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Numerical examples

The proposed techniques have been implemented in Fortran 77, and tested on a computer having a DEC Alpha 21164(600 MHz) chip as the CPU.

In FMM, we truncate the infinite series in (3.105), (3.106), (3.109) and (3.110) taking 10 terms, or 264 moments, into consideration. Also, we set the maximum number of boundary elements in a leaf to be 60, since this number turned out to be the most efficient choice in a separate parameter study. As regards the integration, we calculate the inner integrals on the right-hand side of (3.104) and those on the right-hand side of (3.108) analytically when direct evaluation is needed. The outer integrals in the right-hand side of (3.104) and the integrals in (3.53), (3.54) and (3.106) are computed numerically with Gaussian quadrature. Notice that the definition of the belonging of a global boundary element to a cell in the second step in 3.4.3 makes this calculation easy. Indeed this definition guarantees that a global boundary element is never divided between two or more cells. Because of this we never have to deal with discontinuities of the shape functions in the direct evaluation of the integrals in (3.112), thus making the use of (3.104) possible. If a global boundary element with a basis function $N^J(\mbox{\boldmath$\space x $ })$had been divided between adjacent cells C and C'in (3.112), however, one would have had to deal with an integral of the form

\begin{displaymath}\int_{S\cap C} N^J(\mbox{\boldmath$ x $ }) \
\pfint_{S\cap ...
...ath$ y $ }) N^J(\mbox{\boldmath$ y $ }) dS_{y} dS_x \phi_d^J.
\end{displaymath}

which is actually divergent since the inner integral includes a singularity proportional to the reciprocal of the distance from the line of discontinuity of $N^J(\mbox{\boldmath$\space x $ })$.

In the present implementation of the FM-GBIEM the basis functions are chosen to be piecewise linear. With this choice the square root behaviour of the crack opening displacement near the tip is not taken into account. There would be, however, no essential difficulty in including this effect into the analysis.

To solve the discretised integral equation we use the preconditioned GMRES. As the preconditioner, we use the block diagonal matrix corresponding to the leaves, following the technique proposed by Nishida and Hayami[62]. In GMRES we terminate the iteration when the relative error is less than 10-5. In all the examples to follow this convergence condition was met after less than 20 iterations.



 
next up previous contents
Next: One penny-shaped crack Up: Crack problems for three-dimensional Previous: Algorithm for computing stress
Ken-ichi Yoshida
2001-07-28