Global attractor for the nonlinear strain waves in elastic waveguides

1. IntroductionIn some problems of nonlinear wave propagation in waveguides, the illteraction of waveguides and the external medium and, therefore, the possibility of energy exchange throughlateral surface of waveguide cannot be neglected. When the energy exchange between therod and the medium is considered, for one cajse, there is a dissipation of deformation wavein the viscous external medium. The general cubic double dispersion equation (CDDE) canbe derived from Hamilton principled]:where…     

Two-Way Interactions Between Equatorially-Trapped Waves and the Barotropic Flow

Lateral energy exchange between the tropics and the midlatitudes is a topic of great importance for understanding Earth＇s climate system. In this paper, the authors address this issue in an idealized set up through simple shallow water models for the interactions between equatorially trapped waves and the barotropic mode, which supports Rossby waves that propagate poleward and can excite midlatitude teleconnection patterns. It is found here that the interactions between a Kelvin wave and a fixed meridionai shear （mimicking the jet stream） generates a non-trivial meridional velocity and meridional convergence in phase with the upward motion that can attain a maximum of about 50%, which oscillates on frequencies ranging from one day to 10 days. When, on the other hand, the barotropic flow is forced by slowly propagating Kelvin waves a complex flow pattern emerges; it consists of a phase-locked barotropic response that is equatoriaily trapped and that propagates eastward with the forcing Kelvin wave and a certain number of planetary Rossby waves that propagate westward and toward the poles as seen in nature. It is suggested here that the poleward propagating waves are to some sort of multi-way resonant interaction with the phase locked response. Moreover, it is shown here that a numerical scheme with dispersion properties that depend on the direction perpendicular to the direction of propagation, namely the 2D central scheme of Nessyahu and Tadmor, can artificially alter significantly the topology of the wave fields and thus should be avoided in climate models.     

COUPLING OF FINITE ELEMENT AND BOUNDARY ELEMENT METHODS FOR THE SCATTERING BY PERIODIC CHIRAL STRUCTURES

Consider a time-harmonic electromagnetic plane wave incident on a biperiodic structure in R^3. The periodic structure separates two homogeneous regions. The medium inside the structure is chiral and nonhomogeneous. In this paper, variational formulations coupling finite element methods in the chiral medium with a method of integral equations on the periodic interfaces are studied. The well-posedness of the continuous and discretized problems is established. Uniform convergence for the coupling variational approximations of the model problem is obtained.     

Determining a Multi-layered Fluid-solid Medium from the Acoustic Measurements

Consider the inverse problem of recovering a multi-layered fluid-solid medium from many acoustic measurements corresponding to time-harmonic acoustic plane waves.We prove that both the supports of the embedded solid obstacle and the surrounding layered fluid medium can be uniquely identified by means of acoustic far-field pattern for all incident wave fields at a fixed frequency.Our proof is based on the constructions of some well-posed partial differential equation systems in sufficiently small domains combined with the a priori estimates for the solutions of the forward scattering problem.     

A Time-dependent Interaction Problem Between an Electromagnetic Field and an Elastic Body

This paper considers the scattering of a time-dependent electromagnetic plane wave by a bounded elastic body.An appropriate decomposition of the coupling interface conditions is proposed according to the Voigt’s model between the electromagnetic and elastic medium.The original unbounded scattering problem is equivalently reduced into an initial-boundary value problem in a bounded domain by introducing an exact transparent boundary condition(TBC)on a sufficiently large sphere.Making use of the Lax-Milgram lemma,the abstract inversion theorem of Laplace transform and the energy method,we verify the well-posedness and stability for the reduced problem.Moreover,a priori estimates are established for the electromagnetic field and elastic displacement by taking special test functions directly in the time domain variational formulation.     

STABILITY AND SUPER CONVERGENCE ANALYSIS OF ADI-FDTD FOR THE 2D MAXWELL EQUATIONS IN A LOSSY MEDIUM

Several new energy identities of the two dimensional(2D) Maxwell equations in a lossy medium in the case of the perfectly electric conducting boundary conditions are proposed and proved.These identities show a new kind of energy conservation in the Maxwell system and provide a new energy method to analyze the alternating direction implicit finite difference time domain method for the 2D Maxwell equations(2D-ADI-FDTD).It is proved that 2D-ADI-FDTD is approximately energy conserved,unconditionally stable and second order convergent in the discrete L2 and H1 norms,which implies that 2D-ADI-FDTD is super convergent.By this super convergence,it is simply proved that the error of the divergence of the solution of 2D-ADI-FDTD is second order accurate.It is also proved that the difference scheme of 2D-ADI-FDTD with respect to time t is second order convergent in the discrete H1 norm.Experimental results to confirm the theoretical analysis on stability,convergence and energy conservation are presented.     

