Real Wavenumber Research Articles

Lippmann-Schwinger equation representation of Green's function and its preconditioned generalized over-relaxation iterative solution in wavelet domain

The calculation of Green's function is the core of seismic forward and inverse methods based on integral operators. When the Lippmann-Schwinger (L-S) equation is used to calculate Green's function in strongly scattering media, both the Born scattering series and the numerical iterative method encounter issues of slow convergence or divergence. Although the renormalization method derived from quantum mechanics can effectively address the convergence problem of Born scattering series in strong scattering problems, it is acknowledgeed that the convergence conditions and rates of convergence of different reformulation series may vary, and no universal convergence reformulation scattering series exists. Numerical methods for solving integral equations tend to be more general and mathematically robust. In this work, we focus on the numerical solution method of L-S equations. By using a wavelet-domain preconditioner to a reformulated or equivalent Lippmann-Schwinger (L-S) equation, we present an iterative method for numerically solving the equivalent L-S equation aimed at improving the rate of convergence and iteration efficiency in strongly inhomogeneous media. Following Jakobsen et al. (2020), we first introduce a small imaginary component into the background wave number，then rewrite the L-S equation to derive the equivalent complex wave number L-S equation. This reformulation ensures that the coefficient matrix exhibits a banded structure after numerical discretization, allowing the wavelet coefficient matrix to maintain good sparsity. We employ a multi-level fill-in incomplete LU (ILU) factorization method along with a block ILU-based algebraic recursive multilevel solve (ARMS) method in the wavelet domain to generate sparse approximate inverses as preconditioning operators, thereby accelerating the convergence of the generalized successive over-relaxation (GSOR) iterative method. This method is applied to compute numerical Green's functions in strongly inhomogeneous media. Numerical results demonstrate that our method yields simulation outcomes consistent with those obtained from the direct method for solving the original real wave number L-S equation. By testing various preconditioners, we find that the ARMS preconditioner offers significant advantages in operator generation efficiency and non-zero element filling ratio, effectively accelerating the convergence of the GSOR iterative method while achieving higher computational efficiency.

Computation of Green’s Function in a Strongly Heterogeneous Medium Using the Lippmann–Schwinger Equation: A Generalized Successive Over-Relaxion plus Preconditioning Scheme

The computation of Green’s function is a basic and time-consuming task in realizing seismic imaging using integral operators because the function is the kernel of the integral operators and because every image point functions as the source point of Green’s function. If the perturbation theory is used, the problem of the computation of Green’s function can be transformed into one of solving the Lippmann–Schwinger (L–S) equation. However, if the velocity model under consideration has large scale and strong heterogeneity, solving the L–S equation may become difficult because only numerical or successive approximate (iterative) methods can be used in this case. In the literature, one of these methods is the generalized successive over-relaxation (GSOR) iterative method, which can effectively solve the L–S equation and obtain the desired convergent iterative series. However, the GSOR iterative method may encounter slow convergence when calculating the high-frequency Green’s function. In this paper, we propose a new scheme that utilizes the GSOR iterative with a precondition to solve the complex wavenumber L–S equation in a slightly attenuated medium. The complex wavenumber with imaginary components localizes the energy of the background Green’s function and reduces its singularity by enabling exponential decay. Introduction of the preconditioning operator can further improve the convergence speed of the GSOR iterative series. Then, we provide a preconditioned generalized successive over-relaxation (pre-GSOR) iterative format. Our numerical results show that if an appropriate damping factor and a proper preconditioning operator are selected, the method presented here outperforms the GSOR iterative for the real wavenumber L–S equation in terms of the convergence speed, accuracy, and adaptation to high frequencies.

