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It is well known that acoustic emission (AE) signals, generated by external impacts or damages such as crack initiation, mainly propagate in the form of Lamb waves in plate-like structures. In this work, MEMS-based resonant capacitive micro-machined ultrasonic transducers (CMUTs), which are designed for sensing out-of-plane displacements, have been verified by finite element method (FEM) modeling and theoretical analysis for their feasibility of detecting low-order Lamb waves (A0 and S0). First, combining the propagation theory of Lamb waves and the "spring-mass-damper" model of CMUTs, the out-of-plane sensing mechanism has been explained, together with the analytical expression of sensitivity. Then, simulations based on FEM have been carried out to show that the designed CMUTs are sensitive to out-of-plane displacements, while extremely insensitive to in-plane displacements. Meanwhile, a transient analysis has found the potential abilities of CMUTs for sensing A0 and S0 lamb waves. Besides, the sensing characteristics of CMUTs have also been investigated, including the influence of squeezed-film damping, the amplitude of the input signal, the cell number, and cell space. Finally, the ball drop impact is simulated to show the potential of identifying the location of the AE source by CMUTs. Our studies reveal the out-of-plane sensing behaviors of CMUTs for Lamb waves and may have the potential in promoting the miniaturization and integration of AE sensors. [ABSTRACT FROM AUTHOR]

A new type of stepwise radial metamaterial (SRM) with ultralow-frequency and broadband characteristics is proposed in this study. In contrast to the traditional radial metamaterial (TRM), the proposed structure is periodically arranged in a stepwise shape along the radial direction. The propagation characteristics of Lamb waves in the SRM were investigated using the finite element method. For the numerical analysis, the degeneracy between the bands of the SRM was separated, resulting in the opening of the bandgaps in the ultralow-frequency range. The total bandwidth was 75 times that of the TRM, and the wave attenuation ability was increased by more than 70%. The introduction of a stepwise array in the SRM opened up the local resonance and Bragg scattering bandgaps, and as a result, the SRM exhibited ultralow-frequency broadband characteristics. Furthermore, the influences of the structural parameters of the SRM on the bandgap characteristics were discussed. With the increase in the stepped angle, the coupling relationship between the Lamb wave mode and the local resonance was enhanced, which caused the band structure to shift to a lower frequency. In addition, the hole rotation and shape played important roles in the bandgap tuning. Finally, the experimental sample was processed based on the model, and the vibration propagation characteristics were tested to prove its ultralow-frequency broadband characteristics. The proposed shielding approach could provide a better alternative in the field of ultralow-frequency noise reduction and vibration reduction. [ABSTRACT FROM AUTHOR]

During production or service life, lightweight components made of fibre-reinforced polymers (FRP) may suffer from different types of defects such as cracks or delamination, which may be detected, amongst others, with the help of ultrasonic lamb waves (LW). The study intends to illustrate the potentials of numerical models to assist the design of LW-based non-destructive testing (NDT) processes. For that, the complete NDT chain of an LW driven inspection system was supported by a Finite Element Analysis (FEA) using stringer-stiffened C FRP panels as an exemplary component. The study was divided in two publications: The first part focused on the description of the numerical model, illustration of its functionalities and – most importantly – the determination of LW related effects that were needed to design the inspection system (dispersion characteristics, excitation frequency, propagating modes etc.). In this part, the FEA driven design approach was first validated with self measured experimental data as well as experiments provided in literature. In the second step, the numerical model was applied for the performance of parameter studies in order to identify component specific aspects that proved important for the NDT process. Finally, general advices and FEA related working steps were derived from all findings to assist practitioners in the implementation of the technique. [ABSTRACT FROM AUTHOR]

High-density polyethylene (HDPE) pipes are becoming increasingly popular, being used in various fields, such as construction, marine, petroleum, water transfer, process water, methane gas collection, oil and gas gathering, gas distribution systems, mining, acid and wet gas lines, offshore oil and gas and in nuclear power plants. Higher-order guided Lamb wave (UGW) modes can be used to detect various defects in complex structures. We will apply this methodology to one of the types of plastic—the structure of high-density polyethylene (HDPE). However, the excitation of UGW modes faces numerous challenges, especially when there is a need to identify which mode is excited. It is essential to note that, in the higher frequency range, multiple different higher-order modes can usually be excited. This can make it difficult to determine which modes have actually been excited. The objective of this research was to successfully excite and receive various higher-order UGW modes in high-density polyethylene structures using both ultrasonic single-element transducers and a phased array. Theoretical calculations were performed using a variety of methods: semi-analytical finite element (SAFE) method, 2D spatial–temporal spectrum analysis and finite element modeling (FEM). The results obtained from both measurements and simulations clearly demonstrate the possibility of efficiently exciting and receiving different Lamb wave modes possessing different phase velocities. [ABSTRACT FROM AUTHOR]

Components made of fibre-reinforced polymers (FRP) may suffer from various types of damages, such as cracks or delamination. In order to monitor the structures' material condition, non destructive testing (NDT) methods have emerged – among them, the use of ultrasonic waves. Inter alia, research focused on Lamb waves (LW), which offer promising conditions for NDT processes. However, prior implementation in a specific production environment, practitioners are faced with many open questions such as excitation frequencies to be used or which mode to select for defect identification. Therefore, the study aimed at demonstrating how a finite element analysis (FEA) can assist the experimental design of an LW driven NDT process. For that, a numerical model was developed, capable of computing LW propagations in an exemplary component: Stringer-stiffened C-FRP panels. The study was divided into two parts: In this part, all information regarding the model building is presented and principal functionality demonstrated. In the second part, the FEA is validated and then used to perform parameter studies, highlighting important conditions for practical application. The results of the present paper showed that stiffeners provoke varying interactions with LW modes – effects that increase wave field complexity but do not limit defect identification within or behind the stringer. [ABSTRACT FROM AUTHOR]

