Conformal Mapping on Rough Boundaries I: Applications to harmonic problems

TL;DR

This paper introduces a conformal mapping method to analyze harmonic fields near rough boundaries, enabling efficient computation of fields and Green functions, with applications to smooth interface approximation and surface flux problems.

Contribution

A novel conformal mapping technique tailored for 2D rough boundary harmonic problems, including an efficient algorithm and applications to Green functions and surface field analysis.

Findings

Efficient algorithm for conformal map computation for arbitrary boundaries

Ability to derive Green functions for localized flux on rough surfaces

Insights into the correlation between surface topography and harmonic fields

Abstract

The aim of this study is to analyze the properties of harmonic fields in the vicinity of rough boundaries where either a constant potential or a zero flux is imposed, while a constant field is prescribed at an infinite distance from this boundary. We introduce a conformal mapping technique that is tailored to this problem in two dimensions. An efficient algorithm is introduced to compute the conformal map for arbitrarily chosen boundaries. Harmonic fields can then simply be read from the conformal map. We discuss applications to "equivalent" smooth interfaces. We study the correlations between the topography and the field at the surface. Finally we apply the conformal map to the computation of inhomogeneous harmonic fields such as the derivation of Green function for localized flux on the surface of a rough boundary.