Gradient-index Phononic Crystal Research Articles

Structurally embedded gradient index lens for guided wave amplification in polymers

Guided wave inspection is a commonly used technique in structural health monitoring, yet there are challenges associated with ultrasonic inspection of polymeric structures due to high material damping resulting in wave attenuation. In this work, we present a structurally embedded gradient-index phononic crystal (GRIN-PC) lens for elastic wave amplification in polyvinyl chloride (PVC) structures to overcome the attenuation problem in long-range guided wave inspection. The GRIN-PC lens is made up of cylindrical brass stubs of varying heights or diameters embedded in the PVC base structure, providing a hyperbolic secant profile of refractive index distribution transverse to the wave propagation direction. We first demonstrate flexural wave focusing in a PVC plate by implementing the embedded lens design via numerical simulations and experiments. Then, the embedded GRIN-PC lens concept is implemented to PVC pipe for multi-mode wave focusing using a single lens design which is studied through numerical simulations. The proposed GRIN-PC lens design reduces the amplitude decay due to material damping by amplifying the wave energy at the focal regions of the lens. The embedded lens concept developed for polymers can also benefit other applications requiring the transmission of wave energy in structures with high material damping.

Graphene-based phononic crystal lenses: Machine learning-assisted analysis and design

The advance of artificial intelligence and graphene-based composites brings new vitality into the conventional design of acoustic lenses which suffers from high computation cost and difficulties in achieving precise desired refractive indices. This paper presents an efficient and accurate design methodology for graphene-based gradient-index phononic crystal (GGPC) lenses by combing theoretical formulations and machine learning methods. The GGPC lenses consist of two-dimensional phononic crystals possessing square unit cells with graphene-based composite inclusions. The plane wave expansion method is exploited to obtain the dispersion relations of elastic waves in the structures and then establish the data sets of the effective refractive indices in structures with different volume fractions of graphene fillers in composite materials and filling fractions of inclusions. Based on the database established by the theoretical formulation, genetic programming, a superior machine learning algorithm, is introduced to generate explicit mathematical expressions to predict the effective refractive indices under different structural information. The design of GGPC lenses is conducted with the assistance of the machine learning prediction model, and it will be illustrated by several typical design examples. The proposed design method offers high efficiency, accuracy as well as the ability to achieve inverse design of GGPC lenses, thus significantly facilitating the development of novel phononic crystal lenses and acoustic energy focusing.

Machine learning-enabled development of high performance gradient-index phononic crystals for energy focusing and harvesting

Gradient-index (GRIN) phononic crystals (PnCs) offer an excellent platform for various applications, including energy harvesting via wave focusing. Despite its versatile wave manipulation capability, the conventional design of GRIN PnCs has thus far been limited to relatively simple shapes, such as circular holes or inclusions. In this study, we propose a GRIN PnC comprising of unconventional unit cell designs derived from machine learning-based optimization for maximizing elastic wave focusing and harvesting. A deep neural network (NN) is trained to learn the complicated relationship between the hole shape and intensity at the focal point. By leveraging the fast inference of the trained NN, the genetic optimization approach derives new hole shapes with improved focusing performance, and the NN is updated by augmenting the new dataset to enhance the prediction accuracy over a gradually extended range of performance via active learning. The optimized GRIN PnC design exhibits 3.06 times higher wave energy intensity compared to the conventional GRIN PnC with circular holes. The performance of the best GRIN PnC within the allowable range of our machining tools was validated against experimental measurements, which shows 1.35 and 2.35 times higher focused wave energy intensity and energy harvesting output, respectively.

Gradient-index phononic crystal and Helmholtz resonator coupled structure for high-performance acoustic energy harvesting

