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Next: Concluding remarks Up: Numerical examples Previous: One penny-shaped crack

Many penny-shaped cracks

We now consider an infinite space which contains an array of penny-shaped cracks, each having the same radius a0, subjected to the same asymptotic condition as in the previous example. The centroids of these cracks are located at the same interval of 4a0 in each coordinate direction, but the direction of each crack is taken random. First, we consider an array of $8\times
8\times 8(=512)$ penny-shaped cracks in the infinite domain. Fig.4.10 plots the total CPU time (sec) required by the original FM-BIEM and the new FM-BIEM with this array. In Fig.4.10 the lines marked `Tfmm' and `Tfmmnew' indicate the CPU times required with the original FM-BIEM and the new FM-BIEM, respectively. Next we consider an infinite space which contains an array of $12\times 12\times 12(=1728)$ penny-shaped cracks (total DOF=1,285,632) in the infinite domain. Fig.4.11 plots the non-dimensional crack opening displacement ( $\mu \mbox{\boldmath$\space \phi $ }/a_0 p_0$) on the non-dimensional mesh $\mbox{\boldmath$\space x $ }/a_0$. The required CPU times with the original FM-BIEM and the new FM-BIEM are 13954(sec) and 8290(sec), respectively. In this example the 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*}


is $9.09 \times 10^{-4}$, where $\widetilde{\mbox{\boldmath$\space \phi $ }}$ is the numerical solution obtained with the new FM-BIEM, $\mbox{\boldmath$\space \phi $ }$ the one obtained with the original FM-BIEM and $\vert\vert\cdot\vert\vert$ denotes the L2-norm. These results show that the new FM-BIEM is more efficient than the original FM-BIEM when distribution of the boundary elements is dense in the domain.










  
Figure 4.8: Crack opening displacement
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/NEWFMM:ES3D/open.eps,scale=1.0}\end{center}\end{figure}


  
Figure 4.9: CPU time (sec)
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/NEWFMM:ES3D/time.eps,scale=1.0}\end{center}\end{figure}


  
Figure 4.10: CPU time (sec)
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/NEWFMM:ES3D/time_multi.eps,scale=1.0}\end{center}\end{figure}


  
Figure 4.11: Crack opening displacement (DOF=1,285,632)
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/NEWFMM:ES3D/uge.ps,scale=0.8}\end{center}\end{figure}


next up previous contents
Next: Concluding remarks Up: Numerical examples Previous: One penny-shaped crack
Ken-ichi Yoshida
2001-07-28