Keywords: boundary element method, elastodynamics, fictitious eigenvalues.

It is well known that when the boundary element method (BEM) is applied to exterior acoustic problems, then unrealistic fictitious eigenfrequencies appear; thus the overall numerical solution becomes inaccurate in certain large regions of frequencies [1]. In brief, the fictitious eigenvalues of an exterior problem under Neumann boundary conditions are those of its dual internal problem under Dirichlet boundary conditions and vice-versa. An effective remedy to this shortcoming is the dominating technique of Burton and Miller [2] while a detailed literature survey of most other known techniques has been previously reported by the authors [3].

Similar fictitious eigenvalues appear also in the BEM solution of exterior time-harmonic elastodynamics [4], which is the topic of this paper. However, apart from older theoretical comments, a detailed report on computational results related to the numerical treatment of this particular shortcoming in exterior elastodynamics is still lacking in the literature. Even for the simplest case of the combined Helmholtz integral equation formulation (CHIEF) method [1], which is well known in acoustics, numerical results in elastodynamics were not available in the literature. The disadvantage of CHIEF is that it uses additional equations that are written for internal nodes that should be carefully selected. As a result of to the larger number of equations than the boundary unknowns, a least squares scheme should be applied.

Within this context, this paper proposes a new technique to circumvent the fictitious eigenfrequencies in the BEM solution of the exterior time-harmonic elastodynamics and thoroughly investigates its performance. The novel feature of this paper is the derivation of a new elastodynamical operator for the derivation of the initial integral equation, which was obtained through the following two steps.

First, the Naviér-Cauchy equation was written in vector form and was compared with the equivalent acoustic equation. From this comparison a correspondence of partial differential operators between elastodynamics and acoustics was obtained. Furthermore, by expressing in the acoustical analogue the sound velocity in terms of the bulk modulus and mass density , the original internal equation and its derivative were written in a modified way so that material properties appear as well. As a result, the comparison yielded that the elasticity matrix corresponds to the bulk modulus .

Second, in order to avoid the hypersingular integrals involved in the differentiated integral equations, in this work it was decided to apply a modified scheme previously proposed by Cunefare et al. [5], according to which the differentiation is performed perpendicularly to an internal surface, for example an offset of the real domain boundary.

By intuitively extending the meaning of the normal derivative over the internal boundary to a more complex operator that includes both geometrical and material quantities, the aforementioned two steps were successfully combined, thus a new elastodynamical operator suitable for the differentiation of the initial integral equation was identified.

The proposed technique was applied to a cubic enclosure of unit size excited by a singular source. The numerical study includes many alternative choices (real and imaginary) of the coupling coefficient between the original and the differentiated integral equation and finally proposes a specific formula for which the most accurate numerical solution is achieved. It was also found that the proposed method is more accurate than the usual CHIEF method [1]. This is an important conclusion that suggests that the new method might replace the CHIEF method; an additional reason for the replacement might be the instabilities coming from the least squares procedure used in the CHIEF method.

1

H.A. Schenck, "Improved integral formulation for acoustic radiation problems", J Acoust Soc Am, 44, 45-58, 1968. doi:10.1121/1.1911085

2

A.J. Burton, and G.F. Miller, "The application of integral equation methods to the numerical solution of some exterior boundary-value problems", Proc Roy Soc London Ser A., 323, 201-210, 1971. doi:10.1098/rspa.1971.0097

3

C.G. Provatidis, and N.K. Zafiropoulos, "On the 'interior Helmholtz integral equation formulation' in sound radiation problems", Engineering Analysis with Boundary Elements, 26, 29-40, 2002. doi:10.1016/S0955-7997(01)00079-0

4

J.D. Achenbach, G.E. Kechter, Y.L. Xu, "Off-boundary approach to the Boundary Element Method", Comp Meth Appl Meth Eng, 70, 191-201, 1988. doi:10.1016/0045-7825(88)90157-0

5

K.A. Cunefare, G. Koopmann, K. Brod, "A boundary element method for acoustic radiation valid for all wavenumbers", J.Acoust.Soc.Am., 85(1), 39-48, 1989. doi:10.1121/1.397691

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