To begin with we consider a penny-shaped crack having the radius of
*a*_{0} and the unit normal vector of
.
The asymptotic
field
is given by
.
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 ()
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.