Treatment of Singular Integrals for Boundary Element Methods in Hydro- and Aerodynamics - A Short Review

Boundary element methods are widely employed in engineering and research to solve partial differential equations in the form of boundary integral equations numerically. In hydro- and aerodynamics, these methods provide fast solution to potential flow problems, especially useful in early design phases of ships or aircraft, e.g. for the computation of ship motion and added wave resistance or of aircraft loads and acoustic analyses. However, the fundamental solution of the underlying boundary integral equations as well as their derivatives are characterized by an increasingly singular nature, so the assembly of the required boundary integral operators is non-trivial. Therefore, suitable treatment of the singular integrals is crucial for the boundary element method, since it requires the evaluation of the singular boundary integrals in the near- and self-influence regimes. In this paper, we classify the different methods for singular integration, detail the theory behind these techniques, give examples of existing approaches and sort them according to the presented classification. The present review for singular integration methods for Laplace boundary element methods aims to give an overview of existing frameworks and the related theory, intended as a starting point for choosing appropriate methods by considering the advantageous characteristics or identifying fields of further research.