Acoustic energy harvesting (AEH) has emerged as a promising powering process for the sustainable self-powered operation of wireless sensors in wearable, biomedical, and industrial applications. However, despite the abundance of sound sources in the environment, intrinsically low acoustic energy density still poses a critical challenge; therefore, the development of various sound amplification structures, such as Helmholtz resonators (HRs), phononic crystals, and acoustic metamaterials is required. In this study, we combined two distinct sound amplification mechanisms, Helmholtz resonance and omnidirectional acoustic wave focusing, to generate a sufficiently high power output within an ambient sound environment at practically low frequencies (< 1 kHz). A two-degree-of-freedom model was developed to systematically design a HR integrated with a piezoelectric device (HR-PEH). Subsequently, the HR-PEH was incorporated with a circularly symmetric gradient index phononic crystal (GRIN PnC) structure that enabled omnidirectional sound focusing. Finally, we constructed a coupled acoustic system GRIN-HR-PEH. Analytical modeling and experimental characterization revealed the existence of two distinct dual operating frequencies for power maximization: mechanical and acoustic resonances, which offers design flexibility in coupled AEH systems. As a result, the coupled GRIN-HR-PEH system yielded an output power of up to 4.1 mW under an ambient incident sound pressure of 47 dB, satisfying the power requirements of practical applications.

Mechanically Tuned Gradient-Index Phononic Crystal Lens

A two-dimensional gradient-index phononic crystal composed of aluminum scatterers in air matrix for frequency ofλ=6.1a is designed and simulated. We gradually change the radiuses to make a hyperbolic secant refractive index profile. Then the path of waves inside the structure is shown by analytical solutions. In this work, filling factor is mechanically increased which results in a shift in the focal point whereas the spot size remains almost constant. In order to enable the structure to be mechanically tuned, the conservators have been utilized and changed the circular cross section of scatterers slightly. However, due to this fact that the results of circular and non-circular structures do not show any differences, we used the circular scatterers for calculating the refractive indices. However, the amount of focal pressure can be slightly changed. The calculation of band structure as well as the simulation of acoustic waves’ propagation inside the structure has been accomplished using the finite element method.

Controlling the thickness dependence of torsional wave mode in pipe-like structures with the gradient-index phononic crystal lens

The fundamental torsional wave mode, T(0,1), is typically preferred in monitoring defects in long-range pipe-like structures due to non-dispersive and low attenuation characteristics such that the wave packet is not distorted with distance. However, the sensitivity of T(0,1) wave mode depends on the defect type and orientation. While it is highly sensitive to cracks along the axis of pipes, the sensitivity to detect thickness change is low as T(0,1) wave mode generates tangential displacement motion around the pipe circumference. In this study, a gradient-index phononic crystal (GRIN-PC) lens is integrated into a steel pipe to manipulate its dispersion characteristics such that the phase velocity of T(0,1) wave mode is affected by the overall change in pipe thickness. The modified behavior of T(0,1) wave mode within the GRIN-PC lens region increases the damage detection capability of T(0,1) wave mode. Numerical models involving a parametric unit cell study indicate that the phase velocity of T(0,1) wave mode decreases as the thickness of the pipe with the GRIN-PC lens decreases, in contrast with the conventional pipe. The signal difference coefficient (SDC) is applied to the acquired signal from the focal point of GRIN-PC lens to quantify the uniform thickness change. Full-scale wave propagation simulations involving solid mechanics coupled with the piezoelectric actuation demonstrate that the SDC increases as the thickness decreases. The numerical results are validated with experiments using three steel pipes with different wall thicknesses.

Viscous and thermal dissipation during the sound propagation in the continuously graded phononic crystals

The viscous and thermal effects contribute to the acoustic energy dissipation during sound propagation in complicated structures. In this study, the contributions of viscous and thermal effects, including either single viscous or thermal effects, to the entire acoustic energy dissipation during the sound propagation within graded structures have been investigated. The results show that the integral average of absorption capability coefficient of continuously graded phononic crystal (CGPC) increases by 40% than that of uniform phononic crystal (UPC), which demonstrates that the thermal and viscous effects in graded structures, compared with uniform structures, can improve the sound absorption performance. The thermal effects during the sound propagation in the CGPC account for about 12%-26% of the total thermoviscous dissipation in the studied frequency range (1–6 kHz). Therefore, the thermal effects should be considered an important factor during sound propagation. The viscous effects in CGPC account for about 74%-88% of the total thermoviscous dissipation in the studied frequency, which is slightly higher than that (72%–86%) in UPC due to graded structures. The understanding of the contribution of viscous and thermal effects in CGPC can serve for the next generation of gradient-index phononic crystals.

