Band Gaps Of Phononic Crystals Research Articles

Isogeometric topology optimization for maximizing band gap of two-dimensional phononic crystal structures

A smooth and efficient isogeometric topology optimization (ITO) method for maximizing band gaps in two-dimensional phononic crystals is developed in the paper. The band gaps of phononic crystals are computed by the NURBS-based isogeometric analysis, and the material distributions of two-dimensional phononic crystals are represented by the smooth, high-order continuity of the NURBS surface. The densities defined at each control point of the NURBS surface are employed as optimized design variables. The isogeometric analysis optimization using the same spline technique for both geometry and numerical analysis can benefit the optimization procedure and obtain smooth optimized structures, which can be easily used in 3D printing. The maximizing band gaps of phononic crystals for both out-of-plane and in-plane wave modes are obtained here using the present ITO approach. Numerical examples validated the effectiveness and reliability of the ITO method in broadening the band gaps and finding the optimal phononic crystals. And the numerical results show that the smooth optimized structures are obtained with fewer iterations.

Maximizing acoustic band gap in phononic crystals via topology optimization

Designing phononic crystals (PnCs) to exhibit the widest attainable band gap, or to encompass a designated frequency range, is imperative for the realization of PnCs tailored to specific functionalities. However, achieving a wide band gap of acoustic waves centered around a specified frequency remains a significant challenge. The challenge arises from the fact that the optimization objective function may be enhanced by disconnected air regions. Nonetheless, these disconnected regions obstruct the circulation of air, rendering PnCs devoid of practical engineering utility. In this study, we propose a topology optimization methodology aimed at crafting air/solid PnCs that maximize the band gap of acoustic waves at a specified central frequency. To ensure adequate air permeability within the PnCs, the optimization model incorporates the virtual temperature technique in conjunction with the minimum length scale technique. The topology of the PnC unit cell is represented with a low number of design variables through the material-field series expansion, and the Kriging-based optimization algorithm is used to solve the complicated optimization problem. Diverse optimized configurations are presented, and experimental findings compellingly underscore the efficacy of the proposed optimization approach in the design of phononic band gap crystals geared towards acoustic waves at the specified central frequency.

Phononic crystal bandgap optimization based on a multistage grid-pixel refinement method

Phononic crystal (PnC) is a kind of periodic distribution structure with elastic constants and densities, which prevents the propagation of waves in the bandgap by the interaction of internal microstructures. A slow convergence process and topological ill-conditioned structure cannot fail to have noticed during the fine grid optimization process of PnC. In this paper, a multistage grid-pixel refinement method (MGPRM) is proposed to quickly obtain the high-quality PnC topology based on the fast plane wave expansion method. Combined with the MGPRM and genetic algorithm, the bandgap of PnC is used to optimize. Results show that the MGPRM combined with the optimization algorithm can provide high-quality original configurations for the tunable parameterized microstructures. Compared with invariant grids of different densities and traditional refinement methods, the MGPRM has higher quality and a faster convergence rate for the process of optimization. Using the MGPRM for multi-objective optimization, the highly nonlinear correspondence between the characteristic bandgap and the topological morphology is obtained. In addition, the dynamic response of the finite PnC microstructure plate composed of the optimized topology and the calculated transmission spectrum are in perfect agreement with the bandgap of PnC. The MGPRM is further applied to the structure design and performance optimization of PnC.

Imaging phonon eigenstates and elucidating the energy storage characteristics of a honeycomb-lattice phononic crystal cavity

We extend gigahertz time-domain imaging to a wideband investigation of the eigenstates of a phononic crystal cavity. Using omnidirectionally excited phonon wave vectors, we implement an ultrafast technique to experimentally probe the two-dimensional acoustic field inside and outside a hexagonal cavity in a honeycomb-lattice phononic crystal formed in a microscopic crystalline silicon slab, thereby revealing the confinement and mode volumes of phonon eigenstates—some of which are clearly hexapole in character—lying both inside and outside the phononic-crystal band gap. This allows us to obtain a quantitative measure of the spatial acoustic energy storage characteristics of a phononic crystal cavity. We also introduce a numerical approach involving toneburst excitation and the monitoring of the acoustic energy decay together with the integral of the Poynting vector to calculate the Q factor of the principal in-gap eigenmode, showing it to be limited by ultrasonic attenuation rather than by phonon leakage to the surrounding region.

