Nonlinear Dynamics

, Volume 72, Issue 4, pp 813–835 | Cite as

Isogeometric analysis of nonlinear Euler–Bernoulli beam vibrations

  • Oliver Weeger
  • Utz Wever
  • Bernd Simeon
Original Paper


In this paper we analyze the vibrations of nonlinear structures by means of the novel approach of isogeometric finite elements. The fundamental idea of isogeometric finite elements is to apply the same functions, namely B-Splines and NURBS (Non-Uniform Rational B-Splines), for describing the geometry and for representing the numerical solution. In case of linear vibrational analysis, this approach has already been shown to possess substantial advantages over classical finite elements, and we extend it here to a nonlinear framework based on the harmonic balance principle. As application, the straight nonlinear Euler–Bernoulli beam is used, and overall, it is demonstrated that isogeometric finite elements with B-Splines in combination with the harmonic balance method are a powerful means for the analysis of nonlinear structural vibrations. In particular, the smoother k-method provides higher accuracy than the p-method for isogeometric nonlinear vibration analysis.


Isogeometric analysis Finite element method Nonlinear vibration Harmonic balance Nonlinear beam 



The authors were supported by the 7th Framework Programme of the European Union, project TERRIFIC (FP7-2011-NMP-ICT-FoF 284981) [40].


