基于哈密顿动力系统新变分原理的保辛算法之一：变分原理和算法构造

提出了哈密顿动力系统的一个新变分原理,并基于此变分原理构造了四类保辛算法。通过新的变分原理定义修正作用量,然后将位移和动量采用拉格朗日多项式近似,并采用高斯积分对时间近似积分得到近似的修正作用量。在修正作用量的基础上,通过选择时间步两端不同的位移或动量作为独立变量,可构造四种不同类型的保辛算法。     

基于对偶变量变分原理和两端位移独立变量的保辛方法

将广义位移和动量同时用拉格朗日多项式近似,并选择积分区间两端位移为独立变量,然后基于对偶变量变分原理导出了哈密顿系统的离散正则变换和对应的数值积分保辛算法。当位移和动量的拉格朗日多项式近似阶数满足一定条件时,可以自然导出保辛算法的不动点格式。通过数值算例分析了位移和动量采用不同阶次插值所需最少Gauss积分点个数,并讨论了位移插值阶数、动量插值阶数以及Gauss积分点个数对保辛算法精度的影响,说明了上述不动点格式恰好是一种最优格式。     

非线性轨迹优化问题的保辛自适应求解方法

非线性轨迹优化问题一般是一个非线性最优控制问题。将非线性系统的最优控制问题导入到哈密顿体系的辛几何空间当中,基于对偶变量变分原理提出了求解非线性最优控制问题的一种保辛自适应方法。以时间区段两端协态作为独立变量,在时间区段内采用拉格朗日插值近似状态和协态变量,并利用对偶变量变分原理将非线性最优控制问题转化为非线性方程组的求解,保持了哈密顿系统的辛几何结构。并进一步,提出了基于多层次迭代的自适应算法,提高了非线性最优控制问题的求解效率。数值实验验证了该算法在求解非线性轨迹优化问题中的有效性。     

基于非传统哈密顿变分原理的高阶辛算法

给出了非传统哈密顿变分原理的一种简化形式,并在此基础上利用拉格朗日多项式近似位移和动量,采用高斯积分法对时间积分,建立了针对动力学初值问题的一类高阶辛算法。在建立高阶辛算法的过程中,本文方法与基于传统哈密顿变分原理的辛算法不同,无需由端值问题向初值问题转换,因此更加简捷有效。此外,给出了线性动力问题中本文算法保辛性的证明。当位移、动量的插值次数和高斯积分点个数均为m时,本文算法是具有2m阶精度的辛算法,且是线性无条件稳定的。通过数值算例结果表明,本文算法与辛算法性质吻合,并且计算效率比同阶辛龙格库塔法提高了约50%。     

基于哈密顿动力系统新变分原理的保辛算法之二：算法保辛性质证明

文献[1]给出了哈密顿系统的一个新的变分原理,并基于此变分原理,通过选择一个时间步长两端不同广义位移或广义动量为独立变量,给出了四种不同类型的求解哈密顿动力系统的数值方法。本文将分别证明这四类数值方法都是保辛的数值方法。     

电磁共振腔辛有限元法

将电磁场的基本方程导向了对偶方程形式。给出了推导电磁场有限元所需相应的对偶变量变分原理。为了有限元列式的保辛，交分原理被积函数可导向对于对偶变量为对称的形式。交分原理的边界积分项对于相邻单元互相抵消。对偶变量有限元推导可避免所谓的C1连续性问题。采用对偶变量离散分析了共振腔本征值问题，离散后再消去一类变量可导出普通的广义本征值问题而求解。算例表明了对偶变量有限元分析的有效性。     

基于拉格朗日插值的无网格直接配点法和稳定配点法

由于无网格法中大多数近似函数均为有理式,不具有Kronecker delta性质,因此难以精确地施加本质边界条件.边界误差较大容易导致整个求解域求解结果精度低,甚至引起数值不稳定现象.文章在无网格直接配点法和稳定配点法中引入拉格朗日插值函数作为形函数,构建了拉格朗日插值配点法(LICM)和拉格朗日插值稳定配点法(SLICM).由于拉格朗日插值具有Kronecker delta性质,可以像有限元法一样简单而精确地施加本质边界条件,提高这两种方法的数值求解精度.稳定配点法基于子域对强形式方程进行积分,可以满足高阶积分约束,即可以保证形函数在积分形式下也满足高阶一致性条件,实现精确积分.同时,进行子域积分还可以减少离散矩阵的条件数,从而提高算法的稳定性.进一步提高拉格朗日插值稳定配点法的精度和稳定性.通过数值算例验证这两种方法的精度、收敛性和稳定性,结果表明基于拉格朗日插值的配点法的精度优于基于重构核近似的配点法,拉格朗日插值稳定配点法的精度和稳定性均优于拉格朗日插值配点法.     

