局部流型分析的哈密顿系统小参数正则变换

表达二维不可压缩流动的流速分量与流函数关系的微分方程组是典型的具有一个自由度的哈密顿系统.将流函数用Taylor级数展开,应用非线性系统动力学方法对流型及其分岔进行了分析.对退化临界点,基于流动平面的小参数正则变换,导出了流函数的正形表达式和简化的微分方程,并对简化系统的一般特性进行了分析.     

彩色多普勒图像心脏流场流线可视化方法

为了解决心脏流体运动状态的可视化描述问题,根据心脏流场超声图像的特点,提出一种基于彩色多普勒图像信息的心脏流体运动平面流线可视化描述方法.该方法采用点源和点汇表示二维观测区域的三维流动,通过拓展流函数计算原理实现多普勒流函数值的计算;以点源和点汇作为平面流线的起点和终点,根据同一流线上多普勒流函数值相等的原理绘制平面流线;选取平面流线中的封闭曲线获得心脏流场内涡流流线图.实验结果表明,文中方法能够对彩色多普勒图像中的流体运动状态进行有效的可视化描述,并能有效地描述心脏流体的涡流运动状态,为心脏流体运动状态的有效可视化观察和精确量化评价提供了一种新方法.     

导出变系数非线性动力学系统拉格朗日函数的两种方法

利用从运动微分方程出发和从第一积分出发导出拉格朗日函数的两种直接方法,构造变系数非线性动力学系统¨x+b(x)x~2+c(x)x=0的拉格朗日函数和c(x)=0特殊情况的拉格朗日函数族.另外,讨论了这种非保守系统广义能量守恒的物理意义.     

不可达系统的鲁棒贝叶斯估计方法

本文针对模型扰动下的不可达系统, 提出了一种新的针对退化分布下的极大极小博弈问题的求解和证明方法. 首先, 文章将有相对熵约束的极大极小博弈问题转换成了一个无约束的拉格朗日函数, 并找到其在均值和奇异的方差矩阵方向上都为严格凹函数的条件; 其次, 本文通过求解其均值和方差的极大值, 得到所对应的鲁棒贝叶斯估计器和奇异的扰动状态误差协方差矩阵; 最后, 文章证明存在一个唯一的拉格朗日乘子满足其约束条件. 微机电系统加速度计漂移估计仿真结果表明对所提算法的有效性     

Stokes外问题的一种重叠型区域分解法

本文讨论了平面无界区域上Stokes问题的重叠型区域分解法.利用混合元方法求解内子区域问题得到速度和压力,再用Poisson积分公式解出外子区域的速度和压力,如此交替迭代克服区域无界性并按原始变量求出原问题的数值解.根据投影理论证明重叠型区域分解法的几何收敛性.最后给出数值例子.     

矩形腔内Stokes流动混合过程追踪模拟

基于涡量-速度方法建立了矩形腔上盖拖动的数学模型,采用交错网格,对腔内Stokes流动进行了有限体积数值模拟研究,得到了不同长高比的矩形腔内速度场及流函数分布.发现随着长高比的增大,中垂线的水平速度分布逐渐向无限大长高比得到解析解抛物线分布靠近.采用4阶Runge-Kutta方法对示踪剂混合过程进行前锋追踪模拟,得到了不同时刻示踪剂的混合图像.结果表明,示踪剂界面随时间呈线性增长,而且长高比越大,示踪剂界面的增长越快.     

隧道内细水雾抑制火灾事故的数值仿真

以工业灾害的抑制技术为背景,针对隧道内发生的火灾事故,采用数值仿真技术研究连续喷射水雾阻挡气相物质燃烧的过程.并讨论水雾抑制燃烧的机理.相比于欧拉/欧拉方法,欧拉/拉格朗日方法能较好地描述水雾液滴引起的瞬时流动特性及变化经历,利用欧拉/拉格朗日方法,对气相反应流及水雾扩散过程进行数值研究.应用颗粒随机轨道模型,来考察湍流对水雾液滴扩散的影响.比较了有、无水雾抑制时和水雾不同初始流量下,隧道内流场的温度分布情况,揭示了水雾液滴的运动特性,为火灾事故抑制技术的发展提供理论依据.     

RLC串联电路与运动介质板综合分析

考虑磁路非线性特点,基于磁共能、电场能和电系统耗散函数,通过引入拉格拉日函数得到基尔霍夫定律的拉格朗日方程表示法.应用此方法建立了RLC串联电路与运动介质板系统数学模型.给出介质板所受的电磁力和插入介质板电容器电容.根据电路特点进行理论分析和数值计算,结果表明电荷经过几秒震荡后达到稳定运动.     