2D phononic-crystal Luneburg lens for all-angle underwater sound localization

Phononic crystals are well known for acoustic wave manipulation which may have potential application in an underwater acoustic detection system. In this work, we design and simulate a two-dimensional Luneburg lens based on gradient-index (GRIN) phononic crystal that is composed of PLA-Air inclusion, and a novel application of GRIN phononic crystals is proposed to sound localization. The Luneburg lens has a broadband working range, from 1500 Hz to 7500 Hz, for acoustic wave focusing with sensitive directivity and signal-to-noise improvement. By searching maximum wave intensity’s position of the focusing beam, the propagating direction of an unknown sound wave can be directly recognized covering 360°. Besides, we redesign the conventional square-lattice Luneburg lenses using annular lattices for better performance. The annular-lattice Luneburg lens overcomes the weakness of configuration defect due to the square lattice. The numerical results show that the redesign Luneburg lenses have high accuracy for distance measurement from 5 m to 35 m through the triangulation location. In a word, this work tries to explore a novel application of phononic crystals in underwater acoustic positioning and navigation technology.

Double phononic crystal lens-based enhancement of underwater power transfer

Acoustic waves are widely used for underwater communication and also investigated for power transfer owing to their low attenuation compared to electromagnetic waves. We investigate enhancing the performance of an underwater Acoustic Power Transfer (APT) system using Gradient-Index Phononic Crystals (GRIN-PCs). The GRIN-PCs consist of air inclusions in a 3D-printed polymer and are used to construct 3D acoustic lenses capable of collimating and focusing acoustic waves. The lenses are used to improve the directivity of a point source transmitter and focus the acoustic waves at a point receiver to reduce spreading losses in the APT system. The developed lenses allow for using lower frequency transducers to transmit power, hence reducing wave attenuation, and enhancing the efficiency of the system. The enhancement is verified experimentally using hydrophones as point transmitters and receivers in a double-lens system. Additionally, the lenses were used to enhance the performance of 100 kHz piezoelectric transducers with limited directivity, increasing the output power by an order of magnitude. Numerical models based on the finite element method were used to study the effect of varying the separation distance, lens aperture, and frequency on the pressure field. The developed APT concept could find applications for power delivery to wireless devices used in ocean monitoring, underwater navigation, marine life tracking, among others.

Focus of ultrasonic underwater sound with 3D printed phononic crystal

Ultrasound is widely used in various applications, ranging from ultrasound imaging to particle manipulation. Acoustic materials, such as phononic crystal (PC) and metamaterials, are designed to control the propagation and concentration of ultrasound. While traditional metal-based underwater PCs are usually limited to large 2D structures and used for low-frequency sound wave manipulation, it is difficult to reach the ultrasonic frequency region of the order of 1 MHz with cumbersome metallic structures. Here, in this work, we proposed a 3D printed gradient-index phononic crystal (GRIN PC) lens based on the hyperbolic secant index profile. With a huge impedance difference between air and the 3D printing materials, the GRIN PC lens adhered to the ultrasonic transducers can easily manipulate the propagation of acoustic waves and achieve wave focusing. This transparent and flat lens demonstrates the beam focusing in water even at a high frequency. The integration and miniaturization of transducer and lens make particle capture convenient in relevant medical applications.

Parametric acoustic array lensed by a gradient-index phononic crystal.

This work presents a theoretical study of a parametric transmitter employing a small ultrasonic transducer and an acoustic lens for the collimation of the non-directional primary ultrasonic waves into a highly-directional beam. The acoustic lens is represented by a gradient-index phononic crystal (GRIN PC) composed of an array of toroidal scatterers. Parameters of the GRIN PC lens are determined employing an optimization procedure that maximizes the minimum value of the primary-wave amplitude over a wide frequency range at a distant point in front of the transducer-lens system. The Westervelt equation is used as a wave equation taking into account diffraction, nonlinearity, and thermoviscous attenuation. The wave equation is solved numerically in the quasi-linear approximation in the frequency domain employing the finite element method. The numerical results show that employing a simple GRIN PC lens, a highly-directional low-frequency beam can be parametrically radiated from a small ultrasonic transducer.

