Non-Hyper-Singular Boundary Integral Equations for Acoustic Problems, Implemented by the Collocation-Based Boundary Element Method

The weak-form of Helmholtz differential equation, in conjunction with vector test-functions (which are gradients of the fundamental solutions to the Helmholtz differential equation in free space) is utilized as the basis in order to directly derive non-hyper-singular boundary integral equations for the velocity potential, as well as its gradients. Thereby, the presently proposed boundary integral equations, for the gradients of the acoustic velocity potential, involve only O(r−2) singularities at the surface of a 3-D body. Several basic identities governing the fundamental solution to the Helmholtz differential equation for velocity potential, are also derived for the further desingularization of the strongly singular integral equations for the potential and its gradients to be only weakly-singular. These weakly-singular equations are denoted as R-φ-BIE, and R-q-BIE, respectively. [i.e., containing singularities of O(r−1) only at the boundary] Collocation-based boundary-element numerical approaches [denoted as BEM-R-φ-BIE, and BEMR-q−BIE] are implemented to solve these R-φ-BIE, and R-q-BIE. The lower computational costs of BEM-R-φBIE and BEM-R-q-BIE, as compared to the previously published Symmetric Galerkin BEM based solutions of R-φ&q-BIE [Qian, Han and Atluri (2004)] are demonstrated through examples involving acoustic radiation as well as scattering from 3-D bodies. keyword: Boundary integral equations, hypersingularity, collocation