Viscoelastic Wave Propagation Research Articles

Cubic B-spline based elastic and viscoelastic wave propagation method

The requirement for high computational efficiency in inversion imaging poses challenges to further enhancing the accuracy of imaging large elastic and viscoelastic models. Considering that forward modeling serves as a prerequisite for inversion imaging, it is essential to introduce a wave propagation simulation method that can enhance simulation accuracy without significantly compromising computational speed. In our paper, cubic B-spline is introduced into the forward modeling of the wave equation. The B-spline collocation method offers a linear complexity for computation and exhibits fourth-order convergence in terms of computational errors. We also find that it can perfectly adapt to the boundary conditions of the perfectly matched layer in the forward modeling solution of the wave simulation. Several typical numerical examples, including 2-D acoustic wave equation, 2-D elastic wave propagation, and 2-D viscoelastic propagation in homogeneous media, are provided to validate its convergence, accuracy, and computational efficiency.

Compressing the memory variables in constant-Q viscoelastic wave propagation via an improved sum-of-exponentials approximation

Earth introduces strong attenuation and dispersion to propagating waves. The time-fractional wave equation with very small fractional exponent, based on Kjartansson's constant-Q theory, is widely recognized in the field of geophysics as a reliable model for frequency-independent Q anelastic behavior. Nonetheless, the numerical resolution of this equation poses considerable challenges due to the requirement of storing a complete time history of wavefields. To address this computational challenge, we present a novel approach: a nearly optimal sum-of-exponentials (SOE) approximation to the Caputo fractional derivative with very small fractional exponent, utilizing the machinery of generalized Gaussian quadrature. This method minimizes the number of memory variables needed to approximate the power attenuation law within a specified error tolerance. We establish a mathematical equivalence between this SOE approximation and the continuous fractional stress-strain relationship, relating it to the generalized Maxwell body model. Furthermore, we prove an improved SOE approximation error bound to thoroughly assess the ability of rheological models to replicate the power attenuation law. Numerical simulations on constant-Q viscoacoustic equation in 3D homogeneous media and variable-order P- and S- viscoelastic wave equations in 3D inhomogeneous media are performed. These simulations demonstrate that our proposed technique accurately captures changes in amplitude and phase resulting from material anelasticity. This advancement provides a significant step towards the practical usage of the time-fractional wave equation in seismic inversion.

Convolution quadrature time-domain boundary element method for antiplane anisotropic viscoelastic wave propagation

This paper presents a Convolution Quadrature Time-Domain Boundary Element Method (CQBEM) for antiplane anisotropic viscoelastodynamics. The proposed CQBEM formulation employs the standard linear viscoelastic model and the fundamental solution developed by Wang and Achenbach for expressing viscoelastic and anisotropic properties, respectively. This fundamental solution, which involves integration over the unit circle, is evaluated numerically. After offering a comprehensive overview of the proposed CQBEM formulation, we demonstrate the effectiveness of the proposed method by analyzing the scattering of elastic waves by cavities in certain types of anisotropic viscoelastic materials. Additionally, the use of group velocity curves further validates the numerical results, confirming the anisotropic and viscoelastic effects and thus substantiates the proposed CQBEM for antiplane anisotropic viscoelastic wave propagation.

Thermodynamically extended symplectic numerical simulation of viscoelastic, thermal expansion and heat conduction phenomena in solids

AbstractSymplectic numerical schemes for reversible dynamical systems predict the solution reliably over large times as well, and are a good starting point for extension to schemes for simulating irreversible situations like viscoelastic wave propagation and heat conduction coupled via thermal expansion occuring in rocks, plastics, biological samples etc. Dissipation error (artificial nonpreservation of energies and amplitudes) of the numerical solution should be as small as possible since it should not be confused with the real dissipation occurring in the irreversible system. In addition, the other well-known numerical artefact, dispersion error (artificial oscillations emerging at sharp changes), should also be minimal to avoid confusion with the true wavy behavior. The continuum thermodynamical aspects (respect for balances with fluxes, systematic constitutive relationships between intensive quantities and fluxes, the second law of thermodynamics with positive definite entropy production, and the spacetime-based kinematic viewpoint) prove valuable for obtaining such extended schemes and for monitoring the solutions. Generalizing earlier works in this direction, here, we establish and investigate such a numerical scheme for one-dimensional viscoelastic wave propagation in the presence of heat conduction coupled via thermal expansion, demonstrating long-term reliability and the applicability of thermodynamics-based quantities in supervising the quality of the solution.

