WAVELET METHOD FOR CALCULATING THE DEFECT STATES OF TWO-DIMENSIONAL PHONONIC CRYSTALS

Based on the variational theory, a wavelet-based numerical method is developed to calculate the defect states of acoustic waves in two-dimensional phononic crystals with point and line defects. The supercell technique is applied. By expanding the displacement field and the material constants （mass density and elastic stiffness） in periodic wavelets, the explicit formulations of an eigenvalue problem for the plane harmonic bulk waves in such a phononic structure are derived. The point and line defect states in solid-liquid and solid-solid systems are calculated. Comparisons of the present results with those measured experimentally or those from the plane wave expansion method show that the present method can yield accurate results with faster convergence and less computing time.

ELASTIC WAVE LOCALIZATION IN TWO-DIMENSIONAL PHONONIC CRYSTALS WITH ONE-DIMENSIONAL QUASI-PERIODICITY AND RANDOM DISORDER

The band structures of both in-plane and anti-plane elastic waves propagating in two-dimensional ordered and disordered （in one direction） phononic crystals are studied in this paper. The localization of wave propagation due to random disorder is discussed by introducing the concept of the localization factor that is calculated by the plane-wave-based transfer-matrix method. By treating the quasi-periodicity as the deviation from the periodicity in a special way, two kinds of quasi phononic crystal that has quasi-periodicity （Fibonacci sequence） in one direction and translational symmetry in the other direction are considered and the band structures are characterized by using localization factors. The results show that the localization factor is an effective parameter in characterizing the band gaps of two-dimensional perfect, randomly disordered and quasi-periodic phononic crystals. Band structures of the phononic crystals can be tuned by different random disorder or changing quasi-periodic parameters. The quasi phononic crystals exhibit more band gaps with narrower width than the ordered and randomly disordered systems.

Band structure calculation of scalar waves in two-dimensional phononic crystals based on generalized multipole technique

A multiple monopole （or multipole） method based on the generalized mul- tipole technique （GMT） is proposed to calculate the band structures of scalar waves in two-dimensional phononic crystals which are composed of arbitrarily shaped cylinders embedded in a host medium. In order to find the eigenvalues of the problem, besides the sources used to expand the wave field, an extra monopole source is introduced which acts as the external excitation. By varying the frequency of the excitation, the eigenvalues can be localized as the extreme points of an appropriately chosen function. By sweeping the frequency range of interest and sweeping the boundary of the irreducible first Brillouin zone, the band structure is obtained. Some numerical examples are presented to validate the proposed method.

Interface-guided mode of Lamb waves in a two-dimensional phononic crystal plate

We investigate the interface-guided mode of Lamb waves in a phononic crystal heterostructures plate, which is com- posed of two different semi-infinite phononic crystal （PC） plates. The interface-guided modes of the Lamb wave can be obtained by the lateral lattice slipping or by the interface longitudinal gliding. Significantly, it is observed that the condition to generate the interface-guided modes of the Lamb wave is more demanding than that of the studied fluid-fluid system. The interface-guided modes are strongly affected not only by the relative movement of the two semi-infinite PCs but also by the thickness of the PC plate.

Point defect states of a hollow cylinder in two-dimensional phononic crystal

Point defect states in two-dimensional phononic crystal of a hollow mercury cylinder in a water host are studied. An improved plane expansion method combined with the supercell technique is used to calculate the band gaps and the pressure distribution at the defect position. The sonic pressure of defect modes shows that the waves are localized at or near the defect. As the filing fraction increases, more defect modes appear in the band gaps.

Tuning of band gaps for a two-dimensional piezoelectric phononic crystal with a rectangular lattice

In this paper, the elastic wave propagation in a two-dimensional piezoelectric phononic crystal is studied by considering the mechanic-electric coupling. The generalized eigenvalue equation is obtained by the relation of the mechanic and electric fields as well as the Bloch-Floquet theorem. The band structures of both the in-plane and anti-plane modes are calculated for a rectangular lattice by the planewave expansion method. The effects of the lattice constant ratio and the piezoelectricity with different filling fractions are analyzed. The results show that the largest gap width is not always obtained for a square lattice. In some situations, a rectangular lattice may generate larger gaps. The band gap characteristics are influenced obviously by the piezoelectricity with the larger lattice constant ratios and the filling fractions.

