The paper presents the complex variables-based approach for analytical evaluation of three-dimensional integrals involved in boundary integral representations (potentials) for the Helmholtz equation. The boundary element is assumed to be planar bounded by an arbitrary number of straight lines and/or circular arcs. The integrals are re-written in local (element) coordinates, while in-plane components of the fields are described in terms of certain complex combinations. The use of Cauchy-Pompeiu formula (a particular case of Bochner-Martinelli formula) allows for the reduction of surface integrals over the element to the line integrals over its boundary. By considering the requirement of the minimum number of elements per wavelength and using an asymptotic analysis, analytical expressions for the line integrals are obtained for various density functions. A comparative study of numerical and analytical integration for particular integrals over two types of elements is performed.
Bibliographical noteFunding Information:
The support from the Theodore W. Bennett Chair, University of Minnesota, is kindly acknowledged. Special thanks are extended to Professor Bojan Guzina for the constructive discussions during the course of this investigation.
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- Analytical integration
- Complex variables
- Helmholtz equation