2D Phononic Crystals Research Articles

Surface acoustic waves in 2D-phononic crystal of laminated pillars on a semi-infinite ZnO substrate

In this paper a phononic crystal of laminated pillars which are composed of two materials on a semi-infinite ZnO substrate is proposed. The unit cell can be regarded as a double-resonator model, which explicates the variation of band gaps. It is revealed that the band structure of the PC is highly influenced by the proportion of PZT-4 in laminated pillars. Besides, the high-frequency band gaps are the result of combined actions of the local resonance and the Bragg scattering, and the former dominates. Furthermore, the influences of the structural damping are investigated. It is observed that the bandgap-ratio remain unchanged and the band gaps translate to the right with isotropic structural loss factor increasing. Additionally, by introducing linear defects inside the laminated pillars, elastic energy at surface is strongly confined within defect pillars, and the coupling between defective pillars and SAW is governed by the physical parameter of resonators.

Locally resonant phononic crystals band-gap analysis on a two dimensional phononic crystal with a square and a triangular lattice

Locally resonant phononic crystals (LRPC) are a new type of sound insulating material. Using the plane wave expansion method based on the Bloch theorem, we compute the band structure of two dimensional (2D) phononic crystals (PC) with square and triangular lattices. Such PC typically consists of infinitely long carbon rods coated with silicon rubber and embedded in an elastic background. Computational results show that gaps appear at the lower frequency range, which are lower than those expected from the Bragg mechanism. Those gaps are generated due to local resonances; the optimum gap is obtained by tuning the thickness ratio of the coating layer. The gap created by the LRPC depends on the filling fraction of the coating cylinders.

DETECTION AND DISCRIMINATION BETWEEN ALPHA PARTICLES AND PROTONS BASED ON PHONONIC CRYSTALS MATERIALS

Ionizing particles detection based on phonons counting are considered as a growing research point of great interest. Phononic crystal (PnC) detectors have a higher resolution than other detectors. In the present work, we shall prepare a setup of a radiation detector based on a 1D PnC. The PnC detector can be used in detection and discrimination between protons and alpha particles with incident energy 1[Formula: see text]MeV. We have proposed a model capable of filtering the energies of two different ionizing particles (proton and alpha particle) of specific lattice frequencies in steps. Firstly, the high probability of phonons production was found at transmitted energy 5[Formula: see text]KeV from the whole path of protons and alpha particles through a vertical thin sheet made from Mylar and Polymethyl methacrylate (PMMA), respectively. The outgoing elastic waves are subjected to propagate through the proposed PnCs structure (Teflon-Polyethylene)2 that shows the different transmission percentage to each particle. Therefore, the detection and discrimination between ionizing ions were achieved.

Neural networks for inverse design of phononic crystals

Intelligent design of one-dimensional (1D) phononic crystals (PCs) by neural networks (NNs) is proposed. Two neural network models, supervised neural network (S-NN) and unsupervised neural network (U-NN), are used to realize the inverse design of PCs, concerning both geometric and physical parameter designs. Performances of the two models are compared and discussed. The results show that the bandgaps of the designed PCs by the two NNs are highly consistent with the target bandgaps. For the design of single or two parameters, the performances of the two NNs are excellent; while for the case of three-parameter design, U-NN works much better than S-NN due to the impact of non-uniqueness on S-NN. The present work confirms the feasibility of inverse design of PCs by NNs, and provides a useful reference for the application of NNs in the intelligent inverse design of 2D or 3D PCs.

Topological design of 3D phononic crystals for ultra-wide omnidirectional bandgaps

As a unique kind of artificial periodic composite materials, phononic crystals have attracted the wide attention in recent years. Unremitting efforts have been made to investigate the potential existence of phononic bandgaps, which can be used to manipulate the propagation of acoustic/elastic waves. This paper proposes a new topology optimization algorithm to achieve the optimised bandgap structures of 3D phononic crystals based on fixed-grid finite element method (FEM) and evolutionary procedure. The maximum normalised gap ratio obtained is as large as 97% between the 12th and 13th order of bands, with the optimised structure of tungsten carbide spheres embraced in epoxy matrix. Symmetric unit cells for simple cubic (SC) lattice are presented for the specified bandgaps to validate the effectiveness of the proposed optimization algorithm for designing 3D phononic bandgap crystals. Meanwhile, the proposed topology optimization represents structures with clear topologies and smooth boundaries.