COMPREHENSIVE ANALYSIS OF ENERGY DISSIPATION IN SOIL-EMBEDDED PILES: AN ANALYTICAL APPROACH

The study delves into a comprehensive examination of the total energy dissipation associated with a pile entrenched in the soil, with the soil being conceptualized as viscoelastic springs. "Dissipation" can be utilized to gauge the decay rate of displacement amplitude over time. Understanding dissipation in piles is pivotal for discerning the pace at which a pile achieves its equilibrium state when exposed to impact or free vibrations. Elevated dissipation levels indicate a swifter attenuation of transient vibrations, a crucial aspect for effective vibration mitigation during significant occurrences such as earthquakes, severe weather conditions, explosions, and impacts. The dispersion relation, reliant on wavenumber and free wave frequency, is derived for the pile-soil system using the Bloch's Floquet theorem. This relation is approached via the free-wave method, where a real wavenumber produces a sequence of complex frequencies. We propose an analytical closed-form expression in this study to forecast the overall dissipation from these complex frequency values. Additionally, we validate the accuracy of the predicted dissipation against a finite element-based numerical model of the soil-pile system. Furthermore, we examine the influence of factors such as material damping, Young's modulus, and shear modulus of the soil on the dissipation characteristics of the pile-soil system.

COMPREHENSIVE ANALYSIS OF ENERGY DISSIPATION IN SOIL-EMBEDDED PILES: AN ANALYTICAL APPROACH

The study delves into a comprehensive examination of the total energy dissipation associated with a pile entrenched in the soil, with the soil being conceptualized as viscoelastic springs. "Dissipation" can be utilized to gauge the decay rate of displacement amplitude over time. Understanding dissipation in piles is pivotal for discerning the pace at which a pile achieves its equilibrium state when exposed to impact or free vibrations. Elevated dissipation levels indicate a swifter attenuation of transient vibrations, a crucial aspect for effective vibration mitigation during significant occurrences such as earthquakes, severe weather conditions, explosions, and impacts. The dispersion relation, reliant on wavenumber and free wave frequency, is derived for the pile-soil system using the Bloch's Floquet theorem. This relation is approached via the free-wave method, where a real wavenumber produces a sequence of complex frequencies. We propose an analytical closed-form expression in this study to forecast the overall dissipation from these complex frequency values. Additionally, we validate the accuracy of the predicted dissipation against a finite element-based numerical model of the soil-pile system. Furthermore, we examine the influence of factors such as material damping, Young's modulus, and shear modulus of the soil on the dissipation characteristics of the pile-soil system.

Analytical solutions for dispersions and waveforms of acoustic logging in a cased hole

Acoustic logging is one of the most promising methods for the quantitative evaluation of cement bond conditions in cased holes. However, inefficient use of full-wave information yields unsatisfactory interpretation accuracy. Fundamentally, this is because the wavefield characteristics have not been thoroughly investigated under various cement bonding conditions. Thus, this study derives analytical solutions of wavefields for a single-cased-hole model and emphasizes the dispersion calculation algorithm. To solve the dispersion equation when solving for the poles of the propagating modes with real wavenumbers, we renormalize the Bessel function related to the borehole fluid by multiplying it with an attenuation factor. For leaky modes with complex wavenumbers, we develop a novel method to find local peaks of the matrix condition number (LPMCN) in the frequency domain to determine dispersion poles, avoiding the local optimization issues resulting from the traditional Gauss-Newton iteration method. Combining these two methods, we establish a fast and accurate workflow for evaluating the dispersion of all modes in cased holes using a relatively fast bisection method to manage the dispersion of the propagating modes and using the LPMCN method to derive the dispersion curves of leaky modes. Furthermore, all propagating modes are individually investigated in the monopole measurement by evaluating the residues of the real poles in a casing-free model. The analysis demonstrates that the first-order pseudo-Rayleigh wave (PR1) and inner Stoneley wave (ST1) are the two strongest modes. Finally, we focus on the waveforms and dispersion characteristics of the outer Stoneley wave (ST2) related to the fluid channel in the cement annulus. The results reveal that as the fluid thickness increases, the phase velocity of the ST2 mode decreases, and its amplitude increases. Therefore, the ST2 mode can potentially evaluate the thickness of the fluid channel in a cement annulus if an effective weak-signal-extraction method is used.

