Skip to main content

Stochastic finite cell method for structural mechanics

Abstract

Finite cell method is known as a combination of finite element method and fictitious domain approach in order to reduce the difficulties associated with mesh generation so that it can successfully handle complex geometries. This study proposes a stochastic extension of finite cell method, as a novel computational framework, for uncertainty quantification of structures. For this purpose, stochastic finite cell method (SFCM) is developed as a new efficient method, including the features of finite cell method, for computational stochastic mechanics considering complicated geometries arising from computer-aided design (CAD). Firstly, finite cell method is formulated for solving the Fredholm integral equation of the second kind used for Karhunen-Loève expansion in order to decompose the random field within a physical domain having arbitrary boundaries. Then, the SFCM is formulated based on Karhunen-Loève and polynomial chaos expansions for the stochastic analysis. Several numerical examples consisting of benchmark problems are provided to demonstrate the efficiency, accuracy and capability of the proposed SFCM.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

References

  1. 1.

    Ghanem RG, Spanos PD (2003) Stochastic Finite elements: a spectral approach. Courier Dover Publications, Mineola

  2. 2.

    Stefanou G (2009) The stochastic finite element method: past, present and future. Comput Methods Appl Mech Eng 198:1031–1051. https://doi.org/10.1016/j.cma.2008.11.007

    Article  MATH  Google Scholar 

  3. 3.

    Kaminski M (2013) The stochastic perturbation method for computational mechanics. Wiley, Hoboken

    Book  Google Scholar 

  4. 4.

    Anders M, Hori M (2001) Three-dimensional stochastic finite element method for elasto-plastic bodies. Int J Numer Meth Eng 51:449–478. https://doi.org/10.1002/nme.165

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Mishra S, Schwab C, Šukys J (2016) Multi-level Monte Carlo finite volume methods for uncertainty quantification of acoustic wave propagation in random heterogeneous layered medium. J Comput Phys 312:192–217. https://doi.org/10.1016/j.jcp.2016.02.014

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Zakian P, Khaji N (2018) A stochastic spectral finite element method for wave propagation analyses with medium uncertainties. Appl Math Model 63:84–108. https://doi.org/10.1016/j.apm.2018.06.027

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Zakian P, Khaji N (2019) A stochastic spectral finite element method for solution of faulting-induced wave propagation in materially random continua without explicitly modeled discontinuities. Comput Mech 64:1017–1048. https://doi.org/10.1007/s00466-019-01692-5

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Shang S, Yun GJ (2013) Stochastic finite element with material uncertainties: implementation in a general purpose simulation program. Finite Elem Anal Des 64:65–78. https://doi.org/10.1016/j.finel.2012.10.001

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Khaji N, Zakian P (2017) Uncertainty analysis of elastostatic problems incorporating a new hybrid stochastic-spectral finite element method. Mech Adv Mater Struct 24:1030–1042. https://doi.org/10.1080/15376494.2016.1202359

    Article  Google Scholar 

  10. 10.

    Pokusiński B, Kamiński M (2019) Lattice domes reliability by the perturbation-based approaches vs. semi-analytical method. Comput Struct 221:179–192. https://doi.org/10.1016/j.compstruc.2019.05.012

    Article  Google Scholar 

  11. 11.

    Szafran J, Juszczyk K, Kamiński M (2020) Reliability assessment of steel lattice tower subjected to random wind load by the stochastic finite-element method. ASCE-ASME J Risk Uncertain Eng Syst Part A: Civil Eng 6:04020003. https://doi.org/10.1061/AJRUA6.0001040

    Article  Google Scholar 

  12. 12.

    Ghanem R, Dham S (1998) Stochastic finite element analysis for multiphase flow in heterogeneous porous media. Transp Porous Media 32:239–262. https://doi.org/10.1023/A:1006514109327

    MathSciNet  Article  Google Scholar 

  13. 13.

    Maitre OL, Knio OM (2010) Spectral methods for uncertainty quantification: with applications to computational fluid dynamics. Springer, Netherlands

    Book  Google Scholar 

  14. 14.

    Laz PJ, Browne M (2010) A review of probabilistic analysis in orthopaedic biomechanics. Proc Inst Mech Eng [H] 224:927–943. https://doi.org/10.1243/09544119jeim739

    Article  Google Scholar 

  15. 15.

    Arregui-Mena JD, Margetts L, Mummery PM (2016) Practical application of the stochastic finite element method. Archives Comput Methods Eng 23:171–190. https://doi.org/10.1007/s11831-014-9139-3

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Zakian P, Khaji N (2016) A novel stochastic-spectral finite element method for analysis of elastodynamic problems in the time domain. Meccanica 51:893–920. https://doi.org/10.1007/s11012-015-0242-9

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Komatitsch D, Tromp J (1999) Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys J Int 139:806–822. https://doi.org/10.1046/j.1365-246x.1999.00967.x

    Article  Google Scholar 

  18. 18.

