Bloch Floquet Research Articles

Active feedback control of effective mass density and sound transmission on elastic wave metamaterials

With the active feedback control system on elastic wave metamaterials, this research is concentrated on the effective mass density and sound transmission by a harmonic incident sound pressure. The elastic wave metamaterials consist of double plates which are attached by the upper and lower four-link mechanisms being bonded with four lateral resonators. The vertical resonator in every unit cell is jointed by the active feedback control system which is connected by two four-link mechanisms. Using the dynamic effective medium method, the expressions of the effective mass density are obtained and the dynamic responses are presented. Based on the Bloch–Floquet theorem and Poisson summation formula, the sound transmission loss (STL) of this elastic wave metamaterial is shown by the principle of virtual work. This research also shows that the characteristics of the effective mass density and STL can be tuned by the acceleration and displacement feedback in the active control system. Furthermore, it is found that the dynamic response and STL are also changed obviously by different incident angles (including the elevation and azimuth angles), the periodicity spacing of local resonators and the structural damping.

High-frequency homogenization in periodic media with imperfect interfaces.

In this work, the concept of high-frequency homogenization is extended to the case of one-dimensional periodic media with imperfect interfaces of the spring-mass type. In other words, when considering the propagation of elastic waves in such media, displacement and stress discontinuities are allowed across the borders of the periodic cell. As is customary in high-frequency homogenization, the homogenization is carried out about the periodic and antiperiodic solutions corresponding to the edges of the Brillouin zone. Asymptotic approximations are provided for both the higher branches of the dispersion diagram (second-order) and the resulting wave field (leading-order). The special case of two branches of the dispersion diagram intersecting with a non-zero slope at an edge of the Brillouin zone (occurrence of a so-called Dirac point) is also considered in detail, resulting in an approximation of the dispersion diagram (first-order) and the wave field (zeroth-order) near these points. Finally, a uniform approximation valid for both Dirac and non-Dirac points is provided. Numerical comparisons are made with the exact solutions obtained by the Bloch–Floquet approach for the particular examples of monolayered and bilayered materials. In these two cases, convergence measurements are carried out to validate the approach, and we show that the uniform approximation remains a very good approximation even far from the edges of the Brillouin zone.

Plate arrays as a perfectly-transmitting negative-refraction metamaterial

A closely-spaced periodic array of identical thin rigid plates illuminated by incident waves is shown to exhibit properties of a negative refraction metamaterial under certain conditions. The close-spacing assumption is used as a basis for an approximation in which the region occupied by the plate array acts as an effective medium. Effective matching conditions on the plate array boundary are also derived. The approximation allows explicit expressions to be derived to wave scattering problems involving tilted plate arrays. This approximation is tested for its accuracy against an exact treatment of the problem based on Bloch–Floquet theory.Both the exact and effective medium theory predict perfect wave transmission at all wave frequencies through the array when the tilt angle of plates in the array is the reverse of the incident wave direction: the array acts as an all-frequency perfectly-transmitting negative-refraction medium. For certain frequencies the array is also shown to act as an all-angle perfectly-transmitting negative-refraction material.

77/79-GHz Forward-Wave Directional Coupler Component Based on Microstrip and SIW for FMCW Radar Application

A low-cost 77/79-GHz ladder-type forward-wave directional coupler (FWDC) is proposed based on microstrip and substrate integrated waveguide (SIW) for frequency modulated continuous wave (FMCW) radar application in this article. The SIW structures are periodically loaded between two parallel microstrip lines. In the circuit topology of the proposed FWDC, the impedance matching between the even/odd modes is achieved. Meanwhile, an approximate constant phase difference (about 12°) between the even and odd modes is designed within 76–81 GHz. The proposed FWDC is modeled by the Bloch–Floquet theorem, and the design procedure is given. Besides, the full-microstrip ladder-type FWDC is discussed and its theoretical analysis demonstrates that such an FWDC with a weak coupling cannot be obtained using the full-microstrip topology. Finally, the proposed FWDC is fabricated by the PCB technology. To measure its performances, $W$ -band fin-line microstrip-waveguide transition and waveguide package are utilized, and three waveguide packaging cases are measured and discussed. Good agreement between the simulations and measurements is achieved.