Estimation of the dissipation due to radiation damping for the pile embedded in soil: A closed-form solution

In the present work, the overall energy dissipation of a pile embedded in the soil, modeled as visco-elastic springs, is thoroughly investigated. The term dissipation is used to measure the rate of decay of amplitude of displacement response in the time domain. The estimation of dissipation in piles is necessary to understand how quickly a pile reaches its steady state or equilibrium under dynamic loads or in cases of free vibration, respectively. A higher level of dissipation indicates a shorter duration of transient vibration, which is a deserving trait for vibration mitigation in extreme events, such as earthquakes, storms, blasts, shocks, etc. The dispersion relation which is a function of wavenumber and free wave frequency is obtained for the pile embedded in the soil medium by implementing the generalized form of Bloch’s theorem. The dispersion relationship is solved in the free-wave approach, in which a real wave number yields a set of complex frequencies. For predicting the overall dissipation of the system from the complex frequency values, an analytical closed-form expression is proposed in this paper. Further, the predicted dissipation is validated with a finite element-based numerical model of the soil–pile system.

COMPUTATION OF DISPERSION CURVES OF RAYLEIGH-LAMB WAVES BY NEWTON-RAPHSON METHOD

The paper treats the numerical construction of dispersion curves which belong to the propagation of Rayleigh-Lamb waves in isotropic elastic media. The Newton-Raphson method.is applied for computation of real, imaginary and complex wavenumbers in dependence on the frequency of the wave. Due to multivaluedness, unexpected jumps between modes of dispersion curves occur in branch tracing, which prevents successful direct application of the method. This shortcoming is eliminated in the article by introducing the arc-length continuation method. The efficiency of this supplement to the Newton-Raphson method is shown on the example of an aluminum plate with constructions of dispersion curves for real, imaginary and complex wave numbers.

Exterior diffraction problems for a triangular lattice

Exterior Dirichlet problems for two-dimensional (2D) lattice waves on the infinite triangular lattice are considered. Namely, we study Dirichlet problems for the 2D discrete Helmholtz equation in a plane with a hole. New results are obtained for the existence and uniqueness of the solution in the case of the real wave number without passing to a complex wave number. Besides, Green’s representation formula for the solution is derived with the help of difference potentials. To demonstrate the results, we propose a method for numerical calculation.

Spatial–temporal transformation for primary and secondary instabilities in weakly non-parallel shear flows

When studying instability of weakly non-parallel flows, it is often desirable to convert temporal growth rates of unstable modes, which can readily be computed, to physically more relevant spatial growth rates. This has been performed using the well-known Gaster's transformation for primary instability and Herbert's transformation for the secondary instability of a saturated primary mode. The issue of temporal–spatial transformation is revisited in the present paper to clarify/rectify the ambiguity/misunderstanding that appears to exist in the literature. A temporal mode and its spatial counterpart may be related by sharing either the real frequency or wavenumber, and the respective transformations between their growth rates are obtained by a simpler consistent derivation than the original one. These transformations, which consist of first- and second-order versions, are valid under conditions less restrictive than those for Gaster's and Herbert's transformations, and reduce to the latter under additional conditions, which are not always satisfied in practice. The transformations are applied to inviscid Rayleigh instability of a mixing layer and a jet, secondary instability of a streaky flow as well as general detuned secondary instability (including subharmonic and fundamental resonances) of primary Mack modes in a supersonic boundary layer. Comparison of the transformed growth rates with the directly calculated spatial growth rates shows that the transformations derived in this paper outperform Gaster's and Herbert's transformations consistently. The first-order transformation is accurate when the growth rates are small or moderate, while the second-order transformations are sufficiently accurate across the entire instability bands, and thus stand as a useful tool for obtaining spatial instability characteristics via temporal stability analysis.