Torsional wave focusing in cylindrical structures with the conformal gradient-index phononic crystal lens

We present torsional wave focusing in cylindrical structures integrated with conformal gradient-index phononic crystal (GRIN-PC) lenses. The mechanical lens modifies the refractive characteristics of propagating waves along the axial direction such that the torsional wave energy focuses at a desired point with an increased amplitude. In this paper, our main goal is to exploit the GRIN-PC concept for amplification of the fundamental torsional wave mode, namely, T(0,1), which is especially favorable in the guided wave ultrasonic testing of pipelines and can extend important structural information over long-range pipelines. To this end, we first study d35 piezoelectric transducer arrays through multiphysics simulations coupling piezoelectric and solid mechanics models and determine the optimal number of transducers to achieve the excitation of torsional plane wave. Two-dimensional Fourier transform as well as the tangential displacement histories around the pipe circumference at various axial positions verify the pure T(0,1) excitation, which is validated with experiments. Next, we design, fabricate, and numerically and experimentally study the conformal GRIN-PC for T(0,1) mode focusing in a range of ultrasonic frequencies, from 30 to 50 kHz. Measurements show that the amplitude of the torsional wave mode is amplified by 39% at the focal point with the presence of the GRIN-PC lens.

Partitioned gradient-index phononic crystals for full phase control

Gradient-index phononic crystals (GRIN-PC), characterized by layers with spatially changing refractive indices, have recently been investigated as part of the effort to realize flat lenses in acoustic and elastic regimes. Such gradient-index lens must be inversely designed from the corresponding refractive indices in order to manipulate the target wave. Unfortunately, estimating the index of this type of lens is not straightforward and requires substantial iterative computation in general, which greatly limits the applicability of GRIN-PC to flat lenses. In this work, we propose a novel design of a GRIN-PC in which neighboring layers are separated by partitions, thus preventing waves in each layer from interacting with other layers. This partitioned GRIN-PC design enables us readily to control the phase gradient accurately at the lens’ end, resulting in direct calculation of indices for target wave manipulation. A detailed methodology for partitioned GRIN-PC based collimator and Bessel-beam generator is proposed and experimentally validated to confirm the versatile use of our design in wave engineering applications.

Achromatic gradient-index phononic crystal lenses with unique acoustic applications

Thanks to a method known as topology optimization, gradient-index phononic crystal lenses may have applications such as imaging, energy harvesting and wireless power transfer.

Achromatic acoustic gradient-index phononic crystal lens for broadband focusing

The aim of this study is to realize an achromatic acoustic gradient-index (GRIN) phononic crystal (PC) lens system with a spatially invariant focal length over a broad operating frequency range. To this end, we propose an approach of introducing thin achromatic coating layers that can be easily assembled into the front and rear regions of the acoustic GRIN PC lens. A systematic design method based on topology optimization (TO) is developed to inversely design the achromatic coating components. The topology-optimized achromatic coating components are fabricated using 3D printing and coupled with the acoustic GRIN PC lens for acoustic characterization. Both numerical simulation and experimental characterization demonstrate the achromatic focusing capabilities of the GRIN PC lens with the designed achromatic coating layers in a wide range of frequencies (2.5 kHz–5.5 kHz). The proposed concept of applying achromatic coating layers along with the TO-based design method is expected to provide remarkable versatility to design GRIN PC lens-based applications such as energy harvesting, acoustic imaging, and acoustic wireless power transfer in broadband operation.

Gradient-index phononic crystals for omnidirectional acoustic wave focusing and energy harvesting

The design, fabrication, and analysis of omnidirectional gradient-index (GRIN) phononic crystals (PnCs) for acoustic wave focusing and energy harvesting have been demonstrated both numerically and experimentally. Despite that omnidirectional functionality is a key factor to alleviate the directivity dependence issues, the concept has not yet been incorporated into acoustic energy harvesting. In this work, a symmetrical GRIN PnC structure consisting of cylinders with variation in filling fractions has been presented to tailor the spatial acoustic refractive index, thus enforcing the acoustic waves in any direction toward the targeted center area for focusing purposes. Both a numerical simulation and experimental validation confirm substantial sound energy amplification of the designed GRIN PnC over a broad frequency range from 250 Hz to 1 kHz. Notably, the maximum sound amplification occurs at the hybrid resonant frequency of the GRIN PnC structure and the acoustic duct system used to generate incident plane waves. Numerical simulation reveals that the cavity resonance and the refraction of the GRIN PnC mainly contribute to enhanced sound amplification in addition to the reflection from the acoustic duct. The GRIN PnC structure coupled with the acoustic duct system leads to enhanced harvesting output performance when integrated with a piezoelectric energy harvesting device.

