A weakly singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems

The composite boundary integral equation (BIE) formulation, using a linear combination of the conventional BIE and the hypersingular BIE, emerges as the most effective and efficient formula for acoustic wave problems in an exterior medium which is free of the well-known fictitious eigen-frequency difficulty. The crucial part in implementing this formulation is dealing with the hypersingular integrals. Various regularization procedures in the literature give rise, in general, to integrals which are still difficult and/or extremely time consuming to evaluate or are limited to the use of special, usually flat, boundary elements. In this paper, a general form of the hypersingular BIE is developed for 3-D acoustic wave problems, which contains at most weakly singular integrals. This weakly singular form can be derived by employing certain integral identities involving the static Green's function. It is shown that the discretization of this weakly singular form of the hypersingular BIE is straightforward and the same collocation procedures and regular quadrature as that used for conventional BIEs are sufficient to compute all the integrals involved. Computing times are only slightly longer than with conventional BIEs. The C1 smoothness requirement imposed on the density function for existence of the hypersingular BIEs and the possibility of relaxing this requirement are discussed. Three kinds of boundary elements, having different smoothness features, are employed. Numerical results are given for scattering from a rigid sphere at the fictitious frequencies, for values of wavenumber from π to 5π. In essence, with the methodology in this paper the fictitious eigenfrequency difficulty, long associated with the BIE for exterior problems, should no longer be a troublesome issue.