FMM can be used also in the computation of the stress or displacement within the domain . The algorithm used to this end is essentially the same as the one presented above, except that two tree structures of cells are used. We here describe necessary modifications to the FMM presented in 3.4.3 in the computation of the stress.
In step 2, the cell of the level 0 is taken as a cube which contains all the global boundary elements and all the points of observation where the stresses are computed. From this cell, we construct two tree structures of cells, i.e. the boundary element tree and observation point tree. We construct the former (latter) by subdividing parent cells into 8 sub cubes and by retaining those to which some boundary elements (observation points) belong. The boundary element tree has already been used in the FMM for the discretised BIE in 3.4.3. In the stress computation we use (3.53)-(3.58) to obtain the multipole moments in cells in the boundary element tree, and use (3.60)-(3.65) to calculate the coefficients of local expansion at the centroids of cells in the observation point tree. Hence the cells in step 3 are now understood to be from the boundary element tree, while cells denoted by C (C') in steps 4 and 5 are taken from the observation point tree (boundary element tree). The rest of the calculation proceeds in an obvious manner as one replaces (3.100), (3.104), (3.106), etc. by (3.107), (3.108), (3.110), etc., respectively, in the algorithm described in 3.4.3.