The boundary element method applied to static and dynamic crack problems using hypersingular boundary integral equations
Frank J. Rizzo
The need for hypersingular boundary integral equations in crack problems is motivated through acoustic and elastic wave scattering from a thin screen and crack. By integrating over a small (not necessarily symmetric) neighborhood about the singular point, and the rest of the boundary, and identifying terms from the integrals over the two surfaces which cancel each other, the finite-part of the hypersingular integral is defined for curved surfaces in both two and three dimensions. Stokes' theorem is used to regularize the hypersingular integrals to a form conducive to simple numerical integration techniques. With no prior assumptions on the discretizasion or integration by parts, this method results in integrals which are at most weakly singular. The equivalence of this approach to the finite-part of the hypersingular integral is established;The necessary condition on the density function for the hypersingular integral equation to have meaning and the consequences on the solution of not satisfying the necessary conditions is discussed. This new formulation places restrictions on the choice of shape functions and the possible location of the collocation points within elements due to the smoothness requirement on the density function. Such restrictions for regular boundary integral equations with Cauchy principal value integrals are also discussed. The different kinds of integrals encountered in a hypersingular boundary integral equation such as weakly singular integrals, nearly singular integrals and regular area and line integrals are studied. Discretization considerations for precise and efficient numerical computation of these integrals in the context of the boundary element method is established and the influence of discretization on the solution is highlighted through numerical examples. Examples are chosen from problems of acoustic and elastic wave scattering from thin screens and cracks in three dimensions.