精细辛有限元方法及其相位误差研究

哈密顿系统是一类重要的动力系统,针对哈密顿系统,设计出多类辛方法:SRK、SPRK、辛多步法、生成函数法等.长久以来数值方法在求解哈密顿系统过程中辛特性和保能量特性不能得到同时满足,近年来提出的有限元方法,对于线性系统具有保辛和保能量的优良特性.但是,以上方法都存在相位漂移(轨道偏离)现象,长时间仿真,计算效果会大打折扣.提出精细辛有限元方法(HPD-FEM)求解哈密顿系统,该方法继承时间有限元方法求解哈密顿系统所具有的保哈密顿系统的辛结构和哈密顿函数守恒性的优良特性,同时,通过精细化时间步长极大地减小了时间有限元方法的相位误差.HPD-FEM相较与针对相位误差专门设计的计算格式FSJS、RKN以及SRPK方法具有更好的纠正效果,几乎达到机器精度,误差为O(10-13),同时,HPD-FEM克服了FSJS、RKN和SPRK方法不能保证哈密顿函数守恒的缺点.对于高低混频系统和刚性系统,常规算法很难在较大步长下,同时实现对高低频精确仿真,HPD-FEM通过精细计算时间步长,在大步长情况下,实现高低混频的精确仿真.HPD-FEM方法在计算过程中精细方法没有额外增加计算量,计算效率高.数值结果显示本文提出的方法切实有效.      

单步辛算法的相位误差分析及修正

若一个算法的幅值误差和相位误差都不累加，则该算法就是最理想的算法， 但这样的算法难以构造. 辛几何算法解决了幅值误差的累加问题，但相位误差累加问题仍然 存在. 给出了单步隐式辛算法相位误差的精确估计公式，提出了简单而实用的修正方法. 以 Euler中点隐式辛差分格式为例，针对几个线性动力学系统，对相位误差进行了数值分析和 修正.     

基于对偶变量变分原理的完整约束系统保辛算法

基于对偶变量变分原理，选择积分区间两端位移为独立变量，构造了求解完整约束哈密顿动力系统的高阶保辛算法。首先，利用拉格朗日多项式对作用量中的位移、动量及拉格朗日乘子进行近似；然后，对作用量中不包含约束的积分项采用Gauss积分近似，对作用量中包含约束的积分项采用Lobatto积分近似，从而得到近似作用量；最后，在此近似作用量的基础上，利用对偶变量变分原理，将求解完整约束哈密顿动力系统问题转化为一组非线性方程组的求解。算法具有保辛性和高阶收敛性，能够在位移的插值点处高精度地满足完整约束。算法的收敛阶数及数值性质通过数值算例验证。     

哈密顿系统正则变换在时变最优控制中的应用

利用哈密顿系统正则变换和生成函数理论求解线性时变最优控制问题,构造了新的最优控制律形式并提出了控制增益计算的保结构算法. 利用生成函数求解最优控制导出的哈密顿系统两端边值问题,并构造线性时变系统的最优控制律,由第2类生成函数所构造的最优控制律避免了末端时刻出现无穷大反馈增益. 控制系统设计中需求解生成函数满足的时变矩阵微分方程组. 根据生成函数与哈密顿系统状态转移矩阵之间的关系,从正则变换的辛矩阵描述出发,导出了求解这组微分方程组的保结构递推算法.为了保持递推计算中的辛矩阵结构,哈密顿系统状态转移矩阵的计算中利用了Magnus级数.      

随机和区间非齐次线性哈密顿系统的比较研究及其应用

随着计算机技术的飞速发展,更高效、更稳定和长时间模拟能力更强的数值算法需求迫切.哈密顿系统辛算法与传统算法相比在稳定性和长期模拟方面具有显著优越性.但动力系统中不可避免地存在大量不同程度的不确定性,动力学分析中需要考虑这些不确定性的影响以确保合理有效性. 然而,目前考虑参数不确定性的哈密顿系统响应分析的研究基础还比较薄弱. 为此,本文考虑随机和区间参数不确定性,对两种不确定性非齐次线性哈密顿系统分析计算结果进行了比较研究,从而突破了传统哈密顿系统的局限性, 并应用于结构动力响应评估中. 首先,针对确定性非齐次线性哈密顿系统, 提出了考虑确定性扰动的参数摄动法;在此基础上, 分别提出了随机、区间非齐次线性哈密顿系统的参数摄动法,得到了它们响应界限的数学表达; 随后,用数学理论推导得到了区间响应范围包含随机响应范围的相容性结论; 最后,两个数值算例在较小时间步长下验证了所提方法在结构动力响应中的可行性和有效性,体现了随机、区间哈密顿系统响应结果之间的包络关系,并在较大时间步长下与传统方法相比较凸显了哈密顿系统辛算法的数值计算优势、与蒙特卡洛模拟方法相比较验证了所提方法的精度.      

