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Comparison of formulations for the single- and double-layer potentials

In this section we compare our FMM formulation with the formulation proposed by Fu et al.[18] and attempt to clarify the difference between these formulations. In this section we deal with both single- and double-layer potentials.

Fu et al. noted that the fundamental solution in (3.45) is expressed as

 
$\displaystyle \Gamma_{ij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ })= P_{ij...
...eft(\frac{y_j}{\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert}\right),$     (3.86)

where $P_{ij}(\mbox{\boldmath$\space x $ })$ and $Q_{i}(\mbox{\boldmath$\space x $ })$ are given by

\begin{eqnarray*}P_{ij}(\mbox{\boldmath$ x $ })=\frac{1}{8\pi\mu}\left(\frac{\la...
...mu}\frac{\lambda+\mu}{\lambda+2\mu}\frac{\partial}{\partialx_i}.
\end{eqnarray*}


To interpret their expression for the double layer kernel Tijdefined by

 \begin{displaymath}T_{ij}(\mbox{\boldmath$ x $ }, \mbox{\boldmath$ y $ }) = \fra...
...- \mbox{\boldmath$ y $ })C_{klpj} n_p(\mbox{\boldmath$ y $ }),
\end{displaymath} (3.87)

we see that the following alternative to (3.87) is available:

 \begin{displaymath}T_{ij}(\mbox{\boldmath$ x $ }, \mbox{\boldmath$ y $ }) = -\fr...
...- \mbox{\boldmath$ y $ })C_{klpj} n_p(\mbox{\boldmath$ y $ }).
\end{displaymath} (3.88)

From (3.86) and (3.88) we obtain

 \begin{displaymath}T_{ij}(\mbox{\boldmath$ x $ }, \mbox{\boldmath$ y $ })=-n_p(\...
...x{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert}\right)\right),
\end{displaymath} (3.89)

which can further be reduced to

 \begin{displaymath}T_{ij}(\mbox{\boldmath$ x $ }, \mbox{\boldmath$ y $ })= \Thet...
...ert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert}\right),
\end{displaymath} (3.90)

where $\Theta_{ijk}$ and $\Lambda_{ij}$ are certain operators defined in Fu et al.[18]. In this way they could maintain the term $1/\vert\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ }\vert$in their expression for Tij, thus making the use of the FMM for Laplace's equation possible in their formulation. The single- and double-layer potentials
$\displaystyle U_i(\mbox{\boldmath$ x $ }):=\int_S \Gamma_{ij}(\mbox{\boldmath$ x $ },\mbox{\boldmath$ y $ }) \varphi_j(\mbox{\boldmath$ y $ }) dS_y$     (3.91)
$\displaystyle W_i(\mbox{\boldmath$ x $ }):=\int_S T_{ij}(\mbox{\boldmath$ x $ }, \mbox{\boldmath$ y $ }) \varphi _j(\mbox{\boldmath$ y $ }) dS_y$     (3.92)

are now written as
  
$\displaystyle U_i(\mbox{\boldmath$ x $ })$ = $\displaystyle P_{ij}(\mbox{\boldmath$ x $ })
\int_S \left(\frac{\varphi_j(\mbox...
...h$ y $ })}{\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert}\right) dS_y$ (3.93)
$\displaystyle W_i(\mbox{\boldmath$ x $ })$ = $\displaystyle \Theta_{ijp}(\mbox{\boldmath$ x $ })\int _S\left(\frac{n_p(\mbox{...
...h$ y $ })}{\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert}\right)dS_y.$ (3.94)

Substituting (3.17) into (3.93) and (3.94), one obtains

\begin{eqnarray*}U_i(\mbox{\boldmath$ x $ }) &=&\sum_{n=0}^{\infty}\sum_{m=-n}^{...
...errightarrow{O\mbox{\boldmath$ x $ }})\acute{M}^{2W}_{p,n,m}(O),
\end{eqnarray*}


where

\begin{eqnarray*}\acute{M}^{1U}_{j,n,m}(O)=\int_S \varphi_j(\mbox{\boldmath$ y $...
...h$ y $ })R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }}) dS_y.
\end{eqnarray*}


The multipole moments $\acute{M}^{1U}_{j,n,m}(O)$ has three components and $\acute{M}^{2U}_{n,m}(O)$ has one. Also, the multipole moments $\acute{M}^{1W}_{jp,n,m}(O)$ has nine components and $\acute{M}^{2W}_{p,n,m}(O)$ has three. Thus, one can see that Fu et al.[18] use 4 and 12 multipole moments for the single- and double-layer potentials, respectively.

