Calculated Dispersion Curves Research Articles

Significantly anisotropic Raman response of the (110) surface in quasi-one-dimensional system (TaSe4)2I by polarized Raman spectroscopy.

Quasi-one-dimensional materials are usually characterized by optical response spectroscopy methods, which show significant polarization dependence. Herein, we report a systematical investigation of polarized Raman scattering on the (110) crystal surface of the layered (TaSe4)2I compound. Taking into account group theory analysis of the crystal structure and the Raman tensor transformation technique, the vibrational mode of the Raman peaks can be differentiated by the angular dependence of the Raman peak intensity in parallel and vertical polarization Raman scattering tests. Moreover, density functional perturbation theory (DFPT) calculation confirmed the form of the Raman tensor of the (110) crystal surface, which was consistent with the result of the Raman tensor transformation technique, and the Raman spectrum and phonon dispersion curve calculations were also performed based on the Vienna ab initio simulation package (VASP). This new method provides useful insight for accurately identifying the lattice vibration behavior in new 2D layered structures.

In-situ laser-ultrasonic visualization with the use of regenerated fiber Bragg grating sensors at elevated temperatures

The authors have developed a high-temperature ultrasonic sensing system with a regenerated fiber Bragg grating (RFBG) sensor, that was fabricated by annealing a fiber-optic Bragg grating (FBG). Different from the common FBG, the RFBG ultrasonic sensor is resistant to temperature as high as 900 °C. In this study, taking advantage of the RFBG sensor, we attempted to establish a laser-ultrasonic visualization method for in-situ damage diagnosis in heat-resistant materials at elevated temperatures. In the first part of this paper, after reporting the fabrication of the RFBG in detail, we validated that the RFBG sensor has reasonable sensitivity to ultrasonic measurement over a broad frequency range at high temperatures. The RFBG was then incorporated into a laser-ultrasonic visualizing inspector to visualize the ultrasonic wave propagating in a SiC plate with an artificial defect at 800 °C. An improved data processing method was employed to verify the reliability of the visualization result. In the processing method, the visualization result was transformed to the spectrum in the wavenumber-frequency domain with the help of three-dimensional fast Fourier transform. Referring to theoretically calculated dispersion curves of the Lamb wave propagating in the SiC plate, we identified S0 and A0 modes from the spectral results. In addition, the agreement between the theoretical results and the experimental results also verified the feasibility of our proposed system to visualize the ultrasonic wavefield in high-temperature environment. Moreover, we developed a window function to extract the wave modes from the visualization result based on the dispersion curves. The artificial defect in the SiC plate could be then clearly localized from the visualization results for the separated individual modes.

Lamb wave mode conversion and multiple-reflection mechanisms for simply and reliably evaluating delamination in composite laminates

Lamb wave propagation must be understood comprehensively for simply evaluating delamination during ultrasonic testing. However, the difference between wave propagation, visualized using laser Doppler vibrometer and pulsed-laser scanners, has not been sufficiently investigated, and knowledge of optimal conditions for evaluating delamination is limited. Thus, in this study, the mode conversion and multiple reflections of Lamb waves propagating in a delaminated cross-ply laminate were visualized using different laser scanners, delamination depths, and wave incident angles. Delamination was characterized using maximum-amplitude map postprocessing under specific conditions. Further numerical analysis revealed that owing to multiple reflections of the antisymmetric mode in incident and mode-converted waves, standing waves were generated in the delaminated sublaminate. Dispersion curve and flexural stiffness calculations confirmed the conditions required for high-amplitude standing waves, thereby providing guidelines for simply and reliably evaluating delamination during inspections.

Estimating dispersion relations of ultrasonic guided waves in bone using a modified matrix pencil algorithm

Guided wave ultrasound technology is well recognized for non-destructive testing. The technology is increasingly applied in bone characterization and imaging to evaluate bone strength and fracture risk. Cortical bone with porous microstructure induces substantial dispersion and attenuation effects on ultrasonic guided waves (UGW). Estimating frequency-dependent propagation characteristics of co-excited wave modes is significant to studying UGW propagation and developing wave-based approaches. This work implements a modified matrix pencil method to simultaneously compute modal wavenumber and attenuation coefficient from dispersive bone UGW signals with improved convergence rate and noise reduction ability. The dispersion estimation is formulated as a matrix pencil or generalized eigenvalue problem with Loewner matrices. The extracted eigenvalues are estimated complex wavevectors, in which the wavenumber and attenuation can be deduced from the real and imaginary components respectively. The performance of the proposed algorithm is demonstrated with low signal-to-noise-ratio synthetic and experimental datasets acquired using axial-transmission measurement settings. The computed dispersive features are validated via comparison with the theoretically calculated dispersion curves by semi-analytical finite-element simulation. The dispersive wave properties are accurately reconstructed in a computationally efficient manner and can be further utilized for bone parametric analyses and subsequently clinical bone health assessment.

