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首页> 外文期刊>Journal of Computational and Nonlinear Dynamics >Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations
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Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations

机译：多体动力学仿真中约束条件的缩放和增广的拉格朗日公式

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摘要

This paper addresses practical issues associated with the numerical enforcement of constraints in flexible multibody systems, which are characterized by index-3 differential algebraic equations (DAEs). The need to scale the equations of motion is emphasized; in the proposed approach, they are scaled based on simple physical arguments, and an augmented Lagrangian term is added to the formulation. Time discretization followed by a linearization of the resulting equations leads to a Jacobian matrix that is independent of the time step size, h; hence, the condition number of the Jacobian and error propagation are both O(h{sup}0): the numerical solution of index-3 DAEs behaves as in the case of regular ordinary differential equations (ODEs). Since the scaling factor depends on the physical properties of the system, the proposed scaling decreases the dependency of this Jacobian on physical properties, further improving the numerical conditioning of the resulting linearized equations. Because the scaling of the equations is performed before the time and space discretizations, its benefits are reaped for all time integration schemes. The augmented Lagrangian term is shown to be indispensable if the solution of the linearized system of equations is to be performed without pivoting, a requirement for the efficient solution of the sparse system of linear equations. Finally, a number of numerical examples demonstrate the efficiency of the proposed approach to scaling.
机译：本文讨论了与柔性多体系统中约束的数值执行相关的实际问题，这些问题以index-3微分代数方程（DAE）为特征。强调了缩放运动方程的必要性;在提出的方法中，基于简单的物理参数对它们进行缩放，并在公式中添加一个扩展的拉格朗日项。时间离散化后的结果方程式线性化，得出的雅可比矩阵与时间步长h无关;因此，雅可比行列式的条件数和误差传播都为O（h {sup} 0）：索引3 DAE的数值解的行为与正则常微分方程（ODE）相同。由于缩放因子取决于系统的物理属性，因此建议的缩放比例减小了该Jacobian对物理属性的依赖性，从而进一步改善了所得线性化方程的数值条件。由于方程式的缩放是在时间和空间离散化之前执行的，因此它的优点可用于所有时间积分方案。如果要在不旋转的情况下执行线性化方程组的解，则必须显示增强的拉格朗日项，这是有效解决线性方程组稀疏系统的要求。最后，许多数值示例证明了所提出的缩放方法的效率。

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