A Fast Multipole Boundary Element Method for Three-Dimensional Half-Space Acoustic Wave Problems Over an Impedance Plane

A high-frequency fast multipole boundary element method (FMBEM) based on the Burton-Miller formulation is proposed for three-dimensional acoustic wave problems over an infinite plane with impedance boundary conditions. The Green's function for the sound propagation over an impedance plane is employed explicitly in the boundary integral equation (BIE). To deal with the integral appearing in the half-space Green's function, the downward pass in the FMBEM is divided into two parts to compute contributions from the real domain to the real and image domains, respectively. A piecewise analytical method is proposed to compute the moment-to-local (M2L) translator from the real domain to the image domain accurately. An algorithm based on the multi-level tree structure is designed to compute the M2L translators efficiently. Correspondingly, the direct coefficient can also be computed efficiently by taking advantage of the algorithm of the efficient M2L. A flexible generalized minimal residual (fGMRES) is applied to accelerating the solution when the convergence is very slow. Numerical examples are presented to demonstrate the accuracy and efficiency of the developed FMBEM. Good solutions and high acceleration ratios compared with the conventional boundary element method clearly show the potential of the FMBEM for large-scale 3D acoustic wave problems over an infinite impedance plane which are of practical significance.