Keywords: applied mechanics, wave motion, infinite domains, artificial boundary, halfspace condition, finite element.

This paper presents an iterative method for solving two dimensional wave problems in infinite domains. The method is based on an iterative variation of fictive boundary conditions on an artificial finite boundary. The finite computational domain is in each iteration subjected to actual boundary conditions and to different (Dirichlet or Neumann) fictive boundary conditions on the artificial boundary. The halfspace Dirichlet to Neumann (DtN) operator is used only to determinate new fictive boundary conditions and is not included into a finite element formulation. Thus any finite elements can be used and the method is especially applicable for computing higher harmonics.

In solving wave problems in infinite domains the main problem is to satisfy the radiation condition - the boundary condition at infinity. Radiation condition is satisfied automatically as a part of the fundamental solution in the boundary element method. Unfortunately, the fundamental solution is not always available. Although the boundary-element method is regarded as the most powerful procedure for modeling the unbounded medium, it requires a strong analytical and numerical background.

More flexible is the finite element method. The infinite domain is first truncated by introducing an artificial finite boundary (). On this boundary some fictive boundary conditions must now be imposed. This is the critical step because they must totally eliminate all reflected waves and they must be simple enough. We want that the presented method will be applicable also for computing higher harmonics. In this way the halfspace operator will not be introduced into the finite computational domain as in references [1,2]. Therefore any finite elements can be used in case of numerical solving of the problem (by using FEM).

In the presented method an iterative solution for solving wave problems in infinite domains is obtained. The infinite domain outside of the artificial boundary is represented with halfspace operator. The exact non-local operator () is a solution for tractions due to prescribed some Dirichlet boundary conditions on the artificial boundary. The radiation condition is exactly satisfied. In the exact formulation is the operator in the integral form and it is usually not simple enough to use in the finite element formulation on the artificial boundary. Thus using a local one usually approximates a non-local operator. To obtain local operators () some asymptotic expansions for Hankel functions according to the independent value of the product of the wave number () and of the location of the artificial boundary () must be introduced. In the presented method asymptotic local operators obtained by Bayliss and Turkel[4] are used. The finite computational domain is subjected in each iteration to actual boundary conditions and to different (Dirichlet or Neumann) fictive boundary conditions on the artificial boundary (). Tractions on as a result of Dirichlet boundary conditions first must be computed. Analogously displacements on the artificial boundary as a result of fictive Neumann boundary conditions must be obtained. The line which connects the two obtained interior values is then projected on the Dirichlet to Neumann operator ( or ). The new fictive boundary conditions on for the next iteration are so obtained. In the method the line practically represents the solution for all actual boundary conditions in the space inside of the artificial boundary, except radiation boundary condition, which is approximately represented with the line . While the halfspace operators are radial dependent accuracy of the method of course depends on the location of the artificial boundary. The prescribed procedure is iterative but it was presented in reference [4] that there are practically no difference in results already between the third and the fourth iteration. In many cases the procedure after the second iteration can be stopped. In the method it is also possible to use different fictive boundary conditions in the first iteration. Thus better results can be obtained in the second and also in the third iteration.

The method in this paper was tested on two dimensional out of plane problems. Future work will include the extension of these ideas to two dimensional in plane problems. Finally we expect that the method will be applicable for solving three dimensional problems.

1

D. Givoli, J.B. Keller, "Non-reflecting Boundary Conditions for Elastic Waves", Wave Motion, 12, 261-279, 1990. doi:10.1016/0165-2125(90)90043-4

2

P.M. Pinsky, L.L. Thompson, "Local high-order radiation boundary conditions for the time-dependent structural acoustic problem", Journal for Acoustical Society of America, 91 (3), 1320-1335,1992. doi:10.1121/1.402461

3

A. Bayliss, E. Turkel, "Radiation Boundary Conditions for Wave Like Equations ", Communications on Pure and Applied Mathematics, XXXIII, 707-725, 1980. doi:10.1002/cpa.3160330603

4

M. Premrov , A. Umek, "An alternative method for solving elastodynamics in infinite domains", Proceedings of the Fourth conference on structural dynamics Eurodyn 99, Prague, 7.-10. June 1999.

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