具有精确色散性的非线性波浪数学模型

以完全非线性的自由表面边界条件为基础,以波面升高\eta和自由表面速度势\phi _\eta为待求变量,建立了新的波浪方程．方程在色散性上是完全精确的,非线性近似至三阶．与缓坡方程相比较,两者都具有精确的色散性,但该方程属于非线性模型,可模拟波浪的非线性效应,且适用于不规则波．方程的特点是属于微分-积分方程,对如何处理方程中积分项进行了讨论,并数值模拟了不同周期的线性波和二阶Stokes波,也模拟了波群的非线性演化,以对模型进行验证．      

黏弹夹芯板的有限元建模及实验研究

从有限元分析和数值模拟及实验验证的角度研究了黏弹夹芯板的频率依赖振动特性。夹芯板中间层为黏弹性材料，其刚度和阻尼的频率依赖性行为直接影响系统的模态频率和阻尼，并导致振动模式求解的复杂化。采用三阶七参数Biot模型描述黏弹性材料频率相关的黏弹性行为。开发了三层四节点28自由度的夹芯板单元，基于经典板理论和哈密顿原理建立了黏弹夹芯板的有限元动力学方程。通过引入辅助耗散坐标，将Biot模型和黏弹夹芯板的有限元动力学模型结合起来，并将其转化为常规二阶线性系统形式，极大简化了求解非线性振动特性的过程。对一边固定、另三边自由的黏弹夹芯板进行了前三阶固有频率和损耗因子的预测，并与实验结果对比。数值模拟结果和实验结果吻合良好，说明所提有限元方法是正确有效的。     

旋转薄壁圆柱壳振型进动的非线性振动特性

选取在工程上常用的悬臂旋转薄壁圆柱壳为研究模型,首先推导出考虑阻尼的振型进动因子,然后根据Donnell's简化壳理论建立考虑科氏力,阻尼与几何大变形的非线性波动方程,采用Galerkin方法对波动方程进行离散化,得到模态坐标中相互耦合的三阶非线性微分方程组.应用Runge-Kutta法求解获得非线性幅频特性曲线,分析了不同模态组合下系统主模态(m=1,n=6,k=1)的共振响应.应用谐波平衡法对系统三阶非线性微分方程组解析分析,与数值解比较验证了解析解的正确性和有效性.最后分析了动力系统的运动稳定性.结果表明,节径数n和频率倍数k对于主模态共振响应的影响很小,而轴向半波数m对主共振的影响则相对较大,因此只需选取相邻的两个轴向模态(M=2)即可较为简洁,准确的描述主共振响应;谐波平衡法可以很好的解决三阶微分方程组的非线性问题,并且能够达到较为满意的精度.     

悬索的多重内共振研究

本文对谐波激励的悬索的非线性响应进行了研究,同时考虑了如下问题(1):面内第三阶对称模态的主共振:(2):面内第一阶、第三阶对称模态和面外第五阶模态之间的内共振.本方首先针对考虑大变形的悬索动力学方程,由线性理论求得各阶频率,考察可能出现的内共振.然后利用直接法对悬索的运动学方程和边界条件进行非线性求解.由多尺度法得到系统的平均方程和悬索响应的二阶近似解.随后利用Newton-Raphson 方法和弧长法对特定张拉索进行数值仿真计算,得到面内第一阶对称模态、面内第三阶对称模态和面外第五阶模态的稳态解,并分析了解的稳定性.绘制幅频响应曲线,发现了关于悬索响应的多种分叉现象,并且对各种分叉现象周期解、混沌解进行了讨论.     

基于势流理论的内孤立波与立管作用数值研究

内孤立波是一种发生在水面以下的在世界各个海域广泛存在的大幅波浪, 其剧烈的波面起伏所携带的巨大能量对以海洋立管为代表的海洋结构物产生严重威胁, 分析其传播演化过程的流场特征及立管在内孤立波作用下的动力响应规律对于海洋立管的设计具有重要意义. 本文基于分层流体的非线性势流理论, 采用高效率的多域边界单元法, 建立了内孤立波流场分析计算的数值模型, 可以实时获得内孤立波的流场特征. 根据获得的流场信息, 采用莫里森方程计算内孤立波对海洋立管作用的载荷分布. 将内孤立波流场非线性势流计算模型与动力学有限元模型结合来求解内孤立波作用下海洋立管的动力响应特征, 讨论了内孤立波参数、顶张力大小以及内部流体密度对立管动力响应的影响. 发现随着内孤立波波幅的增大, 海洋立管的流向位移和应力明显增大. 由于上层流体速度明显大于下层, 且在所研究问题中拖曳力远大于惯性力, 因此管道顺流向的最大位移发生在上层区域. 顶张力通过改变几何刚度阵的值进而对立管的响应产生明显影响. 对于弱约束立管, 内部流体的密度对管道的流向位移影响较小.      

