The University of Southampton
University of Southampton Institutional Repository

The radial basis integral equation method for solving the Helmholtz equation

The radial basis integral equation method for solving the Helmholtz equation
The radial basis integral equation method for solving the Helmholtz equation
A meshless method for the solution of Helmholtz equation has been developed by using the radial basis integral equation method (RBIEM). The derivation of the integral equation used in the RBIEM is based on the fundamental solution of the Helmholtz equation, therefore domain integrals are not encountered in the method. The method exploits the advantage of placing the source points always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green’s identities and the remaining equations are the derivatives of the first equation with respect to space coordinates. Radial basis function (RBF) interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). The accuracy and robustness of the method has been tested on some analytical solutions of the problem. Two different RBFs have been used, namely augmented thin plate spline (ATPS) in 2D and f(R)=R4ln(R)f(R)=R4ln(R) augmented by a second order polynomial. The latter has been found to produce more accurate results
meshless methods, RBIEM, helmholtz equation, multigrid, radial basis functions, eigenfrequencies
0955-7997
934-943
Dogan, Hakan
a1e136a9-aab8-4942-a977-0ae3440758cc
Popov, Viktor
e4c470fd-8a77-43ee-84d4-8a3c95e4f4f3
Ooi, Ean Hin
63ebcf57-32be-471a-af96-57e3f01e5fe2
Dogan, Hakan
a1e136a9-aab8-4942-a977-0ae3440758cc
Popov, Viktor
e4c470fd-8a77-43ee-84d4-8a3c95e4f4f3
Ooi, Ean Hin
63ebcf57-32be-471a-af96-57e3f01e5fe2

Dogan, Hakan, Popov, Viktor and Ooi, Ean Hin (2012) The radial basis integral equation method for solving the Helmholtz equation. Engineering Analysis with Boundary Elements, 36 (6), 934-943. (doi:10.1016/j.enganabound.2011.12.003).

Record type: Article

Abstract

A meshless method for the solution of Helmholtz equation has been developed by using the radial basis integral equation method (RBIEM). The derivation of the integral equation used in the RBIEM is based on the fundamental solution of the Helmholtz equation, therefore domain integrals are not encountered in the method. The method exploits the advantage of placing the source points always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green’s identities and the remaining equations are the derivatives of the first equation with respect to space coordinates. Radial basis function (RBF) interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). The accuracy and robustness of the method has been tested on some analytical solutions of the problem. Two different RBFs have been used, namely augmented thin plate spline (ATPS) in 2D and f(R)=R4ln(R)f(R)=R4ln(R) augmented by a second order polynomial. The latter has been found to produce more accurate results

This record has no associated files available for download.

More information

Accepted/In Press date: 10 December 2011
Published date: 31 January 2012
Keywords: meshless methods, RBIEM, helmholtz equation, multigrid, radial basis functions, eigenfrequencies
Organisations: Inst. Sound & Vibration Research

Identifiers

Local EPrints ID: 376831
URI: http://eprints.soton.ac.uk/id/eprint/376831
ISSN: 0955-7997
PURE UUID: babc0e27-6db8-4a02-ae5c-6e044df9bfd2

Catalogue record

Date deposited: 07 May 2015 09:29
Last modified: 14 Mar 2024 19:50

Export record

Altmetrics

Contributors

Author: Hakan Dogan
Author: Viktor Popov
Author: Ean Hin Ooi

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×