多体系统动力学的常用积分器算法  

IMPLEMENTATION DETAILS OF DAE INTEGRATORS FOR MULTIBODY SYSTEM DYNAMICS

作　　者：任辉 周平 Ren Hui;Zhou Ping(School of Astronautics,Harbin Institute of Technology,Heilongjiang 150001,China)

机构地区：[1]哈尔滨工业大学航天学院,黑龙江150001

出　　处：《动力学与控制学报》2021年第1期1-28,共28页Journal of Dynamics and Control

基　　金：国家自然科学基金资助项目(11772101)。

摘　　要：本文将以单步法中的广义α族积分器和多步法中的BDF族积分器为主要讨论对象,详细介绍大型多体系统动力学软件中常见类型的积分器的算法细节.每族积分器都给出了不止一套计算公式,而且其对应求解微分代数方程组(DAE)的index可以为1、2或者3.除此以外,本文还着重介绍了微分代数方程组的误差估计、变阶变步长策略等关键技术;并讨论了大型DAE问题求解过程中的初始条件分析、Jacobian矩阵复用等重要环节的算法实现;对于BDF积分器族,文中还详细描述了高阶格式的非绝对稳定性、速度变量的误差估计等瓶颈问题的解决方案.全文以多体系统动力学软件的积分器程序实现为目标,强调在满足给定精度的条件下,如何提高计算效率和保证仿真运行的鲁棒性.另外,本文也简要介绍了在某些应用场合中有很大潜力的显式积分器族.通过分析和比较,文中还将指出各种算法的优缺点以及可能的改进方向,希望能够为研究人员和程序开发者提供一定的参考.由于篇幅限制,本文只列出了几个标准的算例比较,作为文中内容的补充;并给出了几种积分器性能比较的一般性结论.文中几乎所有方法都经由作者程序实现、测试和比较,并且相关算法的实现细节也都已尽量列出,可以很容易地编程实现并应用到实际问题的求解中去.The governing equations of multibody system(MBS)dynamics are differential algebraic equations(DAEs),and those equations must be solved by DAE integrators.In this paper,the details of two families of DAE integrators,the backward difference formula(BDF)family and the generalizedαmethod family,were amply described and discussed.At least two formulations were provided in the description of each family of integrators;the numerical schemes of each formulation for index-3,index-2 and index-1 DAEs were depicted in details;and the accuracy,efficiency,and robust⁃ness of all those integrators were discussed.Adaptive step-size techniques were implemented in all the integrators,based on error estimations,and adaptive order technique was implemented in the BDF integrators.The index-3 integrators were usually more efficient than the other schemes,but the calculated accelerations and Lagrange multipliers in those in⁃dex-3 integrators endure spike phenomena,while the corresponding results in index-2 and index-1 schemes were usually smooth.Critical computational procedures,such as the initial condition analysis and reutilizations of Jacobian matrices,were discussed.Moreover,for BDF integrators,an error filtering method was introduced to improve the error estimations for velocity variables in the index-3 integrators,and several stabilization techniques were introduced to solve the issues caused by high order≥3 BDF schemes,which were not absolutely stable.Furthermore,explicit DAE integrators were briefly introduced,which do not require iterations and might be significant in some specific applications.Two bench⁃mark examples were calculated using those integrators,and the pros and cons of each integrator were depicted and dis⁃cussed.Typical integrator algorithms were provided in details in the appendix,which can be directly adopted to practical problems.

关 键 词：微分代数方程组 积分器 向后差分公式 广义α 方法 显式积分器 

分 类 号：O313.7[理学—一般力学与力学基础]