On hypersingular boundary integral equations for certain problems in mechanics

Abstract

Improper or weakly singular integrals, once thought to be a handicap in computations, are now accepted as a source of effectiveness and stability in the numerical solution of many problems in mechanics. Such integrals naturally arise, for example, in the Boundary Integral Equation (BIE) method, and the literature contains many examples of successful computations based on Boundary Element-type (EEM) solutions of the BIEs (e.g. [1,2]). For some vector problems, the BIE/BEM process gives rise to a stronger type of singular integral which exists in the sense of the Cauchy Principal Value (CPV) (cf. [3]). This integral also has been treated numerically with success. Less commonly, but with growing frequency it seems, the gradient or normal derivative of such boundary integrals is taken, especially in the formulation of mechanics problems involving cracks. Then integrals more (hyper) singular than the CPV can explicitly arise. Usually, however, rather than confront such hypersingular integrals directly, a process of regularization (e.g. [4,5,6,7]) is employed to lower the singularity of the integrands. Such regularizatlon usually carries a formulational complexity and computational cost if it is even possible. However, the alternatives to regularization seem to be divergent integrals or numerical computation with integrals more singular than the CPV with, perhaps, even questionable definition.