Due to its multi-mode and dispersion characteristics, Lamb waves cause interference to signal processing, which profoundly limits their application in nondestructive testing. To resolve this issue, firstly, based on the traditional EMAT, a horizontal polarization periodic permanent magnet electromagnetic acoustic transducer (HP-PPM-EMAT) was proposed. A 2-D finite element model was then developed to compare magnetic flux density, Lorentz force, and signal strength between the traditional EMAT and the HP-PPM-EMAT. The simulation results show that the HP-PPM-EMAT enhances the A0 mode Lamb wave (A0 wave) and suppresses the S0 mode Lamb wave (S0 wave). Finally, the influence of structural parameters of the HP-PPM-EMAT on the total displacement amplitude ratio of A0 and S0 was investigated using orthogonal test theory, and the width of magnet units was improved based on the orthogonal test. The results show that the total displacement amplitude ratio of A0 to S0 of the improved HP-PPM-EMAT can be improved by a factor of 7.74 compared with that of the traditional Lamb wave EMAT, which can produce higher-purity A0 mode Lamb waves. [ABSTRACT FROM AUTHOR]

Most cracks do not grow straight due to the inhomogeneity of material properties. Furthermore, quantifying structural damage using Lamb waves is difficult due to uncertain crack shape. Here, we present a newly developed finite element model to simulate the propagation of Lamb waves in a cracked plate to quantify the influence of crack size and orientation on the characteristics of Lamb wave propagation. This model includes a piezoelectric actuator and sensor to generate and receive the tone-burst signals of Lamb wave. The damping coefficient of the finite element model was calibrated with the experimental data of plates without a crack. The Lamb wave propagation simulations were performed with various crack sizes, crack orientations, and distances between the actuator and sensor. Our model was utilized to investigate the influence of crack orientation on damage quantification based on the signal phase change and normalized amplitude. We successfully evaluated the effect of the incident angle of the crack, crack size, and distance between the sensors on the change of the plate wave signal characteristics. The signal characteristics obtained from our novel finite element model can be used to develop a quantification model to estimate the crack size, direction, and distance from the sensor. [ABSTRACT FROM AUTHOR]

New designs based on S0 Lamb modes in AlN thin layer resonating structures coupled with the implementation of structural elements in SiO2, are theoretically analyzed by the Finite Element Method (FEM). This study compares the typical characteristics of different interdigital transducer (IDTs) configurations, involving either a continuous SiO2 cap layer, or structured SiO2 elements, showing their performance in the usual terms of electromechanical coupling coefficient (K2), phase velocity, and temperature coefficient of frequency (TCF), by varying structural parameters and boundary conditions. This paper shows how to reach temperature-compensated, high-performance resonator structures based on ribbon-structured SiO2 capping. The addition of a thin diamond layer can also improve the velocity and electromechanical coupling coefficient, while keeping zero TCF and increasing the solidity of the membranes. Beyond the increase in performance allowed by such resonator configurations, their inherent structure shows additional benefits in terms of passivation, which makes them particularly relevant for sensing applications in stern environments. [ABSTRACT FROM AUTHOR]

Recently Lamb wave devices have originated as an alternate acoustic device for high‐frequency wireless sensing applications. Their potential for sensing devices, including biomedical diagnostics and environmental monitoring as a wireless and passive device, calls for further analysis of the device with higher sensitivity. ZnO‐based sensor has always been of research interest due to its biocompatibility, sensing abilities, and yet cost‐effectivity. A Lamb wave device based on ZnO/SiO2/Si membrane has been theoretically simulated using finite element analysis method (FEM) to study the higher modes, as higher working frequency leads to higher sensitivity of the device toward sensing applications. Optimized properties are identified and utilized for the fabrication of Lamb wave devices. It is observed that for optimized parameters of ZnO/SiO2/Si Lamb device, higher antisymmetric modes are not only advantageous for high sensitivity applications but also more stable within the given operating conditions. Experimentally obtained results indicate close matching with theoretical results. [ABSTRACT FROM AUTHOR]

Lamb waves are extensively used in non-destructive tests (NDT) to detect corrosion and another defects in thin plates. Several NDT methods require to compute the phase velocity of the Lamb wave, which depends on the frequency. In this work we show the potential of the combination of the phase velocity method (PVM) with the finite element method (FEM) for computing the phase velocity dispersion curve of an ultrasonic pulse traveling in a transversally annular isotropic thin plate. The FEM-PVM is based on the numerical solution of the wave propagation equations for several selected frequencies. To solve these equations, a second order difference scheme is used to discretize the temporal variable, while spatial variables are discretized with FEM. The open software FreeFem++ is used with quadratic triangular elements to compute the displacements. The phase velocity for a given frequency is obtained from the computed displacements at few points on the top of the plate. Repeating the procedure for several frequencies we compute a sample of points that are fitted to obtain an approximation of the phase velocity dispersion curve. A deeper understanding on the behavior of the phase velocity dispersion curve with respect to the plate curvature is obtained as result of an extensive experimentation with annular isotropic plates of two materials. [ABSTRACT FROM AUTHOR]

A 400 MHz magnetoelectric (ME) Lamb wave antenna design to function in the medical implant communication service band is proposed. The antenna employs a heterostructure of piezoelectric and magnetostrictive membranes to acoustically excite standing shear bulk wave and radiate as a magnetic dipole. Multiphysics finite element analysis simulations are performed for transmission and reception modes. In these simulations, three aspects are investigated: piezoelectricity, micromagnetic precession, and magnetic dipole radiation. An experimental demonstration of the antenna is also conducted and shows mechanical resonance with a Q-factor of 500 and ME coupling. These results indicate that the design can be operated in zero-order antisymmetric (A0) mode as a tunable oscillator or sensor. This ME approach provides a solution to the miniaturization problem of traditional current-based implantable antennas. [ABSTRACT FROM AUTHOR]

Lamb wave-based damage detection technology shows great potential for structural integrity assessment. However, conventional damage features based damage detection methods and data-driven intelligent damage detection methods highly rely on expert knowledge and sufficient labeled data for training, for which collecting is usually expensive and time-consuming. Therefore, this paper proposes an automated fatigue crack detection method using Lamb wave based on finite element method (FEM) and adversarial domain adaptation. FEM-simulation was used to obtain simulated response signals under various conditions to solve the problem of the insufficient labeled data in practice. Due to the distribution discrepancy between simulated signals and experimental signals, the detection performance of classifier just trained with simulated signals will drop sharply on the experimental signals. Then, Domain-adversarial neural network (DANN) with maximum mean discrepancy (MMD) was used to achieve discriminative and domain-invariant feature extraction between simulation source domain and experiment target domain, and the unlabeled experimental signals samples will be accurately classified. The proposed method is validated by fatigue tests on center-hole metal specimens. The results show that the proposed method presents superior detection ability compared to other methods and can be used as an effective tool for cross-domain damage detection. [ABSTRACT FROM AUTHOR]

