Viscoelastic Equation Research Articles

Global existence and decay of a viscoelastic wave equation with logarithmic source under acoustic, fractional, and nonlinear delay conditions

This study is concerned with a nonlinear viscoelastic wave equation with logarithmic source term. By considering the nonlinear distributed delay feedback influenced by acoustic and fractional boundary conditions at the boundary coupling, we establish the global existence and general decay results under appropriate assumptions, even in the case of a general kernel. This result extends and complements some previous results.

A 1D micromechanics-based constitutive model for the thermoviscoelastic behavior of crosslinked semicrystalline shape memory polymers: Numerical simulation and experimental validation

Abstract The paper develops a 1D thermoviscoelastic constitutive model for the crystallization- and melting-induced one-way and two-way shape memory effects, as well as isothermal yielding behaviors, of crosslinked semi-crystalline polymers. A micromolecular chain model is proposed to characterize the transition between the amorphous and crystalline phases. Structural equations including a modified Eying model that combine phase transition and viscoelasticity equations are employed to predict the shape memory effects. An extensive experimental campaign has been carried out on poly (ethylene-co-vinyl acetate) (EVA) based semi-crystalline elastomers to characterize the thermoviscoelastic temperature-stress-strain relations of the material under different loading and rate conditions. Some results guide the determination of the model parameters, while the rest validate the model capabilities. Comparisons with the experimental results show that the model can well reproduce the stress-strain-temperature responses, providing valuable insights for application development.

Stabilisation of linear waves with inhomogeneous Neumann boundary conditions

ABSTRACT We study linear damped and viscoelastic wave equations evolving on a bounded domain. For both models, we assume that waves are subject to an inhomogeneous Neumann boundary condition on a portion of the domain's boundary. The analysis of these models presents additional interesting features and challenges compared to their homogeneous counterparts. In the present context, energy depends on the boundary trace of velocity. It is not clear in advance how this quantity should be controlled based on the given data, due to regularity issues. However, we establish global existence and also prove uniform stabilisation of solutions with decay rates characterised by the Neumann input. We supplement these results with numerical simulations in which the data do not necessarily satisfy the given assumptions for decay. These simulations provide, at a numerical level, insights into how energy could possibly change in the presence of, for example, improper data.

Enhancing Learning to Solve Multicomponent Fractional Viscoelastic Equations with U-net Fourier Neural Operators

Abstract The research of viscoelastic media is currently a hot topic in the interpretation and processing of seismic data. To accurately simulate the propagation of seismic waves in viscoelastic media, the fractional viscoelastic equation has emerged as an indispensable method. However, solving this equation numerically has proven to be challenging due to the complexity introduced by its fractional Laplacian operators. Recently, deep learning, especially Fourier neural operators (FNO), has shown excellent performance in learning to fast solve partial differential equations. Traditional FNO methods may face crosstalk problem and make it difficult to achieve satisfactory accuracy, when solving the multi-component fractional order viscoelastic equation. To solve this problem, we introduce a novel approach based on U-net Fourier neural operator (U-FNO). As an enhanced learning method to the traditional FNO-based method, the U-FNO-based method integrates a U-Fourier layer following the standard Fourier layer as a form of regularization, thereby achieving superior prediction accuracy for multi-component equations. Specifically, both the Fourier layers and U-Fourier layers in U-FNO are trained with the solutions of the equation from previous time steps as inputs. This training process enables the U-FNO to efficiently produce more accurate solutions for subsequent wavefield. Numerical simulations reveal that the U-FNO-based method efficiently learns to solve the fractional viscoelastic wave equation independent of fractional Laplacian operators. Additionally, U-FNO-based method offers superior prediction accuracy in comparison with the traditional FNO-based method.