Multispectral transmission through phoxonic crystal slot-waveguide at midwave infrared frequencies

We design a multispectral transmission waveguide based on phoxonic crystals at midwave infrared (MWIR) frequencies. The phoxonic crystal slot-waveguide architecture is realized using a germanium (Ge)-slot waveguide, surrounded by a supercell array of oxide holes in silicon–germanium (SiGe) membrane tailored photonic and phononic crystal bandgap. The plane wave simulations for both photonic and phononic crystal unit cells were performed to confirm the geometry of the phoxonic supercell. The bandgap analysis shows the capability of the proposed architecture to confine photons of the terahertz frequency range within the slot waveguide by isolating them from the phonons of gigahertz frequency range. The phononic and photonic bandgaps were simultaneously engineered by varying the periodic variation of the density function and dielectric permittivity, respectively. The computational approach shows the suppression in photon-phonon scattering as validated by a uniform transmission of ∼99.8 % over a broad range of 3 to 5 μm wavelengths. The designed phoxonic crystal waveguide can be fabricated with planar processing technology and used in many applications where multispectral control of mid-IR signals is required.

Influence of spatial distribution and the type of material on the occurrence of bandgaps in phononic crystals

Influence of spatial distribution and the type of material on the occurrence of bandgaps in phononic crystals

Phononic crystals’ band gap manipulation via displacement modes

Phononic crystal band gaps (BGs), which are realized by Bragg scattering, have a central frequency and width related to the unit cell’s size and the impedance mismatch between material phases. BG tuning has generally been performed by either trial and error or by computational tools such as topology optimization. In either case, understanding how to systematically change the design for a particular band structure is missing. This paper addresses this by closely studying the displacement modes within the wavebands that are responsible for the BG. We look at the variation in different displacement modes due to the changes in the geometry and correlate these changes to their corresponding band structures. We then use this insight to design the unit cell for a particular application, for instance, for generating partial BGs.

Study of 1-3 piezoelectric composites based on phononic crystals theory

To further improve the working performance of 1-3 piezoelectric composites, in this paper, they are regarded as two-dimensional phononic crystals, the band structures, transmission characteristics and vibration modes are firstly calculated by the finite element method, and the suppression of lateral vibration by 1-3 piezoelectric composite structure is investigated based on band gap theory to obtain a purer thickness vibration. Secondly, the influence of the lattice system on the band gap as well as the overall working performance of piezoelectric composites is analyzed, finite element models of square and hexagonal lattice piezoelectric composites are established, and their operating parameters such as transmitting voltage response level and total radiated power are calculated by simulation. This study shows that the working frequency of 1-3 piezoelectric composites can be set in the band gap of phononic crystals to obtain better lateral vibration suppression; compared with square lattice, 1-3 piezoelectric composites with piezoelectric phase hexagonal lattice arrangement can obtain larger working bandwidth and higher total radiated power in the thickness mode, the overall working performance is better. This study is a guideline for the design and application of piezoelectric components in the fields of acoustics and medicine.

Pneumatic soft phononic crystals with tunable band gap

Tuning the band gap by mechanically generating deformation of soft phononic crystals provides an effective means to manipulate the propagation of acoustic waves. In this paper, a pneumatic soft phononic crystal is computationally designed to realize tunable band gap. It consists of a line of soft inflatable silicone cylinders surrounded by air. In detail, there is no band gap in the soft phononic crystal when it is unpressurized. By applying the air pressure, the scatterer in the soft phononic crystal is deformed, then a new reversible band gap appears in the phononic crystal. By adjusting the air pressure, the opening and closing of the band gap can be controlled. In addition, the width of the band gap can be tuned by the application of air pressure. We obtained the mechanical parameters of the rubber-like material from the experimental data. On this basis, the variation of the band gap of phononic crystals with applied different air pressures to the soft scatterer is investigated numerically. The effect of scatterer geometry on the band gap evolution of phononic crystals is also parametrically studied. Compared with traditional mechanical deformation means, such as tension and compression, the pneumatic manipulation is low-cost, fast in response and easy to integrate. This work introduces a feasible design of tunable soft phononic crystals.

