Keywords: discrete Green's functions, unbounded domain, periodic material.

In this paper we present a numerical method suitable for solution of steady state wave problems in which the material properties vary periodically throughout the domain. Discrete Green's functions, in finite element sense, are selected as the representatives of the problems. The Green's functions are evaluated on unbounded domains and this involves satisfaction of radiation conditions in the solutions. Having found the discrete Green's functions, one can use them in solution of more general bounded or unbounded problems. To this end a simple superposition algorithm is required, especially for general unbounded problems where the effects of the radiation condition should be taken into account and the discrete Green's functions can play a prominent role. For problems with homogenous materials, there are a number of approaches for considering the effect of the radiation on the unbounded domain. The boundary element method (BEM) is one of the approaches. Apart from the problem of dealing with the inherent singularity of the functions in the BEM, the Green's functions are not always available. An example is a model comprising a set of periodic mechanical properties that we shall deal with in this paper.

Like many other engineering problems the finite element method (FEM) may also be used for general unbounded domains. Then the problem is considered to consist of a bounded part, discretized with elements, and an unbounded part. Nevertheless, for the unbounded part some special treatments are usually required. The use of infinite elements and artificial boundary conditions are recommended by many researchers during the past thirty years. For the such categories of treatments, the reader may consult review papers [1,2,4] or reference [3].

The aforementioned treatments for using the FEM are basically devised for models with homogenous mechanical properties. For models with periodic properties the available information, there is little to be found in the literature. Dealing with such problems needs behaviour prediction of the FEM when the entire unbounded domain is discretized with elements. For modelling unbounded domains with homogenous properties using the FEM, an early work by Thatcher is for Laplace's equations [5]. Somewhat similar approaches have been introduced by other scientists. The history can be found in reference [6]. A breakthrough has recently been introduced by the authors in [7] assuming that the entire unbounded domain is discretized into similar patterns of elements. The model is capable of solving problems with periodic material properties.

The formulation given in this paper is the extension of the one given in [7]. Here, the principles of Floquet theory [8] for the solution of partial differential equations with periodic coefficients are used. First, the fundamental exponential-like wave bases are obtained through the dispersion relations and then the radiation conditions are satisfied by selecting the wave bases. For selecting the wave bases, a quadrant of the main unbounded domain together with a set of appropriate boundary conditions is considered. For satisfaction of the boundary conditions a discrete transformation proposed by the authors in [7,9] is used.

Application of the method is shown on a sample problem and the results are compared with those obtained from solution of a problem with homogenized material. Excellent agreement is seen between the amplitudes and the wavelengths of the solutions. However, the details of the deformation are just seen in the results of the numerical solution proposed. The method we present in this paper is suitable for more general complicated cases where homogenization cannot easily be used.

1

K. Gerdes, "A summary of Infinite Element formulations for exterior Helmholtz problems", Comput. Methods Appl. Mech. Engrg., 164, 95-105, 1998. doi:10.1016/S0045-7825(98)00048-6

2

R.J. Astley, "Infinite elements for wave problems: a review of current formulations and assessment of accuracy", Int. J. Numer. Methods Eng., 49, 951-976, 2000. doi:10.1002/1097-0207(20001110)49:7<951::AID-NME989>3.0.CO;2-T

3

O.C. Zienkiewicz, R.L. Taylor, "The finite element method", 5th edition, Butterworth-Heineman, 2000.

4

S. Tsynkov, "Numerical solution of problems on unbounded domains. A review". Appl. Numer. Math., 27, 465-532, 1998. doi:10.1016/S0168-9274(98)00025-7

5

T.W. Thatcher, "On the finite element method for unbounded regions", SIAM J. Numer. Anal., 15(5), 466-477, 1978. doi:10.1137/0715030

6

J.P. Wolf, "The Scaled Boundary Finite Element Method", Wiley, 2003.

7

B. Boroomand, F. Mossaiby, "Dynamic solution of unbounded domains using finite element method: Discrete Green's functions in frequency domain", Int. J. Numer. Methods Eng., 2006, in press. doi:10.1002/nme.1670

8

P. Kuchment, "Floquet Theory for Partial Differential Equations", Operator Theory, Advances and Applications; Vol. 60; Boston, Birkhäuser, 1993.

9

B. Boroomand, F. Mossaiby, "Generalization of robustness test procedure for error estimators. Part I: formulation for patches near kinked boundaries", Int. J. Numer. Methods Eng., 2005, 64, 427-460. doi:10.1002/nme.1377

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