Implicit boundary integral methods for the Helmholtz equation in exterior domains

We propose a new algorithm for solving Helmholtz equations in the exterior domain. The algorithm not only combines the advantages of implicit surface representation and the boundary integral method, but also provides a new way to compute a class of the so-called hypersingular integrals. The keys to the proposed algorithm are the derivation of the volume integrals which are equivalent to any given integrals on smooth closed hypersurfaces, and the ability to approximate the natural limit of the singular integrals via seamless extrapolation. We present numerical results for both two and three dimensional scattering problems at near resonant frequencies as well as with non-convex scattering surfaces.