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One penny-shaped crack

In the beginning we consider an infinite space which contains a penny-shaped crack having the radius of a0 and the unit normal vector of $\mbox{\boldmath$\space n $ }=(0,0,1)$. The function $\mbox{\boldmath$\space t $ }^{\infty}(\mbox{\boldmath$\space x $ })$ is given by $\mbox{\boldmath$\space t $ }^{\infty}(\mbox{\boldmath$\space x $ })=\mbox{\bold...
...oldmath$\space x $ })\mbox{\boldmath$\space n $ }(\mbox{\boldmath$\space x $ })$ where
 
$\displaystyle \mbox{\boldmath$ \sigma $ }^{\infty}(\mbox{\boldmath$ x $ }) =
\l...
...
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & p_0
\end{array}\right),$     (4.68)

and, hence, $\mbox{\boldmath$\space t $ }^{\infty}=(0,0,p_0)$. The asymptotic condition (4.68) indicates that the domain is subjected to a uniform uniaxial tension. Also, Poisson's ratio is set to be 0.25; i.e. $\lambda=\mu$. This problem is solved with the conventional BIEM, the original FM-BIEM and the new FM-BIEM. Fig.4.8 plots the non-dimensional crack opening displacement $\mu \phi_3 / a_0 p_0$ obtained with this mesh. In Fig.4.8 the symbols marked ``conv'', ``fmm'' and ``newfmm'' indicate numerical results computed with the conventional BIEM, the original FM-BIEM and the new FM-BIEM, respectively. This figure shows a good agreement between these numerical results. Fig.4.9 plots the total CPU time (sec) vs the number of unknowns. In Fig.4.9 the lines marked ``Tdir'', ``Tfmm'' and ``Tfmmnew'' indicate the CPU time required with the conventional BIEM, the original FM-BIEM and the new FM-BIEM, respectively. This figure shows that the new FM-BIEM is only slightly faster than the original FM-BIEM. As in the Laplace case, this is because this example is essentially a two-dimensional one where the computational cost for the M2L translation is not dominant. In order to show the efficiency of the new FMM more clearly we need to consider an example where boundary elements are distributed three-dimensionally. Therefore we consider many crack problems in the next example.
next up previous contents
Next: Many penny-shaped cracks Up: Numerical examples Previous: Numerical examples
Ken-ichi Yoshida
2001-07-28