Waves In Phononic Crystals Research Articles

Band gap study of periodic piezoelectric micro-composite laminated plates by finite element method and its application in feedback control

Phononic crystals are materials or structures with periodic variations of elastic constants and density. Elastic waves in phononic crystals cannot propagate within a certain frequency range, called the band gap, due to the interference of their internal structure. This article investigates how fiber orientation affects the band gap properties of periodic micro-composite laminated plates. We examined the impact of five common composite laminate configurations on the system’s band gap and discovered that the antisymmetric configuration had superior high-frequency band gap performance. Based on this finding, we further explored how the antisymmetric angle influenced the system’s band gap. Since the study was conducted at a micro scale, micro structural effects were considered. Therefore, we used the modified coupled stress elastic dynamics and a four-node quadrilateral non-conforming element to discretize the micro composite laminated plate model, which had the nodal compatibility of high-order elasticity theory. Moreover, we employed feedback control to the proposed structure to dynamically adjust the band gap width according to different practical needs. This research supports the design of periodic micro-composite laminated plates and the piezoelectric feedback control system, which can control the vibration and wave propagation behaviors of micro structures using coupled piezoelectric sensors and actuators. The piezoelectric feedback control employed two control methods: direct proportional control and acceleration control. This article compared the single and multiple control methods of these two approaches to determine the optimal control strategy.

Simultaneous pseudospin and valley topological edge states of elastic waves in phononic crystals made of distorted Kekulé lattices

Topological metamaterials protected by the spatial inversion symmetry mainly support single type edge state, interpreted by either the quantum valley Hall effect or the quantum spin Hall effect. However, owing to the existence of the complicated couplings and waveform conversions during elastic wave propagation, realizing topologically protected edge states that support both pseudospin and valley degrees of freedom in elastic system remains a great challenge. Here, we propose a two-dimensional Kekulé phononic crystal (PC) that can simultaneously possess pseudospin- and valley-Hall edge states in different frequency bands. By inhomogeneously changing the elliptical direction in a Kekulé lattice of elliptical cylinders, three complete phononic bandgaps exhibiting distinct topological phase transitions can be obtained, one of which supports a pair of pseudospin-Hall edge states and the other hosts valley-Hall edge states in the low and high frequency regime. Furthermore, a sandwiched PC heterostructure and a four-channel cross-waveguide splitter are constructed to achieve selective excitation and topological robust propagation of pseudospin- and valley-momentum locking edge states in a single configuration. These results provide new possibilities for manipulating in-plane bulk elastic waves with both pseudospin and valley degrees of freedom in a single configuration, which has potential applications for multiband and multifunctional waveguiding.

Realization of Topological Valley Hall Edge States of Elastic Waves in Phononic Crystals Based on Material Differences

In this study, the topological valley Hall edge states of elastic waves in phononic crystals are realized based on material differences. A phononic crystal structure with lantern rings is proposed. The results show that differences in the Young's modulus or density of lantern-ring materials can cause the destruction of the effective spatial inversion symmetry, which causes the valley Hall phase transition; thus, topological edge states emerge at the interface of two domains with topologically dissimilar phases. For valley topological transport, the traditional method to break the spatial inversion symmetry is by changing the dimensional parameters of scatterers in phononic crystals. However, in many cases, the space between scatterers is extremely small, making it difficult to obtain a sufficiently large band gap. Moreover, it is difficult to change the transport path and operating-frequency range of elastic waves after sample fabrication is finished. Here, the above problems are solved by the realization of valley topological transport based on changes to lantern-ring materials. The waveguide path is reconfigurable, that is, it can be altered conveniently, and the operating frequency can be tuned by changing the lantern-ring materials. These factors are very important for the topological transport of elastic waves. Topological valley transport achieved by this method has a good backscattering-suppression ability and obvious robustness to defects. Realization of topological valley Hall edge states by utilizing material differences provides an effective method for obtaining the topological transport of elastic waves, breaking the limitations of those based only on structural parameter changes, and this has good application prospects in elastic wave manipulation and communication.

Features of the Propagation of Long Waves in Phononic Crystals, the Influence of the Concentration and Polydispersity of the Components

The relevance of the research is determined by the need to simulate wave propagation in heterogeneous media based on phononic crystals. These studies are necessary for the tasks of flaw detection, the development of non-destructive testing methods, geophysics, engineering geophysics and other methods of acoustic research. During the study, it was determined that the fragmentation of the components of two-component phononic crystals into 2, 3, … N equal parts in the unit cell leads to a catastrophic rearrangement of the dispersion relations and the appearance of new elementary wave packets for acoustic waves. Using numerical-analytical methods, it is shown that in two-component phononic crystals, the polydispersity of the components can significantly affect the average characteristics of the propagation of long acoustic waves when the polydisperse crystal is close to monodisperse with a smaller unit cell size. For polydisperse crystals close to monodisperse with the same period (unit cell size), their averaged characteristics for long waves coincide.