基于布尔函数的网络可达性验证方法

针对SDN中由于不同应用的转发路径交叠等导致的数据平面配置问题,提出一种基于布尔函数的网络可达性验证方法。首先,将网络拓扑抽象为端口拓扑并计算端口邻接矩阵;之后,生成网络的路径空间和各端口的转发函数并计算每条路径的路径函数;最后通过判断路径函数的可满足性来确定路径的可达性。通过仿真实验,对网络拓扑和流规则规模等因素对算法验证效率的影响进行研究,并将所提方法与APV和DASDA进行性能比较。实验结果表明,所提方法能够有效检测SDN中的流规则配置问题。随着网络中环路的增加和流规则规模的增长,验证网络所需的时间开销逐渐增加。其中,网络拓扑对路径生成时间影响较大,而转发函数的生成时间则主要受流规则规模的影响。方法的验证时间相较于APV和DASDA分别平均缩短约53.76%和27.74%。     

基于球摆模型的离散变分积分子算法研究

在动力学系统长时间的仿真计算中,力学系统固有的结构将影响到计算精度及稳定性.离散变分积分子能够保持力学系统的能量,动量及辛结构的守恒.结合离散变分原理,通过对系统的拉格朗日函数进行离散化以及求变分和积分的过程,可以得到力学系统的离散变分积分子算法.该算法是一种递归算法,给定初始条件便可得到系统的动力学参数的时间历程.使用该原理可以构造具有完整约束的拉格朗日系统的辛-动量积分子方法.与连续算法相比,离散变分积分子算法能够直接在离散拉格朗日函数的基础上得到姿态与角速度的递推公式,而不需要复杂的迭代计算.本文研究是基于第一类拉格朗日函数的离散变分积分子算法.球摆模型是一个具有完整约束的拉格朗日系统.仿真结果表明,系统的能量值在长时间的仿真中得到保持,且计算的精度与步长的数量级呈现二次方的关系,系统角速度和姿态的仿真结果都符合球摆的运动规律.     

Meshless Conservative Scheme for Multivariate Nonlinear Hamiltonian PDEs

For multivariate nonlinear Hamiltonian equations, we propose a meshless conservative method by using radial basis approximation. Based on the method of lines, we first discretize the Hamiltonian functional using radial basis function interpolation, and then obtain a finite-dimensional semi-discrete Hamiltonian system. Moreover, we define a discrete symplectic form and verify that it is an approximation to the continuous one and is conserved with respect to time. For time discretization, two conservative methods (symplectic method and energy-conserving method) are employed to derive the full-discretized system. Approximation errors together with conservation properties including symplecticity and energy are discussed in detail. Finally, we present several numerical examples to illustrate that our method is accurate and effective when processing nonlinear Hamiltonian equations with scattered nodes. Besides, the numerical results also confirm the excellent conservation properties of the proposed method.     

Sixth-order symplectic and symmetric explicit ERKN schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations

This paper is devoted to the analysis of the sixth-order symplectic and symmetric explicit extended Runge–Kutta–Nyström (ERKN) schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. Fourteen practical sixth-order symplectic and symmetric explicit ERKN schemes are constructed, and their phase properties are investigated. The paper is accompanied by five numerical experiments, including a nonlinear two-dimensional wave equation. The numerical results in comparison with the sixth-order symplectic and symmetric Runge–Kutta–Nyström methods and a Gautschi-type method demonstrate the efficiency and robustness of the new explicit schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations.     

Symplectic algorithms with mesh refinement for a hypersensitive optimal control problem

A symplectic algorithm with nonuniform grids is proposed for solving the hypersensitive optimal control problem using the density function. The proposed method satisfies the first-order necessary conditions for the optimal control problem that can preserve the structure of the original Hamiltonian systems. Furthermore, the explicit Jacobi matrix with sparse symmetric character is derived to speed up the convergence rate of the resulting nonlinear equations. Numerical simulations highlight the features of the proposed method and show that the symplectic algorithm with nonuniform grids is more computationally efficient and accuracy compared with uniform grid implementations. Besides, the symplectic algorithm has obvious advantages on optimality and convergence accuracy compared with the direct collocation methods using the same density function for mesh refinement.     

Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation

In this paper, two methods for approximating the stabilizing solution of the Hamilton–Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton–Jacobi equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.      

