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FM-BIEM

In FM-BIEM our first step is to expand the fundamental solution (3.139) into a series of products of functions of $\mbox{\boldmath$\space x $ }$ and those of $\mbox{\boldmath$\space y $ }$. To this end we notice that the fundamental solution (3.139) can be rewritten as
 
$\displaystyle \Gamma_{ip}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }) =
\fra...
...partial}{\partialx_i}\frac{\partial}{\partialy_p}
\frac{e^{i k_L R}}{R}\right),$     (3.141)

where we have used the following identity:

\begin{eqnarray*}(\Delta+k_T^2)\frac{e^{i k_T R}}{R}=0 \quad (R \ne 0).
\end{eqnarray*}


Using (3.120) and (3.142), we obtain a series expansion of the fundamental solution as follows:

 
$\displaystyle \Gamma_{ip}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }) =
\sum...
...tialy_p} \overline{{\cal I}}_n^m(k_L,\overrightarrow{O\mbox{\boldmath$ y $ }}),$      

where the function ${\cal I}_n^m$ is given by (3.121) and the functions $D^{T,{\cal O}}_{ri,n,m}(\overrightarrow{O\mbox{\boldmath$\space x $ }})$ and $D^{L,{\cal
O}}_{i,n,m}(\overrightarrow{O\mbox{\boldmath$\space x $ }})$ are defined as
  
$\displaystyle D^{T,{\cal O}}_{ri,n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})$ = $\displaystyle \frac{i(2n+1) }{4 \pi \mu k_T}
e_{rqi}\frac{\partial}{\partialx_q} {\cal O}_n^m(k_T,\overrightarrow{O\mbox{\boldmath$ x $ }}),$ (3.142)
$\displaystyle D^{L,{\cal O}}_{i,n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})$ = $\displaystyle \frac{i k_L(2n+1) }{4 \pi \mu k_T^2}
\frac{\partial}{\partialx_i}{\cal O}_n^m(k_L,\overrightarrow{O\mbox{\boldmath$ x $ }}).$ (3.143)

In (3.144) and (3.145) the function ${\cal O}_n^m$ is given by (3.122).

Now we compute the integral on the right-hand side of (3.140) over a subset of S denoted by Sy for an $\mbox{\boldmath$\space x $ }$which is away from Sy. Using (3.143) we obtain

 
$\displaystyle {\int_{S_y}
C_{jktp} \frac{\partial^2}{\partial x_{l}\partial
y_{...
... y $ }) n_{k}(\mbox{\boldmath$ y $ })
\phi_{j}(\mbox{\boldmath$ y $ }) dS_{y} }$
  = $\displaystyle \sum_{n=0}^{\infty} \sum_{m=-n}^{n}
\frac{\partial}{\partialx_l} ...
... D^{L,{\cal O}}_{i,n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})
M^L_{n,m}(O),$ (3.144)

where MTr,n,m(O) and MLn,m(O) are the multipole moments centred at O defined as
  
MTr,n,m(O) = $\displaystyle \int_{S_y}
C_{jktp}e_{rsp}\frac{\partial^2}{\partial y_t \partial...
... y $ }})
\phi_{j}(\mbox{\boldmath$ y $ }) n_{k}(\mbox{\boldmath$ y $ }) dS_{y},$ (3.145)
MLn,m(O) = $\displaystyle \int_{S_y}
C_{jktp}\frac{\partial^2}{\partial y_t \partial y_p}
\...
... y $ }})
\phi_{j}(\mbox{\boldmath$ y $ }) n_{k}(\mbox{\boldmath$ y $ }) dS_{y}.$ (3.146)

In (3.146) 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. Notice that MTr,n,m(O) has three components and MLn,m(O) has one component for a fixed pair of nand m and therefore the number of the multipole moments is 4. The number of the multipole moments is thus seen to be the same as that in three-dimensional elastostatics. Fujiwara [22] uses FMM to compute the double-layer potential in three-dimensional elastodynamics and in Fujiwara's formulation the number of the multipole moments is 6. If one adopts our formulation, the number of the multipole moments for the double-layer potential is 4 (details of the difference between our formulation and Fujiwara's one are given later). Also, Fukui and Inoue [24] use 4 multipole moments in FM-BIEM for ordinary problems in two-dimensional elastodynamics. However, replacing erspwith e3sp in (3.147), one can find that the number of the multipole moments is 2 in our formulation for two-dimensional elastodynamics.

