To begin with we consider a simple problem where the exact solution is
available. Namely, we consider a penny-shaped crack having the radius
of *a*_{0} opened by a uniform remote normal tension having the
magnitude of *p*_{0}. Taking the 3rd axis of a Cartesian coordinate
system into the direction of the unit normal vector
on *S*, we write the boundary condition in the
following form:
on *S*. The Lame
constants are chosen to be
,
hence Poisson's ratio is
1/4.

Fig.3.29 shows the 471 DOF mesh and the bird's-eye view of
the non-dimensional crack opening displacement
obtained with FM-GBIEM using this mesh. Fig.3.30 shows
the crack opening displacement vs the distance from the centre of the
crack. In this figure the line and symbols marked ``galerkin'',
``collocation'' and ``analytic'' indicate numerical results obtained
with FM-GBIEM, those obtained with FM-BIEM with collocation and the
analytic solution (Green and Zerna [32]), respectively. The BIE
results are obtained with the mesh shown in Fig.3.29, and the basis
functions are piecewise linear in FM-GBIEM and piecewise constant in
collocation. This figure shows the high accuracy of the FM-GBIEM. Also
noteworthy is the poor quality of the collocation results. This
inaccuracy is due to the combination of the collocation and the
piecewise constant basis functions, but not to FMM. This is because the
collocation FMM results were essentially identical with those of the
conventional collocation BIEM obtained with the same mesh and basis
functions. Fig.3.31 compares the normal stress (
)
near the
crack tip obtained with FM-GBIEM using a 471 DOF mesh, the same stress
obtained with a 981 DOF mesh, and the analytical solution. Here again
the agreement is satisfactory. We expect further improvement of the
results near the tip with basis functions having the square root
behaviour since this introduces the correct singularity in the stress at
the tip while the piecewise linear basis function leads to a weaker
logarithmic singularity. Fig.3.32 plots the total CPU time (sec) vs
the number of unknowns *N*. In Fig.3.32 the solid (broken) line
marked Tfmm (Tdir) indicates the CPU time required with FM-GBIEM
(conventional GBIEM). We use Crout's method as the solver for
conventional GBIEM. We did not compare FM-GBIEM and conventional
collocation BIEM results in this figure because the conventional
collocation BIEM will not work with piecewise linear base functions
since the *C*^{1} continuity of the base function does not hold, and
because the accuracies of the Galerkin and collocation BIEMs are not
identical with the same number of unknowns. This figure shows that
FM-GBIEM is faster than the conventional GBIEM when *N* is more than
about 1200. Fig.3.33 shows the CPU time per iteration in GMRES vs
*N*. The slope of this graph is seen to approach the theoretical value
of 1 as *N* increases.