Rayleigh wave is an interference wave of longitudinal wave and transverse wave which propagate along the free surface of solids. There remains a dispute about the number of Rayleigh waves in viscoelastic media until n...Rayleigh wave is an interference wave of longitudinal wave and transverse wave which propagate along the free surface of solids. There remains a dispute about the number of Rayleigh waves in viscoelastic media until now, which is an essential problem of Rayleigh wave propagation. The purpose of this study is to propose a brief way of handling this essential problem within half-space Kelvin viscoelastic media. Starting from the dynamic equations of transverse wave and longitudinal wave based on Kelvin viscoelastic model, this study sets the complex wave number as a variable, introduces complex moduli and complex exponential factors, then a characteristic equation of Kelvin viscoelastic Rayleigh wave in half space is derived and simplified support for analysis of its uniqueness. After reviewing mathematical models describing phenomena of having multiple solutions but uniqueness when a natural condition is taken into account, a conjecture is given that the Rayleigh wave in Kelvin viscoelastic media must be unique if we assume a natural condition in accordance with the natural phenomena.展开更多
The nonlinear viscoelastic wave equation ｜μt｜^pμtt-△μ-μutt＋∫^t0g（t-s）△μ（s）ds＋｜μ｜^pU=0,in a bounded domain with initial conditions and Dirichlet boundary conditions is consid- ered. We prove that, fo...The nonlinear viscoelastic wave equation ｜μt｜^pμtt-△μ-μutt＋∫^t0g（t-s）△μ（s）ds＋｜μ｜^pU=0,in a bounded domain with initial conditions and Dirichlet boundary conditions is consid- ered. We prove that, for a class of kernels 9 which is singular at zero, the exponential decay rate of the solution energy. The result is obtained by introducing an appropriate Lyapounov functional and using energy method similar to the work of Tatar in 2009. This work improves earlier results.展开更多
In this paper, we consider the following viscoelastic wave equation with delay|u_t|~ρu_(tt)-△u-△u_(tt)+∫_0~t g(t-8)△u(8)d8 + μ_1 u_t(x,t) + μ_2 u_t(x,t-τ) = b|u|^(p-2) u in a bounded domain. Under appropriate ...In this paper, we consider the following viscoelastic wave equation with delay|u_t|~ρu_(tt)-△u-△u_(tt)+∫_0~t g(t-8)△u(8)d8 + μ_1 u_t(x,t) + μ_2 u_t(x,t-τ) = b|u|^(p-2) u in a bounded domain. Under appropriate conditions on μ1, μ2, the kernel function g, the nonlinear source and the initial data, there are solutions that blow up in finite time.展开更多
The authors study decay properties of solutions for a viscoelastic wave equation with variable coefficients and a nonlinear boundary damping by the differential geometric approach.
This study develops an optimized finite difference iterative （OFDI） scheme for the two-dimensional （2D） viscoelastic wave equation. The OFDI scheme is obtained using a proper orthogonal decomposition （POD） metho...This study develops an optimized finite difference iterative （OFDI） scheme for the two-dimensional （2D） viscoelastic wave equation. The OFDI scheme is obtained using a proper orthogonal decomposition （POD） method. It has sufficiently high accuracy with very few unknowns for the 2D viscoelastic wave equation. Existence, stability, and convergence of the OFDI solutions are analyzed. Numerical simulations verify efficiency and feasibility of the proposed scheme.展开更多
The stability problem is a very important aspect in seismic wave numerical modeling. Based on the theory of seismic waves and constitutive equations of viscoelastic models, the stability problems of finite difference ...The stability problem is a very important aspect in seismic wave numerical modeling. Based on the theory of seismic waves and constitutive equations of viscoelastic models, the stability problems of finite difference scheme for Kelvin- Voigt and Maxwell models with rectangular grids are analyzed. Expressions of stability conditions with arbitrary spatial accuracies for two viscoelastic models are derived. With approximation of quality factor Q≥5, simplified expressions are developed and some numerical models are given to verify the validity of the corresponding theoretical results. Then this paper summarizes the influences of seismic wave velocity, frequency, size of grid and difference coefficients, as well as quality factor on stability condition. Finally the prerequisite conditions of the simplified stability equations are given with error analysis.展开更多
文摘Rayleigh wave is an interference wave of longitudinal wave and transverse wave which propagate along the free surface of solids. There remains a dispute about the number of Rayleigh waves in viscoelastic media until now, which is an essential problem of Rayleigh wave propagation. The purpose of this study is to propose a brief way of handling this essential problem within half-space Kelvin viscoelastic media. Starting from the dynamic equations of transverse wave and longitudinal wave based on Kelvin viscoelastic model, this study sets the complex wave number as a variable, introduces complex moduli and complex exponential factors, then a characteristic equation of Kelvin viscoelastic Rayleigh wave in half space is derived and simplified support for analysis of its uniqueness. After reviewing mathematical models describing phenomena of having multiple solutions but uniqueness when a natural condition is taken into account, a conjecture is given that the Rayleigh wave in Kelvin viscoelastic media must be unique if we assume a natural condition in accordance with the natural phenomena.
文摘The nonlinear viscoelastic wave equation ｜μt｜^pμtt-△μ-μutt＋∫^t0g（t-s）△μ（s）ds＋｜μ｜^pU=0,in a bounded domain with initial conditions and Dirichlet boundary conditions is consid- ered. We prove that, for a class of kernels 9 which is singular at zero, the exponential decay rate of the solution energy. The result is obtained by introducing an appropriate Lyapounov functional and using energy method similar to the work of Tatar in 2009. This work improves earlier results.
文摘In this paper, we consider the following viscoelastic wave equation with delay|u_t|~ρu_(tt)-△u-△u_(tt)+∫_0~t g(t-8)△u(8)d8 + μ_1 u_t(x,t) + μ_2 u_t(x,t-τ) = b|u|^(p-2) u in a bounded domain. Under appropriate conditions on μ1, μ2, the kernel function g, the nonlinear source and the initial data, there are solutions that blow up in finite time.
基金supported by the National Science Foundation of China under Grant Nos.60225003,60334040,60221301,60774025,10831007,61104129,11171195the Excellent PhD Adviser Program of Beijing under Grant No.YB20098000101
文摘The authors study decay properties of solutions for a viscoelastic wave equation with variable coefficients and a nonlinear boundary damping by the differential geometric approach.
基金Project supported by the National Natural Science Foundation of China(No.11671106)the Fundamental Research Funds for the Central Universities(No.2016MS33)
文摘This study develops an optimized finite difference iterative （OFDI） scheme for the two-dimensional （2D） viscoelastic wave equation. The OFDI scheme is obtained using a proper orthogonal decomposition （POD） method. It has sufficiently high accuracy with very few unknowns for the 2D viscoelastic wave equation. Existence, stability, and convergence of the OFDI solutions are analyzed. Numerical simulations verify efficiency and feasibility of the proposed scheme.
文摘The stability problem is a very important aspect in seismic wave numerical modeling. Based on the theory of seismic waves and constitutive equations of viscoelastic models, the stability problems of finite difference scheme for Kelvin- Voigt and Maxwell models with rectangular grids are analyzed. Expressions of stability conditions with arbitrary spatial accuracies for two viscoelastic models are derived. With approximation of quality factor Q≥5, simplified expressions are developed and some numerical models are given to verify the validity of the corresponding theoretical results. Then this paper summarizes the influences of seismic wave velocity, frequency, size of grid and difference coefficients, as well as quality factor on stability condition. Finally the prerequisite conditions of the simplified stability equations are given with error analysis.
