This section prepares formulae needed for FM-GBIEM in elastostatics.

We now compute a part of the double integral in the right-hand side of
(3.100). Namely, we consider the contribution to this
integral from
,
where *S*_{x} and *S*_{y} are
disjoint parts of *S*. Using (3.49) and taking the origin
*O* near *S*_{y}, we obtain

where we have assumed that the inequality ( , ) holds. Notice that the multipole moments

With (3.105) the evaluation of the left-hand
side of (3.100) is made efficient since the multipole
moments are common to many evaluation points
and can be
reused. This efficiency is further enhanced by using the local
expansion as

where we have assumed that the inequality holds. Notice that the coefficients of the local expansion and are the same as (3.60) and (3.61) and L2L translation formulae are given by (3.64) and (3.65).

One may be interested in obtaining stress and displacement
distributions within the domain
after the
crack opening displacements are computed. In the rest of this section we
shall prepare formulae needed for the fast computation of the stress
components with the help of FMM. One may similarly compute
displacements, but the detail is omitted here.
It is immediate to obtain the following integral representation for the
stress from (3.44):

where is the stress associated with . The integral on the right-hand side of (3.107) is regularised with the help of the stress function in (3.101) as

As a matter of fact, (3.108) holds identically for any continuous having support within

where we have assumed that the origin is near

where we have assumed that the inequality holds.