Solution of the 3D-Helmholtz equation in exterior domains using spherical harmonic decomposition

This work is devoted to a finite element formulation for the Helmholtz equation in exterior domains. The proposed formulation uses a separation of variables, combining a 2D FE discretization on an intermediate spherical boundary and an ‘a priori’ analytical pattern for the radial direction. Using the analytical radial pattern and the series expansion of trial and test functions in terms of spherical harmonics, an efficient semi-analytical technique is obtained for the direct calculation of the global FE matrices. The accuracy and reliability of the formulation are illustrated through numerical examples of radiation and scattering in the exterior domain.