Two accelerated isogeometric boundary element method formulations: fast multipole method and hierarchical matrices method

Abstract

This work presents two fast isogeometric formulations of the Boundary Element Method (BEM) applied to heat conduction problems, one accelerated by Fast Multipole Method (FMM) and other by Hierarchical Matrices. The Fast Multipole Method uses complex variables and expansion of fundamental solutions into Laurant series, while the Hierarchical Matrices are created by low rank CUR approximations from the k−Means clustering technique for geometric sampling. Both use Non-Uniform Rational B-Splines (NURBS) as shape functions. To reduce computational cost and facilitate implementation, NURBS are decomposed into Bézier curves, making the isogeometric formulation very similar to the conventional BEM. A description of the hierarchical structure of the data and the implemented algorithms are presented. Validation is performed by comparing the results of the proposed formulations with those of the conventional BEM formulation. The computational cost of both formulations is analyzed showing the advantages of the proposed formulations for large scale problems.