Band structure of love wave in a one-dimensional piezoelectric layered phononic crystal with imperfect interface

The propagation of Love wave in a phononic crystal consisting of superlattice lithium niobate substrate carrying a piezoelectric waveguide layer is analyzed. The interface between the waveguide layer and the substrate is imperfectly bonded, which is described by means of a shear spring model. Numerical results show that the bandgap shifts to a lower frequency range and the bandwidth decreases as the interface imperfection increases. Moreover, the transmission coefficient calculations using finite element simulation revealed that a perfect interface results in more effective bandgap attenuation.

Propagation Characteristics of Flexural Wave in One-Dimensional Phononic Crystals Based on Lattice Dynamics Model

In this paper, we establish discrete flexural lattice chain models of Bragg and locally resonant phononic crystals by setting mass defect atoms and local resonant elements on the flexural lattice chain. The bandgap characteristics of flexural wave in phononic crystals are studied by establishing the governing equations of the model. The results from models show that with the change of the mass ratio of defective atoms to normal atoms, the bandgap of the flexural wave produced by Bragg scattering shows a certain rule. When the local resonant bandgap and Bragg scattering bandgap are close to each other, the two bandgaps will be coupled to form a wider flexural wave bandgap. The effect of axial strain on bending wave propagation is only the shift of bandgap position. The effect of material damping on the propagation of a bending wave is only energy dissipation at high frequency. In addition, we use finite element simulation to calculate the bandgap of flexural wave in phononic crystals with mass defects, and the results are consistent with lattice chain model. This shows that lattice chain model can effectively guide the bandgap design of phononic crystals. This comprehensive study may help to elucidate the rule of bandgap generation of flexural wave in one-dimensional phononic crystals.

Experimental realization of negative refraction and subwavelength imaging for flexural waves in phononic crystal plates

In this paper, we numerically and experimentally demonstrate negative refraction of flexural waves in phononic crystal (PC) plates which is employed for designing flat elastic lenses. We propose a thickness contrast-based plate design to achieve refractive index equal to -1 at the interface of the PC and host plate. The thickness contrast between the PC and host plate enables matching their wave numbers in the all angle negative refraction (AANR) frequency regime. The PC-lens design is then numerically and experimentally verified to achieve the image of an omnidirectional subwavelength excitation source. By changing the thickness contrast between the plate and PC, the PC-lens can be tuned for wave focusing and subwavelength imaging at a desired frequency.

Valley-Chiral Edge States of Antisymmetric Plate Wave in Phononic Crystals with Linear Defect

Valley-Chiral Edge States of Antisymmetric Plate Wave in Phononic Crystals with Linear Defect

A localized meshless collocation method based on semi‐analytical basis functions for bandgap calculation of elastic waves in phononic crystals

AbstractThis paper makes the first attempt to develop an efficient localized meshless collocation method (LMCM) based on the semi‐analytical T‐complete functions to calculate the bandgaps of elastic waves in solid phononic crystals. In comparison with the FEM (COMSOL software), the proposed LMCM can provide the similar accurate results with less computational resources for band structure calculation.

A Chebyshev collocation method for band structure calculations of the longitudinal elastic waves in phononic crystals

AbstractIn this study, a Chebyshev collocation method (CCM) is developed and applied to calculate the band structures of longitudinal elastic waves in periodically multi‐layered phononic crystals. The general form of the CCM for a unit‐cell is derived, in which the periodic boundary conditions and continuity conditions on the interface between different component materials are imposed directly to the CCM scheme. The band structures or dispersion relations can be obtained by solving the corresponding linear eigenvalue problem, where the unknown frequencies are the eigenvalues and the components of the Bloch wave vector are given. The proposed CCM is verified by using the corresponding results obtained by the transfer matrix method. Due to the advantages of the CCM, the CCM presented in this paper can be easily generalized and applied to the high‐dimensional such as two‐dimensional (2D) problems.

