Encyclopedia of Earthquake Engineering

2015 Edition
| Editors: Michael Beer, Ioannis A. Kougioumtzoglou, Edoardo Patelli, Siu-Kui Au

Stochastic Finite Elements

  • Carsten ProppeEmail author
Reference work entry
DOI: https://doi.org/10.1007/978-3-642-35344-4_337

Introduction

The finite element method (FEM) has become the dominant computational method in structural engineering. In general, the input parameters in the standard FEM assume deterministic values. In earthquake engineering, at least the excitation is often random. However, considerable uncertainties might be involved not only in the excitation of a structure but also in its material and geometric properties. A rational treatment of these uncertainties needs a mathematical concept similar to that underlying the standard FEM. Thus, FEM as a numerical method for solving boundary value problems has to be extended to stochastic boundary value problems. The extension of the FEM to stochastic boundary value problems is called stochastic finite element method (SFEM).

The first developments of the SFEM can be traced back at least to Cornell (1970), who studied soil settlement problems, and to Shinozuka (Astill et al. 1972), who combined FEM with Monte Carlo simulation for reliability...

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References

  1. Acharjee S, Zabaras N (2007) A non-intrusive stochastic Galerkin approach for modeling uncertainty propagation in deformation processes. Comput Struct 85(5–6):244–254CrossRefGoogle Scholar
  2. Astill CJ, Noseir SB, Shinozuka M (1972) Impact loading on structures with random properties. J Struct Mech 1:63–77CrossRefGoogle Scholar
  3. Babuška IM, Tempone R, Zouraris GE (2005) Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation. Comput Methods Appl Mech Eng 194(1):1251–1294MathSciNetzbMATHCrossRefGoogle Scholar
  4. Babuška I, Nobile F, Tampone R (2007) A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J Numer Anal 45:1005–1034MathSciNetzbMATHCrossRefGoogle Scholar
  5. Baecher GB, Ingra TS (1981) Stochastic FEM in settlement predictions. J Geotech Eng Div 107:449–463Google Scholar
  6. Baroth J, Bresslotette P, Chauvière C, Fogli M (2007) An efficient SFE method using Lagrange polynomials: application to nonlinear mechanical problems with uncertain parameters. Comput Methods Appl Mech Eng 196:4419–4429zbMATHCrossRefGoogle Scholar
  7. Berveiller M, Sudret B, Lemaire M (2006) Stochastic finite element: a non intrusive approach by regression. Rev Eur Mècanique Numèrique 15:81–92zbMATHCrossRefGoogle Scholar
  8. Brenner CE, Bucher CG (1995) A contribution to the SFE-based reliability assessment of nonlinear structures under dynamic loading. Probab Eng Mech 10:265–273CrossRefGoogle Scholar
  9. Contreras H (1980) The stochastic finite-element method. Comput Struct 12:341–348MathSciNetzbMATHCrossRefGoogle Scholar
  10. Cornell CA (1970) First order uncertainty analysis in soils deformation and stability. In: Proceedings of first international conference of statistics and probability in soil and structural engineering, Hong KongGoogle Scholar
  11. Deb MK, Babuška IM, Oden JT (2001) Solution of stochastic partial differential equations using Galerkin finite element techniques. Comp Methods Appl Mech Eng 190:6359–6372MathSciNetzbMATHCrossRefGoogle Scholar
  12. Deodatis G (1991) Weighted integral method I: stochastic stiffness matrix. J Eng Mech 117:1851–1864CrossRefGoogle Scholar
  13. Deodatis G, Micaletti RC (2001) Simulation of highly skewed non-Gaussian stochastic processes. Trans ASCE, J Eng Mech 127:1284–1295CrossRefGoogle Scholar
  14. Der Kiureghian A, Ke JB (1988) The stochastic finite element method in structural reliability. Probab Eng Mech 3:83–91CrossRefGoogle Scholar
  15. Field RV Jr, Grigoriu M (2004) On the accuracy of the polynomial chaos approximation. Probab Eng Mech 19(1–2):65–80CrossRefGoogle Scholar
