多体系统指标2运动方程HHT方法违约校正1）

摘要

Equations of motion for multibody system with holonomic constraints in Cartesian absolute coordinates mod-eling method are index 3 differential-algebraic equations (DAEs). It is high index problem for numerical integration of index 3 DAEs. The index can be reduced to 2 by taking the derivative of position constraint equations, and velocity con-straint equations can be obtained. During the integration of index 3 equations of motion, the velocity constraint equations are violated, and there are some problems in the integration of high index DAEs. Firstly, HHT (Hilber-Hughes-Taylor) direct integration method is used to the numerical integration of index 2 equations of motion. The velocity constraint equations involved in the integration, and they are satisfied in the view of computer precision. However, the position constraint equations are violated. Secondly, in order to eliminate the violation, the correction method based on Moore-Penrose generalized inverse theory is adopted. HHT method with constraints violation correction for index 2 equations of motion is the combination of HHT and correction method. There are no position and velocity constraints violation during the integration in the view of computer precision. No new unknown variables are introduced, and the quantity of equations in nonlinear equations from discretization is the same as index 2 equations of motion. The new integration method is vali-dated by numerical experiments. In addition, some characteristics of HHT method, such as controlled numerical damping and second-order accuracy, are persisted by the new integration method. Finally, the quantity of nonlinear equations from discretization and computational efficiency are compared with some other methods. The advantages of the new method are illustrated.%采用Cartesian绝对坐标建模方法，完整约束多体系统运动方程是指标3的微分-代数方程(differential-algebraic equations, DAEs)，数值求解指标3的DAEs属于高指标问题，通过对位置约束方程求导，可使运动方程的指标降为2.位置约束方程求导得到的是速度约束方程.直接求解指标3的运动方程，速度约束方程得不到满足，而且高指标DAEs的数值求解存在一些问题.论文首先采用HHT (Hilber-Hughes-Taylor)直接积分方法求解降指标得到的指标2运动方程，此时速度约束方程参与离散计算，从机器精度上讲速度约束自然得到满足，而位置约束方程没有参与计算，存在“违约”.针对违约问题，采用基于Moore-Penrose广义逆理论的违约校正方法，消除位置约束方程的违约.指标2运动方程HHT方法违约校正，将HHT方法和违约校正方法很好地结合，在数值求解指标2运动方程的过程中，位置约束方程和速度约束方程都不存在违约问题，而且新方法没有引入新的未知数向量，离散得到的非线性方程组的方程数量与原指标2运动方程的方程数量相同，求解规模没有扩大.新方法的实用和有效性通过算例的数值实验得到验证，数值实验也说明新方法保持了HHT方法本身具有的数值阻尼可以控制和二阶精度的特性.最后从非线性方程组的求解规模和计算速度上与其他方法进行了比较分析，说明新方法的优势所在.