Homogenization approach and Floquet-Bloch theory for wave analysis in fluid-saturated porous media with mesoscopic heterogeneities

We consider fluid-saturated poroelastic media whose the mechanical response is governed by the Biot model relevant to a mesoscopic scale. Assuming the material properties being described by periodic functions, to analyze wave propagation in such heterogeneous and anisotropic media, we derive a formulation based on the Floquet-Bloch (FB) wave decomposition which enables to analyze waves within the whole first Brillouin zone associated with the periodic structure. The wave dispersion results obtained by the FB approach are compared with those computed using a model derived by the homogenization based on the asymptotic analysis with respect to the scale parameter. As another new ingredient, the homogenized model is extended to describe media saturated simultaneously by multiple different fluids, so that the model involves new permeability tensors and differs in structure from the model derived earlier. The dispersion analysis by the FB approach leads to a cumbersome quadratic eigenvalue problem to be solved for complex wave numbers. We suggest an efficient filtration strategy to identify the principle propagating modes (the fast and slow compressional waves and the shear waves). For comparison with results of the FB transformation applied at the mesoscopic heterogeneity scale, the homogenized model responses are reconstructed using the corrector results of the homogenization with fixing a finite scale. Numerical examples illustrate very good correspondence of the dispersion results, as computed by both the approaches.

On the Failure of Classic Elasticity in Predicting Elastic Wave Propagation in Gyroid Lattices for Very Long Wavelengths

In this work we investigate the properties of elastic waves propagating in gyroid lattices. First, we rigorously characterize the lattice from the point of view of crystallography. Second, we use Bloch–Floquet analysis to compute the dispersion relations for elastic waves. The results for very long wavelengths are then compared to those given by classic elasticity for a cubic material. A discrepancy is found in terms of the polarization of waves and it is related to the noncentrosymmetry of the gyroid. The gyroid lattice results to be acoustically active, meaning that transverse waves exhibit a circular polarization when they propagate along an axis of rotational symmetry. This phenomenon is present even for very long wavelengths and is not captured by classic elasticity.

Reduction of Bloch-Floquet bending waves via annular acoustic black holes in periodically supported cylindrical shell structures

In recent years, acoustic black holes (ABHs) have revealed as a very effective method for the passive reduction of vibrations in flat plates and straight beams. Those essentially consist of circular indentations and/or side boundaries, whose thicknesses decrease to zero following a power-law profile. Nonetheless, many structures in the aeronautical and naval sectors, among others, involve cylindrical structures with periodic stiffeners or supports. Such periodic structures admit the propagation of Bloch-Floquet (BF) bending waves at some frequency passbands, which may result in strong vibration levels and noise radiation. In this work, we propose the design of annular ABHs to mitigate the problem. A wave finite element model is first used to determine the frequency passbands on an ideal, periodically simply supported cylindrical shell of infinite length, with embedded annular ABHs. Then, a long, but now finite simply supported cylindrical shell is presented to show how the transmissibility between two sections strongly diminishes with the inclusion of ABHs. That confirms annular ABHs as a very effective means of reducing BF wave propagation. Unfortunately, embedding ABHs on the shell weakens its structural rigidity so a thorough analysis is made on the effects of inserting stiffeners in the longitudinal direction to partially remedy that inconvenient. Quite surprisingly, it is shown that the inclusion of stiffeners can even enhance the performance of the ABHs for some configurations. To finally foresee the potential of annular ABHs for practical applications, a finite element simulation is made on a slightly more realistic geometry, resembling a very simplified version of the hull of a submarine vehicle or a gas tank.

Generalized analytical dispersion equations for guided Rayleigh-Lamb (RL) waves and shear horizontal (SH) waves in corrugated waveguides

