D. Rosen et De. Cormack, THE CONTINUATION APPROACH - A GENERAL FRAMEWORK FOR THE ANALYSIS AND EVALUATION OF SINGULAR AND NEAR-SINGULAR INTEGRALS, SIAM journal on applied mathematics, 55(3), 1995, pp. 723-762
The continuation approach, presented in this paper, is a simple and un
ified framework for analyzing and computing a large class of singular
and near-singular integrals such as those that arise in the boundary e
lement method. The analysis formulates singular and near-singular inte
grals consistently as instances of a single phenomenon, with all types
of algebraic singularities treated in a similar way for domains of ar
bitrary dimension. Singular integrals are Viewed merely as ''continuat
ions'' of nonsingular (but perhaps near-singular) ones, by placing the
singularity of the integrand outside the integration domain, and taki
ng the limit as it approaches the domain. The technique exploits the f
unctional 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 evaluat
ed by the flat surface techniques. Moreover, corners are handled very
effectively, without the complication of algebraic approaches. This al
so leads to an efficient, and strictly numerical, method for directly
computing the jumps that commonly arise. Also arising from the analysi
s are the gauge conditions, which are necessary and sufficient for the
existence of singular integrals with strong singularities (Cauchy typ
e and hypersingular). If these conditions are satisfied, the continuat
ion singular integral is bounded and coincides with the classical defi
nitions of either the Cauchy principal value (with a jump), or the Had
amard finite part.