Cover interpolation functions and h-enrichment in finite element method

Document Type : Research

Authors

1 Assistant professor, Faculty of civil engineering, Shahid Rajaee Teacher Training University, Lavizan, Tehran, Iran

2 Ph.D. Student., Department of Civil Engineering, Shahid Rajaee Teacher Training University, Lavizan, Tehran, Iran

3 M.Sc. Graduate Student, Department of Civil Engineering, Shahid Rajaee Teacher Training University, Lavizan, Tehran, Iran.

Abstract

This paper presents a method to improve the generation of meshes and the accuracy of numerical solutions of elasticity problems, in which two techniques of h-refinement and enrichment are used by interpolation cover functions. Initially, regions which possess desired accuracy are detected. Mesh improvment is done through h-refinement for the elements existing in those regions. Total error of the domain is thus reduced and limited to the allowable range. In order to increase the accuracy of solutions to an excellent level, the results of mesh refinement are reassessed in the next steps and the nodes exceeding the value of allowable error are determined. The method automatically improves the subdomain by increasing the order of  interpolation cover  functions which  yields to solutions of appropriate accuracy. A comparison of solutions achieved by the proposed method with that of other methods and also the accurate solutions for linear elasticity examples proves acceptable efficiency and accuracy of the proposed method. In this research, we illustrate the power of the strategy through the solutions obtained for various problems.

