Advertisement

Adaptive Algorithm Based on Functional-Type A Posteriori Error Estimate for Reissner-Mindlin Plates

  • Maxim FrolovEmail author
  • Olga Chistiakova
Chapter
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 128)

Abstract

This research is devoted to numerical justification of an adaptive mesh refinement algorithm based on functional-type a posteriori local error indicator for Reissner-Mindlin plates. Four stages of this algorithm (solver, estimator, marker and refiner) and its implementation are discussed. A number of numerical experiments for L-shape and skew Reissner-Mindlin plates is provided for verification and efficiency demonstration. It is also shown that efficiency index for functional-type a posteriori error estimate is stable and has acceptable value. As such a technique can be used with in-house implementation of finite element solver as well as with commercial software packages with closed sources, proposed algorithm may be applicable for engineering practice.

Notes

Acknowledgements

This research is supported by the Grant of the President of the Russian Federation MD-1071.2017.1.

References

  1. 1.
    Arnold, D., Boffi, D., Falk, R.: Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42, 2429–2451 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bathe, K., Dvorkin, E.: A formulation of general shell elements—the use of mixed interpolation of tensoral components. Int. J. Numer. Methods Eng. 22, 697–722 (1986)CrossRefGoogle Scholar
  3. 3.
    Beirão da Veiga, L., Chinosi, C., Lovadina, C., Stenberg, R.: A-priori and a-posteriori error analysis for a family of Reissner-Mindlin plate elements. BIT Numer. Math. 48, 189–213 (2008)Google Scholar
  4. 4.
    Beirão da Veiga, L., Mora, D., Rivera, G.: Virtual elements for a shear-deflection formulation of Reissner-Mindlin plates. Math. Comput. (2018)Google Scholar
  5. 5.
    Carstensen, C., Xie, X., Yu, G., Zhou, T.: A priori and a posteriori analysis for a locking-free low order quadrilateral hybrid finite element for Reissner-Mindlin plates. Comput. Methods Appl. Mech. Eng. 200, 1161–1175 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cen, S., Shang, Y.: Developments of Mindlin-Reissner plate elements. Math. Probl. Eng. 2015, 12 (2015). Article ID 456740Google Scholar
  7. 7.
    Dörfler, W.: A convergent adaptive algorithm for poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Frolov, M.: Reliable a posteriori error control for solutions to problems of Reissner-Mindlin plates bending. In: Procceding of 10th International Conference. Mesh methods for boundary-value problems and applications, Kazan, Russia, pp. 610–615 (2014)Google Scholar
  9. 9.
    Frolov, M., Chistiakova, O.: A functional-type a posteriori error estimate of approximate solutions for Reissner-Mindlin plates and its implementation. IOP Conf. Ser. Mater. Sci. Eng. 208, 012043 (2017)CrossRefGoogle Scholar
  10. 10.
    Frolov, M., Neittaanmäki, P., Repin, S.: Guaranteed functional error estimates for the Reissner-Mindlin plate problem. J. Math. Sci. 132, 553–561 (2006). Translated from Problemy Matematicheskogo Analiza 31, 159–167 (2005)zbMATHGoogle Scholar
  11. 11.
    Karavaev, A., Kopysov, S.: A refinement of unstructured quadrilateral and mixed meshes. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauk. 4, 62–78 (2013)CrossRefGoogle Scholar
  12. 12.
    Ko, Y., Lee, P.S., Bathe, K.: A new 4-node MITC element for analysis of two-dimensional solids and its formulation in a shell element. Comput. Struct. 192, 34–49 (2017)CrossRefGoogle Scholar
  13. 13.
    Mekchay, K., Nochetto, R.H.: Convergence of adaptive finite element methods for general second order linear elliptic PDEs. SIAM J. Numer. Anal. 43, 1803–1827 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pechstein, A.S., Schöberl, J.: An analysis of the TDNNS method using natural norms. Numer. Math. 139, 93–120 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Repin, S.: A posteriori estimates for partial differential equations. Radon Series on Computational and Applied Mathematics, vol. 4. de Gruyter, Berlin (2008)Google Scholar
  16. 16.
    Repin, S., Frolov, M.: Estimation of deviations from the exact solution for the Reissner-Mindlin plate problem. J. Math. Sci. 132, 331–338 (2006). Translated from Zapiski Nauchnykh Seminarov POMI 310, 145–157 (2004)Google Scholar
  17. 17.
    Song, S., Niu, C.: A mixed finite element method for the Reissner-Mindlin plate. Bound. Value Probl. 1, 194 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Verfürth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques. Wiley/B.G. Teubner, Chichester/Stuttgart (1996)Google Scholar
  19. 19.
    Wu, C.T., Wang, H.P.: An enhanced cell-based smoothed finite element method for the analysis of Reissner–Mindlin plate bending problems involving distorted mesh. Int. J. Numer. Methods Eng. 95, 288–312 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Peter the Great St Petersburg Polytechnic UniversitySt. PetersburgRussia

Personalised recommendations