基于向后差分法求解多体系统动力学微分-代数方程组的双循环隐式积分方法

Two-loop Implicit Integration Method Based on Backward Differential Formulation for Differential-algebraic Equations of Multibody System Dynamics

在利用坐标缩并方法求解多体系统动力学指标3的微分-代数方程组的过程中,由隐式积分方法进行积分时需要进行迭代求解,采用牛顿法进行迭代时需要利用数值微分求得雅可比矩阵。通过引入固定点迭代以避免用于计算雅可比矩阵的数值微分。非线性代数约束方程组的求解也需要进行迭代,两组迭代一起构成一种双循环的格式。双循环中隐式积分方法的数值精度影响外层循环的迭代次数。将向后差分法引入双循环隐式积分方法中作为积分方法,并针对向后差分法的特点提出新的迭代求解策略,构造一种新的双循环隐式积分方法。这一新的双循环隐式积分方法中外层循环的迭代次数减少,计算效率得到了显著提高。这一方法能够很好地解决指标3的多体系统动力学微分-代数方程组,具有良好的通用性。给出了数值算例。

An iterative method is needed when implicit integration method is used to integrate the independent coordinates of differential-algebraic equations（DAEs） which come from multibody system dynamics. If the iterative method is Newton＇s method, numerical differentiation is needed to obtain the Jacobian matrix. A fixed-point iterative method without the using of the Jacobian matrix can simplify the progress. The nonlinear algebraic constraint equations in DAEs are also solved by iteration to obtain the dependent coordinates. A two-loop structure is designed to manage those two iterations. The numerical accuracy of the integration method influences the numbers of iteration of implicit integration method which is called as the outer loop. Backward differential formulation（BDF） is introduced as the integration method into the two-loop method. New strategy is proposed to constitute the new two-loop method. The numbers of iteration of outer loop in the new two-loop method are reduced, and the computational efficiency is improved. The new two-loop method can solve the DAEs of multibody system dynamics well, and its universality is good. Numerical examples are provided.