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One penny-shaped crack

To begin with we consider a simple problem where the exact solution is available. Namely, we consider a penny-shaped crack having the radius of a0 opened by a uniform remote normal tension having the magnitude of p0. Taking the 3rd axis of a Cartesian coordinate system into the direction of the unit normal vector $\mbox{\boldmath$\space n $ }=(0,0,1)$ on S, we write the boundary condition in the following form: $\mbox{\boldmath$\space t $ }^{\infty}=(0,0,p_0)$ on S. The Lame constants are chosen to be $\lambda=\mu$, hence Poisson's ratio is 1/4.

Fig.3.29 shows the 471 DOF mesh and the bird's-eye view of the non-dimensional crack opening displacement $\mu\phi_3/(a_0
p_0)$ obtained with FM-GBIEM using this mesh. Fig.3.30 shows the crack opening displacement vs the distance from the centre of the crack. In this figure the line and symbols marked ``galerkin'', ``collocation'' and ``analytic'' indicate numerical results obtained with FM-GBIEM, those obtained with FM-BIEM with collocation and the analytic solution (Green and Zerna [32]), respectively. The BIE results are obtained with the mesh shown in Fig.3.29, and the basis functions are piecewise linear in FM-GBIEM and piecewise constant in collocation. This figure shows the high accuracy of the FM-GBIEM. Also noteworthy is the poor quality of the collocation results. This inaccuracy is due to the combination of the collocation and the piecewise constant basis functions, but not to FMM. This is because the collocation FMM results were essentially identical with those of the conventional collocation BIEM obtained with the same mesh and basis functions. Fig.3.31 compares the normal stress ( $\sigma_{33}$) near the crack tip obtained with FM-GBIEM using a 471 DOF mesh, the same stress obtained with a 981 DOF mesh, and the analytical solution. Here again the agreement is satisfactory. We expect further improvement of the results near the tip with basis functions having the square root behaviour since this introduces the correct singularity in the stress at the tip while the piecewise linear basis function leads to a weaker logarithmic singularity. Fig.3.32 plots the total CPU time (sec) vs the number of unknowns N. In Fig.3.32 the solid (broken) line marked Tfmm (Tdir) indicates the CPU time required with FM-GBIEM (conventional GBIEM). We use Crout's method as the solver for conventional GBIEM. We did not compare FM-GBIEM and conventional collocation BIEM results in this figure because the conventional collocation BIEM will not work with piecewise linear base functions since the C1 continuity of the base function does not hold, and because the accuracies of the Galerkin and collocation BIEMs are not identical with the same number of unknowns. This figure shows that FM-GBIEM is faster than the conventional GBIEM when N is more than about 1200. Fig.3.33 shows the CPU time per iteration in GMRES vs N. The slope of this graph is seen to approach the theoretical value of 1 as N increases.


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