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

 
$\displaystyle M_{r,n,m}^T(O')
= \sum_{n'=0}^{\infty}\sum_{m'=-n'}^{n'}
\sum_{\s...
...W_{n,n',m,m',l}
{\cal I}_l^{-m-m'}(k_T,\overrightarrow{O'O}) M_{r,n',-m'}^T(O),$     (3.147)


 
$\displaystyle M_{n,m}^L(O')= \sum_{n'=0}^{\infty}\sum_{m'=-n'}^{n'}
\sum_{\scri...
...'}W_{n,n',m,m',l}
{\cal I}_l^{-m-m'}(k_L,\overrightarrow{O'O}) M_{n',-m'}^L(O).$     (3.148)

where we have used (3.123).

One can evaluate the integral on the right-hand side of (3.140) with the local expansion in the following manner:

 
$\displaystyle {\int_{S_y}
C_{jktp} \frac{\partial^2}{\partial x_{l}\partial
y_{...
... y $ }) n_{k}(\mbox{\boldmath$ y $ })
\phi_{j}(\mbox{\boldmath$ y $ }) dS_{y} }$
  = $\displaystyle \sum_{n=0}^{\infty} \sum_{m=-n}^{n}
\frac{\partial}{\partialx_l} ...
...\boldmath$ x $ }_0\mbox{\boldmath$ x $ }})
L^L_{n,m}(\mbox{\boldmath$ x $ }_0),$ (3.149)

where $L^T_{r,n,m}(\mbox{\boldmath$\space x $ }_0)$ and $L^L_{n,m}(\mbox{\boldmath$\space x $ }_0)$ are the coefficients of the local expansion centred at $\mbox{\boldmath$\space x $ }_0$ defined as
 
$\displaystyle L_{r,n,m}^T(\mbox{\boldmath$ x $ }_0)=\sum_{n'=0}^{\infty}\sum_{m...
...O}}_l^{-m-m'}(k_T,\overrightarrow{O\mbox{\boldmath$ x $ }_0})
M_{r,n',m'}^T(O),$     (3.150)


 
$\displaystyle L_{n,m}^L(\mbox{\boldmath$ x $ }_0)=\sum_{n'=0}^{\infty}\sum_{m'=...
...l O}}_l^{-m-m'}(k_L,\overrightarrow{O\mbox{\boldmath$ x $ }_0})
M_{n',m'}^L(O),$     (3.151)

and the functions $D^{T,\overline{\cal I}}$ and $D^{L,\overline{\cal I}}$ are obtained by replacing ${\cal O}_{n,m}$ with $\overline{\cal I}_{n,m}$ in (3.144) and (3.145). In the derivation of (3.152) and (3.153) we have used (3.124) 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.

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

 
$\displaystyle L_{r,n,m}^T(\mbox{\boldmath$ x $ }_1)=\sum_{n'=0}^{\infty}\sum_{m...
...th$ x $ }_0 \mbox{\boldmath$ x $ }_1}) L_{r,n',m'}^T(\mbox{\boldmath$ x $ }_0),$     (3.152)


 
$\displaystyle L_{n,m}^L(\mbox{\boldmath$ x $ }_1)=\sum_{n'=0}^{\infty}\sum_{m'=...
...math$ x $ }_0 \mbox{\boldmath$ x $ }_1}) L_{n',m'}^L(\mbox{\boldmath$ x $ }_0).$     (3.153)

where we have used (3.123).

Notice that M2M translations in (3.149) and (3.150), M2L translations in (3.152) and (3.153) and L2L translations in (3.154) and (3.155) have the same form as the M2M translation in (3.129), M2L translation in (3.131) and L2L translation in (3.132) in Helmholtz's equation. Also, the derivation of translation formulae in elastodynamics is similar to that in Helmholtz's equation.

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: Three-dimensional scattering of elastic Previous: Integral equation
Ken-ichi Yoshida
2001-07-28