Skip to main content
SpringerLink
Log in

On the Capabilities of the Polynomial Chaos Expansion Method within SFE Analysis—An Overview

  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Abstract

This paper addresses the most recent developments concerning the application of the P-C expansion method within the Stochastic Finite Element (SFE) analysis, in particular considering computational solid mechanics. More specifically, the focus has been on the use of the method for the propagation of the stochastic structural responses due to the extensive amount of contributions in this context. Numerical examples presented in the literature are listed in this regard, in order to shed some light on the range of applications, especially in terms of the probabilistic dimensionality of the problem. Furthermore, the recently emerging utilization of the method within the modeling of uncertain input parameters is covered briefly. With this contribution, it is aimed to present an overview on the state-of-art of the P-C literature and the capabilities of the method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz M, Stegun IA (1965) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York

    Google Scholar 

  2. Acharjee S, Zabaras N (2007) A non-intrusive stochastic Galerkin approach for modeling uncertainty propagation in deformation processes. Comput Struct 85(5–6):244–254

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  4. Anders M, Hori M (2001) Three-dimensional stochastic finite element method for elasto-plastic bodies. Int J Numer Methods Eng 51(4):449–478

    Article  MATH  MathSciNet  Google Scholar 

  5. Arnst M, Ghanem R (2008) Probabilistic equivalence and stochastic model reduction in multiscale analysis. Comput Methods Appl Mech Eng 197(43–44):3584–3592

    Article  MATH  MathSciNet  Google Scholar 

  6. Arnst M, Ghanem R, Soize C (2010) Identification of Bayesian posteriors for coefficients of chaos expansions. J Comput Phys. Corrected proof (in press). doi:10.1016/j.jcp.2009.12.033

  7. Babuska I, Nobile F, Tempone R (2007) A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J Numer Anal 45(3):1005–1034

    Article  MATH  MathSciNet  Google Scholar 

  8. Bah MT, Nair PB, Bhaskar A, Keane AJ (2002) Statistical analysis of the forced response of mistuned bladed disks using stochastic reduced basis methods. In: 43rd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, Denver, Colorado, Apr 22–25, vol 1534

    Google Scholar 

  9. Bai Z (2002) Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl Numer Math 43(1–2):9–44

    Article  MATH  MathSciNet  Google Scholar 

  10. Berveiller M, Sudret B, Lemaire M (2006) Stochastic finite elements: a non intrusive approach by regression. Eur J Comput Mech 1–3(15):81–92

    Google Scholar 

  11. Blatman G, Sudret B (2008) Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. C R, Méc 336(6):518–523

    MATH  Google Scholar 

  12. Blatman G, Sudret B (2010) An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Probab Eng Mech 25(2):183–197

    Article  Google Scholar 

  13. Chen NZ, Guedes Soares C (2008) Spectral stochastic finite element analysis for laminated composite plates. Comput Methods Appl Mech Eng 197(51–52):4830–4839

    Article  MATH  Google Scholar 

  14. Cheng H, Sandu A (2009) Efficient uncertainty quantification with the polynomial chaos method for stiff systems. Math Comput Simul 79(11):3278–3295

    Article  MATH  MathSciNet  Google Scholar 

  15. Chih-jen L, Jorge JM (1999) Incomplete Cholesky factorizations with limited memory. SIAM J Sci Comput 21:24–45

    Article  MATH  MathSciNet  Google Scholar 

  16. Choi SK, Grandhi RV, Canfield RA (2004) Structural reliability under non-Gaussian stochastic behavior. Comput Struct 82(13–14):1113–1121

    Article  Google Scholar 

  17. Chung DB, Gutiérrez MA, Graham-Brady L, Lingen FJ (2005) Efficient numerical strategies for spectral stochastic finite element models. Int J Numer Methods Eng 64(10):1334–1349

    Article  MATH  Google Scholar 

  18. Crestaux T, Le Maitre O, Martinez JM (2009) Polynomial chaos expansion for sensitivity analysis. Reliab Eng Syst Saf 94(7):1161–1172

