Hypersingular boundary integral equations: Some applications in acoustic and elastic wave scattering.

*(English)*Zbl 0729.73251Summary: 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.

##### MSC:

74S15 | Boundary element methods applied to problems in solid mechanics |

74J20 | Wave scattering in solid mechanics |

45E05 | Integral equations with kernels of Cauchy type |

74R99 | Fracture and damage |