Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 9651430, 15 pages

http://dx.doi.org/10.1155/2016/9651430

## Periodic Properties of 1D FE Discrete Models in High Frequency Dynamics

Faculty of Electrical and Control Engineering, Gdańsk University of Technology, Ulice Narutowicza 11/12, 80-234 Gdansk, Poland

Received 16 September 2015; Revised 11 December 2015; Accepted 13 December 2015

Academic Editor: Gen Qi Xu

Copyright © 2016 A. Żak et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Finite element discrete models of various engineering 1D structures may be considered as structures of certain periodic characteristics. The source of this periodicity comes from the discontinuity of stress/strain field between the elements. This behaviour remains unnoticeable, when low frequency dynamics of these structures is investigated. At high frequency regimes, however, its influence may be strong enough to dominate calculated structural responses distorting or even falsifying them completely. In this paper, certain computational aspects of structural periodicity of 1D FE discrete models are discussed by the authors. In this discussion, the authors focus their attention on an exemplary problem of 1D rod modelled according to the elementary theory.

#### 1. Introduction

Periodic structures due to their unique dynamic properties are widely applied in many real engineering structures, such as isolation devices, acoustic filters, and dampers. The dynamic behaviour of periodic structures, especially the existence of frequency band gaps and negative refraction indexes, has been under investigation by many researchers [1–5]. Many analytical and discrete methods have been developed and employed for the analysis of dynamic characteristics of periodic structures.

Based on the literature of the subject, the following list of the applied methods can be made: the lump mass method [6], the multiple scattering method [7], the transfer matrix method [8], the plane wave expansion (PWE) method [9], the finite difference method (FDM) [10], the finite element method (FEM) [11], the wavelet method [12], the Dirichlet-to-Neumann map method [13], the boundary element method (BEM) [14], as well as the spectral finite element method in the frequency domain (FD-SFEM) [15], and the spectral finite element method in the time domain (TD-SFEM) [16].

It should be realised that the methods mentioned above have numerous limitations. Analytical methods such as the lump mass method, multiple scattering method, and the FD-SFEM are not suitable for investigation of geometrically complex structures. These limitations are nonexistent in the case of discrete methods such as the FEM and the TD-SFEM. On the other hand, the results of numerical simulations obtained by the use of the FEM or the TD-SFEM are not as accurate as the results obtained by the use of analytical methods. Their accuracy can be improved by an adaptive increase in the number of FEs (-method) or in the degree of approximation polynomials (-method) [17]. It is worth noting that the accuracy of - and -methods has been studied extensively by many researchers in the context of global, local, pollution, or dispersion errors [18–22]. Interesting results on the comparison between adaptive - and -methods and nonuniform rational B-splines (NURBS) can be found in [23, 24]; however, periodic properties of the numerical models used have stayed beyond such interest.

It is known that an increase in the number of FEs (-method) is not as effective as an increase in the degree of approximation polynomials (-method). In the case of equidistant node, distribution within FEs a significant increase in the degree of approximation polynomials may lead to their uncontrollable oscillations near the edges of FEs, known as Runge’s phenomenon [25, 26]. For this reason in numerical simulations by use of the FEM, the degree of approximation polynomials is typically not higher than 3. Contrary to this, Runge’s phenomenon is not observed in the case of the TD-SFEM, which is based on nonequidistant node distributions. The degree of approximation polynomials used by the TD-SFME is theoretically unlimited, but in numerical simulations mainly Chebyshev or Legendre polynomials of the fifth degree are applied. A further increase in the degree of approximation polynomials improves the accuracy of numerical results, but visible frequency band gaps appear at the high end of the natural frequency spectra [26]. This effect is very similar to the behaviour of periodic structures, but the source of structural periodicity has a different nature.

The source of this periodicity comes from the discontinuity of stress/strain field between FEs [17, 27]. Thus, the silent assumption about the representation of FE discrete models of continuous media/structures is not valid, despite the fact that it is not always visible based on their structural dynamic responses at low frequency regimes. However, there are certain regimes (i.e., high frequency dynamics and wave propagation) where this silent assumption may lead to significant errors due to the fact that overlooked structural periodicity of FE discrete models strongly manifests itself, as presented in this paper.

In this paper certain computational aspects of structural periodicity of 1D FE discrete models are discussed by the authors. Discrete models characterised by equidistant node distribution and nonequidistant distributions based on the roots of Chebyshev and Legendre polynomials were investigated. As approximation functions, Chebyshev and Legendre as well as cubic B-spline polynomials were tested. In this discussion, the authors focus their attention on an exemplary problem of 1D rod modelled according to the elementary theory of rods.

#### 2. Finite Element Models as Periodic Structures

Bloch’s theorem is a very powerful tool used by physicists to investigate the behaviour of electrons in crystalline structures. However, Bloch’s theorem is applicable in a much broader context. This is thanks to the dual nature of electrons that can be treated as matter waves, also known as de Broglie waves. In fact, Bloch’s theorem can be successfully applied not only to explain the behaviour of wave related phenomena in periodic media, such as photonic and phononic crystals, but also to explain the behaviour of other types of mechanical structures characterised by certain periodic properties.

In a 1D case, Bloch’s theorem states that solution to a wave propagation problem in a periodic medium, also known as a Bloch wave, can be split up into the product of two waves: periodic wave of the same periodicity as the medium and plane-wave [28]:where denotes the wave number and is the imaginary unit.

A 1D bimaterial periodic medium built up from periodically spaced cells is presented in Figure 1. The lengths of particular material regions within a single cell can be denoted as and . These regions are characterised by different values of elastic moduli and as well as different values of material densities and . Thus, these two regions are also characterised by different values of wave propagation phase velocities and or, alternatively, by different values of wave numbers and , but they are associated with common frequency through the relation:where phase velocities and are expressed as