    Article  Google Scholar 

  19. Debusschere BJ, Najm HN, Matta A, Knio OM, Ghanem R, Maitre OPL (2003) Protein labeling reactions in electrochemical microchannel flow: Numerical simulation and uncertainty propagation. Phys Fluids 15(8):2238–2250

    Article  Google Scholar 

  20. Debusschere BJ, Najm HN, Pebay PP, Knio OM, Ghanem R, Maitre OPL (2004) Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J Sci Comput 26(2):698–719

    Article  MATH  MathSciNet  Google Scholar 

  21. Desceliers C, Ghanem R, Soize C (2006) Maximum likelihood estimation of stochastic chaos representations from experimental data. Int J Numer Methods Eng 66(6):978–1001

    Article  MATH  MathSciNet  Google Scholar 

  22. Desceliers C, Soize C, Ghanem R (2007) Identification of chaos representations of elastic properties of random media using experimental vibration tests. Comput Mech 39(6):831–838

    Article  MATH  Google Scholar 

  23. Doostan A, Ghanem R, Red-Horse J (2007) Stochastic model reduction for chaos representations. Comput Methods Appl Mech Eng 196(37–40):3951–3966

    Article  MATH  MathSciNet  Google Scholar 

  24. Doostan A, Iaccarino G (2009) A least-squares approximation of partial differential equations with high-dimensional random inputs. J Comput Phys 228(12):4332–4345

    Article  MATH  MathSciNet  Google Scholar 

  25. Eldred MS (2009) Recent advances in non-intrusive polynomial chaos and stochastic collocation methods for uncertainty analysis and design. In: 50th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, 4–7 May 2009, Palm Springs, California

    Google Scholar 

  26. Eldred MS, Agarwal H, Perez V, Wojtkiewicz S, Renaud J (2007) Investigation of reliability formulations in DAKOTA/UQ. Struct Infrast Eng Maintenance, Manag Life Cycle Des Perform 3:199–312

    Google Scholar 

  27. Elman HC, Ernst OG, O’Leary DP, Stewart M (2005) Efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering. Comput Methods Appl Mech Eng 194(9–11):1037–1055

    Article  MATH  MathSciNet  Google Scholar 

  28. Field RV, Grigoriu M (2004) On the accuracy of the polynomial chaos approximation. Probab Eng Mech 19(1–2):65–80

    Article  Google Scholar 

  29. Field RV (2002) Numerical methods to estimate the coefficients of the polynomial chaos expansion. In: 15th ASCE engineering mechanics conference, June 2–5, 2002

    Google Scholar 

  30. Ghanem R, Doostan A (2006) On the construction and analysis of stochastic models: Characterization and propagation of the errors associated with limited data. J Comput Phys 217(1):63–81

    Article  MATH  MathSciNet  Google Scholar 

  31. Ghanem R, Ghosh D (2007) Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition. Int J Numer Methods Eng 72(4):486–504

    Article  MATH  MathSciNet  Google Scholar 

  32. Ghanem R, Kruger RM (1996) Numerical solution of spectral stochastic finite element systems. Comput Methods Appl Mech Eng 129(3):289–303

    Article  MATH  Google Scholar 

  33. Ghanem R, Saad G, Doostan A (2007) Efficient solution of stochastic systems: Application to the embankment dam problem. Struct Saf 29(3):238–251

    Article  Google Scholar 

  34. Ghanem R, Spanos PD (2002) Stochastic finite elements: a spectral approach. Dover, New York

    Google Scholar 

  35. Ghosh D, Avery P, Farhat C (2009) A FETI-preconditioned conjugate gradient method for large-scale stochastic finite element problems. Int J Numer Methods Eng 80(6–7):914–931

    Article  MATH  MathSciNet  Google Scholar 

  36. Ghosh D, Farhat C (2008) Strain and stress computations in stochastic finite element methods. Int J Numer Methods Eng 74(8):1219–1239

