An effective method for solving the hypersingular integral equations in 3-D acoustics

The application of the boundary integral methods to the problem of acoustics, exterior to a three-dimensional surface, suffers, in general, from the nonexistence of nonuniqueness of its solutions at frequencies that are characteristic of the associated interior problem. The formulation that is most suitable for numerical implementation still appears to be that proposed by Burton and Miller [Proc. R. Soc. London Ser. A 323, 201–210 (1971)]. However, the hypersingular kernels present in such a formulation render it computationally unattractive. Previous attempts to regularize such hypersingular kernels involved the use of double surface integrals, or implicit use of tangential operators, or closed-form evaluations of hypersingular integrals. The method of double surface integrals is computationally highly inefficient, even though it allows higher-order interpolation schemes on the surface. The other two approaches are more conducive to the assumption of a constant value for each of thes variables (pressure and its normal gradient) over each of the discretized surface subdomains. The present work is concerned with the development of a procedure to regularize the hypersingular integral found in the Burton and Miller formulation, through a novel method by employing certain identities for the hypersingular integrals arising in an associated integral equation for the Laplace equation in the interior domain. It is feasible to use arbitrary-order C° interpolations for pressure and its normal derivative on the three-dimensional surface in the present approach. Even in such a case, the contribution to the hypersingular integrals over a small patch near the source point can be evaluated indirectly without using any special quadrature schemes, through a consideration of simple solutions to the associated Laplace equation in the interior domain. It should be noted here that there is no extra effort required in computing the potential kernels since they are a subset of the kernels found in the acoustic problem. In order to demonstrate the simplicity, robustness, and efficiency of the proposed method, numerical results with quadratic isoparametric C° surface elements are presented for both radiation and scattering problems. Comparisons are made with closed-form solutions and other existing formulations.