The Continuation Approach: A General Framework for the Analysis and Evaluation of Singular and Near-Singular Integrals

Abstract

The continuation approach, presented in this paper, is a simple and unified framework for analyzing and computing a large class of singular and near-singular integrals such as those that arise in the boundary element method. The analysis formulates singular and near-singular integrals consistently as instances of a single phenomenon, with all types of algebraic singularities treated in a similar way for domains of arbitrary dimension. Singular integrals are viewed merely as “continuations” of nonsingular (but perhaps near-singular) ones, by placing the singularity of the integrand outside the integration domain, and taking the limit as it approaches the domain. The technique exploits the functional homogeneity shared by the Green’s functions of many physical problems. For flat surfaces, this allows the integral to be mapped to the contour of the integration domain. Curved surfaces are handled by projecting the integration domain to a tangent hyperplane, where the singular components of the integral can be easily isolated and evaluated by the flat surface techniques. Moreover, corners are handled very effectively, without the complication of algebraic approaches. This also leads to an efficient, and strictly numerical, method for directly computing the jumps that commonly arise. Also arising from the analysis are the gauge conditions, which are necessary and sufficient for the existence of singular integrals with strong singularities (Cauchy type and hypersingular). If these conditions are satisfied, the continuation singular integral is bounded and coincides with the classical definitions of either the Cauchy principal value (with a jump), or the Hadamard finite part.