Viscoelastic Wave Equation Research Articles

NEW GENERAL DECAY RATES OF SOLUTIONS FOR TWO VISCOELASTIC WAVE EQUATIONS WITH INFINITE MEMORY

We consider in this paper the problem of asymptotic behavior of solutions for two viscoelastic wave equations with infinite memory. We show that the stability of the system holds for a much larger class of kernels and get better decay rate than the ones known in the literature. More precisely, we consider infinite memory kernels satisfying, where and are given functions. Under this very general assumption on the behavior of g at infinity and for each viscoelastic wave equation, we provide a relation between the decay rate of the solutions and the growth of g at infinity, which improves the decay rates obtained in [15, 16, 17, 19, 40]. Moreover, we drop the boundedness assumptions on the history data considered in [15, 16, 17, 40].

Decay of energy for viscoelastic wave equations with Balakrishnan-Taylor damping and memories

In this article, we consider a viscoelastic wave equation with Balakrishnan-Taylor damping, and finite and infinite memory terms in a bounded domain. Under suitable assumptions on relaxation functions and with certain initial data, by adopting the perturbed energy method, we establish a decay of energy which depends on the behavior of the relaxation functions.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/42/abstr.html

Weak Galerkin finite element method for viscoelastic wave equations

In this article, we consider a weak Galerkin finite element method (WG-FEM) for solving one type of viscoelastic wave equation. A discrete weak gradient operator on discontinuous piecewise polynomials is used in the numerical scheme. We introduce both the semi- and fully-discrete WG-FEMs and establish the corresponding stability estimates. Optimal order error estimate in H1-norm is derived. Numerical experiments are preformed to verify the theoretical results.

Logaritmik Kaynak Terimli Yüksek Mertebeden Viskoelastik Dalga Denkleminin Çözümlerinin Üstel Büyümesi

This paper deals with a higher-order viscoelastic wave equation with logarithmic source term. We prove, for suitable conditions, the exponential growth of solutions.

Two meshless methods based on local radial basis function and barycentric rational interpolation for solving 2D viscoelastic wave equation

In this paper, 2D viscoelastic wave equation is solved numerically both on regular and irregular domains. For spatial approximation of viscoelastic wave equation two meshless methods based on local radial basis function and barycentric rational interpolation are proposed. Both of the spatial approximation methods do not need mesh, node connectivity or integration process on local subdomains so they are truly meshless. For local radial basis function method we used an existing algorithm in literature to choose an acceptable shape parameter. Time marching is performed with fourth order Runge Kutta method. L2 and L∞ error norms for some test problems are reckoned to indicate efficiency and performance of the proposed two methods. Also, stability of the methods is discussed. Acquired results confirm the applicability of the proposed methods for 2D viscoelastic wave equation. We have performed some comparisons between the proposed two methods in the sense of accuracy and computational cost. From the comparisons, we have observed that performance of the barycentric rational interpolation in the sense of accuracy is slightly better than the performance of local radial basis function however computational cost of the local radial basis function is less than the computational cost of barycentric rational interpolation.

A Novel Decay Rate for a Coupled System of Nonlinear Viscoelastic Wave Equations

The main goal of the present paper is to study the existence, uniqueness and behavior of a solution for a coupled system of nonlinear viscoelastic wave equations with the presence of weak and strong damping terms. Owing to the Faedo-Galerkin method combined with the contraction mapping theorem, we established a local existence in [ 0 , T ] . The local solution was made global in time by using appropriate a priori energy estimates. The key to obtaining a novel decay rate is the convexity of the function χ , under the special condition of the initial energy E ( 0 ) . The condition of the weights of weak and strong damping has a fundamental role in the proof. The existence of both three different damping mechanisms and strong nonlinear sources make the paper very interesting from a mathematics point of view, especially when it comes to unbounded spaces such as R n .

A comparative analysis on wave characteristics of viscoelastic rods coated with piezoelectric layer

In this study, longitudinal stress waves of viscoelastic rods bonded with piezoelectric layer are tackled comparatively. The analysis is based on one-dimensional wave theory and various rheological models are considered. The new viscoelastic wave equations that include the effect of piezoelectric material layer are developed. The numerical illustrations of the wave characteristics are presented in detail. Especially, the attenuation illustrations can be regarded as significant novelties of this study for the present literature. This study can be helpful in modelling laminated piezoelectric structures having viscoelastic core as suitable to desired operating conditions.

