Viscoelastic Wave Equation Research Articles

On a Kelvin--Voigt Viscoelastic Wave Equation with Strong Delay

An initial-boundary value problem for a viscoelastic wave equation subject to a strong time-localized delay in a Kelvin--Voigt-type material law is considered. After transforming the equation to an abstract Cauchy problem on the extended phase space, a global well-posedness theory is established using the operator semigroup theory both in Sobolev-valued $C^{0}$- and $\mathrm{BV}$-spaces. Under appropriate assumptions on the coefficients, a global exponential decay rate is obtained and the stability region in the parameter space is further explored using Lyapunov's indirect method. The singular limit $\tau \to 0$ is studied with the aid of the energy method. Finally, a numerical example from a real-world application in biomechanics is presented.

A Crank–Nicolson collocation spectral method for the two‐dimensional viscoelastic wave equation

In this paper, we first establish the Crank–Nicolson collocation spectral (CNCS) method for two‐dimensional (2D) viscoelastic wave equation by means of the Chebyshev polynomials. And then, we analyze the existence, uniqueness, stability, and convergence of the CNCS solutions. Finally, we use some numerical experiments to verify the correctness of theoretical analysis. This implies that the CNCS model is very effective for solving the 2D viscoelastic wave equations.

Seismic wave propagation in nonlinear viscoelastic media using the auxiliary differential equation method

In previous studies, the auxiliary differential equation (ADE) method has been applied to 2-D seismic-wave propagation modelling in viscoelastic media. This method is based on the separation of the wave propagation equations derived from the constitutive law defining the stress–strain relation. We make here a 3-D extension of a finite-difference (FD) scheme to solve a system of separated equations consisting in the stress–strain rheological relation, the strain–velocity and the velocity–stress equations. The current 3-D FD scheme consists in the discretization of the second order formulation of a non-linear viscoelastic wave equation with a time actualization of the velocity and displacement fields. Compared to the usual memory variable formalism, the ADE method allows flexible implementation of complex expressions of the desired rheological model such as attenuation/viscoelastic models or even non-linear behaviours, with physical parameters that can be provided from dispersion analysis. The method can also be associated with optimized perfectly matched layers-based boundary conditions that can be seen as additional attenuation (viscoelastic) terms. We present the results obtained for a non-linear viscoelastic model made of a Zener viscoelastic body associated with a non-linear quadratic strain term. Such non-linearity is relevant to define unconsolidated granular model behaviour. Thanks to a simple model, but without loss of generality, we demonstrate the accuracy of the proposed numerical approach.

General decay of solutions to a viscoelastic wave equation with linear damping, nonlinear damping and source term

ABSTRACTThis paper is concerned with the decay property of a nonlinear viscoelastic wave equation with linear damping, nonlinear damping and source term. Under weaker assumption on the relaxation function, we establish a general decay result, which extends the result obtained in Messaoudi [Exponential decay of solutions of a nonlinearly damped wave equation. Nodea-Nonlinear Differ Equat Appl. 2005;12:391–399].

Blow up and asymptotic behavior for a system of viscoelastic wave equations of Kirchhoff type with a delay term

The focus of the current paper is to investigate the initial boundary value problem for a system of viscoelastic wave equations of Kirchho type with a delay term in a bounded domain. At first,the energy decay rate is proved by Nakao's technique and expressed polynomially and exponentially depending on the parameter m. However and in the unstable set, for certain initial data, the blow-upof solutions is obtained.

General decay and blow-up of solutions for a nonlinear viscoelastic wave equation with strong damping

This article is concerned with the decay and blow-up properties of a nonlinear viscoelastic wave equation with strong damping. We first show a local existence theorem. Then, we prove the global existence of solutions and establish a general decay rate estimate. Finally, we show the finite time blow-up result for some solutions with negative initial energy and positive initial energy.

A global nonexistence of solutions for a quasilinear viscoelastic wave equation with acoustic boundary conditions

In this paper, we consider a quasilinear viscoelastic wave equation with acoustic boundary conditions. Under some appropriate assumption on the relaxation function g, the function Φ, p > max { rho +2, m, q,2}, and the initial data, we prove a global nonexistence of solutions for a quasilinear viscoelastic wave equation with positive initial energy.

