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One 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 $\sigma_{33}=p_0$ and $\sigma_{ij}=0$ (otherwise) and, hence, $\mbox{\boldmath$\space t $ }^{\infty}=(0,0,p_0)$. This asymptotic condition 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 (Fast Multipole-BIEM) and the new FM-BIEM. The numerical results obtained with the original FM-BIEM and the new FM-BIEM should be identical with those obtained with the conventional BIEM if calculations were carried out without errors caused by truncations of the infinite series. Fig.1 shows the 5736 DOF mesh and Fig.2 plots the non-dimensional crack opening displacement $\mu \phi_3 /a_0 p_0$ obtained with this mesh. In Fig.2 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.2 shows good agreement in numerical results. Fig.3 plots the total CPU time (sec) vs the number of unknowns. In Fig.3 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. 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
Next: Many cracks Up: Numerical Examples Previous: Numerical Examples
Ken-ichi Yoshida
2001-03-26