  1. 1.
    Ginsberg, J.H.: Mechanical and Structural Vibrations: Theory and Applications. Wiley, New York (2001) Google Scholar
  2. 2.
    Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover, Mineola, New York (2000) MATHGoogle Scholar
  3. 3.
    Farin, G.E.: Curves and Surfaces for CAGD: A Practical Guide. Morgan Kaufmann, San Francisco (2002) Google Scholar
  4. 4.
    Piegl, L., Tiller, W.: The NURBS Book. Springer, London (1995) MATHCrossRefGoogle Scholar
  5. 5.
    Rogers, D.F.: An Introduction to NURBS with Historical Perspective. Academic Press, San Diego (2001) Google Scholar
  6. 6.
    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) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Szabó, B., Babuška, I.: Finite Element Analysis. Wiley, New York (1991) MATHGoogle Scholar
  8. 8.
    Bazilevs, Y., Calo, V.M., Hughes, T.J.R., Zhang, Y.: Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput. Mech. 43(1), 3–37 (2008) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Cottrell, J.A., Reali, A., Bazilevs, Y., Hughes, T.J.R.: Isogeometric analysis of structural vibrations. Comput. Methods Appl. Mech. Eng. 195(41–43), 5257–5296 (2006) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Vuong, A.-V., Heinrich, C., Simeon, B.: Isogat: a 2d tutorial Matlab code for isogeometric analysis. Comput. Aided Geom. Des. 27, 644–655 (2010) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Ziani, M., Duvigneau, R., Dörfel, M.: On the role played by NURBS weights in isogeometric structural shape optimization. In: International Conference on Inverse Problems, Control and Shape Optimization, Cartagena, Spain, April 2010 Google Scholar
  12. 12.
    Aigner, M., Heinrich, C., Jüttler, B., Pilgerstorfer, E., Simeon, B., Vuong, A.-V.: Swept volume parameterization for isogeometric analysis. In: Mathematics of Surfaces XIII. Lecture Notes in Computer Science, vol. 5654, pp. 19–44. Springer, Berlin/Heidelberg (2009) CrossRefGoogle Scholar
  13. 13.
    Bazilevs, Y., Beirão de Veiga, L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci. 16(7), 1031–1090 (2006) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Dörfel, M., Jüttler, B., Simeon, B.: Adaptive isogeometric analysis by local h-refinement with t-splines. Comput. Methods Appl. Mech. Eng. 199(5–8), 264–275 (2010) MATHCrossRefGoogle Scholar
  15. 15.
    Hughes, T.J.R., Reali, A., Sangalli, G.: Efficient quadrature for NURBS-based isogeometric analysis. Comput. Methods Appl. Mech. Eng. 199(5–8), 301–313 (2010) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Beirão da Veiga, L., Buffa, A., Rivas, J., Sangalli, G.: Some estimates for h-p-k-refinement in isogeometric analysis. Numer. Math. 118, 271–305 (2011) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Vuong, A.-V., Gianelli, C., Jüttler, B., Simeon, B.: A hierarchical approach to adaptive local refinement in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 200(49–52), 3554–3567 (2011) MATHCrossRefGoogle Scholar
  18. 18.
    Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, New York (2009) CrossRefGoogle Scholar
  19. 19.
    Ferri, A.A.: On the equivalence of the incremental harmonic balance method and the harmonic balance-Newton–Raphson method. J. Appl. Mech. 53(2), 455–457 (1986) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical Computational, and Experimental Methods. Wiley Series in Nonlinear Science. Wiley, New York (1995) MATHCrossRefGoogle Scholar
  21. 21.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley Classics Library. Wiley, New York (1995) CrossRefGoogle Scholar
  22. 22.
    Szemplinska-Stupnicka, W.: The Behaviour of Nonlinear Vibrating Systems. Kluwer Academic, Dordrecht, Boston, London (1990) MATHCrossRefGoogle Scholar
  23. 23.
    Wagg, D., Neild, S.: Nonlinear Vibration with Control: For Flexible and Adaptive Structures. Solid Mechanics and Its Applications. Springer, Berlin (2010) CrossRefGoogle Scholar
  24. 24.
    Worden, K., Tomlinson, G.R.: Nonlinearity in Structural Dynamics: Detection, Identification and Modelling. Institute of Physics, Bristol (2001) CrossRefGoogle Scholar
  25. 25.
    Lewandowski, R.: Non-linear, steady-state vibration of structures by harmonic balance/finite element method. Comput. Struct. 44(1–2), 287–296 (1992) MATHCrossRefGoogle Scholar
  26. 26.
    Lewandowski, R.: Computational formulation for periodic vibration of geometrically nonlinear structures, part 1: theoretical background; part 2: numerical strategy and examples. Int. J. Solids Struct. 34(15), 1925–1964 (1997) MATHCrossRefGoogle Scholar
  27. 27.
    Ribeiro, P., Petyt, M.: Non-linear vibration of beams with internal resonance by the hierarchical finite element method. J. Sound Vib. 224(15), 591–624 (1999) CrossRefGoogle Scholar
  28. 28.
    Ribeiro, P.: Hierarchical finite element analyses of geometrically non-linear vibration of beams and plane frames. J. Sound Vib. 246(2), 225–244 (2001) MathSciNetCrossRefGoogle Scholar
  29. 29.
    Ribeiro, P.: Non-linear forced vibrations of thin/thick beams and plates by the finite element and shooting methods. Comput. Struct. 82(17–19), 1413–1423 (2004) CrossRefGoogle Scholar
  30. 30.
    Cheung, Y.K., Chen, S.H., Lau, S.L.: Application of the incremental harmonic balance method to cubic non-linearity systems. J. Sound Vib. 140(2), 273–286 (1990) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Chen, S.H., Cheung, Y.K., Xing, H.X.: Nonlinear vibration of plane structures by finite element and incremental harmonic balance method. Nonlinear Dyn. 26, 87–104 (2001) MATHCrossRefGoogle Scholar
  32. 32.
    Reddy, J.N.: An Introduction to Nonlinear Finite Elements. Oxford University Press, New York (2004) CrossRefGoogle Scholar
  33. 33.
    Gross, D., Hauger, W., Wriggers, P.: Technische Mechanik 4—Hydromechanik, Elemente der Höheren Mechanik, Numerische Methoden, 7. auflage edition. Springer, Berlin Heidelberg (2009) Google Scholar
  34. 34.
    Bobylev, N.A., Burman, Y.M., Korovin, S.K.: Approximation Procedures in Nonlinear Oscillation Theory. De Gruyter Series in Nonlinear Analysis and Applications. W. de Gruyer, Berlin (1994) MATHCrossRefGoogle Scholar
  35. 35.
    Schneider, M., Wever, U., Zheng, Q.: Parallel harmonic balance. In: VLSI 93, Proceedings of the IFIP TC10/WG 10.5 International Conference on Very Large Scale Integration, Grenoble, France, 7–10 September, 1993, pp. 251–260 (1993) Google Scholar
  36. 36.
    Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods. Colorado State University, Fort Collins (1990) CrossRefGoogle Scholar
  37. 37.
    Haisler, W.E., Stricklin, J.A., Key, J.E.: Displacement incrementation in non-linear structural analysis by the self-correcting method. Int. J. Numer. Methods Eng. 11(1), 3–10 (1977) MATHCrossRefGoogle Scholar
  38. 38.
    Belytschko, T., Liu, W.K., Moran, B.: Nonlinear Finite Elements for Continua and Structures. Wiley, New York (2000) MATHGoogle Scholar
  39. 39.
    Oden, J.T.: Finite Elements of Nonlinear Continua. Dover Civil and Mechanical Engineering Series. Dover, New York (2006) MATHGoogle Scholar
  40. 40.
    TERRIFIC: Towards enhanced integration of design and production in the factory of the future through isogeometric technologies. EU Project FP7, FoF-ICT-2011.7.4 Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.TU KaiserslauternFelix-Klein-Center for MathematicsKaiserslauternGermany
  2. 2.Siemens AG, Corporate TechnologyMunichGermany

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