Boundary conditions for the solution of the 3-dimensional Poisson equation in open metallic enclosures

TL;DR

This paper investigates boundary conditions for solving the 3D Poisson equation in open metallic enclosures, proposing methods that improve accuracy for various aspect ratios and charge distributions in practical electromagnetic applications.

Contribution

It introduces a hybrid approach combining local asymptotic boundary conditions and a non-local matching method for better solutions in open metallic enclosures.

Findings

Second and third order ABCs are effective for large aspect ratios.

Non-local matching method performs well near unity aspect ratio.

The two methods complement each other for diverse charge densities.

Abstract

Numerical solution of the Poisson equation in metallic enclosures, open at one or more ends, is important in many practical situations such as High Power Microwave (HPM) or photo-cathode devices. It requires imposition of a suitable boundary condition at the open end. In this paper, methods for solving the Poisson equation are investigated for various charge densities and aspect ratios of the open ends. It is found that a mixture of second order and third order local asymptotic boundary condition (ABC) is best suited for large aspect ratios while a proposed non-local matching method, based on the solution of the Laplace equation, scores well when the aspect ratio is near unity for all charge density variations, including ones where the centre of charge is close to an open end or the charge density is non-localized. The two methods complement each other and can be used in electrostatic calculations where the computational domain needs to be terminated at the open boundaries of the metallic enclosure.