It is seen that the expression used by Fu et al.[18] is not the only possibility in elastostatics which allows the use of the FMM for Laplace's equation. For example one uses (3.89) instead of (3.90) to have

 
$\displaystyle W_i(\mbox{\boldmath$ x $ }) = -\frac{\partial}{\partialx_l} P_{ik...
...$ y $ })}{\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert}
\right)dS_y.$     (3.95)

Substituting (3.17) into (3.95) one obtains

  \begin{eqnarray*}W_i(\mbox{\boldmath$ x $ }) = \sum_{n=0}^{\infty}\sum_{m=-n}^{n...
...overrightarrow{O\mbox{\boldmath$ x $ }})
\check{M}^2_{l,n,m}(O)
\end{eqnarray*}


where $\check{M}^1_{j,l,N,M}(O)$ and $\check{M}^2_{l,N,M}(O)$ are given by

\begin{eqnarray*}\check{M}^1_{kl,n,m}(O) = \int_S C_{klpj} R_{n,m}(\overrightarr...
...mbox{\boldmath$ y $ }) \varphi_j(\mbox{\boldmath$ y $ }) dS_{y}.
\end{eqnarray*}


The moments $\check{M}^1_{kl,n,m}(O)$ have 6 components (since $\check{M}^1_{kl,n,m}(O)=\check{M}^1_{lk,n,m}(O)$) and $\check{M}^2_{l,n,m}(O)$ have 3. Hence in this case the number of multipole moments for the double-layer potential is 9 for a given pair of n and m.

On the other hand, in order to obtain our formulations for the single- and double-layer potentials we first substitute (3.17) into (3.86):

 
$\displaystyle \Gamma_{ij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ })= \sum_...
...{\boldmath$ x $ }})\right)R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }})y_j.$     (3.96)

Noting the following identities

\begin{displaymath}P_{ij}(\mbox{\boldmath$ x $ })S_{n,m}(\overrightarrow{O\mbox{...
...8\pi\mu}G^S_{i,n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }}),
\end{displaymath}

one sees that (3.96) is identical with (3.49).

In order to obtain our expression for the double-layer kernel we use (3.87), instead of (3.88), with (3.49) to have

 
$\displaystyle {T_{ij}(\mbox{\boldmath$ x $ }, \mbox{\boldmath$ y $ }) =}$
    $\displaystyle \frac{1}{8\pi\mu}\sum_{n=0}^{\infty} \sum_{m=-n}^{n}
\left(\overl...
...th$ y $ }})\right)}
{\partial y_l}C_{klpj} n_p(\mbox{\boldmath$ y $ }) \right).$ (3.97)

Substituting (3.96) and (3.97) into (3.93) and (3.94) we obtain
  
$\displaystyle U_i(\mbox{\boldmath$ x $ })$ = $\displaystyle \frac{1}{8\pi\mu}\sum_{n=0}^{\infty}\sum_{m=-n}^{n}
\overline{F^S...
...ne{G^S_{i,n,m}}(\overrightarrow{O\mbox{\boldmath$ x $ }}) \hat{M}^{2U}_{n,m}(O)$ (3.98)
$\displaystyle W_i(\mbox{\boldmath$ x $ })$ = $\displaystyle \frac{1}{8\pi\mu}\sum_{n=0}^{\infty}\sum_{m=-n}^{n}
\overline{F^S...
...ne{G^S_{i,n,m}}(\overrightarrow{O\mbox{\boldmath$ x $ }}) \hat{M}^{2W}_{n,m}(O)$ (3.99)

where

\begin{eqnarray*}\hat{M}^{1U}_{j,n,m}(O)= \int_S R_{n,m}(\overrightarrow{O\mbox{...
...x{\boldmath$ y $ }}) y_j\varphi_j(\mbox{\boldmath$ y $ }) dS_y,
\end{eqnarray*}



\begin{eqnarray*}\hat{M}^{1W}_{k,n,m}(O)=\int_S \frac{\partial R_{n,m}(\overrigh...
...p(\mbox{\boldmath$ y $ })\varphi_j(\mbox{\boldmath$ y $ }) dS_y.
\end{eqnarray*}


The multipole moments $\hat{M}^{1U}_{j,n,m}(O)$ has three components and $\hat{M}^{2U}_{n,m}(O)$ has one. Also, the multipole moments $\hat{M}^{1W}_{j,n,m}(O)$ has three components and $\hat{M}^{2W}_{n,m}(O)$ has one. In our formulations for both the single- and double-layer potentials the number of the multipole moments is 4 and, moreover, $\hat{M}^{1W}_{j,n,m}(O)$ and $\hat{M}^{2W}_{n,m}(O)$ are the same as the multipole moments (3.53) and (3.54) for crack problems.

Notice that in this case the 4 moment formulation may not necessarily be three times faster than the 12 moment formulation. However, it is seen that our formulation is probably more efficient than Fu et al.'s one because the reduction of the number of the multipole moments leads to the reduction of M2L translations which dominate the performance of FMM.


next up previous contents
Next: Crack problems for three-dimensional Up: Crack problems for three-dimensional Previous: Many penny-shaped cracks
Ken-ichi Yoshida
2001-07-28