Band structure calculation of 2D fluid/solid and solid/fluid phononic crystal using a modified smoothed finite element method with fluid–solid interaction

Recently, a novel modified smoothed finite element method (M-SFEM) is proposed to calculate the band structures of two-dimensional (2D) in-plane elastic waves in phononic crystals (PCs) by the authors. In present work, the proposed algorithm is extended to analyze the band structures of 2D phononic crystals (PCs) with fluid/structure interaction (FSI). In order to study the phononic crystals (PCs) with FSI, this method extends the gradient smoothing technique to soften the stiffness matrix of fluid domain and solid domain. Meanwhile, according to the generalized integration rules, this method simply shifts these integration points of elements to an unconventional location in the mass matrix. In addition, different Bloch boundary conditions, viz. fluid/solid and solid/fluid, including the square and triangular lattice using M-SFEM are also investigated in the computation of PCs with FSI. Several PCs models with FSI are studied to verify the high accuracy of the M-SFEM in the analysis of band structures. In order to emerge the superiority of the proposed method, the results of finite element method (integration points on conventional location), modified finite element method (integration points on unconventional location), smoothed finite element method are all presented. Numerical analysis shows that the M-SFEM can be used to predict the band structures in PCs with FSI, and can obtained more precise results as compared to FEM and SFEM.

Dispersion features of elastic waves in phononic crystals: finite element analysis

In this report, we present an investigation of the dispersion properties of the phononic crystal in different lattice, geometries for various piezoelectric materials. We study dispersion properties of the epoxy resin/piezoelectric based phononic structures in the frequency range by using the finite element method for various lattices, piezoelectric materials and cross-sectional inclusions. The effect of the cross-section of piezoelectric inclusions, lattice geometry and filling factor on the phononic band gap is illustrated. A numerical approach is used to obtain the dispersion properties of the 2D piezoelectric phononic crystals which yields further insight into the phenomenon of the reflection from different families of piezoelectric in relation to the presence of phononic gaps.

A Petrov-Galerkin finite element interface method for interface problems with Bloch-periodic boundary conditions and its application in phononic crystals

In this paper, we propose a Petrov-Galerkin finite element interface method (PGFEIM) to solve the elliptic and elastic interface problems with Bloch-periodic boundary conditions. The main idea of this method is to choose the standard finite element basis independent of the interface to be the test function basis, and choose a piecewise linear function satisfying the jump conditions across the interface to be the solution basis. The grid we use is a non-body-fitted grid. The PGFEIM is capable of dealing with sharp-edged interface problems with matrix coefficients and nonhomogeneous jump conditions. Further, we extend this method to compute the band structure of anti-plane transverse waves and in-plane elastic waves in phononic crystals. Different acoustic impedance ratios, arbitrary complex scatterer geometries and various material properties are considered and discussed. Numerical experiments demonstrate that the PGFEIM is nearly second order accurate in the L∞ norm for interface problems, and it is accurate and efficient for computing the band structure of phononic crystals.

A modified smoothed finite element method (M-SFEM) for analyzing the band gap in phononic crystals

The present work proposes a novel modified smoothed finite element method (M-SFEM) to calculate the band structures of two-dimensional in-plane elastic waves in phononic crystals (PCs). Using the gradient smoothing technique over the cell-based smoothing domains, the cell-based gradient smoothing operation can offer ‘proper softening effects’ in SFEM modeling. According to the generalized integration rules, by simply shifting integration points to an unconventional location in the mass matrix, the accuracy of the M-SFEM model can be further improved. Numerical examples are presented for PCs using the proposed method for the computation of band gap (BG) frequency regions. The accuracy and efficiency of the modified SFEM are compared with those of the corresponding FEM and SFEM. The advantages of the modified SFEM for computing the BGs in PCs are discussed as compared to the conventional FEM. The results show the performance improvement of the M-SFEM compared to the FEM and SFEM.

Dispersion Features of the Sound Waves in Phononic Crystals: Finite Element Analysis

Bu çalışmada farklı tasarımlardaki fononik kristallerin dispersiyon özellikleri deneysel olarak FFT algoritması ve nümerik olarak sonlu elemanlar yöntemi kullanılarak araştırılmıştır. Periyodik olarak yerleştirilmiş kompozit silindirlerden oluşan kare örgü ve yarı-periyodik (Fibonacci dizilimi) yapıdaki iki boyutlu fononik kristal de, birinci Brillouin bölgesinde dalga vektörüne (k) karşılık frekanslar elde edilerek M–Г–X–M yolu boyunca fononik bant yapıları çizdirildi. Periyodik fononik kristal tasarımlarından dolu kompozit silindirlerden oluşan yapıda 4 kHz–7 kHz aralığında Г–X yönünde kısmi bant, nümerik olarak incelenen c-şekilli fononik kristalde 2 kHz–3 kHz aralığında tam bant, Fibonacci dizilimindeki fononik kristalde ise 3 kHz–4 kHz ve 3,5 kHz–6,7 kHz aralıklarına Г–X yönünde kısmi bantlar gözlendi. Sonuçların geçerliliğini test etmek için sonlu yapıda iletim kayıpları nümerik olarak hesaplandı ve deneysel olarak ölçüldü. Fononik bant yapısı ile elde edilen iletim kayıpları karşılaştırıldığında sonuçların uyumlu oldukları gözlendi.