Characterization of the full complex-valued stiffness tensor of orthotropic viscoelastic plates using 3D guided wavefield data

A two-stage inversion scheme is proposed to determine the full set of 9 complex-valued stiffness properties of orthotropic viscoelastic plates using their 3D surface velocity response to a broadband vibrational excitation. A combination of the hybrid TLS-ESPRIT and IWC method is employed to the extract complex-valued wavenumber–frequency pairs corresponding to relevant Lamb wave and shear horizontal plate waves from the data from surface velocity response. Particle swarm optimization is chosen to inversely determine the plate’s orthotropic viscoelastic properties using the semi-analytical finite element method as a forward model to compute the real and imaginary parts of the wavenumbers. The inversion procedure first identifies the elastic parameters by matching the real wavenumber values, followed by the characterization of the viscoelastic properties which are merely linked to the imaginary wavenumber values. To validate the accuracy of the proposed method, a series of numerical simulations of 3D wavefield data using a finite element model (COMSOL) is conducted, as well as broadband experimental measurements obtained by means of a 3D Infrared Scanning Laser Doppler Vibrometer. It is shown that the inverted orthotropic viscoelastic properties based on the virtual wavefield data are in close agreement with the target values, showing a mean relative error of less than 2% and 5% error on the elastic and viscous properties, respectively. Additionally, the median absolute deviations are negligible, showing the robust convergence of the inversion procedure to a global minimum, and giving confidence to apply the method on experimental wavefield data in order to reveal the full (unknown) C-tensor for material characterization. The proposed characterization method can be used for various orthotropic viscoelastic materials, including metal sheets, carbon/epoxy laminates, and wooden plates.

Half-plane diffraction problems on a triangular lattice

We investigate thin-slit diffraction problems for two-dimensional lattice waves. Namely, Dirichlet problems for the two-dimensional discrete Helmholtz equation on the triangular lattice in a half-plane are studied. Using the notion of the radiating solution, we prove the existence and uniqueness of a solution for the real wave number $$k\in (0,3)\backslash \{2\sqrt{2}\}$$ without passing to the complex wave number. Besides, an exact representation formula for the solution is derived. Here, we develop a numerical calculation method and demonstrate by example the effectiveness of our approach related to the propagation of wavefronts in metamaterials through two small openings.

Active exterior cloaking for the two-dimensional Helmholtz equation with complex wavenumbers and application to thermal cloaking.

We design sources for the two-dimensional Helmholtz equation that can cloak an object by cancelling out the incident field in a region, without the sources completely surrounding the object to hide. As in previous work for real positive wavenumbers, the sources are also determined by the Green identities. The novelty is that we prove that the same approach works for complex wavenumbers which makes it applicable to a variety of media, including media with dispersion, loss and gain. Furthermore, by deriving bounds on Graf's addition formulas with complex arguments, we obtain new estimates that allow to quantify the quality of the cloaking effect. We illustrate our results by applying them to achieve active exterior cloaking for the heat equation. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)'.

A topology optimization of open acoustic waveguides based on a scattering matrix method

This study presents a topology optimization scheme for realizing a bound state in the continuum along an open acoustic waveguide comprising a periodic array of elastic materials. First, we formulate the periodic problem as a system of linear algebraic equations using a scattering matrix associated with a single unit structure of the waveguide. The scattering matrix is numerically constructed using the boundary element method. Subsequently, we employ the Sakurai–Sugiura method to determine resonant frequencies and the Floquet wavenumbers by solving a nonlinear eigenvalue problem for the linear system. We design the shape and topology of the unit elastic material such that the periodic structure has a real resonant wavenumber at a given frequency by minimizing the imaginary part of the resonant wavenumber. The proposed topology optimization scheme is based on a level-set method with a novel topological derivative. We demonstrate a numerical example of the proposed topology optimization and show that it realizes a bound state in the continuum through some numerical experiments.

