A Fast-Multipole Unified Technique for the Analysis of Potential Problems with the Boundary Element Methods

The proposed developments are based on a consistent implementation of the conventional, collocation boundary element method (BEM). A scheme is used to expand a generic (not problem-dependent) fundamental solution about hierarchical levels of source and field poles, which is particularly advantageous to make the technique seamlessly applicable to 2D and 3D problems of elasticity or potential, in terms of different types of curved elements for generally complicated geometry and topology. The proposed compact algorithm is more straightforward to lay out and seems to be more efficient than the ones available in the technical literature – particularly because the outermost loop refers to field nodes and geometry, in what may be called a reverse implementation. Some numerical results are shown for the conventional BEM, with validation and assessment for a few simple, but very large-scale, 2D potential problems with complicated geometry and topology for constant, linear and quadratic elements. Since iterative solvers are not required in this first step of numerical simulations, an isolated assessment of accuracy, computational effort and storage allocation of the proposed fast multipole technique becomes possible.