The present paper comprehensively investigates the complex interaction between ultrasonic guided waves and local structural discontinuities, such as damages, through highly sensitive features: so-called wave damage interaction coefficients (WDICs). These WDICs are unique for each structural discontinuity and depend solely on their characteristics for a given structure and condition. Thus, they can be particularly useful for advanced assessment of lightweight structures in the context of non-destructive evaluation and structural health monitoring. However, the practical application of WDICs entails significant difficulties due to their sensitivity and complex patterns. Therefore, this study attempts to guide researchers and practitioners in the estimation of WDICs from numerical simulations and physical experiments. Detailed investigations are made for an aluminum host plate modified by artificial structural discontinuities, i.e., surface-bonded steel sheets. The numerical simulations are performed to predict WDICs and study sensitivities using a sophisticated finite element model. The experimental setup uses piezoelectric transducers to excite guided waves in the host plate. A single scanning laser Doppler vibrometer measures the scattered guided waves caused by the surface-bonded steel sheets, and the resulting WDICs with possible influences are investigated. In both cases, the orientation and thickness of the attached steel sheets were varied to create 12 different damage scenarios. In general, the comparison between numerical and experimental WDICs show good agreement. This underpins the applicability of the general methodology for simulating and measuring WDICs over all scenarios. Furthermore, the WDIC scattering patterns reveal a clear dependency of the peaks in the back-scattered reflections for both the numerical and experimental amplitude coefficients on the damage orientation, basically following the law of reflection. However, some discrepancies between both studies were observed. Numerical sensitivity analysis identified the adhesive layer as one reason for such differences. Additionally, misalignment errors in the experimental measurements were also found to affect WDICs. Therefore, an improved baseline subtraction method is proposed, which clearly enhances the experimental WDICs. Finally, an experimental sensitivity study of WDICs for selected sensing radii revealed only a minor influence. All these investigations were made for the amplitude as well as the phase representation of WDICs. Thus, these findings may open the way to future research and development of techniques employing WDICs for advanced applications of non-destructive evaluation and structural health monitoring. [ABSTRACT FROM AUTHOR]

The present paper assessed the use of variable kinematic two-dimensional elements in the dynamic analysis of Lamb waves propagation in an isotropic plate with piezo-patches. The multi-field finite element model used in this work was based on the Carrera Unified Formulation which offers a versatile application enabling the model to apply the desired order theory. The used variable kinematic model allowed for the kinematic model to vary in space, thereby providing the possibility to implement a classical plate model in collaboration with a refined kinematic model in selected areas where higher order kinematics are needed. The propagation of the symmetric ( S 0 ) and the antisymmetric ( A 0 ) fundamental lamb waves in an isotropic strip was considered in both mechanical and piezo-elastic plate models. The convergence of the models was discussed for different kinematics approaches, under different mesh refinement, and under different time steps. The results were compared to the exact solution proposed in the literature in order to assess and further determine the effects of the different parameters used when dynamically modeling a Lamb wave propagating in such material. It was shown that the higher order kinematic models delivered a higher accuracy of the propagating wave evaluated using the corresponding Time Of Flight (TOF). Upon using the appropriate mesh refinement of 2000 elements and sufficient time steps of 4000 steps, the error between the TOF obtained analytically and numerically using a high order kinematics was found to be less than 1% for both types of fundamental Lamb waves S 0 and A 0 . Node-dependent kinematics models were also exploited in wave propagation to decrease the computational cost and to study their effect on the accuracy of the obtained results. The obtained results show, in both the mechanical and the piezo-electric models, that a reduction in the computational cost of up to 50% can be easily attained using such models while maintaining an error inferior to 1%. [ABSTRACT FROM AUTHOR]

Scanning laser Doppler vibrometry is a widely adopted method to measure the full-field out-of-plane vibrational response of materials in view of detecting defects or estimating stiffness parameters. Recent technological developments have led to performant 3D scanning laser Doppler vibrometers, which give access to both out-of-plane and in-plane vibrational velocity components. In the present study, the effect of using (i) the in-plane component; (ii) the out-of-plane component; and (iii) both the in-plane and out-of-plane components of the recorded vibration velocity on the inverse determination of the stiffness parameters is studied. Input data were gathered from a series of numerical simulations using a finite element model (COMSOL), as well as from broadband experimental measurements by means of a 3D infrared scanning laser Doppler vibrometer. Various materials were studied, including carbon epoxy composite and wood materials. The full-field vibrational velocity response is converted to the frequency-wavenumber domain by means of Fourier transform, from which complex wavenumbers are extracted using the matrix pencil decomposition method. To infer the orthotropic elastic stiffness tensor, an inversion procedure is developed by coupling the semi-analytical finite element (SAFE) as a forward method to the particle swarm optimizer. It is shown that accounting for the in-plane velocity component leads to a more accurate and robust determination of the orthotropic elastic stiffness parameters. [ABSTRACT FROM AUTHOR]

This paper deals with guided wave-based structural health monitoring of composite overwrapped pressure vessels adopted for space application. Indeed, they are well suited for this scope thanks to their improved performance compared with metallic tanks. However, they are characterized by a complex damage mechanics and suffer from impact induced damage, e.g., due to space debris. After reviewing the limited progress in this specific application, the paper thoroughly covers all the steps needed to design and verify guided wave structural health monitoring system, including methodology, digital modelling, reliability, and noise estimation for a correct decision-making process in a virtual environment. In particular, propagation characteristics of the fundamental anti-symmetric mode are derived experimentally on a real specimen to validate a variety of finite element models useful to investigate wave interaction with damage. Different signal processing techniques are demonstrated sensitive to defect and linearly dependent upon damage severity, showing promising reliability. Those features can be implemented in a probability-based diagnostic imaging in order to detect and localized impact induce damage. A multi-parameter approach is achieved by metrics fusion demonstrating increased capability in damage detection with promising implication in enhancing probability of detection. [ABSTRACT FROM AUTHOR]