Fractional Laplacians viscoacoustic wavefield modeling with k-space-based time-stepping error compensating scheme

The spatial derivatives in decoupled fractional Laplacian (DFL) viscoacoustic and viscoelastic wave equations are the mixed-domain Laplacian operators. Using the approximation of the mixed-domain operators, the spatial derivatives can be calculated by using the Fourier pseudospectral (PS) method with barely spatial numerical dispersions, whereas the time derivative is often computed with the finite-difference (FD) method in second-order accuracy (referred to as the FD-PS scheme). The time-stepping errors caused by the FD discretization inevitably introduce the accumulative temporal dispersion during the wavefield extrapolation, especially for a long-time simulation. To eliminate the time-stepping errors, here, we adopted the [Formula: see text]-space concept in the numerical discretization of the DFL viscoacoustic wave equation. Different from existing [Formula: see text]-space methods, our [Formula: see text]-space method for DFL viscoacoustic wave equation contains two correction terms, which were designed to compensate for the time-stepping errors in the dispersion-dominated operator and loss-dominated operator, respectively. Using theoretical analyses and numerical experiments, we determine that our [Formula: see text]-space approach is superior to the traditional FD-PS scheme mainly in three aspects. First, our approach can effectively compensate for the time-stepping errors. Second, the stability condition is more relaxed, which makes the selection of sampling intervals more flexible. Finally, the [Formula: see text]-space approach allows us to conduct high-accuracy wavefield extrapolation with larger time steps. These features make our scheme suitable for seismic modeling and imaging problems.

Optimal decay rates of a nonlinear time-delayed viscoelastic wave equation

This paper concerns a nonlinear viscoelastic wave equation with time-dependent delay. Under suitable relation between the weight of the delay and the weight of the term without delay, we prove the global existence of weak solutions by the combination of the Galerkin method and potential well theory. In addition, by assuming the minimal conditions on the $L^1(0,\infty)$ relaxation function $g$, namely, $g'(t)\leq-\xi(t)H(g(t))$, where $H$ is an increasing and convex function and $\xi$ is a nonincreasing differentiable function, and by using some properties of convex functions, we establish optimal explicit and general energy decay results. This result is new and substantially improves existing results in the literature.

Local existence and blow-up of solutions for coupled viscoelastic wave equations with degenerate damping terms

Local existence and blow-up of solutions for coupled viscoelastic wave equations with degenerate damping terms

Blow-up of a nonlinear viscoelastic wave equation with distributed delay combined with strong damping and source terms

Blow-up of a nonlinear viscoelastic wave equation with distributed delay combined with strong damping and source terms

Stability results for an elastic–viscoelastic wave equation with localized Kelvin–Voigt damping and with an internal or boundary time delay

We investigate the stability of a one-dimensional wave equation with non smooth localized internal viscoelastic damping of Kelvin–Voigt type and with boundary or localized internal delay feedback. The main novelty in this paper is that the Kelvin–Voigt and the delay damping are both localized via non smooth coefficients. Under sufficient assumptions, in the case that the Kelvin–Voigt damping is localized faraway from the tip and the wave is subjected to a boundary delay feedback, we prove that the energy of the system decays polynomially of type t − 4 . However, an exponential decay of the energy of the system is established provided that the Kelvin–Voigt damping is localized near a part of the boundary and a time delay damping acts on the second boundary. While, when the Kelvin–Voigt and the internal delay damping are both localized via non smooth coefficients near the boundary, under sufficient assumptions, using frequency domain arguments combined with piecewise multiplier techniques, we prove that the energy of the system decays polynomially of type t − 4 . Otherwise, if the above assumptions are not true, we establish instability results.

General decay and blow-up results of solutions for a viscoelastic wave equation with Robin-Dirichlet conditions

General decay and blow-up results of solutions for a viscoelastic wave equation with Robin-Dirichlet conditions

BLOW UP OF SOLUTIONS FOR A NONLINEAR VISCOELASTIC WAVE EQUATIONS WITH VARIABLE EXPONENTS

The aim of this work is to study the blow up of solutions for the viscoelastic wave equation with variable exponents in a bounded domain. Our result extends the one in Messaoudi1 to problems with variable exponent nonlinearities.