Deep-learning-based inverse design of phononic crystals for anticipated wave attenuation

Bandgaps of phononic crystals dominating the propagation of evanescent waves have received significant attention recently, which can be determined and tuned by the topology of a unit cell. Predicting a band structure and designing topological structures with desirable characteristics have become a research hotspot. In this study, a data-driven deep learning framework is applied to arrive at the prediction of the band structure and the inverse design of topology. A convolutional neural network is trained to predict band structures of phononic crystals. After training a generative adversarial network, the generator is concatenated with the convolutional neural network for inverse design. Meanwhile, a complex band structure of phononic crystals is computed by the periodic spectral finite element method to present the spatial decay of evanescent waves. The topology with the greater spatial attenuation is screened from the ground truth topology and the inversely designed topology. Finally, an optimized topological phononic crystal with an anticipated bandgap is obtained, which has the potential for better acoustic insulation and vibration isolation.

Vibration Isolation and Noise Reduction Method Based on Phononic Crystal.

Phononic crystal is a new kind of sound insulation material, which has an elastic wave gap. When the elastic wave falls within the band gap, it will attenuate strongly in phononic crystal, and its attenuation degree is far greater than the predicted value of the mass density theorem. In this paper, the practical application of phononic crystals in low frequency sound insulation is taken as the breakthrough point. Firstly, the theory of phononic crystal band gap generation is calculated and analyzed, and the band structure of one-dimensional two-component Bragg scattering phononic crystals is calculated. Hypermesh is used to build the model, and the sound insulation performance of phononic crystals is simulated and analyzed through Nastran. A sound isolation test platform was built for the local resonant phononic crystal samples to verify its sound isolation ability.

Influence of the Core Pillar Height on the Bandgap Characteristics of Piezoelectric Phononic Crystals with Ring-Shaped Grooves

Dispersion profiles and surface acoustic wave attenuation characteristics of ring-shaped phononic crystals are investigated as a function of the core pillar height. Finite element method simulations are carried out for both band analyses and transmission spectra calculations. The results reveal that increasing core pillar height results in a decrement in the local resonance band frequency and the corresponding transmission peaks. The obtained dispersion profiles show that the phononic crystal bandgap also expands from 6 MHz to 11 MHz while the pillar height increases from 5 um to 7 um. Similar characteristics are also seen in the transmission spectra for the varying core pillar heights of the ring-shaped periodic grooves. In addition, surface acoustic wave attenuation competency depends on the core pillar height. The frequencies where the investigated phononic crystals are functional can be tuned by adjusting the core pillar height.

Planes approximation method for investigating the physical origins of deep, wide phononic bandgaps

The objective of this paper is to formulate a simple analytical model that describes Bragg and Mie scattering leading to the creation of bandgaps in phononic crystals (PnCs). The model is a homogenized, mean-field 1D model, dubbed the Planes Approximation Method (PAM). Bragg and Mie scattering are decoupled into independent 1D models and then recombined graphically to understand better and visualize bandgap creation. Due to the model's simplicity, it allows for rapid solutions compared to other computational methods. This paper argues that acoustic impedance mismatches yield bandgaps based on the formulation. PAM is demonstrated using a square lattice of microscale circular inclusions of tungsten with a lattice constant of 45 μm in a SiO2 matrix and then compared to results from finite difference time domain and plane wave expansion simulations.

Deep-subwavelength interface states in mechanical systems

Topological interface states in mechanical systems are the analogues of the edge modes in topological insulators from the field of condensed matter physics. The interface states produced and reported in the existing literature are located in the Bragg Scattering band gaps of phononic crystals. The corresponding frequencies of the interface states are thus high. This paper proposes a novel mechanical system that can produce an ultra-low frequency interface state in the deep-subwavelength region. Due to the periodic spring connections to the ground, a quasi-static band gap with a starting frequency of 0 Hz can be produced in the proposed system. The spring connections can be tuned to alter the polarization direction of the mode shape. Through a mass-spring model, the formation of an interface state in the quasi-static band gap is demonstrated. Moreover, it is found that by carefully configuring the spring connections to the ground, the interface state in the quasi-static band gap can be tuned to an arbitrarily low frequency. In addition, practical systems based on rod and beam structures are constructed following a similar design methodology. Theoretical analyses demonstrate that for the longitudinal/transverse wave mode in the topological rod/beam structure, a deep-subwavelength interface state can always form in the quasi-static band gap. The possibility of producing the deep-subwavelength interface state at an arbitrarily low frequency is also confirmed in a rod and a beam lattice system.