Transcranial focused ultrasound (FUS) is a noninvasive treatment for brain tumors and neuromodulation. Based on normal incidence, conventional FUS techniques use a focused or an array of ultrasonic transducers to overcome the attenuation and absorption of ultrasound in the skull; however, this remains the main limitation of using FUS. A dual-mode conversion technique based on Lamb waves is proposed to achieve high transmission efficiency. This concept was validated using the finite element analysis (FEA) and experiments based on changes in the incident angle. Aluminum, plexiglass, and a human skull were used as materials with different attenuations. The transmission loss was calculated for each material, and the results were compared with the reflectance function of the Lamb waves. Oblique incidence based on dual-mode conversion exhibited a better transmission efficiency than that of a normal incidence for all of the specimens. The total transmission losses for the materials were 13.7, 15.46, and 3.91 dB less than those associated with the normal incidence. A wedge transducer was designed and fabricated to implement the proposed method. The results demonstrated the potential applicability of the dual-mode conversion technique for the human skull. [ABSTRACT FROM AUTHOR]

This paper presents a new framework for stochastic updating of a finite element model for a composite plate, considering the influence of temperature on Lamb wave propagation. The framework involves deterministic updating to optimize mechanical properties and stochastic updating to derive probability density functions for key parameters. It utilizes sensitivity analysis and Bayesian inference with Markov-Chain Monte Carlo simulations and the Metropolis–Hastings sampling algorithm. This paper proposes a machine learning surrogate model based on artificial neural networks to improve computational efficiency. This surrogate modeling approach allows parallelized Monte Carlo simulations, reducing updating time significantly without compromising the accuracy of the resulting probability density functions for model parameters. These advancements show a promising way to enhance composite plate modeling and Lamb wave propagation studies, providing a more efficient and accurate approach to verify and validate finite element models with potential applications in engineering simulations. [ABSTRACT FROM AUTHOR]

• Construction of transparent boundary conditions (TBCs) for exural wave cavity scattering in two dimensions, ensuring the satisfaction of the sommerfeld radiation conditions. • Deduction of a decomposed problem for the biharmonic plate wave equation using two auxiliary functions and establishment of the unique-ness of the solution for the decomposed problem. • Proposal of the linear _nite element method with interior penalty term (IP-FEM) and boundary penalty term (BP-FEM) to solve the coupled boundary helmholtz and modi_ed helmholtz equations, leading to stable numerical solutions. • Introduction of an analytical solution for exural wave scattering by a circular cavity with a clamped boundary, facilitating comparative analysis. Flexural wave scattering plays a crucial role in optimizing and designing structures for various engineering applications. Mathematically, the flexural wave scattering problem on an infinite thin plate is described by a fourth-order plate wave equation on an unbounded domain, making it challenging to solve directly using the regular linear finite element method (FEM). In this paper, we propose two numerical methods, the interior penalty FEM (IP-FEM) and the boundary penalty FEM (BP-FEM) with a transparent boundary condition (TBC), to study flexural wave scattering by an arbitrary-shaped cavity on an infinite thin plate. Both methods decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary. A TBC is then constructed based on the analytical solutions of the Helmholtz and modified Helmholtz equations in the exterior domain, effectively truncating the unbounded domain into a bounded one. Using linear triangular elements, the IP-FEM and BP-FEM successfully suppress the oscillation of the bending moment of the solution on the cavity boundary, demonstrating superior stability and accuracy compared to the regular linear FEM when applied to this problem. [ABSTRACT FROM AUTHOR]

An ultrasonic phased array scanning method was used for lamb wave evaluation of surface and in-depth defects in a steel plate. The ultrasonic guided wave modes resulting from a phased array transducer were modeled using finite element method. The experimental results showed that the high-frequency A0 mode gives acceptable sensitivity and resolution for small diameter through-thickness and surface defects. The sectorial scanning showed better axial resolution rather than spatial resolution. [ABSTRACT FROM AUTHOR]

The conventional finite element method can be used for modeling the multiscale phenomenon of the interaction of Lamb waves with matrix cracks, but it requires a lot of computational resources. In the present investigation, appropriate damage models are used to model the reduction in the stiffness of the ply in which the matrix crack has occurred. Moreover, the interactions of Lamb waves with matrix cracks, which have been modeled as a local reduction in stiffness, are being studied. Furthermore, in a Glass Fiber Reinforced Plastic (GFRP) laminate with a [0/90/90/0] lay-up, Lamb waves of various frequencies and with So (symmetric) and Ao (asymmetric) modes are transmitted along the fiber and across the fiber directions, respectively. As a result, such waves are recorded at two points, 75 and 225 mm from the site of excitation. For two different crack densities, such as 1 crack/mm and 4 cracks/mm, the healthy and damaged laminate signals of similar mode are compared. The amplitudes of the received signals are reduced upon interaction of these waves with damage (with degraded stiffness values), indicating that So modes of Lamb waves are much more susceptible to damages at higher frequencies of 100, 150, and 200 kHz compared to Ao modes. Furthermore, for a given crack density, equivalent damage models are validated by comparing these signals to cracked laminate signals of similar frequencies and modes. During the study, it was revealed that equivalent damage models are perfectly appropriate for modeling matrix cracks of any density as a local loss in stiffness in the damaged crossply GFRP laminate. [ABSTRACT FROM AUTHOR]

The propagation of laser-generated Lamb waves in a two-dimensional acoustic black-hole structure was studied numerically and experimentally. The geometrical acoustic theory has been applied to calculate the beam trajectories in the region of the acoustic black hole. The finite element method was also used to study the time evolution of propagating waves. An optical system based on the laser-Doppler vibration method was assembled. The effect of the focusing wave and the reduction in wave speed of the acoustic black hole has been validated. [ABSTRACT FROM AUTHOR]

Broadband metamaterial absorbers in the terahertz (THz) range are of great importance because of their applications, especially in the field of THz detection and imaging. Characteristics of an ideal absorber include wide bandwidth and insensitivity to polarization and incidence angle. Here we propose a new structure of an ultra-broadband THz absorber by adding an excess dielectric layer on a metamaterial structure with two stacked chromium square resonators. The excess layer besides protecting chromium resonator from oxidation creates an extra Fabry–Pérot cavity that extends the bandwidth of the structure. The relative bandwidth of this structure for 90% absorption exceeds 150% and covers frequencies ranging from 0.52 to 3.66 THz, which is an advantage over the works reported in the literature review. The number of resonating layers is decreased related to the earlier works too. The proposed structure is not sensitive to the polarization of the incident radiation, at the same time its absorption is almost insensitive to the incident angle too. For both TE and TM polarization the absorption stays more than 80% up to 60° of the incident angles. Furthermore, the insensitivity of the proposed device to misalignment between layers is evaluated for misalignment up to 28 μm. The results show that the structure is insensitive to misalignment which makes its fabrication easy. We have simulated the proposed structure using the numerical finite element method (FEM). Also, we have extracted a circuit model for the suggested absorber which is in accordance to the numerical model and can be used to reduce the designing process time. Correspondingly, we efficiently explain the absorption mechanism using the circuit model and magnetic field analysis in the structure. We have shown that the broad bandwidth of the structure is the result of coupling between various modes of Fabry–Perot (FP) cavity resonances and surface plasmons (SPs). The FP resonances are coupled through the gap resonators between the different layers. This means absorbers are more compact than the convectional $\text{n}\lambda $ /4 wave plate absorbers. [ABSTRACT FROM AUTHOR]

