Laplace Green's functions for infinite ground planes with local roughness

Abstract

The Green's functions for the Laplace equation satisfying the Dirichlet and Neumann boundary conditions on the upper side of the infinite plane with a circular hole are introduced and studied. These functions enable solutions of the boundary value problems in domains where the hole is closed by an arbitrary mesh (locally rough surfaces). The developed approach accounts for arbitrary positive and negative ground elevations inside the domain of interest, which is not possible to achieve using the regular method of images. Such problems appear in electrostatics, however, the methods developed apply to other domains where the Laplace or Poisson equations govern. Integral and series representations of the Green's functions are provided. Using these Green's functions, an efficient computational technique based on the boundary element method with fast multipole acceleration is developed. A numerical study of some benchmark problems is presented.