On the gradually vanishing bandgaps of periodic locally resonant metamaterial Timoshenko beams in the high frequency limit

Contrary to the perpetual-existed Bragg bandgaps in phononic crystals, the Bragg (BG) bandgaps in a periodic locally resonant (LR) metamaterial will gradually vanish in the high frequency limit. In this short discussion, from a mathematical perspective, we explain the physical phenomenon of the vanishing high-frequency bandgaps. To allow the model to be more practical while being general, a beam with periodic local resonators (i.e., LR+BG bandgaps) and a beam with concentrated masses (i.e., BG bandgaps only) are considered. In addition, to avoid high-frequency discrepancies caused by the unreasonably rapid increasing wave velocity in a Euler beam, a Timoshenko beam model considering shear deformation and rotary inertia is considered. Based on our analytical derivations, the transition behavior between the LR and Bragg bandgaps is also explored.

Precise and target-oriented control of the low-frequency Lamb wave bandgaps

Precise and target-oriented control of acoustic/elastic wave bandgaps in phononic crystals or acoustic metamaterials according to particular requirements is of both theoretical and practical importance. A very challenging and also interesting task is to tune the widths and the overall positions of the bandgaps of interest independently and arbitrarily. This work presents a simple and smart acoustic metamaterial structure, which can achieve the above mentioned goal by considering the symmetric Lamb waves. The proposed acoustic metamaterial structure consists of a homogenous piezoelectric plate which is connected to the LC circuits through an array of periodic surface electrodes. The control of the resonance bandgaps (RBGs), which are generated by the coupled resonance of the piezoelectric plate with the external LC circuits, is the focus of this study. To solve the problem at hand, we develop a two-dimensional (2D) spectral element method (SEM) considering the full electro-mechanical coupling, and the high tunability of the RBGs is demonstrated numerically in the low-frequency range. Especially, we find that the upper/lower boundary of a RGB can be intentionally fixed at any specified frequency while changing its’ bandwidth. To explore the corresponding physical phenomena and reveal the regulation mechanism, a simplified theory is developed based on the assumption of thin piezoelectric plates. Several simple analytical formulas are established, which are essential to the precise control of the RBGs for both the thin and thick piezoelectric plates. A numerical example is given to demonstrate the detailed procedure of precisely tuning a RBG in a targeted frequency range, which is also verified by the transmission spectrum for the corresponding finite piezoelectric acoustic metamaterial plate.

Characteristics of Band Gap and Low-Frequency Wave Propagation of Mechanically Tunable Phononic Crystals With Scatterers in Periodic Porous Elastomeric Matrices

Abstract The characteristics of passive responses and fixed band gaps of phononic crystals (PnCs) limit their possible applications. For overcoming this shortcoming, a class of tunable PnCs comprised multiple scatterers and soft periodic porous elastomeric matrices are designed to manipulate the band structures and directionality of wave propagation through the applied deformation. During deformation, some tunable factors such as the coupling effect of scatterer and hole in the matrix, geometric and material nonlinearities, and the rearrangement of scatterer are activated by deformation to tune the dynamic responses of PnCs. The roles of these tunable factors in the manipulation of dynamic responses of PnCs are investigated in detail. The numerical results indicate that the tunability of the dynamic characteristic of PnCs is the result of the comprehensive function of these tunable factors mentioned earlier. The strong coupling effect between the hole in the matrix and the scatterer contributes to the formation of band gaps. The geometric nonlinearity of matrix and rearrangement of scatterer induced by deformation can simultaneously tune the band gaps and the directionality of wave propagation. However, the matrix’s material nonlinearity only adjusts the band gaps of PnCs and does not affect the directionality of wave propagation in them. The research extends our understanding of the formation mechanism of band gaps of PnCs and provides an excellent opportunity for the design of the optimized tunable PnCs and acoustic metamaterials (AMMs).

