Boundary element method for hypersingular integral equations: Implementation and applications in potential theory

The main objective of this paper is to develop effective numerical methods to solve hypersingular integral equations arising in various physical and mechanical applications. Both surface and contour integrals are considered. The novelty of the proposed approach lies in the exact formulas obtained for an arbitrary planar polygon in hypersingular integral estimations. A one-dimensional hypersingular integral equation is derived for axially symmetrical configurations, and analytical formulas are established for calculating the hypersingular parts. It is proved that the hypersingular component of the surface integral is equal to its hypersingular component along the tangent plane. These exact formulas enable the development of an effective numerical method based on boundary element implementation. Benchmark tests are considered, and the convergence of the proposed methods is demonstrated. Problems in crack analysis are formulated and solved using both surface and contour hypersingular integral equations. A comparison of the results is made between boundary element methods and finite element methods for penny-shaped cracks. Boundary value problems in fluid-structure interaction are considered, and numerical simulations are performed. An estimation of modes and frequencies of panel and blade vibrations when interacting with liquids is carried out.