A mixed finite element approach for a factional viscoelastic wave propagation in-time-domain

A mixed finite element approach for a factional viscoelastic wave propagation in-time-domain

An Implementation Method of the Complex Frequency-Shifted Uniaxial/Multi-Axial PML Technique for Viscoelastic Seismic Wave Propagation

ABSTRACT A numerical implementation method of the complex frequency-shifted uniaxial/multi-axial PML (CFS UPML/MPML) is developed for modeling half-space viscoelastic wave propagation. A mixed CFS PML combining CFS UPML and CFS MPML is designed to improve PML’s performances in avoiding evanescent waves at grazing incidence and stabilizing long-time simulation of low-frequency viscoelastic wave propagation. Two numerical examples’ results showed that the developed method for the mixed CFS PML can implement elimination of evanescent waves at grazing incidence near the PML interface, long-time accurate simulation of low-frequency viscoelastic wave propagation and efficient absorption of the near-fault outgoing seismic waves in homogeneous earth medium.

3D Viscoelastic Finite-Difference Analysis of the Monopole Acoustic Logs in Cylindrical Coordinates

In this paper, a model of the heterogeneous anelastic seismic wave problem is proposed in three-dimensional (3D) cylindrical coordinates. We use the velocity-stress formula to describe the realistic attenuation properties of viscoelastic materials, derived from a rheological model of the generalized standard linear solid (GSLS). The equation system is completed by additional equations for the anelastic functions including the strain history of the material. We apply the staggered grid finite-difference (FD) method in 3D cylindrical coordinates to solve the equations. Moreover, to avoid the effect of gradual expansion of the grid size as the radius increases, we use a variable grid method to achieve compensation. In real drilling operations, the mud injected in the borehole is a fluid with viscous properties. The actual formation is also not elastic. In the synthetic data of acoustic logging while drilling (LWD), we find that the drill collar wave is not affected by the viscoelastic parameters of the formation. In contrast, the Stoneley wave is more sensitive to the viscosity of the drilling fluid. The phase and amplitude of the received waveform are affected by the drilling fluid as well as the formation viscoelasticity. Therefore, the development of the cylindrical variable-grid FD method provides a flexible and efficient numerical technique to solve 3D viscoelastic wave propagation problems, including realistic attenuation and complex geometry.

On the propagation of intrinsic viscoelastic waves in a cylindrical shell

The article illustrates that viscoelastic wave in an extended cylindrical shell of circular cross-section. To solve boundary value problems used classical methods of mathematical physics. The unwavering quality of the comes about of the work is guaranteed by the rightness of the detailing of issues, the utilize of the classical device of scientific material science and other demonstrated strategies for tackling issues of wave engendering, the consistent legitimacy of the conclusions and the great assentation of the comes about with the uncommon cases known from the references.

Viscoelastic transcranial wave propagation modeling for neuronavigated focused ultrasound procedures in humans

Transcranial focused ultrasound (FUS) has gained significant attention as a non-invasive therapeutic technique that can reach both cortical and subcortical targets in the human brain. However, the skull barrier distorts the ultrasound beam and introduces significant losses that need to be accounted for. In this talk, I will present our efforts to develop high precise transcranial ultrasound modeling based on the open-source viscoelastic wave propagation solver BabelViscoFDTD and its application in neuronavigated procedures for FUS-based neuromodulation. The viscoelastic wave equation is expressed in isotropic stress tensors and displacement vectors, and their nodes are placed in a staggered grid arrangement. The solver uses a Cartesian O(4)-space O(2)-time FDTD scheme. A narrow-band quality factor models attenuation. A reduction of staircase artifacts can be enabled using a superposition operator. BabelViscoFDTD is optimized for multiple computing GPU and CPU backends (CUDA, OpenCL, Metal, and X86-64) and includes GPU-accelerated solvers of the Rayleigh-Sommerfeld equation and Bio-Heat thermal equations. We adapted the viscoelastic and thermal solvers to neuronavigate two FUS devices: A 128-element concaved array operable at 250 kHz and 700 kHz and a 4-ring concaved array operable at 500 kHz. The high degree of optimization of the solvers allow that numerical procedures (neuronavigation and modeling) can be performed in an Apple M1 Max (64 GB RAM) laptop computer.