Markov chain Monte Carlo methods applied to the stochastic inversion of 1D viscoelastic parameters

Abstract The characterization of the subsurface profile properties that are susceptible to liquefaction during earthquakes is of great importance in accident prevention. Field data can be used in a stochastic inverse problem to estimate the material properties using Markov chain Monte Carlo Methods (McMC) that incorporates prior knowledge about the unknown parameters as well as the data available. Here, we numerically study the Random Walk (RW) and Differential Evolution (DE) variations of the Metropolis algorithm in the context of seismic modeling considering the viscoelastic wave equation. The algorithms are tested with data from the 1987 Superstition Hills earthquake recorded at the Wildlife Site. The results show the effectiveness and accuracy of these algorithms, with emphasis to DE which reached convergence earlier compared to RW.

A K-Space-Based Temporal Compensating Scheme for a First-Order Viscoacoustic Wave Equation with Fractional Laplace Operators

Inherent constant Q attenuation can be described using fractional Laplacian operators. Typically, the fractional Laplacian viscoacoustic or viscoelastic wave equations are addressed utilizing the staggered-grid pseudo-spectral (SGPS) method. However, this approach results in time numerical dispersion errors due to the low-order finite difference approximation. In order to address these time-stepping errors, a k-space-based temporal compensating scheme is established to solve the first-order viscoacoustic wave equation. This scheme offers the advantage of being nearly free from grid dispersion for homogeneous media and enhances simulation stability. Numerical examples indicate that the proposed k-space scheme aligns well with analytical solutions for homogeneous media. Additionally, this method demonstrates excellent applicability for complex models and is more efficient due to its ability to adopt a larger time step compared with conventional staggered-grid pseudo-spectral methods.

Blow up in a nonlinear viscoelastic wave equation with strong damping and variable sources

Blow up in a nonlinear viscoelastic wave equation with strong damping and variable sources

On the viscoelastic-electromagnetic-gravitational analogy

The analogy between electromagnetism and gravitation was achieved by linearizing the tensorial gravitational equations of general relativity and converting them into a vector form corresponding to Maxwell’s electromagnetic equations. On this basis, we use the equivalence with viscoelasticity and propose a theory of gravitational waves. We add a damping term to the differential equations, which is equivalent to Ohm’s law in electromagnetism and Maxwell’s viscosity in viscoelasticity, to describe the attenuation of the waves. The differential equations in viscoelasticity are those of cross-plane shear waves, commonly referred to as SH waves. A plane-wave analysis gives the phase velocity, the energy velocity, the quality factor and the attenuation factor of the field as well as the energy balance. To obtain these properties, we use the analogy with viscoelasticity; the properties of electromagnetic and gravitational waves are similar to those of shear waves. The presence of attenuation means that the transient field is generally a composition of inhomogeneous (non-uniform) plane waves, where the propagation and attenuation vectors do not point in the same direction and the phase velocity vector and the energy flux (energy velocity) are not collinear. The polarization of cross-plane field is linear and perpendicular to the propagation-attenuation plane, while the polarization of the field within the plane is elliptical. Transient wave fields in the space–time domain are analyzed with the Green function (in homogeneous media) and with a grid method (in heterogeneous media) based on the Fourier pseudospectral method for calculating the spatial derivatives and a Runge–Kutta scheme of order 4 for the time stepping. In the examples, wave propagation at the Sun–Earth and Earth–Moon distances using quadrupole sources is considered in comparison to viscoelastic waves. The Green and grid solutions are compared to test the latter algorithm. Finally, an example of propagation in heterogeneous media is presented.

Global and exponential attractors for viscoelastic Kirchhoff plate equation with memory and time delay

In this paper, we consider the viscoelastic Kirchhoff plate equation with memory and time delay. Under appropriate conditions on the real numbers and , we prove the existence of a compact global attractor with finite fractal dimension through a stabilizability estimate. Additionally, we demonstrate the existence of a fractal exponential attractor. To the best of our knowledge, these findings are novel for for this system.

Viscoelasticity, logarithmic stresses, and tensorial transport equations

Viscoelasticity, logarithmic stresses, and tensorial transport equations

Existence of Solutions for a Viscoelastic Plate Equation with Variable Exponents and a General Source Term

The subject of this study is a nonlinear viscoelastic plate equation with variable exponents and a general source term. Through the application of the Faedo–Galerkin approximation method and a fixed point theorem under appropriate assumptions, we proved the existence of weak solutions.

