Keywords: wave propagation, finite element, plane wave approximation, economic modelling.

In order to solve wave problems using finite elements, it is usually necessary to discretize the domain such that there are at least ten nodal points per wavelength. However, such a procedure is computationally expensive and impractical for high frequency problems. The goal is to develop finite elements capable of containing many wavelengths per nodal spacing and therefore simulating problems with high frequency without refining the mesh. These finite elements are based on the Melenk and Babuska approach of including multiple wave directions in the element shape functions.

The method has appeared in a few authors' published or unpublished work. It was used in the approximation of integral equations in electromagnetic scattering by de La Bourdonnaye [1] under the name of Microlocal Discretization. Melenk and Babuska [2,3] developed it as a subset of the Partition of Unity Finite Element Method and used it to solve the Helmholtz equation in the case of a progressive plane wave. Later, Mayer and Mandel presented it with the name Finite Ray Element Method [4]. They investigated the asymptotic convergence of the method when the number of the approximating plane waves is increased. All authors reported severe ill-conditioning problems.

Farhat et al. [5] have proposed a model in which the standard finite element polynomial field is enriched by plane waves in the case of acoustic problems. The enriched field is added to the polynomial one rather than multiplied by it, as was done by previous authors. Therefore it is not continuous at element boundaries and continuity is enforced weakly by Lagrange multipliers. It seems that such an enrichment yields an adequate conditioning of the formulation. Recently, new special wave boundary elements for short wave problems were developed [6]. They incorporate wave shapes into the element shape functions. These new elements demonstrate reduced errors, for a given number of degrees of freedom, of three to five orders of magnitude.

In a previous work, Laghrouche and Bettess implemented this approach and solved a few practical problems such as the scattering of a plane wave by a rigid circular cylinder [7], a progressive plane wave with a local wave number [8] and scattering problems by elliptical cylinders [9] and spheres [10]. The numerical results are in a good agreement with the available analytical solutions. In the case of scattering by a circular cylinder, the wave number was increased and the mesh remains unchanged until a single finite element contains around wavelengths in each spatial direction and therefore the dimension of the whole problem is greatly reduced.

In this paper, at each finite element node, the horizontal and vertical displacements are, first, written as a summation of displacements due to each of the three waves P, S and R (Compressional, shear and Rayleigh waves), then each of the component is expanded in discrete series of displacements with respect to many directions taken between 0 and . The method is implemented for simple problems where pure P-waves, S-waves and R-waves are propagating in the studied medium. The aim is to create finite elements capable of containing many wavelengths per nodal spacing and to keep the mesh unchanged when the wave number increases.

1

A. de La Bourdonnaye, High frequency approximation of integral equations modeling scattering phenomena, Mathematical Modelling and Numerical Analysis, 28, pp.223-241 (1994).

2

J.M. Melenk, On Generalized Finite Element Methods, PhD Thesis, University of Maryland (1995).

3

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

4

P. Mayer and J. Mandel, The Finite Ray Element Method for the Hemholtz equation of scattering: first numerical experiments, UCD/CCM Report 111 (1997).
URL: http://www-math.cudenver.edu/ccm/reports.html

5

C. Farhat, I. Harari and L. Franca, The discontinuous enrichment method, Research Report CU-CAS-00-20, University of Colorado, Boulder, 2000, 1-38.

6

E. Perrey-Debain, J. Trevelyan and P. Bettess, New special wave boundary elements for short wave problems. Comm. Num. Meth. Eng., 2001 (submitted) doi:10.1002/cnm.492.abs

7

O. Laghrouche and P. Bettess, Short wave modelling using special finite elements, Journal of Computational Acoustics, 8, No.1, 189-210 (2000). doi:10.1142/S0218396X00000121

8

O. Laghrouche and P. Bettess, Solving short wave problems using special finite elements - Towards an adaptive approach. The Mathematics of Finite Elements and Applications X 2000; Chapter 10:181-194, (Ed. J.R.Whiteman), Elsevier.

9

O. Laghrouche, P. Bettess and R. Sugimoto, Short wave scattering by elliptical cylinders, 8th ACME Conference, University of Greenwich, Extended Abstracts, M. Cross (Ed.), pp.70-73, 2000.

10

O. Laghrouche, P. Bttess and R. Sugimoto, Analysis of the conditioning number of the plane wave approximation for he Helmholtz equation. Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineeing, ECCOMAS 2000, 1-14.

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