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Meccanica

, Volume 53, Issue 6, pp 1403–1413 | Cite as

A displacement-free formulation for the Timoshenko beam problem and a corresponding isogeometric collocation approach

  • J. KiendlEmail author
  • F. Auricchio
  • A. Reali
Novel Computational Approaches to Old and New Problems in Mechanics

Abstract

We present a reformulation of the classical Timoshenko beam problem, resulting in a single differential equation with the rotation as the only primal variable. We show that this formulation is equivalent to the standard formulation and the same types of boundary conditions apply. Moreover, we develop an isogeometric collocation scheme to solve the problem numerically. The formulation is completely locking-free and involves only half the degrees of freedom compared to a standard formulation. Numerical tests are presented to confirm the performance of the proposed approach.

Keywords

Timoshenko beam Shear-deformable Locking-free Displacement-free Isogeometric Collocation 

Notes

Acknowledgements

J. Kiendl was partially supported by the Onsager fellowship program of NTNU. J. Kiendl, A. Reali, and F. Auricchio were partially supported by the ERC Starting Grant No. 259229 ISOBIO. A. Reali was partially supported by Fondazione Cariplo–Regione Lombardia through the project “Verso nuovi strumenti di simulazione super veloci ed accurati basati sull’analisi isogeometrica”, within the program “RST–rafforzamento”. This support is gratefully acknowledged.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Anitescu C, Jia Y, Zhang J, Rabczuk T (2015) An isogeometric collocation method using superconvergent points. Comput Methods Appl Mech Eng 284:1073–1097ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Auricchio F, Beirão da Veiga L, Hughes TJR, Reali A, Sangalli G (2010) Isogeometric collocation methods. Math Models Methods Appl Sci 20(11):2075–2107MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Auricchio F, Beirão da Veiga L, Hughes TJR, Reali A, Sangalli G (2012) Isogeometric collocation for elastostatics and explicit dynamics. Comput Methods Appl Mech Eng 249–252:2–14MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Auricchio F, Beirão da Veiga L, Kiendl J, Lovadina C, Reali A (2013) Locking-free isogeometric collocation methods for spatial Timoshenko rods. Comput Methods Appl Mech Eng 263:113–126ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Balduzzi G, Morganti S, Auricchio F, Reali A (2017) Non-prismatic Timoshenko-like beam model: numerical solution via isogeometric collocation. Comput Math Appl, in press. doi: 10.1016/j.camwa.2017.04.025
  6. 6.
    Beirão da Veiga L, Hughes T.J.R., Kiendl J, Lovadina C, Niiranen J, Reali A, Speelers H(2014) A locking-free model for Reissner-Mindlin plates: analysis and isogeometric implementation via NURBS and triangular NURPS. In preparation Google Scholar
  7. 7.
    Beirão da Veiga L, Lovadina C, Reali A (2012) Avoiding shear locking for the Timoshenko beam problem via isogeometric collocation methods. Comput Methods Appl Mech Eng 241–244:38–51MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Casquero H, Liu L, Zhang Y, Reali A, Gomez H (2016) Isogeometric collocation using analysis-suitable T-splines of arbitrary degree. Comput Methods Appl Mech Eng 301:164–186ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  10. 10.
    De Lorenzis L, Evans JA, Hughes TJR, Reali A (2015) Isogeometric collocation: Neumann boundary conditions and contact. Comput Methods Appl Mech Eng 284:21–54ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Echter R, Oesterle B, Bischoff M (2013) A hierarchic family of isogeometric shell finite elements. Comput Methods Appl Mech Eng 254:170–180ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gomez H, De Lorenzis L (2016) The variational collocation method. Comput Methods Appl Mech Eng 309:152–181ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Gomez H, Reali A, Sangalli G (2014) Accurate, efficient, and (iso)geometrically flexible collocation methods for phase-field models. J Comput Phys 262:153–171ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kiendl J, Auricchio F, Beirão da Veiga L, Lovadina C, Reali A (2015) Isogeometric collocation methods for the Reissner-Mindlin plate problem. Comput Methods Appl Mech Eng 284:489–507ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kiendl J, Auricchio F, Hughes TJR, Reali A (2015) Single-variable formulations and isogeometric discretizations for shear deformable beams. Comput Methods Appl Mech Eng 284:988–1004ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Kiendl J, Marino E, De Lorenzis L (2017) Isogeometric collocation for the Reissner–Mindlin shell problem. Comput Methods Appl Mech Eng 325:645–665ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kruse R, Nguyen-Thanh N, De Lorenzis L, Hughes TJR (2015) Isogeometric collocation for large deformation elasticity and frictional contact problems. Comput Methods Appl Mech Eng 296:73–112ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Lin H, Hu Q, Xiong Y (2013) Consistency and convergence properties of the isogeometric collocation method. Comput Methods Appl Mech Eng 267:471–486ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Long Q, Bornemann PB, Cirak F (2012) Shear-flexible subdivision shells. Int J Numer Methods Eng 90(13):1549–1577MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Marino E (2016) Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams. Comput Methods Appl Mech Eng 307:383–410ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Marino E (2017) Locking-free isogeometric collocation formulation for three-dimensional geometrically exact shear-deformable beams with arbitrary initial curvature. Comput Methods Appl Mech Eng 324:546–572ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Montardini M, Sangalli G, Tamellini L (2017) Optimal-order isogeometric collocation at galerkin superconvergent points. Comput Methods Appl Mech Eng 316:741–757ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Reali A, Gomez H (2015) An isogeometric collocation approach for Bernoulli-Euler beams and Kirchhoff plates. Comput Methods Appl Mech Eng 284:623–636ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Reali A, Hughes TJR (2015) An introduction to isogeometric collocation methods. In: Beer G (ed) Isogeometric methods for numerical simulation. Springer, BerlinGoogle Scholar
  26. 26.
    Schillinger D, Borden MJ (2015) Stolarski H (2015) Isogeometric collocation for phase-field fracture models. Comput Methods Appl Mech Eng 284:583–610ADSCrossRefGoogle Scholar
  27. 27.
    Schillinger D, Evans JA, Reali A, Scott MA, Hughes TJR (2013) Isogeometric collocation: cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations. Comput Methods Appl Mech Eng 267:170–232ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Weeger O, Yeung S-K, Dunn ML (2017) Isogeometric collocation methods for Cosserat rods and rod structures. Comput Methods Appl Mech Eng 316:100–122ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Marine TechnologyNorwegian University of Science and TechnologyTrondheimNorway
  2. 2.Department of Civil Engineering and ArchitectureUniversity of PaviaPaviaItaly
  3. 3.Institute for Applied Mathematics and Information Technology (IMATI)National Research Council (CNR)PaviaItaly
  4. 4.Institute for Advanced Study (IAS)Technical University of MunichGarchingGermany

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