    Article  MATH  MathSciNet  Google Scholar 

  37. Ghosh D, Farhat C, Avery P (2007) Uncertainty analysis of large-scale system using domain decomposition. Annual Research Briefs, Center for Turbulence Research, Stanford

  38. Ghosh D, Ghanem R, Red-Horse J (2005) Analysis of eigenvalues and modal interaction of stochastic systems. AIAA J 43(10):2196–2201

    Article  Google Scholar 

  39. Goller B, Pradlwarter HJ, Schuëller GI (2009) Robust modal updating with insufficient data. Comput Methods Appl Mech Eng 198(37–40):3096–3104

    Article  Google Scholar 

  40. Guedri M, Bouhaddi N, Majed R (2006) Reduction of the stochastic finite element models using a robust dynamic condensation method. J Sound Vib 297(1–2):123–145

    Article  Google Scholar 

  41. Gutiérrez MA, Krenk S (2004) Stochastic finite element methods. In: Stein E, de Borst R, Hughes TRJ (eds) Encyclopedia of computational mechanics. Wiley, Chichester, pp 657–681

    Google Scholar 

  42. Huang S, Mahadevan S, Rebba R (2007) Collocation-based stochastic finite element analysis for random field problems. Probab Eng Mech 22(2):194–205

    Article  Google Scholar 

  43. Jones MT, Plassmann PE (1995) An improved incomplete Cholesky factorization. ACM Trans Math Softw 21(1):5–17

    Article  MATH  MathSciNet  Google Scholar 

  44. Keese A, Matthies HG (2003) Numerical methods and Smolyak quadrature for nonlinear stochastic partial differential equations. Technical report, Institute of Scientific Computing, Technical University Braunschweig Brunswick, Germany

  45. Keese A, Matthies HG (2005) Hierarchical parallelisation for the solution of stochastic finite element equations. Comput Struct 83(14):1033–1047

    Article  MathSciNet  Google Scholar 

  46. Le Maitre OP, Knio OM, Debusschere BJ, Najm HN, Ghanem R (2003) A multigrid solver for two-dimensional stochastic diffusion equations. Comput Methods Appl Mech Eng 192(41–42):4723–4744

    MATH  Google Scholar 

  47. Lee S, Chen W (2009) A comparative study of uncertainty propagation methods for black-box-type problems. Struct Multidiscip Optim 37(3):239–253

    Article  MathSciNet  Google Scholar 

  48. Lin G, Karniadakis GE (2009) Sensitivity analysis and stochastic simulations of non-equilibrium plasma flow. Int J Numer Methods Eng 80(6–7):738–766

    Article  MATH  MathSciNet  Google Scholar 

  49. Loève M (1977) Probability theory, 4th edn. Springer, New York

    MATH  Google Scholar 

  50. Mathelin L, Hussaini M, Zang T (2005) Stochastic approaches to uncertainty quantification in cfd simulations. Numer Algorithms 38(1):209–236

    Article  MATH  MathSciNet  Google Scholar 

  51. Maute K, Weickum G, Eldred M (2009) A reduced-order stochastic finite element approach for design optimization under uncertainty. Struct Saf 31(6):450–459

    Article  Google Scholar 

  52. Nair B, Keane AJ (2002) Stochastic reduced basis methods. AIAA J 40(8):1653–1664

    Article  Google Scholar 

  53. Najm HN (2009) Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics. Annu Rev Fluid Mech 41(1):35–52

    Article  MathSciNet  Google Scholar 

  54. Ngah MF, Young A (2007) Application of the spectral stochastic finite element method for performance prediction of composite structures. Compos Struct 78(3):447–456

    Article  Google Scholar 

  55. Nobile F, Tempone R, Webster CG (2008) An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J Numer Anal 46(5):2411–2442

    Article  MATH  MathSciNet  Google Scholar 

  56. Nobile F, Tempone R, Webster CG (2008) A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J Numer Anal 46(5):2309–2345

