Nonlinear Dynamics

, Volume 61, Issue 1–2, pp 193–206 | Cite as

Integration of B-spline geometry and ANCF finite element analysis

  • Peng Lan
  • Ahmed A. Shabana
Original Paper


The goal of this investigation is to introduce a new computer procedure for the integration of B-spline geometry and the absolute nodal coordinate formulation (ANCF) finite element analysis. The procedure is based on developing a linear transformation that can be used to transform systematically the B-spline representation to an ANCF finite element mesh preserving the same geometry and the same degree of continuity. Such a linear transformation that relates the B-spline control points and the finite element position and gradient coordinates will facilitate the integration of computer aided design and analysis (ICADA). While ANCF finite elements automatically ensure the continuity of the position and gradient vectors at the nodal points, the B-spline representation allows for imposing a higher degree of continuity by decreasing the knot multiplicity. As shown in this investigation, a higher degree of continuity can be systematically achieved using ANCF finite elements by imposing linear algebraic constraint equations that can be used to eliminate nodal variables. The analysis presented in this study shows that continuity of the curvature vector and its derivative which corresponds in the cubic B-spline representation to zero knot multiplicity can be systematically achieved using ANCF finite elements. In this special case, as the knot multiplicity reduces to zero, the recurrence B-spline formula causes two segments to automatically blend together forming one cubic segment defined on a larger domain. Similarly in this special case, the algebraic constraint equations required for the C 3 continuity convert two ANCF cubic finite elements to one finite element, demonstrating the strong relationship between the B-spline representation and the ANCF finite element representation. For the same order of interpolation, higher degree of continuity at the finite element interface can lead to a coarser mesh and to a lower dimensional model. Using the B-spline/ANCF finite element transformation developed in this paper, the equations of motion of a finite element mesh that represents exactly the B-spline geometry can be developed. Because of the linearity of the transformation developed in this investigation, all the ANCF finite element desirable features are preserved; including the constant mass matrix that can be used to develop an optimum sparse matrix structure of the nonlinear multibody system dynamic equations.

Finite element method Absolute nodal coordinate formulation (ANCF) B-spline Bezier curves Isogeometric analysis 


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  1. 1.
    Piegl, L., Tiller, W.: The NURBS Book, 2nd edn. Springer, New York (1997) Google Scholar
  2. 2.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cottrell, J.A., Hughes, T.J.R., Reali, A.: Studies of refinement and continuity in the isogeometric analysis. Comput. Methods Appl. Mech. Eng. 196, 4160–4183 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method for Solid and Structural Mechanics, 6th edn. Butterworth/Heinemann, Stoneham/London (2005) zbMATHGoogle Scholar
  5. 5.
    Sanborn, G.G., Shabana, A.A.: On the integration of computer aided design and analysis using the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 22, 181–197 (2009) zbMATHCrossRefGoogle Scholar
  6. 6.
    Dmitrochenko, O.N., Pogorelov, D.Y.: Generalization of plate finite elements for absolute nodal coordinate formulation. Multibody Syst. Dyn. 10(1), 17–43 (2003) zbMATHCrossRefGoogle Scholar
  7. 7.
    Shabana, A.A., Mikkola, A.M.: Use of the finite element absolute nodal coordinate formulation in modeling slope discontinuity. ASME J. Mech. Des. 125(2), 342–350 (2003) CrossRefGoogle Scholar
  8. 8.
    Shabana, A.A.: Computational Continuum Mechanics. Cambridge University Press, Cambridge (2008) zbMATHGoogle Scholar
  9. 9.
    Shabana, A.A.: Dynamics of Multibody Systems, 3rd edn. Cambridge University Press, Cambridge (2005) zbMATHCrossRefGoogle Scholar
  10. 10.
    Tian, Q., Chen, L.P., Zhang, Y.Q., Yang, J.Z.: An efficient hybrid method for multibody dynamics simulation based on absolute nodal coordinate formulation. ASME J. Comput. Nonlinear Dyn. 4, 021009-1–021009-14 (2009) CrossRefGoogle Scholar
  11. 11.
    Yoo, W.S., Lee, J.H., Park, S.J., Sohn, J.H., Pogorelov, D., Dimitrochenko, O.: Large deflection analysis of a thin plate: computer simulation and experiment. Multibody Syst. Dyn. 11, 185–208 (2004) zbMATHCrossRefGoogle Scholar
  12. 12.
    Dierckx, P.: Curve and Surface Fitting with Splines. Oxford University Press, London (1993) zbMATHGoogle Scholar
  13. 13.
    Farin, G.: Curves and Surfaces for CAGD, A Practical Guide, 5th edn. Morgan Kaufmann, San Mateo (1999) Google Scholar
  14. 14.
    Rogers, D.F.: An Introduction to NURBS With Historical Perspective. Academic Press, San Diego (2001) Google Scholar
  15. 15.
    Gerstmayr, J., Shabana, A.A.: Analysis of thin beams and cables using the absolute nodal coordinate formulation. Nonlinear Dyn. 45, 109–130 (2006) zbMATHCrossRefGoogle Scholar
  16. 16.
    Shabana, A.A.: Use of Gradient Deficient ANCF Finite Elements in Modeling Slope Discontinuities and T-Sections. Technical Report # MBS09-9-UIC, Department of Mechanical Engineering, The University of Illinois at Chicago, September 2009, revised October 2009 Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.School of Mechatronic EngineeringHarbin Institute of TechnologyHarbinPeople’s Republic of China
  2. 2.Department of Mechanical and Industrial EngineeringUniversity of Illinois at ChicagoChicagoUSA

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