    Komatitsch D, Vilotte J-P, Vai R, Castillo-Covarrubias JM, Sánchez-Sesma FJ (1999) The spectral element method for elastic wave equations—application to 2-D and 3-D seismic problems. Int J Numer Meth Eng 45:1139–1164. https://doi.org/10.1002/(sici)1097-0207(19990730)45:9%3c1139::aid-nme617%3e3.0.co;2-t

    Article  MATH  Google Scholar 

  19. 19.

    Zakian P, Khaji N, Kaveh A (2017) Graph theoretical methods for efficient stochastic finite element analysis of structures. Comput Struct 178:29–46. https://doi.org/10.1016/j.compstruc.2016.10.009

    Article  Google Scholar 

  20. 20.

    Bathe KJ (1996) Finite element procedures, 1st edn. Prentice Hall; 2nd ed KJ Bathe, Watertown, MA, 2014

  21. 21.

    Kaveh A (2013) Computational structural analysis and finite element methods. Springer, Switzerland

    MATH  Google Scholar 

  22. 22.

    Kaveh A (2006) Optimal structural analysis, 2nd edn. Wiley, Chichester

    MATH  Google Scholar 

  23. 23.

    Kaveh A (2013) Optimal analysis of structures by concepts of symmetry and regularity. Springer, Vienna

    Book  Google Scholar 

  24. 24.

    Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139:3–47. https://doi.org/10.1016/S0045-7825(96)01078-X

    Article  MATH  Google Scholar 

  25. 25.

    Liu GR (2009) Mesh free methods: moving beyond the finite element method. 2nd Edition, CRC Press, Boca Raton

  26. 26.

    Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195. https://doi.org/10.1016/j.cma.2004.10.008

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Parvizian J, Düster A, Rank E (2007) Finite cell method. Comput Mech 41:121–133. https://doi.org/10.1007/s00466-007-0173-y

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Zhang L, Bathe KJ (2017) Overlapping finite elements for a new paradigm of solution. Comput Struct 187:64–76. https://doi.org/10.1016/j.compstruc.2017.03.008

    Article  Google Scholar 

  29. 29.

    Ruess M, Tal D, Trabelsi N, Yosibash Z, Rank E (2012) The finite cell method for bone simulations: verification and validation. Biomech Model Mechanobiol 11:425–437. https://doi.org/10.1007/s10237-011-0322-2

    Article  Google Scholar 

  30. 30.

    Schillinger D, Düster A, Rank E (2012) The hp-d-adaptive finite cell method for geometrically nonlinear problems of solid mechanics. Int J Numer Meth Eng 89:1171–1202. https://doi.org/10.1002/nme.3289

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Duczek S, Joulaian M, Düster A, Gabbert U (2014) Numerical analysis of Lamb waves using the finite and spectral cell methods. Int J Numer Meth Eng 99:26–53. https://doi.org/10.1002/nme.4663

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Joulaian M, Duczek S, Gabbert U, Düster A (2014) Finite and spectral cell method for wave propagation in heterogeneous materials. Comput Mech 54:661–675. https://doi.org/10.1007/s00466-014-1019-z

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Li K, Gao W, Wu D, Song C, Chen T (2018) Spectral stochastic isogeometric analysis of linear elasticity. Comput Methods Appl Mech Eng 332:157–190. https://doi.org/10.1016/j.cma.2017.12.012

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Li K, Wu D, Gao W, Song C (2019) Spectral stochastic isogeometric analysis of free vibration. Comput Methods Appl Mech Eng 350:1–27. https://doi.org/10.1016/j.cma.2019.03.008

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Mohammadi S (2012) XFEM fracture analysis of composites. Wiley, Hoboken

    Book  Google Scholar 

  36. 36.

    Spanos PD, Ghanem R (1989) Stochastic finite element expansion for random media. J Eng Mech 115:1035–1053. https://doi.org/10.1061/(ASCE)0733-9399(1989)115:5(1035)

    Article  Google Scholar 

  37. 37.

    Oliveira SP, Azevedo JS (2014) Spectral element approximation of Fredholm integral eigenvalue problems. J Comput Appl Math 257:46–56. https://doi.org/10.1016/j.cam.2013.08.016

    MathSciNet  Article  MATH  Google Scholar 

  38. 38.

    Xiu D (2010) Numerical methods for stochastic computations: a spectral method approach. Princeton University Press, New Jersey

  39. 39.

    Huang S, Mahadevan S, Rebba R (2007) Collocation-based stochastic finite element analysis for random field problems. Probab Eng Mech 22:194–205. https://doi.org/10.1016/j.probengmech.2006.11.004

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Pooya Zakian.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zakian, P. Stochastic finite cell method for structural mechanics. Comput Mech 68, 185–210 (2021). https://doi.org/10.1007/s00466-021-02026-0

Download citation

Keywords

  • Stochastic finite cell method
  • Finite cell method
  • Karhunen-Loève expansion
  • Polynomial chaos expansion
  • Fredholm integral equation
  • Computational stochastic mechanics