This paper presents a conforming C' boundary integral algorithm based on Hermite interpolation. This work is motivated by the requirement that the surface function multiplying a hypersingular kernel be differentiable at the collocation nodes. The unknown surface derivatives utilized by the Hermite approximation are determined, consistent with other boundary values, by writing a tangential hypersingular equation. Hypersingular equations are primarily invoked for solving crack problems, and the focus herein is on developing a suitable approximation for this geometry. Test calculations for the Laplace equation in two dimensions indicate that the algorithm is a promising technique for three-dimensional problems.
1 . INTRODUCTION
Over the last several years, hypersingular integral equations have been established as an essential tool in boundary element analysis.'' The primary motivation for studying these equations has been for solving problems posed in a domain containing a crack, a situation that arises in important areas of computational engineering, e.g. fracture acoustic and elastic wave ~cattering,~ and potential theory.6 While this application is clearly sufficient justification for studying these equations, hypersingular equations have also proven useful in boundary integral formulations for non-crack geometries.'.
The general form of a hypersingular integral is where P, Q are points on the boundary r, n = n(Q) is the surface normal on r, D is a fixed direction independent of Q and G(P, Q) is Green's function for the differential equation. The singularity of G at P = Q is made worse by the two derivatives, and the integral does not exist under standard definitions. Although early calculations with hypersingular integrals relied on an ad hoe finite part e ~a l u a t i o n , ~ a rigorous mathematical definition is provided by employing a limit process.339 (For a more complete discussion of the various approaches to evaluating hypersingular integrals, see the recent reviews', l o and the references therein.) The analysis of equation (1) using this limit definition reveals that the function 4 (e.g. potential in the Laplace equation,