DOI QR코드

DOI QR Code

NURBS-based isogeometric analysis for thin plate problems

  • Shojaee, S. (Civil Engineering Department, Shahid Bahonar University of Kerman) ;
  • Valizadeh, N. (Civil Engineering Department, Shahid Bahonar University of Kerman)
  • Received : 2011.03.12
  • Accepted : 2012.02.01
  • Published : 2012.03.10

Abstract

An isogeometric approach is presented for static analysis of thin plate problems of various geometries. Non-Uniform Rational B-Splines (NURBS) basis function is applied for approximation of the thin plate deflection, as for description of the geometry. The governing equation based on Kirchhoff plate theory, is discretized using the standard Galerkin method. The essential boundary conditions are enforced by the Lagrange multiplier method. Several typical examples of thin plate and thin plate on elastic foundation are solved and compared with the theoretical solutions and other numerical methods. The numerical results show the robustness and efficiency of the proposed approach.

Keywords

References

  1. Bazilevs, Y., Calo, V.M., Zhang, Y. and Hughes, T.J.R. (2006), "Isogeometric fluid-structure interaction analysis with applications to arterial blood flow", Comput. Mech., 38, 310-322. https://doi.org/10.1007/s00466-006-0084-3
  2. Bazilevs, Y. and Hughes, T.J.R. (2008a), "NURBS-based isogeometric analysis for the computation of flows about rotating components", Comput. Mech., 43, 143-150. https://doi.org/10.1007/s00466-008-0277-z
  3. Bazilevs, Y., Calo, V.M., Hughes, T.J.R. and Zhang, Y. (2008b), "Isogeometric fluid-structure interaction: theory, algorithms and computations", Comput. Mech., 43, 3-37. https://doi.org/10.1007/s00466-008-0315-x
  4. Belytschko, T., Lu, Y.Y. and Gu, L. (1994), "Element free Galerkin methods", Int. J. Numer. Meth. Eng., 37, 229-256. https://doi.org/10.1002/nme.1620370205
  5. Benson, D.J., Bazilevs, Y., Hsu, M.C. and Hughes, T.J.R. (2010), "Isogeometric shell analysis: the Reissner- Mindlin shell", Comput. Meth. Appl. Mech. Eng., 199, 276-289. https://doi.org/10.1016/j.cma.2009.05.011
  6. Ghorashi, S.S., Valizadeh, N. and Mohammadi, S. (2011), "Extended isogeometric analysis for simulation of stationary and propagating cracks", Int. J. Numer. Meth. Eng., doi:10.1002/nme.3277.
  7. Hoschek, J. and Lasser, D. (1993), Fundamentals of Computer Aided Geometric Design, A.K. Peters, Ldt, Wellesley, Massachusetts.
  8. Hughes, T.J.R., Cottrell, J.A. and Bazilevs, Y. (2005), "Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement", Comput. Meth. Appl. Mech. Eng., 194, 4135-4195. https://doi.org/10.1016/j.cma.2004.10.008
  9. Kiendl, J., Bletzinger, K.U., Linhard, J. and Wchner, R. (2009), "Isogeometric shell analysis with Kirchhoff-Love elements", Comput. Meth. Appl. Mech. Eng., 198, 3902-3914. https://doi.org/10.1016/j.cma.2009.08.013
  10. Krysl, P. and Belytschko, T. (1995), "Analysis of thin plates by the element-free Galerkin method", Comput. Mech., 17, 26-35. https://doi.org/10.1007/BF00356476
  11. Liu, Y., Hon, Y.C. and Liew, K.M. (2006), "A meshfree Hermite-type radial point interpolation method for Kirchhoff plate problems", Int. J. Numer. Meth. Eng., 66, 1153-1178. https://doi.org/10.1002/nme.1587
  12. Liu, F.L. and Liew, K.M. (1998), "Differential cubature method for static solutions of arbitrary shaped thick plates", Int. J. Solids Struct., 35, 3655-3674. https://doi.org/10.1016/S0020-7683(97)00215-1
