基于θ1方法的多体动力学数值算法研究

将结构动力学领域的\theta_1方法拓展到数值求解多体系统运动方程------微分--代数方 程(DAEs), 分别求解指标-3 DAEs形式的运动方程和指标-2超定DAEs (ODAEs)形式的运动方程. 通过数值算例验证了方法的有效性, 并得到\theta _1 方法中参数\theta _1的选取与数值耗散量之间的关系. 数值算例还说明对于同 一个多体系统, 采用指标-3的DAEs 描述时存在速度违约, 用指标-2的ODAEs描述时, 从计算机精度上讲, 位置和速度约束方程 同时满足, 并且\theta_1方法在求解非保守系统DAEs和ODAEs形式的运动方程时 都具有2阶精度. 最后\theta_1 方法与其他直接积分法求解DAEs和ODAEs形式运 动方程的CPU时间进行了比较.

NUMERICAL METHOD OF MULTIBODY DYNAMICS BASED ON θ1 METHOD

In the numerical integration of ordinary differential equations (ODEs) in structural dynamics community, \theta _1 method has characteristics of controlled numerical dissipation and second-order accuracy for systems with or without physical damping. Based on these characteristics, \theta _1 method is extended to the numerical integration of motion equations in multibody system dynamics. The solved motion equations are index-3 differential-algebraic equations (DAEs) and index-2 over-determined DAEs (ODAEs). Numerical experiments validate the \theta_1 method, experiments also show the relationship of numerical dissipation with parameter \theta_1. As for the integration of index-3 DAEs by \theta _1 method, it has violation of velocity constraint, while for index-2 ODAEs, there are no violation of position and velocity constraint in the view of computer precision. In addition, experiments illustrate that, for non-conservative system motion equations in the form of index-3 DAEs and index-2 ODAEs, \theta _1 method has second-order accuracy. In the end, \theta _1 methods for motion equations are compared with other direct-time integrations from the CPU time point of view.