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 mo- mentum 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.

A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications

In this paper,a novel symplectic conservative perturbation series expansion method is proposed to investigate the dynamic response of linear Hamiltonian systems accounting for perturbations,which mainly originate from parameters dispersions and measurement errors.Taking the perturbations into account,the perturbed system is regarded as a modification of the nominal system.By combining the perturbation series expansion method with the deterministic linear Hamiltonian system,the solution to the perturbed system is expressed in the form of asymptotic series by introducing a small parameter and a series of Hamiltonian canonical equations to predict the dynamic response are derived.Eventually,the response of the perturbed system can be obtained successfully by solving these Hamiltonian canonical equations using the symplectic difference schemes.The symplectic conservation of the proposed method is demonstrated mathematically indicating that the proposed method can preserve the characteristic property of the system.The performance of the proposed method is evaluated by three examples compared with the Runge-Kutta algorithm.Numerical examples illustrate the superiority of the proposed method in accuracy and stability,especially symplectic conservation for solving linear Hamiltonian systems with perturbations and the applicability in structural dynamic response estimation.

基于辛算法的纳米加工过程的分子动力学仿真研究

近年来,从统计物理学中衍生出来的分子动力学仿真技术(Moleculardynamicssimulation,MDS)做为一种描述微观现象的有效方法,得到人们越来越广泛的重视,并成功应用于包括纳米加工技术在内的许多微观研究领域。随着分子动力学仿真技术的日益成熟,作为提高其计算精度及效率基础的仿真算法的研究也日益引起人们的关注。在运算周期很长的分子动力学仿真研究中,实践证明辛算法(Symplecticalgorithm)是一种高精度的收敛算法。以一维谐振摆为例,分析比较了几种辛算法和非辛算法的优劣,并进行了纳米加工的分子动力学仿真,从中可以看出辛算法有效地控制了动力学计算的误差积累,保证了计算精度。

冲击荷载作用下中心对称薄圆板振动的多辛分析

基于Bridges建立的多辛理论,构造了中心对称薄圆板振动方程的多辛对称形式及其多种局部守恒律,针对振动方程的多辛形式,采用Euler Box差分离散方法构造其多辛格式,利用计算机模拟,研究了冲击荷载作用下中心对称薄圆板的振动问题,并在模拟过程中重点关注多辛算法是否精确保持振动系统的局部几何性质,该研究结果为薄板振动问题提供了新的数值研究途径。

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%。

常微分方程数值解法在动力天文中的应用

在动力天文中,利用常微数值解法求解运动方程时,通常会出现两个问题——解的定性性质受到歪曲和轨道沿迹误差的严重累积。本文将介绍利用运动本身的一些力学性质作为控制条件,能够有效地解决这些问题,特别是对Hamiltom系统,采用保持辛结构的差分格式——半积分器(Symplectic Imtegrators),有它独特的优点,而且可以将相应的差分格式稍作修改,就可用于小耗散系统,这对动力天文而言也是很重要的。