmath$ y $ }}\vert<\vert\overrightarrow{O\mbox{\boldmath$ x $ }}\vert ,$ (3.20)
$\displaystyle R_{n,m}(\overrightarrow{\mbox{\boldmath$ y $ }\mbox{\boldmath$ x $ }})$ = $\displaystyle \sum_{n'=0}^{n}\sum_{m'=-n'}^{n'}
R_{n',m'}(\overrightarrow{\mbox{\boldmath$ y $ }O})R_{n-n',m-m'}(\overrightarrow{O\mbox{\boldmath$ x $ }})$ (3.21)
    $\displaystyle \quad(\mbox{This holds for arbitrary} \ \overrightarrow{O\mbox{\boldmath$ x $ }} \ \mbox{and}
\ \overrightarrow{O\mbox{\boldmath$ y $ }}).$  

In FMM, Greengard uses the spherical harmonics making his formulation somewhat complicated. Therefore we use solid harmonics to make the formulae needed for FMM concise. The use of solid harmonics has been suggested by White and Head-Gordon[81] and Perez-Jorda and Yang[66].

We now compute the integral on the right-hand side of (3.15) over a subset of S denoted by Sy for $\mbox{\boldmath$\space x $ }$ which is away from Sy. Using (3.17) we obtain

 
$\displaystyle \int_{S_y} \frac{\partial^2 G(\mbox{\boldmath$ x $ }-\mbox{\boldm...
...e{S_{n,m}}(\overrightarrow{O\mbox{\boldmath$ x $ }})}
{\partial n_x}M_{n,m}(O),$     (3.22)

where Mn,m(O) is the multipole moment centred at O, defined by
 
$\displaystyle M_{n,m}(O) = \int_{S_y} \frac{\partial R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }})}
{\partial n_y}\phi(\mbox{\boldmath$ y $ }) dS_y.$     (3.23)

In (3.22) 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 according to the following formula when the centre of the multipole moment is translated from O to O'

 
$\displaystyle M_{n,m}(O') = \sum_{n'=0}^{n}\sum_{m'=-n'}^{n'} R_{n',m'}(\overrightarrow{O'O})
M_{n-n',m-m'}(O),$     (3.24)

where we have used (3.21) and (3.23) (See Appendix E.1). The procedure given by (3.24) is called ``M2M translation''. Noting (A.5) and the fact that $\phi$ is real-valued, one finds that the multipole moment has the following property:
 
Mn,-m(O) = $\displaystyle (-1)^m \overline{M_{n,m}}(O)\quad (m \ge 0).$ (3.25)

The integral in (3.22) can be evaluated with the coefficient of the local expansion Ln,m as follows:
 
$\displaystyle \int_{S_y} \frac{\partial^2 G(\mbox{\boldmath$ x $ }-\mbox{\boldm...
...$ }_0\mbox{\boldmath$ x $ }})}{\partial n_x}
L_{n,m}(\mbox{\boldmath$ x $ }_0),$     (3.26)

where $L_{n,m}(\mbox{\boldmath$\space x $ }_0)$ is expressed in terms of Mn,m(O) by
 
$\displaystyle L_{n,m}(\mbox{\boldmath$ x $ }_0) = \sum_{n'=0}^{\infty}\sum_{m'=...
...verline{S_{n+n',m+m'}}(\overrightarrow{O\mbox{\boldmath$ x $ }_0})M_{n',m'}(O).$     (3.27)

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 (See Appendix E.2). The procedure given by (3.27) is called ``M2L translation''. Also, the coefficient of the local expansion has the following property:
 
$\displaystyle L_{n,-m}(\mbox{\boldmath$ x $ }_0) = (-1)^m \overline{L_{n,m}}(\mbox{\boldmath$ x $ }_0)
\quad (m \ge 0).$     (3.28)

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$
 
$\displaystyle L_{n,m}(\mbox{\boldmath$ x $ }_1) = \sum_{n'=n}^{\infty}\sum_{m'=...
...boldmath$ x $ }_0\mbox{\boldmath$ x $ }_1})L_{n',m'}(\mbox{\boldmath$ x $ }_0),$     (3.29)

where we have used (3.21) and (3.26) (See Appendix E.3). The procedure given by (3.29) is called ``L2L translation''.


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