Next: Many pennyshaped cracks
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To begin with we consider a pennyshaped crack having the radius of
a_{0} and the unit normal vector of
.
The function
is given by
where



(3.85) 
and, hence,
.
The asymptotic condition
(3.85) indicates that the domain is subjected to a
uniform uniaxial tension. Also, Poisson's ratio is set to be
0.25. This problem is solved with the conventional BIEM, FMBIEM 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
 FMBIEM with hypersingular integral equation
Compute multipole moments (3.53) and
(3.54) numerically with 1 point Gaussian
quadrature
 NE3
 FMBIEM with hypersingular integral equation
Compute multipole moments (3.53) and
(3.54) numerically with 3 point Gaussian
quadrature
 NE4
 FMBIEM with regularised integral equation:
Fig.3.17 plots the total CPU time (sec) required with
numerical experiments NE1, NE2, NE3 and NE4, respectively vs the
number of unknowns. This figure shows that the FMBIEM is faster
than the conventional BIEM when the number of unknowns is larger
than several thousands. Fig.3.18 shows the CPU time per
iteration (sec). Fig.3.19 shows ratios of the
computational times required with FMBIEM (NE2,NE3,NE4) to the
computational time required with the conventional BIEM
(NE1). Fig.3.20 shows the L_{2}norm error defined as
where
denotes the L_{2}norm,
is the
corresponding numerical results obtained with FMBIEM (NE2,NE3,NE4)
and
is the numerical result obtained with conventional BIEM,
respectively. Fig.3.21, Fig.3.22 and
Fig.3.23 show the nondimensional 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
(Green and Zerna [32]). Also, Fig.3.24 shows
the crack opening displacement when the number of unknowns is 2664,
5736 and 28776 respectively.
The results show that the hypersingular formulation is more efficient
than the regularised one. As in the case of Laplace's equation, the
regularised formulation is inferior to the nonregularised
one in both computational time and precision. Considering results
obtained through numerical experiments in Laplace's equation and
elastostatics, we can see that the nonregularised formulation is
suitable for FMBIEM.
Next: Many pennyshaped cracks
Up: Numerical examples
Previous: Numerical examples
Kenichi Yoshida
20010728