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A novel stochastic-spectral finite element method for analysis of elastodynamic problems in the time domain

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Abstract

This article proposes a stochastically-tuned spectral finite element method (SFEM) which is applied to elastodynamic problems. Stochastic finite element method is an efficient numerical method incorporating randomness for uncertainty quantification of engineering systems. On the other hand, SFEM is an excellent remedy for solving dynamic problems with fine accuracy, which employs Lobatto polynomials leading to reduction of domain discretization and making diagonal mass matrices. The presented method simultaneously collects the advantages of the both methods in order to solve stochastically linear elastodynamic problems with suitable computational efficiency and accuracy. Furthermore, spectral finite element is also proposed for numerical solution of Fredholm integral equation associated with Karhunen–Loève expansion followed by the presented hybrid method which enhances the efficiency of the methodology. Various types of numerical examples are prepared so as to demonstrate advantages of the proposed stochastic SFEM.

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References

  1. Kaminski M (2013) The stochastic perturbation method for computational mechanics. Wiley, New York

    Book  MATH  Google Scholar 

  2. Chang TP, Chang HC (1994) Stochastic dynamic finite element analysis of a nonuniform beam. Int J Solids Struct 31(5):587–597

    Article  MATH  Google Scholar 

  3. Schuëller GI (1997) A state-of-the-art report on computational stochastic mechanics. Probab Eng Mech 12(4):197–321

    Article  Google Scholar 

  4. Takada T (1990) Weighted integral method in stochastic finite element analysis. Probab Eng Mech 5(3):146–156

    Article  Google Scholar 

  5. Ghanem RG, Spanos PD (2003) Stochastic finite elements: a spectral approach. Courier Dover Publications, New York

    MATH  Google Scholar 

  6. Deb MK, Babuška IM, Oden JT (2001) Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput Methods Appl Mech Eng 190(48):6359–6372

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Kamiński M (2008) On stochastic finite element method for linear elastostatics by the Taylor expansion. Struct Multidiscip Optim 35(3):213–223

    Article  Google Scholar 

  8. Anders M, Hori M (1999) Stochastic finite element method for elasto-plastic body. Int J Numer Meth Eng 46(11):1897–1916

    Article  MATH  Google Scholar 

  9. Xu XF (2012) Quasi-weak and weak formulation of stochastic finite elements on static and dynamic problems—a unifying framework. Probab Eng Mech 28:103–109

    Article  Google Scholar 

  10. Muscolino G, Sofi A (2013) Bounds for the stationary stochastic response of truss structures with uncertain-but-bounded parameters. Mech Syst Signal Process 37(1–2):163–181

    Article  ADS  Google Scholar 

  11. Muscolino G, Sofi A (2012) Stochastic analysis of structures with uncertain-but-bounded parameters via improved interval analysis. Probab Eng Mech 28:152–163

    Article  Google Scholar 

  12. Gioffrè M, Gusella V (2002) Numerical analysis of structural systems subjected to non-gaussian random fields. Meccanica 37(1–2):115–128

    Article  MATH  Google Scholar 

  13. Impollonia N, Muscolino G (2002) Static and dynamic analysis of non-linear uncertain structures. Meccanica 37(1–2):179–192

    Article  MATH  Google Scholar 

  14. Arnst M, Ghanem R, Phipps E, Red-Horse J (2012) Dimension reduction in stochastic modeling of coupled problems. Int J Numer Meth Eng 92(11):940–968

    Article  MathSciNet  Google Scholar 

  15. Chevreuil M, Nouy A (2012) Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics. Int J Numer Meth Eng 89(2):241–268

    Article  MathSciNet  MATH  Google Scholar 

  16. Adhikari S (2011) A reduced spectral function approach for the stochastic finite element analysis. Comput Methods Appl Mech Eng 200(21–22):1804–1821

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Chowdhury R, Adhikari S (2010) High dimensional model representation for stochastic finite element analysis. Appl Math Model 34(12):3917–3932