  16. Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer, New YorkzbMATHCrossRefGoogle Scholar
  17. Grigoriu M (1998) Simulation of stationary non-Gaussian translation processes. J Eng Mech 124:121–126CrossRefGoogle Scholar
  18. Grigoriu M (2006) Evaluation of Karhunen-Loève, spectral and sampling representations for stochastic processes. J Eng Mech 132:179–189CrossRefGoogle Scholar
  19. Hisada T, Nakagiri S (1981) Stochastic finite element method developed for structural safety and reliability. In: Proceedings of third international conference on structural safety and reliability, Trondheim, pp 395–408Google Scholar
  20. Huang SP, Quek ST, Phoon KK (2001) Convergence study of the truncated Karhunen-Loève expansion for simulation of stochastic processes. Int J Numer Methods Eng 52:1029–1043zbMATHCrossRefGoogle Scholar
  21. Huang S, Mahadevan S, Rebba R (2007) Collocation-based stochastic finite element analysis for random field problems. Probab Eng Mech 22:194–205CrossRefGoogle Scholar
  22. Lawrence M (1987) Basis random variables in finite element analysis. Int J Numer Methods Eng 24:1849–1863MathSciNetzbMATHCrossRefGoogle Scholar
  23. Li CC, Der Kiureghian A (1993) Optimal discretization of random fields. J Eng Mech 119:1136–1154CrossRefGoogle Scholar
  24. Liu WK, Belytschko T, Mani A (1986) Probabilistic finite elements for nonlinear structural dynamics. Comput Methods Appl Mech Eng 56:61–81zbMATHCrossRefGoogle Scholar
  25. Loève M (1977) Probability theory. Springer, BerlinzbMATHGoogle Scholar
  26. Matthies HG, Brenner CG, Bucher CG, Guedes Soares C (1997) Uncertainties in probabilistic numerical analysis of structures and solids – stochastic finite elements. Struct Saf 19:283–336CrossRefGoogle Scholar
  27. Ostoja-Starzewski M (2011) Stochastic finite elements: where is the physics? Theor Appl Mech 38:379–396MathSciNetzbMATHCrossRefGoogle Scholar
  28. Phoon KK, Huang HW, Quek ST (2002) Simulation of second-order processes using Karhunen-Loève expansion. Comput Struct 80:1049–1160MathSciNetCrossRefGoogle Scholar
  29. Phoon KK, Huang SP, Quek ST (2005) Simulation of strongly non-Gaussian processes using Karhunen-Loève expansion. Probab Eng Mech 20:188–198CrossRefGoogle Scholar
  30. Proppe C (2008) Estimation of failure probabilities by local approximation of the limit state function. Struct Saf 30(4):277–290MathSciNetCrossRefGoogle Scholar
  31. Shinozuka M (1971) Simulation of multivariate and multidimensional random processes. J Acoust Soc Am 49:357–367CrossRefGoogle Scholar
  32. Stefanou G (2009) The stochastic finite element method: past, present and future. Comput Methods Appl Mech Eng 198:1031–1051zbMATHCrossRefGoogle Scholar
  33. Stefanou G, Papadrakakis M (2007) Assessment of spectral representation and Karhunen-Loève expansion methods for the simulation of Gaussian stochastic fields. Comput Methods Appl Mech Eng 196:2465–2477zbMATHCrossRefGoogle Scholar
  34. Sudret B, Der Kiureghian A (2000) Stochastic finite element methods and reliability – state of the art. Technical report, UCB/SEMM-2000/08, Department of Civil & Environmental Engineering, University of California, BerkeleyGoogle Scholar
  35. Todor RA, Schwab C (2006) Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. Research Report No. 2006–05, Seminar für Angwandte Mathematik, ETH ZürichGoogle Scholar
  36. Vanmarcke E, Grigoriu M (1984) Stochastic finite element analysis of simple beams. J Eng Mech 109:1203–1214CrossRefGoogle Scholar
  37. Xiu D, Karniadakis G (2002) The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput 24:619–644MathSciNetzbMATHCrossRefGoogle Scholar
  38. Yamazaki F, Shinozuka M (1988) Digital generation of non-Gaussian stochastic fields. Trans ASCE, J Eng Mech 114:1183–1197CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institut für Technische MechanikKarlsruhe Institute of TechnologyKarlsruheGermany