Compute Dispersion Curves Research Articles

Wavenumber dynamic stiffness formulation for exact dispersion analysis of moderately thick symmetric cross-ply laminated plate built-up waveguides

A wavenumber dynamic stiffness (WDS) formulation for exact dispersion analysis of moderately thick symmetric cross-ply laminated plate built-up waveguides is presented. The effects of shear deformation and rotatory inertia are considered in the formulation by adopting the first order shear deformation theory. The elemental WDS matrix is derived from the exact general solutions of the governing differential equations and can be assembled to model complex plate built-up waveguides using the assembly procedure similar to that of the finite element method. The nonlinear eigen-solution technique, the Wittrick–Williams (WW) algorithm, is employed to compute dispersion curves from the obtained WDS matrices. To enhance computational efficiency, the J0 count problem of the WW algorithm is addressed by developing explicit expressions for the J0 term. This eliminates the need for mesh refinement over the entire frequency/wavenumber range. The accuracy and efficiency of the proposed method are demonstrated through comparisons with analytical solutions based on the classical laminated plate theory and the first order shear deformation theory, as well as numerical results obtained from the wave and finite element method.

Dynamic behavior of a zero-group velocity guided mode in rail structures.

Important characteristics of a zero-group velocity (ZGV) mode in a standard rail are investigated through numerical simulation and experiment. First, the semi-analytical finite element analysis is implemented to compute dispersion curves for the rail structure and the first ZGV point is identified. Backward waves are identified through opposing senses of group and phase velocities. Next, a time-dependent finite element model is used to understand the dynamic response of the rail. Finally, experimental measurements confirm that ZGV modes in rail structures are formed through interferences between two opposite-traveling waves, which is analogous to the S1-S2b ZGV Lamb mode in plate structures.

Geoacoustic Inversion with a Single Vector Sensor and Multichannel Dispersion Curves

This paper discusses the value added by using a single vector sensor over a conventional pressure-only hydrophone for geoacoustic inversions. Inversion methods based on genetic algorithms are used to estimate the seabed properties. Synthetic signals of impulsive arrivals first are modeled using KRAKEN and RAM propagation models, each being modified to predict components of the vector field. While KRAKEN is utilized to directly compute dispersion curves, RAM provides full-field results that require the application of time warping to separate the modal arrivals. Combinations of dispersion curves utilizing all vector sensor channels are compared to curves estimated with the pressure-only channel. Within the time warping analysis, both binary masking and band-pass filter masking methods are applied to compare stability of results. The environment modeled for the synthetic analysis and inversion method utilize sound speed profiles measured during the Monterey Bay 2019 at-sea experiment and assume a sediment layer of constant thickness overlying a deeper sub-bottom type. White noise is added to the synthetic data at different signal-to-noise ratios to evaluate the impact of signal excess on the results. A hybrid optimization approach is used to improve the results of the genetic algorithm method. The analysis with synthetic data is consistent with the analysis of broadband, impulsive data collected from the experiment, indicating that the additional information from the vertical velocity channel further improves the geoacoustic parameter estimates.

On the existence of zero-group velocity modes in free rails: Modeling and experiments

Zero-group velocity (ZGV) modes in rails are studied through simulation and experiments in this work. Local resonances associated with ZGV modes appear as distinct, sharp peaks in the frequency amplitude spectrum, whose resonant frequencies can serve as indicators of the local structural integrity condition of the rail itself, assuming that one can excite, detect, and identify wave mode type with confidence. To better understand these interesting modes, semi-analytical finite element (SAFE) analysis is implemented to compute dispersion curves of a standard rail and to identify potential ZGV points. A fully discretized Finite Element (FDFE) model then simulates responses of a free rail when subjected to impulse-based dynamic testing. Experimental impulse vibration data are collected from a 25-m rail with multiple impact-receiver configurations to understand the detectability and excitability of specific resonances associated with ZGV modes in rails. Spatial sampling of wave disturbance is performed to calculate the dispersion relations experimentally via two-dimensional Fourier Transforms (2D-FFT). The excellent agreement between simulation and experimental results confirms the existence of ZGV modes and cutoff frequency resonances in rails and verifies the feasibility of using impulse-based dynamic tests for the promotion of ZGV modes.

