Rigid body formulation in a finite element context with contact interaction

  • Paulo R. Refachinho de Campos
  • Alfredo Gay Neto
Original Paper


The present work proposes a formulation to employ rigid bodies together with flexible bodies in the context of a nonlinear finite element solver, with contact interactions. Inertial contributions due to distribution of mass of a rigid body are fully developed, considering a general pole position associated with a single node, representing a rigid body element. Additionally, a mechanical constraint is proposed to connect a rigid region composed by several nodes, which is useful for linking rigid/flexible bodies in a finite element environment. Rodrigues rotation parameters are used to describe finite rotations, by an updated Lagrangian description. In addition, the contact formulation entitled master-surface to master-surface is employed in conjunction with the rigid body element and flexible bodies, aiming to consider their interaction in a rigid–flexible multibody environment. New surface parameterizations are presented to establish contact pairs, permitting pointwise interaction in a frictional scenario. Numerical examples are provided to show robustness and applicability of the methods.


Rigid body Flexible body Master–master contact Finite rotations Finite element method 



The authors acknowledge Vale S.A. for the support through Wheel-Rail Chair project and Altair Brazil, for providing full access to the HyperWorks suite. The second author acknowledges FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) under the Grant 2016/14230-6 and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) under the Grant 308190/2015-7.