An efficient two-level preconditioner for multi-frequency wave propagation problems

We consider wave propagation problems that are modeled in the frequency-domain, and that need to be solved simultaneously for multiple frequencies within a fixed range. For this, a single shift-and-invert preconditioner at a so-called seed frequency is applied. The choice of the seed is crucial for the performance of preconditioned multi-shift GMRES and is closely related to the parameter choice for the Complex Shifted Laplace preconditioner. Based on a classical GMRES convergence bound, we present an analytic formula for the optimal seed parameter that purely depends on the original frequency range. The new insight is exploited in a two-level preconditioning strategy: A shifted Neumann preconditioner with minimized spectral radius is additionally applied to multi-shift GMRES. Moreover, we present a reformulation of the multi-shift problem to a matrix equation solved with, for instance, global GMRES. Here, our analysis allows for rotation of the spectrum of the linear operator. Numerical experiments for the time-harmonic visco-elastic wave equation demonstrate the performance of the new preconditioners.

A stable approach for Q-compensated viscoelastic reverse time migration using excitation amplitude imaging condition

The earth [Formula: see text] filtering causes poor illumination of the subsurface. Compensating for the attenuated amplitude and distorted phase is crucial during elastic reverse time migration (ERTM) to improve the imaging quality. Conventional [Formula: see text]-compensated ERTM ([Formula: see text]-ERTM) methods tend to boost the attenuated energy to inverse the [Formula: see text] effects. These methods usually suffer from severe numerical instability because of the unlimited amplification of the high-frequency noise. Low-pass filtering is generally used to stabilize the process, however, at the expense of precision. We have developed a stable compensation approach in this paper. Based on the decoupled fractional Laplacians viscoelastic wave equation, two compensation operators are obtained by extrapolating wavefield in the dispersion-only and viscoelastic media. Because no explicit amplification is included, these two operators are absolutely stable for implementation. To improve the division morbidity for calculating the compensation operators, we use the excitation amplitude criterion and embed the operators into a vector-based [Formula: see text]-compensated excitation amplitude imaging condition. With the derived imaging condition, we could compensate for the absorption accurately without needing to concern the stability issue. The [Formula: see text]-ERTM results for noise-free data are carried out over a simple layered model and a more realistic Marmousi model with an attenuating area to verify the feasibility of the proposed approach. The migration results for noisy data from the Marmousi model further prove that the proposed method performs better fidelity, adaptability, and antinoise performance compared with conventional compensation method with low-pass filtering.

Magnetic resonance elastography in nonlinear viscoelastic materials under load

Characterisation of soft tissue mechanical properties is a topic of increasing interest in translational and clinical research. Magnetic resonance elastography (MRE) has been used in this context to assess the mechanical properties of tissues in vivo noninvasively. Typically, these analyses rely on linear viscoelastic wave equations to assess material properties from measured wave dynamics. However, deformations that occur in some tissues (e.g. liver during respiration, heart during the cardiac cycle, or external compression during a breast exam) can yield loading bias, complicating the interpretation of tissue stiffness from MRE measurements. In this paper, it is shown how combined knowledge of a material’s rheology and loading state can be used to eliminate loading bias and enable interpretation of intrinsic (unloaded) stiffness properties. Equations are derived utilising perturbation theory and Cauchy’s equations of motion to demonstrate the impact of loading state on periodic steady-state wave behaviour in nonlinear viscoelastic materials. These equations demonstrate how loading bias yields apparent material stiffening, softening and anisotropy. MRE sensitivity to deformation is demonstrated in an experimental phantom, showing a loading bias of up to twofold. From an unbiased stiffness of 4910.4 pm 635.8 Pa in unloaded state, the biased stiffness increases to 9767.5 pm ,1949.9 Pa under a load of approx 34% uniaxial compression. Integrating knowledge of phantom loading and rheology into a novel MRE reconstruction, it is shown that it is possible to characterise intrinsic material characteristics, eliminating the loading bias from MRE data. The framework introduced and demonstrated in phantoms illustrates a pathway that can be translated and applied to MRE in complex deforming tissues. This would contribute to a better assessment of material properties in soft tissues employing elastography.

Blow up and asymptotic behavior for a system of viscoelastic wave equations of Kirchhoff type with a delay term

Blow up and asymptotic behavior for a system of viscoelastic wave equations of Kirchhoff type with a delay term

Global nonexistence of solutions for viscoelastic wave equation with delay

In this work, we consider a viscoelastic wave equation with delay urn:x-wiley:mma:media:mma5194:mma5194-math-0001 with Dirichlet boundary condition. There are much literature on the blow‐up result of solutions for the wave equation. However, to the best of our knowledge, there is no blow‐up result of solutions for the viscoelastic wave equation with delay. We study a finite time blow‐up result of solutions with nonpositive initial energy as well as positive initial energy.