Guided wave propagation in buried and immersed fluid-filled pipes: Application of the semi analytic finite element method

Developing an understanding of guided wave propagation in multi-layered systems has important applications in non-destructive evaluation. This article presents a general weighted residual formulation for guided wave propagation in fluid-filled pipes buried in an elastic solid, or immersed in a quiescent fluid. A one-dimensional semi-analytic finite element (SAFE) approach is combined with a perfectly matched layer (PML), to compute dispersion curves for different pipe applications. The speed and accuracy of this approach are compared against the scaled boundary finite element method (SBFEM) and it is shown that for a rod immersed in a liquid the two methods provide very similar computational speeds. The speed and accuracy of the model is then investigated for immersed and buried fluid-filled pipes, and it is shown that when discretising the PML no advantage in computational speed is found when using either quadratic or higher order spectral elements, provided the number of degrees of freedom in the PML is equivalent. Accordingly, it is shown that the SAFE-PML method is capable of obtaining accurately the modal characteristics of buried and immersed fluid-filled pipes, with computational speeds comparable to the SBFEM approach.

The features of three-dimensional photonic crystals with the space-depended dielectric

The features of three-dimensional (3D) photonic crystals (PCs) composed of air and space-depended dielectric constituent are theoretically investigated by the plane wave expansion method, and the formulas to compute dispersion curves are derived. The arrangement for the present PCs is that the space-depended dielectric spheres are surrounded by the air, and the embedded dielectric spheres are arranged according to the simple-cubic lattices. The relative permittivity of dielectric spheres is depend on the control parameters and space coordinates. The relationships between the parameters of proposed PCs and the band structures also are studied. The computed results demonstrate that the achieved photonic band gaps (PBGs) can be adjusted by those parameters as mentioned above. Compared to the conventional dielectric-air PCs with the same topology, the wider bandwidth of PBG can be observed in such PCs. The computed results also shows us another alternative away to realize the reconfigurable applications with the 3D PCs.

A Bloch wave reduction scheme for ultrafast band diagram and dynamic response computation in periodic structures

A variety of Reduced-Order Modelling (ROM) techniques have been developed for the Wave/Finite Element (WFE) Framework. However most of these techniques are not compatible with dynamic response computation or frequency-dependent problems. This paper introduces a new reduction strategy for the WFE method, enabling the computation of both the forced response and the complex dispersion curves of periodic structures modelled using large-sized finite element (FE) models. The method exploits the duality between Inverse and Direct Bloch formulations to build a reduced solution subspace, accounting for both propagating and evanescent behaviours, while ensuring high reduction factors. This reduction strategy therefore enables the resolution of a wider range of problems, including near/far field response computation in finite waveguides subjected to dynamic loadings, or vibroacoustic transmission/reflection problems. First, the method is used to compute dispersion curves and forced response in a duct. Then a large bi-stiffened structure is studied to evaluate the method's performances. The high frequency resolution provided by the proposed ROM allows us to explore a variety of propagation and guided resonances localization effects, hardly accessible otherwise. Furthermore, the considerable reduction factors enable fast wave dispersion analyses in large-scaled periodic structures, complex phononic crystals designs or locally resonant metamaterials.

Numerical Solution of a Wave Propagation Problem Along Plate Structures Based on the Isogeometric Approach

Ultrasonic guided waves propagating along large structures have great potential as a nondestructive evaluation method. In this context, it is very important to obtain the dispersion curves, which depend on the cross-section of the structure. In this paper, we compute dispersion curves along infinite isotropic plate-like structures using the semi-analytical method (SAFEM) with an isogeometric approach based on B-spline functions. The SAFEM method leads to a family of generalized eigenvalue problems depending on the wave number. For a prescribed wave number, the solution of this problem consists of the nodal displacement vector and the frequency of the guided wave. In this work, the results obtained with B-splines shape functions are compared to the numerical SAFEM solution with quadratic Lagrange shape functions. Advantages of the isogeometric approach are highlighted and include the smoothness of the displacement field components and the computational cost of solving the corresponding generalized eigenvalue problems. Finally, we investigate the convergence of Lagrange and B-spline approaches when the number of degrees of freedom grows. The study shows that cubic B-spline functions provide the best solution with the smallest relative errors for a given number of degrees of freedom.

