Quadrature rule for solving the Helmholtz equation in hypersingular BEM formulation

Abstract

Velázquez-Mata et al. recently presented a quadrature rule to accurately evaluate singular and weakly singular integrals in the sense of the Cauchy Principal Value by an exclusively numerical procedure. The procedure was verified by solving engineering problems using the boundary element method with fundamental solutions that have singularities of type log(r) and 1/r. However, that quadrature does not handle the evaluation of the Hadamard Finite Part of hypersingular integrals. These types of singularity appear in several fundamental solutions and, also, when the hypersingular boundary element formulation is applied to the Green functions previously analysed by the authors. In this paper, the quadrature rule presented in Velázquez-Mata et al. is extended to accurately compute integrals with singularities of the type 1/r2. The quadrature weights are derived from a system of equations defined from the finite part of known integrals called generalised moments, which include the element shape functions. This novelty is included in the hypersingular formulation of the boundary element method to solve the Helmholtz equation, taking advantage of this methodology to consider null-thickness boundaries using the Dual BEM.