Consistent quaternion interpolation for objective finite element approximation of geometrically exact beam

We explore an isoparametric interpolation of total quaternion for geometrically consistent, strain-objective and path-independent finite element solutions of the geometrically exact beam. This interpolation is a variant of the broader class known as slerp. The equivalence between the proposed interpolation and that of relative rotation is shown without any recourse to local bijection between quaternions and rotations. We show that, for a two-noded beam element, the use of relative rotation is not mandatory for attaining consistency cum objectivity and an appropriate interpolation of total rotation variables is sufficient. The interpolation of total quaternion, which is computationally more efficient than the one based on local rotations, converts nodal rotation vectors to quaternions and interpolates them in a manner consistent with the character of the rotation manifold. This interpolation, unlike the additive interpolation of total rotation, corresponds to a geodesic on the rotation manifold. For beam elements with more than two nodes, however, a consistent extension of the proposed quaternion interpolation is difficult. Alternatively, a quaternion-based procedure involving interpolation of relative rotations is proposed for such higher order elements. We also briefly discuss a strategy for the removal of possible singularity in the interpolation of quaternions, proposed in [I. Romero, The interpolation of rotations and its application to finite element models of geometrically exact rods, Comput. Mech. 34 (2004) 121–133]. The strain-objectivity and path-independence of solutions are justified theoretically and then demonstrated through numerical experiments. This study, being focused only on the interpolation of rotations, uses a standard finite element discretization, as adopted by Simo and Vu-Quoc [J.C. Simo, L. Vu-Quoc, A three-dimensional finite rod model part II: computational aspects, Comput. Methods Appl. Mech. Engrg. 58 (1986) 79–116]. The rotation update is achieved via quaternion multiplication followed by the extraction of the rotation vector. Nodal rotations are stored in terms of rotation vectors and no secondary storages are required.

[1]  Jing-Song Huang.  Lectures on Representation Theory , 1999 .

[2]  Daniel J. Rixen,et al.  Parametrization of finite rotations in computational dynamics: a review , 1995 .

[3]  Ignacio Romero,et al.  An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy–momentum conserving scheme in dynamics , 2002 .

[4]  J. Mäkinen.  Total Lagrangian Reissner's geometrically exact beam element without singularities , 2007 .

[5]  J. C. Simo,et al.  A three-dimensional finite-strain rod model. Part II: Computational aspects , 1986 .

[6]  Ignacio Romero,et al.  The interpolation of rotations and its application to finite element models of geometrically exact rods , 2004 .

[7]  A. Ibrahimbegovic.  On finite element implementation of geometrically nonlinear Reissner's beam theory: three-dimensional curved beam elements , 1995 .

[8]  Gordan Jelenić,et al.  Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics , 1999 .

[9]  W. Smoleński.  Statically and kinematically exact nonlinear theory of rods and its numerical verification , 1999 .

[10]  J. Kuipers.  Quaternions and Rotation Sequences , 1998 .

[11]  J. C. Simo,et al.  A Geometrically-exact rod model incorporating shear and torsion-warping deformation , 1991 .

[12]  Goto Yoshiaki,et al.  Elastic buckling phenomenon applicable to deployable rings , 1992 .

[13]  Debasish Roy,et al.  A frame-invariant scheme for the geometrically exact beam using rotation vector parametrization , 2009 .

[14]  A. Ibrahimbegovic.  On the choice of finite rotation parameters , 1997 .

[15]  Joan Lasenby,et al.  Simo-Vu Quoc rods using Clifford algebra , 1999 .

[16]  Parviz E. Nikravesh,et al.  Computer-aided analysis of mechanical systems , 1988 .

[17]  M. Crisfield,et al.  Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[18]  P. Betsch,et al.  Frame‐indifferent beam finite elements based upon the geometrically exact beam theory , 2002 .

[19]  Gordan Jelenić,et al.  A kinematically exact space finite strain beam model - finite element formulation by generalized virtual work principle , 1995 .

[20]  Ken Shoemake,et al.  Animating rotation with quaternion curves , 1985, SIGGRAPH.

[21]  Robert L. Taylor,et al.  On the role of frame-invariance in structural mechanics models at finite rotations , 2002 .

[22]  M. Géradin,et al.  Flexible Multibody Dynamics: A Finite Element Approach , 2001 .

[23]  K. Spring.  Euler parameters and the use of quaternion algebra in the manipulation of finite rotations: A review , 1986 .

[24]  J. C. Simo,et al.  A finite strain beam formulation. The three-dimensional dynamic problem. Part I , 1985 .

[25]  M. Géradin,et al.  A beam finite element non‐linear theory with finite rotations , 1988 .

[26]  A. Ibrahimbegovic,et al.  Computational aspects of vector-like parametrization of three-dimensional finite rotations , 1995 .