Global Nonexistence for A Viscoelastic Wave Equation with Acoustic Boundary Conditions

This paper deals with a class of nonlinear viscoelastic wave equation with damping and source terms ${u_{tt}} - \Delta u - \Delta {u_t} - \Delta {u_{tt}} + \int_0^t {g(t - s)\Delta u(s)ds + {u_t}{u_t}{^{m - 2}} = u|u|{^{p - 2}}} $ with acoustic boundary conditions. Under some appropriate assumption on relaxation function g and the initial data, we prove that the solution blows up in finite time if the positive initial energy satisfies a suitable condition.

A sharper decay rate for a viscoelastic wave equation with power nonlinearity

We consider a viscoelastic wave equation with power nonlinearity. First, we construct a local solution by the Faedo‐Galerkin approximation scheme and contraction mapping theorem. Next, we continue the local solution to the global one by a priori estimates obtained from a decreasing energy. Finally, we discuss the decay rate of the global solution by assuming that the kernel function is convex.

Logarithmic viscoelastic wave equation in three-dimensional space

ABSTRACT The initial-boundary value problem of logarithmic viscoelastic wave equation in three space dimensions is studied. The existence of local and global solutions for this problem are proved by means of the contraction mapping criterion and potential well method. Meanwhile, the blow-up of solutions in the unstable set is also obtained.

Modeling the viscoelastic effects in P-waves with modified viscoacoustic wave propagation

Accurate full-waveform seismic modeling is a powerful tool for understanding wave propagation and building subsurface images. However, it can be computationally expensive for viscoelastic media. Viscoacoustic seismic modeling is much cheaper, but at the trade-off of using incomplete physics. We have developed a modified viscoacoustic wave simulation algorithm for modeling the viscoelastic effects of P-waves. The algorithm contains two viscoacoustic forward-modeling steps; the first is the same as the traditional viscoacoustic modeling, whereas the second propagation is generated using a residual error source, which is derived by comparing the viscoacoustic and viscoelastic wave equations in the form of stress-particle velocity formulations. The corrected P-wave particle velocities can be obtained by adding the wavefield from the second simulation step to the original (the first simulation step) viscoacoustic wavefield. Only P-waves are modeled. The overall cost is about twice that of viscoacoustic modeling, but it is significantly less than a viscoelastic propagation because there are fewer calculations, and we can use a coarser grid and larger time steps for the same accuracy. Numerical examples indicate that the P-wave waveforms, after correction, match those from viscoelastic wave modeling better than those from the original viscoacoustic simulation. Our method provides a cost-efficient alternative for approximating the viscoelastic effects in P-wave modeling.

The Long-Time Stability of Solutions for Intermittently Controlled Viscoelastic Wave Equations with Memory Terms

In this paper, we investigate the asymptotic stability of global solutions to the following intermittently controlled semilinear viscoelastic equation with memory term $$\begin{aligned}u(x,t)\in \varOmega \times [0,\infty ), \end{aligned}$$ under the null Dirichlet boundary condition and $$\tau \in \{t,\infty \}$$ . By virtue of appropriate new Lyapunov functional and Łojasiewicz–Simon inequality, we show that any global bounded solution converges to a steady state and get the rate of convergence as well, when damping coefficients are integral positive and positive-negative, respectively. Moreover, under the assumptions of on–off or sign-changing dampings, we derive the asymptotic stability of solutions.

Nearly perfectly matched layer absorber for viscoelastic wave equations

We have applied the nearly perfectly matched layer (N-PML) absorber to the viscoelastic wave equation based on the Kelvin-Voigt and Zener constitutive equations. In the first case, the stress-strain relation has the advantage of not requiring additional physical field (memory) variables, whereas the Zener model is more adapted to describe the behavior of rocks subject to wave propagation in the whole frequency range. In both cases, eight N-PML artificial memory variables are required in the absorbing strips. The modeling simulates 2D waves by using two different approaches to compute the spatial derivatives, generating different artifacts from the boundaries, namely, 16th-order finite differences, where reflections from the boundaries are expected, and the staggered Fourier pseudospectral method, where wraparound occurs. The time stepping in both cases is a staggered second-order finite-difference scheme. Numerical experiments demonstrate that the N-PML has a similar performance as in the lossless case. Comparisons with other approaches (S-PML and C-PML) are carried out for several models, which indicate the advantages and drawbacks of the N-PML absorber in the anelastic case.