基于θ_1方法的多体动力学数值算法研究

NUMERICAL METHOD OF MULTIBODY DYNAMICS BASED ON θ_1 METHOD

将结构动力学领域的θ_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θ_1方法中参数θ_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θ_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θ_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.

In the numerical integration of ordinary differential equations （ODEs） in structural dynamics community, θ1 method has characteristics of controlled numerical dissipation and second-order accuracy for systems with or without physical damping. Based on these characteristics, θ1 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 θ1 method, experiments also show the relationship of numerical dissipation with parameter 01. As for the integration of index-3 DAEs by θ1 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-20DAEs, θ1 method has second-order accuracy. In the end, θ1 methods for motion equations are compared with other direct-time integrations from the CPU time point of view.