Accurate Calculation of Nodal Vectors in Isogeometric Analysis of Shell Structures Using Greville Points

Document Type : Research Article

Authors

1 Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran.

2 فردوسی مشهد-مهندسی- گروه مهندسی مکانیک

Abstract

In this paper, the isogeometric analysis of shell structures along with the optimal method for the accurate calculation of nodal vectors is proposed. The Non-uniform rational B-spline is used for shell mid-surface description. According to the Reisner- Mindlin hypothesis, the director vectors at control points are needed for the interpolation of the rotations. The calculated nodal direct vectors must lead to exact interpolated director vectors on the shell surface. Hence, a method has been proposed in which the components of director vectors at control points are obtained by solving a system of equations on the whole patch. The system of equations is formed using known values of direction vectors at the Greville points. The accuracy of the proposed method has been investigated by using the results of the most common problem in shell analysis. Convergence behavior for displacement at the loading points has been studied in all solved problems for a different order of basis functions and net of control points. The deformation results show better convergence behavior with increasing the regularity and order of basis functions. The Greville points are in a one-to-one correspondence with control points. Thus, the system of equations on these points leads to a unique solution for the nodal direction vectors, and the time to solve equations is significantly reduced. 

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References

[1] J. Hughes, J. A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Eng., 194(2) (2005) 4135-4195.
[2] J. Cottrell, T.J.R. Hughes, A. Reali, Studies of refinement and continuity in isogeometric structural analysis, Comput.Methods Appl. Mech. Engrg. 196 (2007) 4160–4183
[3] Y. Bazilevs, L. Beir˜ao da Veiga, J. Cottrell, T.J.R. Hughes., G. Sangalli, Isogeometric analysis: Approximation, stability and error estimates for h-refined meshes. Math. Mod. Methods Appl. Sci. 16 (2006), 1-60.
[4] T. W. Sederberg, J. Zheng, A. Bakenov, A. Nasri, T-splines and T-NURCCSs, ACM Transactions on Graphics, 22(3) (2003). 477–484.        
[5] Y. Bazilevs, V. M Calo. J. A Cottrell, J. A Evans, T. J. R Hughes, S. Lipton, T. W Sederberg,. Isogeometric analysis using T-splines. Computer Methods in Applied Mechanics and Engineering, 199(5) (2010),  229-263.
[6] R. Echter, M. Bischoff, Numerical efficiency, locking and unlocking of NURBS finite elements, Computer Method sin Applied Mechanics and Engineering, 199(1) 2010 374–382.
[7] L. Beirão Da Veiga, T. J. R. Hughes, J. Kiendl, C. Lovadina, J. Niiranen, A. Reali and H. Speleers, A locking-free model for Reissner–Mindlin plates: Analysis and isogeometric implementation via NURBS and triangular NURPS, Mathematical Models and Methods in Applied Sciences. 25(8) 2015 1519-1551.
[8] R. Bouclier, T. Elguedj, A. Combescure, Efficient isogeometric NURBS-based solid-shell elements: Mixed formulation and B-bar method, Computer Methods in Applied Mechanics and Engineering, 267(1) 2013 86-110.
[9] R. Bouclier, T. Elguedj, A. Combescure, An isogeometric locking-free NURBS-based solid-shell element for geometrically nonlinear analysis, International Journal for Numerical Methods in Engineering, 101(10) 2015 777-808.
[10] J. F. Caseiro, R. A. F. Valente, A. Reali, J. Kiend, F. Auricchio, R. J. Alves de Sousa. On the assumed natural strain method to alleviate locking in solid-shell NURBS-based finite elements, Computational Mechanics, 53 (2014) 1341–1353.