A Kind of Discrete Non-Reflecting Boundary Conditions for Varieties of Wave Equations

Abstract In this paper, a new kind of discrete non-reflecting boundary conditions is developed.It can be usedfor a variety of wave equations such as the acoustic wave equation, the isotropic and anisotropic elastic waveequations and the equations for wave propagation in multi-phase media and so on.In this kind of boundaryconditions,the composition of all artifical reflected waves,but not the individual reflected ones,is consideredand eliminated.Thus, it has a uniform formula for different wave equations.The velocity C_A of the composedreflected wave is determined in the way to make the reflection coefficients minimal,the value of which depends onequations.In this psper,the construction of the boundary conditions illustrated and C_A is found,numericalresults are presented to illustrate the effectiveness of the boundary conditions.     

A Pseudo-Kinetic Approach for Helmholtz Equation

A lattice Boltzmann type pseudo-kinetic model for a non-homogeneous Helmholtz equation is derived in this paper. Numerical results for some model problems show the robustness and efficiency of this lattice Boltzmann type pseudo-kinetic scheme. The computation at each site is determined only by local parameters, and can be easily adapted to solve multiple scattering problems with many scatterers or wave propagation in non-homogeneous medium without increasing the computational cost.     

Analysis of Time-domain Electromagnetic Scattering Problem by Multiple Cavities

The time-domain multiple cavity scattering problem,which arises in diverse scientific areas,has significant industrial and military applications.The multiple cavities,embedded in an infinite ground plane,is filled with inhomogeneous media characterized by variable dielectric permittivities and magnetic permeabilities.Corresponding to the transverse electric,the scattering problem can be studied by the Helmholtz equation in frequency domain and wave equation in time-domain respectively.A novel transparent boundary condition in time-domain is developed to reformulate the cavity scattering problem into an initial-boundary value problem in a bounded domain.The well-posedness and stability of the reduced problem are established.Moreover,a priori energy estimates for the electric field is obtained with minimum regularity requirement for the data and an explicit dependence on the time by studying the wave equation directly.     

The Two-dimensional Electromagnetic Scattering from Periodic Chiral Structures and Its Finite Element Approximation

In this paper, we consider the electromagnetic scattering from periodic chiral structures. The structure is periodic in one direction and invariant in another direction. The electromagnetic fields in the chiral medium are governed by the Maxwell equations together with the Drude-Born-Fedorov equations. We simplify the problem to a two-dimensional scattering problem and we show that for all but possibly a discrete set of wave numbers, there is a unique quasi-periodic weak solution to the diffraction problem. The diffraction problem can be solved by finite element method. We also establish uniform error estimates for the finite element method and the error estimates when the truncation of the nonlocal transparent boundary operators takes place.     

The dual nature of the transverse vibrations of an elastic rod

The transverse vibrations of an elastic rod, to one of which displacements are applied while the other end is free, are investigated. It is assumed that the propagation velocity of the perturbations in the rod is finite. The unperturbed part performed rotational motion around the centre line. The angle of rotation is expressed by the angle of curvature of the centre line of the perturbed part of the rod. Two types of elastic vibrations are obtained: (1) the rod vibrates elastically due to displacements applied at the end, and (2) when performing rotational motion elastic vibrations and additional forces occur in the rod due to elasticity [1].     

A new solution of the equilibrium equations of a system of bodies with elastic joints

Different ways of representing the elastic moments are proposed, which can be used for the finite-dimensional modelling of rod systems using a system of n axisymmetric solids, connected by elastic spherical joints. Using the example of a closed plane rod, possible states of equilibrium of the finite-dimensional model of the rod are analysed for different methods of specifying the elastic torques at the joints. The case when the rod axis has the form of a “figure of eight”, which is modelled by a system of six axisymmetric solids with a relative torsion angle that depends on the bending, is investigated in detail.     

夹层结构杆在弹性基础上的弹塑性变形

对位于弹性基底上的、具有可压缩非线性芯子的3层弹塑性杆的弯曲问题进行了研究.研究分析了由2个受力层和1个芯子层组成的3层构件的力学响应.解决了位于弹性基底上的3层杆弯曲的复杂问题.对所给出的弹性解法进行了收敛性检验,以保证该弹性解是可以接受的.计算结果表明,材料的塑性和物理非线性对位于弹性基底上的夹层结构杆的变形影响很大.     

