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.      

非线性弹性杆中纵波传播过程的数值模拟

基于Hamilton空间体系的多辛理论研究了包含材料非线性效应和几何弥散效应的非线性弹性杆中纵波传播问题。导出了其Bridges意义下的多辛形式及其多种守恒律,并构造了等价于Box多辛格式的隐式多辛格式对不考虑材料非线性效应和几何弥散效应、只考虑材料非线性效应、只考虑几何弥散效应、同时考虑材料非线性效应和几何弥散效应四种情况下不同截面参数的圆杆中的纵波传播过程进行数值模拟,模拟结果不仅全面地反映了非线性效应和几何弥散效应对纵波传播的影响,而且也反映了多辛方法的两大优点:精确的保持多种守恒律和良好的长时间数值行为。     

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.     

动力学平衡方程的Euler中点辛差分求解格式

给出了动力学方程${\pmb M}\ddot {\pmb x} + {\pmb C}\dot {\pmb x} + {\pmb K \pmb x} = {\pmb R}$的二阶Euler中点隐式差分求解格式，分保守系统、无 阻尼受迫振动系统和阻尼系统3种情况, 讨论了算法中Jacobi矩阵${\pmb A}$的性质，譬 如${\pmb A}$是否为辛矩阵以及谱半径等. 对于无阻尼系统，证明了无论是否存在外 载荷，Jacobi 矩阵都是辛矩阵. 证明了辛矩阵的所有本征值的模为1，其谱半径永远 为1, 以及$\delta = 0.5$和$\alpha = 0.25$的Newmark算法就是Euler中点隐式差 分格式，对保守系统它们都是辛算法. 严格证 明了Euler中点辛格式是严格保持系统能量的. 通过算例详细讨论了保辛算法用于求解非保 守系统动态特性的优越性，如广义保结构特性等；分析了保辛算法的相位误差以及由其引起 的系统的附加能量特性；分析了保辛算法和$\delta \ne 0.5$的Newmark算法的精度随着激励频率与系统固有频率比的变化情况等     

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。     

与结构动特性协同的自适应Newmark方法

提出了一种与结构动特性协同的自适应Newmark方法,其参数可基于数值弥散和数值耗散最小化的条件用解析方法求得.对线性单自由度动力学系统,该方法的相位误差精确为零并且谱半径为1.对线性多自由度系统和非线性系统,该方法在所有二阶积分解法中最精确.数值结果验证了新提出格式的高精度和结论.      

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.     

UKF在深组合GPS/INS导航系统中的应用

采用联邦滤波的深组合GPS/INS导航系统预滤波器量测模型具有很强的非线性,导致扩展卡尔曼滤波（EKF）的预滤波器估计精度不高。Unscented卡尔曼滤波（UKF）方法是一种非线性分布近似方法,它使用有限数量的sigma点去逼近整个非线性动态系统的分布可能,从而避免了对非线性测量模型进行线性化,具有较高的精度和较好的鲁棒性。在分析深组合导航系统预滤波器模型和UKF原理的基础上,设计了基于UKF滤波算法的预滤波器,对码相位误差、载波相位误差、载波频率误差、载波频率变化率等参数进行估计,同时将UKF和EKF算法进行了仿真比较。结果表明,在深组合导航系统中使用UKF滤波比EKF有更高的导航定位精度。     

基于混合Lie算子辛算法的不变流形计算

平动点附近周期轨道的不变流形因其在低能轨道转移中起着重要作用而受到广泛关注.在设计低能轨道过程中不变流形要实时进行能量匹配,但利用传统数值积分方法进行积分时能量会耗散.显式辛算法具有比隐式辛算法计算效率高的优势,但其要求Hamilton系统必须分成两个可积的部分,而旋转坐标系下的圆型限制性三体问题是不可分的,因而显式辛算法难以用于求解旋转坐标系下的圆型限制性三体问题.本文通过引入混合Lie算子,成功实现了带三阶导数项的力梯度辛算法对圆型限制性三体问题的求解,并将基于混合Lie算子的带三阶导数项的辛算法与Runge-Kutta78算法和Runge-Kutta45算法进行仿真对比,仿真结果表明基于混合Lie算子的含有三阶导数项的辛算法位置精度高、能量误差小且计算效率高.利用基于混合Lie算子的带三阶导数项的辛算法计算不变流形,可以实现低能轨道转移过程中轨道拼接点的能量精准匹配.     

一种基于惯性测量与自适应滤波的舰船升沉计算方法

针对舰船作业过程中升沉运动精确测量的应用需求,提出了基于惯性导航系统的升沉运动测量方案。针对海况变化条件下,传统固定滤波参数数字滤波器自适应性不足,以及存在相位偏移和幅值误差等问题,提出了一种结合互补方法的自适应数字高通滤波器。该滤波器基于先验知识与实时海况监测结果,可自适应调整滤波器参数。首先通过一滑动监测窗口对海况进行实时监测,并对滤波器参数进行实时自适应调整;随后对加速度测量值进行2次积分与3次自适应高通滤波,滤除低频误差,保留有效高频升沉运动。不同海况条件下的仿真与试验验证了测量与滤波方案的可行性。试验结果表明:新滤波方法相比传统方法,能根据海况实时自适应调整滤波参数,升沉运动相位误差和幅值误差减小,仿真实验中幅值误差缩减约10倍,模拟试验中幅值误差缩减约2倍,测量精度精确到厘米级。     

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.     