Third order modal exceptional degeneracy in waveguides with glide-time symmetry

The dispersion of a three-way waveguide is engineered to exhibit exceptional modal characteristics. Two coupled waveguides with Parity-Time (PT) symmetry have been previously demonstrated to exhibit second order exceptional points of degeneracy (EPDs). In this work, we introduce and investigate a particular class of EPDs, applicable from radio frequency to optical wavelengths, whereby three coupled waveguides satisfy Glide-Time (GT) symmetry to exhibit a third order modal degeneracy with a real-valued wavenumber. GT symmetry involves glide symmetry of lossless/gainless components of the waveguide in addition to changing the sign of passive/active elements while applying a glide symmetry operation. This GT-symmetry condition allows three Floquet-Bloch eigenmodes of the structure to coalesce to a real-valued wavenumber at a single frequency, in addition of having one branch of the dispersion diagram with a purely real wavenumber. The proposed scheme may have applications including but not limited to distributed amplifiers, radiating arrays, and sensors, from radio frequency to optics.

Антиплоские задачи о движении осциллирующей нагрузки по границе упругой изотропной полосы при наличии поверхностных напряжений

In this paper, symmetric and antisymmetric antiplane problems about the motion with a constant subsonic velocity of an oscillating load along the boundary of an elastic isotropic nanothin strip are considered. The nanoscale strip thickness is considered by introducing surface stresses in accordance with the Gurtin-Murdoch theory. According to this theory, it is assumed that, in addition to external loads, surface stresses act on the layer boundaries, which are described by Hooke's surface law. As a result, the properties of the elastic material of the strip with nanoscale thickness become different from the material properties of a regular-sized body. A standard technique was used for the so-lution, including the application of limiting absorption principle, the Fourier transform over infinitely extended co-ordinate and the theory of residues for finding the inverse Fourier transform. For various strip thicknesses, solutions were obtained in the form of series in natural waves, dispersion relations were studied, and graphs of the displace-ment amplitudes along the thickness were plotted. The analysis showed that for fixed values of frequency and velocity of the source, the values of non-negative real wave numbers are greater in the presence of surface stresses than the values of the wave numbers for the classical case of the problem without surface stresses. It is noted that surface stresses have a significant effect only when the strip thickness decreases to nanosizes.

An explicit solution for the design of a target-frequency-customized, piezoelectric-defect-introduced phononic crystal for elastic wave energy harvesting

This paper proposes an explicit solution for the design of a target-frequency-customized, one-dimensional phononic crystal (PNC) with a defect for piezoelectric energy harvesting under longitudinal waves. Due to the innate narrow bandwidth nature of the defect modes of a PNC at the target frequency, there is a great need to generate an electromechanically coupled defect band of a piezoelectric-defect-introduced PNC. This work considers the transfer matrix method which has been widely used in analytical approaches. The need for defect bands to be included in a bandgap inspires the use of a quarter-wave stack as a unit cell to match the bandgap's central frequency with the target frequency. In band structure analysis, considering that the electromechanically coupled defect band corresponds to a set of real wavenumbers despite being within the bandgap, several possible solutions for the piezoelectric defect's length are derived in an explicit fashion. Since switching from a short- to an open-circuit condition causes defect bands to slightly increase due to piezoelectric effects, an explicit solution that reflects the piezoelectric defect's electrical characteristics is finally proposed. Finite-element-based numerical validation studies are conducted to study two aspects, specifically parametric studies (i.e., the natural numbers in the solution to the piezoelectric defect's length, the supercell sizes, and the defect locations) and supporting studies (i.e., the electrical boundary conditions and unit cell designs). At the target frequency, it is demonstrated that the proposed PNC design actualizes the formation of one defect band and the representation of the peak output voltage.

Interactions of drift wave turbulence with streamer flows in wave-kinetic formalism

Interaction between drift wave turbulence and streamer flow in magnetized plasmas is investigated in the framework of wave-kinetic theory. By performing a wave-kinetic simulation of turbulence coupling with the streamer flow, the role of the phase-space dynamics of turbulence is clarified. Here, the phase-space consists of the real space and wavenumber space. As a consequence of the phase-space dynamics, the turbulence is trapped in the flow and propagates with the streamer. Because of the trapping, the asymmetry of the turbulence intensity with respect to the sign of the flow curvature appears. The trajectories of the turbulence wave-packet, which are the characteristic curves of the wave-kinetic solution, can be categorized by the trapping, passing, and scattering. Therefore, the obtained phase-space structure of the turbulence can be understood by their trajectories. The propagation direction and speed are found to be governed by the turbulence spectrum. The obtained characteristics can be explained by the theoretical analysis.