[11] G. Kikis, W. Dornisch, S. Klinkel, Isogeometric Reissner-Mindlin shell analysis - employing different control meshes for displacements and rotations, Appl. Math. Mech. 16(1) (2016) 209 – 210.
[12] Q. Hu, Y. Xia, S. Natarajan, A. Zilian, P. Hu, Stéphane P.A. Bordas, Isogeometric analysis of thin Reissner-Mindlin plates and shells: locking phenomena and B-bar method, Comput Mech. 65  (2020) 1323–1341.
[13]  C. Adam, S. Bouabdallah, M. Zarroug, H. Maitournam, Improved numerical integration for locking treatment in isogeometric structural elements, Part I: Beams, Computer Methods in Applied Mechanics and Engineering 279 (2014) 1- 28.
[14] C. Adam, S. Bouabdallah, M. Zarroug, H. Maitournam, Improved numerical integration for locking treatment in isogeometric structural elements. Part II: Plates and shells, Computer Methods in Applied Mechanics and Engineering, 284(1) (2015) 106-137
[15]  C. Adam, S. Bouabdallah, M. Zarroug, H. Maitournam, A Reduced Integration for Reissner-Mindlin Non-linear Shell Analysis Using T-Splines, in: Isogeometric Analysis and Applications, Springer, 2015 103-125.
[16] J. Kiendl, K. U. Bletzinger, J. Linhard, R. Wüchner, Isogeometric shell analysis with Kirchhoff-Love elements, Computer Methods in Applied Mechanics and Engineering, 198 (2009) 3902–3914.
[17] N. Nguyen-Thanh, J. Kiendl, H. Nguyen-Xuan, R. Wüchner, K.-U. Bletzinger, Y. Bazilevs, T. Rabczuk, Rotation free isogeometric thin shell analysis using PHT-splines, Computer Methods in Applied Mechanics and Engineering, 200 (2011) 3410–3424.
[18] D. J. Benson, Y. Bazilevs, M. C. Hsu, T. J. R.  Hughes, A large deformation, rotation-free, isogeometric shell, Computer Methods in Applied Mechanics and Engineering, 200 (2011) 1367–1378.
[19] D. J. Benson, Y. Bazilevs, M. C. Hsu, T. J. R.  Hughes, Isogeometric shellanalysis: The Reissner–Mindlin shell, Computer Methods in Applied Mechanics and Engineering, 199 (2010) 276–289.
[20] D. J. Benson, S. Hartmann, Y. Bazilevs, M. C. Hsu, T. J. R.  Hughes, Blended isogeometric shells, Computer Methods in Applied Mechanics and Engineering, 255(2013)133–146.
[21] R. Echter, B. Oesterle, M. Bischoff, A hierarchic family of isogeometricshell finite elements, Computer Methods in Applied Mechanics and Engineering, 254(2013)170–180.
[22] T. K. Uhm, S. K. Youn, T‐spline finite element method for the analysis of shell structures, International Journal for Numerical Methods in Engineering,80(4) (2009) 507-536.
[23] W. Dornisch, S. Klinkel, B. Simeon, Isogeometric Reissner–Mindlin shell analysis with exactly calculated director vectors, Computer Methods in Applied Mechanics and Engineering, 253 (2013) 491-504.
[24] W. Dornisch, S. Klinkel, Treatment of Reissner–Mindlin shells with kinks without the need for drilling rotation stabilization in an isogeometric frame-work, Comput. MethodsAppl. Mech. Eng.276(2014)35–66.
[25] L. Piegl, W. Tiller, The NURBS book: Monographs in visual communications, Springer Publisher, Berlin, second edition, ISBN: 1997, 1431-6897.
[26] O. C. Zienkiewicz, R. L. Taylor, The finite element method for solid and structural mechanics. Butterworth-heinemann., (2005).
[27] P. Kang, S. K. Youn, Isogeometric analysis of topologically complex shell structures, Finite Elements in Analysis and Design, 99 (2015) 68-81.
[28] R. H. Macneal, R. L. Harder, A proposed standard set of problems to test finite element accuracy. Finite Elements in Analysis and Design, 1(1) (1985) 3–20.
[29] S. Nikoei, B. Hassani, Isogeometric analysis of Free Form Shells and Accurate calculation of direction vectors using Kirchhoff-Love and Reissner-Mindlin theories, JMEUT,50(4) (2020) 209-218 (in Persian).
Amirkabir Journal of Mechanical Engineering
Volume 53, Issue 10
January 2022
Pages 5185-5202

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