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Next: FM-BIEM with regularised integral Up: Crack problems for three-dimensional Previous: Integral equation

   
FM-BIEM with hypersingluar integral equation

In this section we use the hypersingular integral equation (3.46) for the formulation. In the application of FMM to BIEM our starting point is to expand the fundamental solution $\Gamma_{ij}(\mbox{\boldmath$\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 $ }$. From the expression of (3.45) one finds that it is necessary to expand $\vert\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ }\vert$ into a series. Thus, we expand $\vert\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ }\vert$ as (See Appendix F)
 
$\displaystyle \vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert=\sum_{n=0...
...ox{\boldmath$ x $ }}\vert > \vert\overrightarrow{O\mbox{\boldmath$ y $ }}\vert.$     (3.48)

Using (3.48), we rewrite (3.45) as (See Appendix G)

 
$\displaystyle \Gamma_{ij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }) = \fra...
...boldmath$ y $ }})_{j}R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }}) \right),$     (3.49)

where FSij,n,m and GSi,n,m are functions defined as
  
$\displaystyle F^S_{ij,n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})$ = $\displaystyle \frac{\lambda+3\mu}{\lambda+2\mu}\delta_{ij}S_{n,m}
(\overrightar...
...\frac{\partial}{\partial x_i}S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }}),$ (3.50)
$\displaystyle G^S_{i,n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})$ = $\displaystyle \frac{\lambda+\mu}{\lambda+2\mu}
\frac{\partial}{\partial x_i}S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }}).$ (3.51)

We now compute the integral on the right-hand side of (3.46) over a subset of S denoted by Sy for $\mbox{\boldmath$\space x $ }$ which is away from Sy. Using (3.49), we obtain
 
$\displaystyle { \int_{S_y} \frac{\partial}{\partial x_k}
\frac{\partial}{\parti...
... $ }) C_{cdjl} n_c(\mbox{\boldmath$ y $ })\phi_d(\mbox{\boldmath$ y $ }) dS_y }$
    $\displaystyle = \frac{1}{8\pi\mu} \sum_{n=0}^{\infty} \sum_{m=-n}^{n}
\left( \f...
...line{G^S_{i,n,m}}(\overrightarrow{O\mbox{\boldmath$ x $ }})M^2_{n,m}(O)\right),$ (3.52)

where M1j,n,m(O) and M2n,m(O) are the multipole moments centred at O, expressed as
  
M1j,n,m(O) = $\displaystyle \int_{S_y} C_{cdjl} \frac{\partial}{\partial y_l}
R_{n,m}(\overri...
...th$ y $ }}) \phi_d(\mbox{\boldmath$ y $ }) n_c(\mbox{\boldmath$ y $ }) dS_{y} ,$ (3.53)
M2n,m(O) = $\displaystyle \int_{S_y} C_{cdjl} \frac{\partial}{\partial y_l}
((\overrightarr...
...ath$ y $ }}))\phi_d(\mbox{\boldmath$ y $ }) n_c(\mbox{\boldmath$ y $ }) dS_{y}.$ (3.54)

Notice that M1j,n,m(O) has three components and M2n,m(O) has one component for a given pair of n and m, and therefore the number of multipole moments in this formulation is 4. Fu et al.[18] applied FM-BIEM to three-dimensional elastostatics and used 4 and 12 multipole moments for the single- and double-layer potentials in their formulation, whereas in our formulation we use 4 multipole moments for ordinary problems where the single- and double-layer potentials are used as well as for crack problems (details of formulations for the single- and double-layer potentials are given later). It is seen that the number multipole moments in our formulation is related to the fact that in the Neuber-Papkovich representation the displacement field is expressed in terms of four harmonics functions. Also, Fukui and Kutsumi[27] use 4 moment formulation for ordinary problems.

Noting (A.5) and the fact that $\phi_i$ is real-valued one can find that the multipole moments have the following properties:

  
$\displaystyle M^1_{j,n,-m}(O) = (-1)^m \overline{{M}^1_{j,n,m}}(O)
\quad (m \ge 0),$     (3.55)
$\displaystyle M^2_{n,-m}(O) = (-1)^m \overline{{M}^2_{n,m}}(O)
\quad (m \ge 0).$     (3.56)

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

  
M1j,n,m(O') = $\displaystyle \sum_{n'=0}^{n}\sum_{m'=-n'}^{n'}
R_{n',m'}(\overrightarrow{O'O})M^{1}_{j,n-n',m-m'}(O),$ (3.57)
M2n,m(O') = $\displaystyle \sum_{n'=0}^{n}\sum_{m'=-n'}^{n'} R_{n',m'}(\overrightarrow{O'O})...
...gl(M^2_{n-n',m-m'}(O) - (\overrightarrow{OO'})_{j} M^1_{j,n-n',m-m'}(O) \bigr),$ (3.58)

where we have used (3.21), (3.53) and (3.54) (See Appendix H.1). The procedures given by (3.57) and (3.58) are called ``M2M translation''.

