Use of B-spline surface to model large-deformation continuum plates: procedure and applications. (English) Zbl 1268.65021

Summary: The absolute nodal coordinate formulation (ANCF) has been used in the analysis of large deformation of flexible multibody systems that encompass belt drive, rotor blade, and cable applications. As demonstrated in the literature, the ANCF finite elements are ideal for isogeometric analysis. The purpose of this investigation is to establish a relationship between the B-splines, which are widely used in the geometric modeling, and the ANCF finite elements in order to construct continuum models of large-deformation geometries. This paper proposes a simplified approach to map the B-spline surfaces into ANCF thin plate elements. Matrix representation of the mapping process is established and examined through numerical examples successfully. The matrix representation of the mapping process is used because of its suitability of computer coding and to minimize the calculation time. The error estimation is carried out by analyzing the gap between the points of each ANCF element and the corresponding points of the portion of the B-spline surface. The Hausdorff distance is used to study the effect of the number of control points, the degree of interpolation, and the knot multiplicity on the mapped geometry. It is found that cubic interpolation is recommended for optimizing the accuracy of mapping the B-spline surface to ANCF thin plate elements. It is found that thin plate elements in ANCF are missing a number of basic functions which is considered a source of error between the two surfaces, as well as it does not allow to convert the ANCF thin plate element models to the B-spline surface. In this investigation, an application example of modeling a large-size wind turbine blade with uniform structure is illustrated. The use of the continuum plate elements in modeling flexible blades is more efficient because of the relative scale between the plate thickness and its length and width and the high flexibility of its structure. The numerical results are compared with the results of ANSYS code with a good agreement. The dynamic simulation for the mapped surface model shows a numerical convergence, which ensures the ability of using the proposed approach for applications of dynamics for design and computer-aided design.


65D07 Numerical computation using splines
74K20 Plates
74S99 Numerical and other methods in solid mechanics


Matlab; MESH
Full Text: DOI


[1] Dufva, K., Shabana, A.A.: Analysis of thin plate structure using the absolute nodal coordinate formulation. J. Multibody Dyn. 219, 345-355 (2005)
[2] Dufva, K.E., Kerkkänen, K.S., Maqueda, L., Shabana, A.A.: Nonlinear dynamics of three-dimensional belt drives using the finite-element method. Nonlinear Dyn. 48(4), 449-466 (2007) · Zbl 1177.74363
[3] Yoo, W.-S., Lee, J.-H., Park, S.-J., Sohn, J.-H., Pogorelov, D., Dmitrochenko, O.: Large deflection analysis of a thin plate: computer simulations and experiments. Multibody Syst. Dyn. 11, 185-208 (2004) · Zbl 1143.74339
[4] García-Vallejo, D.; Sugiyama, H.; Shabana, A. A., Finite element analysis of the geometric stiffening effect using the absolute nodal coordinate formulation, Long Beach, CA, USA, 24-28 September
[5] Nada, A.; El-Assal A, A., A non-incremental finite element formulation of large deformation piezoceramic-laminated-plates, Lappeenranta, Finland, 25-27 May
[6] Nada, A.A., Hussein, B.A., Megahed, S.M., Shabana, A.A.: Use of the floating frame of reference formulation in large deformation analysis: experimental and numerical validation. J. Multibody Dyn. 224, 45-58 (2010)
[7] Gerstmayr, J., Shabana, A.A.: Analysis of thin beams and cables using the absolute nodal coordinate formulation. Nonlinear Dyn. 45, 109-130 (2006) · Zbl 1138.74391
[8] Shabana, A.A.: Dynamics of Multibody Systems, 3rd edn. Cambridge University Press, Cambridge (2005) · Zbl 1068.70002
[9] Shabana, A.A.: Computational Continuum Mechanics. Cambridge University Press, Cambridge (2008) · Zbl 1184.74003
[10] 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) · Zbl 1347.65043
[11] Piegl, L., Tiller, W.: The NURBS Book, 2nd edn. Springer, New York (1997) · Zbl 0868.68106
[12] Hussein, B.A., Sugiyama, H., Shabana, A.A.: Absolute nodal coordinate formulation coupled deformation modes: problem definition. J. Comput. Nonlinear Dyn. 2(2), 146-154 (2007)
[13] 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(39-41), 4135-4195 (2005) · Zbl 1151.74419
[14] Rank, E., Ruess, M., Kollmannsberger, S., Schillinger, D., Düster, A.: Geometric modeling, isogeometric analysis and the finite cell method. Comput. Methods Appl. Mech. Eng. (2012). doi:10.1016/j.cma.2012.05.022 · Zbl 1348.74340
[15] Kiendl, J., Bletzinger, K.-U., Linhard, J., Wüchner, R.: Isogeometric shell analysis with Kirchhoff-Love elements. Comput. Methods Appl. Mech. Eng. 198(49-52), 3902-3914 (2009) · Zbl 1231.74422
[16] Benson, D.J., Bazilevs, Y., Hsu, M.C., Hughes, T.J.R.: Isogeometric shell analysis: the Reissner-Mindlin shell. Comput. Methods Appl. Mech. Eng. 199(5-8), 276-289 (2010) · Zbl 1227.74107
[17] Shabana, A.A., Hamed, A.M., Mohamed, A.-N.A., Jayakumar, P., Letherwood, M.D.: Use of B-spline in the finite element analysis: comparison with ANCF geometry. J. Comput. Nonlinear Dyn. 7, 011008 (2012). doi:10.1115/1.4004377
[18] Aspert, N.; Santa-Cruz, D.; Ebrahimi, T., MESH: measuring error between surfaces using the Hausdorff distance, Lausanne, Switzerland
[19] Shabana, A.A.: General method for modeling slope discontinuities and T-sections using ANCF gradient deficient finite elements. J. Comput. Nonlinear Dyn. 6, 024502 (2011)
[20] MatLab Users Guide. MathWorks (c) (2008)
[21] Shabana, A.A.: Computational Dynamics, 3rd edn. Wiley, New York (2010) · Zbl 1182.70002
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