Optimization on ultrasonic plastic welding systems based on two-dimensional phononic crystal

Large-sized tools are needed for plastic welding of large objects. However, the displacement distribution at the radiation surface of the tools is not uniform due to the strong lateral vibration. In this paper, two-dimensional (2D) phononic crystal (PnC) is introduced by processing periodically arranged grooves in large-sized tools. This method is different from the traditional experience-based grooving method. The lateral vibration is suppressed due to the characteristic of PnC. In this paper, six kinds of ultrasonic plastic welding systems based on 2D PnC with different groove width and height are designed. The vibration characteristics of ultrasonic plastic welding systems were simulated and experimented. The band gaps of large-sized tools are analyzed. The results show that the lateral vibration is suppressed when the operating frequency of the ultrasonic plastic welding system is within the band gap of the large tool. The displacement distribution at the radiation surface of the tools is more uniform. A theoretical support is provided for the design of large-sized ultrasonic plastic welding systems.

Band structure computation of two-dimensional and three-dimensional phononic crystals using a finite element-least square point interpolation method

• FE-LSPIM proposed to calculate band structures of elastic waves in 2D and 3D models. • Shape functions in FE-LSPIM exploit advantages of mesh-free/finite element methods. • FE-LSPIM can achieve excellent accuracy, especially in high-frequency modes. • FE-LSPIM achieves higher accuracy when calculating band gaps for phononic crystals. In the present study, a finite element-least square point interpolation method (FE-LSPIM) is proposed for calculating the band structures of in-plane elastic waves in two-dimensional (2D) and three-dimensional (3D) phononic crystals (PCs). This method utilizes new shape functions by combining mesh-free shape functions and finite element shape functions to exploit the specific advantages of the mesh-free method and finite element method (FEM). As a result, FE-LSPIM inherits the completeness properties of the mesh-free method and the compatibility properties of FEM, and thus the solutions obtained tend to be more accurate. Indeed, according to our previous research, the present method obtains excellent accuracy, especially in the high-frequency domain. The proposed FE-LSPIM was combined with Bloch's theory and applied to compute the band gaps (BGs) for 2D PCs and 3D PCs in the present study, where several PCs were investigated to verify the high accuracy when computing the BGs. Numerical analysis showed that the proposed method can predict the BGs more precisely compared with the FEM and modified FEM.

Multiferroic based 2D phononic crystals: Band structure and wave propagations

In the present work the acoustic band structure of a two-dimensional phononic crystal containing an organic ferroelectric (PVDF- polyvinylidene fluoride) and muliferroic material (LiVCuO4) were investigated by the plane-wave-expansion method. A two-dimensional PC with square lattices composed of LiVCuO4 cylindrical rods embedded in the PVDF matrix are studied to find the existence of stop bands for the waves of certain energy. This phononic bandgap – forbidden frequency range – allows sound to be controlled in many useful ways in structures that can act as sonic filters, waveguides or resonant cavities. Phononic band diagram ω = ω(k) for a 2D PC, in which non-dimensional frequencies ωa/2πc (c-velocity of wave) were plotted versus the wavevector k along the Γ-X-M-Γ path in the square Brillouin zone show four stop bands in the frequency range 0.01–8.0 kHz. The ferroelectric properties of PVDF and unusual properties of multiferroic LiVCuO4 give us the ability to control the wave propagation through the PC in over a wide frequency range.

Topological triply degenerate point with double Fermi arcs

Unconventional chiral particles have recently been predicted to appear in certain three dimensional (3D) crystal structures containing three- or more-fold linear band degeneracy points (BDPs). These BDPs carry topological charges, but are distinct from the standard twofold Weyl points or fourfold Dirac points, and cannot be described in terms of an emergent relativistic field theory. Here, we report on the experimental observation of a topological threefold BDP in a 3D phononic crystal. Using direct acoustic field mapping, we demonstrate the existence of the threefold BDP in the bulk bandstructure, as well as doubled Fermi arcs of surface states consistent with a topological charge of 2. Another novel BDP, similar to a Dirac point but carrying nonzero topological charge, is connected to the threefold BDP via the doubled Fermi arcs. These findings pave the way to using these unconventional particles for exploring new emergent physical phenomena.

Global sensitivity analysis of frequency band gaps in one-dimensional phononic crystals

Phononic crystals have been widely employed in many engineering fields, which is because of their unique feature of frequency band gaps. For example, their capability to filter out the incoming elastic waves, which include seismic waves, will have a significant impact on the seismic safety of nuclear infrastructure. To design accurately design the desired frequency band gaps, one must pay attention to how the input parameters and the interaction of the parameters can affect the frequency band gaps. Global sensitivity analysis can decompose the dispersion relationship of the phononic crystals and screen the variance attributed to each of the parameters and the interaction between them. Prior to the application in one-dimensional (1D) phononic crystals, this paper will first review the theory of global sensitivity analysis using variance decomposition (Sobol’ sensitivity analysis). Afterwards, the sensitivity analysis is applied to study a simple mathematical model with three input variables for a better understanding of the concept. Then, the sensitivity analysis is utilized to study the characteristic of the first frequency band gap in 1D phononic crystals with respect to the input parameters. This study reveals the quantified influence of the parameters and their correlation in determining the first frequency band gap. In addition, simple straight-forward design equations based on reduced Sobol’ functions are proposed to estimate easily the first frequency band gap. Finally, the error associated with the proposed design equations is also addressed.

