Development of ANCF tetrahedral finite elements for the nonlinear dynamics of flexible structures

Development of ANCF tetrahedral finite elements for the nonlinear dynamics of flexible structures In this paper, methods for developing isoparametric tetrahedral finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are presented. The proposed ANCF tetrahedral elements have twelve coordinates per node that include three position and nine gradient coordinates. The fundamental differences between the coordinate parametrizations used for conventional finite elements and the coordinate parametrizations employed for the proposed ANCF tetrahedral elements are discussed. Two different parametric definitions are introduced: a volume parametrization based on coordinate lines along the sides of the tetrahedral element in the straight (un-deformed) configuration and a Cartesian parametrization based on coordinate lines directed along the global axes. The volume parametrization facilitates the development of a concise set of shape functions in a closed form, and the Cartesian parametrization serves as a unique standard for the element assembly. A linear mapping based on the Bezier geometry is used to systematically define the cubic position fields of ANCF tetrahedral elements: the complete polynomial-based eight-node mixed-coordinate and the incomplete polynomial-based four-node ANCF tetrahedral elements. An element transformation matrix that defines the relationship between the volume and Cartesian parametrizations is developed and used to convert the parametric gradients to structure gradients, thereby allowing for the use of a standard FE assembly procedure. A general computational approach is employed to formulate the generalized inertia, external, and elastic forces. The performance of the proposed ANCF tetrahedral elements is evaluated by comparison with the conventional linear and quadratic tetrahedral elements and also with the ANCF brick element. In the case of small deformations, the numerical results obtained show that all the tetrahedral elements considered can correctly produce rigid body motion. In the case of large deformations, on the other hand, the solutions of all the elements considered are in good agreement, provided that appropriate mesh sizes are used. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nonlinear Dynamics Springer Journals

Development of ANCF tetrahedral finite elements for the nonlinear dynamics of flexible structures

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Publisher
Springer Netherlands
Copyright
Copyright © 2017 by Springer Science+Business Media B.V.
Subject
Engineering; Vibration, Dynamical Systems, Control; Classical Mechanics; Mechanical Engineering; Automotive Engineering
ISSN
0924-090X
eISSN
1573-269X
D.O.I.
10.1007/s11071-017-3635-6
Publisher site
See Article on Publisher Site

Abstract

In this paper, methods for developing isoparametric tetrahedral finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are presented. The proposed ANCF tetrahedral elements have twelve coordinates per node that include three position and nine gradient coordinates. The fundamental differences between the coordinate parametrizations used for conventional finite elements and the coordinate parametrizations employed for the proposed ANCF tetrahedral elements are discussed. Two different parametric definitions are introduced: a volume parametrization based on coordinate lines along the sides of the tetrahedral element in the straight (un-deformed) configuration and a Cartesian parametrization based on coordinate lines directed along the global axes. The volume parametrization facilitates the development of a concise set of shape functions in a closed form, and the Cartesian parametrization serves as a unique standard for the element assembly. A linear mapping based on the Bezier geometry is used to systematically define the cubic position fields of ANCF tetrahedral elements: the complete polynomial-based eight-node mixed-coordinate and the incomplete polynomial-based four-node ANCF tetrahedral elements. An element transformation matrix that defines the relationship between the volume and Cartesian parametrizations is developed and used to convert the parametric gradients to structure gradients, thereby allowing for the use of a standard FE assembly procedure. A general computational approach is employed to formulate the generalized inertia, external, and elastic forces. The performance of the proposed ANCF tetrahedral elements is evaluated by comparison with the conventional linear and quadratic tetrahedral elements and also with the ANCF brick element. In the case of small deformations, the numerical results obtained show that all the tetrahedral elements considered can correctly produce rigid body motion. In the case of large deformations, on the other hand, the solutions of all the elements considered are in good agreement, provided that appropriate mesh sizes are used.

Journal

Nonlinear DynamicsSpringer Journals

Published: Jul 17, 2017

References

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