Corrugated waveguides are periodic structures that exhibit important acoustic features. Features like acoustic bandgaps exhibit the ability to filter acoustic and ultrasonic frequencies. By varying the mean thickness, corrugation height, and periodicity, one can tune the propagating wave modes in the guide and thus can modify the bandgaps. This study aims to obtain a generalized Rayleigh–Lamb equation for corrugated waveguides such that a single equation is sufficient for both flat waveguides such as plates as well as corrugated and tapered plates. Further, the objective was to understand the effects of the corrugation height, periodicity, and mean thickness such that a physics-based predictive design of wave filters can be achieved. Instead of applying the Bloch–Floquet theorem to the displacement function directly, which is conventional, the theorem is applied to the scalar and vector wave potentials obtained from Helmholtz decomposition. The governing equations from these relationships after applying boundary conditions were then solved using a logical root-finding algorithm. To verify the generalized expressions guided wave band structure for a plate was obtained using Rayleigh–Lamb equation and compared with the generalized form setting the corrugation height to zero. The analytical solutions are then validated through a comparison with the results obtained from a rigorous finite element simulation. Finally, the effects on the propagating and evanescent wave modes due to the corrugation height and periodicity are studied for future reference.

Non-dimensional analysis of the elastic beam having periodic linear spring mass resonators

Band gaps appear in the frequency spectra of an elastic beam having periodic spring-mass attachment, acronymed as metabeam. Two widely used beam theories, i.e. Euler–Bernoulli and Timoshenko, are non-dimensionalized in order to obtain the dimensionless periodicity parameters. In this paper, the location and width of band-gaps are investigated due to the variation of these periodicity parameters of a metabeam. Implementing transfer matrix method in conjunction with Bloch–Floquet theorem, a unified methodology has been adopted to estimate the propagation and attenuation bands in a metabeam. This paper provides a guideline towards the designing of wide-band metabeam by tunning the non-dimensional parameters.

On the unwrapping of dispersion curves in the irreducible Brillouin zone by means of a spatial Fourier transform approach

Using Bloch–Floquet boundary conditions implies a periodicity in both the spatial and wavenumber domains, causing the numerically computed wavenumbers to be folded or aliased within the Brillouin zone. A methodology is presented to unwrap the wavenumber spectrum to extended Brillouin zones using a Fourier transform method in the spatial domain, thus providing new insights on the physical meaning and relative amplitudes of aliased wave components. In addition, a procedure is presented to identify out-of-plane (Lamb wave type) components, which are usually of interest for thin structures.The proposed methodologies were applied to finite element based simulations of a 2D periodic reticulated structure consisting of interconnected rectangular struts with orthorombic symmetry. This reticulated structure was used as a toy model for the skeleton in porous media, focussing on the out-of-plane waves. Laser ultrasonic experiments were conducted to verify the numerical results.Both the simulations and measurement results indicate that, in spite of bandgap features induced by the periodicity of the structure, the dispersion behavior of the out of plane Lamb wave modes in the considered type of structure is similar to the one of the Lamb waves of a homogeneous plate with thickness equal to the one of the struts, both for wavenumbers within and outside the Irreducible Brillouin Zone.

Computation of dispersion diagrams for periodic porous materials modeled as equivalent fluids

The application of shift cell technique is presented and discussed for periodic porous media described with equivalent fluid models: as it can be found in literature, it consists in a reformulation of classical Floquet-Bloch (F-B) conditions, in which the phase shift of the boundary conditions, related to wave propagation, is integrated into the partial derivative operator. Consequently, the periodicity is included in the overall behavior of the structure, while continuity conditions are imposed at the edges of the unit cell. Its major advantage stands in allowing the introduction of a generic frequency dependence of porous material behavior, through the resolution a quadratic eigenvalue problem, providing an efficient way to compute the dispersion curves of a porous material modeled as an equivalent fluid. A validation and a computational cost comparison are performed between the shift cell technique and the classical F-B approach, pointing out that the first can provide, among its other advantages, a sensible computational time reduction for this kind of analyses. The derivation of the equivalent acoustic properties of the unit cell from its dispersion characteristics is also investigated. To this aim, group velocity matrix formulation and a branch-tracking algorithm are described. Some test cases are used for validating the proposed methodology.

Chiral flexural waves in structured plates: Directional localisation and control

A new class of elastic waveforms, referred to as “chiral flexural waves”, is introduced for a multi-structure, which encompasses an elastic plate connected to a system of elastic flexural rods with gyroscopic spinners. The junction conditions describing the connection between the plate and the thin flexural rod require logarithmic asymptotics. The directional preference of the system is governed by the motion of gyroscopic spinners. For doubly-periodic chiral multi-structures studied here, parabolic modes associated with strong dynamic anisotropy of Bloch–Floquet waves are identified. Closed form analytical findings are accompanied by numerical simulations, which identify one-way flexural waves propagating along a straight interface in a flexural chiral system, without requiring the presence of Dirac cones on the dispersion surfaces.