Keywords

References

1. Arzani, H., Kaveh, A., Dehghana, M., "Adaptive node moving refinement in discrete least squares meshless method using charged system search", Scientia Iranica. Transaction A, Civil Engineering, 21, 2014, p. 1529-1538.
2. Babuška, I., Zienkiewicz, O.C., Gago, J.P., "Accuracy estimates and adaptive refinements in finite element Computations", John Wiley & Sons, 1986.
3. Bathe, K. J., "Finite element procedures", Cambridge, MA: Klaus-Jürgen Bathe, 2006.
4. Ghasemzadeh, H., Ramezanpour, M.A., Bodaghpour, S., "Dynamic high order numerical manifold method based on weighted residual method". Internatinal journal for numerical methods in engineering Int. J. Numer. Meth. Engng, Published online in Wiley Online Library, 2014. [DOI:10.1002/nme.4752]
5. Grayeli, R,. Mortazavi, A., "Discontinuous deformation analysis with second-order finite element meshed block", Int J Numer Anal Methods Geomech, 30, 2006, p.1545-61. [DOI:10.1002/nag.538]
6. Jeon, H.M., Lee, S.L., Bathe, K.J., "The MITC3 shell finite element enriched by interpolation covers", Computers & Structures, 134, 2014, p. 128-142. [DOI:10.1016/j.compstruc.2013.12.003]
7. Johnson, C., "Numerical solution of partial differential equations by the finite element method", Studentlitteratur, Lund, Sweden, 1987.
8. Johnson, C., Eriksson, K., "Adaptive finite element methods for parabolic Problems I: A Linear Model Problem", SIAM J, 28, 1991, p. 43-77. [DOI:10.1137/0728003]
9. Kim, J., Bathe, K. J., "The finite element method enriched by interpolation covers", Comput Struct, 116, 2013, pp. 35-49. [DOI:10.1016/j.compstruc.2012.10.001]
10. Kim, J., Bathe, K. J., "Towards a procedure to automatically improve finite element solutions by interpolation covers", Computers & Structures, 131, 2014, p. 81-97. [DOI:10.1016/j.compstruc.2013.09.007]
11. Liu, J., Kong, X., Lin, G., "Formulations of the three-dimensional discontinuous deformation analysis method", Acta Mech Sin, 20, 2004, p. 270-82. [DOI:10.1007/BF02486719]
12. Plaza, A., Padrón, M. A., Suárez, J. P., "Non-degeneracy study of the 8-tetrahedra longest-edge partition", Applied Numerical Mathematics, 55, 2005, pp. 458-472. [DOI:10.1016/j.apnum.2004.12.003]
13. Rosenberg, I. G., Stenger, F., "A lower bound on the angles of triangles constructed by bisecting the longest side", Math. Comp, 29, 1975, p. 390-395. [DOI:10.2307/2005558]
14. Shi, GH., "Block system modeling by discontinuous deformation analysis", Southampton, UK: Computational Mechanics Publications, 1993.
15. Xinmei, An., Guowei, Ma., Yongchang, Cai., Hehua, Zhu., "The numerical manifold method: a review" International Journal of Computational Methods, 07, 2010, p. 1-32. [DOI:10.1142/S0219876210002040]
16. Yang, R., Yuan, G., "h-Refinement for simple corner balance scheme of SN transport equation on distorted meshes", Journal of Quantitative Spectroscopy and Radiative Transfer, 184, 2016, p. 241-253. [DOI:10.1016/j.jqsrt.2016.07.020]
17. Yang, J., Zhang, B., Liang, C., Rong, Y., "A high-order flux reconstruction method with adaptive mesh refinement and artificial diffusivity on unstructured moving/deforming mesh for shock capturing", Computers & Fluids, 139, 2016, p. 17-35. [DOI:10.1016/j.compfluid.2016.03.025]
18. Yershov, D. S,. Frazzoli, E., "Asymptotically optimal feedback planning using a numerical Hamilton-Jacobi-Bellman solver and an adaptive mesh refinement", Int. J. Rob. Res, 35, 2016, p. 565-584. [DOI:10.1177/0278364915602958]
19. Zander, N., Bog, T., Elhaddad, M., Frischmann, F., Kollmannsberger, S., Rank, E., "The multi-level -method for three-dimensional problems: Dynamically changing high-order mesh refinement with arbitrary hanging nodes", Computer Methods in Applied Mechanics and Engineering, 310, 2016, p. 252-277. [DOI:10.1016/j.cma.2016.07.007]
20. Zeng, W., Liu, G. R., Li, D., Dong, X. W., "A smoothing technique based beta finite element method (βFEM) for crystal plasticity modeling", Computers & Structures, 162, 2016, p. 48-67. [DOI:10.1016/j.compstruc.2015.09.007]
21. Zhang, H. H., Li, L. X., An, X. M., Ma, G. W., "Numerical analysis of 2-D crack propagation problems using the numerical manifold method", Engineering Analysis with Boundary Elements, 34, 2010, pp. 41-50. [DOI:10.1016/j.enganabound.2009.07.006]
22. Zienkiewicz, O. C., "Achievements and some unsolved problems of the finite element method", International Journal for Numerical Methods in Engineering, 47, 2000, p. 9-28. https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3<9::AID-NME793>3.0.CO;2-P [DOI:10.1002/(SICI)1097-0207(20000110/30)47:1/33.0.CO;2-P]
23. Zienkiewicz, O.C., Zhu, J. Z., "A simple error estimator and adaptive procedure for practical engineerng analysis", International Journal for Numerical Methods in Engineering, 24, 1987 ,p. 337-357. [DOI:10.1002/nme.1620240206]
24. Zienkiewicz, O.C., Zhu, J. Z., "The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique", International Journal for Numerical Methods in Engineering, 33, 1992, p. 1331-1364. [DOI:10.1002/nme.1620330702]
25. Zienkiewicz, O.C., Zhu, J. Z., "The superconvergent patch recovery and a posteriori error estimates, Part 2: Error estimates and adaptivity", International Journal for Numerical Methods in Engineering, 33, 1992, p. 1365-1382. [DOI:10.1002/nme.1620330703]
Numerical Methods in Civil Engineering
Volume 2, Issue 1
September 2017
Pages 72-79

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History

  • Receive Date: 24 February 2017
  • Revise Date: 17 July 2017
  • Accept Date: 19 September 2017

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