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  19. Stefanou G (2009) The stochastic finite element method: past, present and future. Comput Methods Appl Mech Eng 198(9–12):1031–1051

    Article  ADS  MATH  Google Scholar 

  20. Patera AT (1984) A spectral element method for fluid dynamics: laminar flow in a channel expansion. J Comput Phys 54(3):468–488

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Dauksher W, Emery AF (2000) The solution of elastostatic and elastodynamic problems with Chebyshev spectral finite elements. Comput Methods Appl Mech Eng 188(1–3):217–233

    Article  ADS  MATH  Google Scholar 

  22. Hennings B, Lammering R, Gabbert U (2013) Numerical simulation of wave propagation using spectral finite elements. CEAS Aeronaut J 4(1):3–10

    Article  Google Scholar 

  23. Khaji N, Habibi M, Mirhashemian P (2009) Modeling transient elastodynamic problems using spectral element method. Asian J Civ Eng 10(4):361–380

    Google Scholar 

  24. Witkowski W, Rucka M, Chróścielewski J, Wilde K (2012) On some properties of 2D spectral finite elements in problems of wave propagation. Finite Elem Anal Des 55:31–41

    Article  Google Scholar 

  25. Komatitsch D, Tromp J (1999) Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys J Int 139(3):806–822

    Article  ADS  Google Scholar 

  26. Shafiei M, Khaji N (2014) Simulation of two-dimensional elastodynamic problems using a new adaptive physics-based method. Meccanica 49(6):1353–1366

    Article  MATH  Google Scholar 

  27. Ghanem RG, Spanos PD (1997) Spectral techniques for stochastic finite elements. Arch Comput Methods Eng 4(1):63–100

    Article  MathSciNet  MATH  Google Scholar 

  28. Kudela P, Krawczuk M, Ostachowicz W (2007) Wave propagation modelling in 1D structures using spectral finite elements. J Sound Vib 300(1–2):88–100

    Article  ADS  MATH  Google Scholar 

  29. Adler RJ, Taylor JE (2007) Random fields and geometry, vol 115. Springer, Berlin

    MATH  Google Scholar 

  30. Bobrowski A (2005) Functional analysis for probability and stochastic processes: an introduction. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  31. Abramowitz M, Stegun IA (1964) Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Dover Publications, New York

    MATH  Google Scholar 

  32. Ziari S, Ezzati R, Abbasbandy S (2012) Numerical solution of linear fuzzy Fredholm integral equations of the second kind using fuzzy Haar wavelet. In: Greco S, Bouchon-Meunier B, Coletti G, Fedrizzi M, Matarazzo B, Yager R (eds) Advances in computational intelligence, vol 299., Communications in computer and information scienceSpringer, Berlin, pp 79–89

    Chapter  Google Scholar 

  33. Ikebe Y (1972) The Galerkin method for the numerical solution of Fredholm integral equations of the second kind. SIAM Review 14(3):465–491

    Article  MathSciNet  MATH  Google Scholar 

  34. Bathe KJ (1996) Finite element procedures. Prentice Hall

  35. Saleh MM, El-Kalla IL, Ehab MM (2007) Stochastic finite element technique for stochastic one-dimension time-dependent differential equations with random coefficients. Differ Equ Nonlinear Mech 2007:1–16. doi:10.1155/2007/48527

  36. Kamiński M (2010) Generalized stochastic perturbation technique in engineering computations. Math Comput Model 51(3–4):272–285

    Article  MathSciNet  MATH  Google Scholar 

  37. Ghanem R (1999) Higher-order sensitivity of heat conduction problems to random data using the spectral stochastic finite element method. J Heat Transf 121(2):290–299

    Article  Google Scholar 

  38. Bruch JC, Zyvoloski G (1974) Transient two-dimensional heat conduction problems solved by the finite element method. Int J Numer Meth Eng 8(3):481–494

    Article  MATH  Google Scholar 

  39. Khodakarami MI, Khaji N (2011) Analysis of elastostatic problems using a semi-analytical method with diagonal coefficient matrices. Eng Anal Bound Elem 35(12):1288–1296

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Civil and Environmental Engineering, Tarbiat Modares University, P.O. Box 14115–397, Tehran, Iran

    P. Zakian & N. Khaji

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  1. P. Zakian
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  2. N. Khaji
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Correspondence to N. Khaji.