Numerical Solution of a Wave Propagation Problem Along Plate Structures Based on the Isogeometric Approach

Ultrasonic guided waves propagating along large structures have great potential as a nondestructive evaluation method. In this context, it is very important to obtain the dispersion curves, which depend on the cross-section of the structure. In this paper, we compute dispersion curves along infinite isotropic plate-like structures using the semi-analytical method (SAFEM) with an isogeometric approach based on B-spline functions. The SAFEM method leads to a family of generalized eigenvalue problems depending on the wave number. For a prescribed wave number, the solution of this problem consists of the nodal displacement vector and the frequency of the guided wave. In this work, the results obtained with B-splines shape functions are compared to the numerical SAFEM solution with quadratic Lagrange shape functions. Advantages of the isogeometric approach are highlighted and include the smoothness of the displacement field components and the computational cost of solving the corresponding generalized eigenval...

Efficient computation of dispersion curves for multilayered waveguides and half-spaces

Motivated by the need to compute dispersion curves for layered media in the contexts of geophysical inversion and nondestructive testing, a novel discretization approach, termed complex-length finite element method (CFEM), is developed and shown to be more efficient than the existing finite element approaches. The new approach is exponentially convergent based on two key features: unconventional stretching of the mesh into complex space and midpoint integration for evaluating the contribution matrices. For modeling the layered half-spaces of infinite depth, we couple CFEM with the method of perfectly matched discrete layers (PMDL) to minimize the errors due to mesh truncation. A number of numerical examples are used to investigate the efficiency of the proposed methods. It is shown that the suggested combination of CFEM and PMDL drastically reduces the number of elements, while requiring minor modifications to the existing finite element codes. It is concluded that the methods’ exponential convergence and sparse computation associated with linear finite elements, result in significant reduction in the overall computational cost.

Computation of dispersion curves for embedded waveguides using a dashpot boundary condition

In this paper a numerical approach is presented to compute dispersion curves for solid waveguides coupled to an infinite medium. The derivation is based on the scaled boundary finite element method that has been developed previously for waveguides with stress-free surfaces. The effect of the surrounding medium is accounted for by introducing a dashpot boundary condition at the interface between the waveguide and the adjoining medium. The damping coefficients are derived from the acoustic impedances of the surrounding medium. Results are validated using an improved implementation of an absorbing region. Since no discretization of the surrounding medium is required for the dashpot approach, the required number of degrees of freedom is typically 10 to 50 times smaller compared to the absorbing region. When compared to other finite element based results presented in the literature, the number of degrees of freedom can be reduced by as much as a factor of 4000.

Dispersion Curves Computation for Waveguides Buried in Infinite Space by Semi-Analytical Finite Element Method

A semi-analytical finite element method is presented to compute dispersion curves of plane strain waves in waveguides buried in infinite space system. The waveguide is approximated by finite elements while its surrounding medium by evanescent bulk wave field. The dispersion curves are finally obtained from an eigenvalue equation. The method requires the buried waveguide system has a uniform physical and geometric property along its propagation axis; however, its cross-section may be isotropic, anisotropic, homogeneous or heterogeneous. The formulation and its computer program are verified by comparing with the available results obtained using a global matrix based method.

Straight and Circular Crested Lamb Waves in Generalized Piezothermoelastic Plates

This paper concentrates on the study of propagation of straight and circularly crested Lamb waves in homogeneous, transversely isotropic, piezothermoelastic plates subjected to: stress- and charge-free, thermally insulated/isothermal and stress free, thermally insulated/isothermaland electrically shorted boundary conditions in the context of generalized theories of thermoelasticity. The motion of purely transverse shear horizontal (SH) modes get decoupled from rest of the motion and do not interact with other fields. Secular equations for wave propagation modes in the plate are derived from a coupled system of governing partial differential equations of linear piezothermoelasticity. It is shown that the Rayleigh–Lamb secular equation also governs circular crested piezothermoelastic waves in a plate. Although the frequency wave number relationship holds whether the waves are straight or circularly crested the displacement, stress, electric and temperature fields vary according to Bessel functions rather than trigonometric one as far as the radial coordinate is concerned. At short wavelength limits, the secular equations reduce to Rayleigh surface wave frequency equation. The amplitudes of displacements, temperature change and electrical potential during vibrations of the plate have also been obtained. Finally, in order to illustrate and verify the analytical developments numerical solution of secular equations corresponding to stressfree, thermally insulated, open and closed circuit piezoelectric plates is carried out for cadmium-selenide (6 mm class) material. After obtaining the complex characteristic roots with the help of DesCartes' algorithm, the transcendental secular equations have been solved by functional iteration numerical technique to compute dispersion curves and attenuation coefficients. These are then presented graphically in order to illustrate and compare the results in the context of coupled and generalized theories of thermoelasticity.

