Hypersingular Boundary Integral Equations: Some Applications in Acoustic and Elastic Wave Scattering

The properties of hypersingular integrals, which arise when the gradient of conventional boundary integrals is taken, are discussed. Interpretation in terms of Hadamard finite-part integrals, even for integrals in three dimensions, is given, and this concept is compared with the Cauchy Principal Value, which, by itself, is insufficient to render meaning to the hypersingular integrals. It is shown that the finite-part integrals may be avoided, if desired, by conversion to regular line and surface integrals through a novel use of Stokes’ theorem. Motivation for this work is given in the context of scattering of time-harmonic waves by cracks. Static crack analysis of linear elastic fracture mechanics is included as an important special case in the zero-frequency limit. A numerical example is given for the problem of acoustic scattering by a rigid screen in three spatial dimensions.