三维电磁场辛有限元列式及电磁共振腔的计算

针对三维共振腔的电磁场分析,利用Maxwell方程的对偶方程体系形式,从其相应的对偶变量变分原理出发,导出了三维电磁场辛有限单元的详细列式。为了有限元列式的保辛,变分原理被积函数可导向对于对偶变量为对称的形式。变分原理的边界积分项对于相邻单元相互抵消。由于采用了对偶变量的插值函数,使得电磁场单元构造可以在层面上进行,从而避免了所谓的连续性问题。无物理意义的零本征解可采用奇异值分解加以排除。文末分别对矩形及圆柱形的共振腔做了数值计算并与解析解和棱边元计算结果进行对比,算例表明了列式及算法的有效性。     

非线性最小二乘跟踪的对偶变量变分方法

最小二乘跟踪方法是近几年提出的一种计算动力系统跟踪轨迹的方法.基于最小二乘跟踪的灵敏度分析算法可以有效避免传统的非线性系统灵敏度分析方法中的病态初值问题,因此其在混沌系统灵敏度分析方面有着重要的应用.针对非线性的最小二乘跟踪问题,首先将其重新描述为带有约束的非线性最优控制问题,引入协态变量并将系统的哈密顿函数表示为关于状态变量和协态变量的函数.然后将目标函数的积分时间离散化,根据对偶变量变分原理,以离散区间两端的状态变量作为独立变量,用Lagrange插值多项式近似离散区间内的状态变量和协态变量,进而将非线性最优控制问题转化为求解非线性方程组问题.这种算法无需对原问题做线性化处理,避免了复杂的线性化过程以及可能因此造成的误差,同时为求解非线性最小二乘跟踪问题提供了新的思路.根据最小二乘方法可以得到两条设计参数有微小变化的状态轨迹,基于这两条状态轨迹可进一步计算出系统关于设计参数的灵敏度,范德波振子作为数值算例验证了该方法在求解最小二乘跟踪问题以及计算非线性系统灵敏度时的有效性.      

对偶法求解应力与位移约束不同显式的优化模型

基于响应面法,通过内力将应力约束显式化为正变量的一阶近似,位移约束采用倒变量一阶近似,建立以桁架结构重量最小为目标的截面优化模型,采用对偶规划理论转化为拟无约束模型,采用序列二次规划算法求解。此方法同应力约束零阶近似和一阶近似(使用响应面方法显式化)等方法进行了算例比较,表明本文的优化模型有较好的求解效率。     

Structure-preserving Analysis on Folding and Unfolding Process of Undercarriage

The main idea of the structure-preserving method is to preserve the intrinsic geometric properties of the continuous system as much as possible in numerical algorithm design. The geometric constraint in the multi-body systems, one of the difficulties in the numerical methods that are proposed for the multi-body systems, can also be regarded as a geometric property of the multi-body systems. Based on this idea, the symplectic precise integration method is applied in this paper to analyze the kinematics problem of folding and unfolding process of nose undercarriage. The Lagrange governing equation is established for the folding and unfolding process of nose undercarriage with the generalized defined displacements firstly. And then, the constrained Hamiltonian canonical form is derived from the Lagrange governing equation based on the Hamiltonian variational principle. Finally, the symplectic precise integration scheme is used to simulate the kinematics process of nose undercarriage during folding and unfolding described by the constrained Hamiltonian canonical formulation. From the numerical results, it can be concluded that the geometric constraint of the undercarriage system can be preserved well during the numerical simulation on the folding and unfolding process of undercarriage using the symplectic precise integration method.     

Chebyshev spectral variational integrator and applications

The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rates. The geometric numerical methods are praised for their excellent long-time geometric structure-preserving properties.According to the generalized Galerkin framework, we combine two methods together to construct a variational integrator, which captures the merits of both methods. Since the interpolating points of the variational integrator are chosen as the Chebyshev points,the integration of Lagrangian can be approximated by the Clenshaw-Curtis quadrature rule, and the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of configuration variables and the corresponding derivatives. The numerical float errors of the first-order spectral differentiation matrix can be alleviated by using a trigonometric identity especially when the number of Chebyshev points is large. Furthermore, the spectral variational integrator(SVI) constructed by the Gauss-Legendre quadrature rule and the multi-interval spectral method are carried out to compare with the CSVI, and the interesting kink phenomena for the Clenshaw-Curtis quadrature rule are discovered. The numerical results reveal that the CSVI has an advantage on the computing time over the whole progress and a higher accuracy than the SVI before the kink position. The effectiveness of the proposed method is demonstrated and verified perfectly through the numerical simulations for several classical mechanics examples and the orbital propagation for the planet systems and the Solar system.     