Appendices

Appendix 1: Interpolation functions of a spectral element

This appendix presents interpolation functions of the matrix H in Eq. (18) for a spectral element consisting of 16 nodes as follows

$$ \begin{aligned} N_{1} & = &\,\beta \left( {5r^{2} - 1} \right)\left( {5s^{2} - 1} \right)(r - 1)(s - 1),\quad N_{2} = - \sqrt 5 \beta \left( {5r - \sqrt 5 } \right)\left( {5s^{2} - 1} \right)\left( {r^{2} - 1} \right)(s - 1), \\ N_{3} & = \sqrt 5 \beta \left( {5r + \sqrt 5 } \right)\left( {5s^{2} - 1} \right)\left( {r^{2} - 1} \right)(s - 1),\quad N_{4} = - \beta \left( {5r^{2} - 1} \right)\left( {5s^{2} - 1} \right)(r + 1)(s - 1), \\ N_{5} & = - \sqrt 5 \beta \left( {5r^{2} - 1} \right)\left( {5s - \sqrt 5 } \right)(r - 1)\left( {s^{2} - 1} \right),\quad N_{6} = 5\beta \left( {5r - \sqrt 5 } \right)\left( {5s - \sqrt 5 } \right)\left( {r^{2} - 1} \right)\left( {s^{2} - 1} \right), \\ N_{7} & = - 5\beta \left( {5r + \sqrt 5 } \right)\left( {5s - \sqrt 5 } \right)\left( {r^{2} - 1} \right)\left( {s^{2} - 1} \right),\quad N_{8} = \sqrt 5 \beta \left( {5r^{2} - 1} \right)\left( {5s - \sqrt 5 } \right)(r + 1)\left( {s^{2} - 1} \right), \\ N_{9} & = \sqrt 5 \beta \left( {5r^{2} - 1} \right)\left( {5s + \sqrt 5 } \right)(r - 1)\left( {s^{2} - 1} \right),\quad N_{10} = - 5\beta \left( {5r - \sqrt 5 } \right)\left( {5s + \sqrt 5 } \right)\left( {r^{2} - 1} \right)\left( {s^{2} - 1} \right), \\ N_{11} & = 5\beta \left( {5r + \sqrt 5 } \right)\left( {5s + \sqrt 5 } \right)\left( {r^{2} - 1} \right)\left( {s^{2} - 1} \right),\quad N_{12} = - \sqrt 5 \beta \left( {5r^{2} - 1} \right)\left( {5s + \sqrt 5 } \right)(r + 1)\left( {s^{2} - 1} \right), \\ N_{13} & = - \beta \left( {5r^{2} - 1} \right)\left( {5s^{2} - 1} \right)(r - 1)(s + 1),\quad N_{14} = \sqrt 5 \beta \left( {5r - \sqrt 5 } \right)\left( {5s^{2} - \sqrt 5 } \right)\left( {r^{2} - 1} \right)(s + 1), \\ N_{15} & = - \sqrt 5 \beta \left( {5r + \sqrt 5 } \right)\left( {5s^{2} - 1} \right)\left( {r^{2} - 1} \right)(s + 1),\quad N_{16} = \beta \left( {5r^{2} - 1} \right)\left( {5s^{2} - 1} \right)(r + 1)(s + 1). \\ \end{aligned} $$
(58)

where \( \beta = \tfrac{1}{64} \).

Appendix 2: Flowchart of the StSFEM

A flowchart is designed here to explain fundamental steps of the proposed StSFEM (see Fig. 19), so that one may conveniently follow the procedure of the method.

Fig. 19
figure 19

The flowchart of calculations for the proposed StSFEM

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

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