Maximizing spatial decay of evanescent waves in phononic crystals by topology optimization

The propagation of evanescent waves inside phononic band gaps is important for the design of phononic crystals with desirable functionalities. This paper extends the bi-directional evolutionary structure optimization (BESO) method to the design of phononic crystals for maximizing spatial decay of evanescent waves. The optimization objective is to enlarge the minimum imaginary part of wave vectors at a specified frequency. The study is systematically conducted for both out-of-plane and in-plane waves at various frequencies. Numerical examples demonstrate that the proposed optimization algorithm is effective for designing phononic crystals with maximum spatial decay of evanescent waves. Various topological patterns of optimized phononic crystals are given. The results also show that the proposed optimization algorithm can successfully overcome difficulties in opening band gap at desirable frequencies, especially for the complete band gap and multiple band gaps.

Study on the transmission properties of periodical and quasi-periodical phononic crystal in elastic wave

The propagation of the elastic wave in phononic crystal is different from the normal uniformity medium. The finite-difference time-domain (FDTD) method is irrelevant to structure model and used widely. Moreover, when the numerical stability of FDTD iterations is satisfied, the elastic waves’ transmission property through periodical and quasi-periodical phononic crystal can be achieved. In this paper, the transmission coefficients of elastic wave through two systems are numerically calculated and the results of band gaps are analyzed. The results are helpful to study phononic crystal.

Influences of active control on elastic wave propagation in a weakly nonlinear phononic crystal with a monoatomic lattice chain

Different from the linear phononic crystals, elastic waves can exhibit more interesting phenomena in periodic structures with nonlinear properties. By the active control action, elastic wave characteristics can be tuned based on the demand in practical application. This work discusses the possibility of the active control action on elastic waves in phononic crystals with the weakly nonlinear monoatomic lattice chain. Based on the Lindstedt-Poincaré method, the approximate solution of the dispersion relation is derived. Numerical calculations are performed to discuss the influences of the active control action on wave propagation properties. In order to present the features of nonlinear elastic wave dynamics, it is mainly focused on the effects of active control accompanied by the changes of the elastic wave amplitude, nonlinear degree, hard and soft nonlinearities. It is expected to extend the analysis and design of new phononic crystal devices for both the theory and application fields.

A meshfree local RBF collocation method for anti-plane transverse elastic wave propagation analysis in 2D phononic crystals

In this paper, a meshfree or meshless local radial basis function (RBF) collocation method is proposed to calculate the band structures of two-dimensional (2D) anti-plane transverse elastic waves in phononic crystals. Three new techniques are developed for calculating the normal derivative of the field quantity required by the treatment of the boundary conditions, which improve the stability of the local RBF collocation method significantly. The general form of the local RBF collocation method for a unit-cell with periodic boundary conditions is proposed, where the continuity conditions on the interface between the matrix and the scatterer are taken into account. The band structures or dispersion relations can be obtained by solving the eigenvalue problem and sweeping the boundary of the irreducible first Brillouin zone. The proposed local RBF collocation method is verified by using the corresponding results obtained with the finite element method. For different acoustic impedance ratios, various scatterer shapes, scatterer arrangements (lattice forms) and material properties, numerical examples are presented and discussed to show the performance and the efficiency of the developed local RBF collocation method compared to the FEM for computing the band structures of 2D phononic crystals.

Negative refraction of acoustic waves in phononic crystals using recursive algorithms for block toeplitz matrices

Recent improvements in design and manufacturing have significantly improved our ability to manage phononic crystals (PC). Researchers have been studied the negative refraction in PC experimentally and theoretically using multiple scattering (MS) theory. This problem requires solving a large complex valued linear system that has a special multilevel block Toeplitz (BT) structure. We study negative refraction of acoustic waves in 2D PC by means of MS theory, by taking advantage of the PC structure and using specific recursive algorithms for BT matrices. We present new efficient and accurate algorithms for solving acoustic MS problem by the cluster of closely spaced cylinders to design acoustic negative refraction imaging in a PC. The unit cell of the PC consists, in general, of a solid cylinder in an acoustic medium. The dispersion curves of PC have a negative refraction dispersion branch producing the focusing effect. Particular attention is given to the dynamic behavior in the vicinity of dispersion branches with negative slope, and to the band gaps. We explore the effect of focal point on structural parameters, and employ a parallelization technique that allows efficient application of the proposed recursive algorithms for solving BT systems on high performance computer clusters. Numerical comparisons of CPU time and total elapsed time taken to solve the linear system using the direct LAPACK and TOEPLITZ libraries on Intel FORTRAN show the advantage of high performance recursive algorithms over the Gaussian elimination.