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. 

COMPARATIVE STUDY OF STOCHASTIC AND INTERVAL NON-HOMOGENEOUS LINEAR HAMILTONIAN SYSTEMS AND THEIR APPLICATIONS 1)

With the rapid development of computer technology, there is an urgent need for more efficient and more stable numerical algorithms with more powerful long-term simulation capabilities. Compared with the traditional algorithms, the symplectic algorithms of Hamiltonian systems have significant advantages in stability and long-term simulation. However, a variety of different degrees of uncertainties exist inevitably in the dynamic system, and the impacts of these uncertainties need to be considered in the dynamic analysis to ensure the rationality and effectiveness. Nevertheless, there has been very little research considering parameter uncertainties on the dynamic response analysis of Hamiltonian systems. For this reason, two kinds of uncertain non-homogeneous linear Hamiltonian systems are studied and compared in this paper breaking through the limitations of traditional Hamiltonian systems, where stochastic and interval parameter uncertainties are taken into account, and applied to the evaluation of structural dynamic response. Firstly, for the deterministic non-homogeneous linear Hamiltonian systems, a parameter perturbation method considering deterministic perturbations is proposed. On this basis, the parameter perturbation methods of stochastic and interval non-homogeneous linear Hamiltonian systems are proposed respectively, and the mathematical expressions of the bounds of their response are obtained. Then, the compatibility conclusion that the region of the dynamic response obtained by the interval method contains that obtained by the stochastic one is derived theoretically. Finally, two numerical examples verify the feasibility and effectiveness of the proposed method in structural dynamic response in a smaller time step, and reflect the envelope relationship between the numerical results of the response of stochastic and interval Hamiltonian systems. Also, in a larger time step, the numerical advantages of the symplectic algorithms of Hamiltonian systems are highlighted compared to the traditional algorithms and the accuracy of the proposed method is verified by comparison with the Monte Carlo simulation method.