If the fast multipole method (FMM) is applied in the context of the boundary element method, the efficiency and accuracy of the FMM is significantly influenced by the used hierarchical grouping scheme. Hence, in this paper, a new approach to the grouping scheme is presented to solve numerical examples with problem-oriented meshes and higher order elements accurately and efficiently. Furthermore, with the proposed meshing strategies the efficiency of the FMM can be additionally controlled.
IntroductionThe fast multipole method (FMM) is based on a truncated series expansion of Green's function into spherical harmonics (Greengard and Rokhlin, 1987). Its accuracy depends on both the number of its members, which can be controlled with the chosen order L, and the distance between the domain with sources and the domain with evaluation points. To apply this series expansion to an efficient solution of static problems with the boundary element method (BEM), a hierarchical grouping scheme for all boundary elements is necessary. Starting from the grouping scheme it is decided, whether the interactions between the two domains can be computed with the help of a series expansion or not. The original grouping scheme, which was proposed for the computation of particle interactions (Greengard and Rokhlin, 1987), can be relatively easily adapted to constant elements, since it is unproblematic to split constant elements at the boundaries of the subdomains of the grouping scheme (Nabors and White, 1991). However, this approach is not recommended in combination with linear or higher order elements. Hence, in this paper, a new approach to the grouping scheme is presented, which enables the accurate and efficient treatment of problem-oriented meshes with extremely varying size of the elements, even if only low order series expansions are used.Based on a detailed knowledge of the grouping scheme meshing strategies can be developed to further improve the efficiency of the FMM. The focus in this is on the shape of the boundary elements and the position of the evaluation points in respect to these elements.
Purpose -Various parallelization strategies are investigated to mainly reduce the computational costs in the context of boundary element methods and a compressed system matrix. Design/methodology/approach -Electrostatic field problems are solved numerically by an indirect boundary element method. The fully dense system matrix is compressed by an application of the fast multipole method. Various parallelization techniques such as vectorization, multiple threads, and multiple processes are applied to reduce the computational costs. Findings -It is shown that in total a good speedup is achieved by a parallelization approach which is relatively easy to implement. Furthermore, a detailed discussion on the influence of problem oriented meshes to the different parts of the method is presented. On the one hand the application of problem oriented meshes leads to relatively small linear systems of equations along with a high accuracy of the solution, but on the other hand the efficiency of parallelization itself is diminished.Research limitations/implications -The presented parallelization approach has been tested on a small PC cluster only. Additionally, the main focus has been laid on a reduction of computing time. Practical implications -Typical properties of general static field problems are comprised in the investigated numerical example. Hence, the results and conclusions are rather general. Originality/value -Implementation details of a parallelization of existing fast and efficient boundary element method solvers are discussed. The presented approach is relatively easy to implement and takes special properties of fast methods in combination with parallelization into account.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.