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Archive of Mechanical Engineering

The Journal of Committee on Machine Building of Polish Academy of Sciences

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Volume 61, Issue 2 (Aug 2014)

Geometric Interpretation of a Non-Linear Beam Finite Element on The Lie Group SE(3)

Interpretacja Geometryczna Nieliniowego Belkowego Elementu Skończonego w Formalizmie Grupy Liego SE(3)

Valentin Sonneville
  • University of Liege, Department of Aerospace and Mechanical Engineering (LTAS), Chemin des Chevreuils 1 (B52/3), 4000 Liege, Belgium
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/ Alberto Cardona / Olivier Brüls
  • University of Liege, Department of Aerospace and Mechanical Engineering (LTAS), Chemin des Chevreuils 1 (B52/3), 4000 Liege, Belgium
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Published Online: 2014-08-15 | DOI: https://doi.org/10.2478/meceng-2014-0018

Abstract

Recently, the authors proposed a geometrically exact beam finite element formulation on the Lie group SE(3). Some important numerical and theoretical aspects leading to a computationally efficient strategy were obtained. For instance, the formulation leads to invariant equilibrium equations under rigid body motions and a locking free element. In this paper we discuss some important aspects of this formulation. The invariance property of the equilibrium equations under rigid body motions is discussed and brought out in simple analytical examples. The discretization method based on the exponential map is recalled and a geometric interpretation is given. Special attention is also dedicated to the consistent interpolation of the velocities.

Streszczenie

W ostatnim czasie autorzy zaproponowali geometrycznie dokładne sformułowanie dla belkowego elementu skończonego w oparciu o formalizm grupy Liego SE(3). Otrzymano szereg istotnych wyników numerycznych i teoretycznych prowadzĄcych do efektywnej strategii obliczeniowej. Dla przykładu, formalizm ten pozwala uzyskać niezmiennicze równania równowagi przy ruchach ciała sztywnego i elemencie wolnym od blokowania siłami ścinajĄcymi. W obecnym artykule autorzy zajmujĄ się kilkoma istotnymi aspektami tego formalizmu. Właściwość niezmienniczości równań równowagi w warunkach ruchu ciała sztywnego przedyskutowano i zilustrowano prostymi przykładami analitycznymi. Przypomniano metodę dyskretyzacji opartĄ na mapowaniu wykładniczym i pokazano jej interpretację geometrycznĄ. SpecjalnĄ uwagę poświęcono zgodnej interpolacji prędkości.

Key words:: dynamic beam; finite element; Lie group; special Euclidean group

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About the article

Published Online: 2014-08-15


Citation Information: Archive of Mechanical Engineering, ISSN (Online) 2300-1895, DOI: https://doi.org/10.2478/meceng-2014-0018.

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© 2014 Valentin Sonneville et. al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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