Wave polarization and dynamic degeneracy in a chiral elastic lattice.

This paper addresses fundamental questions arising in the theory of Bloch–Floquet waves in chiral elastic lattice systems. This area has received a significant attention in the context of ‘topologically protected’ waveforms. Although practical applications of chiral elastic lattices are widely appreciated, especially in problems of controlling low-frequency vibrations, wave polarization and filtering, the fundamental questions of the relationship of these lattices to classical waveforms associated with longitudinal and shear waves retain a substantial scope for further development. The notion of chirality is introduced into the systematic analysis of dispersive elastic waves in a doubly-periodic lattice. Important quantitative characteristics of the dynamic response of the lattice, such as lattice flux and lattice circulation, are used in the analysis along with the novel concept of ‘vortex waveforms’ that characterize the dynamic response of the chiral system. We note that the continuum concepts of pressure and shear waves do not apply for waves in a lattice, especially in the case when the wavelength is comparable with the size of the elementary cell of the periodic structure. Special critical regimes are highlighted when vortex waveforms become dominant. Analytical findings are accompanied by illustrative numerical simulations.

Effective Description of Anisotropic Wave Dispersion in Mechanical Band-Gap Metamaterials via the Relaxed Micromorphic Model

In this paper the relaxed micromorphic material model for anisotropic elasticity is used to describe the dynamical behavior of a band-gap metamaterial with tetragonal symmetry. Unlike other continuum models (Cauchy, Cosserat, second gradient, classical Mindlin–Eringen micromorphic etc.), the relaxed micromorphic model is endowed to capture the main microscopic and macroscopic characteristics of the targeted metamaterial, namely, stiffness, anisotropy, dispersion and band-gaps. The simple structure of our material model, which simultaneously lives on a micro-, a meso- and a macroscopic scale, requires only the identification of a limited number of frequency-independent and thus truly constitutive parameters, valid for both static and wave-propagation analyses in the plane. The static macro- and micro-parameters are identified by numerical homogenization in static tests on the unit-cell level in Neff et al. (J. Elast., https://doi.org/10.1007/s10659-019-09752-w, 2019, in this volume). The remaining inertia parameters for dynamical analyses are calibrated on the dispersion curves of the same metamaterial as obtained by a classical Bloch–Floquet analysis for two wave directions. We demonstrate via polar plots that the obtained material parameters describe very well the response of the structural material for all wave directions in the plane, thus covering the complete panorama of anisotropy of the targeted metamaterial.

Reconstruction of a local perturbation in inhomogeneous periodic layers from partial near field measurements

We consider the inverse scattering problem to reconstruct a local perturbation of a given inhomogeneous periodic layer in , , using near field measurements of the scattered wave on an open set of the boundary above the medium, or, the measurements of the full wave in some area. The appearance of the perturbation prevents the reduction of the problem to one periodic cell, such that classical methods are not applicable and the problem becomes more challenging. We first show the equivalence of the direct scattering problem, modeled by the Helmholtz equation formulated on an unbounded domain, to a family of quasi-periodic problems on a bounded domain, for which we can apply some classical results to provide unique existence of the solution to the scattering problem. The reformulation of the problem is also the key idea for the numerical algorithm to approximate the solution, which we will describe in more detail. Moreover, we characterize the smoothness of the Bloch–Floquet transformed solution of the perturbed problem w.r.t. the quasi-periodicity to improve the convergence rate of the numerical approximation. Afterward, we define two measurement operators, which map the perturbation to some measurement data, and show uniqueness results for the inverse problems, and the ill-posedness of these. Finally, we provide numerical examples for the direct problem solver as well as examples of the reconstruction in 2D and 3D.