This article presents the design approach and the first demonstration of a wideband hybrid monolithic acoustic filter in the K-band, which exceeds the limitation of electromechanical coupling on the fractional bandwidth (FBW) of acoustic filters. The hybrid filter utilizes the codesign of electromagnetic (EM) and acoustic to attain wide bandwidth while keeping the advantages of small sizes and high Q in the acoustic domain. The performance trade space and design flow of the hybrid filter are also presented in this article, which allows this technology to be applied for filters with different center frequencies and FBWs. The hybrid filter is simulated by hybridizing the EM and acoustic finite element analysis, which are carried out separately and combined at a system level. The fabricated filter built with resonators having an electromechanical coupling of 0.7% based on the seventh-order antisymmetric Lamb wave mode (A7) has a 3-dB FBW of 2.4% at 19 GHz and a compact footprint of 1.4 mm2. [ABSTRACT FROM AUTHOR]

A theoretical model is derived to extend existing work on the theory of acoustoelasticity in isotropic materials subjected to uniaxial or hydrostatic loadings, up to the case of arbitrary triaxial loading. The model is applied to study guided wave propagation in a plate. The semi-analytical finite element method is adapted to deal with the present theory. Effects of triaxial loading on velocities of Lamb and shear horizontal (SH) modes are studied. They are non-linearly dependent on stress, and this nonlinearity is both frequency-dependent and anisotropic. Velocity changes induced by the effect of stress on the plate thickness are shown to be non-negligible. When a stress is applied, both Lamb and SH modes lose their simple polarization characteristics when they propagate in directions different from the principal directions of stress. The assumption that effects induced by a multiaxial stress equal the sum of effects induced by each of its components independently is tested. Its validity is shown to depend on frequency and propagation direction. Finally, the model is validated by comparing its predictions to theoretical and experimental results of the literature. Its predictions agree very well with measurements and are significantly more accurate than those of existing theories. [ABSTRACT FROM AUTHOR]

We investigate guided (Lamb) waves in a human cadaver skull through experiments and computational simulations. Ultrasonic wedge transducers and scanning laser Doppler vibrometry are used respectively to excite and measure Lamb waves propagating in the cranial bone of a degassed skull. Measurements are performed over a section of the parietal bone and temporal bone spanning the squamous suture. The experimental data are analyzed for the identification of wave modes and the characterization of dispersion properties. In the parietal bone, for instance, the A0 wave mode is excited between 200 and 600 kHz, and higher-order Lamb waves are excited from 1 to 1.8 MHz. From the experimental dispersion curves and average thickness extracted from the skull computed tomography scan, we estimate average isotropic material properties that capture the essential dispersion characteristics using a semi-analytical finite-element model. We also explore the leaky and non-leaky wave behavior of the degassed skull with water loading in the cranial cavity. Successful excitation of leaky Lamb waves is confirmed (for higher-order wave modes with phase velocity faster than the speed of sound in water) from 500 kHz to 1.5 MHz, which may find applications in imaging and therapeutics at the brain periphery or skull-brain interface (e.g., for metastases). The non-leaky A0 Lamb wave mode propagates between 200 and 600 kHz, with or without fluid loading, for potential use in skull-related diagnostics and imaging (e.g., for sutures). [ABSTRACT FROM AUTHOR]

• Semi-analytical approach presenting derivation of the scattering matrix related to the region with adhesively bonded joint. • Effect of the energy balance convergence of local sub-waveguide sections on the global scattering matrix. • Effect of the number of considered complex wavenumbers entering the computational process on the predicted results accuracy. Problems related to scattering phenomena of the antisymmetric fundamental Lamb wave mode with an adhesively bonded lap joint containing a disbond in the adhesive layer is being solved within the framework of this paper. Overall, two methods are used in order to identify energy scattering coefficients of reflected as well as transmitted Lamb waves. These methods include a hybrid approach, which incorporates the finite element method (FEM) with general orthogonality relation, and semi-analytical approach, combining the modal expansion technique and orthogonal property of Lamb waves. With the use of the stress and displacement continuity condition on selected interfaces, it is possible to explicitly derive the global scattering matrix, containing the information related to the scattering process of present Lamb wave modes in a given geometry. Both approaches are then adopted on three configurations of the adhesively bonded lap joint in terms of the length of the disbond, which is then positioned within the overlapped region. The obtained results, among others, clarify the relationship between geometric parameters of the region containing the overlapping with the horizontal disbond and the role of non-propagating modes in the computational process of energy scattering coefficients of propagating modes. [ABSTRACT FROM AUTHOR]

Bolts are commonly applied to lap joints in various industrial fields. Harsh environment, low quality of operation, and other unfavorable conditions may result in bolt looseness. Due to the structural complexity, conventional Lamb wave-based methods just extract statistical characteristics for looseness detection, while neglecting the information contained in the propagation process of Lamb waves. To take full advantage of the information on bolt states carried by array signals during the propagation, this study proposes a detection procedure based on propagation model and imaging methods of Lamb waves. Firstly, the contact status on the macro and micro scales is investigated with theoretical analysis and finite element simulation, and the theoretical model of Lamb wave propagation across bolted joints is established deriving from the analysis of contact. Subsequently, three imaging methods, including delay-and-sum, minimum variance, and sparse reconstruction, are adapted for the identification of bolt state. Since Lamb waves are mainly transmitted into another plate through the contact area near the bolts based on the proposed propagation model, pixel values are inclined to be concentrated around the bolts. On this basis, imaging intensity near each individual bolt can reveal its fastening state given that the transmitted energy around its location is associated with its torque. Finally, experiments are carried out in both single-bolt and double-bolt joints. The proposed propagation model is verified in the single-bolt joint and loose bolts can be identified in the imaging results of the double-bolt joint, which demonstrates the effectiveness of the proposed method for bolt state identification. [ABSTRACT FROM AUTHOR]