李级数算法和显式辛算法的相位分析

以线性可分Hamilton动力学系统为例,研究了李级数算法和显式辛算法的相位精度,研究了李级数算法的保辛精度及其保辛精度的提高方法;指出了显式辛算法相位精度与算法阶次的不协调性,印辛算法的阶次高并不意味着其相位精度也高,李级数算法不存在这种问题,指出了一个算法的相位可能超前也可能滞后.分析结果表明三阶显式辛算法具有比较高的相位精度.     

Symplectic multi-level method for solving nonlinear optimal control problem

By converting an optimal control problem for nonlinear systems to a Hamiltonian system,a symplecitc-preserving method is proposed.The state and costate variables are approximated by the Lagrange polynomial.The state variables at two ends of the time interval are taken as independent variables.Based on the dual variable principle,nonlinear optimal control problems are replaced with nonlinear equations.Furthermore,in the implementation of the symplectic algorithm,based on the 2N algorithm,a multilevel method is proposed.When the time grid is refined from low level to high level,the initial state and costate variables of the nonlinear equations can be obtained from the Lagrange interpolation at the low level grid to improve efficiency.Numerical simulations show the precision and the efficiency of the proposed algorithm in this paper.     

GENERALIZED CANONICAL TRANSFORMATION AND SYMPLECTIC ALGORITHM OF THE AUTONOMOUS BIRKHOFFIAN SYSTEMS

The Birkhoff systems are the generalization of the Hamiltonian systems. Generalized canonical transformations are studied. The symplectic algorithm of the Hamiltonian systems is extended into that of the Birkhoffian systems . Symplectic differential scheme of autonomous Birkhoffian systems was structured and discussed by introducing the Kailey Transformation .     

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.     

Mixed-mode thermal stress intensity factors from the finite element discretized symplectic method

A finite element discretized symplectic method is introduced to find the thermal stress intensity factors (TSIFs) under steady-state thermal loading by symplectic expansion. The cracked body is modeled by the conventional finite elements and divided into two regions: near and far fields. In the near field, Hamiltonian systems are established for the heat conduction and thermoelasticity problems respectively. Closed form temperature and displacement functions are expressed by symplectic eigen-solutions in polar coordinates. Combined with the analytic symplectic series and the classical finite elements for arbitrary boundary conditions, the main unknowns are no longer the nodal temperature and displacements but are the coefficients of the symplectic series after matrix transformation. The TSIFs, temperatures, displacements and stresses at the singular region are obtained simultaneously without any post-processing. A number of numerical examples as well as convergence studies are given and are found to be in good agreement with the existing solutions.     

含裂纹纳米板振动问题的哈密顿体系方法

基于裂纹处范德华力效应,采用非局部弹性理论构造纳米板模型,并通过导入哈密顿体系建立含裂纹纳米板振动问题的对偶正则控制方程组。在全状态向量表示的哈密顿体系下,将含裂纹纳米板的固有频率和振型问题归结为广义辛本征值和本征解问题。利用哈密顿体系具有的辛共轭正交关系,得到问题解的级数解析表达式。结合边界条件,得到固有频率与辛本征值的代数方程关系式,进而直接给出固有频率的表达式。数值结果表明,非局部尺寸参数和裂纹长度对纳米板振动的各阶固有频率有直接的影响。对比表明,辛方法是准确且可靠的,可为工程应用提供依据。     

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．     

An eigenelement method and two homogenization conditions

Under inspiration from the structure-preserving property of symplectic difference schemes for Hamiltonian systems, two homogenization conditions for a representative unit cell of the periodical composites are proposed, one condition is the equivalence of strain energy, and the other is the deformation similarity. Based on these two homogenization conditions, an eigenelement method is presented, which is characteristic of structure-preserving property. It follows from the frequency comparisons that the eigenelement method is more accurate than the stiffness average method and the compliance average method.     

