The Theory of Rods

Antman, Stuart S.

doi:10.1007/978-3-642-69567-4_6

Cited by 87 publications

(83 citation statements)

References 57 publications

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“…The exploitation of these is one reason, why the right hand side of the models can be implemented with extremely few elementary arithmetic operations, see Table 2. Another benefit of quaternions is that they comprise quadratic instead of trigonometric expressions, see (1). So there is no vast blow-up for the Jacobians or Hessians.…”

Section: Discrete Geometrically Exact Rodsmentioning

confidence: 99%

“…Our starting point for the continuous rod models is the exposition in [3,38,39]. For the constitutive material behaviour, we choose a simple linear viscoelastic one that is called 'viscoelastic of complexity one' in [1,2,3]. The elastic parameters can be straightforwardly deduced from material and geometric ones [28,39].…”

Section: Introductionmentioning

confidence: 99%

“…K is based on the presumption that locally the stresses and strains remain small [28,38]. For the dissipative potential, we follow [1,2,3] and choose friction forces resp. moments that are proportional to the strain resp.…”

Section: Introductionmentioning

confidence: 99%

“…For the Kirchhoff model, things are essentially different. Here the internal forces F -in contrast to the moments M -are not determined constitutively, since they are of reactive nature due to the presence of internal constraints [1,3,30]. For the inextensible model for example, where all the components of the strain vector Γ must vanish, these constraint forces must be of the form F = (∇ x,p Γ) ⊤ λ with appropriate Lagrange multipliers.…”

mentioning

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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Section: Discrete Geometrically Exact Rodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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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Mathematical justification of a one-dimensional model for general elastic shallow arches

Álvarez‐Dios¹,

Viaño²

1998

Math. Meth. Appl. Sci.

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Geodesic finite elements for Cosserat rods

Sander¹

2009

Numerical Meth Engineering

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SUMMARYWe introduce geodesic finite elements as a new way to discretize the non-linear configuration space of a geometrically exact Cosserat rod. These geodesic finite elements naturally generalize standard onedimensional finite elements to spaces of functions with values in a Riemannian manifold. For the special orthogonal group, our approach reproduces the interpolation formulas of Crisfield and Jelenić. Geodesic finite elements are conforming and lead to objective and path-independent problem formulations. We introduce geodesic finite elements for general Riemannian manifolds, discuss the relationship between geodesic finite elements and coefficient vectors, and estimate the interpolation error. Then we use them to find static equilibria of hyperelastic Cosserat rods. Using the Riemannian trust-region algorithm of Absil et al. we show numerically that the discretization error depends optimally on the mesh size.

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The Theory of Rods

Cited by 87 publications

References 57 publications

Numerical aspects in the dynamic simulation of geometrically exact rods

Numerical aspects in the dynamic simulation of geometrically exact rods

Mathematical justification of a one-dimensional model for general elastic shallow arches

Geodesic finite elements for Cosserat rods

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