The main aim of this paper is to study the approximation to viscoelasticity type equations with a Crouzeix-Raviart type nonconforming finite element on the anisotropic meshes. The superclose property of the exact solu...The main aim of this paper is to study the approximation to viscoelasticity type equations with a Crouzeix-Raviart type nonconforming finite element on the anisotropic meshes. The superclose property of the exact solution and the optimal error estimate of its derivative with respect to time are derived by using some novel techniques. Moreover, employing a postprocessing technique, the global superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself is studied.展开更多
1 INTRODUCTIONIt is well known that viscoelasticity and thixotropy of fluids are often investigated by measuringhysteresis loops under transient conditions.For some fluids(such as blood),it has been provedthat,an“8”...1 INTRODUCTIONIt is well known that viscoelasticity and thixotropy of fluids are often investigated by measuringhysteresis loops under transient conditions.For some fluids(such as blood),it has been provedthat,an“8”shaped hysteresis loop may be obtained if triangular steps of shear rate wereapplied on it(Fig.l).This kind of hysteresis loop is composed of a viscoelastic loop and athixotropic loop.The viscoelastic and thixotropic properties are unified in the formation ofthe hysteresis loop.How to describe the formation of this complex hysteresis loop and how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
A variable coefficient viscoelastic equation with a time-varying delay in the boundary feedback and acoustic boundary conditions and nonlinear source term is considered.Under suitable assumptions, general decay result...A variable coefficient viscoelastic equation with a time-varying delay in the boundary feedback and acoustic boundary conditions and nonlinear source term is considered.Under suitable assumptions, general decay results of the energy are established via suitable Lyapunov functionals and some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term and the elements of the matrix A and the kernel function g.展开更多
In this paper, we consider a system of coupled quasilinear viscoelastic equa- tions with nonlinear damping. We use the perturbed energy method to show the general decay rate estimates of energy of solutions, which ext...In this paper, we consider a system of coupled quasilinear viscoelastic equa- tions with nonlinear damping. We use the perturbed energy method to show the general decay rate estimates of energy of solutions, which extends some existing results concerning a general decay for a single equation to the case of system, and a nonlinear system of viscoelastic wave equations to a quasilinear system.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
A viscoelastic equation with Balakrishnan-Taylor damping and nonlinear boundary/interior sources is considered in a bounded domain. Under appropriate assumptions ira- posed on the source and the damping, we establish ...A viscoelastic equation with Balakrishnan-Taylor damping and nonlinear boundary/interior sources is considered in a bounded domain. Under appropriate assumptions ira- posed on the source and the damping, we establish uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function.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
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
In this paper, we consider an initial-boundary value problem for a nonlinear viscoelastic wave equation with strong damping, nonlinear damping and source terms. We proved a blow up result for the solution with negativ...In this paper, we consider an initial-boundary value problem for a nonlinear viscoelastic wave equation with strong damping, nonlinear damping and source terms. We proved a blow up result for the solution with negative initial energy if p > m, and a global result for p ≤ m.展开更多
This paper deals with a class of nonlinear viscoelastic wave equation with damping and source terms utt-Δu-Δut-Δutt+∫^t0g(t-s)Δu(s)ds+ut|ut|^m2-=u|u|^p-2 with acoustic boundary conditions.Under some appropriate a...This paper deals with a class of nonlinear viscoelastic wave equation with damping and source terms utt-Δu-Δut-Δutt+∫^t0g(t-s)Δu(s)ds+ut|ut|^m2-=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.展开更多