In the evaluation of the integral on the right-hand side of (3.46) one can use not only the multipole moments but also the coefficients of local expansion in the following manner:

 
$\displaystyle { \int_{S_y} \frac{\partial}{\partial x_k}
\frac{\partial}{\parti...
... $ }) C_{cdjl} n_c(\mbox{\boldmath$ y $ })\phi_d(\mbox{\boldmath$ y $ }) dS_y }$
    $\displaystyle = \frac{1}{8\pi\mu} \sum_{n=0}^{\infty} \sum_{m=-n}^{n}
\left( \f...
...th$ x $ }_0\mbox{\boldmath$ x $ }})
L^2_{n,m}(\mbox{\boldmath$ x $ }_0)\right),$ (3.59)

where $L^1_{j,n,m}(\mbox{\boldmath$\space x $ }_0)$ and $L^2_{n,m}(\mbox{\boldmath$\space x $ }_0)$ are the coefficients of the local expansion centred at $\mbox{\boldmath$\space x $ }_0$ and are expressed with M1j,n,m(O) and M2n,m(O) by
  
$\displaystyle L^1_{j,n',m'}(\mbox{\boldmath$ x $ }_{0})$ = $\displaystyle \sum_{n=0}^{\infty}\sum_{m=-n}^{n}(-1)^{n'}
\overline{S_{n+n',m+m'}}(\overrightarrow{O\mbox{\boldmath$ x $ }_0})M^1_{j,n,m}(O),$ (3.60)
$\displaystyle L^2_{n',m'}(\mbox{\boldmath$ x $ }_{0})$ = $\displaystyle \sum_{n=0}^{\infty}\sum_{m=-n}^{n}(-1)^{n'}
\overline{S_{n+n',m+m...
...(M^2_{n,m}(O)-(\overrightarrow{O\mbox{\boldmath$ x $ }_0})_{j} M^1_{j,n,m}(O)).$ (3.61)

Also, FRij,n,m and GRi,n,m are functions obtained by replacing Sn,m with Rn,m in (3.50) and (3.51). In these formulae 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 H.2). The procedures given by (3.60) and (3.61) are called ``M2L translation''. Notice that $L^1_{j,n,m}(\mbox{\boldmath$\space x $ }_0)$ has three components and $L^2_{n,m}(\mbox{\boldmath$\space x $ }_0)$ has one component for a given pair of n and m, and therefore the number of the coefficients of the local expansion in this formulation is 4. Also, $L^1_{j,n,m}(\mbox{\boldmath$\space x $ }_0)$ and $L^2_{n,m}(\mbox{\boldmath$\space x $ }_0)$ have the following properties:
  
$\displaystyle L^1_{j,n,-m}(\mbox{\boldmath$ x $ }_0) = (-1)^m \overline{L^1_{j,n,m}}(\mbox{\boldmath$ x $ }_0)
\quad (m \ge 0),$     (3.62)
$\displaystyle L^2_{n,-m}(\mbox{\boldmath$ x $ }_0) = (-1)^m \overline{L^2_{n,m}}(\mbox{\boldmath$ x $ }_0)
\quad (m \ge 0).$     (3.63)

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

  
$\displaystyle L^1_{j,n'',m''}(\mbox{\boldmath$ x $ }_1)$ = $\displaystyle \sum_{n'=n''}^{\infty}\sum_{m'=-n'}^{n'}
R_{n'-n'',m'-m''}(\overr...
...ath$ x $ }_0\mbox{\boldmath$ x $ }_1})
L^1_{j,n',m'}(\mbox{\boldmath$ x $ }_0),$ (3.64)
$\displaystyle L^2_{n'',m''}(\mbox{\boldmath$ x $ }_1)$ = $\displaystyle \sum_{n'=n''}^{\infty}\sum_{m'=-n'}^{n'}
R_{n'-n'',m'-m''}(\overr...
...0\mbox{\boldmath$ x $ }_1})_{p}L^1_{p,n',m'}(\mbox{\boldmath$ x $ }_0)
\right),$ (3.65)

where we have used (3.21) and (3.59) (See Appendix H.3). The procedures given by (3.64) and (3.65) are 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