多体动力学的多尺度保辛摄动理论与算法

The efficient, stable and high accuracy numerical algorithm plays an important role in the study of multibody dynamics. In this project, on the basis of analytical structural mechanics, the symplectic algebra theory will be developded in multibody dynamics, and the symplectic-preserving numerical method with high accuracy and high stability will be established. In addition, the theory and algorithm of multi-scale symplectic conservative perturbation will be specially focused. The details of the research are presented as follows. Firstly, a symplectic-preserving integration algorithm for the multi-rigid-body system in terms of quaternions will be proposed, in which the analytical structural mechanics theory is applied and the conventional difference scheme is replaced by the time finite element method, and then an accurate and stable numerical method of symplectic-preserving for the general differential-algebraic equations in multibody system will be established. Secondly, in according to the property of the transfer symplectic matrix group, the multi-scale symplectic conservative perturbation theory and algorithm for the rigid-flexible coupling dynamics will be developed, which would provide an efficient numerical simulation method for the time multi-scale problem in the multibody system. Furthermore, an efficient and stable symplectic-preserving algorithm for the large and complex flexible multibody systems will be proposed by combination of the multi-level substructure method and the symplectic conservative perturbation method.

高效、稳定、高精度数值算法的研究在多体动力学中占有重要地位。本项目以分析结构力学为基础，拟在多体动力学领域展开辛代数理论研究，并建立相应的高精度、高稳定、保辛数值算法，特别是建立多尺度保辛摄动理论与算法。具体研究安排如下：首先，根据分析结构力学理论，拟采用时间有限元代替传统差分格式，构造四元数表述的多刚体系统保辛积分方法，并进一步建立针对一般多体系统微分代数方程的高精度、高稳定性的保辛数值方法；其次，针对刚-柔耦合动力学，根据"传递辛矩阵群"的性质，展开多尺度保辛摄动理论与方法的研究工作，为多体系统时间不同尺度问题的应用研究提供有效的数值分析手段；此外，针对大规模、复杂多柔体系统，将多重子结构算法与保辛摄动方法相结合，构造针对复杂结构的高效、稳定保辛算法。

高效、稳定、高精度数值算法在多体动力学研究中占有重要地位。本项目以分析结构力学为基础，在多体动力学领域开展保辛算法研究，主要研究内容包括：1) 发现了多体动力学新的动能表达形式，并发展了系列高精度、高稳定性的保辛算法；2)动力学子系统与乘法摄动结构，建立了复杂刚-柔动力学问题的多尺度保辛摄动方法。本项目所提出的算法，可以用于机器人、航天器、空间结构、海洋工程等出现的非线性多体动力学问题，为发展自主CAE软件系统提供理论和技术支撑。