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A penny-shaped crack

We consider an infinite space which contains one penny-shaped crack having the radius $ a_0$ and the unit normal vector of $ \mbox{$\mathbf n $}$$ =(0,0,1)$. We compute the crack opening displacement when the crack is subject to a plane longitudinal wave of the normal incidence from the $ -x_3$ direction. The wavenumber and Poisson's ratio are set to be $ k_Ta_0=5$ and $ \nu= 0.25$, respectively. This problem has been solved with the conventional BIEM with Crout's method and with multilevel FM-BIEMs with both Wigner-3j symbols and diagonal forms. The numerical results obtained by these methods were essentially identical. Fig.1 plots the total CPU time vs the number of unknowns. In Fig.1 the lines marked `Crout', `FMM-Wigner3j' and `FMM-diag' indicate the CPU times required by the conventional BIEM with Crout's method, FM-BIEM with Wigner-3j symbols and FM-BIEM with diagonal forms, respectively. This figure shows that FM-BIEM with diagonal forms is faster than that with Wigner-3j symbols.

Figure 1: CPU time (sec)