On finite element implementation of geometrically nonlinear Reissner's beam theory: three-dimensional curved beam elements

Ibrahimbegović, Adnan

doi:10.1016/0045-7825(95)00724-f

Cited by 289 publications

(234 citation statements)

References 25 publications

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“…The incremental rotation vectors, together with the beam axis displacements, are assumed as independent kinematic variables in our displacement-based formulation. Unlike [40,52], no use of quaternion parameters is made.…”

Section: Introductionmentioning

confidence: 99%

Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

Marino

2016

Computer Methods in Applied Mechanics and Engineering

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Highlights• Isogeometric collocation is extended to geometrically exact beams.• Consistent linearization of the strong form of the governing equations is derived.• Incremental rotations are parametrized through Eulerian rotation vectors.• Numerical tests show efficiency and high accuracy. AbstractWe extend the isogeometric collocation method to the geometrically nonlinear beams. An exact kinematic formulation, able to represent three-dimensional displacements and rotations without any restriction in magnitude, is presented without the introduction of the moving frame concept. A displacement-based formulation is adopted. Full linearization of the strong form of the governing equations is derived consistently with the underlying geometric structure of the configuration manifold. Incremental rotations are parametrized through Eulerian rotation vectors and configuration updates are performed by means of the exponential map. Numerical tests demonstrate that the proposed combination of isogeometric collocation method with the chosen rotations parametrization results in an efficient computational scheme able to model complex problems with high accuracy.

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Section: Introductionmentioning

confidence: 99%

Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

Marino

2016

Computer Methods in Applied Mechanics and Engineering

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“…So, for example, the inextensible Kirchhoff rod is simply a chain of rigid cylinders connected with bushings that are consistently derived from the continuum equations. But our discretisation approach stands in contrast to the usual way in computational continuum mechanics, where the finite element (FE) approach is favored [9,11,14,21,23,38,39]. The reason for that is, that the main focus in FE is accuracy, not computational efficiency.…”

Section: Discrete Geometrically Exact Rodsmentioning

confidence: 93%

“…Nevertheless, the dynamical analysis of fully nonlinear beams and rods in 3D is even today a challenging problem, both from the viewpoint of modeling and from the viewpoint of the efficient numerical solution of the resulting model equations [14,21,23,36,38,39]. In the present paper, we combine an objective/frame-indifferent geometrically exact space discretisation of Kirchhoff and Cosserat rods with standard methods for the time integration of the equations of motion for constrained mechanical systems [4,17,22].…”

Section: Introductionmentioning

confidence: 99%

“…system (14) must be solved for (q,v, λ), so that one can choose u = (q, v) in (23). Banded Gaussian LU factorisation [19] is appropriate with complexity O(m 2 N ).…”

Section: Numerical Problems In Time Integrationmentioning

confidence: 99%

“…We distinguish three basic types of classical rod models. In hierarchical descending order, these are the Cosserat, the extensible Kirchhoff and the inextensible Kirchhoff model [1,2,3,9,13,14,21,23,24,25,26,27,28,32,38,39]. Table 1 presents a short overview, including numerical problems in time integration to be discussed in this article.…”

Section: Introductionmentioning

confidence: 99%

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Numerical aspects in the dynamic simulation of geometrically exact rods

Lang

Arnold

2012

Applied Numerical Mathematics

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Classical geometrically exact Kirchhoff and Cosserat models are used to study the nonlinear deformation of rods. Extension, bending and torsion of the rod may be represented by the Kirchhoff model. The Cosserat model additionally takes into account shearing effects. Second order finite differences on a staggered grid define discrete viscoelastic versions of these classical models. Since the rotations are parametrisecl by unit quaternions, the space discretisation results in differential-algebraic equations that are solved numerically by standard techniques like index reduction and projection methods. Using absolute coordinates, the mass and constraint matrices are sparse and this sparsity may be exploited to speed-up time integration. Further improvements are possible in the Cosserat model, because the constraints are just the normalisation conditions for unit quaternions such that the null space of the constraint matrix can be given analytically. The results of the theoretical investigations are illustrated by numerical tests

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Finite rotations in dynamics of beams and implicit time-stepping schemes

Ibrahimbegović

Mikdad

1998

Int. J. Numer. Meth. Engng.

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123

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We examine theoretical and computational aspects of three-dimensional ÿnite rotations pertinent to the dynamics of beams. The model problem chosen for consideration is the Reissner beam theory capable of modelling ÿnite strains and ÿnite rotations in geometrically exact manner. Special emphasis is placed on clarifying the geometry aspects, ÿnite rotation updates and the associated linearization procedure pertaining to di erent choices of rotation parameters. The latter is shown to play an important role in constructing the optimal implementation of a time-stepping scheme. ?

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On finite element implementation of geometrically nonlinear Reissner's beam theory: three-dimensional curved beam elements

Cited by 289 publications

References 25 publications

Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

Numerical aspects in the dynamic simulation of geometrically exact rods

Finite rotations in dynamics of beams and implicit time-stepping schemes

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