基于黎曼几何的非线性振子动力学分析递推解析算法  

A recursive analytical algorithm for dynamics analysis of nonlinear oscillators based on Riemannian geometry

作　　者：杨喆 陈国海[1] 杨迪雄[1] YANG Zhe;CHEN Guo -hai;YANG Di-xiong(State Key Laboratory of Structural Analysis for Industrial Equipment,Department of Engineering Mechanics,Dalian University of Technology,Dalian 116024,China)

机构地区：[1]大连理工大学工程力学系工业装备结构分析国家重点实验室

出　　处：《计算力学学报》2019年第3期310-316,共7页Chinese Journal of Computational Mechanics

基　　金：国家自然科学基金(51478086;11772079);土木工程防灾国家重点实验室开放基金(SLDRCE17-03)资助项目

摘　　要：基于黎曼几何和变分原理,推导了黎曼流形上非线性耗散动力系统的二阶微分动力学方程,并运用流形收缩的概念将动力学方程离散化,进而建立了相应的递推求解格式。选取3个自治非线性阻尼振子系统,分别采用递推解析算法和龙格库塔法求解微分动力学方程,并比较分析了不同的时间步长下两种算法的计算耗时。结果表明,与龙格库塔法相比,基于黎曼几何的递推算法不仅能得到每一时步的解析表达式,而且计算耗时短,计算效率高。基于黎曼流形的动力学方程递推算法为非线性动力学系统的解析求解提供了新思路。Based on Riemannian geometry and a variational principle,this paper derives the second order differential equation of a nonlinear dissipative dynamical system on the Riemannian manifold.The concept of manifold retraction is applied to discretise the dynamic equation,and the corresponding recursive scheme is established.Three autonomous nonlinear damped oscillator systems are taken as examples,and their differential dynamic equations are solved by using the recursive analytical algorithm and the Runge-Kutta algorithm,respectively.The computational time of the two algorithms with different time steps is also compared.Numerical results indicate in comparison with the Runge-Kutta algorithm,the Riemannian geometry-based recursive algorithm can not only achieve the analytical expression of dynamic equations in each time step,but also its running time is shorter than that of the former with higher computational efficiency.The recursive algorithm for dynamic equations based on Riemannian manifolds offers a new idea for analytically solving nonlinear dynamic systems.

关 键 词：非线性振子 黎曼几何 递推解析算法 龙格库塔法 计算效率 

分 类 号：O322[理学—一般力学与力学基础]