Viscoelastic wave propagation in transversely isotropic media based on constant-order fractional polynomial approximations

Accurately simulating viscoelastic wave propagation in the subsurface is essential to image subsurface reservoirs. One established method to investigate constant- Q effects is based on the variable-order fractional viscoelastic wave equation. However, the mixed-domain operators introduced in this approach are cumbersome to calculate and impede a high-efficiency numerical solver. To solve this issue, we approximate the mixed-domain operator by using a polynomial approximation and derive the corresponding viscoelastic wave equation in anisotropic attenuative media. The constant-order fractional polynomial method is further developed to improve the approximation accuracy. Based on these polynomial approximations, only several constant-order Laplacians remain in the new form of the viscoelastic wave equations. Therefore, these equations can be easily solved using Fourier transform and finite-difference methods. In addition, we also perform an accuracy analysis among the conventional quadratic, cubic, and constant-order fractional polynomial approximations. It indicates that the relative error of the constant-order fractional polynomial method is much smaller than that generated by the conventional quadratic and cubic polynomial approximations. Compared to the variable-order fractional viscoelastic wave equation, our constant-order viscoelastic wave equation can simulate seismic wave propagations with comparable accuracy but with high efficiency. Numerical results based on our approach can well match reference solutions in homogeneous and heterogeneous models, demonstrating the accuracy and effectiveness of the new method.

Viscoelastic wave propagation and resonance in a metal–plastic bonded laminate

Stress wave propagation in a metal–plastic bonded laminate immersed in a fluid is investigated based on linear viscoelastic theory. The reflection spectrum is theoretically derived for normal wave incidence. For the incidence from the plastic layer, the amplitude of the reflection spectrum is shown to have local minima at the resonance frequencies of every single layer. Resonance in the metal layer leads to the destructive interference of the reflected wave components, while resonance in the plastic layer appears due to its viscoelastic damping. Experimental results obtained for bonded specimens are discussed from the viewpoint of the characterization for dissimilar joints.

Validation of seismic velocity models in southern California with full-waveform simulations

SUMMARY Crustal seismic velocity models provide essential information for many applications including earthquake source properties, simulations of ground motion and related derivative products. We present a systematic workflow for assessing the accuracy of velocity models with full-waveform simulations. The framework is applied to four regional seismic velocity models for southern California: CVM-H15.11, CVM-S4.26, CVM-S4.26.M01 that includes a shallow geotechnical layer, and the model of Berg et al. For each model, we perform 3-D viscoelastic wave propagation simulations for 48 virtual seismic noise sources (down to 2 s) and 44 moderate-magnitude earthquakes (down to 2 s generally and 0.5 s for some cases) assuming a minimum shear wave velocity of 200 m s–1. The synthetic waveforms are compared with observations associated with both earthquake records and noise cross-correlation data sets. We measure, at multiple period bands for well-isolated seismic phases, traveltime delays and normalized zero-lag cross-correlation coefficients between the synthetic and observed data. The obtained measurements are summarized using the mean absolute derivation of time delay and the mean correlation coefficient. These two metrics provide reliable statistical representations of model quality with consistent results in all data sets. In addition to assessing the overall (average) performance of different models in the entire study area, we examine spatial variations of the models’ quality. All examined models show good phase and waveform agreements for surface waves at periods longer than 5 s, and discrepancies at shorter periods reflecting small-scale heterogeneities and near-surface structures. The model performing best overall is CVM-S4.26.M01. The largest misfits for both body and surface waves are in basin structures and around large fault zones. Inaccuracies generated in these areas may affect tomography and model simulation results at other regions. The seismic velocity models for southern California can be improved by adding better resolved structural representations of the shallow crust and volumes around the main faults.