Phononic band gap and wave propagation on multiferroic-based acoustic metamaterials

In the present work, the acoustic band structure of a two-dimensional (2D) phononic crystal containing a multiferroic and liquid were investigated by the plane-wave-expansion method. 2D PnC with triangular and honeycomb lattices composed of LiCu2O4 cylindrical rods embedded in the seawater matrix are studied to find the existence of stop bands for the waves of certain energy. Phononic band diagram ω=ω(k) for a 2D PC, in which nondimensional frequencies ωa/2πc (c-velocity of wave) were plotted versus the wavevector k along the г-X-M-г path in the Brillouin zone show few stop bands in the frequency range between 10 and 110 kHz.

Deep convolutional neural networks for eigenvalue problems in mechanics

SummaryWe show that deep convolutional neural networks (CNNs) can massively outperform traditional densely connected neural networks (NNs) (both deep or shallow) in predicting eigenvalue problems in mechanics. In this sense, we strike out in a new direction in mechanics computations with strongly predictive NNs whose success depends not only on architectures being deep but also being fundamentally different from the widely used to date. We consider a model problem: predicting the eigenvalues of one‐dimensional (1D) and two‐dimensional (2D) phononic crystals. For the 1D case, the optimal CNN architecture reaches 98% accuracy level on unseen data when trained with just 20 000 samples, compared to 85% accuracy even with 100 000 samples for the typical network of choice in mechanics research. We show that, with relatively high data efficiency, CNNs have the capability to generalize well and automatically learn deep symmetry operations, easily extending to higher dimensions and our 2D case. Most importantly, we show how CNNs can naturally represent mechanical material tensors, with its convolution kernels serving as local receptive fields, which is a natural representation of mechanical response. Strategies proposed are applicable to other mechanics' problems and may, in the future, be used to sidestep cumbersome algorithms with purely data‐driven approaches based upon modern deep architectures.

Simulation of the band structure for scalar waves in 2D phononic crystals by the singular boundary method

In this paper, the singular boundary method (SBM) is applied to compute band structures for scalar waves in two-dimensional (2D) phononic crystals (PCs). By employing the SBM to discretize the boundaries, including the periodic boundary and the continuity boundary in the PCs, we can obtain a linear eigenvalue equation which describes the relationship between the Block wave and frequency. By sweeping the frequencies and solving the eigenvalue equation, the band structures can be achieved. Several numerical examples are presented to validate the performance and efficiency of the proposed method. Compared with the PWE and FEM methods, the SBM has an advantage of quick convergence and low computational cost. Moreover, it can easily deal with various shaped scatters, even with sharp corners. Therefore, the present SBM can be considered to be a fast, accurate and stable alternative for band structure calculation of the PCs.

One-dimensional phononic crystals that incorporate a defective piezoelectric/piezomagnetic as a new sensor

In this paper, we have studied the propagation and localization of acoustic waves in a one-dimensional (1D) phononic crystal at the effects of both electric and magnetic fields. We focused on the sensing and measuring the electric and magnetic fields by adjusting the position of the transmitted peak (defect mode) inside the two piezoelectric/piezomagnetic defect layers (0.7 PMN-0.3PT/Terfenol-D). The defect mode is shifted towards the lower frequencies by applying electric and magnetic fields and towards the higher frequencies by opposing the field’s direction. The bigger the value of the applied field is the greater movement in the peak position is. In addition, our results revealed that the magnetic field has a bigger influence than the electric field on the position of the defect mode. Moreover, we studied such movement of the defect mode position by applying the electric and magnetic field simultaneously. These results can be used to improve and enhance piezoelectric and piezomagnetic energy harvesters.

