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New FM-BIEM

Suppose that we evaluate the contribution from a cell Cs to the coefficients of the local expansion in a cell Ct contained in the interaction list of Cs. The cells Cs and Ct are called a source cell and a target cell, respectively. We divide the interaction list of Cs into 6 lists called the uplist, downlist, northlist, southlist, eastlist and westlist of Cs. The uplist and downlist contain target cells located in +x3 and -x3directions of Cs, respectively. The northlist and southlist contain target cells located in +x2 and -x2 directions of Csexcept those in the uplist or downlist, respectively. The eastlist and westlist contain target cells located in +x1 and -x1directions of Cs except those in the uplist, downlist, northlist or southlist, respectively. In this section we discuss the case where target cells are included in the uplist of Cs and each of the cells is a cube having a volume of d3.
  
Figure 4.2: Uplist of source cell
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/UP3.eps,scale=0.8}\end{center}\end{figure}

Suppose that a source point $\mbox{\boldmath$\space y $ }$ is located at (y1,y2,y3) in Cs and a target point $\mbox{\boldmath$\space x $ }$ at (x1,x2,x3) in Ct. It is easy to see that the following integral representation[61] holds:
 
$\displaystyle { \frac{1}{\sqrt{(x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2}}}$
    $\displaystyle =\frac{1}{2\pi} \int_{0}^{\infty} e^{-\lambda (x_3-y_3)} \int_{0}^{2\pi}
e^{i\lambda((x_1-y_1)\cos\alpha+(x_2-y_2)\sin\alpha)} d\alpha d\lambda$ (4.1)

where we have assumed that the inequality x3 > y3 holds. The inner integral with respect to $\alpha$ in (4.1) is computed with the trapezoidal rule and the outer integral with generalised Gaussian quadrature rules[83], yielding
 
$\displaystyle { \frac{1}{\sqrt{(x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2}}}$
    $\displaystyle = \sum_{k=1}^{s(\varepsilon)} \sum_{j=1}^{M(k)} \frac{\omega_k}{M...
...da_k/d)((x_1-y_1) \cos \alpha_j(k)+ (x_2-y_2) \sin \alpha_j(k))}
+ \varepsilon,$ (4.2)
    $\displaystyle \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (x_3 > y_3)$  

where $\alpha_j(k)$ is given by

\begin{displaymath}\alpha_j(k) = \frac{2\pi j}{M(k)},
\end{displaymath}

$\varepsilon$ is the error term and the numbers $s(\varepsilon)$, M(k), Gaussian weights $\omega_k$ and nodes $\lambda_k$ are given in Yarvin and Rokhlin[83]. These numbers are available from Netlib (http://www.netlib.org/pdes/multipole/). One may determine these parameters considering the required accuracy.

Noting (4.2) and the following formulae: (See (3.19) and Appendix B)

   
$\displaystyle S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})$ = $\displaystyle (-1)^{n}\partial_{+}^{m}\partial_{z}^{n-m}
\left(\frac{1}{\vert\overrightarrow{O\mbox{\boldmath$ x $ }}\vert}\right)\quad (m \ge 0),$ (4.3)
$\displaystyle S_{n,-m}(\overrightarrow{O\mbox{\boldmath$ x $ }})$ = $\displaystyle (-1)^{n+m} \partial_{-}^{m}\partial_{z}^{n-m}
\left(\frac{1}{\vert\overrightarrow{O\mbox{\boldmath$ x $ }}\vert}\right)
\quad (m \ge 0),$ (4.4)
$\displaystyle \partial_{\pm}$ = $\displaystyle \left(\frac{\partial}{\partial x}
\pm i \frac{\partial}{\partial y} \right),$ (4.5)

one can evaluate the integral in (3.22) in the following manner: (See Appendix J.1)

\begin{eqnarray*}\lefteqn{ \int_{S_y} \frac{\partial^2 G(\mbox{\boldmath$ x $ }-...
... (\overrightarrow{O\mbox{\boldmath$ x $ }})_2 \sin \alpha_j(k))}
\end{eqnarray*}


where X(k,j;O) is the coefficient of the exponential expansion at O, defined by
 
$\displaystyle X(k,j;O)=\frac{\omega_k}{M(k)d}\sum_{m=-\infty}^{\infty}(-i)^m e^{-i m
\alpha_j(k)} \sum_{n=\vert m\vert}^{\infty} (\lambda_k/d)^n M_{n,m}(O).$     (4.6)

This formula converts the multipole moments into the exponential expansion coefficients. We call the procedure given by (4.6) ``M2X (Multipole moment to(2) eXponential expansion) translation''. The coefficients of the exponential expansion is translated according to the following formula when the centre of the exponential expansion is shifted from O to $\mbox{\boldmath$\space x $ }_0$: (See Appendix J.2)
 
$\displaystyle X(k,j;\mbox{\boldmath$ x $ }_0) = X(k,j;O)
e^{\scriptstyle -(\lam...
...\alpha_j(k) + (\overrightarrow{O\mbox{\boldmath$ x $ }_0})_2 \sin \alpha_j(k))}$     (4.7)

where $(\mbox{\boldmath$\space x $ })_i$ stand for the components of the vector $\mbox{\boldmath$\space x $ }$. We call the procedure given by (4.7) ``X2X (eXponential expansion to(2) eXponential expansion) translation''.

We next need to convert the coefficients of the exponential expansion into the coefficients of the local expansion (See (3.27)). Noting that the integral in (3.22) is expressed with $X(k,j;\mbox{\boldmath$\space x $ }_0)$ as follows:

 
$\displaystyle {\int_{S_y} \frac{\partial^2 G(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ })}
{\partial n_x \partial n_y} \varphi(\mbox{\boldmath$ y $ }) dS_y }$
  = $\displaystyle \frac{1}{4\pi} \frac{\partial}{\partial n_x}
\sum_{k=1}^{s(\varep...
...ightarrow{\mbox{\boldmath$ x $ }_0\mbox{\boldmath$ x $ }})_2 \sin \alpha_j(k))}$ (4.8)

and using (3.26) and the following formula (See (3.18) and Hobson[43]):
 
$\displaystyle \frac{((\overrightarrow{O\mbox{\boldmath$ x $ }})_3 - i (\overrig...
...-i)^m e^{- i m \alpha_j(k)} \ R_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})$     (4.9)

we obtain the following formula which converts the coefficients of the exponential expansion into the coefficient of the local expansion: (See Appendix J.3)
 
$\displaystyle L_{n,m}(\mbox{\boldmath$ x $ }_0) = \sum_{k=1}^{s(\varepsilon)}\s...
...} X(k,j;\mbox{\boldmath$ x $ }_0)
(-i)^m (-\lambda_k/d)^n e^{-i m \alpha_j(k)}.$     (4.10)

We call the procedure given by (4.10) ``X2L (eXponential expansion to(2) Local expansion) translation''. In the new FM-BIEM we replace M2L translation given by (3.27) with M2X, X2X and X2L translation given by (4.6), (4.7) and (4.10).


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