一类变形Euler棒P-稳定性的研究

借助于特征根法研究Euler弹性棒变形的P稳定性.将广泛存在于应用技术中的一类弹性单元抽象为Euler弹性棒,建立相应变形的物理和数学模型-常微分方程的边值问题,将其嵌入偏微分方程,得到数学模型解的P-稳定性.     

基于精确Cosserat模型的螺旋杆稳定性分析

弹性杆的螺旋线平衡问题在DNA、纤维、海底电缆和输油管线等方面具有应用背景.Kirchhoff动力学比拟是分析弹性细杆平衡稳定性的有效方法.Kirchhoff模型中包括中心线无拉伸变形和截面无剪切变形的基本假定与生物大分子等软物质的实际状况有较大差异.基于精确Cosserat模型,考虑中心线的拉伸压缩变形和截面剪切变形,以及剪切变形引起杆中心线转动导致切线轴相对截面法线轴的偏离,以Euler角表达截面姿态,建立圆截面弹性杆的动力学普遍方程.在静力学范畴内讨论螺旋线平衡状态的Liapunov稳定性和Euler稳定性问题,导出稳定性条件及轴向力和扭矩的Euler临界值.证明螺旋杆平衡的静态Liapunov稳定性和Euler稳定性条件是动态Liapunov稳定性的必要条件.     

Waves induced by the longitudinal impact of a rod against a stepped rod in contact with a rigid barrier

The longitudinal elastic central impact of a rod system, consisting of a uniform rod having a pre-impact velocity against a stepped rod which is in a state of rest and interacts by means of unilateral constraints with a rigid barrier, is modelled.     

The oscillations of a rod in an inhomogeneous elastic medium

The local length-dependence of the natural frequencies and forms of plane transverse oscillations of a thin inhomogeneous rod in an elastic medium with a variable stiffness and arbitrary elastic-fastening boundary conditions is investigated. It is established that the presence of an external elastic medium, described by the Winkler model, can lead to an anomalous effect – an increase in the natural frequencies of lower oscillation modes as the length of the rod increases continuously. The extremely fine properties of this change as a function of the length, the mode number and the method of fastening are revealed. The oscillations in the case of standard methods of fastening are investigated separately. Simple examples, which illustrate the anomalous dependence of the natural oscillation frequencies of the rod in an extremely inhomogeneous elastic medium with different boundary conditions are calculated.     

Zero Leakage Sealings for Reciprocating Compressors

The piston rod of a reciprocating compressor is sealed with elastic cylindrical sealing elements. Across the sealings the pressure drops from the operating pressure to the ambient pressure. The lubrication gap between the elastic sealing and reciprocating piston rod is studied with the aim to find conditions of a leakage free sealing. The flow in the lubrication gap and the elastic deformation of the sealing are determined simultaneously. The netflow during one cycle of the reciprocating piston rod is calculated. It turns out that maintaining zero leakage is very sensible. Indeed the outbound flow during out-stroke has to be equal the inbound flow during the instroke. By prescribing a special shape of the undeformed sealing zero leakage can be attained — at least theoretically for certain operating conditions. It turns out that temperature dependent material data and a model for cavitation is necessary. The model, its numerical implementation and results will be discussed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary control by an elastic force at one end of an inhomogeneous rod with the other end free for the case in which the wave travels either of the homogeneous parts in the same time

We study the boundary control by an elastic force at one end of an inhomogeneous rod that has two parts of different densities and elasticities and whose other end is free. The case in which the wave travels either of the homogeneous parts in the same time is considered. We present a closed-form analytical expression for the boundary control by an elastic force that brings the rod from the initial quiescent state to a given terminal state specified by given terminal displacement and terminal velocity.     

Group Classifications, Symmetry Reductions and Exact Solutions to the Nonlinear Elastic Rod Equations

In this paper, the Lie symmetry analysis are performed on the three nonlinear elastic rod (NER) equations. The complete group classifications of the generalized nonlinear elastic rod equations are obtained. The symmetry reductions and exact solutions to the equations are presented. Furthermore, by means of dynamical system and power series methods, the exact explicit solutions to the equations are investigated. It is shown that the combination of Lie symmetry analysis and dynamical system method is a feasible approach to deal with symmetry reductions and exact solutions to nonlinear PDEs.