** Next:** Many penny-shaped cracks
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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
hypersingular integral equation and the regularised integral equation.
In this example we carry out numerical experiments using the following
four schemes:
- NE1
- Conventional BIEM

- NE2
- FM-BIEM with the hypersingular integral equation
Compute multipole moments (3.23) numerically with 1
point Gaussian quadrature

- NE3
- FM-BIEM with the hypersingular integral equation
Compute multipole moments (3.23) numerically with 3
point Gaussian quadrature

- NE4
- FM-BIEM with the regularised integral equation

Fig.3.5 shows the 9592 DOF mesh. Fig.3.6 plots the
total CPU time (sec) required with numerical experiments NE1, NE2, NE3
and NE4 vs the number of unknowns. This figure shows that FM-BIEM is
faster than the conventional BIEM when the number of unknowns is larger
than several thousands. Fig.3.7 shows the CPU time per
iteration (sec) vs the number of unknowns. Fig.3.8 shows
ratios of the computational times required with FM-BIEM (NE2,NE3,NE4) to
the computational time required with the conventional BIEM
(NE1). Fig.3.9 shows the *L*_{2}-norm error defined as

where
denotes the *L*_{2}-norm,
is the corresponding
numerical result obtained with FM-BIEM (NE2,NE3,NE4) and
is
the numerical result obtained with the conventional BIEM (NE1),
respectively. Fig.3.10, Fig.3.11 and
Fig.3.12 show the crack opening displacement
obtained with numerical experiments when the number of unknowns is
1912. In these figures the line marked ``analytic'' indicates the
analytical solution (Chen and Huang[9]). Also,
Fig.3.13 shows the crack opening displacement when the
number of unknowns is 888, 4344 and 20728, respectively. The inaccuracy
of the numerical results is due to collocation method with piecewise
constant shape functions but not to FMM.
The numerical results show that the FM-BIEM with the hypersingular
integral equation is more efficient than that with the regularised
integral equation. Although in the regularised formulation we can
compute the multipole moments analytically, the results obtained with
the formulation without regularisation are more accurate than that with
regularisation. It seems that these unexpected results are due to
the fact that a canceling may occur in computation of the zero-valued
integral (See (L.3) in Appendix L.1) with
FMM which is used to regularise the hypersingular integral equation
(See Appendix L.1).

** Next:** Many penny-shaped cracks
** Up:** Numerical examples
** Previous:** Numerical examples
*Ken-ichi Yoshida*

*2001-07-28*