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Next: Comparison of formulations for Up: Three-dimensional scattering of elastic Previous: Algorithm

Numerical examples

The proposed techniques have been implemented in Fortran 77, and have been tested on a computer having a DEC Alpha 21264(500 MHz) chip as the CPU. In FMM, we truncate the infinite series via (3.133) with k=kT. Also, we set the maximum number of boundary elements in a leaf to be 100. The integrals in (3.147) and (3.148) are computed numerically with Gaussian quadrature. In GMRES we terminate the iteration when the relative error is less than 10-5. To solve the resulting matrix equation we use the preconditioned GMRES and adopt the block diagonal matrix corresponding to the leaves as the preconditioner according to Nishida and Hayami[62].

We consider an infinite space which contains one penny-shaped crack having the radius of a0 and the unit normal vector of $\mbox{\boldmath$\space n $ }=(0,0,1)$. We compute the crack opening displacement when the crack is subject to a plane longitudinal wave of normal incidence from -x3 direction. The stress magnitude of the incident wave is p0. Also, Poisson's ratio is 0.25. This problem is solved with the conventional BIEM and FM-BIEM. Fig.3.52, Fig.3.54 and Fig.3.56 plot the total CPU time (sec) vs the number of unknowns when kTa0=1.4,3.2,4.4. In these figures lines marked ``Tdir'' and ``Tfmm'' indicate the CPU time required with the conventional BIEM and FM-BIEM, respectively. This figure shows that the FM-BIEM is faster than the conventional BIEM when the number of unknowns is larger than several thousands. Fig.3.53, Fig.3.55 and Fig.3.57 show the crack opening displacement $\vert\phi\vert /
\phi_0$ where $\phi_0$ is the static opening displacement at the centre of the crack and $\phi_0=3p_0a_0/\pi\mu$. In these figures lines marked ``conv'' and ``fmm'' indicate the crack opening displacements obtained with the conventional BIEM and FM-BIEM, respectively. In Fig.3.58 ``numer'' indicates the crack opening displacement $\vert\phi\vert /
\phi_0$ obtained with FM-BIEM numerically, and ``analytic'' stands for analytical solutions (Mal [56]). In Fig.3.59 the lines marked ``Elastostatics'' and ``Elastodynamics'' show the computational times required with FM-BIEM for elastostatics and elastodynamics ( kTa0=1.4), respectively. As in the case of Helmholtz's equation the computational time required by an elastodynamic analysis is longer than that required with an elastostatic one.


  
Figure 3.52: Total CPU time (sec) ( kT a0=1.4)
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ED3D/hikaku_ka=1.4.eps,scale=1.0}\end{center}\end{figure}


  
Figure 3.53: Crack opening displacement ( kT a0=1.4)
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ED3D/disp_ka=1.4.eps,scale=1.0}\end{center}\end{figure}


  
Figure 3.54: Total CPU time (sec) ( kT a0=3.2)
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ED3D/hikaku_ka=3.2.eps,scale=1.0}\end{center}\end{figure}


  
Figure 3.55: Crack opening displacement ( kT a0=3.2)
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ED3D/disp_ka=3.2.eps,scale=1.0}\end{center}\end{figure}


  
Figure 3.56: Total CPU time (sec) ( kT a0=4.4)
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ED3D/hikaku_ka=4.4.eps,scale=1.0}\end{center}\end{figure}


  
Figure 3.57: Crack opening displacement ( kT a0=4.4)
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ED3D/disp_ka=4.4.eps,scale=1.0}\end{center}\end{figure}


  
Figure 3.58: Numerical result and analytical solution ( kTa0=3.2)
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ED3D/analytic/hikaku_disp.eps,scale=1.0}\end{center}\end{figure}


  
Figure 3.59: Comparison of computational times for elastostatics and elastodynamics ( kTa0=1.4)
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/TEST/es3d-ed3d.eps,scale=1}\end{center}\end{figure}


next up previous contents
Next: Comparison of formulations for Up: Three-dimensional scattering of elastic Previous: Algorithm
Ken-ichi Yoshida
2001-07-28