THE CONTINUATION APPROACH FOR SINGULAR AND NEAR-SINGULAR INTEGRATION

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.