The second-harmonic generation of the fundamental antisymmetric Lamb wave at a closed parallel crack in an elastic plate is studied by numerical analysis. The closed crack is modeled as a spring-type interface with quadratic nonlinearity. Based on a perturbation method, the problem of nonlinear Lamb wave scattering is decomposed into two linearized problems, i.e., for the linear reflection/transmission of the incident Lamb wave at the crack and for the generation/radiation of the second-harmonic Lamb waves due to the contact nonlinearity. The reduced problems are solved by the finite element method in the frequency domain. Numerical results demonstrate significant effects of the crack resonance on the linear and nonlinear Lamb wave scattering responses, which appear as sharp peaks/dips in the reflection/transmission spectra and enhanced second-harmonic amplitudes at some frequencies. Two simple frequency selection rules are established which explain the enhanced generation of the second-harmonic Lamb waves. The time-domain analysis is also carried out to supplement the frequency-domain analysis, which confirms that the incident Lamb wave interacts with the crack at some specific frequencies in its bandwidth in a selective manner and enhances the generation of the second-harmonic components. [ABSTRACT FROM AUTHOR]

Purpose: The aim of the study was to show that a new method, using a look-up table technique, can be used to detect the presence and position of an inclusion embedded in a tissue-like material. Due to the time-consuming nature of the finite element (FE) method or FEM, real-time applications involving FEM as part of a control loop, are traditionally limited to slowly varying systems. By using a simplified two-dimensional FE model and a look-up table, we show by simulations and experiments that it is possible to achieve reasonable computational times in a tactile resonance sensor application. Design/methodology/approach: A piezoelectric disk was placed in the center of a silicone rubber disk (SRD) with viscoelastic properties, where it acted as both sensor and actuator and dissipated radial acoustic waves into the silicone. The look-up table was constructed by calculating the radial Lamb wave transition frequencies in the impedance frequency response of the sensor while varying the position of an inclusion. A position-matching algorithm was developed that matched measured and calculated Lamb wave transitions and thereby identified the presence and position of an inclusion. Findings: In an experiment, the position of a hard inclusion was determined by measuring the Lamb transition frequencies of the first radial resonance in two SRDs. The result of the matching algorithm for Disk 1 was that the matched position was less than 3% from the expected value. For Disk 2, the matching algorithm erroneously reported two false positions before reporting a position that was less than 5% from the expected value. An explanation for this discrepancy is presented. In a verifying experiment, the algorithm identified the condition with no inclusion present. Originality/value: The approach outlined in this work, adds to the prospect of developing time-sensitive diagnostic instruments. This approach has the potential to provide a powerful technique to quickly present spatial information on detected tumors. [ABSTRACT FROM AUTHOR]

Propagation of Lamb waves in both laminated and functionally graded (FG) composites is analyzed. Anomalous dispersion at high frequencies is observed revealing substantial discrepancy in the high frequencies asymptotes for quasi-flexural and quasi-symmetric fundamental modes. The applied methodology is based on a variant of sextic complex formalism. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this study is to numerically simulate the Lamb wave propagation through a clamp-on ultrasonic gas flowmeter (UGF) in contact mode, using a new semi three-dimensional approach. Moreover, experimental and analytical modeling results for transit time difference method have been used to confirm the simulation results at different gas flow velocities from 0.3 to 2.4 m/s. Design/methodology/approach: The new semi three-dimensional approach involves the simulation of the flow field of the gas in a three-dimensional model and subsequently the simulation of wave generation, propagation and reception in a two-dimensional (2D) model. Moreover, the analytical model assumes that the wave transitions occur in a 2D mode. Findings: The new approach is a semi three-dimensional approach used in this work, has better accuracy than a complete 2D simulation while maintaining the computing time and costs approximately constant. It is faster and less expensive than a complete 3D simulation and more accurate than a complete 2D simulation. It was concluded that the new approach could be extended to simulate all types of ultrasonic gas and non-gas flowmeters, even under harsh conditions. Originality/value: In this work, a new approach for the numerical simulation of all types of ultrasonic flowmeters is introduced. It was used for simulation of a Lamb wave ultrasonic flow meter in contact mode. [ABSTRACT FROM AUTHOR]

A semi-analytical finite element method algorithm was established to plot the dispersion curves of isotropic aluminum and orthotropic plates. The curves obtained are compared with those plotted by the DISPERSE software and with previous experimental work. The results showed that the accuracy of the method depends on the number of elements for meshing. To ensure good precision and speed of the method, the number of elements per plate thickness must be optimized. [ABSTRACT FROM AUTHOR]

Analytical expressions are derived for the stresses and electric fields induced in a piezoelectric multilayer deposited on a substrate with lattice misfit and thermal expansion coefficient mismatch. The piezoelectric multilayer can be subjected to an externally applied force, moment and electric potential. The derived formulations can model any number of layers using recursive relations that minimize the computation time. A proper rotation matrix is utilized to generalize the derived expressions to accommodate various orientations allowing each layer to have hexagonal crystal symmetry. The influence of lattice misfit and thermal expansion coefficient mismatch in the presence of externally applied force, moment and electric potential on the state of the electroelastic fields in each layer is evaluated for various applications. Comparison with finite element analysis results shows excellent agreement. The analytical expressions developed here can be useful in designing electromechanical sensors, actuators and optoelectronic devices made from piezoelectric multilayers. [ABSTRACT FROM AUTHOR]

Summary: This study presents a new approach for designing optimal sensors layout based on accuracy of defect mapping. It is obtained from combination of the reference‐free damage detection technique and the probability‐based diagnostic imaging method. Considering damage indices based on continuous wavelet transform of sensors signals, the core of this study involves with development of a database of continuous wavelet transform features of a crack. In fact, the database contains the data from 594 different states in crack positions, orientations, and the considered sensing path lengths. Eventually, this database is used for localization of damage by interpolating the stored data collected from the above mentioned 594 states of the crack. The verification tests of defect mapping interpolating the stored data show an excellent match with the finite element method crack positioning performed for three different crack conditions. To generalize the results, here, 134 sensor layouts are considered in each of which 121 crack positions are detected by the help of probability‐based diagnostic imaging method. Then to check upon the accuracy of the results, the distance between the real center of the crack and the detected crack position by the interpolation of the stored data is calculated. Finally, the optimal sensor layouts single out from among the 134 ones based on user‐defined accuracy in localizing damage on an arbitrary monitoring area and the number of sensors. [ABSTRACT FROM AUTHOR]

