Finite Periodic Structures Research Articles

Ultra-wide band gap and wave attenuation mechanism of a novel star-shaped chiral metamaterial

A novel hollow star-shaped chiral metamaterial (SCM) is proposed by incorporating chiral structural properties into the standard hollow star-shaped metamaterial, exhibiting a wide band gap over 1 500 Hz. To broaden the band gap, solid single-phase and two-phase SCMs are designed and simulated, which produce two ultra-wide band gaps (approximately 5 116 Hz and 6 027 Hz, respectively). The main reason for the formation of the ultra-wide band gap is that the rotational vibration of the concave star of two novel SCMs drains the energy of an elastic wave. The impacts of the concave angle of a single-phase SCM and the resonator radius of a two-phase SCM on the band gaps are studied. Decreasing the concave angle leads to an increase in the width of the widest band gap, and the width of the widest band gap increases as the resonator radius of the two-phase SCM increases. Additionally, the study on elastic wave propagation characteristics involves analyzing frequency dispersion surfaces, wave propagation directions, group velocities, and phase velocities. Ultimately, the analysis focuses on the transmission properties of finite periodic structures. The solid single-phase SCM achieves a maximum vibration attenuation over 800, while the width of the band gap is smaller than that of the two-phase SCM. Both metamaterials exhibit high vibration attenuation capabilities, which can be used in wideband vibration reduction to satisfy the requirement of ultra-wide frequencies.

Reduced-order modeling for time domain analysis of finite periodic structures with absorbing boundary conditions

A reduced-order model of finite periodic structures with absorbing boundary conditions (ABCs) and localized time-dependent excitations is proposed. 1D-periodic structures whose cells can represent any 2D or 3D arbitrary substructures are considered. The key steps for modeling an ABC in the time domain are: (i) expression of the impedance matrix with the wave finite element (WFE) method; (ii) partial rational decomposition of the impedance matrix; (iii) introduction of vectors of supplementary variables to describe the ABC in the time domain. In this paper, a model reduction approach is proposed to speed up the computation of the ABCs. The focus is on the reduction of the number of internal degrees of freedom of substructures, and the reduction of the number of degrees of freedom at the substructure interfaces. The approach allows the consideration of complex substructures with large FE models, and the computation of the time response of periodic structures at a low computational cost. Numerical experiments are carried out to highlight the relevance of the proposed approach. Specifically, the analysis of a metamaterial structure containing nonlinear resonant substructures is carried out with the proposed approach. Results show that, when compared to the linear case, band gaps in periodic structures with nonlinear substructures can be significantly improved.

Computation of the dynamic response of finite two-dimensional periodic structures

Computation of the dynamic response of finite two-dimensional periodic structures

Computation of the dynamic scalar response of large two-dimensional periodic and symmetric structures by the wave finite element method

In the past, the study of periodic media mainly focused on one-dimensional periodic structures (meaning periodic along one direction), on the one hand to determine the dispersion curves linking the frequencies to the wavenumbers and on the other hand to obtain the response of a structure to an external excitation, both for bounded or unbounded structures. In the latter case, effective approaches have been obtained, based on methods such as the Wave Finite Element (WFE). Two-dimensional periodic media are more complex to analyse but dispersion curves can be obtained rather easily as in the one-dimensional case. Obtaining the steady state response of two-dimensional periodic structures to time-harmonic excitations is much more difficult than for one-dimensional media and the results mainly concern infinite media. This work is about this last case of the steady state response of finite two-dimensional periodic structures to time-harmonic excitations by limiting oneself to structures described by a scalar variable (acoustic, thermal, membrane behaviour) and having symmetries compared to two orthogonal planes parallel to the edges of a substructure. Using the WFE for a rectangular substructure and imposing the wavenumber in one direction, we can numerically calculate the wavenumbers and mode shapes associated with propagation in the perpendicular direction. By building solutions with null forces on parallel boundaries, we can decouple the waves in the two directions parallel to the sides of the rectangle. The solution of each of these two problems is obtained by a fast Fourier transformation giving the amplitudes associated with the waves. By summing the contributions of all these waves we obtain the global solution for a two-dimensional periodic medium with a large number of substructures and a low computing time. Examples are given for the case of a two-dimensional membrane with many substructures and different types of heterogeneities.

