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Algorithm for the new FM-BIEM

The algorithm for the new FMM is given as follows:

Steps 1-3. Same as the steps 1-3 in the algorithm described in 3.3.
Step 4. Computation of the coefficients of the exponential expansion:

Compute the coefficients of the exponential expansion at each cell using (4.36)-(4.49), taking the origin (O) at the centroid of the cell.

Step 5. Computation of the coefficients of the local expansion:

We compute the coefficients of the local expansion of cells of level l, starting from l=2 and increasing l. Now we consider a cell C and another cell C' which is contained in the interaction list of C. Considering the position of C' relative to C, we translate the coefficients of the exponential expansion via (4.50) and (4.51) as the centre of the exponential expansion is shifted from the centroid of C (O) to that of C' ( $\mbox{\boldmath$\space x $ }_0$) and then use appropriate formulae in (4.52)-(4.65) to convert the coefficients of the exponential expansion to the coefficients of the local expansion. After carrying out these conversions about all cells in the interaction list of C, we add them together via (4.66) and (4.67) to obtain the contribution from the interaction list of C to the coefficients of the local expansion. To these contributions we add the coefficients of the local expansion of the parent of C with the origin shifted from the centroid of the parent $(\mbox{\boldmath$\space x $ }_0)$ to that of C $(\mbox{\boldmath$\space x $ }_1)$ via (3.64) and (3.65) to obtain the coefficients of the local expansion associated with C.

Step 6. Evaluation of the integral in (3.46):

This step is the same as the step 5 in the algorithm described in 3.3.


next up previous contents
Next: Numerical examples Up: Crack problems for three-dimensional Previous: Rotation of coefficients
Ken-ichi Yoshida
2001-07-28