Effects of the magnetic field on the viscoelastic wave propagation and dynamic material properties

In this paper, the longitudinal wave propagation of a thin ferromagnetic (Fe-Cr-Al alloy) rod under the transverse magnetic field is investigated analytically without ignoring the viscoelastic characteristics of this alloy. The analysis is based on the Bernoulli-Euler rod theory and a 4-element viscoelastic model based on the experimental observation. The viscoelastic wave characteristics and the behaviours of dynamic material properties under the magnetic field are investigated. The analytically results show that the longitudinal wave propagation and the dynamic material properties of the rod are affected by the magnetic field existence (for magnetic field magnitude values greater than about 107A/m), except the imaginary part of the complex elasticity modulus. Furthermore, in a large frequency range, depending on the ratio of the group velocity to the phase velocity an abnormal dispersion characteristic is observed.

Fluids Alter Elasticity Measurements: Porous Wave Propagation Accounts for Shear Wave Dispersion in Elastography

In shear wave elastography, rotational wave speeds are converted to elasticity measures using elastodynamic theory. The method has a wide range of applications and is the gold standard for non-invasive liver fibrosis detection. However, the observed shear wave dispersion of in vivo human liver shows a mismatch with purely elastic and visco-elastic wave propagation theory. In a laboratory phantom experiment we demonstrate that porosity and fluid viscosity need to be considered to properly convert shear wave speeds to elasticity in soft porous materials. We extend this conclusion to the clinical application of liver stiffness characterization by revisiting in vivo studies of liver elastography. To that end we compare Biot’s theory of poro-visco-elastic wave propagation to Voigt’s visco-elastic model. Our results suggest that accounting for dispersion due to fluid viscosity could improve shear wave imaging in the liver and other highly vascularized organs.

Removing numerical dispersion from linear evolution equations

We describe a method for removing the numerical errors in the modeling of linear evolution equations that are caused by approximating the time derivative by a finite difference operator. The method is based on integral transforms realized as certain Fourier integral operators, called time dispersion transforms, and we prove that, under an assumption about the frequency content, it yields a solution with correct evolution throughout the entire lifespan. We demonstrate the method on a model equation as well as on the simulation of elastic and viscoelastic wave propagation.

LP or VT signals? How intrinsic attenuation influences volcano seismic signatures constrained by Whakaari volcano parameters

Long Period (LP) seismicity is considered a key indicator of fluid migration in the volcanic edifice, with a source mechanism typically ascribed to fluid resonance in the volcano conduit and fractures. Our viscoelastic numerical wave propagation models at laboratory and field scales are based on quantitative estimates of wave speeds and intrinsic attenuation from Whakaari rocks saturated with a range of fluid types. Such models display path effects in homogeneous and layered media that influence the frequency content and duration of the seismic waveforms. We find that volcano-tectonic (VT) earthquakes can resemble short-duration LP events when propagating through fully or partially gas-saturated ash tuff deposits. Meanwhile, in a shallow volcanic subsurface comprising low-velocity, attenuative ash tuff and high-velocity, low-attenuation lava layers, VT signals may be distorted to resemble long-duration LP events. Such geological models are built based on interpretations of the shallow hydrothermal system at Whakaari volcano, and discussed in the context of other volcanoes around the world.

Elastic and viscoelastic flexural wave motion in woodpecker-beak-inspired structures

This paper comprehensively investigates elastic and viscoelastic flexural wave propagation in structures that are inspired by the unique suture configurations present in a woodpecker’s beak, in order to understand their ability to attenuate velocity amplitudes and wave speeds. Waveguides characterized by sinusoidal depth variations, both plain and variously graded along the length, mimicking the suture geometry, are considered in this work. Elastic and viscoelastic wave propagation analyses, along with prior static and free vibration studies, are carried out using a novel superconvergent finite element formulation. In elastic wave propagation analysis, firstly the attenuation characteristics are appraised in relation to the high amplitude and frequency waves of three different plain waveguides with differing depth profile orientations. This prompted us to next consider waveguides of hybrid configurations derived from them. Further, waveguides with lengthwise graded sinusoidal segments, as observed in nature, are studied for better wave attenuation properties compared to plain waveguides. This is followed by viscoelastic wave propagation analysis. Regarding the important role of the suture geometry, which is the focus of this work, the results from the elastic analyses revealed the nature of the reduction in wave speeds and amplitudes, both qualitatively and quantitatively, in such waveguides, and their dependence on the orientation and magnitude of the sinusoidal depth variation. Some waveguide configurations with remarkable wave attenuation characteristics, in terms of both wave speeds and amplitudes, are presented, along with their implications regarding impact mitigation applications.

