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Next: Conclusions Up: Numerical Examples Previous: One crack

Many 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 $12\times 12\times
12(=1728)$ penny-shaped cracks (total DOF=1,285,632) in the infinite domain. Fig.4 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$. Notice that the originally flat cracks appear curved since the crack opening displacements have been superposed. The required CPU times with FM-BIEM and the new FM-BIEM are 13,954(sec) and 8,290(sec), respectively. In this example the error defined as $\mbox{error} = \vert\vert\widetilde{\mbox{\boldmath$\space \phi $ }}-{\mbox{\bo...
...space \phi $ }}\vert\vert/\vert\vert{\mbox{\boldmath$\space \phi $ }}\vert\vert$ 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. Next we consider an array of $8\times 8\times 8(=512)$ penny-shaped cracks in the infinite domain. Fig.5 plots the total CPU time (sec) required by the original FM-BIEM and the new FM-BIEM with this array. In Fig.5 the lines marked `Tfmm' and `Tfmmnew' indicate the CPU times required with the original FM-BIEM and the new FM-BIEM, respectively. These results show that the new FM-BIEM is more efficient than the original FM-BIEM when the distribution of the boundary elements is more three-dimensional.

  
Figure 1: crack mesh (DOF=5736)
\begin{figure}
\begin{center}
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\epsfile{file=FIG/mesh.eps,scale=0.8}\end{center}\end{figure}


  
Figure 2: crack opening displacement of one penny-shaped crack
\begin{figure}
\begin{center}
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\epsfile{file=FIG/open.eps,scale=0.8}\end{center}\end{figure}

 



  
Figure 3: CPU time (sec) for the numerical exmaples with one penny-shaped crack
\begin{figure}
\begin{center}
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\epsfile{file=FIG/time.eps,scale=0.8}\end{center}\end{figure}

 

  
Figure: crack opening displacement(DOF=1,285,632) of an array of $12\times 12\times 12$ penny-shaped cracks
\begin{figure}
\begin{center}
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\epsfile{file=FIG/uge.ps,scale=0.5}\end{center}\end{figure}


  
Figure: CPU time (sec) for the numerical exmaples with an array of $8\times 8\times 8$ penny-shaped cracks
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/time_multi.eps,scale=0.8}\end{center}\end{figure}


next up previous
Next: Conclusions Up: Numerical Examples Previous: One crack
Ken-ichi Yoshida
2001-03-26