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One penny-shaped crack

To begin with we consider a penny-shaped crack having the radius of a0 and the unit normal vector of $\mbox{\boldmath$\space n $ }=(0,0,1)$. The asymptotic field $u^{\infty}(\mbox{\boldmath$\space x $ })$ is given by $u^{\infty}(\mbox{\boldmath$\space x $ })= u_0 x_3 / a_0$. This problem is solved with the conventional BIEM, FM-BIEM with the original FMM and with FM-BIEM with the new version of FMM. The numerical solutions obtained with these methods were found to be essentially identical. Fig.4.3 shows the 1912 DOF mesh and Fig.4.4 plots the non-dimensional crack opening displacement ($\phi / u_0$) obtained with this mesh. In Fig.4.4 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. Fig.4.4 shows good agreement between numerical results. Fig.4.5 plots the total CPU time (sec) vs the number of unknowns. In Fig.4.5 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 slightly faster than the original FM-BIEM. This small improvement is to be expected since the computational cost for M2L is not dominant in the single crack case, where the cell arrangement is essentially two-dimensional. Hence the improvement of the efficiency achieved with the new FM-BIEM in this particular numerical example may not necessarily be very typical. Therefore we consider many crack problems in the next example to show the efficiency of the new approach more clearly.


next up previous contents
Next: Many penny-shaped cracks Up: Numerical examples Previous: Numerical examples
Ken-ichi Yoshida
2001-07-28