A fast directional boundary element method for solving wideband three-dimensional half-space acoustic problems

This paper presents a novel wideband fast multipole boundary element method (BEM) based on a fast directional algorithm (FDA) for solving large-scale three-dimensional (3-D) half-space acoustic wave problems. The method employs the half-space Green’s function in the boundary integral equations, eliminating the need to discretize the infinite plane and avoiding truncation errors. An improved FDA is developed to expand the kernel functions, enabling efficient matrix-vector product acceleration across a wide range of frequencies. Unlike full-space problems, the translations for half-space problems differ due to the half-space Green’s function. Leveraging the symmetry of the half-space problem, we introduce techniques to reduce the computational cost in fast multipole translations. Specifically, only an additional moment-to-local (M2L) translation for the image part is required, bypassing the computation and storage of other image-related translations. The iterative solver GMRES is used to further accelerate the solution in the proposed FDA-BEM. Numerical examples validate the accuracy of the method and demonstrate its nearly linear computational efficiency in solving large-scale 3-D half-space acoustic problems. Large scale models with the nondimensional wavenumber above 400 and number of elements above 3 million have been solved successfully using the developed method.