Applications of the dual integral formulation in conjunction with fast multipole method in large-scale problems for 2D exterior acoustics

Abstract

In this paper, we solve the large-scale problem for exterior acoustics by employing the concept of fast multipole method (FMM) to accelerate the construction of influence matrix in the dual boundary element method (DBEM). By adopting the addition theorem, the four kernels in the dual formulation are expanded into degenerate kernels, which separate the field point and source point. The separable technique can promote the efficiency in determining the coefficients in a similar way of the fast Fourier transform over the Fourier transform. The source point matrices decomposed in the four influence matrices are similar to each other or only some combinations. There are many zeros or the same influence coefficients in the field point matrices decomposed in the four influence matrices, which can avoid calculating repeatedly the same terms. The separable technique reduces the number of floating-point operations from O( N 2) to O(N log a(N)), where N is number of elements and a is a small constant independent of N. To speed up the convergence in constructing the influence matrix, the center of multipole is designed to locate on the center of local coordinate for each boundary element. This approach enhances convergence by collocating multipoles on each center of the source element. The singular and hypersingular integrals are transformed into the summability of divergent series and regular integrals. Finally, the FMM is shown to reduce CPU time and memory requirement thus enabling us apply BEM to solve for large-scale problems. Five moment FMM formulation was found to be sufficient for convergence. The results are compared well with those of FEM, conventional BEM and analytical solutions and it shows the accuracy and efficiency of the FMM when compared with the conventional BEM.