E-mail: renhui@hit.edu.cnrenhui@hit.edu.cn

Implementation Details of DAE Integrators for Multibody System Dynamics
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The National Natural Science Foundation of China (General Program, Key Program, Major Research Plan)

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摘要:

本文将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， 以单步法中的广义$\alpha$族积分器和多步法中的BDF族积分器为主要内容。每族积分器都给出了不止一套计算方案，而且其对应求解的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、变阶变步长策略等核心技术；并讨论了大型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，可以很容易编程实现并应用到实际问题的求解中去。

Abstract:

The governing equations of multibody system (MBS) dynamics are differential algebraic equations (DAEs), and those equations must be solved by DAE integrators. In this work, the details of two families of DAE integrators, the backward difference formula (BDF) family and the generalized $\alpha$ method family, are amply described and discussed. At least two formulations are provided in the description of each family of integrators; the numerical schemes of each formulation for index-3, index-2, and index-1 DAEs are depicted in details; and the accuracy, efficiency, and robustness of all those integrators are discussed. Adaptive step size techniques are implemented in all the integrators, based on error estimations, and adaptive order technique is also implemented in each BDF integrator. The resultant spikes phenomena in calculations of the accelerations and Lagrange multipliers, which are due to index-3 formulations, are automatically resolved in the corresponding index-2 and index-1 schemes; while the index-3 integrators are usually more efficient than the other schemes in the simulations. Critical computational procedures, such as the initial condition analyses and reuse of Jacobian matrices, are described. Moreover, for BDF integrators, an error filtering method is introduced to improve the error estimations for velocity variables in the index-3 integrators, and several stabilization techniques are introduced to solve the issues caused by high order ($\geq3$) BDF schemes, which are not absolutely stable. Furthermore, explicit DAE integrators are briefly introduced, which do not require iterations and might be significant in some specific applications. Two benchmark examples are calculated using those integrators, and the pros and cons of each integrator are depicted and discussed. Typical integrator algorithms are provided in details in the appendix, which can be directly adopted to practical problems.

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##### 历史
• 收稿日期:2019-05-30
• 最后修改日期:2019-11-06
• 录用日期:2019-11-15
• 在线发布日期: 2021-03-08
• 出版日期: 2021-02-20