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Novel basis functions for the partition of unity boundary element method for Helmholtz problems
Author(s) -
Peake M.J.,
Trevelyan J.,
Coates G.
Publication year - 2012
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.4411
Subject(s) - partition of unity , mathematics , boundary element method , basis function , collocation (remote sensing) , mathematical analysis , helmholtz free energy , quadratic equation , polynomial basis , element (criminal law) , trigonometric functions , helmholtz equation , boundary value problem , finite element method , geometry , physics , computer science , quantum mechanics , machine learning , political science , law , thermodynamics
SUMMARY The BEM is a popular technique for wave scattering problems given its inherent ability to deal with infinite domains. In the last decade, the partition of unity BEM, in which the approximation space is enriched with a linear combination of plane waves, has been developed; this significantly reduces the number of DOFs required per wavelength. It has been shown that the element ends are more susceptible to errors in the approximation than the mid‐element regions. In this paper, the authors propose that this is due to the use of a collocation approach in combination with a reduced order of continuity in the Lagrangian shape function component of the basis functions. It is demonstrated, using numerical examples, that choosing trigonometric shape functions, rather than classical polynomial shape functions (quadratic in this case), provides accuracy benefits. Collocation schemes are investigated; it is found that the somewhat arbitrary choice of collocating at equally spaced points about the surface of a scatterer is better than schemes based on the roots of polynomials or consideration of the Fock domain. Copyright © 2012 John Wiley & Sons, Ltd.