Metric deformation and boundary value problems in 2D

TL;DR

This paper introduces an analytical method using metric deformation and diffeomorphism to solve the 2D Helmholtz equation with arbitrary boundaries, simplifying boundary conditions while handling complex geometries.

Contribution

It presents a novel perturbative approach that transforms arbitrary boundaries into circles via diffeomorphism, enabling closed-form solutions for the Helmholtz equation with complex boundaries.

Findings

Method works well for boundaries with large deviations from a circle

Perturbation series converges due to Fourier boundary representation

Solutions compare favorably with numerical results

Abstract

A new analytical formulation is prescribed to solve the Helmholtz equation in 2D with arbitrary boundary. A suitable diffeomorphism is used to annul the asymmetries in the boundary by mapping it into an equivalent circle. This results in a modification of the metric in the interior of the region and manifests itself in the appearance of new source terms in the original homogeneous equation. The modified equation is then solved perturbatively. At each order the general solution is written in a closed form irrespective of boundary conditions. This method allows one to retain the simple form of the boundary condition at the cost of complicating the original equation. When compared with numerical results the formulation is seen to work reasonably well even for boundaries with large deviations from a circle. The Fourier representation of the boundary ensures the convergence of the perturbation series.