Let G he a hounded domain in E Consider the following quasi-linear elliptic equationAlthough the houndedness of generalized solutions of the equation is proved for very general structural conditions, it does not suppl...Let G he a hounded domain in E Consider the following quasi-linear elliptic equationAlthough the houndedness of generalized solutions of the equation is proved for very general structural conditions, it does not supply a priori estimate for maximum modulus of solutions. In this paper an estimate to the maximum modulus is made firstly for a special case of quasi-linear elliptic equations, i.e. the A and B satisfy the following structural conditions展开更多
A constitutive equation theory of Oldroyd fluid B type,i.e.the co-rotational derivative type,is developed for the anisotropic-viscoelastic fluid of liquid crystalline(LC)polymer.Analyzing the influence of the orientat...A constitutive equation theory of Oldroyd fluid B type,i.e.the co-rotational derivative type,is developed for the anisotropic-viscoelastic fluid of liquid crystalline(LC)polymer.Analyzing the influence of the orientational motion on the material behavior and neglecting the influence,the constitutive equation is applied to a simple case for the hydrodynamic motion when the orientational contribution is neglected in it and the anisotropic relaxation,retardation times and anisotropic viscosi- ties are introduced to describe the macroscopic behavior of the anisotropic LC polymer fluid.Using the equation for the shear flow of LC polymer fluid,the analytical expressions of the apparent viscosity and the normal stress differences are given which are in a good agreement with the experimental results of Baek et al.For the fiber spinning flow of the fluid,the analytical expression of the extensional viscosity is given.展开更多
In this paper an equation of motion is presented for a general thick viscoelastic plate, including the effects of shear deformation, extrusion deformation and rotatory inertia. This equation is the generalization of e...In this paper an equation of motion is presented for a general thick viscoelastic plate, including the effects of shear deformation, extrusion deformation and rotatory inertia. This equation is the generalization of equations of motion for the corresponding thick elastic plate, and it can be degenerated into several types of equations for various special cases.展开更多
This paper is concerned with a nonlinear viscoelastic equation with strong damping: . The objective of the present paper is to provide some results on the long-time behavior to this equation with acoustic boundary con...This paper is concerned with a nonlinear viscoelastic equation with strong damping: . The objective of the present paper is to provide some results on the long-time behavior to this equation with acoustic boundary conditions. By using the assumptions on the relaxation function due to Tatar [1], we show an arbitrary rate of decay with not necessary of an exponential or polynomial one and without the assumption condition. The result extends and improves some results given in Cavalcanti [2].展开更多
This paper deals with the singular perturbation of the boundary value problem of the systems for quasi-linear ordinary differential equationswhere x,f, y , h, A, B and C all belong to Rn , and g is an n×n matrix ...This paper deals with the singular perturbation of the boundary value problem of the systems for quasi-linear ordinary differential equationswhere x,f, y , h, A, B and C all belong to Rn , and g is an n×n matrix function. Under suitable conditions we prove the existence of the solutions by diagonalization and the fixed point theorem and also estimate the remainder.展开更多
This paper is concerned with the quasi-linear equation with critical Sobolev-Hardy exponentwhere Ω(?) RN(N(?)3) is a smooth bounded domain, 0∈Ω,0(?)s<p, 1<p< N,p(s) := p(N-s)/N-p is the critical Sobolev-Ha...This paper is concerned with the quasi-linear equation with critical Sobolev-Hardy exponentwhere Ω(?) RN(N(?)3) is a smooth bounded domain, 0∈Ω,0(?)s<p, 1<p< N,p(s) := p(N-s)/N-p is the critical Sobolev-Hardy exponent, λ>0,p(?)r < p ,p := Np/N-p is the critical Sobolev exponent, μ>, 0(?)t < p,p (?) q < p (t) =p(N-t)/N-p. The existence of a positive solution is proved by Sobolev-Hardy inequality and variational method.展开更多