PROPAGATION OF LINEAR WAVES IN MULTILAYERED STRUCTURAL - INHOMOGENEOUS CYLINDRICAL SHELLS

In this paper, axisymmetric viscoelastic waves in an extended cylindrical shell of circular cross section are considered. The study of axisymmetric viscoelastic waves in multilayer cylindrical systems with the maximum possible use of analytical methods is relevant for the development of a new generation of sounding techniques for wells. The aim of the work is to study the propagation of axisymmetric viscoelastic waves in extended multilayer cylindrical structures, to determine the volumetric structure variant from the responses of one-way sounding using a priori information about the possible structure of the system. The paper uses classical methods of mathematical physics, applied to solving boundary value problems in a cylindrical coordinate system, the spatial Fourier transform, the method of complex amplitudes for the variable components of the displacement vector and the stress tensor. The low-frequency resonances of a cylindrical shell are investigated. The physical laws of the formation of the pitch of its fundamental tone are determined sounding. The impossibility of propagation along the casing of wells is proved low-frequency surface waves, as well as SH-polarization waves at axisymmetric excitation. The thin-walled cylindrical shell in the low-frequency region has two fundamental resonance associated with the resonance of the shell as a whole and resonance of each individual section as a ring. Beats of these frequencies the basic tone of the sound of an extended shell of this type is determined. At frequencies below the cutoff of higher modes, all the features of the dispersion natural waves in the wall of a circular cylinder shell are described approximate algebraic equation of the third degree. One the real root of this equation corresponds to the zero mode antisymmetric Lamb wave, and one of the other two complex conjugate, describes the zero mode of the symmetric wave.

Spalling of a one-dimensional viscoelastic bar induced by stress wave propagation

As a stress wave propagates in rock, the amplitude of the stress wave will attenuate, and the waveform will dissipate, which can be equivalently characterized by the stress wave propagation in a viscoelastic material. The stress wave propagation may induce rock damage when the stress is greater than the rock strength. In this study, viscoelastic wave propagation in rock and the damage and failure process of rock induced by viscoelastic wave propagation, which are usually examined separately in existing theoretical and numerical models, were investigated under a unified framework. Based on viscoelastic damage theory and the theory of stress wave propagation in a one-dimensional viscoelastic bar, theoretical equations for spalling fracture of a viscoelastic bar were obtained. Then, the numerical model implemented by the one-dimensional wave propagation program was validated by reproducing the dynamic spalling of green sandstone bar; after that step, the program was employed to study the behavior of viscoelastic bar spalling. Based on the numerical simulation, the condition, position and time of the dynamic spalling induced by quarter sine wave propagation in the viscoelastic bar were examined, and the influence of the amplitude and frequency of the quarter sine wave and the viscosity coefficient and length of the viscoelastic bar on the spalling were quantified.

Enriched homogenized model for viscoelastic plane wave propagation in periodic layered composites

An enriched homogenized model is developed based on a proposed homogenization strategy, to describe the wave propagation behaviour through periodic layered composites. The intrinsic parameters characterising the micro-inertia effect and non-local interactions are defined transparently in terms of the constituent materials’ properties and volume fractions. The framework starts with the introduction of an additional kinematic field to characterise the displacement of the stiff layer, before setting up macro kinematic fields to account for the average deformation of the constituent materials within a segmented unit cell. Relationships between these macro average strain fields are determined based on suitable micro-mechanical arguments. The Hill–Mandel condition is next applied to translate the energy statements from micro to macro. A system of coupled governing equations of motion is finally extracted naturally at the macro level via Hamilton’s Principle. Through a series of benchmark examples, it is shown that the proposed model exhibits excellent predictive capabilities over a broad range of loading frequencies.