多体动力学的几何积分方法研究进展

PROGRESS IN GEOMETRIC INTEGRATION METHOD FOR MULTIBODY DYNAMICS

动力系统的几何积分研究是近20年来工程计算领域非常活跃的方向.多体动力学方程(微分方程,微分代数方程)是一类典型的动力系统,将其从Lagrange体系向Hamilton系统过渡,目的在于从欧氏几何过渡到辛几何形态,将对偶变量引入到力学研究中,然后利用辛几何的数学框架对多体系统动力学方程进行数值计算,可以预知多体动力学系统的一些定性信息,并在数值离散时能保持这些定性性质特征,尤其在表示关键的物理意义时需要强调保持这些几何性质.简要介绍多体系统(无约束多刚体系统、完整约束多刚体系统和柔性多体系统)的Hamilton正则方程的建立和几何积分方法的构造,着重介绍了在多体动力学计算中非常有应用前景的高阶辛算法(合成辛算法、分裂合成辛算法和辛精细积分法)、多辛算法,以及广义Hamilton系统与Lie群积分方法等计算几何力学方法,并对Lie群积分的投影方法、流形局部坐标法等方法进行了阐述.

The geometric integration method of dynamical systems has been an attractive direction in the last two decades. Dynamic equations of multibody systems, such as differential equation and differential-algebraic equation, are a kind of representative dynamical systems. The significance of the transformation from Lagrange framework to Hamilton framework is the configuration transformation from Euclidian to Hamiltonian. The symplectic variable is introduced into the mechanics system, and thus the symplectic integration method can be adopted to solve the dynamic equations. Certain qualitative information of the multibody dynamic system can be predicted and is expected to be maintained in the process of discretization, which is of particular signif- icance when this qualitative information indicates key physical meaning. How to establish proper Hamiltonian canonical equations of the multibody system （multi-rigid body system without constraint or with holonomic constraint, flexible multibody system） is briefly described, with emphases laid on the formulation of geometric integral methods, especially the computational geometric mechanics methods with promising applications, such as the high-order symplectic algorithm （synthesized algorithm, Partition-synthesized algorithm, symplectic precise integration algorithm）, multi-symplectic algorithm and Lie group algorithm （projected method and located coordination method）.