The constant Q property in viscoelastic media assumes that the quality factor Q does not change with frequency(i.e.,the Q value is independent of the frequency).For seismic waves propagating in viscoelastic media,the ...The constant Q property in viscoelastic media assumes that the quality factor Q does not change with frequency(i.e.,the Q value is independent of the frequency).For seismic waves propagating in viscoelastic media,the wave equation is determined by the viscoelastic media model.Equivalence relations exist between various frequency domain mathematical models and physical rheological models for the constant Q property.Considering two elastic moduli and three attenuation variables,24 kinds of wave equations based on diff erent generalized rheological models are divided into six classes in this study,and the 12 kinds of specifi c representation for the wave equations in the time domain are derived.On the basis of the equivalence relations between the generalized rheological models,the diff erence and equivalence relation between diff erent wave equations are proven and clarifi ed.Results show that the high-order generalized rheological model can accurately characterize the attenuation characteristics of seismic waves and has advantages in characterizing the dispersion characteristics in viscoelastic media.Lastly,the seismic refl ection characteristics caused by the diff erence of Q value are verifi ed by the forward modeling of the constant Q wave equation in this study,thereby providing a theoretical basis for the analysis and inversion of the formation Q value from refl ection seismic data.展开更多
基金This research is supported by the NSF of China (10371113 10471133),SF of Henan ProvinceSF of Education Committee of Henan Province (2006110011)
文摘The main aim of this paper is to study the approximation to viscoelasticity type equations with a Crouzeix-Raviart type nonconforming finite element on the anisotropic meshes. The superclose property of the exact solution and the optimal error estimate of its derivative with respect to time are derived by using some novel techniques. Moreover, employing a postprocessing technique, the global superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself is studied.
基金Supported by NNSF(10601022,11061021)Supported by NSF of Inner Mongolia Au-tonomous Region(200607010106)Supported by SRP of Higher Schools of Inner Mongolia(NJ10006)
文摘1 INTRODUCTIONIt is well known that viscoelasticity and thixotropy of fluids are often investigated by measuringhysteresis loops under transient conditions.For some fluids(such as blood),it has been provedthat,an“8”shaped hysteresis loop may be obtained if triangular steps of shear rate wereapplied on it(Fig.l).This kind of hysteresis loop is composed of a viscoelastic loop and athixotropic loop.The viscoelastic and thixotropic properties are unified in the formation ofthe hysteresis loop.How to describe the formation of this complex hysteresis loop and how
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.
文摘A variable coefficient viscoelastic equation with a time-varying delay in the boundary feedback and acoustic boundary conditions and nonlinear source term is considered.Under suitable assumptions, general decay results of the energy are established via suitable Lyapunov functionals and some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term and the elements of the matrix A and the kernel function g.
基金supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education,Science and Technology (2011-0007870)
文摘In this paper, we consider a system of coupled quasilinear viscoelastic equa- tions with nonlinear damping. We use the perturbed energy method to show the general decay rate estimates of energy of solutions, which extends some existing results concerning a general decay for a single equation to the case of system, and a nonlinear system of viscoelastic wave equations to a quasilinear system.
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A viscoelastic equation with Balakrishnan-Taylor damping and nonlinear boundary/interior sources is considered in a bounded domain. Under appropriate assumptions ira- posed on the source and the damping, we establish uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function.
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.
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In this paper, we consider an initial-boundary value problem for a nonlinear viscoelastic wave equation with strong damping, nonlinear damping and source terms. We proved a blow up result for the solution with negative initial energy if p > m, and a global result for p ≤ m.
基金supported by the NSF of China(11626070,11801108)the Scientific Program of Guangdong Provience(2016A030310262)+1 种基金the College Scientific Research Project of Guangzhou City(1201630180)the Program for the Innovation Research Grant for the Postgraduates of Guangzhou University(2017GDJC-D08)。
文摘This paper deals with a class of nonlinear viscoelastic wave equation with damping and source terms utt-Δu-Δut-Δutt+∫^t0g(t-s)Δu(s)ds+ut|ut|^m2-=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.
