D. Rosen et De. Cormack, THE CONTINUATION APPROACH FOR SINGULAR AND NEAR-SINGULAR INTEGRATION, Engineering analysis with boundary elements, 13(2), 1994, pp. 99-113
This work provides an overview of the continuation approach, a unified
framework for understanding and directly computing a large class of s
ingular and near-singular integrals. Singular surface integrals are vi
ewed merely as limits, of 'continuations', of non-singular (but perhap
s near-singular) ones. The analysis presents a clear and intuitive pic
ture of the behaviour of these integrals, which leads to general formu
lae for singular and near-singular integral evaluations, the former be
ing a special case of the latter. We also obtain necessary and suffici
ent boundedness conditions for the singular integrals. When these cond
itions are met, the continuation singular integral encompasses the cla
ssical cases of a Cauchy principal value, jump terms, and Hadamard fin
ite part. The analysis exploits the functional homogeneity of many Gre
en's functions, and it covers integrals on smooth (flat and curved) as
well as non-smooth surfaces. Some numerical integration examples are
presented.