WAVE PROPAGATION IN TWO-DIMENSIONAL DISORDERED PIEZOELECTRIC PHONONIC CRYSTALS

The wave propagation is studied in two-dimensional disordered piezoelectric phononic crystals using the finite-difference time-domain （FDTD） method. For different cases of disorder, the transmission coefficients are calculated. The influences of disorders on band gaps are investigated. The results show that the disorder in the piezoelectric phononic crystals has more significant influences on the band gap in the low frequency regions than in the high frequency ones. The relation between the width of band gap and the direction of position disorder is also discussed. When the position disorder is along the direction perpendicular to the wave transmission, the piezoelectric phononic crystals have wider band gaps at low frequency regions than the case of position disorder being along the wave transmission direction. It can also be found that the effect of. size disorder on band gaps is analogous to that of location disorder. When the perturbation coefficient is big, it has more pronounced effects on the pass bands in the piezoelectric phononic crystals with both size and location disorders than in the piezoelectric phononic crystals with single disorder. In higher frequency regions the piezoelectric effect reduces the transmission coefficients. But for larger disorder degree, the effects of the piezoelectricity will be reduced.

Formation mechanism of the low-frequency locally resonant band gap in the two-dimensional ternary phononic crystals

The low-frequency band gap and the corresponding vibration modes in two-dimensional ternary locally resonant phononic crystals are restudied successfully with the lumped-mass method. Compared with the work of C. Goffaux and J. Sánchez-Dehesa （Phys. Rev. B 67 14 4301（2003））, it is shown that there exists an error of about 50% in their calculated results of the band structure, and one band is missing in their results. Moreover, the in-plane modes shown in their paper are improper, which results in the wrong conclusion on the mechanism of the ternary locally resonant phononic crystals. Based on the lumped-mass method and better description of the vibration modes according to the band gaps, the locally resonant mechanism in forming the subfrequency gaps is thoroughly analysed. The rule used to judge whether a resonant mode in the phononic crystals can result in a corresponding subfrequency gap is also verified in this ternary case.

Complete low-frequency bandgap in a two-dimensional phononic crystal with spindle-shaped inclusions

A two-dimensional phononic crystal （PC） structure possessing a relatively low frequency range of complete bandgap is presented. The structure is composed of periodic spindle-shaped plumbum inclusions in a rubber matrix which forms a square lattice. The dispersion relation, transmission spectrum and displacement field are studied using the finite element method in conjunction with the Bloch theorem. Numerical results show that the present PC structure can achieve a large complete bandgap in a relatively low frequency range compared with two inclusions of different materials, which is useful in low-frequency noise and vibration control and can be designed as a low frequency acoustic filter and waveguides. Moreover, the transmission spectrum and effective mass are evaluated to validate the obtained band structure. It is interesting to see that within the band gap the effective mass becomes negative, resulting in an imaginary wave speed and wave exponential attenuation. Finally, sensitivity analysis of the effect of geometrical parameters of the presented PC structure on the lowest bandgap is performed to investigate the variations of the bandgap width and frequency.

Tunability of Band Gaps in Two-Dimensional Phononic Crystals with Magnetorheological and Electrorheological Composites

The elastic wave propagation properties of phononic crystals(PnCs)composed of an elastic matrix embedded in magnetorheological and electrorheological elastomers are studied in this paper.The tunable band gaps and transmission spectra of these materials are calculated using the finite element method and supercell technology.The variations in the band gap characteristics with changes in the electric/magnetic fields are given.The numerical results show that the electric and magnetic fields can be used in combination to adjust the band gaps effectively.The start and stop frequencies of the band gap are obviously affected by the electric field,and the band gap width is regulated more significantly by the magnetic field.The widest and highest band gap can be obtained by combined application of the electric and magnetic fields.In addition,the band gaps can be moved to the low-frequency region by drilling holes in the PnC,which can also open or close new band gaps.These results indicate the possibility of multi-physical field regulation and design optimization of the elastic wave properties of intelligent PnCs.

Boundary element method for calculation of elastic wave transmission in two-dimensional phononic crystals

A boundary element method(BEM) is presented to compute the transmission spectra of two-dimensional(2-D) phononic crystals of a square lattice which are finite along the x-direction and infinite along the y-direction.The cross sections of the scatterers may be circular or square.For a periodic cell,the boundary integral equations of the matrix and the scatterers are formulated.Substituting the periodic boundary conditions and the interface continuity conditions,a linear equation set is formed,from which the elastic wave transmission can be obtained.From the transmission spectra,the band gaps can be identified,which are compared with the band structures of the corresponding infinite systems.It is shown that generally the transmission spectra completely correspond to the band structures.In addition,the accuracy and the efficiency of the boundary element method are analyzed and discussed.

