| The computation of element integrals is of major importance to the Boundary Element Method (BEM). This thesis provides a unified framework for understanding and computing a large class of singular and near-singular integrals. In the continuation approach, the singular surface integrals, of the type which arise in the BEM, are viewed merely as "continuations" of non-singular (but perhaps near-singular) ones. Thus, in this approach, singular integrals are obtained from a limiting process in which the singularity of the integrand originally lies outside the integration domain and is gradually moved towards it. The analysis presents a clear and intuitive picture of the behaviour of these integrals, which leads to general efficient formulae for singular and near-singular integral evaluation, the former being a special case of the latter. In addition, we obtain necessary and sufficient conditions, herein referred to as "gauge conditions", that guarantee the boundedness of the continuation singular integrals. It is further shown that, when these conditions are met, the continuation singular integral encompasses the classical cases of Cauchy principal value integrals, jump terms, and Hadamard finite part. Numerical examples, which illustrate the computational advantages of the continuation formulae, are presented.;For flat integration surfaces, the continuation approach exploits the functional homogeneity of many Green's functions. The analysis is expanded to more general integrands and to curved integration domains by using the series expansion/subtraction ideas, popular in the BEM literature. It is shown that this expansion is greatly simplified by first mapping the integral to the tangent plane at the (near-)singularity, instead of to the parametric element. The continuation approach is also presented for the case when the singularity lies on, or close to, a corner (or non-smooth part) of the domain. This analysis is of considerable practical importance since it overcomes the difficulties and limitations of strictly analytical techniques. moreover, it is shown that the gauge conditions of the singular integrals impose some constraints on the functional approximation of the source's densities, in BEM discretizations. A systematic method to obtain these constraints and to incorporate them into the formulation of general problems is presented. |