  1. 1.
    Shabana AA (1997) Flexible multibody dynamics: review of past and recent developments. Multibody Sys Dyn 1:189–222MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Mchenry MJ (2012) When skeletons are geared for speed: the morphology, biomechanics, and energetics of rapid animal motion. Integr Comp Biol 52(5):588–596.
  3. 3.
    Patek SN, Nowroozi BN, Baio JE, Caldwell RL, Summers AP (2007) Linkage mechanics and power amplification of the mantis Shrimp’s strike. J Exp Biol 210:3677–3688CrossRefGoogle Scholar
  4. 4.
    Schweizerhof K, Nilsson L, Hallquist JO (1992) Crashworthiness analysis in the automotive industry. Int J Comput Appl Technol 5:134–156Google Scholar
  5. 5.
    Fountain M, Happee R, Wismans J, Lupker H, Koppens W (1996) Hybrid modelling for crash dummies for numerical simulation. In: Proceedings international conference on the biomechanics of impact (IRCOBI), DublinGoogle Scholar
  6. 6.
    Puso MA (2002) An energy and momentum conserving method for rigid-flexible body dynamics. Int J Numer Methods Eng 53:1393–1414MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Göttlicher B, Schweizerhof K (2005) Analysis of flexible structures with occasionally rigid parts under transient loading. Comput Struct 83:2035–2051CrossRefGoogle Scholar
  8. 8.
    Schiehlen W (1997) Multibody system dynamics: roots and perspectives. Multibody Sys Dyn 1:149–188MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Eberhard P, Schiehlen W (2005) Computational dynamics of multibody systems: history, formalisms, and applications. J Comput Nonlinear Dyn 1:3–12CrossRefGoogle Scholar
  10. 10.
    Wasfy TM, Noor AK (2003) Computational strategies for flexible multibody systems. Appl Mech Rev 56:553–613CrossRefGoogle Scholar
  11. 11.
    Shabana AA (2005) Dynamics of multibody systems. Cambridge University Press, New YorkCrossRefzbMATHGoogle Scholar
  12. 12.
    De Jalón JG, Bayo E (1994) Kinematic and dynamic simulation of multibody systems the real-time challenge. Springer, New YorkCrossRefGoogle Scholar
  13. 13.
    Dassault Systemes Deutschland GmbH, SIMPACK (online). Accessed 24 July 2017
  14. 14.
    Schwertassek R, Wallrapp O, Shabana AA (1999) Flexible multibody simulation and choice of shape functions. Nonlinear Dyn 20:361–380MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dietz S, Wallrapp O, Wiedemann S (2003) Nodal vs modal representation in flexible multibody system dynamics. In: Proceedings of Multibody Dynamics 2003, International Conference on Advances in Computational Multibody Dynamics Lisbon, Portugal, July 1–4, 2003Google Scholar
  16. 16.
    Géradin M, Cardona A (2001) Flexible multibody dynamics a finite element approach. Wiley, New YorkGoogle Scholar
  17. 17.
    Benson DJ, Hallquist JO (1986) A simple rigid body algorithm for structural dynamics programs. Int J Numer Methods Eng 22:723–749CrossRefzbMATHGoogle Scholar
  18. 18.
    Ibrahimbegovic A, Mamouri S (2000) On rigid components and joint constraints in nonlinear dynamics of flexible multibody systems employing 3D geometrically exact beam model. Comput Methods Appl Mech Eng 188:805–831CrossRefzbMATHGoogle Scholar
  19. 19.
    Bauchau O, Choi JY, Bottasso CL (2002) On the modeling of shells in multibody dynamics. Multibody Syst Dyn 8:459–489CrossRefzbMATHGoogle Scholar
  20. 20.
    Bauchau OA, Bottasso CL, Nikishkov YG (2001) Modeling rotorcraft dynamics with finite element multibody procedures. Math Comput Modell 33:1113–1137CrossRefzbMATHGoogle Scholar
  21. 21.
    Betsch P, Sänger N (2009) On the use of geometrically exact shells in a conserving framework for flexible multibody dynamics. Comput Methods Appl Mech Eng 198:1609–1630MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Neto AG (2017) Simulation of mechanisms modeled by geometrically-exact beams using Rodrigues rotation parameters. Comput Mech 59:459–481.
  23. 23.
    Wriggers P (2002) Computational contact mechanics. Wiley, West SussexzbMATHGoogle Scholar
  24. 24.
    Konyukhov A, Schweizerhof K (2014) On some aspects for contact with rigid surfaces: surface-to-rigid surface and curves-to-rigid surface algorithms. Comput Methods Appl Mech Eng 283:74–105MathSciNetCrossRefGoogle Scholar
  25. 25.
    Pfister J, Eberhard P (2002) Frictional contact of flexible and rigid bodies. Granul Matter 4:25–36CrossRefGoogle Scholar
  26. 26.
    Pfeiffer F, Glocker C (2004) Multibody dynamics with unilateral contacts. Wiley-VCH Verlag GmbH & Co, KGaA, WeinheimzbMATHGoogle Scholar
  27. 27.
    Hippmann G (2003) An algorithm for compliant contact between complexly shaped surfaces in multibody dynamics. Multibody Syst Dyn 12:345–362MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ambrósio JAC (2003) Impact of rigid and flexible multibody systems: deformation description and contact models. in: Schiehlen W, Valášek M (eds) Virtual nonlinear multibody systems. NATO ASI Series (Series II: Mathematics, Physics and Chemistry), vol 103Google Scholar
  29. 29.
    Lin MC, Gottschalk S (1998) Collision detection between geometric models: a survey. Proceedings of IMA Conference on Mathematics of Surfaces San Diego (CA), May 1998Google Scholar
  30. 30.
    Lim X, Ng T (1995) Contact detection algorithms for three-dimensional ellipsoids in discrete element modelling. Int J Numer Anal Methods Geomech 19:653–659CrossRefzbMATHGoogle Scholar
  31. 31.
    Wellmann C, Lillie C, Wriggers P (2008) A contact detection algorithm for superellipsoids based on the common-normal concept. Eng Comput Int J Comput Aided Eng Softw 25:432–442CrossRefzbMATHGoogle Scholar
  32. 32.
    Shinar T, Schroeder C, Fedkiw R (2008) Two-way coupling of rigid and deformable bodies. Eurographics/ ACM SIGGRAPH Symposium on Computer AnimationGoogle Scholar
  33. 33.
    Neto AG, Pimenta PM, Wriggers P (2016) A master-surface to master-surface formulation for beam to beam contact. Part I: frictionless interaction. Comput Methods Appl Mech Eng 303:400–429MathSciNetCrossRefGoogle Scholar
  34. 34.
    Neto AG, Pimenta PM, Wriggers P (2017) A master-surface to master-surface formulation for beam to beam contact. Part II: Frictional interaction. Comput Methods Appl Mech Eng 319:146–174MathSciNetCrossRefGoogle Scholar
  35. 35.
    Neto AG, Martins CA, Pimenta PM (2014) Static analysis of offshore risers with a geometrically-exact 3D beam model subjected to unilateral contact. Comput Mech 53:125–145MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Neto AG (2016) Dynamics of offshore risers using a geometrically-exact beam model with hydrodynamic loads and contact with the seabed. Eng Struct 125:438–454CrossRefGoogle Scholar
  37. 37.
    Campello EMB, Pimenta PM, Wriggers P (2003) A triangular finite shell element based on a fully nonlinear shell formulation. Comput Mech 31:505–518CrossRefzbMATHGoogle Scholar
  38. 38.
    Pimenta PM, Campello EMB, Wriggers P (2008) An exact conserving algorithm for nonlinear dynamics with rotational DOF’s and general hyperelasticity. Part 1: rods. Comput Mech 42:715–732Google Scholar
  39. 39.
    Campello EMB, Pimenta PM, Wriggers P (2011) An exact conserving algorithm for nonlinear dynamics with rotational dofs and general hyperelasticity. Part 2: shells. Comput Mech 48:195–211MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Pimenta PM, Campello EMB (2001) Geometrically nonlinear analysis of thin-walled space frames. In: Proceedings of the second European conference on computational mechanics, II ECCM, KrakowGoogle Scholar
  41. 41.
    Ota NSN, Wilson L, Neto AG, Pellegrino S, Pimenta PM (2016) Nonlinear dynamic analysis of creased shells. Finite Elem Anal Des 121:64–74MathSciNetCrossRefGoogle Scholar
  42. 42.
    Korelc J, Wriggers P (2016) Automation of finite element methods. Springer, New YorkCrossRefzbMATHGoogle Scholar
  43. 43.
    Neto AG (2014) Giraffe User’s Manual—Generic Interface Readily Accessible for Finite Elements. (online).
  44. 44.
    Wriggers P, Zavarise G (1997) On contact between three-dimensional beams undergoing large deflections. Commun Numer Methods Eng 13:429–438MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Wriggers P, Zavarise G (2000) Contact with friction between beams in 3-D space. Int J Numer Methods Eng 49:977–1006CrossRefzbMATHGoogle Scholar
  46. 46.
    Neto AG, Pimenta PM, Wriggers P (2014) Contact between rolling beams and flat surfaces. Int J Numer Methods Eng 97:683–706MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Neto AG, Pimenta PM, Wriggers P (2018) Contact between spheres and general surfaces. Comput Methods Appl Mech Eng 328:686–716.
  48. 48.
    Pimenta PM, Yojo T (1993) Geometrically exact analysis of spatial frames. Appl Mech Rev 46(11S):S118–S128CrossRefGoogle Scholar
  49. 49.
    Goicolea JM, Orden JCG (2000) Dynamic analysis of rigid and deformable multibody systems with penalty methods and energy-momentum schemes. Comput Methods Appl Mech Eng 188:789–804MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Simo JC, Wong KK (1991) Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum. Int J Numer Methods Eng 31:19–52MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Paulo R. Refachinho de Campos
    • 1
  • Alfredo Gay Neto
    • 1
  1. 1.Polytechnic School at the University of São PauloSão PauloBrazil

Personalised recommendations