Solving the Helmholtz equation in terms of amplitude and phase; revisited

Y. H. Wijnant, F. H. De Vries

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Abstract

In a paper presented at the ISMA conference in 2008 [2], we proposed to solve the Helmholtz equation in terms of the pressure amplitude and phase instead of the complex pressure itself. The major advantage of this approach is the large reduction of the number of degrees of freedom, needed to describe predominantly propagating waves at large wavenumbers. This is due to the fact that, for propagating waves, both amplitude and phase are smooth functions for any wavenumber. A drawback of the method is the resulting non-linear, coupled set of differential equations, as well as the resulting non-linear boundary equations (as opposed to boundary conditions). Despite this increased complexity, in 2008, we did present 2D solutions on meshes of approximately 3 wavelengths per element. Solutions for higher wavenumber values could not be obtained (the non-linear solver did not converge). In this paper, we give an explanation and a solution for this non-convergence problem. First, we present accurate finite element solutions for a 1D benchmark problem, obtained on meshes having an arbitrary number of waves per element. In addition, we present solutions for the 2D problem of 2008, also for meshes having an arbitrary number of waves per element. As a prelude to more practical problems, we show the solution for a vibrating cylinder and a preliminary solution for a plane wave scattering on a rigid cylinder. Unluckily, the scattering problem revealed convergence problems which need further investigation. We will discuss these problems and end with conclusions.

Original languageEnglish
Title of host publication26th International Conference on Noise and Vibration Engineering (ISMA 2014)
Subtitle of host publicationProceedings of a meeting held 15-17 September 2014, Leuven, Belgium. Held in Conjunction with the 5th International Conference on Uncertainty in Structural Dynamics (USD 2014)
EditorsPaul Sas
PublisherCurran Associates Inc.
Pages4339-4348
Number of pages10
ISBN (Electronic)9781634395069
Publication statusPublished - 2014
Event26th International Conference on Noise and Vibration Engineering, ISMA 2014 - Leuven, Belgium
Duration: 15 Sep 201417 Sep 2014
Conference number: 26
http://past.isma-isaac.be/isma2014

Conference

Conference26th International Conference on Noise and Vibration Engineering, ISMA 2014
Abbreviated titleISMA
CountryBelgium
CityLeuven
Period15/09/1417/09/14
Internet address

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