Non-Hyper-Singular Boundary Integral Equations for Acoustic Problems

This dissertation deals with sound radiated from vibrating structures, both internal as well as external to the structure. Novel non-hyper-singular [i.e., only stronglysingular] boundary-integral-equations for the gradients of the acoustic velocity potential, involving only O(r<super>-2</super>) singularities at the surface of a 3-D body, are derived, for solving problems of acoustics governed by the Helmholtz dierential equation. The gradients of the fundamental solution to the Helmholtz dierential equation for the velocity potential, are used in this derivation. Several basic identities governing the fundamental solution to the Helmholtz dierential 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 -BIE, and q-BIE, respectively], are rendered to be only weakly-singular [i.e., possessing singularities of O(r<super>-1</super>) at the surface of a 3-D body]. These weakly-singular equations are denoted as R-phi-BIE, and R-q-BIE, respectively. General Petrov-Galerkin weak-solutions of R-phi-BIE, and R-q-BIE are discussed; and special cases of collocation-based boundary-element numerical approaches [denoted as BEM-R-phi-BIE, and BEM-R-q-BIE], Symmetric Galerkin Boundary Element approaches [denoted as SGBEM-R-phi-BIE and SGBEM-R-q-BIE], as well as Meshless Local Petrov Galerkin approaches [denoted as MLPG-R-phi-BIE and MLPG-R-q-BIE, respectively] are also presented as a family. The superior accuracy and efficiency of the BEM-R-phi-BIE and BEM-R-q-BIE, SGBEM-R-phi-BIE and SGBEM-R-q-BIE, MLPG-R-phi-BIE and MLPG-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. In addition, a kernel independent fast multipole method is introduced further to overcome the drawback of fully populated system matrices in BEM, and denoted here as FMM-BEM. The computational costs of FMM-BEM are at the scale of O(nN), which makes it much faster than the matrix based operation, and suitable for large practical problems of acoustics. The method relies on Chebyshev polynomials for the interpolation part to further reduce the computational cost. Examples are used to demonstrate the improvement.