Application of periodic structural wave impeding block for vibration isolation in foundation

The wave impeding board (WIB) is frequently integrated beneath dynamic machinery, tracks, and subgrades to counteract vibrations emanating from artificial sources. However, conventional WIBs have exhibited a limited isolation frequency band due to their dependence on the soil cut-off frequency of soil. Furthermore, the vibration sources typically encompass intricate frequency components spanning low, medium, and high frequencies. To overcome the technical limitations of WIBs relying on the cut-off frequency of soil, a new periodic structural wave impeding board (PSWIB) is proposed based on the principles of phononic crystals. Theoretical and numerical analyses demonstrate that PSWIB exhibits bandgap characteristics, with the attenuation range achieved by finite periodic structures aligning with the bandgap of an infinite PSWIB. Maximum amplitude reductions of 47 dB and 65 dB are achieved within the vibration attenuation range. Compared to traditional WIB, PSWIB surpasses the constraints imposed by the cut-off frequency of soil and allow for the design of constituent parameters based on the characteristics of the vibration source, enabling effective isolation of the target frequency vibrations.

Wave propagation, bi-directional reflectionless, and coherent perfect absorption-lasing in finite periodic PT-symmetric photonic systems.

With consideration of parity-time (PT) symmetry, Lorentz reciprocity theorem, and real Bloch phase, we propose a generalized parametric space for any PT-symmetric unit cells that can comprehensively illustrate the PT phase transition, Bloch phase, and necessary conditions of exotic wave scattering in the general finite periodic PT photonic structures. We put forward rigorous and formal expressions of bi-directional reflectionless and coherent perfect absorption and lasing (CPAL) for the finite one-dimensional PT photonic structures. With a new concept of the parametric space, we demonstrate the necessary PT phases of general unit cells, which result in the abnormal bi-directional reflectionless and CPAL effects. Moreover, thanks to parametrization, analytical formulas for complex relative permittivities of the unit cells composed of subwavelength gain-loss heterostructure are derived to provide a guideline for manipulating different PT scattering events. We accordingly study several one-dimensional PT photonic systems to achieve exotic wave scattering enabled by PT-symmetry. We believe this work may offer a theoretical underpinning for studying extraordinary wave phenomena of PT-symmetric photonics and may open avenues for manipulation of light.

Resonance mode analysis of finite plate strip with acoustic black holes: the gap between bandgap and attenuation band

Acoustic Black Hole (ABH) lattice structures show promise for achieving broadband bandgap in lightweight design. Existing ABH lattice research usually assumes that the bandgaps of infinite periodic structure are roughly the same as the attenuation band with strong energy attenuation in the finite counterpart. This work is concerned with comparison of the real attenuation bandwidth of finite periodic ABH structures and the bandgap of the infinite counterpart. The plate strips consisting of different numbers of periodic ABH elements are considered. It is observed that the bandgap-predicted attenuation band is split into two narrow attenuation bands, so that the periodic ABHs fails to ensure a broadband and continuous attenuation in the finite scenarios. Results show that there are resonance modes of finite plate strips with ABHs falling into the bandgap due to the boundary reflection and would result in high transmission peaks within the bandgap-predicted attenuation band. This unexpected phenomenon suggests the gap between the attenuation bandwidth of finite periodic structure and the bandgap of the corresponding infinite one. Analysis shows that the resonance mode of finite plate strips, which reduces the bandgap-predicted attenuation bandwidth, can be tuned by geometrically disturbing the periodicity and symmetry of the finite periodic configuration.