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。     

Implicit time accurate simulation of unsteady flow

Implicit time integration was studied in the context of unsteady shock‐boundary layer interaction flow. With an explicit second‐order Runge–Kutta scheme, a reference solution to compare with the implicit second‐order Crank–Nicolson scheme was determined. The time step in the explicit scheme is restricted by both temporal accuracy as well as stability requirements, whereas in the A‐stable implicit scheme, the time step has to obey temporal resolution requirements and numerical convergence conditions. The non‐linear discrete equations for each time step are solved iteratively by adding a pseudo‐time derivative. The quasi‐Newton approach is adopted and the linear systems that arise are approximately solved with a symmetric block Gauss–Seidel solver. As a guiding principle for properly setting numerical time integration parameters that yield an efficient time accurate capturing of the solution, the global error caused by the temporal integration is compared with the error resulting from the spatial discretization. Focus is on the sensitivity of properties of the solution in relation to the time step. Numerical simulations show that the time step needed for acceptable accuracy can be considerably larger than the explicit stability time step; typical ratios range from 20 to 80. At large time steps, convergence problems that are closely related to a highly complex structure of the basins of attraction of the iterative method may occur. Copyright © 2001 John Wiley & Sons, Ltd.     

Multiresolution Symplectic Scheme for wave propagation in complex media

A fast adaptive symplectic algorithm named Multiresolution Symplectic Scheme (MSS) was first presented to solve the problem of the wave propagation (WP) in complex media, using the symplectic scheme and Daubechies‘ compactly supported orthogonal wavelet transform to respectively discretise the time and space dimension of wave equation. The problem was solved in multiresolution symplectic geometry space under the conservative Hamiltonian system rather than the traditional Lagrange system. Due to the fascinating properties of the wavelets and symplectic scheme, MSS is a promising method because of little computational burden, robustness and reality of long-time simulation.     

Structure-preserving approach for infinite dimensional nonconservative system

The current structure-preserving theory, including the symplectic method and the multisymplectic method, pays most attention on the conservative properties of the continuous systems because that the conservative properties of the conservative systems can be formulated in the mathematical form. But, the nonconservative characteristics are the nature of the systems existing in engineering. In this letter, the structure-preserving approach for the infinite dimensional nonconservative systems is proposed based on the generalized multi-symplectic method to broaden the application fields of the current structure-preserving idea. In the numerical examples,two nonconservative factors, including the strong excitation on the string and the impact on the cantilever, are considered respectively. The vibrations of the string and the cantilever are investigated by the structure-preserving approach and the good long-time numerical behaviors as well as the high numerical precision of which are illustrated by the numerical results presented.     

Numerical solutions of linear quadratic control for time-varying systems via symplectic conservative perturbation

Optimal control system of state space is a conservative system, whose approximate method should be symplectic conservation. Based on the precise integration method, an algorithm of symplectic conservative perturbation is presented. It gives a uniform way to solve the linear quadratic control (LQ control) problems for linear time-varying systems accurately and efficiently, whose key points are solutions of differential Riccati equation (DRE) with variable coefficients and the state feedback equation. The method is symplectic conservative and has a good numerical stability and high precision. Numerical examples demonstrate the effectiveness of the proposed method.     

Numerical solutions of linear quadratic control for time-varying systems via symplectic par conservative perturbation

Optimal control system of state space is a conservative system, whose approximate method should be symplectic conservation. Based on the precise integration method, an algorithm of symplectic conservative perturbation is presented.It gives a uniform way to solve the linear quadratic control (LQ control) problems for linear time-varying systems accurately and efficiently, whose key points are solutions of differential Riccati equation (DRE) with variable coefficients and the state feedback equation.The method is symplectic conservative and has a good numerical stability and high precision. Numerical examples demonstrate the effectiveness of the proposed method.     

Structure-preserving properties of three differential schemes for oscillator system

A numerical method for the Hamiltonian system is required to preserve some structure-preserving properties. The current structure-preserving method satisfies the requirements that a symplectic method can preserve the symplectic structure of a finite dimension Hamiltonian system, and a multi-symplectic method can preserve the multi-symplectic structure of an infinite dimension Hamiltonian system. In this paper,the structure-preserving properties of three differential schemes for an oscillator system are investigated in detail. Both the theoretical results and the numerical results show that the results obtained by the standard forward Euler scheme lost all the three geometric properties of the oscillator system, i.e., periodicity, boundedness, and total energy,the symplectic scheme can preserve the first two geometric properties of the oscillator system, and the St¨ormer-Verlet scheme can preserve the three geometric properties of the oscillator system well. In addition, the relative errors for the Hamiltonian function of the symplectic scheme increase with the increase in the step length, suggesting that the symplectic scheme possesses good structure-preserving properties only if the step length is small enough.