含电学边界的压电层合梁的非线性弯曲波

非线性科学己成为近代科学发展的一个重要标志, 特别是非线性动力学和非线性波的研究对于解决自然科学各领域中遇到的复杂现象和问题有着极其重要的意义. 本文研究了含电学边界条件的压电层合梁的非线性弯曲波传播特性.首先, 考虑几何非线性效应和压电耦合效应, 利用哈密顿原理建立了一维无限长矩形压电层合梁弯曲波的非线性方程.其次, 采用Jacobi椭圆函数展开法对非线性弯曲波方程进行求解, 得到了非线性弯曲波动方程在近似情况下对应的冲击波解和孤波解.最后, 利用约化摄动法得到了非线性薛定谔方程, 进一步得到了亮孤子和暗孤子解.基于两种方法具体研究了外加电压、压电层厚度等参数对冲击波和孤立波以及亮孤子和暗孤子特性的影响. 研究结果表明, 在波速较小时, 外加电压对冲击波的影响较大, 波速较大时, 外加电压对孤立波影响减弱.通过调整作用在压电层合梁上的电压发现了存在亮孤子和暗孤子, 分析结果表明随着外加电压值的增大, 亮孤子和暗孤子的振幅都增大.      

复合材料层合板1:1参数共振的分岔研究

针对复合材料对称铺设各向异性矩形层合板的物理模型,在同时考虑了材料、阻尼和几何等非线性因素后,建立了二自由度非线性参数振动系统动力学控制方程,并应用多尺度法求得基本参数共振下的近似解析解,利用数值模拟分析了系统的分岔和混沌运动.指出了伽辽金截断对系统动力学分析的影响,以及系统进入混沌的途径.     

考虑阻尼的无限长小垂度缆索弹性波的传播

缆索结构被广泛应用于电气、土木、海洋和航空工程等领域,随着缆索在工程中的应用长度越来越长,高阶振动越来越明显,研究时应该考虑扰动沿着缆索的传播.现有对缆索弹性波传播的研究中,通常不考虑阻尼项,然而阻尼对于波的传播有着重要影响.文章考虑阻尼的影响,发展了包含阻尼项的三维弹性缆索运动方程.通过求解上述含阻尼项的运动方程,分别考察了面内面外弹性波的频率关系、相速度和群速度等自由传播特性,进而通过计算无限长弹性缆索在初始余弦型脉冲作用下的位移响应,分析扰动沿着该缆索的传播规律,考察波的色散现象以及阻尼对于缆索弹性波传播的影响.结果表明,考虑阻尼后,面内波和面外波均为色散波,面内波在曲率的作用下,为高度色散波.此外,在阻尼的影响下,波的峰值在传播过程不断减小,且波的后缘端点响应总是高于前缘端点响应.     

贮腔类三维自由液面动力学问题数值研究

讨论了贮腔类三维自由液面动力学问题的数值研究,将任意的拉格朗日-欧拉运动学描述关系引入到系统的控制方程中,采用任意的拉格朗日-欧拉描述跟踪自由液面,推导了自由面上结点的法向矢量计算公式。采用Galerkin余量法推导了Navier-Stokes方程的空间离散有限元方程,采用三维自由液面上微分几何理论推导了表面张力计算公式。数值研究中考虑了接触角效应,最后进行了三维数值算例分析。     

简谐载荷下嵌入式单壁碳纳米管动态响应的Magnus方法

基于两端固支的弹性梁模型,研究嵌入式单壁碳纳米管在横向简谐载荷作用下的非线性振动问题。利用Galerkin方法对运动微分方程进行近似处理,将原方程从非线性动力学系统转化到二阶动力学系统,对于二阶动力学方程采用Magnus级数方法进行求解。通过数值实验,分析了嵌入式单壁碳纳米管非线性振动幅频特性,根据非线性动力学理论分析了碳纳米管动态响应,结果表明倍周期分岔产生混沌。     

Solitary wave packets beneath a compressed ice cover

A family of plane solitary wave packets of a small (but finite) amplitude on the surface of an ideal incompressible fluid of finite depth beneath an ice cover is described. The solitary wave trains correspond to solutions of the two-dimensional system of Euler’s equations of an ideal incompressible fluid of the type of a traveling wave which decreases at infinity and has identical phase and group velocities. The ice cover is simulated by an elastic Kirchhoff-Love plate freely floating on the fluid surface in the compressed state.     

