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The University of Southampton
Engineering

Wave-based computational methods

(in collaboration with Jeremy Astley (ISVR), Pablo Gamallo (University of Vigo, Spain), Tomi Huttunen (University of Kuopio, Finland)) Funded by EPSRC and Rolls-Royce (CASE Award)

Computational methods for aero-acoustic can become very expensive at high frequencies due to the dispersion error of standard numerical schemes (finite difference, finite element...). With wave-based computational methods, the solution is locally approximated as a sum of plane waves (or Bessel functions). The dispersion properties of the sound waves are directly built into the plane wave basis and offer a simple way to drastically reduce the dispersion error.

Wave-based discontinuous Galerkin methods have been developed, first for the wave equations [1] and then for a general system of linear hyperbolic equations [2]. These methods have proved able to provide very accurate solutions at only a fraction of the costs of standard CAA methods. Examples of applications include solving the linearized Euler equations to predict sound refraction by two-dimensional jets (see below). Current work involves developing impedance models and non-reflecting conditions for wave-based methods.

Sound field radiated from a point source in a jet.

 

 

 

 

 

 

 

 Comparison of theoretical and numerical results.

 

 

 

 

 

 

 

In addition, wave-based methods often require integrating highly oscillatory functions. A set of exact formulae for the evaluation of such integrals has been proposed in [3].

[1] Gabard, G. Discontinuous Galerkin methods with plane waves for the displacement acoustic equation.  International Journal for Numerical Methods in Engineering, 66, 2006, 549-69.

[2] Gabard, G. Discontinuous Galerkin methods with plane waves for time-harmonic problems.  Journal of Computational Physics, 225, 2007, 1961-84.

[3] Gabard, G. Exact integration of polynomial-exponential products with application to wave-based numerical methods. Communications in Numerical Methods in Engineering, 25(3), 2009, 237-46.

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