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Regularized numerical integration of multibody dynamics with the generalized alpha method

机译：用广义alpha方法对多体动力学进行正则化数值积分

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摘要

This paper discusses the consistent regularization property of the generalized a method when applied as an integrator to an initial value high index and singular differential-algebraic equation model of a multibody system. The regularization comes from within the discretization itself and the discretization remains consistent over the range of values the regularization parameter may take. The regularization involves increase of the smallest singular values of the ill-conditioned Jacobian of the discretization and is different from Baumgarte and similar techniques which tend to be inconsistent for poor choice of regularization parameter. This regularization also helps where pre-conditioning the Jacobian by scaling is of limited effect, for example, when the scleronomic constraints contain multiple closed loops or singular configuration or when high index path constraints are present. The feed-forward control in Kane's equation models is additionally considered in the numerical examples to illustrate the effect of regularization. The discretization presented in this work is adopted to the first order DAE system (unlike the original method which is intended for second order systems) for its A-stability and same order of accuracy for positions and velocities.
机译：本文讨论了将这种方法作为积分器应用到多体系统的初始值高指数和奇异微分-代数方程模型时的一致正则化性质。正则化来自离散化本身，并且离散化在正则化参数可能取的值范围内保持一致。正则化涉及离散化病态雅可比行列式的最小奇异值的增加，并且不同于Baumgarte和类似技术，后者对于正则化参数选择不佳往往会不一致。这种正则化还有助于在通过缩放对雅可比行进式进行预处理的情况下效果有限，例如，当硬化约束包含多个闭环或奇异配置时，或者存在高索引路径约束时。数值示例中还考虑了凯恩方程模型中的前馈控制，以说明正则化的效果。由于A稳定性和位置和速度的精确度相同，因此在本工作中提出的离散化被采用到一阶DAE系统中（不同于用于二阶系统的原始方法）。

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