Fast Subentire-Domain Basis Functions Method for Analysis of Composite Finite Periodic Structures With Dielectric-Conductor Cells

The subentire-domain (SED) basis functions method based on the multiple plane waves (MPWs) has achieved the efficient analysis of the scattering problem of the large-scale finite periodic structures (LFPSs) with interconnected perfect electric conducting cells. However, for the composite LFPSs with dielectric-conductor cells, the enormous unknowns in dielectric volume become a huge burden to construct matrix equations and obtain the SED basis functions. In addition, it is hard to manually set the match points of the volume meshes on the boundaries between adjacent cells. In this letter, the fast SED basis functions method has been proposed to analyze the composite LFPSs with dielectric-conductor cells. In the fast SED basis functions method, the discontinuous Galerkin technique has been employed to guarantee the continuity of the current on the boundaries of the adjacent cells without the match points. Moreover, the fast dipole method has been introduced to accelerate the calculation of the coupling among the SED basis functions. In the scattering and the radiation problems, the MPWs and the ports excited in turn have been applied to construct SED basis functions. Numerical experiments prove the accuracy and the efficiency of the fast SED basis functions method.

Fast Forced Vibration Response Calculations of Finite Periodic Structures Using a Wave Based Reduced Order Modelling Approach

Fast Forced Vibration Response Calculations of Finite Periodic Structures Using a Wave Based Reduced Order Modelling Approach

Attenuation band splitting in a finite plate strip with two-dimensional acoustic black holes

Acoustic black holes (ABH) lattice is an effective way to realize the bandgaps (BGs) through lightweight design. The predicted BGs using infinitely large periodic structures are commonly believed to enable corresponding attenuation bands (ABs) even the structure only contains a limited number of unit cells. This paper reports an unusual splitting phenomenon in the ABs of a finite plate strip with two-dimensional (2D) ABHs. A wide AB predicted by BG is split into two ABs in the finite strip separated by a high resonance peak, which challenges the aforementioned common belief. Results show that the increasing number of periodic unit cells has a negligible effect on the frequency and amplitude of the resonance peak. Its underlying mechanism is clarified by investigating the relationship between the BGs and ABs, obtained from infinite and finite periodic plate strips with the same 2D ABHs respectively. Complex wavenumber analysis results show the existence of two flexural wave modes in the considered frequency range due to the finite width of the strip. Two types of BGs, in which one satisfies the Bragg condition, are found when both wave modes fail to propagate. Analysis of structural modes further uncover the physical reason for the emerging AB splitting: a certain natural frequency of the boundary portion of the 2D ABHs strip falling into the theoretical BG. Due to the boundary reflection and energy concentration, this structural portion can loosely be regarded as a clamped-free plate strip. It is demonstrated that the resonance peak splitting the AB becomes tunable via disrupting the structural details of finite ABH strips. In particular, extending the terminal end of a finite ABH strip allows the change in boundary portion, which is conducive to eliminating the splitting peak. The reported study is helpful for the design of finite periodic structures with broadband attenuation bandgap.

Rapid Subentire-Domain Basis Functions Method Based on Adaptive Artificial Neural Networks

The subentire-domain (SED) basis functions method has shown its efficiency in solving the electromagnetic problem of large-scale finite periodic structures (LFPSs). However, calculating the expansion coefficients of SED basis functions is very time-consuming due to the consideration of the mutual coupling between all the elements in LFPSs, even after accelerated by conjugate-gradient fast Fourier transform (CG-FFT) and/or fast multipole method (FMM). In this article, based on the physics locations of observation cells, the adaptive artificial neural networks (AANNs) have been employed to rapidly predict the expansion coefficients of SED basis functions on interior cells, edge cells, and corner cells. By involving the AANNs, the mutual coupling between all the elements can be accounted into the neural networks without the construction of the mutual coupling matrix, and the expansion coefficients of SED basis functions can be obtained rapidly. Numerical experiments prove the accuracy and efficiency of the AANN-assisted SED basis functions method.

