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A new adaptive mesh refinement strategy based on a probabilistic error estimation

  • Ziaei, H. (Department of Civil Engineering, Shahed University) ;
  • Moslemi, H. (Department of Civil Engineering, Shahed University)
  • Received : 2019.04.02
  • Accepted : 2020.01.03
  • Published : 2020.05.25

Abstract

In this paper, an automatic adaptive mesh refinement procedure is presented for two-dimensional problems on the basis of a new probabilistic error estimator. First-order perturbation theory is employed to determine the lower and upper bounds of the structural displacements and stresses considering uncertainties in geometric sizes, material properties and loading conditions. A new probabilistic error estimator is proposed to reduce the mesh dependency of the responses dispersion. The suggested error estimator neglects the refinement at the critical points with stress concentration. Therefore, the proposed strategy is combined with the classic adaptive mesh refinement to achieve an optimal mesh refined properly in regions with either high gradients or high dispersion of the responses. Several numerical examples are illustrated to demonstrate the efficiency, accuracy and robustness of the proposed computational algorithm and the results are compared with the classic adaptive mesh refinement strategy described in the literature.

Keywords

References

  1. Babuska, I. and Rheinboldt, C. (1987), "A-posteriori error estimates for the finite element method", Int. J. Numer. Methods Engrg., 12, 1597-1615. https://doi.org/10.1002/nme.1620121010.
  2. Bespalov, A. and Rocchi, L. (2018), "Efficient adaptive algorithms for elliptic PDEs with random data", SIAM/ASA J. Uncertain. Quantif., 6, 243-272. https://doi.org/10.1137/17M1139928.
  3. Boroomand. B. and Zienkiewicz, O.C. (1997), "Recovery by equilibrium: In patches (REP)", J. Num. Meth. Eng., 40, 137-164. https://doi.org/10.1002/(SICI)1097-0207(19970115)40:1%3C137::AID-NME57%3E3.0.CO;2-5.
  4. Deb, M.K., Babuska, I.M. and Oden, J.T. (2001), "Solution of stochastic partial differential equations using Galerkin finite element techniques", Comput. Methods Appl. Mech. Eng,. 190, 6359-6372. https://doi.org/10.1016/S0045-7825(01)00237-7.
  5. Eigel, M., Gittelson, C.J., Schwab, C. and Zander, E. (2014), "Adaptive stochastic Galerkin FEM", Comput. Methods Appl. Mech. Eng., 270, 247-269. https://doi.org/10.1016/j.cma.2013.11.015.
  6. Erdogan, F. and Sih G.C. (1963), "On the extension of plates under plane loading and transverse shear", J. Basic Engng., 4, 519-527. https://doi.org/10.1115/1.3656897.
  7. Gonzalez-Estrada O.A., Nadal E., Rodenas J.J., Kerfriden P., Bordas S.P.A. and Fuenmayor F.J. (2013), "Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery", Comput. Mech., 53, 957-976. https://doi.org/10.1007/s00466-013-0942-8.
  8. Gratsch, T. and Bathe, K.J. (2005), "A posteriori error estimation techniques in practical finite element analysis", Comput. Struct., 83, 235-265. https://doi.org/10.1016/j.compstruc.2004.08.011.
  9. Guignard, D., Nobile, F. and Picasso, M. (2016), "A posteriori error estimation for elliptic partial differential equations with small uncertainties", Numer. Methods Partial Differential Equations, 32, 175-212. https://doi.org/10.1002/num.21991.
  10. Kumar, M., Kvamsdal, T. and Johannessen, K.A. (2017), "Superconvergent patch recovery and a posteriori error estimation technique in adaptive isogeometric analysis", Comput. Methods Appl. Mech. Eng., 316, 1086-1156. https://doi.org/10.1016/j.cma.2016.11.014.
  11. Lee, D.K., Park, S.S. and Shin, S.M. (2008), "Non-stochastic interval arithmetic-based finite element analysis for structural uncertainty response estimate", Struct. Eng. Mech., 29, 469-488. https://doi.org/10.12989/sem.2008.29.5.469.
  12. Lee, D.K and Shin, S.M. (2008), "Non-stochastic interval factor method-based FEA for structural stress responses with uncertainty", Struct. Eng. Mech., 62, 703-708. https://doi.org/10.12989/sem.2017.62.6.703.
  13. Li, C.C. and Der Kiureghian, A. (1993), "Optimal discretization of random fields", J. Engrg. Mech., 119(6) 1136-1154. https://doi.org/10.1061/(ASCE)0733-9399(1993)119:6(1136).
  14. Mathelin, L. and Le Maitre, O. (2007), "Dual-based a posteriori error estimate for stochastic finite element methods", Commun. Appl. Math. Comput. Sci., 2, 83-115. http://dx.doi.org/10.2140/camcos.2007.2.83.
  15. Moslemi, H. and Khoei, A.R. (2009), "3D adaptive finite element modeling of non-planar curved crack growth using the weighted superconvergent patch recovery method", Eng. Fracture Mech., 76, 1703-1728. https://doi.org/10.1016/j.engfracmech.2009.03.013.
  16. Moslemi, H. and Khoei, A.R. (2010), "3D Modeling of damage growth and crack initiation using adaptive finite element technique", Scientia Iranica, 17, 372-386.
  17. Moslemi, H. and Tavakkoli A. (2018), "A Statistical Approach for Error Estimation In Adaptive Finite Element Method", J. Comput. Methods Eng. Sci. Mech., 19, 440-450. https://doi.org/10.1080/15502287.2018.1558424.
  18. Ozakca, M. (2003), "Comparison of error estimation methods and adaptivity for plane stress/strain problems", Struct. Eng. Mech., 15, 579-608. https://doi.org/10.12989/sem.2003.15.5.579
  19. Richardson, L. (1910), "The approximate arithmetical solution by finite differences of physical problems", Transactions Royal Soc. London, 210, 307-357. https://doi.org/10.1098/rsta.1911.0009.
  20. Rodenas, J.J., Gonzalez-Estrada, O.A., Tarancon, J.E. and Fuenmayor, F.J. (2008), "A recovery-type error estimator for the extended finite element method based on singular+smooth stress field splitting", J. Numerical Methods Eng., 76, 545-571. https://doi.org/10.1002/nme.2313.
  21. Zienkiewicz, O.C. (2006), "The background of error estimation and adaptivity in finite element computation", Comput. Methods Appl. Mech. Eng., 195, 207-213. https://doi.org/10.1016/j.cma.2004.07.053.
  22. Zienkiewicz, O.C. and Zhu, J.Z. (1987), "A simple error estimator and adaptive procedure for practical engineering analysis", J. Num. Meth. Eng., 24, 337-357. https://doi.org/10.1002/nme.1620240206.
  23. Zienkiewicz, O.C. and Zhu, J.Z. (1992), "The super convergent patch recovery (SPR) and adaptive finite element refinement", Comp. Meth. Appl. Mech. Eng., 101, 207-224. https://doi.org/10.1016/0045-7825(92)90023-D.