A New Adaptive Algorithm for the Fast Multipole Boundary Element Method

A new definition of the interaction list in the fast multipole method (FMM) is introduced in this paper, which can reduce the moment-to-local (M2L) translations by about 30-40% and therefore improve the efficiency for the FMM. In addition, an adaptive tree structure is investigated, which is potentially more efficient than the oct-tree structure for thin and slender domains as in the case of micro-electro-mechanical systems (MEMS). A combination of the modified interaction list (termed L2 modification in the adaptive fast multipole BEM) and the adaptive tree structure in the fast multipole BEM has been implemented for both 3D potential and 3-D acoustic wave problems. In the potential theory case, the code is based on the earlier adaptive algorithm proposed in (Shen, L. and Y. J. Liu (2007). “An adaptive fast multipole boundary element method for three-dimensional potential problems.” Computational Mechanics 39(6): 681-691) with the so called “new FMM” where the M2L translations are replaced by the exponential (M2X, X2X, and X2L) translations. Suitable changes are proposed in the algorithm for the new adaptive fast multipole BEM. Finally, a new adaptive algorithm which encompasses all these modifications and the established algorithms is presented (that is, combining the original adaptive fast multipole BEM, L2 modification and adaptive tree for slender structures). Numerical results are presented to demonstrate the efficiencies of the new adaptive fast multipole BEM for solving both potential and acoustic wave problems. About 30-40% improvements in the computational efficiency are achieved with the L2 modification for all cases, and additional improvements are observed with the adaptive tree for some large-scale thin structures (MEMS models), without the lost of accuracy.