A concept for the development of spectrum-clean S1 Lamb wave resonator (LWR) utilizing an aluminum nitride (AlN) plate with damped edge reflectors is demonstrated. Numerical analysis is performed on the dispersive characteristics of the Lamb modes propagating in AlN plates, and the design parameters to enable zero group velocity (vg) for the S1 mode are identified. The simulation results based on finite element analysis verify that the use of the damped edge reflectors, instead of the conventional edge reflectors, can efficiently de-couple most of the Lamb waves such as the S0 mode by mode conversion upon edges, but not the S1 mode characterizing zero vg. Specifically, the measured frequency response of a 1.38-GHz AlN S1 LWR with damped edge reflectors yields a clean spectrum up to 6 GHz and excellent resonance passband performance. [ABSTRACT FROM AUTHOR]

In the present study, the asymptotic stress and displacement solutions related to mixed mode I/II loading are calculated in the vicinity of the V-notches with end holes (VO-notches). After deriving the asymptotic stress and displacement fields, in order to calculate the coefficients of the series, a numerical technique called the over-deterministic method is utilized. According to the over-deterministic method, the unknown coefficients are calculated by fitting the asymptotic displacement field to the nodal displacement values. To show the accuracy of the derived asymptotic stress series and the effectiveness of higher order terms, the Brazilian disk specimen is modeled using finite element analysis. Different examples under pure mode I, pure mode II and mixed mode I/II loading for three different notch opening angles and four notch tip radii are considered and the coefficients of stress series are computed for each example. Finally, the obtained coefficients are used to compare the stress distribution of the truncated stress series with its relevant finite element values. The results show that considering the singular terms alone will generate large amounts of errors while very good accuracy could be achieved by considering the higher order terms. [ABSTRACT FROM AUTHOR]

This work presents an extensive study on the temperature coefficient of frequency (TCF) for thin-film lithium niobate (LiNbO3) resonators. To capture the temperature-frequency behavior for each vibration mode, a one-dimensional (1D) multi-section TCF model is proposed and verified by both the finite-element method (FEM) and experiments. By partitioning the metallized and non-metallized sections of the interdigitated transducers (IDT) as distinct TCF blocks and properly considering the spatial energy distribution, the TCF of each mode can be accurately predicted. Two higher lateral-order resonators based on the fundamental shear-horizontal mode (SH0) are designed and fabricated on a LiNbO3-on-SiO2 wafer to provide odd- and even-order harmonics. TCFs of the first 6 overtones (from 84 to 515 MHz) are calculated using the material properties given in the literature, exhibiting a maximum TCF difference of 3 ppm/K between the theoretical prediction and measurement results. Although only SH0 resonators are selected for experimental verification, the proposed multi-section TCF model can be applied to other plate wave MEMS resonators. [2019-0112] [ABSTRACT FROM AUTHOR]

• A non-contact ultrasonic method of gas flow metering in a duct is described. • Wave propagation was visualized by COMSOL Multiphysics®. • Multipath signals with different time shift were successively received by a receiver in this method. • Receiver could collect the signal in a wide region (from 50 mm to 250 mm in the x direction in this work). • Simulations and experiments show good agreement with the analytical model (RMSE are 0.0708 and 0.0767 respectively). This work investigated the signal reception from multipath including direct transmission and reflection paths in a non-contact gas flow measurement system based on air-coupled leaky Lamb waves. Analytical model was established to provide the multiple reflective signal propagation time differences between upstream and downstream paths at different flow velocities. A multiphysics model based on finite element methods was established to facilitated the visualisation of the leaky Lamb waves reflected between the plates and the Lamb waves excited by multiple reflective paths waves. The propagation time differences were evaluated by performing simulations and experiments, which are in good agreement with the analytical values (RMSE are 0.0708 and 0.0767 respectively). The receiver can collect multipath signals at a large distance separated from the transmitter. Time resolution is enhanced by using received signals from reflection paths, as the transit time increased with the ultrasound path between the transducers. [ABSTRACT FROM AUTHOR]

The wall pressure generated by near-field underwater explosion bubble involves significant nonlinear problems. This has been the focus and difficulty of the research on the anti-explosion design of battleships. To quantify the load characteristics of bubble pulsation and water jet pressure, underwater explosion tests were conducted beneath a clamped square plate subjected to 2.5g TNT at different distances (R = 10, 15, 20, 22.5, and 25 cm). Results showed that as the stand-off distance increased to the maximum radius of the bubble, there was a peak value of the impulse for bubble pulsation and water jet. A finite element model was then established and benchmarked. The benchmarked model was used to further explore the load characteristics of the bubble pulsation and water jet, considering the effect of the stand-off distance, plate dimension, horizontal distribution, and explosive equivalent. It was found that: a) as the plate dimension increased, the pressure of the shock wave to the plate was basically unchanged, while the period and impulse of the bubble pulsation and water jet decreased; b) the impact of the water jet pressure was concentrated within 6.72 times the radius of the explosive. Finally, an empirical formula was proposed to predict the impulse of the bubble pulsation and water jet. This research is able to provide guidance to investigate the blast resistance of ships. • The wall pressure-time history curves of bubble jet were obtained. • The bubble jet load was accurately predicted by the CEL method. • The effect of the stand-off distance, plate dimension, horizontal distribution, and explosive equivalent on bubble jet load. • An empirical formula was proposed to predict the specific impulse of the bubble jet load. [ABSTRACT FROM AUTHOR]

Interface debonding may happen in steel structures repaired or strengthened by Carbon Fiber Reinforced Polymers (CFRP), which will reduce ultimate loads and fatigue life of the structures. In this paper, a non-destructive method to test the interface debonding using Lamb waves is proposed. The electromechanical finite element detection model of a steel plate strengthened by CFRP is constructed, and the results have good consistency with experimental results. The effects of debonding position and debonding degree on the received signals are discussed in this paper, the signal waves are analyzed by the wavelet time-frequency conversion method and wavelet packet analysis method, and the relationship between the signal waves and the debonding parameters are obtained, which could support the damage detection of the repaired or strengthened structures by CFRP in the future. • The signal of Lamb wave in steel plates strengthened by CFRP with interface debonding were analyzed. • The experimental model and the FE model of interface debonding analysis in CFRP strengthening steel plate were constructed. • The effects of debonding position and degree on the characteristics of received signals were quantitatively analyzed. [ABSTRACT FROM AUTHOR]