Dispersion dependence for Rayleigh‐like vibrations in a low‐density channel

Although it is in principle possible to compute dispersion curves for a buried low‐density channel by numerical means, the interpretation of such results is fraught with uncertainty relating to numerical approximations, inaccuracies, and a poorly developed theoretical framework for Rayleigh waves. It is thus interesting to investigate by analytical techniques some simple cases. One of these is a study of the Rayleigh wave in an idealization of a low‐density channel. Such a channel has at least two stationary vibration points and numerous modes with complex wave number as well as the usual four propagating Rayleigh modes. The stationary points are interesting because they correspond to resonant vibrations in an unbounded medium. The algebra required to investigate even this idealized case of a layer bounded above and below by high‐density plates and overlying an elastic half‐space is tedious. It can be automated quite effectively with Mathematica. This algebraic manipulation program also provides simple effective graphics tools to study the qualitative behavior of the results.

OPTO-ACOUSTICAL INSPECTION OF SURFACE LAYERS

A non-contacting and non-destructive testing method using laser techniques was evaluated by theoretical and experimental research. As the inspection of thin layers was of main interest, the opto-acoustical generation and detection of surface acoustic waves were studied. As surface waves are sensitive to minute changes in the elastic properties of a surface they can be effectively used as a tool to characterize surfaces. Variation of the frequency varies the penetration depth, thus enabling depth profiling. An instrumental set-up was realized to make it possible to study the propagation effects of Rayleigh waves of surfaces of layered media and to be able to study the possibilities of determining layer thickness from dispersion data. Rayleigh-wave theory was implemented in a computer program to be able to compute the dispersion curves for some material combinations by numerical methods. Experimental results show that it is possible to measure phase velocities with an accuracy of about 0.2%. This makes it possible to compute dispersion curves from measured data lo determine the layer thickness (by means of curve fitting) with an accuracy better than about 10%. However, this can only be accomplished by using objects with well-reflecting surfaces; for objects with poorly-reflecting surfaces the signal-to-noisc ratio is poor and the errors will rapidly increase. In that case one has to make surface modifications, thereby more or less loosing the non-contacting property, or one has to use more sensitive, and therefore more complicated, interferometrical detectors

Crystal equilibrium and lattice dynamics of face-centered cubic metals

A lattice dynamical model that assumes short range pairwise forces effective up to second nearest neighbours and electron–ion interaction on the lines of Bhatia is considered to study the crystal dynamics of fcc metals. The authors consider an appropriate value of the screening parameter and the band structure effects have been introduced in a satisfactory manner in the present calculations. The volume force is averaged over the whole Wigner–Seitz sphere. The ionic lattice is in equilibrium in a medium of electrons. The present theory has been satisfactorily applied to compute dispersion curves, Debye characteristic temperatures, and temperature dependence on the Debye–Waller factors of thorium.

Dynamic Interaction of a Layer and a Half-Space

The equations of the theory of linear elasticity are used to compute dispersion curves for free time-harmonic waves propagating in a layer and a supporting half-space of different material properties. Both welded and smooth contacts are considered, and the layer may be acoustically softer or stiffer than the half-space. An infinite number of modes exists if the velocity of shear waves in the layer is less than the velocity of shear waves in the half-space. Generally, only one mode exists in the opposite case. The phase velocities of the two lowest modes approach, with decreasing wave length, the velocities of Rayleigh waves in the layer and the velocity of Stoneley waves, if the latter waves are possible. The phase velocities of all higher modes, if these modes exist, approach the velocity of shear waves in the layer. The conditions for existence of Stoneley waves are established.

Computation of surface wave dispersion for multilayered anisotropic media

ABSTRACT With the program described in this paper it is now possible to compute surface wave dispersion in a solid heterogeneous halfspace containing up to 200 anisotropic layers. Certain discrepancies in surface wave observations, such as disagreement between Love and Rayleigh wave data and other independent evidence, suggest that anisotropy may be important in some seismological problems. In order to study the effect of anisotropy on surface wave dispersion a program was written for an IBM 7090 computer which will compute dispersion curves and displacements for Rayleigh waves in a layered halfspace in which each layer is transversely isotropic. A simple redefinition of parameters makes it possible to use existing programs to compute Love wave dispersion.