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Sat, 10 Nov 2012 16:59:06 +0200Sat, 10 Nov 2012 16:59:06 +0200Construction of discrete shell models by geometric finite differences
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3322
In the presented work, we make use of the strong reciprocity between kinematics and geometry to build a geometrically nonlinear, shearable low order discrete shell model of Cosserat type defined on triangular meshes, from which we deduce a rotation–free Kirchhoff type model with the triangle vertex positions as degrees of freedom. Both models behave physically plausible already on very coarse meshes, and show good
convergence properties on regular meshes. Moreover, from the theoretical side, this deduction provides a
common geometric framework for several existing models.C. Weischedel; A. Tuganov; T. Hermansson; J. Linn; M. Wardetzkyreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3322Thu, 11 Oct 2012 16:59:06 +0200Geometrically exact Cosserat rods with Kelvin-Voigt type viscous damping
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3320
We present the derivation of a simple viscous damping model of Kelvin–Voigt type for geometrically exact
Cosserat rods from three–dimensional continuum theory. Assuming a homogeneous and isotropic material,
we obtain explicit formulas for the damping parameters of the model in terms of the well known stiffness
parameters of the rod and the retardation time constants defined as the ratios of bulk and shear viscosities to
the respective elastic moduli. We briefly discuss the range of validity of our damping model and illustrate
its behaviour with a numerical example.J. Linn; H. Lang; A. Tuganovreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3320Thu, 11 Oct 2012 16:58:20 +0200A discrete mechanics approach to Cosserat rod theory – Part 1: static equilibria
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2206
A theory of discrete Cosserat rods is formulated in the language of discrete Lagrangian mechanics. By exploiting Kirchho's kinetic analogy, the potential energy density of a rod is a function on the tangent bundle of the conguration manifold and thus formally corresponds to the Lagrangian function of a dynamical system. The equilibrium equations are derived from a variational principle using a formulation that involves null{space matrices. In this formulation, no Lagrange multipliers are necessary to enforce orthonormality of the directors. Noether's theorem relates rst integrals of the equilibrium equations to Lie group actions on the conguration bundle, so{called symmetries. The symmetries relevant for rod mechanics are frame{indierence, isotropy and uniformity. We show that a completely analogous and self{contained theory of discrete rods can be formulated in which the arc{length is a discrete variable ab initio. In this formulation, the potential energy density is dened directly on pairs of points along the arc{length of the rod, in analogy to Veselov's discrete reformulation of Lagrangian mechanics. A discrete version of Noether's theorem then identies exact rst integrals of the discrete equilibrium equations. These exact conservation properties confer the discrete solutions accuracy and robustness, as demonstrated by selected examples of application. Copyright c 2010 John Wiley & Sons, Ltd.P. Jung; S. Leyendecker; J. Linn; M. Ortizreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2206Wed, 21 Jul 2010 13:45:54 +0200Discrete Lagrangian mechanics and geometrically exact Cosserat rods
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2093
Inspired by Kirchhoff’s kinetic analogy, the special Cosserat theory of rods is formulatedin the language of Lagrangian mechanics. A static rod corresponds to an abstract Lagrangian system where the energy density takes the role of the Lagrangian function. The equilibrium equations are derived from a variational principle. Noether’s theorem relates their first integrals to frame-indifference, isotropy and uniformity. These properties can be formulated in terms of Lie group symmetries. The rotational degrees of freedom, present in the geometrically exact beam theory, are represented in terms of orthonormal director triads. To reduce the number of unknowns, Lagrange multipliers associated with the orthonormality constraints are eliminated using null-space matrices. This is done both in the continuous and in the discrete setting. The discrete equilibrium equations are used to compute discrete rod configurations, where different types of boundary conditions can be handled.P. Jung; S. Leyendecker; J. Linn; M. Ortizreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2093Wed, 20 May 2009 14:51:18 +0200Multibody dynamics simulation of geometrically exact Cosserat rods
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2092