Investigation of a new metamaterial magnetorheological elastomer isolator with tunable vibration bandgaps

Semi-active isolators with laminated magnetorheological elastomer (MRE) structures draw increasing attentions for vibration control because of their resonance-frequency-shift property. Their vibration attenuation capacity can be further improved by integrating the acoustic metamaterial characteristics with tunable vibration bandgaps at designed frequencies. On this basis, a novel metamaterial MRE isolator with controllable vibration bandgaps was designed and prototyped. The mechanism of the formation of vibration bandgaps was analysed theoretically using the method of mass-spring model. Both infinite and finite periodic structures were discussed respectively. The equivalent negative stiffness of the metamaterial MRE isolator was studied. The evaluation experiments were conducted to study bandgaps of the metamaterial MRE isolator and its vibration isolation capacity. Results demonstrate that the new metamaterial MRE isolator possesses controllable bandgaps and can provide improved vibration isolation performance.

Fast multipole boundary element method for the acoustic analysis of finite periodic structures

In this work, two fast multipole boundary element formulations for the linear time-harmonic acoustic analysis of finite periodic structures are presented. Finite periodic structures consist of a bounded number of unit cell replications in one or more directions of periodicity. Such structures can be designed to efficiently control and manipulate sound waves and are referred to as acoustic metamaterials or sonic crystals. Our methods subdivide the geometry into boxes which correspond to the unit cell. A boundary element discretization is applied and interactions between well separated boxes are approximated by a fast multipole expansion. Due to the periodicity of the underlying geometry, certain operators of the expansion become block Toeplitz matrices. This allows to express matrix–vector products as circular convolutions which significantly reduces the computational effort and the overall memory requirements. The efficiency of the presented techniques is shown based on an acoustic scattering problem. In addition, a study on the design of sound barriers is presented where the performance of a wall-like sound barrier is compared to the performance of two sonic crystal sound barriers.

Test-Driven Development of a Substructuring Technique for the Analysis of Electromagnetic Finite Periodic Structures

In this paper, we follow the Test-Driven Development (TDD) paradigm in the development of an in-house code to allow for the finite element analysis of finite periodic type electromagnetic structures (e.g., antenna arrays, metamaterials, and several relevant electromagnetic problems). We use unit and integration tests, system tests (using the Method of Manufactured Solutions—MMS), and application tests (smoke, performance, and validation tests) to increase the reliability of the code and to shorten its development cycle. We apply substructuring techniques based on the definition of a unit cell to benefit from the repeatability of the problem and speed up the computations. Specifically, we propose an approach to model the problem using only one type of Schur complement which has advantages concerning other substructuring techniques.

Linking Time-Domain Vibration Behaviors to Spatial-Domain Propagating Waves in a Leaf-like Gradient-Index Phononic Crystal Lens

It is known that a propagating wave at a certain spatial point can be decomposed into plane waves propagating at different angles. In this work, by designing a gradient index phononic crystal lens (GRIN PCL) with transverse-continuous leaf-like unit cells, we theoretically and experimentally show that the spatial-domain propagating waves in finite periodic structures can be linked to their time-domain vibration behaviors. The full-field instantaneous focusing behaviors of Lamb waves in the proposed leaf-like GRIN PCL give an example of the wave-vibration linkage in finite periodic structures while allowing a certain complexity. The conclusion in this paper can help one skip iterative time-consuming finite element analysis (e.g., time-stepping solutions) to avoid possible numerical instabilities occurred in calculating transient wave field on practical finite metamaterials or phononic crystals having unit cells with complicated configurations.

A multi-scale model order reduction scheme for transient modelling of periodic structures

In this paper, a Floquet Model Order Reduction (MOR) method is proposed for the modelling of finite periodic structures subjected to harmonic or transient loads. This Floquet MOR generates frequency-independent projection matrices of the full-scale structure using a Floquet expansion of the local state-vectors computed on a single unit-cell with the Forced Wave Finite Element framework. This two-step MOR method can handle large multi-scale waveguides regardless of the scale and the number of unit-cells. A comparison with a standard sub-structuring approach based on unit-cell wave-mode reduction demonstrates that Floquet MOR is able to create reduced models of finite periodic structures with an unequalled computational efficiency. The method’s ability to produce small and accurate models suitable for time-domain simulations presents a high potential for virtual sensing, real-time Structural Health Monitoring (SHM) and digital twins for large periodic structures and metamaterials.

