Evaluation of the Helmholtz boundary integral equation and its normal and tangential derivatives in two dimensions

This paper presents a means of evaluating singular integrals in the Helmholtz boundary integral equation and its normal and tangential derivatives in two dimensions. The subtraction–addition technique is applied to the singular integral equations to convert the singular integrals to either ordinary integrals with bounded integrands or modified singular integrals, including hypersingular integrals, with exact integration values. This regularization is performed before any discretization. The modified integral equations can be calculated by directly applying standard quadrature rules over the entire integration domain. Numerical computations involve evaluating the acoustic field associated with a radiating inverse elliptic cylinder. The velocity potential is obtained by applying the Burton–Miller method, which linearly combines the Helmholtz boundary integral equation with its normal derivative, to treat the fictitious characteristic frequencies. Further substituting the velocity potential into the regularized tangential derivative yields the surface tangential velocity. Comparing the numerical results with the analytical solutions verifies the effectiveness of the presented approach.