This paper presents a review of basic concepts of the boundary element method (BEM) for solving 3D half-space problems in a homogeneous medium and in frequency domain. The usual BEM for exterior problems can be extended easily for half-space problems only if the infinite plane is either rigid or soft, since the necessary tailored Green’s function is available. The difficulties arise when the infinite plane has finite impedance. Numerous expressions for the Green’s function have been found which need to be computed numerically. The practical implementation of some of these formulas shows that their application depends on the type of impedance of the plane. In this work, several formulas in frequency domain are discussed. Some of them have been implemented in a BEM formulation and results of their application in specific numerical examples are summarized. As a complement, two formulas of the Green’s function in time domain are presented. These formulas have been computed numerically and after the application of the Fourier Transformation compared with the frequency domain formulas and with a FEM calculation.
The numerical stability of different formulas for the correction term of the half-space Green’s function is investigated. The formula with complex monopoles is taken as a reference. The expressions of Koh and Yook, tested in a previous publication [R. Piscoya and M. Ochmann, Acoustical Green’s function and boundary element techniques for 3D half-space problems, J. Comput. Acoust. (2017), https://doi.org/10.1142/S0218396X17300018 ], are rewritten to improve their range of application. The formulas of Sommerfeld and Thomasson are analyzed and its suitability for a BEM implementation is evaluated by comparing their accuracy against our reference. For the sake of completeness, the first and second derivatives of the formulas are explicitly written.
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