    Article  MATH  MathSciNet  Google Scholar 

  57. Noor A (1994) Recent advances and applications of reduction methods. Appl Mech Rev 47(5):125–146

    Article  Google Scholar 

  58. Nouy A (2008) Generalized spectral decomposition method for solving stochastic finite element equations: Invariant subspace problem and dedicated algorithms. Comput Methods Appl Mech Eng 197(51–52):4718–4736

    Article  MATH  MathSciNet  Google Scholar 

  59. Pellissetti M, Ghanem R (2000) Iterative solution of systems of linear equations arising in the context of stochastic finite elements. Adv Eng Softw 31(8–9):607–616

    Article  MATH  Google Scholar 

  60. Powell CE, Elman HC (2008) Block-diagonal preconditioning for spectral stochastic finite-element systems. IMA J Numer Anal

  61. Pradlwarter HJ, Schuëller GI (2008) The use of kernel densities and confidence intervals to cope with insufficient data in validation experiments. Comput Methods Appl Mech Eng 197(29–32):2550–2560

    Article  MATH  Google Scholar 

  62. Puig B, Poirion F, Soize C (2002) Non-Gaussian simulation using Hermite polynomial expansion: convergences and algorithms. Probab Eng Mech 17(3):253–264

    Article  Google Scholar 

  63. Rosseel E, Boonen T, Vandewalle S (2008) Algebraic multigrid for stationary and time-dependent partial differential equations with stochastic coefficients. Numer Linear Algebra Appl 15(2–3):141–163

    Article  MATH  MathSciNet  Google Scholar 

  64. Saad Y, van der Vorst HA (2000) Iterative solution of linear systems in the 20th century. J Comput Appl Math 123(1–2):1–33

    Article  MATH  MathSciNet  Google Scholar 

  65. Sachdeva SK, Nair PB, Keane AJ (2006) Comparative study of projection schemes for stochastic finite element analysis. Comput Methods Appl Mech Eng 195(19–22):2371–2392

    Article  MATH  MathSciNet  Google Scholar 

  66. Sachdeva SK, Nair PB, Keane AJ (2006) Hybridization of stochastic reduced basis methods with polynomial chaos expansions. Probab Eng Mech 21(2):182–192

    Article  Google Scholar 

  67. Sakamoto S, Ghanem R (2002) Polynomial chaos decomposition for the simulation of non-Gaussian nonstationary stochastic processes. J Eng Mech 128(2):190–201

    Article  Google Scholar 

  68. Sarkar A, Ghanem R (2002) Mid-frequency structural dynamics with parameter uncertainty. Comput Methods Appl Mech Eng 191(47–48):5499–5513

    Article  MATH  Google Scholar 

  69. Smolyak S (1963) Quadrature and interpolation formulas for tensor products of certain classes of functions. Sov Math Dokl 4:240–243

    Google Scholar 

  70. Soize C (2010) Generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions. Int J Numer Methods Eng 81(8):939–970

    MATH  MathSciNet  Google Scholar 

  71. Soize C (2010) Identification of high-dimension polynomial chaos expansions with random coefficients for non-Gaussian tensor-valued random fields using partial and limited experimental data. Comput Methods Appl Mech Eng. Accepted manuscript (in press). doi:10.1016/j.cma.2010.03.013

  72. Soize C, Ghanem R (2004) Physical systems with random uncertainties: Chaos representations with arbitrary probability measure. SIAM J Sci Comput 26(2):395–410

    Article  MATH  MathSciNet  Google Scholar 

  73. Soize C, Ghanem R (2009) Reduced chaos decomposition with random coefficients of vector-valued random variables and random fields. Comput Methods Appl Mech Eng 198(21–26):1926–1934

    Article  MathSciNet  Google Scholar 

  74. Stefanou G, Nouy A, Clement A (2009) Identification of random shapes from images through polynomial chaos expansion of random level set functions. Int J Numer Methods Eng 79(2):127–155