Numerical simulations of inviscid and viscous free‐surface flows with application to nonlinear water waves

The purpose of the present study is to establish a numerical model appropriate for solving inviscid/viscous free‐surface flows related to nonlinear water wave propagation. The viscous model presented herein is based on the Navier–Stokes equations, and the free‐surface is calculated through an arbitrary Lagrangian–Eulerian streamfunction‐vorticity formulation. The streamfunction field is governed by the Poisson equation, and the vorticity is obtained on the basis of the vorticity transport equation. For computing the inviscid flow the Laplace streamfunction equation is used. These equations together with the respective (appropriate) fully nonlinear free‐surface boundary conditions are solved using a finite difference method. To demonstrate the model feasibility, in the present study we first simulate collision processes of two solitary waves of different amplitudes, and compute the phenomenon of overtaking of such solitary waves. The developed model is subsequently applied to calculate (both inviscid and the viscous) flow field, as induced by passing of a solitary wave over submerged rectangular structures and rigid ripple beds. Our study provides a reasonably good understanding of the behavior of (inviscid/viscous) free‐surface flows, within the framework of streamfunction‐vorticity formulation. The successful simulation of the above‐mentioned test cases seems to suggest that the arbitrary Lagrangian–Eulerian/streamfunction‐vorticity formulation is a potentially powerful approach, capable of effectively solving the fully nonlinear inviscid/viscous free‐surface flow interactions. Copyright © 2009 John Wiley & Sons, Ltd.     

Plotnikov——Toland板模型中水弹性孤立波的迎撞

通过奇异摄动方法研究了在薄冰层覆盖的不可压缩理想流体表面上传播的两个水弹性孤立波之间的迎面碰撞.借助特殊的 Cosserat 超弹性壳 理论以及Kirchhoff--Love 板理论,冰层由 Plotnikov--Toland板模型描述.流体运动采用浅水假设和Boussinesq 近似. 应用Poincaré--Lighthill--Kuo 方法进行坐标变形,进而渐近求解控制方程及边界条件, 给出了三阶解的显式表达. 可以观察到碰撞后的孤立波不会改变它们的形状和振幅. 波浪轮廓在碰撞之前是对称的, 而在碰撞之后变成不对称的并且在波传播方向上向后倾斜. 弹性板和流体表面张力减小了波幅. 图示比 较了本文与已有结果可知线性板模型可作为本文的一个特例.      

A universal bifurcation mechanism arising from progressive hydroelastic waves

A unidirectional, weakly dispersive nonlinear model is proposed to describe the supercritical bifurcation arising from hydroelastic waves in deep water. This model equation, including quadratic, cubic, and quartic nonlinearities, is an extension of the famous Whitham equation. The coefficients of the nonlinear terms are chosen to match with the key properties of the full Euler equations, precisely, the associated cubic nonlinear Schr?dinger equation and the amplitude of the solitary wave at the bifurcation point. It is shown that the supercritical bifurcation, rich with Stokes, solitary, generalized solitary, and dark solitary waves in the vicinity of the phase speed minimum, is a universal bifurcation mechanism. The newly developed model can capture the essential features near the bifurcation point and easily be generalized to other nonlinear wave problems in hydrodynamics.     

Damping of large-amplitude solitary waves

We consider the damping of large-amplitude solitary waves in the framework of the extended Korteweg-de Vries equation (that is, the usual Korteweg-de Vries equation supplemented with a cubic nonlinear term) modified by the inclusion of a small damping term. The damping of a solitary wave is studied for several different forms of friction, using both the analytical adiabatic asymptotic theory and numerical simulations. When the coefficient of the cubic nonlinear term has the opposite sign to the coefficient of the linear dispersive term, the extended Kortweg-de Vries equation can support large-amplitude “thick” solitary waves. Under the influence of friction, these “thick” solitary waves decay and may produce one or more secondary solitary waves in this process. On the other hand, when the coefficient of the cubic nonlinear term has the same sign as the coefficient of the linear dispersive term, but the opposite sign to the coefficient of the quadratic nonlinear term, the action of friction may cause a solitary wave to decay into a wave packet.     

