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Solvability of Geometric Integrators for Multi-body Systems

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Multibody Dynamics

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 35))

Abstract

This chapter is concerned with the solvability of implicit time-stepping methods for simulating the dynamics of multi-body systems. The standard approach is to select a time-step based on desired level of accuracy and computational efficiency of integration. Implicit methods impose an additional but often overlooked requirement that the resulting nonlinear root-finding problem is solvable and has a unique solution. Motivated by empirically observed integrator failures when using large time-steps this work develops bounds on the chosen time-step which guarantee convergence of the root-finding problem solved with Newton’s method. Second-order geometric variational integrators are used as a basis for the numerical scheme due to their favorable numerical behavior. In addition to developing solvability conditions for systems described by local coordinates, this work initiates a similar discussion for Lie group integrators which are a favored choice for floating-base systems such as robotic vehicles or molecular structures.

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Notes

  1. 1.

    \(A\ge B\) for any matrices \(A,B\in \mathbb {R}^{n\times n}\) if and only if \(x^TAx \ge x^T B x\) for all \(x\in \mathbb {R}^n\).

  2. 2.

    Any matrix (including non-symmetric) \(A\in \mathbb {R}^{n\times n}\) is positive definite if \(x^TAx > 0\) for all \(x\in \mathbb {R}^n\) such that \(x\ne 0\).

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Authors and Affiliations

  1. Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD, 21208, USA

    Marin Kobilarov

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  1. Marin Kobilarov
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Correspondence to Marin Kobilarov .

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  1. Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia

    Zdravko Terze

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Kobilarov, M. (2014). Solvability of Geometric Integrators for Multi-body Systems. In: Terze, Z. (eds) Multibody Dynamics. Computational Methods in Applied Sciences, vol 35. Springer, Cham. https://doi.org/10.1007/978-3-319-07260-9_7

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  • DOI: https://doi.org/10.1007/978-3-319-07260-9_7

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