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A note on integral equations

We restrict the discussion to the case of elastostatics. In elastostatics we mainly deal with (3.9) and efficiently compute $\mbox{\boldmath$\space \phi $ }$ in (3.9) with FMM. FMM gives us the method for efficient evaluation of integrals which contain products of kernel functions such as single- and double-layer kernels and density functions (e.g. displacement, traction). Now, we consider two integral representations (3.4) and (3.7) for example. Letting $\mbox{\boldmath$\space x $ }$ approach S in (3.4), one can deal with ordinary boundary value problems. Also, one can compute displacement fields with (3.7) if $\mbox{\boldmath$\space \phi $ }$ is given. Moreover, applying the operator $P_{ijk}(\mbox{\boldmath$\space x $ })$ defined as

\begin{eqnarray*}P_{ijk}(\mbox{\boldmath$ x $ }) := C_{ijkl} \frac{\partial}{\partialx_l},
\end{eqnarray*}


on both sides of (3.7), one obtains the integral representation for $\mbox{\boldmath$\space \sigma $ }$ given by
 
$\displaystyle \sigma_{ij}(\mbox{\boldmath$ x $ }) = \sigma^{\infty}_{ij}(\mbox{...
...$ y $ })
\phi_j(\mbox{\boldmath$ y $ }) dS , \quad \mbox{\boldmath$ x $ }\in D,$     (3.10)

Suppose that $\mbox{\boldmath$\space \phi $ }$ is given, one can compute stress fields with (3.10). One can see that the integrals in (3.4), (3.7) and (3.10) contain the product of the single-layer kernel (or the double-layer kernel) and the density function. Namely, if once one implements FM-BIEM for crack problems, one can simultaneously deal with ordinary boundary value problems, displacement fields and stress fields with FM-BIEM. In view of this the analysis of crack problems with FMM is considered to be more versatile than it appears, since techniques proposed in this thesis are not restricted to crack problems. Although we have restricted the discussion to elastostatics, the same is true in other governing equations. This is one of the reasons why we deal with crack problems in this thesis.


next up previous contents
Next: Crack problems for three-dimensional Up: Crack Previous: Integral equation for crack
Ken-ichi Yoshida
2001-07-28