Band structures of elastic waves in two-dimensional eight-fold solid–solid quasi-periodic phononic crystals

Combined with the supercell method, band structures of the anti-plane and in-plane modes of two-dimensional （2D） eight-fold solid-solid quasi-periodic phononic crystals （QPNCs） are calculated by using the finite element method. The influences of the supercell on the band structure and the wave localization phenomenon are discussed based on the modal distributions. The reason for the appearance of unphysical bands is analyzed. The influence of the incidence angle on the transmission spectrum is also discussed.

The Brillouin zones and band gaps of a two-dimensional phononic crystal with parallelogram lattice structure

We present a detailed theoretical study on the acoustic band structure of two-dimensional （2D） phononic crystal. The 2D pho- nonic crystal with parallelogram lattice structure is considered to be formed by rigid solid rods embedded in air. For the circu- lar rods, some of the extrema of the acoustic bands appear in the usual high-symmetry points and, in contrast, we find that some of them are located in other specific lines. For the case of elliptic rods, our results indicate that it is necessary to study the whole first Brillouin zone to obtain rightly the band structure and corresponding band gaps. Furthermore, we evaluate the first and second band gaps using the plane wave expansion method and find that these gaps can be tuned by adjusting the side lengths ratio R, inclined angle 0 and filling fraction F of the parallelogram lattice with circular rods. The results show that the largest value of the first band gap appears at θ=90° and F--0.7854. In contrast, the largest value of the second band gap is at θ=60° and F=0.9068. Our results indicate that the improvement of matching degree between scatterers and lattice pattern, ra- ther than the reduction of structural symmetry, is mainly responsible for the enhancement of the band gaps in the 2D phononic crystal.

Wavelet-Based Boundary Element Method for Calculating the Band Structures of Two-Dimensional Phononic Crystals

A wavelet-based boundary element method is employed to calculate the band structures of two-dimensional phononic crystals,which are composed of square or triangular lattices with scatterers of arbitrary cross sections.With the aid of structural periodicity,the boundary integral equations of both the scatterer and the matrix are discretized in a unit cell.To make the curve boundary compatible,the second-order scaling functions of the B-spline wavelet on the interval are used to approximate the geometric boundaries,while the boundary variables are interpolated by scaling functions of arbitrary order.For any given angular frequency,an effective technique is given to yield matrix values related to the boundary shape.Thereafter,combining the periodic boundary conditions and interface conditions,linear eigenvalue equations related to the Bloch wave vector are developed.Typical numerical examples illustrate the superior performance of the proposed method by comparing with the conventional BEM.

The effects of point defects on transmission coefficients in two-dimensional piezoelectric phononic crystals

Based on finite-difference time-domain(FDTD) method,the wave propagation and localization in two-dimensional defect-containing piezoelectric phononic crystals are investigated when the mechanical-electrical coupling is taken into account.The characteristics of localized defect modes are studied,and the effects of the number and direction of defects on the defect modes and transmission coefficients are discussed.Numerical results of defect modes and transmission coefficients are presented for BaTiO3/polymer piezocomposite,and from which we can see that the number and direction of defects have pronounced effects on the defect modes and transmission coefficients.The results also show the existence of elastic wave localization in piezoelectric phononic crystals containing defects.

Ultrasonic scalpel based on fusiform phononic crystal structure

In response to the ultrasonic scalpels with the vibrational modal coupling which leads to a decrease in efficiency,an ultrasonic scalpel based on fusiform phononic crystals(PnCs)is proposed.An accurate theoretical model is constructed,which is mainly composed of electromechanical equivalent circuit models to analyze the frequency response function and the frequency response curves of the admittance.Bragg band gaps exist in the fusiform PnCs owing to the periodic constraint,which can suppress the corresponding vibrational modes.The vibration characteristics(vibration mode,frequency,and displacement distribution)of the ultrasonic scalpel are analyzed,and the validity of the electromechanical equivalent circuit method is verified.The results indicate that other vibration modes near the working frequency can be isolated.In addition,blades based on fusiform PnCs have a function akin to that of the horn,which enables displacement amplification.