Accelerated dispersion curve calculations for periodic vibro-acoustic structures

Over the years, metamaterials have shown their potential in a wide range of different disciplines, e.g. optics, electromagnetics, dynamics etc. Metamaterials are, often periodic, engineered structures made of conventional materials but which exhibit properties not encountered in nature. In the field of noise and vibration, metamaterials have received increasing interest since they can obtain frequency ranges of high noise and vibration attenuation, called stop bands. Their performance is often investigated by means of dispersion curves, which are calculated based on a single unit cell and assuming a structure of infinite periodic extent. Nowadays, the attenuation of acoustic and structural waves is commonly tackled as two separate problems, whereby either acoustic or structural dispersion curves are used. Recently, vibro-acoustic unit cell designs have come to the fore which can exhibit appealing characteristics, such as simultaneous structural and acoustic stop bands. To consider the vibro-acoustic coupling in these unit cell designs during the performance predictions, vibro-acoustic dispersion curve calculations are thus required. However, these computations are typically cumbersome to perform due to the associated high computational cost and therefore, often, uncoupled dispersion curves are used during the performance assessment. Although several unit cell model order reduction approaches have recently been proposed to accelerate the dispersion curve computations, such as the Bloch mode synthesis (BMS) and Generalized Bloch mode synthesis (GBMS), they are not readily applicable to vibro-acoustic unit cells. To accelerate vibro-acoustic dispersion curve calculations, this work extends the BMS and GBMS techniques towards 2D and 3D periodic vibro-acoustic systems. To balance accuracy versus speed, the extended BMS reduction basis is constructed using a split set of vibro-acoustic coupled modes, while the extended GBMS reduction basis uses the uncoupled modes. Several verification cases demonstrate that strongly accelerated vibro-acoustic dispersion curve computations are achieved whereby the vibro-acoustic coupling inside the unit cell is accurately accounted for.

Modified Generalized R/T Coefficient Method for Surface-Wave Dispersion-Curve Calculation in Elastic and Viscoelastic Media

ABSTRACT This article presents an efficient and effective modified generalized reflection and transmission (R/T) (mGRT) coefficient method for dispersion-curve calculation of the Rayleigh and Love waves. We construct a novel dispersion function based on the kernel function for calculating synthetic seismograms in an elastic layered half-space model. It improves the root searching using the monotonicity of the dispersion function in sections between roots. We introduce an auxiliary function to accelerate the root-searching procedure by approximating the number of roots of the dispersion equation. To solve the low-velocity-layer problem for the generalized R/T coefficient method, we turn to the dispersion function of multiple depths. Numerical results indicate that the proposed method is slightly slower than the other most efficient methods including the delta-matrix method and the fast delta-matrix method. Then, we present a technique for the velocity and attenuation dispersion curves in viscoelastic media. The technique is based on jumping features of the dispersion function and converts 2D root-searching problem to 1D minimization problem. The mGRT method is expected to be a useful candidate for calculating dispersion curves in both elastic and viscoelastic media.

Surface-wave tomography using SeisLib: a Python package for multiscale seismic imaging

