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One penny-shaped crack

To begin with we consider a penny-shaped crack having the radius of a0 and the unit normal vector of $\mbox{\boldmath$\space n $ }=(0,0,1)$. The function $\mbox{\boldmath$\space t $ }^{\infty}(\mbox{\boldmath$\space x $ })$ is given by $\mbox{\boldmath$\space t $ }^{\infty}(\mbox{\boldmath$\space x $ })=\mbox{\bold...
...oldmath$\space x $ })\mbox{\boldmath$\space n $ }(\mbox{\boldmath$\space x $ })$ where
 
$\displaystyle \mbox{\boldmath$ \sigma $ }^{\infty}(\mbox{\boldmath$ x $ }) =
\l...
...
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & p_0
\end{array}\right),$     (3.85)

and, hence, $\mbox{\boldmath$\space t $ }^{\infty}=(0,0,p_0)$. 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, 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 hypersingular integral equation Compute multipole moments (3.53) and (3.54) numerically with 1 point Gaussian quadrature
NE3
FM-BIEM with hypersingular integral equation Compute multipole moments (3.53) and (3.54) numerically with 3 point Gaussian quadrature

NE4
FM-BIEM 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 FM-BIEM 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 FM-BIEM (NE2,NE3,NE4) to the computational time required with the conventional BIEM (NE1). Fig.3.20 shows the L2-norm error defined as

\begin{eqnarray*}\mbox{error} = \frac{\vert\vert\widetilde{\mbox{\boldmath$ \phi...
... }}\vert\vert}{\vert\vert{\mbox{\boldmath$ \phi $ }}\vert\vert}
\end{eqnarray*}


where $\vert\vert\cdot\vert\vert$ denotes the L2-norm, $\widetilde\phi$ is the corresponding numerical results obtained with FM-BIEM (NE2,NE3,NE4) and $\phi$ is the numerical result obtained with conventional BIEM, respectively. Fig.3.21, Fig.3.22 and Fig.3.23 show the non-dimensional crack opening displacement $\mu \phi_3 / a_0 p_0$ 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 non-regularised one in both computational time and precision. Considering results obtained through numerical experiments in Laplace's equation and elastostatics, we can see that the non-regularised formulation is suitable for FM-BIEM.


next up previous contents
Next: Many penny-shaped cracks Up: Numerical examples Previous: Numerical examples
Ken-ichi Yoshida
2001-07-28