Nonlinear Dynamics

, 58:565 | Cite as

A rational finite element method based on the absolute nodal coordinate formulation

Original Paper

Abstract

The NURBS method is widely used for representing the geometry of engineering and physics systems in computer aided design (CAD) software. However, no finite element formulations currently used in computer aided analysis (CAA) can produce complex geometry identical to NURBS representations. Most contemporary finite element methods, in fact, are at best only crudely capable of representing the complex shapes that can be represented using NURBS. However, finite elements based on the nonlinear absolute nodal coordinate formulation (ANCF) are known to utilize similar shape functions to those employed in Bezier and B-spline representations of geometry. In order to facilitate the integration of computer aided design and analysis (ICADA), this investigation proposes the creation of a new finite element formulation based on the ANCF that utilizes rational polynomials for shape functions. This will lead to the new rational absolute nodal coordinate formulation (RANCF). To demonstrate the feasibility of developing RANCF finite elements, it is shown that rational cubic Bezier curve elements can be converted through a simple coordinate transformation into a rational ANCF form, in which the coordinates are the nodal positions at the end points of the element and the gradient vectors at those nodal points. Furthermore, it is shown that because a cubic NURBS curve can be converted to a series of rational cubic Bezier elements that such a NURBS curve can be converted into a series of these RANCF elements. This investigation only proposes the kinematic description of this new RANCF element. The full development of the nonlinear dynamic equations of motion as well as the development of fully parameterized RANCF finite elements will be explored in future investigations.

Keywords

Rational finite element method Absolute nodal coordinate formulation B-spline NURBS Isogeometric analysis 

References

  1. 1.
    Piegl, L., Tiller, W.: The NURBS Book, 2nd edn. Springer, New York (1997) Google Scholar
  2. 2.
    Dierckx, P.: Curve and Surface Fitting with Splines. Oxford University Press, New York (1993) MATHGoogle Scholar
  3. 3.
    Farin, G.: Curves and Surfaces for CAGD, A Practical Guide, 5th edn. Morgan Kaufmann, San Francisco (1999) Google Scholar
  4. 4.
    Crisfield, M.A.: Nonlinear Finite Element Analysis of Solids and Structures, vol. 1: Essentials. Wiley, New York (1991) Google Scholar
  5. 5.
    Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method for Solid and Structural Mechanics, 6th edn. Butterworth Heinemann, London (2005) MATHGoogle Scholar
  6. 6.
    Bazilevs, Y., Calo, V.M., Cottrell, J.A., Evans, J., Hughes, T.J.R., Lipton, S., Scott, M.A., Sederberg, T.W.: Isogeometric analysis: toward unification of computer aided design and finite element analysis. In: Papadrakakis, M., Topping, B.H.V. (eds.) Trends in Engineering Computational Technology. Saxe-Coburg, Stirlingshire (2008) Google Scholar
  7. 7.
    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) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Sanborn, G.G., Shabana, A.A.: Relationship between B-splines, NURBS, and the finite element absolute nodal coordinate formulation in our document. Technical report #MBS08-5-UIC, UIC, Department of Mechanical Engineering, University of Illinois at Chicago (2008) Google Scholar
  9. 9.
    Shabana, A.A.: An absolute nodal coordinate formulation for the large rotation and deformation analysis of flexible bodies. Technical report #MBS96-1-UIC, Department of Mechanical Engineering, University of Illinois at Chicago (1996) Google Scholar
  10. 10.
    Shabana, A.A.: Dynamics of Multibody Systems, 3rd edn. Cambridge University Press, Cambridge (2005) MATHGoogle Scholar
  11. 11.
    Shabana, A.A.: Computational Continuum Mechanics. Cambridge University Press, Cambridge (2008) MATHGoogle Scholar
  12. 12.
    Dmitrochenko, O.N., Pogorelov, D.Y.: Generalization of plate finite elements for absolute nodal coordinate formulation. Multibody Syst. Dyn. 10(1), 17–43 (2003) MATHCrossRefGoogle Scholar
  13. 13.
    Garcia-Vallejo, D., Mayo, J., Escalona, J.L., Dominguez, J.: Efficient evaluation of the elastic forces and the Jacobian in the absolute nodal coordinate formulation. Nonlinear Dyn. 35(4), 313–329 (2004) MATHCrossRefGoogle Scholar
  14. 14.
    Gerstmayr, J., Shabana, A.A.: Analysis of thin beams and cables using the absolute nodal coordinate formulation. Nonlinear Dyn. 45, 109–130 (2006) MATHCrossRefGoogle Scholar
  15. 15.
    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
  16. 16.
    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(2), 185–208 (2004) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of Illinois at ChicagoChicagoUSA

Personalised recommendations