General techniques for constructing variational integrators

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi’s solution of the Hamilton-Jacobi equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method determine the order of accuracy and momentum-conservation properties of the associated variational integrators. We also illustrate these systematic methods for constructing variational integrators with numerical examples.     

基于辛理论的载流碳纳米管能带分析

基于连续介质力学理论和辛弹性理论,将载流碳纳米管等效为铁木辛柯梁,采用哈密顿变分原理建立了载流碳纳米管的振动控制方程;引入对偶变量将振动控制方程从拉格朗日体系导入到哈密顿体系下;通过波传播方法分析了载流碳纳米管的能带结构;研究了流体密度、流速对载流碳纳米管能带结构的影响;同时计算了载流碳纳米管的散射矩阵. 研究发现:管内流速以及流体密度对剪切频率和弯曲频率有着非常重要的影响. 研究结果表明:载流碳纳米管的剪切频率和弯曲频率因流体的加入而减小,并随流速及流体密度的增大而减小;通过对数值结果的分析发现:载流碳纳米管由于管内流体、流速以及流体密度的作用,会使得载流碳纳米管变的更“软”. 其中,哈密顿体系下所得出的载流碳纳米管弯曲频率随管内流体密度的增加而变小,有别于在拉格朗日体系下非局部梁理论所得的结论. 同时,数值结果表明散射矩阵是酉矩阵,辛体系下的入射波功率流与反射波功率流相等,即功率流守恒,体现了辛弹性力学理论的优越性.      

Birkhoffian symplectic algorithms derived from Hamiltonian symplectic algorithms

In this paper, we focus on the construction of structure preserving algorithms for Birkhoffian systems, based on existing symplectic schemes for the Hamiltonian equations. The key of the method is to seek an invertible transformation which drives the Birkhoffian equations reduce to the Hamiltonian equations. When there exists such a transformation,applying the corresponding inverse map to symplectic discretization of the Hamiltonian equations, then resulting difference schemes are verified to be Birkhoffian symplectic for the original Birkhoffian equations. To illustrate the operation process of the method, we construct several desirable algorithms for the linear damped oscillator and the single pendulum with linear dissipation respectively. All of them exhibit excellent numerical behavior, especially in preserving conserved quantities.     

Analytical and numerical methods of symplectic system for Stokes flow in two-dimensional rectangular domain

In this paper,a new analytical method of symplectic system.Hamiltonian system,is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain.In the system,the fundamental problem is reduced to all eigenvalue and eigensolution problem.The solution and boundary conditions call be expanded by eigensolutions using ad.ioint relationships of the symplectic ortho-normalization between the eigensolutions.A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space.The results show that fundamental flows can be described by zero eigenvalue eigensolutions,and local effects by nonzero eigenvalue eigensolutions.Numerical examples give various flows in a rectangular domain and show effectivenees of the method for solving a variety of problems.Meanwhile.the method can be used in solving other problems.     

基于界带模型的碳纳米管声子谱的辛分析

针对碳纳米管声子谱的数值计算方法研究,基于对偶体系和辛几何算法提出了一套全新的计算方法和相应的界带结构模型,通过将碳纳米管模拟成不同的结构力学模型,利用分析结构力学中的振动理论来计算碳纳米管的色散关系.理论框架包括:周期结构的变分原理、周期结构中波的传播分析、子结构方法、界带理论和声子色散关系的基本算法.数值算例验证了理论和算法的有效性,而且也指出了针对碳纳米管的声子谱的计算,界带模型相对于其它传统模型存在着一定的优势.     

车辆-轨道系统垂向随机振动的辛方法分析

将轨道视为无限长的周期结构,建立车辆轨道垂向耦合模型. 使用虚拟激励法将随机的轨道不平顺激励转化为确定性的简谐激励,再用辛数学方法求解轨道结构的频率响应特性和耦合系统的响应功率谱. 整个计算模型只有26个自由度,求解过程快速而精确.数值算例中,将该方法与常规有限元方法进行了比较,验证了方法的高效性和正确性,讨论了车辆速度对系统随机响应的影响.      

高阶辛算法的稳定性与数值色散性分析

利用Maxwell方程的哈密尔顿函数,导出对应的欧拉-哈密尔顿方程.利用辛积分技术与高阶交错差分技术,建立求解三维时域Maxwell方程的高阶辛算法;结合电磁场中的物理概念,借助矩阵分析和张量分析理论,获得高阶时域方法及高阶辛算法的稳定性和数值色散性的统一处理新方法.用数值结果证实方法的正确性,与FDTD算法和其它时域高阶方法相比,高阶辛算法具有较大的计算优势,为电磁计算提供了新的途径.     

影响函数与有限元应力计算

用有限元法得到位移场后,总要计算应力场。通常的做法是对位移进行微商计算应变,再根据应力-应变关系计算应力。有限元位移计算的精度比较高,但通过用位移微商来计算应力,精度会大大降低。本文利用Hamilton对偶体系的已有成果,解析求解位移和应力的影响函数,利用有限元法计算得到的位移和节点力,通过功的互等定理,可以求得一点的应力值。因影响函数是分析解,而且计算应力时不必进行微商,应力精度大幅提高。数值结果表明该方法是可行的和有效的。由该方法编制成的计算程序,可作为有限元通用程序应力计算的一个模块,将较大地提高有限元应力计算的精度和稳定性。     

A Note on Symplectic Algorithm

We present the symplectic algorithm in the Lagrangian formalism for the Hamiltonian systems by virtue of the noncommutative differential calculus with respect to the discrete time and the Euler-Lagrange cohomological concepts. We also show that the trapezoidal integrator is symplectic in certain sense.``     

二维非定常Sine-Gordon方程辛算法及其孤子数值模拟

在矩形域[-a,a]×[-a,a]内对微分算子L=(ə²)/(əx²)+(ə²)/(əy²)用5点差分格式将二维非定常Sine Gordon方程离散化为一个2×799²阶非线性Hamilton系统.对该系统使用Euler中心格式,得到一个非线性方程组.对此方程组建立迭代解法并给出了这个迭代方法的收敛条件和收敛速度.Sine Gordon方程单孤子和双孤子的数值模拟试验显示该辛算法是有效的.