Inverse design of nonlinear phononic crystal configurations based on multi-label classification learning neural networks

Phononic crystals,as artificial composite materials,have sparked significant interest due to their novel characteristics that emerge upon the introduction of nonlinearity.Among these properties,second-harmonic features exhibit potential applications in acoustic frequency conversion,non-reciprocal wave propagation,and non-destructive testing.Precisely manipulating the harmonic band structure presents a major challenge in the design of nonlinear phononic crystals.Traditional design approaches based on parameter adjustments to meet specific application requirements are inefficient and often yield suboptimal performance.Therefore,this paper develops a design methodology using Softmax logistic regression and multi-label classification learning to inversely design the material distribution of nonlinear phononic crystals by exploiting information from harmonic transmission spectra.The results demonstrate that the neural network-based inverse design method can effectively tailor nonlinear phononic crystals with desired functionalities.This work establishes a mapping relationship between the band structure and the material distribution within phononic crystals,providing valuable insights into the inverse design of metamaterials.

Sensing the heavy water concentration in an H_(2)O-D_(2)O mixture by solid-solid phononic crystals

A new method based on phononic crystals is presented to detect the concentration of heavy water(D_(2)O)in an H_(2)O-D_(2)O mixture.Results have been obtained and analyzed in the concentration range of 0%-10%and 90%-100%D_(2)O.A proposed structure of tungsten scatterers in an aluminum host is studied.In order to detect the target material,a cavity region is considered as a sound wave resonator in which the target material with different concentrations of D_(2)O is embedded.By changing the concentration of D_(2)O in the H_(2)O-D_(2)O mixture,the resonance frequency undergoes a frequency shift.Each 1%change in D_(2)O concentration in the H_(2)O-D_(2)O mixture causes a frequency change of about 120 Hz.The finite element method is used as the numerical method to calculate and analyze the natural frequencies and transmission spectra of the proposed sensor.The performance evaluation index shows a high Q factor up to 1475758 and a high sensitivity up to 13075,which are acceptable values for sensing purposes.The other figures of merit related to the detection performance also indicate high-quality performance of the designed sensor.

Application and optimization design of non-obstructive particle damping-phononic crystal vibration isolator in viaduct structure-borne noise reduction

The problems associated with vibrations of viaducts and low-frequency structural noise radiation caused by train excitation continue to increase in importance.A new floating-slab track vibration isolator-non-obstructive particle damping-phononic crystal vibration isolator is proposed herein,which uses the particle damping vibration absorption technology and bandgap vibration control theory.The vibration reduction performance of the NOPD-PCVI was analyzed from the perspective of vibration control.The paper explores the structure-borne noise reduction performance of the NOPD-PCVIs installed on different bridge structures under varying service conditions encountered in practical engineering applications.The load transferred to the bridge is obtained from a coupled train-FST-bridge analytical model considering the different structural parameters of bridges.The vibration responses are obtained using the finite element method,while the structural noise radiation is simulated using the frequency-domain boundary element method.Using the particle swarm optimization algorithm,the parameters of the NOPD-PCVI are optimized so that its frequency bandgap matches the dominant bridge structural noise frequency range.The noise reduction performance of the NOPD-PCVIs is compared to the steel-spring isolation under different service conditions.

Phonon Confinement Effect in Two-dimensional Nanocrystallites of Monolayer MoS_2 to Probe Phonon Dispersion Trends Away from Brillouin-Zone Center

The fundamental momentum conservation requirement q - 0 for the Raman process is relaxed in the nanocrystal- lites （NCs）, and phonons away from the Brillouin-zone center will be involved in the Raman scattering, which is well-known as the phonon confinement effect in NCs. This usually gives a downshift and asymmetric broadening of the Raman peak in various NCs. Recently, the A1 mode of 1L MoS2 NCs is found to exhibit a blue shift and asymmetric broadening toward the high-frequency side [Chem. Soc. Rev. 44 （2015） 2757 and Phys. Rev. B 91 （2015） 195411]. In this work, we carefully check this issue by studying Raman spectra of lL MoS2 NCs prepared by the ion implantation technique in a wide range of ion-implanted dosage. The same confinement coefficient is used for both E＇ and A＇1 modes in 1L MoS2 NCs since the phonon uncertainty in an NC is mainly determined by its domain size. The asymmetrical broadening near the A＇1 and E＇ modes is attributed to the appearance of defect-activated phonons at the zone edge and the intrinsic asymmetrical broadening of the two modes, where the anisotropy of phonon dispersion curves along Г-K and Г- M is also considered. The photoluminescence spectra confirm the formation of small domain size of 1L MoS2 nanocrystallites in the ion-implanted 1L MoS2. This study provides not only an approach to quickly probe phonon dispersion trends of 2D materials away from Г by the Raman scattering of the corresponding NCs, but also a reference to completely understand the confinement effect of different modes in various nanomaterials.