Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering

Abstract

In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods. * Colour online for monochrome figures available at journals.cambridge.org/anu.

The Wave-Matching Boundary Integral Equation — An energy approach to Galerkin BEM for acoustic wave propagation problems

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An Energy Interpretation of the Kirchhoff-Helmholtz Boundary Integral Equation and its Application to Sound Field Synthesis

“…There is however an emerging variant called Numerical-Asymptotic Hybrid BEM [22], which has potential to be extremely useful for acoustic simulation since its computational cost grows only very slowly (if at all) with frequency (unlike conventional BEM where the computational cost and storage grows with frequency to the power four). This method uses elements which are large with respect to wavelength, on which the interpolation functions are usually solutions to the Helmholtz equation (e.g.…”

Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems

“…However, the inclusion of the plane wave enrichment does have some implications on the required number of integration points, in that it changes the apparent wavelength of the oscillatory integrand toλ, whereλ ∈ (0, 2λ). For this reason, although some authors have presented novel integration schemes that offer promise for rapid evaluation of these boundary integrals [27,28,29], in the current work, which is aimed at demonstrating the XIBEM formulation for the first time,…”