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One penny-shaped crack

To begin with 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 subject to a plane wave of normal incidence. The incident wave field uI is given by

\begin{eqnarray*}u_I(\mbox{\boldmath$ x $ }) = u_0 e^{ikz}, \quad u_0 : \rm {const}.
\end{eqnarray*}



  
Figure 3.40: Scattering problem
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\epsfile{file=FIG/Helm3D/problem.eps,scale=0.7}\end{center}\end{figure}

In this example we call the side where the incident wave reaches directly the ``bright side (-)'' and the other side the ``shadow side (+)'' (See Fig.3.40). This problem is solved with the conventional BIEM and FM-BIEM when ka0=1,2,3,4,5. In GMRES we use the block diagonal matrix corresponding to the leaves as the preconditioner according to Nishida and Hayami[62]. Fig.3.41 plots the total CPU time (sec) vs the number of unknowns. In Fig.3.41 the lines marked ``Tdir_ka=5'' and ``Tfmm_ka=1,2,3,4,5'' indicate the CPU time required with the conventional BIEM when ka0=5 and FM-BIEM when ka0=1,2,3,4,5, respectively. Because the computational times required with the conventional BIEM are almost identical when ka0=1,2,3,4,5 we plot only the ``Tdir_ka=5'' result in Fig.3.41. This figure shows that the FM-BIEM is faster than the conventional BIEM when the number of unknowns is larger than several thousands. Fig.3.42 shows the CPU time per iteration. In Fig.3.42 the lines marked ``Tfmm_per_itr_ka=1,2,3,4,5'' show the CPU time per iteration when ka0=1,2,3,4,5. Fig.3.43, Fig.3.45 and Fig.3.47 show the crack opening displacement $\vert\phi\vert/u_0$ when ka0=1,2,3. In these figures ``conv'' and ``fmm'' indicate the numerical solutions with the conventional BIEM and FM-BIEM, respectively. Fig.3.44, Fig.3.46 and Fig.3.48 show |u| on both sides and the analytical solution (Leitner[53]) when ka0=1,2,3. In these figures ``bside_analy'' and ``bside_numer'' denote the analytical solution and the numerical one on the bright side and ``sside_analy'' and ``sside_numer'' denote the analytical solution and the numerical one on the shadow side. In Fig.3.49 the lines marked ``Laplace'' and ``Helmholtz'' show the computational times required with FM-BIEM for Laplace's equation and Helmholtz's equation (ka0=1), respectively. We note that these computational times are obtained on a DEC Alpha 21264(600MHz). This figure shows that the computational time required with Helmholtz's equation is slower than that required with Laplace's equation.
next up previous contents
Next: Many penny-shaped cracks Up: Numerical examples Previous: Numerical examples
Ken-ichi Yoshida
2001-07-28