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New FM-BIEM

Now we consider a source point $\mbox{\boldmath$\space y $ }$ located at (y1,y2,y3) and a target point $\mbox{\boldmath$\space x $ }$ at (x1,x2,x3) and denote a cell including a source point by Cs and a cell including a target point by Ct. Suppose that each of the cells Cs and Ct has a volume of d3. Here we use the integral representation (4.1) again. The integral in (4.1) converges under an assumption that the inequality x3 > y3 is valid. This is why we restrict the discussion to the case where Ct is located in +x3 direction of Cs for the present.

Noting (4.2) and (4.3)-(4.5), one can evaluate the integral in (3.14) in the following manner: (See Appendix K.1)

\begin{eqnarray*}\lefteqn{\int_{S_y} \frac{\partial}{\partial x_k}
\frac{\parti...
... (\overrightarrow{O\mbox{\boldmath$ x $ }})_2 \sin \alpha_q(p))}
\end{eqnarray*}


where X1j(p,q;O) and X2(p,q;O) are the coefficients of the exponential expansion at O, defined in terms of the multipole moments (3.53) and (3.54) as
  
$\displaystyle X^1_j(p,q;O)=\frac{\omega_p}{M(p)d}\sum_{m=-\infty}^{\infty}(-i)^...
...-i m
\alpha_q(p)} \sum_{n=\vert m\vert}^{\infty} (\lambda_p/d)^n M^1_{j,n,m}(O)$     (4.28)
$\displaystyle X^2(p,q;O)=
\frac{\omega_p}{M(p)d}\sum_{m=-\infty}^{\infty}(-i)^m e^{-i m
\alpha_q(p)} \sum_{n=\vert m\vert}^{\infty} (\lambda_p/d)^n M^2_{n,m}(O)$     (4.29)

and $\cal F,G$ are operators defined as

\begin{eqnarray*}{\cal F}_{kij}(\overrightarrow{O\mbox{\boldmath$ x $ }}) &=& \f...
...2\mu}\frac{\partial}{\partial x_k}\frac{\partial}{\partial x_i}
\end{eqnarray*}


These formulae (4.28) and (4.29) convert the multipole moments into the coefficients of the exponential expansion. We call the procedures given by (4.28) and (4.29) ``M2X translation''. The coefficients of the exponential expansion is translated according to the following formulae when the centre of the exponential expansion is shifted from O to $\mbox{\boldmath$\space x $ }_0$: (See Appendix K.2)
  
$\displaystyle X^1_j(p,q;\mbox{\boldmath$ x $ }_0)$ = $\displaystyle X^1_j(p,q;O)
e^{\scriptstyle -(\lambda_p/d) (\overrightarrow{O\mb...
...alpha_q(p) + (\overrightarrow{O\mbox{\boldmath$ x $ }_0})_2 \sin \alpha_q(p))},$ (4.30)
$\displaystyle X^2(p,q;\mbox{\boldmath$ x $ }_0)$ = $\displaystyle (X^2(p,q;O)-(\overrightarrow{O\mbox{\boldmath$ x $ }_0})_k X^1_k(...
...alpha_q(p) + (\overrightarrow{O\mbox{\boldmath$ x $ }_0})_2 \sin \alpha_q(p))},$ (4.31)

where $X^1_j(p,q;\mbox{\boldmath$\space x $ }_0)$ and $X^2(p,q;\mbox{\boldmath$\space x $ }_0)$ are the coefficients of the exponential expansion at $\mbox{\boldmath$\space x $ }_0$ and $(\mbox{\boldmath$\space x $ })_i$ stand for the components of the vector $\mbox{\boldmath$\space x $ }$. We call the procedures given by (4.30) and (4.31) ``X2X translation''.

We next need to convert the coefficients of the exponential expansion into the coefficients of the local expansion. Noting that the integral in (2.3) is evaluated with $X^1_j(p,q;\mbox{\boldmath$\space x $ }_0)$ and $X^2(p,q;\mbox{\boldmath$\space x $ }_0)$ as follows: (See Appendix K.3)

\begin{eqnarray*}\lefteqn{\int_{S_y} \frac{\partial}{\partial x_k}
\frac{\parti...
...{\boldmath$ x $ }_0\mbox{\boldmath$ x $ }})_2 \sin \alpha_q(p))}
\end{eqnarray*}


and using (2.9) and (4.9), we obtain the following formulae which convert the coefficients of the exponential expansion into the coefficients of the local expansion (3.60) and (3.61):
  
$\displaystyle L^1_{j,n,m}(\mbox{\boldmath$ x $ }_0)$ = $\displaystyle \sum_{p=1}^{s(\varepsilon)}\sum_{q=1}^{M(p)} X^1_j(p,q;\mbox{\boldmath$ x $ }_0)
(-i)^m (-\lambda_p/d)^n e^{-i m \alpha_q(p)}$ (4.32)
$\displaystyle L^2_{n,m}(\mbox{\boldmath$ x $ }_0)$ = $\displaystyle \sum_{p=1}^{s(\varepsilon)}\sum_{q=1}^{M(p)} X^2(p,q;\mbox{\boldmath$ x $ }_0)
(-i)^m (-\lambda_p/d)^n e^{-i m \alpha_q(p)}$ (4.33)

We call the procedures given by (4.32) and (4.33) ``X2L translation''.

The procedures given by (4.28)-(4.33) correspond to the M2L translation given by (3.60) and (3.61) in the original FMM described in 3.3.


next up previous contents
Next: Rotation of coefficients Up: Crack problems for three-dimensional Previous: Crack problems for three-dimensional
Ken-ichi Yoshida
2001-07-28