Singular and Near Singular Integrals in the BEM: A Global Approach

Abstract

A unified approach for analyzing and evaluating singular and near singular integrals, relevant to the boundary element method (BEM), is obtained. The original n-dimensional integral is mapped to one that is performed exclusively on the boundary of the integration domain. Therefore, the dimensionality of the integration is reduced by l, and integrand evaluations close to the singularity are avoided. The new formulae are obtained through a straightforward one-dimensional analytical integration, regardless of the dimensionality of the problem and can be used generally for numerical integration. The formulae for evaluating near singular integrals are a generalization of those obtained by the invariant imbedding technique for singular integrals and reduce to the singular integral formulae in the limit. In this sense, the singular integrals are treated as limits, or “continuations,”of nonsingular ones. This leads to a general interpretation that includes the singular integrals as a special case. In particular, the method provides new insights into the different limiting values of the Cauchy integrals. For this case, the definition of the continuation integrals encompasses not only the Cauchy principal value but also the jump term and an even finite term. Although the present results are applicable only to integrals of homogeneous functions over flat domains, the analysis can be extended to curved and nonsmooth surfaces, as well as to more general integrands.