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In the beginning we consider an infinite space which contains a
penny-shaped crack having the radius of *a*_{0} and the unit normal vector
of
.
The function
is given
by
where
and
(otherwise)
and, hence,
.
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.
.
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
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:** Many cracks
** Up:** Numerical Examples
** Previous:** Numerical Examples
*Ken-ichi Yoshida*

*2001-03-26*