Abstract
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.
Similar content being viewed by others
References
Piegl, L., Tiller, W.: The NURBS Book, 2nd edn. Springer, New York (1997)
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)
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)
Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method for Solid and Structural Mechanics, 6th edn. Butterworth/Heinemann, Stoneham/London (2005)
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)
Dmitrochenko, O.N., Pogorelov, D.Y.: Generalization of plate finite elements for absolute nodal coordinate formulation. Multibody Syst. Dyn. 10(1), 17–43 (2003)
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)
Shabana, A.A.: Computational Continuum Mechanics. Cambridge University Press, Cambridge (2008)
Shabana, A.A.: Dynamics of Multibody Systems, 3rd edn. Cambridge University Press, Cambridge (2005)
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)
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)
Dierckx, P.: Curve and Surface Fitting with Splines. Oxford University Press, London (1993)
Farin, G.: Curves and Surfaces for CAGD, A Practical Guide, 5th edn. Morgan Kaufmann, San Mateo (1999)
Rogers, D.F.: An Introduction to NURBS With Historical Perspective. Academic Press, San Diego (2001)
Gerstmayr, J., Shabana, A.A.: Analysis of thin beams and cables using the absolute nodal coordinate formulation. Nonlinear Dyn. 45, 109–130 (2006)
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
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lan, P., Shabana, A.A. Integration of B-spline geometry and ANCF finite element analysis. Nonlinear Dyn 61, 193–206 (2010). https://doi.org/10.1007/s11071-009-9641-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-009-9641-6