Nonlinear Dynamics

, Volume 91, Issue 2, pp 1213–1227 | Cite as

Isogeometric collocation for nonlinear dynamic analysis of Cosserat rods with frictional contact

  • Oliver Weeger
  • Bharath Narayanan
  • Martin L. Dunn
Original Paper


We present a novel isogeometric collocation method for nonlinear dynamic analysis of three-dimensional, slender, elastic rods. The approach is based on the geometrically exact Cosserat model for rod dynamics. We formulate the governing nonlinear partial differential equations as a first-order problem in time and develop an isogeometric semi-discretization of position, orientation, velocity and angular velocity of the rod centerline as NURBS curves. Collocation then leads to a nonlinear system of first-order ordinary differential equations, which can be solved using standard time integration methods. Furthermore, our model includes viscoelastic damping and a frictional contact formulation. The computational method is validated and its practical applicability shown using several numerical applications of nonlinear rod dynamics.


Isogeometric analysis Collocation method Cosserat rod model Nonlinear dynamics Frictional contact 



The authors were supported by the SUTD Digital Manufacturing and Design (DManD) Centre, supported by the Singapore National Research Foundation.


  1. 1.
    Cao, D.Q., Tucker, R.W., Wang, C.: Cosserat dynamics and cable-stayed bridge vibrations. In: Cheng, L., Li, K.M., So, R.M.C. (eds), Proceedings of the Eighth International Congress on Sound and Vibration: 2–6 July 2001, the Hong Kong Polytechnic University, Hong Kong, China, pp. 1139–1146. Hong Kong Polytechnic University, Department of Mechanical Engineering (2001)Google Scholar
  2. 2.
    Tucker, R.W., Wang, C.: Torsional vibration control and Cosserat dynamics of a drill-rig assembly. Meccanica 38(1), 145–161 (2003)CrossRefMATHGoogle Scholar
  3. 3.
    Maurin, F., Dedè, L., Spadoni, A.: Isogeometric rotation-free analysis of planar extensible-elastica for static and dynamic applications. Nonlinear Dyn. 81, 77–96 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Durville, D.: Simulation of the mechanical behaviour of woven fabrics at the scale of fibers. Int. J. Mater. Form. 3, 1241–1251 (2010)CrossRefGoogle Scholar
  5. 5.
    Weeger, O., Y.S.B, Kang, Yeung, S.-K., Dunn, M.L.: Optimal design and manufacture of active rod structures with spatially variable materials. 3D Print. Addit. Manuf. 3(4), 204–215 (2016)CrossRefGoogle Scholar
  6. 6.
    Bertails, F., Audoly, B., Querleux, B., Leroy, F., Leveque, J.-L., Cani, M.-P.: Predicting natural hair shapes by solving the statics of flexible rods. In: Dingliana, J., Ganovelli, F. (eds.) Eurographics Short Papers. Eurographics, Lyon (2005)Google Scholar
  7. 7.
    Bertails-Descoubes, F., Cadoux, F., Daviet, G., Acary, V.: A nonsmooth Newton solver for capturing exact coulomb friction in fiber assemblies. ACM Trans. Graph. (TOG) 30(1), 6:1–6:14 (2011)Google Scholar
  8. 8.
    Rubin, M.B.: Cosserat Theories: Shells, Rods and Points, of Solid Mechanics and Its Applications, Chapter Cosserat Rods, vol. 79, pp. 191–310. Springer, Amsterdam (2000)CrossRefGoogle Scholar
  9. 9.
    Bathe, K.-J., Bolourchi, S.: Large displacement analysis of three-dimensional beam structures. Int. J. Numer. Methods Eng. 14(7), 961–986 (1979)CrossRefMATHGoogle Scholar
  10. 10.
    Simo, J.C.: A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput. Methods Appl. Mech. Eng. 49(1), 55–70 (1985)CrossRefMATHGoogle Scholar
  11. 11.
    Antman, S.S.: Nonlinear Problems of Elasticity. Volume of 107 Applied Mathematical Sciences. Springer, New York (2005)MATHGoogle Scholar
  12. 12.
    Antman, S.S.: Dynamical problems for geometrically exact theories of nonlinearly viscoelastic rods. J. Nonlinear Sci. 6(1), 1–18 (1996)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lang, H., Linn, J., Arnold, M.: Multi-body dynamics simulation of geometrically exact Cosserat rods. Multibody Syst. Dyn. 25(3), 285–312 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Linn, J., Lang, H., Tuganov, A.: Geometrically exact Cosserat rods with Kelvin–Voigt type viscous damping. Mech. Sci. 4, 79–96 (2013)CrossRefGoogle Scholar
  15. 15.