Gradient-index phononic crystals for highly dense flexural energy harvesting

Gradient-index (GRIN) refers to a system where the refractive index changes spatially within a specific region. GRIN phononic crystals are capable of not only amplifying the magnitude of wave energies but also controlling the directional nature of the wave propagation, thus offering substantial benefits with regard to energy harvesting (EH) improvements. Here, we propose a systematic design method for GRIN phononic crystals which combine the two-dimensional Reissner–Mindlin plate model and a genetic algorithm for optimization. This design process allows us to design a GRIN phononic crystal with any arbitrary refractive index profile or complex shape of the unit cells. The experimentally verified focusing capability of the GRIN phononic crystals led to the realization of piezoelectric energy harvesting with a maximum areal power density value of up to 240.4 mW/m2, considerably outperforming the existing non-GRIN-based EH systems without direction controllability.

3D-printed phononic crystal lens for elastic wave focusing and energy harvesting

We explore elastic wave focusing and enhanced energy harvesting by means of a 3D-printed Gradient-Index Phononic Crystal Lens (GRIN-PCL) bonded on a metallic host structure. The lens layer is fabricated by 3D printing a rectangular array of cylindrical nylon stubs with varying heights. The stub heights are designed to obtain a hyperbolic secant distribution of the refractive index to achieve the required phase velocity variation in space, hence the gradient-index lens behavior. Finite element simulations are performed on composite unit cells with various stub heights to obtain the lowest antisymmetric mode Lamb wave band diagrams, yielding a correlation between the stub height and refractive index. The elastic wave focusing performance of lenses with different design parameters (gradient coefficient and aperture size) is simulated numerically under plane wave excitation. It is observed that the focal points of the wider aperture lens designs have better consistency with the analytical beam trajectory results. Experiments are conducted using a PA2200 nylon lens bonded to an aluminum plate to demonstrate wave focusing and enhanced energy harvesting within the 3D-printed GRIN-PCL domain. The piezoelectric energy harvester at the focal region of the GRIN-PCL produces 3 times more power output than the baseline harvester at the same distance in the flat plate region. The results show that 3D printing can provide a simple and practical method for implementing phononic crystal concepts with minimal modification of the host structure.

Phononic crystal lens with an asymmetric scatterer

A novel 2D gradient index phononic crystal (GRIN PC) lens was designed by rotating scatterers that theoretically differs from the traditional GRIN PC lens by altering fill fraction, lattice constant or material parameters. Three-petal epoxy cylinders were designed as scatterers and embedded into a homogenous steel matrix. Rotation of the three-petal cylinder results in an asymmetry about x axis, and there is a change in the first dispersion band which is related to group velocity and the refractive index. The equivalent refraction index along the y direction was modulated to satisfy the distribution of a hyperbolic secant profile using a specific rotation angle of the scatterer. The focus of the acoustic wave in the PC lens with a rotating three-petal scatterer was investigated using a numerical method, and the results showed that the focus length agreed with the trajectories derived from the analytical solution. Results also showed that the range of the operation reduced frequency is 0.18. The influence of the operation frequency on the focusing efficiency and focal length was also discussed. The work provides a convenient method for fabricating a phononic crystal lens and makes the lens suitable for relevant applications.

High-transmission acoustic self-focusing and directional cloaking in a graded perforated metal slab

A design strategy and its modeling for high-transmission acoustic self-focusing and directional cloaking in a two-dimensional (2D) and an axisymmetric three-dimensional (3D) gradient-index phononic crystal (GRIN-PC) are reported in this paper. A gradient perforated aluminum slab sandwiched by water is considered. A low-loss directional cloaking device is achieved by controlling the matching coefficient between the slab and the water. The anisotropy coefficient that affects the scattering properties is also introduced. Furthermore, the phase discontinuity for directional cloaking inside and outside the slab is overcome by introducing a non-gradient slab having a lower group velocity behind the GRIN slab as an acoustic delay device. In addition, an axisymmetric 3D directional cloaking structure is obtained by rotating the corresponding 2D structure around the slab axis.