Reliability-based topology optimization framework of two-dimensional phononic crystal band-gap structures based on interval series expansion and mapping conversion method

• The NRBTO scheme is presented for PnCs where the interval series expansion and adjoint vector methods are then utilized to obtain design sensitivity expressions. • A mapping conversion method is proposed to eliminate the gray-scale elements when relatively coarse grids are used. • The maximum band-gap with a specific frequency region and a filling fraction is obtained while ensuring reliability requirements considering the uncertainty in the optimization parameters. In this study, a non-probabilistic reliability-based topological optimization (NRBTO) method is investigated for the optimized design of the band-gap structure of two-dimensional phononic crystals (PnCs), in which unknown but bounded (UBB) uncertainty issues from the material uncertainty are taken into account. A measurement metric is first defined based on the interval set-theoretical model, which may transform the band-gap setting into a non-probabilistic reliability format. The interval series expansion and adjoint vector methods are then utilized to obtain design sensitivity expressions between frequency measures and design variables. The method of moving asymptote (MMA) is adopted as well to iteratively solve the reliability-oriented band-gap optimization problem for PnCs. To effectively tackle the gray-scale units, a mapping conversion method is involved eventually. The usage and validity of the present methodology are verified through three examples, and numerical results show that considering the uncertainty effect of UBB throughout the topology optimization process has an important impact on the final layout scheme. Additionally, the optimization of the out-of-plane modes of PnCs (both the band-gap upper bound and the filling rate constraint are considered) indicates that the topological configuration may always change accompanied with the filling rate increases.

Analysis of the effects of viscosity on the SH-wave band-gaps of 2D viscoelastic phononic crystals by Dirichlet-to-Neumann map method

The effects of viscosity on the band-baps of two-dimensional (2D) viscoelastic phononic crystals are studied by the Dirichlet-to-Neumann (DtN) map method. The viscoelastic phononic crystals are composed of square or triangular lattices of elastic circular solid cylinders in a viscoelastic solid matrix. Time-harmonic anti-plane transverse (SH) waves are considered and the standard linear solid model is used to describe the viscoelastic behavior of the matrix material, in which the shear modulus depends on the frequency. Based on the DtN map constructed by using the cylindrical wave expansions, a linear eigenvalue problem related to the Bloch wave vector is formulated, which involves relatively small matrices. By using real frequencies and complex-valued wave vectors, the complex band structures are computed for different viscosity parameters, and the wave propagation and attenuation characteristics in dependence of the viscosity are analyzed. Then based on the storage shear modulus, real band structures for the Pb/epoxy phononic crystals with viscosity in the square and triangular lattices are studied, and the effects of the different viscoelastic material parameters and other relevant influencing parameters on the band-gaps are discussed. The results show that the viscosity of the matrix material affects the location and the width of the band-gaps. This fact provides an alternative way to tune the band-gaps of phononic crystals by adjusting the viscoelastic parameters combined with other influencing parameters.

Characterization of ceramic phononic crystals prepared with additive manufacturing: Ultrasonic technique and finite element analysis

Phononic crystals (PCs) are composed of two or more elastic materials arranged periodically. One of the important characteristics of PCs is the band gap. Band gap is the frequency range within which elastic waves are not allowed to transmit. With the band gap, applications of PCs include filters for body waves and surface acoustic waves. In the recent years, the additive manufacturing technique has been widely employed to fabricate objects with complicated geometries like PCs. In this study, samples of ceramic PCs were fabricated with the additive manufacturing technique. In addition to that Finite Element Analysis (FEA) was employed to predict the band gaps of PCs fabricated with the traditional and the additive manufacturing technique. Experimental measurements with pulsed ultrasound technique together with fast Fourier Transform (FFT) were employed to measure the band gaps. In general, measurement results agree well with the FEA results. The outcomes of this study suggest a new PCs fabrication method based on the additive manufacturing technique.