Fast Multipole Singular Boundary Method for Large-Scale Plane Elasticity Problems

The singular boundary method (SBM) is a recent meshless boundary collocation method that remedies the perplexing drawback of fictitious boundary in the method of fundamental solutions (MFS). The basic idea is to use the origin intensity factor to eliminate singularity of the fundamental solution at source. The method has so far been applied successfully to the potential and elasticity problems. However, the SBM solution for large-scale problems has been hindered by the operation count of O(N3)with direct solvers or O(N2)with iterative solvers, as well as the memory requirement of O(N2). In this study, the first attempt was made to combine the fast multipole method (FMM) and the SBM to significantly reduce CPU time and memory requirement by one degree of magnitude, namely, O(N). Based on the complex variable representation of fundamental solutions, the FMM-SBM formulations for both displacement and traction were presented. Numerical examples with up to hundreds of thousands of unknowns have successfully been tested on a desktop computer. These results clearly illustrated that the proposed FMM-SBM was very efficient and promising in solving large-scale plane elasticity problems.