文摘Let G he a hounded domain in E Consider the following quasi-linear elliptic equationAlthough the houndedness of generalized solutions of the equation is proved for very general structural conditions, it does not supply a priori estimate for maximum modulus of solutions. In this paper an estimate to the maximum modulus is made firstly for a special case of quasi-linear elliptic equations, i.e. the A and B satisfy the following structural conditions
基金The project supported by the National Natural Science Foundation of China(19832050 and 10372100)
文摘A constitutive equation theory of Oldroyd fluid B type,i.e.the co-rotational derivative type,is developed for the anisotropic-viscoelastic fluid of liquid crystalline(LC)polymer.Analyzing the influence of the orientational motion on the material behavior and neglecting the influence,the constitutive equation is applied to a simple case for the hydrodynamic motion when the orientational contribution is neglected in it and the anisotropic relaxation,retardation times and anisotropic viscosi- ties are introduced to describe the macroscopic behavior of the anisotropic LC polymer fluid.Using the equation for the shear flow of LC polymer fluid,the analytical expressions of the apparent viscosity and the normal stress differences are given which are in a good agreement with the experimental results of Baek et al.For the fiber spinning flow of the fluid,the analytical expression of the extensional viscosity is given.
文摘In this paper an equation of motion is presented for a general thick viscoelastic plate, including the effects of shear deformation, extrusion deformation and rotatory inertia. This equation is the generalization of equations of motion for the corresponding thick elastic plate, and it can be degenerated into several types of equations for various special cases.
文摘This paper is concerned with a nonlinear viscoelastic equation with strong damping: . The objective of the present paper is to provide some results on the long-time behavior to this equation with acoustic boundary conditions. By using the assumptions on the relaxation function due to Tatar [1], we show an arbitrary rate of decay with not necessary of an exponential or polynomial one and without the assumption condition. The result extends and improves some results given in Cavalcanti [2].
文摘This paper deals with the singular perturbation of the boundary value problem of the systems for quasi-linear ordinary differential equationswhere x,f, y , h, A, B and C all belong to Rn , and g is an n×n matrix function. Under suitable conditions we prove the existence of the solutions by diagonalization and the fixed point theorem and also estimate the remainder.
基金This research is supported by the National Natural Science Foundation of China(l0171036) and the Natural Science Foundation of South-Central University For Nationalities(YZZ03001).
文摘This paper is concerned with the quasi-linear equation with critical Sobolev-Hardy exponentwhere Ω(?) RN(N(?)3) is a smooth bounded domain, 0∈Ω,0(?)s<p, 1<p< N,p(s) := p(N-s)/N-p is the critical Sobolev-Hardy exponent, λ>0,p(?)r < p ,p := Np/N-p is the critical Sobolev exponent, μ>, 0(?)t < p,p (?) q < p (t) =p(N-t)/N-p. The existence of a positive solution is proved by Sobolev-Hardy inequality and variational method.
基金This work was supported by National Natural Science Foundation of China(No.41774137)111 project(No.B18055),and the Fundamental Research Funds for the Central Universities(No.19CX02002A).
文摘The constant Q property in viscoelastic media assumes that the quality factor Q does not change with frequency(i.e.,the Q value is independent of the frequency).For seismic waves propagating in viscoelastic media,the wave equation is determined by the viscoelastic media model.Equivalence relations exist between various frequency domain mathematical models and physical rheological models for the constant Q property.Considering two elastic moduli and three attenuation variables,24 kinds of wave equations based on diff erent generalized rheological models are divided into six classes in this study,and the 12 kinds of specifi c representation for the wave equations in the time domain are derived.On the basis of the equivalence relations between the generalized rheological models,the diff erence and equivalence relation between diff erent wave equations are proven and clarifi ed.Results show that the high-order generalized rheological model can accurately characterize the attenuation characteristics of seismic waves and has advantages in characterizing the dispersion characteristics in viscoelastic media.Lastly,the seismic refl ection characteristics caused by the diff erence of Q value are verifi ed by the forward modeling of the constant Q wave equation in this study,thereby providing a theoretical basis for the analysis and inversion of the formation Q value from refl ection seismic data.