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To begin with we consider an infinite space which contains one
pennyshaped crack having the radius of a_{0} and the unit normal vector
of
.
We compute the crack opening displacement subject
to a plane wave of normal incidence. The incident wave field u_{I} is
given by
Figure 3.40:
Scattering problem

In this example we call the side where the incident wave reaches directly
the ``bright side ()'' and the other side the ``shadow side (+)'' (See
Fig.3.40). This problem is solved with the conventional BIEM
and FMBIEM when
ka_{0}=1,2,3,4,5. In GMRES we use the block diagonal
matrix corresponding to the leaves as the preconditioner according to
Nishida and Hayami[62]. Fig.3.41 plots the total CPU
time (sec) vs the number of unknowns. In Fig.3.41 the lines
marked ``Tdir_ka=5'' and ``Tfmm_ka=1,2,3,4,5'' indicate the CPU time
required with the conventional BIEM when ka_{0}=5 and FMBIEM when
ka_{0}=1,2,3,4,5, respectively. Because the computational times required
with the conventional BIEM are almost identical when
ka_{0}=1,2,3,4,5 we
plot only the ``Tdir_ka=5'' result in Fig.3.41. This figure
shows that the FMBIEM is faster than the conventional BIEM when the
number of unknowns is larger than several
thousands. Fig.3.42 shows the CPU time per iteration. In
Fig.3.42 the lines marked ``Tfmm_per_itr_ka=1,2,3,4,5''
show the CPU time per iteration when
ka_{0}=1,2,3,4,5. Fig.3.43,
Fig.3.45 and Fig.3.47 show the crack opening
displacement
when
ka_{0}=1,2,3. In these figures ``conv''
and ``fmm'' indicate the numerical solutions with the conventional BIEM
and FMBIEM, respectively. Fig.3.44, Fig.3.46 and
Fig.3.48 show u on both sides and the analytical solution
(Leitner[53]) when
ka_{0}=1,2,3. In these figures
``bside_analy'' and ``bside_numer'' denote the analytical solution and
the numerical one on the bright side and ``sside_analy'' and
``sside_numer'' denote the analytical solution and the numerical one on
the shadow side. In Fig.3.49 the lines marked ``Laplace''
and ``Helmholtz'' show the computational times required with FMBIEM for
Laplace's equation and Helmholtz's equation (ka_{0}=1),
respectively. We note that these computational times are obtained on a DEC
Alpha 21264(600MHz). This figure shows that the computational time
required with Helmholtz's equation is slower than that required with
Laplace's equation.
Next: Many pennyshaped cracks
Up: Numerical examples
Previous: Numerical examples
Kenichi Yoshida
20010728