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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 asymptotic field $u^{\infty}(\mbox{\boldmath$\space x $ })$ is given by $u^{\infty}(\mbox{\boldmath$\space x $ })= u_0 x_3 / a_0$. 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 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 result obtained with FM-BIEM (NE2,NE3,NE4) and $\phi$ 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 $\phi / u_0$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 up previous contents
Next: Many penny-shaped cracks Up: Numerical examples Previous: Numerical examples
Ken-ichi Yoshida
2001-07-28