Symplectic integrators for discrete nonlinear Schrödinger systems

Symplectic methods for integrating canonical and non-canonical Hamiltonian systems are examined. A general form for higher order symplectic schemes is developed for non-canonical Hamiltonian systems using generating functions and is directly applied to the Ablowitz–Ladik discrete nonlinear Schrödinger system. The implicit midpoint scheme, which is symplectic for canonical systems, is applied to a standard Hamiltonian discretization. The symplectic integrators are compared with an explicit Runge–Kutta scheme of the same order. The relative performance of the integrators as the dimension of the system is varied is discussed.     

A structure-preserving algorithm for the minimum H ∞ norm computation of finite-time state feedback control problem

The algorithm being developed here is based on the generating function approach for finite-time H _∞ control and application of canonical transformation of linear Hamiltonian system. First, an equivalent finite-time H _∞ control law in terms of the third-type generating function is derived. Then, by using symplectic structure of the Hamiltonian system's state transition matrix, a group of matrix recursive formulae are deduced for the evaluation of the finite-time H _∞ control law. Combining with a matrix singularity testing procedure, this recursive algorithm verifies the existence condition of sub-optimal H _∞ controllers and gives the minimum H _∞ norm of finite-time control systems. Inherited from the canonical transformation, the matrix recursive formulae have a standard symplectic form; this structure-preserving property helps facilitate reliable and effective computation. Numerical results show the effectiveness of the proposed algorithm.     

A Jacobi-like method for solving algebraic Riccati equations onparallel computers

An algorithm to solve continuous-time algebraic Riccati equations through the Hamiltonian Schur form is developed. It is an adaption for Hamiltonian matrices of an asymmetric Jacobi method of Eberlein (1987). It uses unitary symplectic similarity transformations and preserves the Hamiltonian structure of the matrix. Each iteration step needs only local information about the current matrix, thus admitting efficient parallel implementations on certain parallel architectures. Convergence performance of the algorithm is compared with the Hamiltonian-Jacobi algorithm of Byers (1990). The numerical experiments suggest that the method presented here converges considerably faster for non-Hermitian Hamiltonian matrices than Byers' Hamiltonian-Jacobi algorithm. Besides that, numerical experiments suggest that for the method presented here, the number of iterations needed for convergence can be predicted by a simple function of the matrix size     

Symplectic integrators for classical spin systems

We suggest a numerical integration procedure for solving the equations of motion of certain classical spin systems which preserves the underlying symplectic structure of the phase space. Such symplectic integrators have been successfully utilized for other Hamiltonian systems, e.g., for molecular dynamics or non-linear wave equations. Our procedure rests on a decomposition of the spin Hamiltonian into a sum of two completely integrable Hamiltonians and on the corresponding Lie-Trotter decomposition of the time evolution operator. In order to make this method widely applicable we provide a large class of integrable spin systems whose time evolution consists of a sequence of rotations about fixed axes. We test the proposed symplectic integrator for small spin systems, including the model of a recently synthesized magnetic molecule, and compare the results for variants of different order.     

Explicit Symplectic Geometric Algorithms for Quaternion Kinematical Differential Equation

Solving quaternion kinematical differential equations (QKDE) is one of the most significant problems in the automation, navigation, aerospace and aeronautics literatures. Most existing approaches for this problem neither preserve the norm of quaternions nor avoid errors accumulated in the sense of long term time. We present explicit symplectic geometric algorithms to deal with the quaternion kinematical differential equation by modelling its time-invariant and time-varying versions with Hamiltonian systems and adopting a three-step strategy. Firstly, a generalized Euler's formula and Cayley-Euler formula are proved and used to construct symplectic single-step transition operators via the centered implicit Euler scheme for autonomous Hamiltonian system. Secondly, the symplecticity, orthogonality and invertibility of the symplectic transition operators are proved rigorously. Finally, the explicit symplectic geometric algorithm for the time-varying quaternion kinematical differential equation, i.e., a non-autonomous and non-linear Hamiltonian system essentially, is designed with the theorems proved. Our novel algorithms have simple structures, linear time complexity and constant space complexity of computation. The correctness and efficiencies of the proposed algorithms are verified and validated via numerical simulations.      

Families of third and fourth algebraic order trigonometrically fitted symplectic methods for the numerical integration of Hamiltonian systems

The numerical integration of Hamiltonian systems by symplectic and trigonometrically fitted (TF) symplectic method is considered in this work. We construct new trigonometrically fitted symplectic methods of third and fourth order. We apply our new methods as well as other existing methods to the numerical integration of the harmonic oscillator, the 2D harmonic oscillator with an integer frequency ratio and an orbit problem studied by Stiefel and Bettis.