A blow‐up result of a nonlinear viscoelastic wave equations system with variable dampings and sources

A blow‐up result of a nonlinear viscoelastic wave equations system with variable dampings and sources

Decoupled approximate qP‐ and qSV‐wave equations in attenuated transversely isotropic media

AbstractAccurate seismic models with anisotropy and attenuation characteristics are crucial to accurately imaging subsurface structures. However, the anisotropic viscoelastic equations are complex and require significant computational resources. In addition, the single‐mode waves have been sufficient for most practical exploration needs. However, separating the qP‐ and qSV‐waves in anisotropic viscoelastic wavefields is challenging. Thus, we propose a new method to approximate and efficiently separate the qP‐ and qSV‐waves in attenuated transversely isotropic media. First, we obtain the decoupled approximate phase velocities of qP‐ and qSV‐waves by a curve‐fitting method. Consequently, based on the average and maximum relative error analysis, our approximate qP‐ and qSV‐wave phase velocities are more accurate than the existing approximations. Additionally, our approximations have broader applicability, resulting in acceptable errors during their application. Second, based on the approximate qP‐ and qSV‐wave phase velocities, we derive the corresponding qP‐ and qSV‐wave equations for a complete decoupling of the qP‐ and qSV‐wave components in transversely isotropic media. Third, to combine the attenuation and anisotropy characteristics, we incorporate the Kelvin–Voigt attenuation model and obtain the decoupled qP‐ and qSV‐wave equations in attenuated transversely isotropic media. Then, we use an efficient and stable hybrid finite‐difference and pseudo‐spectral method to solve the new decoupled qP‐ and qSV‐wave equations. Finally, several numerical examples demonstrate the separability and high accuracy of the proposed qP‐ and qSV‐wave equations. We obtain a qP‐wave wavefield entirely devoid of SV‐wave artefacts. In addition, the decoupled approximate qP‐ and qSV‐wave equations are accurate and stable in heterogeneous media with different velocities and attenuation. The decoupled, approximated qP‐wave and qSV‐wave equations proposed in this paper can effectively separate the qP‐wave and qSV‐wave components, resulting in fully decoupled qP‐ and qSV‐wave wavefields in attenuated transversely isotropic media.

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.

On a Viscoelastic Plate Equation with Logarithmic Nonlinearity and Variable-Exponents: Global Existence, General Decay and Blow-Up of Solutions

On a Viscoelastic Plate Equation with Logarithmic Nonlinearity and Variable-Exponents: Global Existence, General Decay and Blow-Up of Solutions

Optimal convergence analysis of the virtual element methods for viscoelastic wave equations with variable coefficients on polygonal meshes

Optimal convergence analysis of the virtual element methods for viscoelastic wave equations with variable coefficients on polygonal meshes

A new splitting mixed finite element analysis of the viscoelastic wave equation

A new splitting mixed finite element analysis of the viscoelastic wave equation

Inverse Problem of Determining a Kernel of the Viscoelasticity Equation with Distributed Data in a Limited Domain

Inverse Problem of Determining a Kernel of the Viscoelasticity Equation with Distributed Data in a Limited Domain

Blow-up of solutions for a system of nonlocal singular viscoelastic equations with sources and distributed delay terms

In this paper, we investigate a scenario concerning a coupled nonlocal singular viscoelastic equation with sources and distributed delay terms. By establishing suitable conditions, we have proved that a finite-time blow-up occurs in the solution.

Existence and decay estimate of global solution for a viscoelastic wave equation with nonlinear boundary source term

In this paper, we consider a viscoelastic wave equation with nonlinear boundary source term. By the combination of Galerkin approximation and potential well methods, we prove the global existence of solutions. Then, we give an decay rate estimate of the energy by making use of the perturbed energy method.