  13. Nitsche, J. (1971), "Uber ein variation zur lsung von Dirichlet-problemen bei Verwendung von teilrumen die keinen randbedingungen unterworfen sind", Abh. Math. Se. Univ. Hamburg, 36, 9-15. https://doi.org/10.1007/BF02995904
  14. Piegl, L. and Tiller, W. (1997), The NURBS Book (Monographs in Visual Communication), Springer-Verlag, Second edition, New York.
  15. Qian, X. (2010), "Full analytical sensitivities in NURBS based isogeometric shape optimization", Comput. Meth.Appl. Mech. Eng., 199, 2059-2071. https://doi.org/10.1016/j.cma.2010.03.005
  16. Roh, H.Y. and Cho, M. (2004), "The application of geometrically exact shell elements to B-spline surfaces", Comput. Meth. Appl. Mech. Eng., 193, 2261-2299. https://doi.org/10.1016/j.cma.2004.01.019
  17. Shen, P.C. (1991), Spline Finite Element Methods in Structural Analysis, Hydraulic and Electric Press, Beijing.
  18. Timoshenko, S.P. and Woinowsky-Krieger, S. (1995), Theory of Plates and Shells, Second edition, McGraw Hill, New York.
  19. Uhm, T.K. and Youn, S.K. (2009), "T-spline finite element method for the analysis of shell structures", Int. J. Numer. Meth. Eng., 80, 507-536. https://doi.org/10.1002/nme.2648
  20. Ventsel, E. and Krauthammer, T. (2001), Thin Plates and Shells: Theory, Analysis and Applications, Marcel Dekker, New York.
  21. Verhoosel, C.V., Scott, M.A., Hughes, T.J.R. and de Borst, R. (2011a), "An isogeometric analysis approach to gradient damage models", Int. J. Numer. Meth. Eng., 86, 115-134. https://doi.org/10.1002/nme.3150
  22. Verhoosel, C.V., Scott, M.A., de Borst, R. and Hughes, T.J.R. (2011b), "An isogeometric approach to cohesive zone modeling", Int. J. Numer. Meth. Eng., 87, 336-360. https://doi.org/10.1002/nme.3061
  23. Wall, W.A., Frenzel, M.A. and Cyron, C. (2008), "Isogeometric structural shape optimization", Comput. Meth. Appl. Mech. Eng., 197, 2976-2988. https://doi.org/10.1016/j.cma.2008.01.025
  24. Xiang, J., Chen, X., He, Y. and He, Zh. (2007), "Static and vibration analysis of thin plates by using finite element method of B-spline wavelet on the interval", Struct. Eng. Mech., 25, 613-629. https://doi.org/10.12989/sem.2007.25.5.613
  25. Zhang, X., Chen, X., Wang, X. and He, Zh. (2010), "Multivariable finite elements based on B-spline wavelet on the interval for thin plate static and vibration analysis", Finite Elem. Anal. Des., 46, 416-427. https://doi.org/10.1016/j.finel.2010.01.002
  26. Zhu, T. and Atluri, S.N. (1998), "A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method", Comput. Mech., 21, 211-222. https://doi.org/10.1007/s004660050296

Cited by

  1. A Fortran implementation of isogeometric analysis for thin plate problems with the penalty method vol.33, pp.7, 2016, https://doi.org/10.1108/EC-10-2015-0306
  2. Application of isogeometric method to free vibration of Reissner–Mindlin plates with non-conforming multi-patch vol.82, 2017, https://doi.org/10.1016/j.cad.2016.04.006
  3. Hybrid of topological derivative-based level set method and isogeometric analysis for structural topology optimization vol.21, pp.6, 2016, https://doi.org/10.12989/scs.2016.21.6.1389
  4. Nitsche method for isogeometric analysis of Reissner–Mindlin plate with non-conforming multi-patches vol.35-36, 2015, https://doi.org/10.1016/j.cagd.2015.03.005