Keywords: Helmholtz equation, finite elements, Bessel functions, plane waves, diffraction problem, degrees of freedom per wavelength.

Conventional finite elements have been used for many years for the solution of wave problems. To ensure accurate simulation, each wavelength is discretised into around ten nodal points, with the finite element mesh being updated for each frequency to ensure adequate resolution of the wave pattern. This technique works well when the wavelength is long or the model domain is small. However, when the converse applies and the wavelength is small or the domain of interest is large, the finite element mesh requires a large number of elements, and the procedure becomes computationally expensive and impractical.

Many authors have developed finite elements for wave problems in both 2D and 3D (see for example [1,2,3]) which permitted relaxation of the traditional requirement cited above. The new elements are capable of containing many wavelengths per nodal spacing and have been very successful in reducing the computing effort by up to 90%.

This paper presents a finite element method for the solution of Helmholtz problems at high wave numbers that offers the potential of capturing many wavelengths per nodal spacing. This is done by constructing oscillatory shape functions as the product of polynomial shape functions and either Bessel functions or planar waves. The resulting elementary matrices obtained from the Galerkin-Bubnov formulation contain oscillatory terms and are evaluated using high order Gauss-Legendre integration.

The problem of interest deals with the diffraction of an incident plane wave by a rigid circular cylinder. Numerical experiments are carried out on a square computational domain for which the analytical solution of the problem is imposed on its boundary. The obtained results using the proposed finite element models are compared for different locations of the computational domain with respect to the diffracting object. It is shown that in the near field, the plane wave basis finite elements provide more accurate results. However, far from the scattering object, the Bessel function approximating model provides better accuracy. It is believed that a combination of both models; plane wave basis finite elements in the near field domain and Bessel function basis finite elements in the far field domain, would make an efficient tool for solving Helmholtz wave problems in large domains at reasonable computational cost.

1

J.M. Melenk, I. Babuška, "The Partition of Unity Finite Element Method. Basic Theory and Applications", Comput. Meth. Appl. Mech. Engng., 139, 289-314, 1996. doi:10.1016/S0045-7825(96)01087-0

2

P. Bettess, "Short wave scattering: problems and techniques", Phil. Trans. Royal Society London A., 362, 1816, 421-443,2004. doi:10.1098/rsta.2003.1329

3

O. Laghrouche, P. Bettess, E. Perrey-Debain, J. Trevelyan, "Wave interpolation finite elements for Helmholtz problems with jumps in the wave speed", Comput. Meth. Appl. Mech. Engng., 194, 367-381, 2005. doi:10.1016/j.cma.2003.12.074

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