Computationally efficient finite element method for predicting wave propagation in periodic structures

The wave propagations in the periodic structure composed of auxetic star-shaped honeycombs are investigated. The matrix of the periodic structure is filled with polymer. The effective material properties of the unit cell of the periodic structure such as the Poisson’s ratio, the Young’s modulus, and the shear modulus are varied with different filling materials. The finite element method is used to solve the wave propagation within the auxetic star-shaped honeycombs. The dispersion analysis of the band structures and iso-frequency contour are presented. The auxeticity is found to have a significant effect on the wave propagation in the honeycomb structure, and the collimation of the auxetic star-shaped honeycomb structure is also studied. Effects of different types of filling on the wave propagation are also investigated. Among three filling types, the outer-filled honeycomb has the best self-collimation effect. The self-collimation of the polymer-filled auxetic honeycomb structure presented in this study can be applied to various acoustic devices.

In a companion paper the phenomenon of flexural-longitudinal wave coupling in an infinite dual-beam periodic structure with transverse connection is investigated. The remarkable finding obtained that there was such a periodic structure which conveyed fundamentally three symmetric and three antisymmetric characteristic coupled waves. In this paper, a methodology is proposed to realize the direct identification of characteristic waves from the responses of a finite periodic structure. This represents a considerable departure from traditional methods, in which the assumption of infinite dimension of a periodic structure is implied. It is the inverse process of the traditional studies which obtain characteristic wave-types based on the assumption and then continue the studies with the wave-types. A general expression for the individual transition matrix of one periodic element is defined and derived from two adjacent junction-mobilities of a finite periodic structure. A common transition matrix for all elements is then constructed using responses based on the mathematical analysis of the relations between responses or junction-mobilities, individual transition matrices and characteristic wave-types. Finally, a specific experimental structure and approach was designed to extract the characteristic waves from the responses. All the symmetrical and antisymmetrical characteristic wave-types, except the pairing of near-field wave-types, are extracted from the common transition matrix of the finite periodic structure, demonstrating the feasibility of the method presented.

A theory of wave propagation in periodic structures with smoothly varying parameters is developed. Consideration is provided for the case when both periodicity and modulation are in one dimension. The theory is based on the Wannier-function expansion. As an example, it is shown that electromagnetic waves can propagate along effective waveguides, which are interpreted in terms of defect modes. Related effects such as anomalous scattering, self-channeling (gap solitons) in nonlinear periodic media), and miniband spectra are discussed.

In this article, a method for wave propagation analysis of textile composites at a mesoscopic level is presented for the first time in the vibro-acoustic and ultrasonic frequency range. The method combines a reduction method, allowing for computational cost saving, and the Wave Finite Element Method (WFEM), associating the Periodic Structure Theory (PST) and standard Finite Element Method (FEM). The PST allows to model only a unit cell of a periodic structure for analyzing the wave propagation properties of the entire structure. A comparison of standard vibrational analysis of textile composite structures, using homogenized properties, is compared with the presented method. It is shown for different types of weaves that using the standard method results in errors in the dispersion curves.

This review covers the propagation of acoustic surface waves (ASW) in periodic structures. The authors discuss the main types of ASW: Rayleigh waves, Gulyaev?Bleustein waves, Love waves. Shear surface waves which can propagate along a periodically uneven surface of an elastic solid are described in detail. The authors rigorously treat the Bragg reflection of ASW from periodic arrays of grooves on the substrate for the cases of normal, oblique, and side incidence. Reciprocal conversion effects between bulk and surface waves in an array whose period is close to the ASW wavelength are discussed, together with such related effects as the interaction between Rayleigh waves and Lamb modes, acoustic wave transmission through a gap in a piezoelectric material, etc. Laser excitation of ASW and magnetostatic wave propagation in periodic structures are also reviewed.

New results in the theory of periodic structures (PSs), obtained on the basis of stepped transmission line model, are presented. It is shown that PSs peculiarly transform the wave impedance, inducing unusual properties of the wave medium. Stepped transmission line model makes it possible to determine the fundamental relationships of physics and mathematics for PSs. Model of impedance delta-inhomogeneities allows to get simple analytical expressions for PS characteristics. Two-phase PSs for which band gap frequencies are twice lower than those for traditional single-phase PSs are considered.

Vichnevetsky Robert. Wave propagation in periodic structures and quantum mechanics. In: Bulletin de la Classe des sciences, tome 75, 1989. pp. 135-163.

Recently significant advances have been made in the development of magnetostatic wave devices for microwave signal processing. This technology, based on slow dispersive magnetostatic wave propagation in magnetically biased films of Yttrium iron garnet, grown by liquid phase epitaq on substrates of gadolinium gallium garnet, has many similarities to SAW technology, and comparisons will be made as appropriate, will be briefly reviewed, together with methods for efficient microstrip transduction, weighting, controlled tapping, and terminations. Wave propagation in periodic structures and application of these structures to the development of tunable high Q two-port resonators will be discussed. The status oblique incidence reflective array devices will be discussed including possible reflecting structures, weighting, and dispersion modification. The paper concludes with a survey of possible magnetostatic devices for GHz signal processing. Wave characteristics

The impact of damping on the sound transmission loss of locally resonant metamaterial plates

Rayleigh wave manipulation based on phase shift between two periodic surface corrugations

This paper addresses the question of damage detection of cylindrical structures using finite element and periodic structure theory. The structure is modeled considering two waveguides connected through a coupling element, simulated as the defect. Wave characteristics in the waveguides are evaluated using a Wave Finite Element method (W.F.E.M) based on the analysis of the anomalies which affect the elastic wave travelling in the structure. Reflection and transmission coefficients of the wave modes which are incident to the coupling element are provided by a diffusion matrix. Then forced response of the pipe with and without transversal defect is predicted using wave decomposition. Numerical examples are given. These concern an isotropic pipe, for which dispersion curves are provided and diffusion coefficients in presence of a transversal and a longitudinal defect are predicted.

Multi-scale wave propagation modelling for two-dimensional periodic textile composites

An alternative definition of the wavelet transform matrix used for the sparsification of the moment matrix in the method of moments (MoM) is developed. In scattering problems, if the scatterer is a finite periodic structure, the moment matrix has a periodic structure. By using the periodicity, we establish a new approach to construct the wavelet transform matrix that yields more efficient sparsification. The scattering of plane waves from finite periodic structures, e.g., linear conductor array, is considered to demonstrate the effectiveness of the proposed method. Numerical results indicate that the proposed method yields a sparser coefficient matrix and achieves more significant saving in CPU time compared to the conventional approach, with sufficiently accurate solutions.

Locally resonant metamaterials have attracted lots of research interests for the application of vibration suppression which is a fundamental problem but remains a big challenge in the engineering field. The transverse wave propagation in a beam is through the transmission of the shear force and bending moment. Most designs of metamaterials in the existing literature exploit translational local resonators to induce reaction force to prevent the transmission of the shear force, hence the wave propagation. This paper studies a metamaterial beam attached with torsional local resonators. The reaction moments generated by the torsional resonators are expected to neutralize the bending moment in the beam, thus preventing the wave propagation. The existence of torsional resonators leads to the moment discontinuity conditions which cannot be directly taken into account using the Euler beam theory. Based on the Timoshenko beam theory, the band structure analysis is developed through a modal analysis based on the infinite periodic local resonator structure. The numerical results reveal that the locally resonant frequency corresponds to the upper bound of the band gap. Both infinitely long and finitely long beams are also modeled using finite element method. The transmittance is calculated to verify the band structure analysis.

Interactions beyond nearest neighbors in a periodic structure: Force analysis