Acta Mechanica Solida Sinica

, Volume 31, Issue 2, pp 207–228 | Cite as

Modeling Method and Application of Rational Finite Element Based on Absolute Nodal Coordinate Formulation

  • Chao Ma
  • Cheng Wei
  • Jing Sun
  • Bin Liu


In multibody system dynamics, the absolute nodal coordinate formulation (ANCF) uses power functions as interpolating polynomials to describe the displacement field. It can get accurate results for flexible bodies that undergo large deformation and large rotation. However, the power functions are irrational representation which cannot describe the complex shapes precisely, especially for circular and conic sections. Different from the ANCF representation, the rational absolute nodal coordinate formulation (RANCF) utilizes rational basis functions to describe geometric shapes, which allows the accurate representation of complicated displacement and deformation in dynamics modeling. In this paper, the relationships between the rational surface and volume and the RANCF finite element are provided, and the generalized transformation matrices are established correspondingly. Using these transformation matrices, a new four-node three-dimensional RANCF plate element and a new eight-node three-dimensional RANCF solid element are proposed based on the RANCF. Numerical examples are given to demonstrate the applicability of the proposed elements. It is shown that the proposed elements can depict the geometric characteristics and structure configurations precisely, and lead to better convergence in comparison with the ANCF finite elements for the dynamic analysis of flexible bodies.


Multibody system dynamics Absolute nodal coordinate formulation Flexible deformation Spline representation Rational finite element 


  1. 1.
    Shabana AA. Flexible multibody dynamics: review of past and recent developments. Multibody Syst Dyn. 1997;1(2):189–222.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Shabana AA. Definition of the slopes and the finite element absolute nodal coordinate formulation. Multibody Syst Dyn. 1997;1(3):339–48.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Sanborn GG, Shabana AA. A rational finite element method based on the absolute nodal coordinate formulation. Nonlinear Dyn. 2009;58(3):565–72.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Sanborn GG, Shabana AA. On the integration of computer aided design and analysis using the finite element absolute nodal coordinate formulation. Multibody Syst Dyn. 2009;22(2):181–97.CrossRefzbMATHGoogle Scholar
  5. 5.
    Lan P, Shabana AA. Integration of B-spline geometry and ANCF finite element analysis. Nonlinear Dyn. 2010;61(1):193–206.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lan P, Shabana AA. Rational finite elements and flexible body dynamics. J Vib Acoust. 2010;132(4):0410071–9.CrossRefGoogle Scholar
  7. 7.
    Yamashita H, Sugiyama H. Numerical convergence of finite element solutions of nonrational B-spline element and absolute nodal coordinate formulation. Nonlinear Dyn. 2012;67(1):177–89.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mikkola AM, Shabana AA, Sanchez-Rebollo C, et al. Comparison between ANCF and B-spline surfaces. Multibody Syst Dyn. 2013;30(2):119–38.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Nada AA. Use of B-spline surface to model large-deformation continuum plates: procedure and applications. Nonlinear Dyn. 2013;72(1):243–63.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chang HJ, Liu C, Tian Q, Hu HY, et al. Three new triangular shell elements of ANCF represented by Bézier triangles. Multibody Syst Dyn. 2015;35(4):321–51.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Yu ZO, Shabana AA. A mixed-coordinate ANCF rectangular plate finite element. J Comput Nonlinear Dyn. 2015;10(6):061003.CrossRefGoogle Scholar
  12. 12.
    Pappalardo CM, Yu ZO, Zhang XS, et al. Rational ANCF thin plate finite element. J Comput Nonlinear Dyn. 2016;11(5):051009.CrossRefGoogle Scholar
  13. 13.
    Shabana AA. Dynamics of multibody systems. 3rd ed. Cambridge/New York: Cambridge University Press; 2005. p. 267–304.CrossRefzbMATHGoogle Scholar
  14. 14.
    Shabana AA. Computational continuum mechanics. 2nd ed. Cambridge/New York: Cambridge University Press; 2012. p. 146–217.zbMATHGoogle Scholar
  15. 15.
    Piegl L, Tiller W. The NURBS book. New York: Springer; 1997. p. 47–138.zbMATHGoogle Scholar
  16. 16.
    Belytschko T, Liu WK, Moran B, et al. Nonlinear finite elements for continua and structures. 2nd ed. New York: Wiley; 2014. p. 346–53.Google Scholar
  17. 17.
    Mikkola AM, Shabana AA. A non-incremental finite element procedure for the analysis of large deformation of plates and shells in mechanical system applications. Multibody Syst Dyn. 2003;9(3):283–309.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dmitrochenko ON, Pogorelov DY. Generalization of plate finite elements for absolute nodal coordinate formulation. Multibody Syst Dyn. 2003;10(1):17–43.CrossRefzbMATHGoogle Scholar
  19. 19.
    Dufva K, Shabana AA. Analysis of thin plate structures using the absolute nodal coordinate formulation. Proc Inst Mech Eng Part K J Multibody Dyn. 2005;219(4):345–55.Google Scholar
  20. 20.
    Olshevskiy A, Dmitrochenko O, Kim CW. Three-dimensional solid brick element using slopes in the absolute nodal coordinate formulation. J Comput Nonlinear Dyn. 2014;9(2):021001.CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2018

Authors and Affiliations

  1. 1.Beijing Key Laboratory of Intelligent Space Robotic Systems Technology and ApplicationsBeijing Institute of Spacecraft System EngineeringBeijingChina
  2. 2.Department of Aerospace EngineeringHarbin Institute of TechnologyHarbinChina

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