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In this chapter we apply new Fast Multipole Boundary
Integral Equation Method (new FMBIEM) to crack problems for
threedimensional Laplace's equation and threedimensional
elastostatics. We compare the results obtained with the new FMM with
those obtained in chapter 3. We shall call the FMBIEM described in
chapter 3 the ``original FMBIEM'' to distinguish the ``FMBIEM'' from
the ``new FMBIEM'' clearly. The new FMM provides techniques to reduce
the computational cost for M2L translation which is the bottleneck in
the original FMM. We achieve reduction of the computational cost by
replacing procedures for M2L translation in the original FMBIEM
obtained in the chapter 3 with three procedures called M2X, X2X and X2L
translations (details of M2X, X2X and X2L are given later).
Figure 4.1:
Computational cost for M2L and M2X+X2X+X2L

Here we briefly describe the outline of the new FMM (See
Greengard and Rokhlin[36] for more details). In the original
FMM in three dimensions, if one truncates the infinite series in the
multipole expansion taking p terms then the computational costs for
M2M, M2L and L2L translations are proportional to O(p^{4}), O(189p^{4})in the worst case and O(p^{4}), respectively. In particular, M2L
translation deteriorates the performance of FMM in dealing with
threedimensional problems where boundary elements are distributed
densely. In order to reduce the cost for M2M, M2L and L2L translations,
Greengard and Rokhlin proposed the new FMM on the basis of an integral
representation for a fundamental solution. In this chapter we use
techniques to reduce the computational cost for M2L translation because
the complexity for M2L translation is dominant in FMM. In the new FMM we
use the exponential expansion besides the multipole expansion and the
local expansion. The M2X, X2X and X2L translation formulae are to
convert the multipole expansion to the exponential expansion, the
exponential expansion to another exponential expansion and the
exponential expansion to the local expansion, respectively. The total
computational costs for M2X, X2X and X2L translations are O(p^{3}),
O(p^{2}) and O(p^{3}), respectively, and therefore the total cost is
O(p^{3}) (See Fig.4.1). This is why the new FMM is more efficient
than the original FMM.
Next: Crack problems for threedimensional
Up: Applications of Fast Multipole
Previous: Concluding remarks
Kenichi Yoshida
20010728