• Anisotropic wave skew effects can cause an apparent offset in guided wave velocities. • Difference from theory of UD CFRP plate experiments and FE simulations. • Reduced accuracy for damage detection and localization in composite structures. • Skew angle correction for point source excitation, implemented experimentally. Guided ultrasonic wave propagation in anisotropic structures results in directional dependency of velocity and wave skewing effects that can impact the accuracy of damage detection. Phase and group velocities of the A 0 guided wave mode, propagating in a unidirectional carbon fiber reinforced laminate, were investigated experimentally and through finite element analysis. A correction for the significant offset in phase and group velocities due to wave skewing effects is illustrated for both point and short line sources, achieving good agreement with theoretical calculations assuming planar wave fronts. The influence of the line excitation length on velocity measurements is discussed. [ABSTRACT FROM AUTHOR]

A method is investigated to calculate the vertical force on a horizontal plate above still water level under the action of waves in this paper. The wave uplift force is divided into buoyancy force and impact force. Based on the discrete finite element analysis method, the calculation formula of buoyancy force is given and the calculation formula of impact force under the action of regular waves is derived through impulse theory. Then, a formula is proposed to solve the additional mass of liquid impact on the plate. Taking model tests in literature as an example, two different approaches based on the Finite Element Method are presented to verify the calculation method. The comparative analysis shows that the results of the two methods are both consistent with the test results shown in the literature, computation theories of wave uplift force and calculation formula of added mass of impact liquid are verified. On the basis of verifying the calculation theories, the distributions of buoyancy and impact force on the horizontal plate, the time history curves of buoyancy and impact force of the differential element and the relationships between wave uplift forces and wave periods, wave height and plate clearances are discussed. • A method to calculate the vertical force on a horizontal plate above still water level under the action of waves is studied. • Based on the discretization, the calculation formula of buoyancy force is given after correcting the gravity acceleration. • A new formula is proposed to solve the additional mass of liquid impacting the plate. • Two different approaches based on the Finite Element Method are presented to verify calculation method. • The distributions of buoyancy and impact force on the horizontal plate are discussed. [ABSTRACT FROM AUTHOR]

• A method for liquid level measurement based on leaky lamb waves was presented to improve the sensitivity and accuracy of liquid level detection. • Liquid level detection with lamb waves was simulated using the finite element method. • The field experiment of liquid level detection was performed, and the comparison tests of the detection of lamb waves and longitudinal waves were conducted. The accurate detection of liquid-level plays an important role in industrial process. This paper presents a method for liquid-level measurement based on Lamb waves. The dispersion equation of Lamb waves under unilateral liquid load is deduced to obtain the dispersion curves. The structure and attenuation curves are analyzed under a certain frequency and thickness, and the modes with more energy loss are selected. The ratio of the signal energy of the liquid-level at the upper and lower positions is defined as the discrimination to indicate the change of the liquid-level. The models of the liquid storage tank are built and the Lamb waves of different modes are simulated using the finite element method. The experimental results reveal that it is easy to detect the difference in ultrasonic signals when the liquid-level changes. Compared with the longitudinal wave detection method, the proposed method improves the discrimination of liquid-level detection. The experimental results are consistent with the theoretical calculations. [ABSTRACT FROM AUTHOR]

The scope of this study is the monitoring of the adhesion integrity of composite Single Lap Joints (SLJs) through the assessment of the M U L 2 CODE, software developed by the M U L 2 Research Group - Department of Mechanical and Aerospace Engineering of Politecnico di Torino. The M U L 2 CODE is implemented through the Carrera Unified Formulation (CUF) for 2D structures based on Hierarchical Legendre Expansion (HLE) polynomials. An efficient method for the Structural Health Monitoring (SHM) of bonded composite joints is experimentally verified and simulated by commercial FE software and by the CUF approach, in order to reduce the computational cost of analyses: using transient excitations via toneburst signals, the structural health of damaged SLJ is analytically, experimentally and numerically evaluated. The interaction mechanism between the waves traveling through the investigated specimens is numerically simulated with a simple Finite Elements (FE) model and it is solved via M U L 2 CODE and commercial software Ansys Workbench (with 2D and 3D models), respectively. Experimental campaigns data are compared with analytical, CUF and Ansys results demonstrating the consistence of the M U L 2 formulation that is computationally simpler, but very efficient for the joint analysis. The presented and discussed CUF application is able to quantify with a high accuracy the debonding extension in the damaged SLJ, simply tuning the excitation frequency of the SHM technique. • Elastic waves interference is evaluated by using analytical approach, experimental data, CUF and Ansys simulations. • The method is tested on a composite single lap joint, simple experimental set-up representative of an actual scenario. • A good agreement between experiments and simulations confirmed the validity of the proposed method. • The CUF approach provides finer results with lower elapsed computational time respect to the Ansys FE model. [ABSTRACT FROM AUTHOR]

Lamb wave microresonators with wavelengths λ of 5–8 μm, vibrating in the S0 mode, and having 75 electrode pairs were fabricated and characterized. The results were compared to theoretical predictions obtained by finite element simulation. The active material was a 1 μm-thick Al0.85Sc0.15N thin film. Two types of acoustic isolation solutions were implemented: the first one with freestanding plates fixed by two bridges to a device frame [freestanding Lamb wave resonator (FS-LWR)] and the second one containing an acoustic W/SiO2 5-layer reflector [solidly mounted Lamb wave resonator (SM-LWR)]. All devices showed excellent agreement with FEM predictions, regarding resonance frequency and piezoelectric coupling. The quality factors of the SM-LWR devices were 5–6 times larger than the ones of the freestanding structures fabricated by the same Al0.85Sc0.15N deposition process: we achieved a figure of merit of 12–18 (Qp = 771, Qs = 507, k2 = 2.29%) at an operation frequency of 1430 MHz, which is so far the best performance realized with a MEMS Lamb wave resonator having a large number of electrode fingers. This performance opens up perspectives for filter applications. [ABSTRACT FROM AUTHOR]