Summary To improve our understanding of the Earth’s interior, seismologists often have to deal with enormous amounts of data, requiring automatic tools for their analyses. It is the purpose of this study to present SeisLib, an open-source Python package for multiscale seismic imaging. At present, SeisLib includes routines for carrying out surface-wave tomography tasks based on seismic ambient noise and teleseismic earthquakes. We illustrate here these functionalities, both from the theoretical and algorithmic point of view and by application of our library to seismic data from North America. We first show how SeisLib retrieves surface-wave phase velocities from the ambient noise recorded at pairs of receivers, based on the zero crossings of their normalized cross-spectrum. We then present our implementation of the two-station method, to measure phase velocities from pairs of receivers approximately lying on the same great-circle path as the epicentre of distant earthquakes. We apply these methods to calculate dispersion curves across the conterminous United States, using continuous seismograms from the transportable component of USArray and earthquake recordings from the permanent networks. Overall, we measure 144 272 ambient-noise and 2055 earthquake-based dispersion curves, that we invert for Rayleigh-wave phase-velocity maps. To map the lateral variations in surface-wave velocity, SeisLib exploits a least-squares inversion algorithm based on ray theory. Our implementation supports both equal-area and adaptive parametrizations, with the latter allowing for a finer resolution in the areas characterized by high density of measurements. In the broad period range 4–100 s, the retrieved velocity maps of North America are highly correlated (on average, 96 per cent) and present very small average differences (0.14 ± 0.1 per cent) with those reported in the literature. This points to the robustness of our algorithms. We also produce a global phase-velocity map at the period of 40 s, combining our dispersion measurements with those collected at global scale in previous studies. This allows us to demonstrate the reliability and optimized computational speed of SeisLib, even in presence of very large seismic inverse problems and strong variability in the data coverage. The last part of the manuscript deals with the attenuation of Rayleigh waves, which can be estimated through SeisLib based on the seismic ambient noise recorded at dense arrays of receivers. We apply our algorithm to produce an attenuation map of the United States at the period of 4 s, which we find consistent with the relevant literature.

A comparative study for calculating dispersion curves in viscoelastic multi-layered plates

This study examines and critically compares various commonly used models for calculating complex wavenumbers in viscoelastic orthotropic multi-layer materials (e.g. fiber reinforced polymer laminates). Both exact 3D elasticity theory based methods and approximate methods are considered for wave propagation analysis: (i) global matrix method (GMM), (ii) stiffness matrix method (SMM), (iii) hybrid compliance-stiffness matrix method (HCSMM), (iv) semi analytical finite element (SAFE) method, (v) Legendre polynomial method (LPM), and (vi) fifth order shear deformation theory (5-SDT). A detailed mathematical formulation is provided for each of these techniques.The accuracy and the computational efficiency for calculating complex wavenumbers over a wide frequency range are analyzed for both homogenized and multi-layered viscoelastic orthotropic media. This information is helpful to select the proper method to be used in inversion schemes for material characterization. A comparative analysis is also performed on the computed through thickness displacement fields within the solid medium. The different implemented methods have been compiled in a toolbox with a graphical user interface, called “The Dispersion Box”, and can be freely downloaded from Orta et al. (2022).

A combined helium atom scattering and density-functional theory study of the Nb(100) surface oxide reconstruction: Phonon band structures and vibrational dynamics

Helium atom scattering and density-functional theory (DFT) are used to characterize the phonon band structure of the (3 × 1)-O surface reconstruction of Nb(100). Innovative DFT calculations comparing surface phonons of bare Nb(100) to those of the oxide surface show increased resonances for the oxide, especially at higher energies. Calculated dispersion curves align well with experimental results and yield atomic displacements to characterize polarizations. Inelastic helium time-of-flight measurements show phonons with mixed longitudinal and shear-vertical displacements along both the ⟨1̄00⟩, Γ̄X̄ and ⟨11̄0⟩, Γ̄M̄ symmetry axes over the entire first surface Brillouin zone. Force constants calculated for bulk Nb, Nb(100), and the (3 × 1)-O Nb(100) reconstruction indicate much stronger responses from the oxide surface, particularly for the top few layers of niobium and oxygen atoms. Many of the strengthened bonds at the surface create the characteristic ladder structure, which passivates and stabilizes the surface. These results represent, to our knowledge, the first phonon dispersion data for the oxide surface and the first ab initio calculation of the oxide's surface phonons. This study supplies critical information for the further development of advanced materials for superconducting radiofrequency cavities.

Semi-Analytical Finite Element (SAFE) method for plotting Lamb waves dispersion curves of an aluminum plate and comparison with Disperse software

Investigation into ultrasonic Non-destructive evaluation (NDE) methods for assessing complex designs depends firmly on displaying apparatuses that calculate dispersion curves. These expectations are vital for empowering the best assessment methodologies to be distinguished and their aversions to being assessed. This study, the main purpose of this work is to do a quantitative evaluation of the SAFE technique by calculating dispersion curves (wavenumber, phase velocity, and group velocity) and comparing them to those generated by Disperse Software.

