The use of complex integral representations for analytical evaluation of three-dimensional BEM integrals—potential and elasticity problems

The article presents a new complex variables-based approach for analytical evaluation of three-dimensional integrals involved in boundary element method (BEM) formulations. The boundary element is assumed to be planar and its boundary may contain an arbitrary number of straight lines and/or circular arcs. The idea is to use BEM integral representations written in a local coordinate system of an element, separate in-plane components of the fields involved, arrange them in certain complex combinations, and apply integral representations for complex functions. These integral representations, such as Cauchy–Pompeiu formula (a particular case of Bochner–Martinelli formula) are the corollaries of complex forms of Gauss's theorem and Green's identity. They reduce the integrals over the area of the domain to those over its boundary. The latter integrals can be evaluated analytically for various density functions. Analytical expressions are presented for basic integrals involved in the single- and double-layer potentials for potential (harmonic) and elasticity problems.