A new σ‐transform based Fourier‐Legendre‐Galerkin model for nonlinear water waves

This paper presents a new spectral model for solving the fully nonlinear potential flow problem for water waves in a single horizontal dimension. At the heart of the numerical method is the solution to the Laplace equation which is solved using a variant of the $\sigma$‐transform. The method discretizes the spatial part of the governing equations using the Galerkin method and the temporal part using the classical fourth‐order Runge‐Kutta method. A careful investigation of the numerical method's stability properties is carried out, and it is shown that the method is stable up to a certain threshold steepness when applied to nonlinear monochromatic waves in deep water. Above this threshold artificial damping may be employed to obtain stable solutions. The accuracy of the model is tested for: (i) highly nonlinear progressive wave trains, (ii) solitary wave reflection, and (iii) deep water wave focusing events. In all cases it is demonstrated that the model is capable of obtaining excellent results, essentially up to very near breaking.     

浅水孤立波在三维浮体上的绕射

浅水域中非线性水波运动的控制方程通常是经过深度平均的Boussinesq方程。然而,这一方程在浮体近旁或水下障碍物附近不再适用,在这些区域,流动在水深方向的变化不容忽略,本文应用匹配渐近展开法和边缘层(edge layer)思想,建立了浅水弱非线性波与三维浮体相互作用的数学模型,作为算例,求解了浅水孤立波在垂直圆柱形浮体上的绕射.本方法可以推广到波在一般浮体上绕射的情况。     

Nonlinear analysis and solitary waves for two superposed streaming electrified fluids of uniform depths with rigid boundaries

The nonlinear modulation of the interfacial waves of two superposed dielectric fluids with uniform depths and rigid horizontal boundaries, under the influence of constant normal electric fields and uniform horizontal velocities, is investigated using the multiple-time scales method. It is found that the behavior of small perturbations superimposed on traveling wave trains can be described by a nonlinear Schrödinger equation in a frame of reference moving with the group velocity. Wave-like solutions to this equation are examined, and different types of localized excitations (envelope solitary waves) are shown to exist. It is shown that when these perturbations are neutrally stable and sufficiently long, solutions to the nonlinear Schrödinger equation may be approximated by the well-known Korteweg-de Vries equation. The speed of the solitary on the interface is seen to be reduced by the electric field. It is found that there are two critical values of the applied voltage that lead to (i) breaking up of the solitary waves, and (ii) bifurcation of solutions of the governing equations. On the other hand, the complex amplitude of standing wave trains near the marginal state is governed by a similar type of nonlinear Schrödinger equation in which the roles of time and space are interchanged. This equation, under a suitable transformation, is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equations admit a solitary wave type of solutions with variable speed. Using the tangent hyperbolic method, it is observed that the wave speed increases as well as decreases, with the increase of electric field values, according to the chosen wavenumbers range. Finally, the nonlinear stability analysis is discussed in view of the coefficients of nonlinear Schrödinger equation to show the effects of various physical parameters, and also to recover the some limiting cases studied earlier in the literature.     

Solitary Waves in Nonlinear Beam Equations: Stability,Fission and Fusion

We continue work by the second author and co-workers onsolitary wave solutions of nonlinear beam equations and their stabilityand interaction properties. The equations are partial differentialequations that are fourth-order in space and second-order in time.First, we highlight similarities between the intricate structure ofsolitary wave solutions for two different nonlinearities; apiecewise-linear term versus an exponential approximation to thisnonlinearity which was shown in earlier work to possess remarkablystable solitary waves. Second, we compare two different numericalmethods for solving the time dependent problem. One uses a fixed griddiscretization and the other a moving mesh method. We use these methodsto shed light on the nonlinear dynamics of the solitary waves. Earlywork has reported how even quite complex solitary waves appear stable,and that stable waves appear to interact like solitons. Here we show twofurther effects. The first effect is that large complex waves can, as aresult of roundoff error, spontaneously decompose into two simplerwaves, a process we call fission. The second is the fusion of twostable waves into another plus a small amount of radiation.