    Article  MATH  MathSciNet  Google Scholar 

  75. Stroud A, Secrest D (1966) Gaussian quadrature formulas. Englewood Cliffs, Prentice-Hall

    MATH  Google Scholar 

  76. Sudret B (2008) Global sensitivity analysis using polynomial chaos expansions. Reliab Eng Syst Saf 93(7):964–979

    Article  Google Scholar 

  77. Sudret B, Der Kiureghian A (2002) Comparison of finite element reliability methods. Probab Eng Mech 17(4):337–348

    Article  Google Scholar 

  78. Surya Mohan P, Nair PB, Keane AJ (2008) Multi-element stochastic reduced basis methods. Comput Methods Appl Mech Eng 197(17–18):1495–1506

    Article  MATH  MathSciNet  Google Scholar 

  79. Tootkaboni M, Graham-Brady L (2009) A multi-scale spectral stochastic method for homogenization of multi-phase periodic composites with random material properties. Int J Numer Methods Eng (accepted, in press). doi:10.1002/nme.2829

  80. Tootkaboni M, Graham-Brady L, Schafer BW (2009) Geometrically non-linear behavior of structural systems with random material property: An asymptotic spectral stochastic approach. Comput Methods Appl Mech Eng 198(37–40):3173–3185

    Article  MathSciNet  Google Scholar 

  81. Tuma M, Benzi M (1999) A comparative study of sparse approximate inverse preconditioners. Appl Numer Math 30:305–340

    Article  MATH  MathSciNet  Google Scholar 

  82. van der Vorst HA (2002) Efficient and reliable iterative methods for linear systems. J Comput Appl Math 149(1):251–265

    Article  MATH  MathSciNet  Google Scholar 

  83. Verhoosel CV, Gutiérrez MA, Hulshoff SJ (2006) Iterative solution of the random eigenvalue problem with application to spectral stochastic finite element systems. Int J Numer Methods Eng 68(4):401–424

    Article  MATH  Google Scholar 

  84. Wei DL, Cui ZS, Chen J (2008) Uncertainty quantification using polynomial chaos expansion with points of monomial cubature rules. Comput Struct 86(23–24):2102–2108

    Article  Google Scholar 

  85. Wiener N (1938) The homogeneous chaos. Am J Math 60:897–936

    Article  MathSciNet  Google Scholar 

  86. Xiu D, Hesthaven JS (2005) High-order collocation methods for differential equations with random inputs. SIAM J Sci Comput 27(3):1118–1139

    Article  MATH  MathSciNet  Google Scholar 

  87. Xiu D, Karniadakis GE (2002) The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput 24(2):619–644

    Article  MATH  MathSciNet  Google Scholar 

  88. Xiu D, Karniadakis GE (2003) Modeling uncertainty in flow simulations via generalized polynomial chaos. J Comput Phys 187(1):137–167

    Article  MATH  MathSciNet  Google Scholar 

  89. Xiu D, Lucor D, Su CH, Karniadakis GE (2003) Performance evaluation of generalized polynomial chaos. In: Computational science ICCS 2003, pp 723–723

    Google Scholar 

  90. Zou Y, Cai Y, Zhou Q, Hong X, Tan SXD, Kang L (2007) Practical implementation of stochastic parameterized model order reduction via Hermite polynomial chaos. In: Asia and South Pacific design automation conference, pp 367–372

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Engineering Mechanics, University of Innsbruck, Technikerstrasse 13, 6020, Innsbruck, Austria

    H. M. Panayirci & G. I. Schuëller

Authors
  1. H. M. Panayirci
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. I. Schuëller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. M. Panayirci.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Panayirci, H.M., Schuëller, G.I. On the Capabilities of the Polynomial Chaos Expansion Method within SFE Analysis—An Overview. Arch Computat Methods Eng 18, 43–55 (2011). https://doi.org/10.1007/s11831-011-9058-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11831-011-9058-5

Keywords

Search

Navigation