    Cao, D.Q., Liu, D., Wang, C.H.-T.: Three-dimensional nonlinear dynamics of slender structures: Cosserat rod element approach. Int. J. Solids Struct. 43, 760–783 (2006)CrossRefMATHGoogle Scholar
  16. 16.
    Cao, D.Q., Tucker, R.W.: Nonlinear dynamics of elastic rods using the Cosserat theory: modelling and simulation. Int. J. Solids Struct. 45, 460–477 (2008)CrossRefMATHGoogle Scholar
  17. 17.
    Spillmann, J., Teschner, M.: CoRdE: Cosserat rod elements for the dynamic simulation of one-dimensional elastic objects. In: Metaxas, D., Popovic, J. (eds.) Proceedings of the 2007 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, SCA ’07, pp. 63–72, Aire-la-Ville, Switzerland, Switzerland, Eurographics Association (2007)Google Scholar
  18. 18.
    Stoykov, S., Ribeiro, P.: Stability of nonlinear periodic vibrations of 3D beams. Nonlinear Dyn. 66, 335–353 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gratus, J., Tucker, R.W.: The dynamics of Cosserat nets. J. Appl. Math. 2003(4), 187–226 (2003)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Spillmann, J., Teschner, M.: Cosserat nets. IEEE Trans. Vis. Comput. Graph. 15(2), 325–338 (2009)CrossRefGoogle Scholar
  21. 21.
    Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Bouclier, R., Elguedj, T., Combescure, A.: Locking free isogeometric formulations of curved thick beams. Comput. Methods Appl. Mech. Eng. 245–246, 144–162 (2012)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Raknes, S.B., Deng, X., Bazilevs, Y., Benson, D.J., Mathisen, K.M., Kvamsdal, T.: Isogeometric rotation-free bending-stabilized cables: statics, dynamics, bending strips and coupling with shells. Comput. Methods Appl. Mech. Eng. 263, 127–143 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Greco, L., Cuomo, M.: B-Spline interpolation of Kirchhoff–Love space rods. Comput. Methods Appl. Mech. Eng. 256, 251–269 (2013)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Kiendl, J., Bletzinger, K.-U., Linhard, J., Wüchner, R.: Isogeometric shell analysis with Kirchhoff–Love elements. Comput. Methods Appl. Mech. Eng. 198, 3902–3914 (2009)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Dornisch, W., Klinkel, S., Simeon, B.: Isogeometric Reissner–Mindlin shell analysis with exactly calculated director vectors. Comput. Methods Appl. Mech. Eng. 253, 491–504 (2013)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Nguyen-Thanh, N., Kiendl, J., Nguyen-Xuan, H., Wüchner, R., Bletzinger, K.-U., Bazilevs, Y., Rabczuk, T.: Rotation free isogeometric thin shell analysis using PHT-Splines. Comput. Methods Appl. Mech. Eng. 200, 3410–3424 (2011)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Cottrell, J.A., Reali, A., Bazilevs, Y., Hughes, T.J.R.: Isogeometric analysis of structural vibrations. Comput. Methods Appl. Mech. Eng. 195, 5257–5296 (2006)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Weeger, O., Wever, U., Simeon, B.: Isogeometric analysis of nonlinear Euler–Bernoulli beam vibrations. Nonlinear Dyn. 72(4), 813–835 (2013)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Hughes, T.J.R., Evans, J.A., Reali, A.: Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems. Comput. Methods Appl. Mech. Eng. 272, 290–320 (2014)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Auricchio, F., Beirão da Veiga, L., Hughes, T.J.R., Reali, A., Sangalli, G.: Isogeometric collocation methods. Math. Models Methods Appl. Sci. 20(11), 2075–2107 (2010)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Reali, A., Hughes, T.J.R.: An introduction to isogeometric collocation methods. In: Beer, G., Bordas, S. (eds.) Isogeometric Methods for Numerical Simulation, of CISM International Centre for Mechanical Sciences, vol. 561, pp. 173–204. Springer, Berlin (2015)Google Scholar
  33. 33.
    Schillinger, D., Evans, J.A., Reali, A., Scott, M.A., Hughes, T.J.R.: Isogeometric collocation: cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations. Comput. Methods Appl. Mech. Eng. 267, 170–232 (2013)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Auricchio, F., Beirão da Veiga, L., Hughes, T.J.R., Reali, A., Sangalli. G.: Isogeometric collocation for elastostatics and explicit dynamics. Comput. Methods Appl. Mech. Eng. 249–252, 2–14 (2012)Google Scholar
  35. 35.
