Discrete Cosserat Rod Models Based on the Difference Geometry of Framed Curves for Interactive Simulation of Flexible Cables

For software tools currently used in industry for computer aided design (CAD), digital mock-up and virtual assembly there is an increasing demand to handle not only rigid geometries, but to provide also capabilities for realistic simulations of large deformations of slender flexible structures in real time (i.e.: at interactive rates). The theory of Cosserat rods provides a framework to perform physically correct simulations of arbitrarily large spatial deformations of such structures by stretching, bending and twisting. The kinematics of Cosserat rods is described by the differential geometry of framed curves, with the differential invariants of rod configurations corresponding to the strain measures of the mechanical theory. We utilize ideas from the discrete differential geometry of framed curves in combination with the variational framework of Lagrangian mechanics to construct discrete Cosserat rod models that behave qualitatively correct for rather coarse discretizations, provide a fast computational performance at moderate accuracy, and thus are suitable for interactive simulations. This geometry based discretization approach for flexible 1D structures has industrial applications in design and digital validation. We illustrate this with some application examples from automotive industry.

[1]  R. E. Jones,et al.  Nonlinear finite elements , 1978 .

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

[3]  J. Linn,et al.  Kinetic Aspects of Discrete Cosserat Rods , 2017 .

[4]  J. Linn,et al.  Towards Viscoplastic Constitutive Models for Cosserat Rods , 2016 .

[5]  Sigrid Leyendecker,et al.  A discrete mechanics approach to the Cosserat rod theory—Part 1: static equilibria , 2011 .

[6]  John M. Sullivan,et al.  Curvatures of Smooth and Discrete Surfaces , 2007, 0710.4497.

[7]  S. Antman.  Nonlinear problems of elasticity , 1994 .

[8]  M. Wardetzky,et al.  Construction of discrete shell models by geometric finite differences , 2012 .

[9]  C. Truesdell,et al.  The Non-Linear Field Theories of Mechanics , 1965 .

[10]  Henri Paul Cartan.  Formes différentielles : applications élémentaires au calcul des variations et a la théorie des courbes et des surfaces , 1967 .

[11]  E. Dill,et al.  Kirchhoff's theory of rods , 1992 .

[12]  A. Love.  A treatise on the mathematical theory of elasticity , 1892 .

[13]  Martin Arnold,et al.  Integration of Nonlinear Models of Flexible Body Deformation in Multibody System Dynamics , 2014 .

[14]  C. Weischedel.  A discrete geometric view on shear-deformable shell models , 2012 .

[15]  E. Reissner,et al.  On One‐Dimensional Large‐Displacement Finite‐Strain Beam Theory , 1973 .

[16]  P. Podio-Guidugli.  A Primer in Elasticity , 2000 .

[17]  J. Milnor.  Topology from the differentiable viewpoint , 1965 .

[18]  W. Blaschke.  Vorlesungen über Differentialgeometrie , 1912 .

[19]  Martin Arnold,et al.  Numerical aspects in the dynamic simulation of geometrically exact rods , 2012 .

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

[21]  G. Kirchhoff,et al.  Ueber das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes. , 1859 .

[22]  E. Grinspun,et al.  Discrete elastic rods , 2008, SIGGRAPH 2008.

[23]  A. Bobenko,et al.  Discrete Time Lagrangian Mechanics on Lie Groups,¶with an Application to the Lagrange Top , 1999 .

[24]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[25]  M. Gurtin,et al.  An introduction to continuum mechanics , 1981 .

[26]  H. Lang,et al.  Multi-body dynamics simulation of geometrically exact Cosserat rods , 2011 .

[27]  Holger Lang,et al.  Geometrically exact Cosserat rods with Kelvin–Voigt type viscous damping , 2013 .

[28]  Robert Bohlin,et al.  Fast simulation of quasistatic rod deformations for vr applications , 2008 .