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Numerical aspects in the dynamic simulation of geometrically exact rods

: Lang, H.; Arnold, M.


Verwer, J.G.:
Selected papers from NUMDIFF-12 : 12th Seminar on Numerical Solution of Differential and Differential-Algebraic Equations, 4 - 18 September 2009, Halle, Germany
Amsterdam: Elsevier, 2012 (Applied numerical mathematics 62.2012, Nr.10)
ISSN: 0168-9274
Seminar on Numerical Solution of Differential and Differential-Algebraic Equations (NUMDIFF) <12, 2009, Halle>
Fraunhofer ITWM ()

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.