Directly Derived Non-Hyper-Singular Boundary Integral Equations for Acoustic Problems, and Their Solution through Petrov-Galerkin Schemes

Novel non-hyper-singular [i.e., only strongly-singular] boundary-integral-equations for the gradients of the acoustic velocity potential, involving only O(r−2) singularities at the surface of a 3-D body, are derived, for solving problems of acoustics governed by the Helmholtz differential equation. The gradients of the fundamental solution to the Helmholtz differential equation for the velocity potential, are used in this derivation. Several basic identities governing the fundamental solution to the Helmholtz differential equation for velocity potential, are also derived. Using these basic identities, the strongly singular integral equations for the potential and its gradients [denoted here as φ-BIE, and q-BIE, respectively], are rendered to be only weakly-singular [i.e., possessing singularities of O(r−1) at the surface of a 3-D body]. These weakly-singular equations are denoted as R-φ-BIE, and R-q-BIE, respectively. General Petrov-Galerkin weak-solutions of R-φ-BIE, and R-q-BIE are discussed; and special cases of collocation-based boundary-element numerical approaches [denoted as BEM-R-φ-BIE, and BEM-R-q-BIE], as well as Symmetric Galerkin Boundary Element approaches [denoted as SGBEMR-φ-BIE and SGBEM-R-q-BIE, respectively] are also presented. The superior accuracy and efficiency of the SGBEM-R-φ-BIE and SGBEM-R-q-BIE are illustrated, through examples involving acoustic radiation as well as scattering from 3-D bodies possessing smooth surfaces, as well as surfaces with sharp corners. keyword: Boundary integral equations, hypersingularity, Petrov-Galerkin