In this paper, we present a viscoelastic rod model that is suitable for fast and sufficiently accurate dynamic simulations. It is based on Cosserat’s geometrically exact theory of rods and is able to represent extension, shearing (’stiff ’ dof), bending and torsion (’soft’ dof). For inner dissipation, a consistent damping potential from Antman is chosen. Our discrete model is based on a finite difference discretisation on a staggered grid. The right-hand side function f and the Jacobian ∂f/∂(q, v, t) of the dynamical system q˙ = v, v˙ = f(q, v, t) – after index reduction from three to zero – is free of higher algebraic (e.g. root) or transcendent (e.g. trigonometric or exponential) functions and is therefore cheap to evaluate. For the time integration of the system, we use well established stiff solvers like RADAU5 or DASPK. As our model yields computation times within milliseconds, it is suitable for interactivemanipulation in ’virtual reality’ applications. In contrast to fast common VR rod models, our model reflects the structural mechanics solutions sufficiently correct, as comparison with ABAQUS finite element results shows.H. Lang; J. Linn; M. Arnoldreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2092Wed, 20 May 2009 14:51:05 +0200Simulation of quasistatic deformations using discrete rod models
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2012
Recently we developed a discrete model of elastic rods with symmetric cross section suitable for a fast simulation of quasistatic deformations [33]. The model is based on Kirchhoff’s geometrically exact theory of rods. Unlike simple models of “mass & spring” type typically used in VR applications, our model provides a proper coupling of bending and torsion. The computational approach comprises a variational formulation combined with a finite difference discretization of the continuum model. Approximate solutions of the equilibrium equations for sequentially varying boundary conditions are obtained by means of energy minimization using a nonlinear CG method. As the computational performance of our model yields solution times within the range of milliseconds, our approach proves to be sufficient to simulate an interactive manipulation of such flexible rods in virtual reality applications in real time.J. Linn; T. Stephanreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2012Wed, 23 Jul 2008 15:16:02 +0200Fast simulation of quasistatic rod deformations for VR applications
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2011
Summary. We present a model of exible rods | based on Kirchhoff\\\'s geometrically exact theory | which is suitable for the fast simulation of quasistatic deformations within VR or functional DMU applications. Unlike simple models of \\\"mass & spring\\\" type typically used in VR applications, our model provides a proper coupling of bending and torsion. The computational approach comprises a variational formulation combined with a nite dierence discretization of the continuum model. Approximate solutions of the equilibrium equations for sequentially varying boundary conditions are obtained by means of energy minimization using a nonlinear CG method. The computational performance of our model proves to be sucient for the interactive manipulation of exible cables in assembly simulation.J. Linn; T. Stephan; J. Carlsson; R. Bohlinreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2011Wed, 23 Jul 2008 15:15:31 +0200On the Performance of Certain Iterative Solvers for Coupled Systems Arising in Discretization of Non-Newtonian Flow Equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1555
Iterative solution of large scale systems arising after discretization and linearization of the unsteady non-Newtonian Navier–Stokes equations is studied. cross WLF model is used to account for the non-Newtonian behavior of the fluid. Finite volume method is used to discretize the governing system of PDEs. Viscosity is treated explicitely (e.g., it is taken from the previous time step), while other terms are treated implicitly. Different preconditioners (block–diagonal, block–triangular, relaxed incomplete LU factorization, etc.) are used in conjunction with advanced iterative methods, namely, BiCGStab, CGS, GMRES. The action of the preconditioner in fact requires inverting different blocks. For this purpose, in addition to preconditioned BiCGStab, CGS, GMRES, we use also algebraic multigrid method (AMG). The performance of the iterative solvers is studied with respect to the number of unknowns, characteristic velocity in the basic flow, time step, deviation from Newtonian behavior, etc. Results from numerical experiments are presented and discussed.O. Iliev; J. Linn; M. Moog; D. Niedziela; V. Starikoviciusreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1555Mon, 26 Jul 2004 11:03:55 +0200On the frame-invariant description of the phase space of the Folgar-Tucker equation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1498
The Folgar-Tucker equation is used in flow simulations of fiber suspensions to predict fiber orientation depending on the local flow. In this paper, a complete, frame-invariant description of the phase space of this differential equation is presented for the first time.J. Linnreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1498Tue, 10 Feb 2004 09:52:47 +0100