多体动力学的几何积分方法研究进展

动力系统的几何积分研究是近20年来工程计算领域非常活跃的方向.多体动力学方程(微分方程, 微分代数方程)是一类典型的动力系统,将其从Lagrange体系向Hamilton系统过渡,目的在于从欧氏几何过渡到辛几何形态, 将对偶变量引入到力学研究中,然后利用辛几何的数学框架对多体系统动力学方程进行数值计算,可以预知多体动力学系统的一些定性信息,并在数值离散时能保持这些定性性质特征,尤其在表示关键的物理意义时需要强调保持这些几何性质.简要介绍多体系统(无约束多刚体系统、完整约束多刚体系统和柔性多体系统)的Hamilton正则方程的建立和几何积分方法的构造,着重介绍了在多体动力学计算中非常有应用前景的高阶辛算法(合成辛算法、分裂合成辛算法和辛精细积分法)、多辛算法,以及广义Hamilton 系统与Lie 群积分方法等计算几何力学方法, 并对Lie群积分的投影方法、流形局部坐标法等方法进行了阐述.      

粘弹性厚壁筒问题的辛本征解方法

将哈密顿体系引进到粘弹性力学厚壁筒问题中,在辛体系下重新描述了基本问题,即建立了正则方程组。借助于积分变换,得到了拉伸、扭转和弯曲等问题的解以及有边界局部效应的解。将原问题归结为辛几何空间中的零本征值本征解和非零本征值本征解问题,从而建立了一种有效的分析问题方法和数值方法。为解决同类问题提供了一条可行的路径。     

Hamilton体系下介电弹性体圆形薄膜的动力学建模与辛求解

采用辛算法研究了Hamilton体系下介电弹性体圆形薄膜的动力学响应。首先，将该问题引入Hamilton对偶变量体系，借助Legendre变换，给出系统的广义动量和Hamilton函数，通过对Hamilton函数作用量的变分，得到Hamilton体系下的正则方程。其次，对于得到的正则方程给出了辛Runge-Kutta的计算格式。最后，采用二级四阶辛Runge-Kutta算法对动力学系统进行了数值求解，和四级四阶经典Runge-Kutta算法进行对比，结果表明，二级四阶辛Runge-Kutta算法具有保能量以及长时间数值稳定的优势，同时说明四级四阶经典Runge-Kutta算法对于步长依赖的局限性。     

EIGENVALUE PROBLEM OF A LARGE SCALE INDEFINITE GYROSCOPIC DYNAMIC SYSTEM

Gyroscopic dynamic system can be introduced to Hamiltonian system.Based on an adjoint symplectic subspace iteration method of Hamiltonian gyroscopic system, an adjoint symplectic subspace iteration method of indefinite Hamiltonian function gy- roscopic system was proposed to solve the eigenvalue problem of indefinite Hamiltonian function gyroscopic system.The character that the eigenvalues of Hamiltonian gyroscopic system are only pure imaginary or zero was used.The eigenvalues that Hamiltonian function is negative can be separated so that the eigenvalue problem of positive definite Hamiltonian function system was presented,and an adjoint symplectic subspace iteration method of positive definite Hamiltonian function system was used to solve the separated eigenvalue problem.Therefore,the eigenvalue problem of indefinite Hamiltonian function gyroscopic system was solved,and two numerical examples were given to demonstrate that the eigensolutions converge exactly.     

动力学平衡方程的辛两步求解算法

基于线性多步方法的构造格式和辛变换,给出了动力学方程的两种辛两步法求解格式,它们分别具有四阶精度和二阶精度,但都只有二阶格式的计算量,因此四阶辛两步法具有较大的应用价值。对两种辛两步法和解析解进行了数值比较,证明了二阶精度辛两步格式在一定条件下就是欧拉中点保辛算法,或δ=0.5和α=0.25的Newmark辛格式。     

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.     

平面刚体系统的参数预调节保辛算法

保辛积分方法在约束哈密顿系统中有着重要的应用,是因为其在长时间仿真中表现出极好的稳定性。然而随着仿真时长增加,保辛格式通常具有较大的相位误差累积。本文提出了一种平面多刚体系统的参数预调节保辛积分方法。通过推导具有待定参数的改进的拉格朗日方程,并将其与已有保辛格式相结合并预先调节相关参数取值,可以大幅降低数值解的相位误差。理论分析与数值结果表明参数预调节保辛积分方法不仅保持了辛结构,而且具有很低的相位误差累积。因此,参数预调节保辛积分方法可应用于长时间仿真分析。