Fast Multipole Boundary Element Method for Large-Scale Steady-State Sound Field Analysis. Part I: Setup and Validation

In order to reduce computational complexity and memory requirements for three-dimensional acoustical analysis, using the boundary element method (BEM), a new method for steady-state sound fields is developed based on the fast multipole algorithm. This method, called the Fast Multipole BEM (FMBEM), drastically accelerates an iterative solution of large-scale linear systems, without composing the dense influence coefficient matrices used for the conventional BEM. An efficient computational scheme is presented for the boundary integral equations in the basic form and in the normal derivative form, where the fast multipole algorithm is introduced over the multiple levels, by employing a concept of cells clustering boundary elements and hierarchical cell structure. Theoretical analysis shows how efficiently the FMBEM can reduce the computational complexity and the memory requirements compared with the BEM, in two typical problems assuming uniform distributions of nodes within a space and on its surface, respectively. A numerical test solving an acoustic tube problem confirms the validity of this method with respect to computational accuracy and efficiency.