A structure-preserving algorithm for time-scale non-shifted Hamiltonian systems

The variational calculus of time-scale non-shifted systems includes both the traditional continuous and traditional significant discrete variational calculus. Not only can the combination of $\Delta$ and $?$ derivatives be beneficial to obtaining higher convergence order in numerical analysis, but also it prompts the time-scale numerical computational scheme to have good properties, for instance, structure-preserving. In this letter, a structure-preserving algorithm for time-scale non-shifted Hamiltonian systems is proposed. By using the time-scale discrete variational method and calculus theory, and taking a discrete time scale in the variational principle of non-shifted Hamiltonian systems, the corresponding discrete Hamiltonian principle can be obtained. Furthermore, the time-scale discrete Hamilton difference equations, Noether theorem, and the symplectic scheme of discrete Hamiltonian systems are obtained. Finally, taking the Kepler problem and damped oscillator for time-scale non-shifted Hamiltonian systems as examples, they show that the time-scale discrete variational method is a structure-preserving algorithm. The new algorithm not only provides a numerical method for solving time-scale non-shifted dynamic equations but can be calculated with variable step sizes to improve the computational speed.     

New way to construct high order Hamiltonian variational integrators

This paper develops a new approach to construct variational integrators. A simplified unconventional Hamilton’s variational principle corresponding to initial value problems is proposed, which is convenient for applications. The displacement and momentum are approximated with the same Lagrange interpolation. After the numerical integration and variational operation, the original problems are expressed as algebraic equations with the displacement and momentum at the interpolation points as unknown variables. Some particular variational integrators are derived. An optimal scheme of choosing initial values for the Newton-Raphson method is presented for the nonlinear dynamic system. In addition, specific examples show that the proposed integrators are symplectic when the interpolation point coincides with the numerical integration point, and both are Gaussian quadrature points. Meanwhile, compared with the same order symplectic Runge-Kutta methods, although the accuracy of the two methods is almost the same, the proposed integrators are much simpler and less computationally expensive.     

非线性水波Hamilton系统理论与应用研究进展

概述了辛几何理论与辛算法在Hamilton力学中的应用，综述非线性水波的Hamilton理论研究进展．阐述非线性水波Hamilton变分原理与方法的优越性与局限性，探讨KdV方程和BBM方程的Hamilton描述、对称性与守恒律，提出非线性水波Hamilton描述研究中有待进一步研究的问题和解法设想．     

计算最优控制辛数值方法

针对最优控制问题（OCP）的辛数值方法研究及应用进行综述。主要涉及内容包括,动力学系统为常微分方程描述的一般无约束、含不等式约束和状态时滞的最优控制问题,微分代数方程描述的一般无约束、含不等式约束和含切换系统的最优控制问题,以及闭环最优控制问题。从间接法和直接法两个求解框架出发,重点介绍本课题组在保辛算法方面的研究工作。在间接法框架下,首先基于生成函数和变分原理,将OCP保辛离散为非线性方程组,再数值求解方程组。在直接法框架下,将OCP保辛离散为有限维的非线性规划问题（NLP）,再数值求解。针对闭环最优控制问题,提出了保辛模型预测控制、滚动时域估计和瞬时最优控制算法。研究表明,保辛算法具有高精度和高效率的特点,在航空航天和机器人等领域有着广泛应用前景和价值。     

有多余坐标完整系统的自由运动

对于完整力学系统,若选取的参数不是完全独立的,则称为有多余坐标的完整系统. 由于完整力学系统的第二类Lagrange 方程中没有约束力,故为研究完整力学系统的约束力,需采用有多余坐标的带乘子的Lagrange方程或第一类Lagrange 方程. 一些动力学问题要求约束力不能为零,而另一些问题要求约束力很小. 如果约束力为零,则称为系统的自由运动问题. 本文提出并研究了有多余坐标完整系统的自由运动问题. 为研究系统的自由运动,首先,由d'Alembert-Lagrange 原理, 利用Lagrange 乘子法建立有多余坐标完整系统的运动微分方程;其次,由多余坐标完整系统的运动方程和约束方程建立乘子满足的代数方程并得到约束力的表达式;最后,由约束系统自由运动的定义,令所有乘子为零,得到系统实现自由运动的条件. 这些条件的个数等于约束方程的个数,它们依赖于系统的动能、广义力和约束方程,给出其中任意两个条件,均可以得到实现自由运动时对另一个条件的限制. 即当给定动能和约束方程,这些条件会给出实现自由运动时广义力之间的关系. 当给定动能和广义力,这些条件会给出实现自由运动时对约束方程的限制. 当给定广义力和约束方程,这些条件会给出实现自由运动时对动能的限制. 文末,举例并说明方法和结果的应用.