Nonlocal coupled thermoelastic wave propagation band structures of nano-scale phononic crystal beams based on GN theory with energy dissipation: An analytical solution

This paper deals with the band structures of thermoelastic waves in nano-scale phononic crystal or metamaterial beams considering nano-size effects. The nonlocal coupled thermoelastic governing equations are derived using the Green–Naghdi theory of the generalized coupled thermoelasticity with energy dissipation and Eringen’s nonlocal theory. The derived governing equations are analytically solved and the field quantities including the temperature and the deflection are obtained in the closed forms. Using the proposed analytical solution, the transfer matrix between two unit-cells are obtained using the thermal and mechanical continuity conditions on the interfaces between the unit-cells and between the two sections of each unit-cell. The band structures of the phononic crystals are obtained using the Bloch–Floquet theorem. The detailed discussions are presented for the band structures of nonlocal thermoelastic waves in nano-scale aluminum/epoxy phononic crystal or metamaterial beams. Also, the effects of the small-scale parameter and the thickness of the epoxy layers on the band structures are studied and discussed by using the derived analytical solution.

Fabry-Pérot Huygens' metasurfaces: On homogenization of electrically thick composites

Realization of the anomalous refraction effects predicted by Huygens' metasurfaces (HMS) have required tedious and time-consuming trial-and-error numerical full-wave computations. It is shown herein that these requirements can be alleviated for transverse magnetic (TM) propagation by a periodic dielectric-based HMS consisting of an electrically thick array of cascaded Fabry-P\'erot etalons. This "Fabry-P\'erot HMS" (FP-HMS) is easily designed to mimic the local scattering coefficients of a standard zero-thickness HMS (ZT-HMS) which, according to homogenization theory, should result in the desired anomalous refraction. To probe the characteristics of this practical FP-HMS, a method based on Floquet-Bloch (FB) analysis is derived for predicting the fields scattered from it for arbitrary angles of incidence. This method produces simple closed-form solutions for the FB wave amplitudes and the resulting fields are shown to agree well with full-wave simulations. These predictions and full-wave simulations verify the applicability of homogenization and scattering properties of zero-thickness HMS's to thick structures. They also verify the proposed semi-analytical microscopic design procedure for such structures, offering an effective alternative path to implementation of theoretically envisioned intricate field manipulating devices.

Finite element modeling of micropolar-based phononic crystals

The performance of a Cosserat/micropolar solid as a numerical vehicle to represent dispersive media is explored. The study is conducted using the finite element method with emphasis on Hermiticity, positive definiteness, principle of virtual work and Bloch–Floquet boundary conditions. The periodic boundary conditions are given for both translational and rotational degrees of freedom and for the associated force- and couple-traction vectors. Results in terms of band structures for different material cells and mechanical parameters are provided.

Nested Bloch waves in elastic structures with configurational forces.

Small axial and flexural oscillations are analysed for a periodic and infinite structure, constrained by sliding sleeves and composed of elastic beams. A nested Bloch-Floquet technique is introduced to treat the nonlinear coupling between longitudinal and transverse displacements induced by the configurational forces generated at the sliding sleeve ends. The action of configurational forces is shown to play an important role from two perspectives. First, the band gap structure for purely longitudinal vibration is broken so that axial propagation may occur at frequencies that are forbidden in the absence of a transverse oscillation and, second, a flexural oscillation may induce axial resonance, a situation in which the longitudinal vibrations tend to become unbounded. The presented results disclose the possibility of exploiting configurational forces in the design of mechanical devices towards longitudinal actuation from flexural vibrations of small amplitude at given frequency. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 1)'.

An FDTD Method for Fully Anisotropic Periodic Structures Impinged by Obliquely Incident Plane Waves

A finite-difference time-domain (FDTD) method is developed to analyze electromagnetic scattering from 3-D fully anisotropic periodic structures impinged by obliquely incident plane waves. Starting from Maxwell’s curl equations, we employ material transformation matrices to link the update of the electric and magnetic fields in the FDTD method. The problem under consideration is Bloch–Floquet periodic in the horizontal directions but finite in the vertical direction. At the FDTD truncation boundaries in the horizontal directions, the Bloch–Floquet periodic boundary conditions (BPBCs) are applied, while the perfectly matched layer (PML) absorbing boundary condition is implemented for the vertical direction. We design three different anisotropic models in 3-D simulations to validate our method with a commercial software package COMSOL. These examples show the accuracy and efficiency of the FDTD method for analyzing the propagation characteristics, including reflectance, transmittance, absorptance, and complex reflection and transmission coefficients for the fully anisotropic periodic structures.