    De Lorenzis, L., Evans, J.A., Hughes, T.J.R., Reali, A.: Isogeometric collocation: Neumann boundary conditions and contact. Comput. Methods Appl. Mech. Eng. 284, 21–54 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kruse, R., Nguyen-Thanh, N., De Lorenzis, L., Hughes, T.J.R.: Isogeometric collocation for large deformation elasticity and frictional contact problems. Comput. Methods Appl. Mech. Eng. 296, 73–112 (2015)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Beirão da Veiga, L., Lovadina, C., Reali, A.: Avoiding shear locking for the Timoshenko beam problem via isogeometric collocation methods. Comput. Methods Appl. Mech. Eng. 241–244, 38–51 (2012)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Auricchio, F., Beirão da Veiga, L., Kiendl, J., Lovadina, C., Reali, A.: Locking-free isogeometric collocation methods for spatial Timoshenko rods. Comput. Methods Appl. Mech. Eng. 263, 113–126 (2013)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Kiendl, J., Auricchio, F., Hughes, T.J.R., Reali, A.: Single-variable formulations and isogeometric discretizations for shear deformable beams. Comput. Methods Appl. Mech. Eng. 284, 988–1004 (2015)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Reali, A., Gomez, H.: An isogeometric collocation approach for Bernoulli–Euler beams and Kirchhoff plates. Comput. Methods Appl. Mech. Eng. 284, 623–636 (2015)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Weeger, O., Yeung, S.-K., Dunn, M.L.: Isogeometric collocation methods for Cosserat rods and rod structures. Comput. Methods Appl. Mech. Eng. 316, 100–122 (2017)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Durville, D.: Contact-friction modeling within elastic beam assemblies: an application to knot tightening. Comput. Mech. 49(6), 687–707 (2012)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Meier, C., Popp, A., Wall, W.A.: A finite element approach for the line-to-line contact interaction of thin beams with arbitrary orientation. Comput. Methods Appl. Mech. Eng. 308, 377–413 (2016)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Meier, C., Wall, W.A., Popp, A.: A unified approach for beam-to-beam contact. Comput. Methods Appl. Mech. Eng. 315, 972–1010 (2017)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Meier, C., Grill, M.J., Wall, W.A., Popp, A.: Geometrically exact beam elements and smooth contact schemes for the modeling of fiber-based materials and structures. Int. J. Solids Struct. (2017).
  46. 46.
    Weeger, O., Narayanan, B., De Lorenzis, L., Kiendl, J., Dunn, M.L.: An isogeometric collocation method for frictionless contact of Cosserat rods. Comput. Methods Appl. Mech. Eng. 321, 361–382 (2017)Google Scholar
  47. 47.
    Bîrsan, M., Altenbach, H., Sadowski, T., Eremeyev, V.A., Pietras, D.: Deformation analysis of functionally graded beams by the direct approach. Compos. B Eng. 43(3), 1315–1328 (2012)CrossRefGoogle Scholar
  48. 48.
    Ding, Z., Weeger, O., Qi, H.J., Dunn, M.L.: 4D rods: 3D structures via programmable 1D composite rods. Mater. Des. 137, 256–265 (2018)Google Scholar
  49. 49.
    Piegl, L.A., Tiller, W.: The NURBS Book. Monographs in Visual Communication. Springer, Berlin (1997)CrossRefMATHGoogle Scholar
  50. 50.
    Gomez, H., De Lorenzis, L.: The variational collocation method. Comput. Methods Appl. Mech. Eng. 309, 152–181 (2016)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Montardini, M., Sangalli, G., Tamellini, L.: Optimal-order isogeometric collocation at Galerkin superconvergent points. (pre-print) (2016)
  52. 52.
    Simeon, B.: Computational Flexible Multibody Dynamics—A Differential-Algebraic Approach. Differential-Algebraic Equations Forum. Springer, Berlin (2013)MATHGoogle Scholar
  53. 53.
    Lang, H., Arnold, M.: Numerical aspects in the dynamic simulation of geometrically exact rods. Appl. Numer. Math. 62(10), 1411–1427 (2012)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Oden, J.T., Martins, J.A.C.: Models and computational methods for dynamic friction phenomena. Comput. Methods Appl. Mech. Eng. 52(1), 527–634 (1985)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.SUTD Digital Manufacturing and Design CentreSingapore University of Technology and DesignSingaporeSingapore

Personalised recommendations