Designing a tunable acoustic resonator based on defect modes, stimulated by selectively biased PZT rods in a 2D phononic crystal

Reconfigurable phononic crystals (PnCs) and related devices are highly attractive because of their flexibility for different applications. We present the design procedure for a tunable acoustic resonator based on a 2D PnC, consisting of a periodic array of piezoelectric rods of radii 175 μm as inclusions arranged in air background. A single point defect devised by a rod of radius 161 μm, replacing one of the inclusions, plays the role of the acoustic resonator, leading to a defect frequency in the phononic band gap (fd ≈ 432 kHz). Applying a ∼1% strain to the defect rod, via an external voltage, tunes the defect resonant frequency within the phononic band gap. It is shown that the maximum tunability and the frequency shift depends on the defect size, and is achieved about Δfd = 440 Hz for the defect with the expense of descending quality factor. Considering the pattern of the localized pressure field, we introduce a multi-defect structure with five symmetric defect rods, corresponding to the maxima of field distribution. It is shown that maximum frequency shift of the dominant defect frequency is achieved about Δfd = 1.14 kHz for defect radius of 161 μm, when all five defect rods are strained. The proposed tunable filter based on multi-defect structure results in an enhancement of about 2.6 times in the maximum frequency shift, in comparison with the single defect structure, and introduces a promising approach for realizing tunable acoustic devices.

Imaging localized phononic cavity modes with laser interferometer

In this work, we experimentally confirmed an effective Lamb-wave guiding method, which utilizes a localized phononic cavity mode in a 2D phononic crystal. The phononic crystal is created by a periodic distribution of solid pillars built on an elastic plate, while the cavity mode is induced by replacing part of the pillars by the hollow pillars. In the experiments, transmission characteristics of the Lamb-waves within the waveguides are studied comprehensively. Elastic field distributions of the cavity mode are mapped in the deep-subwavelength regime. In geometrical shapes, both a straight waveguide and a bended one are made for practical measurement, demonstrating visually that the elastic energies are well localized around the hollow pillars and can be transported around the bends.

Wave Propagations in Metamaterial Based 2D Phononic Crystal: Finite Element Analysis

In the present work, the acoustic band structure of a two-dimensional (2D) phononic crystal (PnC) containing composite material were investigated by the finite element method. Two-dimensional PC with triangular and honeycomb lattices composed of composite cylindrical rods are in the air and liquid matrix. The existence of stop bands are investigated for the waves of certain frequency ranges. This phononic band gap - forbidden frequency range - allows sound to be controlled in many useful ways in structures. These structures can be used as sonic filters, waveguides or resonant cavities. Phononic band diagrams ω=ω(k) for a 2D PnC were plotted versus the wavevector k along the Γ-X-M- Γ path in the first Brillouin zone. The calculated phonon dispersion results indicate the existence of full acoustic modes in the proposed structure along the high symmetry points.

A BEM for band structure and elastic wave transmission analysis of 2D phononic crystals with different interface conditions

In this paper, a boundary element method (BEM) is developed and applied to compute the band structures and the elastic wave transmission of 2D phononic crystals which have different interface conditions between the scatterers and the matrix. The 2D phononic crystals consist of circular or square solid cylinders periodically embedded in a solid matrix forming a square lattice. For a periodic unit-cell, the boundary integral equations for both the scatterer and the matrix are formulated. After discretizing the boundary integral equations and for an infinite periodic structure, a linear eigenvalue equation related to the Bloch wave vector is obtained by substituting the periodic boundary conditions and the interface conditions. While for the corresponding semi-infinite periodic structure, a linear equation set is obtained from which the elastic wave transmission can be obtained. To verify the present BEM, some representative numerical examples are presented for the phononic crystals with different acoustic impedance ratios. The numerical results indicate that the present BEM can efficiently provide accurate results for phononic crystals with arbitrary shapes and different interface conditions, and the interface conditions have important effects on the band structures and the elastic wave transmission.

Acoustic Wave Sensor Based on Piezomagnetic Phononic Crystal

The propagation of longitudinal acoustic waves in a 1D phononic crystal (PnC) contains a piezomagnetic material (Terfenol-D) as a defect layer is analyzed and discussed theoretically. A resonant mode was generated inside the phononic band gap when the piezomagnetic material thickness becomes around its half wavelength. The present results revealed that the frequency of the generated resonant mode can be controlled by applying an external magnetic field. In addition, the resonant mode is shifted towards higher and lower frequencies by applying an external pressure on the PnC structure. The simulated results are very valuable in the application of sensors measuring magnetic fields and in the early detection of building cracks.

Band gap structures for 2D phononic crystals with composite scatterer

We investigated the band gap structures in two-dimensional phononic crystals with composite scatterer. The composite scatterers are composed of two materials (Bragg scattering type) or three materials (locally resonance type). The finite element method is used to calculate the band gap structure, eigenmodes and transmission spectrum. The variation of the location and width of band gap are also investigated as a function of material ratio in the scatterer. We have found that the change trends the widest band gap of the two phononic crystals are different as the material ratio changing. In addition to this, there are three complete band gaps at most for the Bragg-scattering-type phononic crystals in the first six bands; however, the locally resonance-type phononic crystals exist only two complete band gap at most in the first six bands. The gap-tuning effect can be controlled by the material ratio in the scatterer.