Exponential Stability for a Transmission Problem of a Viscoelastic Wave Equation

This paper is concerned with the study of a transmission problem of viscoelastic waves with hereditary memory, establishing the existence, uniqueness and exponential stability for the solutions of this problem. The proof of the stabilization result combines energy estimates and results due to Gerard (Commun Partial Differ Equ 16:1761–1794 (1991)) on microlocal defect measures.

Energy decay of solutions for viscoelastic wave equations with a dynamic boundary and delay term

Energy decay of solutions for viscoelastic wave equations with a dynamic boundary and delay term

Stability result for viscoelastic wave equation with dynamic boundary conditions

In this paper we consider wave viscoelastic equation with dynamic boundary condition in a bounded domain, we establish a general decay result of energy by exploiting the frequency domain method which consists in combining a contradiction argument and a special analysis for the resolvent of the operator of interest with assumptions on past history relaxation function.

Energy decay of a viscoelastic wave equation with supercritical nonlinearities

This paper presents a study of the asymptotic behavior of the solutions for the history value problem of a viscoelastic wave equation which features a fading memory term as well as a supercritical source term and a frictional damping term: $$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt}- k(0) \Delta u - \int \limits _0^{\infty } k'(s) \Delta u(t-s) \hbox {d}s +|u_t|^{m-1}u_t =|u|^{p-1}u, \text { in } \Omega \times (0,T), \\ u(x,t)=u_0(x,t), \quad \text { in } \Omega \times (-\infty ,0], \end{array}\right. } \end{aligned}$$ where $$\Omega $$ is a bounded domain in $$\mathbb R^3$$ with a Dirichlét boundary condition and $$u_0$$ represents the history value. A suitable notion of a potential well is introduced for the system, and global existence of solutions is justified, provided that the history value $$u_0$$ is taken from a subset of the potential well. Also, uniform energy decay rate is obtained which depends on the relaxation kernel $$-k'(s)$$ as well as the growth rate of the damping term. This manuscript complements our previous work (Guo et al. in J Differ Equ 257:3778–3812, 2014, J Differ Equ 262:1956–1979, 2017) where Hadamard well-posedness and the singularity formulation have been studied for the system. It is worth stressing the special features of the model, namely the source term here has a supercritical growth rate and the memory term accounts to the full past history that goes back to $$-\infty $$ .

General Decay Rates for a Viscoelastic Wave Equation with Dynamic Boundary Conditions and Past History

In this paper, we study a viscoelastic wave equation with dynamic boundary conditions, source term and a nonlinear weak damping localized on a part of the boundary and past history. Under suitable assumptions, we establish an explicit and general decay result of energy by introducing suitable energy and perturbed Lyapunov functionals and some properties of convex functions.

Simulating Seismic Wave Propagation in Viscoelastic Media with an Irregular Free Surface

In seismic numerical simulations of wave propagation, it is very important for us to consider surface topography and attenuation, which both have large effects (e.g., wave diffractions, conversion, amplitude/phase change) on seismic imaging and inversion. An irregular free surface provides significant information for interpreting the characteristics of seismic wave propagation in areas with rugged or rapidly varying topography, and viscoelastic media are a better representation of the earth’s properties than acoustic/elastic media. In this study, we develop an approach for seismic wavefield simulation in 2D viscoelastic isotropic media with an irregular free surface. Based on the boundary-conforming grid method, the 2D time-domain second-order viscoelastic isotropic equations and irregular free surface boundary conditions are transferred from a Cartesian coordinate system to a curvilinear coordinate system. Finite difference operators with second-order accuracy are applied to discretize the viscoelastic wave equations and the irregular free surface in the curvilinear coordinate system. In addition, we select the convolutional perfectly matched layer boundary condition in order to effectively suppress artificial reflections from the edges of the model. The snapshot and seismogram results from numerical tests show that our algorithm successfully simulates seismic wavefields (e.g., P-wave, Rayleigh wave and converted waves) in viscoelastic isotropic media with an irregular free surface.

Blow-up of solutions for a nonlinear viscoelastic wave equation with initial data at arbitrary energy level

ABSTRACTIn this paper, we study a three-dimensional (3D) viscoelastic wave equation with nonlinear weak damping, supercritical sources and prescribed past history , : where the relaxation function k is monotone decreasing with , and . When the source is stronger than dissipations, i.e. , we obtain some finite time blow-up results with positive initial energy. In particular, we obtain the existence of certain solutions which blow up in finite time for initial data at arbitrary energy level.

Stochastic quasilinear viscoelastic wave equation with degenerate damping and source terms

In this paper, we consider a stochastic quasilinear viscoelastic wave equation with degenerate damping and source term. We prove the blow-up of solution for stochastic quasilinear viscoelastic wave equation with positive probability or explosive in energy sense.