Concluding remarks

- In chapter 3 we have successfully applied FM-BIEM to crack
problems in Laplace's equation, elastostatics, Helmholtz's
equation and elastodynamics and have shown the efficiencies of
FMM in BIEM. We have shown the potential of the FM-BIEM for
large-scale problems in fractures mechanics.
- It can be seen that the regularised formulation is inferior to the non-regularised formulation through numerical examples in 3.2 and 3.3.
- The numerical results obtained with collocation method are
considerably inferior to those obtained with Galerkin's method in
both performance and precision in elastostatic crack
problems. Indeed, comparing Fig.3.24 with
Fig.3.30, one can see that the numerical results obtained with
Galerkin's method with 471 DOF mesh is more accurate than those
obtained with collocation method with 28776 DOF mesh. This is
because the solution of the Galerkin boundary integral equation
minimises the potential energy in the space of displacement
fields spanned by double layer potentials with given set of shape
functions[87]. This is in contrast to the collocation
solution with discontinuous shape functions where the approximate
solution does not even belong to the space with finite energy. In
view of this it would be interesting to investigate FM-GBIEM for
other crack problems.
- The inefficiency of the evaluation of interior quantities has been one of major shortcomings of BIEM. The FMM approach of evaluating interior stresses proposed in 3.4 is expected to resolve this problem at least when one computes field quantities at many points in the domain.
- In 3.4 we have not utilised the symmetry of the matrices
in GBIEM. Hence, we shall use iterative solvers for symmetric
matrices in the future.
- In dynamic problems dealing with Helmholtz's equation and
elastodynamics the performance of FM-BIEM is considerably worse
than that in static problems. In order to apply FM-BIEM to
large-scale dynamic problems we need to improve the performance
using Rokhlin's diagonal form or the new FMM. As a matter of
fact, M2L translation is not the only bottleneck in dynamic
problems. Besides M2L translation, the direct computation also
deteriorates the performance. For example, one can easily see that
the direct computation in (3.157) is time-consuming
because we must evaluate the four integrals in (3.159).
This could be reduced by using numerical integrations properly.
- In this chapter we use the block diagonal matrix corresponding to
the leaves as the preconditioner, following techniques proposed
by Nishida and Hayami[62] and succeed in reduction of
the number of iterations to achieve convergence. However, in
problems considered in [60] the preconditioner of this
type has been shown to increase the number of iterations. The
preconditioner is considered to be one of the keys to improve the
performance of FM-BIEM and, therefore, we need to study the
preconditioners further.
- In engineering the singularity of the solution near the crack tip is very important. However, we have not taken it into consideration for simplicity. Because the use of singular elements is not very difficult, we plan to use singular elements in order to consider the behaviour of the solution near the crack tip and compute the stress intensity factor.
- Growths of cracks lead to failures of structures and therefore
behaviour of them is very significant. Hence, we plan to apply
FM-BIEM to growths of many cracks in the future.
- One can use techniques proposed in this chapter in ordinary
problems as well as in crack problems. Techniques obtained in
elastodynamics have a great utility value because BIEM is
particularly suitable for wave analyses. Applications of BIEM to
elastodynamics are often found in analyses of dynamic response
characteristics of structures. Dynamic response characteristic of
a structure is a very important factor in the
earthquake-resistant design. Also, recent underground structures
are crossing three-dimensionally and, therefore, analyses of such
structures inevitably become large-scale. FM-BIEM is considered
to be a powerful tool in such cases. Hence, we plan to apply
FM-BIEM to such problems.