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FM-GBIEM

This section prepares formulae needed for FM-GBIEM in elastostatics.

We now compute a part of the double integral in the right-hand side of (3.100). Namely, we consider the contribution to this integral from $(\mbox{\boldmath$\space x $ },\mbox{\boldmath$\space y $ }) \in S_x \times S_y$, where Sx and Sy are disjoint parts of S. Using (3.49) and taking the origin O near Sy, we obtain

 
$\displaystyle {-\int_{S_x} \psi_a(\mbox{\boldmath$ x $ }) \int_{S_y} n_{b}(\mbo...
...) C_{cdjl} n_c(\mbox{\boldmath$ y $ })\phi_d(\mbox{\boldmath$ y $ }) dS_y dS_x}$
  = $\displaystyle -\frac{1}{8\pi\mu} \sum_{n=0}^{\infty} \sum_{m=-n}^{n}
\left( \in...
...{ij,n,m}}(\overrightarrow{O\mbox{\boldmath$ x $ }}) M^1_{j,n,m}(O) dS_x
\right.$  
    $\displaystyle \left. + \int_{S_x} \psi_a(\mbox{\boldmath$ x $ }) n_b(\mbox{\bol...
...S_{i,n,m}}(\overrightarrow{O\mbox{\boldmath$ x $ }}) M^2_{n,m}(O) dS_x \right),$ (3.105)

where 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 x $ }\in S_x$, $\mbox{\boldmath$\space y $ }\in S_y$) holds. Notice that the multipole moments M1j,n,m(O) and M2n,m(O) are identical with (3.53) and (3.54) and M2M translation formulae are given by (3.57) and (3.58).

With (3.105) the evaluation of the left-hand side of (3.100) is made efficient since the multipole moments are common to many evaluation points $\mbox{\boldmath$\space x $ }$ and can be reused. This efficiency is further enhanced by using the local expansion as

 
$\displaystyle {-\int_{S_x} \psi_a(\mbox{\boldmath$ x $ }) \int_{S_y} n_{b}(\mbo...
...) C_{cdjl} n_c(\mbox{\boldmath$ y $ })\phi_d(\mbox{\boldmath$ y $ }) dS_y dS_x}$
    $\displaystyle = -\frac{1}{8\pi\mu} \sum_{n=0}^{\infty} \sum_{m=-n}^{n}
\left( \...
... }_0\mbox{\boldmath$ x $ }}) L^1_{j,n,m}(\mbox{\boldmath$ x $ }_0) dS_x \right.$  
    $\displaystyle \left. + \int_{S_x} \psi_a(\mbox{\boldmath$ x $ }) n_b(\mbox{\bol...
...$ }_0\mbox{\boldmath$ x $ }}) L^2_{n,m}(\mbox{\boldmath$ x $ }_0) dS_x \right),$ (3.106)

where we have assumed that the inequality $\vert\overrightarrow{O\mbox{\boldmath$\space x $ }_0}\vert>
\vert\overrightarrow{\mbox{\boldmath$\space x $ }\mbox{\boldmath$ x $ }_0}\vert$ $(\mbox{\boldmath$\space x $ }\in S_x)$ holds. Notice that the coefficients of the local expansion $L^1_{j,n,m}(\mbox{\boldmath$\space x $ }_0)$ and $L^2_{n,m}(\mbox{\boldmath$\space x $ }_0)$ are the same as (3.60) and (3.61) and L2L translation formulae are given by (3.64) and (3.65).

One may be interested in obtaining stress and displacement distributions within the domain $D=R^3\setminus \overline{S}$ after the crack opening displacements are computed. In the rest of this section we shall prepare formulae needed for the fast computation of the stress components with the help of FMM. One may similarly compute displacements, but the detail is omitted here. It is immediate to obtain the following integral representation for the stress from (3.44):

 
$\displaystyle \sigma_{ab}(\mbox{\boldmath$ x $ }) = \sigma^{\infty}_{ab}(\mbox{...
...dmath$ y $ }) dS_{y}, \quad \mbox{\boldmath$ x $ }\in R^3\setminus \overline{S}$     (3.107)

where $\sigma_{ab}^{\infty}$ is the stress associated with $u_i^{\infty}$. The integral on the right-hand side of (3.107) is regularised with the help of the stress function in (3.101) as
 
$\displaystyle {\int_{S} C_{abik} \frac{\partial}{\partial
x_k}\frac{\partial}{\...
...C_{cd\/jl}
n_c(\mbox{\boldmath$ y $ }) \phi_{d}(\mbox{\boldmath$ y $ }) dS_{y}}$
  = $\displaystyle - \int_{S} e_{aip} e_{bjq} e_{ckr}
e_{dls} \frac{\partial}{\parti...
...\partial y_k}dS_{y}, \quad \mbox{\boldmath$ x $ }\in
R^3\setminus \overline{S},$ (3.108)

As a matter of fact, (3.108) holds identically for any continuous $\phi_i$ having support within S. We now compute the contribution to the integral on the right-hand side of (3.107) from Sy, where Sy is a part of S located sufficiently far from the evaluation point $\mbox{\boldmath$\space x $ }$. Using (3.49), (3.53) and (3.54), we rewrite the integral on the right-hand side of (3.107) as
 
$\displaystyle { \int_{S_y} C_{abik}
\frac{\partial}{\partial x_k}\frac{\partial...
...
n_c(\mbox{\boldmath$ y $ }) \phi_{d}(\mbox{\boldmath$ y $ }) dS_{y}} \nonumber$
    $\displaystyle = \frac{1}{8\pi\mu} \sum_{n=0}^{\infty} \sum_{m=-n}^{n}
\left( C_...
...ne{G^S_{i,n,m}}(\overrightarrow{O\mbox{\boldmath$ x $ }}) M^2_{n,m}(O) \right),$ (3.109)

where we have assumed that the origin is near Sy so 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. The local expansion of (3.109) around $\mbox{\boldmath$\space x $ }_0$ is written in terms of the coefficients of the local expansion in (3.60) and (3.61) as follows:
 
$\displaystyle { \int_{S_y} C_{abik}
\frac{\partial}{\partial x_k}\frac{\partial...
..._{cd\/jl}
n_c(\mbox{\boldmath$ y $ }) \phi_{d}(\mbox{\boldmath$ y $ }) dS_{y}} $
    $\displaystyle = \frac{1}{8\pi\mu} \sum_{n=0}^{\infty} \sum_{m=-n}^{n}
\left( C_...
...$ x $ }_0\mbox{\boldmath$ x $ }}) L^2_{n,m}(\mbox{\boldmath$ x $ }_0) \right) ,$ (3.110)

where we have assumed that the inequality $\vert\overrightarrow{O\mbox{\boldmath$\space x $ }_0}\vert>
\vert\overrightarrow{\mbox{\boldmath$\space x $ }\mbox{\boldmath$ x $ }_0}\vert$ holds.


next up previous contents
Next: Algorithm Up: Crack problems for three-dimensional Previous: Variational Boundary Integral Equation
Ken-ichi Yoshida
2001-07-28