First-Principles Calculation of Kh2po4 Anharmonic Force Constants Phonon Dispersion Curves

First-Principles Calculation of Kh2po4 Anharmonic Force Constants Phonon Dispersion Curves

Plotting Lamb waves dispersion curves of an aluminum plate by the Semi-Analytical Finite Element (SAFE) method and comparison with analytical curves

It is well known that the propagation of Lamb waves in elastic solid medias depends on the frequency and on the material properties, but it does not depend on the propagation direction in isotropic medias and gives rise to two basic types of modes, symmetric and antisymmetric ones. This paper presents an application of the semi-analytical finite element (SAFE) method on an isotropic aluminum plate of thickness e = 2mm. The main goal of this paper is to make a quantitative evaluation of the SAFE method by carrying out the following tasks: a) calculation of the dispersion curves (wavenumber, phase velocity, group velocity, b) calculation of the relative error and validation of the proposed algorithm, c) comparison between the curves obtained by SAFE method on those obtained by Bisection method. It should be noted here that our study is restricted on the propagative modes.

Broad Learning Framework for Search Space Design in Rayleigh Wave Inversion

Search space design is a fundamental procedure for the inverse problem in Rayleigh wave exploration. In practice, such a crucial task mainly depends on researchers’ experience-based judgment. However, intricate near-surface materials may lead to erroneous search space design, and consequently, accurate inversion results cannot always be ensured. By the forward calculation, it is found that there is a strong relationship between the fundamental dispersion curves of a given earth model and its miniature model; namely, if the parameters (layer thicknesses and S-wave velocities) of the given earth model are shrunk at a certain scale, the phase velocities on the fundamental dispersion curve will decrease at the same scale. Taking advantage of this relation, we propose a broad learning framework for search space design in a data-driven manner. First, a training set of minified models is generated using the forward calculation of dispersion curves. Then, a mapping relationship between dispersion curves and minified earth models is built via a broad learning network. Finally, a search space for the actual earth model is designed using the minified model based on the relation. As the ranges of parameters in the minified model are much smaller, the network can find the model parameters for the corresponding dispersion curve quickly and easily. Numerical simulations and field data applications demonstrate the reliability and effectiveness of the proposed method for search space design. It is concluded that the proposed method can promote accurate estimation of S-wave velocities by Rayleigh wave inversion without experience-based judgment.

Semi-Analytical Finite Element Method for calculating dispersion curves of a CFRP plate

Composite Materials are widely used thanks to their mechanical characteristics. Non destructive evaluation requires knowing the exact dispersion curves to determine the propagative waves and to resolve the phase velocity of symmetrical and antisymmetrical modes. The aim of this paper is to plot the dispersion curves of a Carbon Fibre Reinforced Polymer plate using Semi Analytical Finite Element Method algorithm. This method combines the analytical expression of the displacement and the finite element method procedure. The resultant advantage is both simplicity and rapidity. The obtained results showed that the method accuracy depends on the elements number of meshing. To ensure good precision and fast computation of the method, the elements number and the order of the interpolation functions must be optimized.

Geometry optimization for tunable band gap and wave guiding in periodic grid structures

A proper lattice structure consisting of homogeneous material is designed in this paper to investigate the maximum bandwidth of perfect lattice structures and tunable property of waveguide with linear geometric defect by means of selecting optimal geometric lattice cell. A simulation model based on finite element method is used to calculate dispersion curves and transmission spectrums of lattice structures with different geometric parameters. Meanwhile, a simplified theoretical model of unit cell, which considers the mass of grid bar and stiffness of node area, is applied to validate the accuracy of simulation result and may provide an effective approach for prediction of band gap lower boundary. Then, the validated numerical results show different orders of widest band gap that can be realized by different optimal geometric structures. Moreover, waveguide property can be effectively controlled and manipulated by changing defect parameters. The present study may establish theoretical and simulation foundation to control and manipulate band structures and other acoustic propagation characteristics of waveguide devices.

Spectral Fourier-microscopy of the periodic structures based on Ge2Sb2Te5

The method of spectral Fourier microscopy was used to study the reflection spectra with an angular resolution of submicron periodic gratings based on amorphous and crystalline Ge2Sb2Te5. The form of the dispersion curves of quasi-waveguide modes in the structures under study was established. The experimental data were compared with the calculations of dispersion curves in synthesized diffraction gratings. Reasonable agreement between theoretical and experimental data was obtained.