Center Impedance Method for Estimating Complex Modulus with Wide Frequency Range and Large Loss Factors

A simple method for determining viscoelasticity over a wide frequency range using the frequency response function (FRF) mobility obtained by the center impedance method is presented. As user data comprise the FRF between the velocity of the excitation rod and excitation force, it is challenging to separate the signal and noise. Our proposed method is based on the FRF obtained from the analytical solution of the equation of motion of the viscoelastic beam and relationship between the complex wavenumber (real wavenumber and attenuation constant) of flexural wave and viscoelasticity. Furthermore, a large loss factor can be handled over a wide frequency range without using the half-power bandwidth. In this study, actual FRF mobility data containing noise were processed using preprocessing, inverse calculation, and postprocessing. Preprocessing removed low-coherence data, compensates for the effects of instrument gain, and transformed the FRF into its dimensionless equivalent. Then, inverse calculations were used to solve the mobility equation and determine the complex wavenumber. In postprocessing, the complex wavenumber obtained by the inverse calculation was curve fitted using functions with mechanical significance. Consequently, the storage modulus based on the curve-fitted complex wavenumber was a monotonically increasing frequency function. The loss factor had a smooth frequency dependence such that it has the maximum value at a single frequency. The proposed method can be applied to composite materials, where the application of time-temperature superposition is challenging. We utilized the measured FRF mobility data obtained over a duration of several seconds, and this method can also be applied to materials with large loss factors of 1 or more.

Ultra-hard rhombohedral carbon by crystal chemistry and ab initio investigations

A new ultra-hard rhombohedral carbon rh-C4 (or hexagonal h-C12) is reported as derived from 3R graphite through crystal chemistry construction and ground state energy within the density functional theory. A resulting cohesive extended hexagonal three-dimensional network of h-C12 is identified as formed of C4 tetrahedra like in cubic and hexagonal diamond. The electronic band structure of rh-C4 is characteristic of insulator with Egap ​= ​4 ​eV similarly to diamond. From the energy-volume equations of state and the set of elastic constants a larger value of bulk modulus versus lonsdaleite, and the largest Vickers hardness (HV) versus both forms of diamond were revealed. The phonon band structure showing only real wave numbers further confirms the stability of the new carbon phase.

Propagation and attenuation characteristics of free flexural waves in multi-stepped periodic beams by the method of reverberation-ray matrix

This paper aims to present better physical understanding of flexural wavenumber spectral behaviors of multi-stepped periodic beams by the method of reverberation-ray matrix (MRRM) based on the Timoshenko beam theory. According to the Bloch theorem, the formulation of MRRM is presented and the complex flexural wavenumbers of multi-stepped periodic beams are solved by numerical techniques. The validation of MRRM in analyzing flexural wave behaviors of multi-stepped periodic beams is verified by comparison with the analytical wavenumber spectrum of a homogeneous beam. Numerical examples are provided to illustrate the general wavenumber spectral characteristics of an infinite periodic beam with alternate variation of cross-sections. The effects of the material properties and geometrical parameters of the beam segments are evaluated. The most remarkable finding inspired from the wavenumber spectral curves is that the real wavenumber of the multi-stepped periodic beam does not vary with frequency in the attenuation bands, which indicates that the wavelength of the flexural wave is constant in the attenuation bands. Highlights A multi-stepped periodic beam model with arbitrary finite number of beam segments is established. Complex wavenumber spectrum analysis is performed to reveal the propagation and attenuation behaviors of multi-stepped periodic beams. Similarity and comparative studies on the propagation and attenuation characteristics of beams of a uniform cross-section and of variable cross-sections are carried out to provide better physical understanding. An innovative finding that the wavelength of the flexural wave is constant in the attenuation bands is achieved.