The Radial Basis Integral Equation Method For
2D Helmholtz Problems
Author(s)
H. Dogan & V. Popov
Abstract
A meshless method for the solution of 2D Helmholtz equation has been
developed by using the Boundary Integral Equation (BIE) combined with Radial
Basis Function (RBF) interpolations. BIE is applied by using 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 point 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 in respect to space coordinates. 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 in 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 are used, namely ( ) 2 ln( ) 1
1 f R R R x y and
( ) 4 ln( ) 2 2 1
2 f R R R x y xy x . The latter has been found to produce
more accurate results.
Keywords: meshless method, 2D Helmholtz equation, circular sub-domains,
radial basis functions.
1 Introduction
The 2-D modelling of acoustics has been of significant importance for a variety
of engineering problem in areas such as ultrasonics, microfluidics, aeoracoustics,
etc. The numerical modelling of the problem is essential in case a complete
Keywords
meshless method, 2D Helmholtz equation, circular sub-domains,
radial basis functions
Related Book
Other papers in this volume
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