Spatial estimation of fundamental mode dispersion curves using geostatistical techniques

SUMMARY This paper proposes the use of geostatistical techniques to estimate dispersion curves between other known ones. To do it, we introduce two novel methodologies: the stacking method and the group-velocity mapping method. We obtain our set of group-velocity fundamental mode dispersion curves from seismic noise correlation. Consequently, we first assign their attribution point at the mid-distance between the stations used for the dispersion curves calculation. The stacking method uses the range of the omnidirectional semivariogram of a regionalized variable that quantifies the similarity between dispersion curves to stack them according to their spatial correlation. We test this technique with dispersion curves obtained in Mexico City and get a range of ∼400 m for the omnidirectional semivariogram. We also calculate directional semivariograms and observe a maximum range (∼500 m) in the NW-SE direction, agreeing with the city's spatial distribution of natural periods. On the other hand, the group-velocity mapping method uses the ordinary kriging estimator in the group velocities for all the ranges of periods to generate maps and then dispersion curves. Estimated dispersion curves retrieved from both, the stacking and the group-velocity mapping method, were compared with those obtained with the fast marching tomographic method. We also establish analogies between getting group-velocity maps with the tomographic method and with the group-velocity mapping method. Finally, we observe that the range of the omnidirectional semivariogram used in the stacking method may be related to the tomographic method resolution.

Nonlinear Ultrasound Crack Detection with Multi-Frequency Excitation-A Comparison.

Nonlinear ultrasound crack detection methods are used as modern, non-destructive testing tools for inspecting early damages in various materials. Nonlinear ultrasonic wave modulation, where typically two or more frequencies are excited, was demonstrated to be a robust method for failure indicators when using measured harmonics and modulated response frequencies. The aim of this study is to address the capability of multi-frequency wave excitation, where more than two excitation frequencies are used, for better damage identification when compared to single and double excitation frequencies without the calculation of dispersion curves. The excitation frequencies were chosen in such a way that harmonic and modulated response frequencies meet at a specific frequency to amplify signal energy. A new concept of nonlinearity parameter grouping with multi-frequency excitation was developed as an early failure parameter. An analytical solution of the one-dimensional wave equation was derived with four fundamental frequencies, and a total of 64 individual and 30 group nonlinearity parameters. Experimental validation of the approach was conducted on metal plates with different types of cracks and on turbine blades where cracks originated under service conditions. The results showed that the use of multi-frequency excitation offers advantages in detecting cracks.

(Generalized) Bloch mode synthesis for the fast dispersion curve calculation of 3D periodic metamaterials

Metamaterials, i.e. artificial structures with unconventional properties, have shown to be highly potential lightweight and compact solutions for the attenuation of noise and vibrations in targeted frequency ranges, called stop bands. In order to analyze the performance of these metamaterials, their stop band behavior is typically predicted by means of dispersion curves, which describe the wave propagation in the corresponding infinite periodic structure. The input for these calculations is usually a finite element model of the corresponding unit cell. Most common in literature are 2D plane metamaterials, which often consist of a plate host structure with periodically added masses or resonators. In recent literature, however, full 3D metamaterials are encountered which are periodic in all three directions and which enable complete, omnidirectional stop bands. Although these 3D metamaterials have favorable vibro-acoustic characteristics, the computational cost to analyze them quickly increases with unit cell model size. Model order reduction techniques are important enablers to overcome this problem. In this work, the Bloch Mode Synthesis (BMS) and generalized BMS (GBMS) reduction techniques are extended from 2D to 3D periodic structures. Through several verifications, it is demonstrated that dispersion curve calculation times can be strongly reduced, while accurate stop band predictions are maintained.

Impact of the Unit Cell Choice on the Efficiency of Dispersion Curve Calculations Using Generalized Bloch Mode Synthesis

Abstract Recently, the potential of metamaterials and phononic crystals to cope with conflicting requirements for obtaining lightweight structures with desirable noise and vibration properties has been demonstrated. These, often periodic, structures are commonly studied based on their representative unit cell (UC) of which the vibro-acoustic performance is examined by means of their wave propagation, visualized by dispersion curves. Typically, the UC is discretized using a finite element technique to capture the possibly complex geometry. This leads to a high computation cost for the dispersion curve calculation which can be strongly reduced by applying modal-based model order reduction techniques such as the (generalized) Bloch mode synthesis (GBMS). In this paper, the choice of the UC is shown to have an impact on the dispersion curve calculation time. Moreover, the efficiency of GBMS strongly depends on the UC choice. The highest reduction in computation time is accomplished when the number of boundary degrees-of-freedom is limited.