Application of Parallel CM-MLFMA Method to the Analysis of Array Structures

The characteristic mode (CM) approach has been applied to analyze the periodic and quasi-periodic structures efficiently. In this communication, an acceleration technique of the multilevel fast multipole algorithm (MLFMA) and a parallel scheme for the CM approach have been proposed to handle large-scale finite periodic and quasi-periodic structures in free space. The CM serves as a kind of entire-domain basis function to reduce unknowns in the method of moments (MoM). The MLFMA is used to accelerate the matrix-vector products (MVPs) between impedance matrix and eigenvectors, while the parallel technique is to reduce the time of generating a reduced matrix. The numerical examples have shown that, whether it is a connected structure or a nonconnected structure, the proposed method has good accuracy and efficiency.

Predicting the stop-band behaviour of finite mono-coupled periodic structures from the transmissibility of a single element

The study of mono-coupled periodic structures has gained renewed interest by the scientific community due to the new applications of metamaterials and meta-structures. Much research has focused on the wave propagation properties of infinite structures. However, this paper focuses on finite periodic structures, in particular the parameters that govern the behaviour of a low frequency stop-band of such a structure. From an engineering perspective, these are the lower and upper cut-off frequencies, i.e., the bandwidth, and the minimum transmission of vibration within the band. Using the Caley-Hamilton theorem, analytical expressions are derived for the receptance, dynamic stiffness and transmissibility of a finite mono-coupled structure. It is shown that the properties of the whole structure can be determined from the transmissibility of a single element. If the element is symmetric, then the expressions describing the stop-band are particularly simple. An approximate analytical expression has been derived that allows the number of elements needed for a given maximum attenuation in a low frequency stop-band to be determined. To illustrate the approach, lumped parameter systems are considered, in which the stop-band behaviour is governed by the addition of mass, stiffness and a vibration absorber. Expressions are derived for the maximum vibration attenuation within the first stop-band, for each case, enabling clear physical insight into the controlling parameters. Expressions are provided for the lower and upper cut-off frequencies of the stop-band. Some experimental results are also presented to support the theoretical analysis.

Vibration energy flow analysis of periodic nanoplate structures under thermal load using fourth-order strain gradient theory

A nonlocal Kirchhoff plate model with fourth-order strain gradient theory is firstly proposed to study variations with band gap frequencies and vibration energy distribution. The temperature rise is supposed to vary linearly through the thickness of the periodic nanoplate structures. The dynamic equations of finite periodic nanoplate structures under thermal load with the small-scale effect and the nonlinear membrane strain taken into consideration are derived based on the finite element method. The structural intensity approach is developed to predict the vibration energy flow of the periodic nanoplates. Effects of the temperature rise on band gaps and the structural intensity are divided into two parts. One is nonlinear effects of temperature rise, and the other is effects of the thermal load. In the numerical calculation, the natural frequencies of single-layer graphene sheets computed by the nonlocal finite element method with fourth-order strain gradient agree well with analytical results, which validate the effectiveness of the present method. The influences of nonlocal parameters and thermal load on the band gap and structural intensity are considered, respectively. The boundary value has been achieved to determine the critical mechanical load or thermal load for analyzing in depth the effects of the mechanical load and thermal load on the structural intensity. The proposed method shows that the vibration energy flow pattern may be controlled by adjusting the magnitude of the mechanical or thermal load.

A wave-based optimization framework for 1D and 2D periodic structures

This paper presents a second order optimization method based on the WFEM framework that enables the optimization of finite 1D periodic structures and 2D infinite ones. While optimization at the unit cell level has been done in previous studies, it did not account for the boundary conditions and excitation on the system, which might have an important influence on its dynamics. The proposed methodology exploits semi-analytical derivatives in an optimization algorithm that combines line search and trust region methods. It is tested and validated in a parameter identification procedure and subsequently used to minimize the mean square velocity of metabeams with clamped free boundary conditions. Finally, it is applied to the optimization of the sound transmission loss of a metapanel in the structural-acoustic coincidence region. The proposed scheme is versatile and can be used in a wide range of applications including, model updating, homogenization, design optimization and possibly damage detection.