Analytic and semi-analytic integration of logarithmic and Newtonian potentials and their gradients over line segments and rectilinear panels

Abstract

The integrals are considered that arise when solving boundary integral equations, which kernels are logarithmic or Newtonian potentials or their gradients, in the case when the solution is considered to be piecewise-constant over panels, which are rectilinear segments in plane problems, and flat triangles in spatial problems. Integrals over one panel are considered which are calculated in the framework of collocations method, and the calculation technique is developed for repeated integrals over two panels arising in the Galerkin method. In plane problems for all the integrals exact analytical expressions suitable for practical usage are presented; the same applies to the integrals over one panel in three-dimensional problems. For repeated integrals in 3D case, a hybrid numericalanalytical scheme is proposed, which involves the extraction of singularities in the integrands and their analytical integration, as well as the numerical integration of smooth functions.