A fast-multipole unified technique for the analysis of continuum mechanics problems with the boundary element methods

Hélvio de Farias Costa Peixoto¹; Larissa Simões Novelino¹; Ney Augusto Dumont¹

¹ Pontifical Catholic University of Rio de Janeiro

doi:10.20906/CPS/CILAMCE2015-0688

Resumo

This paper presents a fast-multipole implementation of the boundary element method for problems with many millions of degrees of freedom. It can lower the number of operations from O(N²) to O(N logN) in the iterative solution of a problem with N degrees of freedom. The memory allocation is also very small, as there is no need to store large matrices such as required by other numerical methods. The proposed implementation is based on a consistent development of the conventional, collocation boundary element method (BEM), which leads to a computationally less intensive (and conceptually more accurate) formulation for large-scale 2D and 3D problems of potential and elasticity. This formulation is especially advantageous for problems that require complicated fundamental solutions and curved boundaries. A scheme is used for expanding a generic fundamental solution about hierarchical levels of source and field poles, which is particularly advantageous to make the fast multipole technique seamlessly applicable to different fundamental solutions. The hierarchical tree of poles is built upon a topological concept of superelements inside superelements, which in part circumvents the need of evaluating geometrical distances between elements. The formulation is assessed and validated in terms of some simple, although very large, 2D potential problems with general geometry and topology for constant, linear and quadratic elements. Since iterative solvers are not addressed in this first step of numerical simulations, an isolated efficiency assessment of the implemented fast multipole technique becomes possible.

Palavras-chave: Boundary elements; fast multipole method; numerical methods