Comptes Rendus
Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach
Comptes Rendus. Mécanique, Volume 336 (2008) no. 6, pp. 518-523.

A method is proposed to build a sparse polynomial chaos (PC) expansion of a mechanical model whose input parameters are random. In this respect, an adaptive algorithm is described for automatically detecting the significant coefficients of the PC expansion. The latter can thus be computed by means of a relatively small number of possibly costly model evaluations, using a non-intrusive regression scheme (also known as stochastic collocation). The method is illustrated by a simple polynomial model, as well as the example of the deflection of a truss structure.

Dans cette communication, on propose un algorithme permettant de construire une représentation par chaos polynomial creux de la réponse d'un modèle mécanique dont les paramètres d'entrée sont aléatoires. L'algorithme construit de façon adaptative une représentation creuse en détectant automatiquement les termes importants et en supprimant ceux qui sont négligeables. A chaque étape, le calcul des coefficients s'effectue par minimisation au sens des moindres carrés (méthode non-intrusive dite de régression). L'algorithme est déroulé pas à pas sur un modèle polynomial, puis appliqué à l'étude de la fiabilité d'un treillis élastique.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2008.02.013
Keywords: Solids and structures, Adaptive stochastic finite elements, Sparse polynomial chaos, Stochastic collocation, Regression, Structural reliability
Mot clés : Solides et structures, Eléments finis stochastiques adaptatifs, Chaos polynomial creux, Collocation stochastique, Régression, Fiabilité

Géraud Blatman 1, 2; Bruno Sudret 2

1 IFMA-LaMI, Campus des Cézeaux, BP 265, 63175 Aubière cedex, France
2 EDF R&D, Département matériaux et mécanique des composants, site des Renardières, 77250 Moret-sur-Loing cedex, France
@article{CRMECA_2008__336_6_518_0,
     author = {G\'eraud Blatman and Bruno Sudret},
     title = {Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {518--523},
     publisher = {Elsevier},
     volume = {336},
     number = {6},
     year = {2008},
     doi = {10.1016/j.crme.2008.02.013},
     language = {en},
}
TY  - JOUR
AU  - Géraud Blatman
AU  - Bruno Sudret
TI  - Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach
JO  - Comptes Rendus. Mécanique
PY  - 2008
SP  - 518
EP  - 523
VL  - 336
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crme.2008.02.013
LA  - en
ID  - CRMECA_2008__336_6_518_0
ER  - 
%0 Journal Article
%A Géraud Blatman
%A Bruno Sudret
%T Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach
%J Comptes Rendus. Mécanique
%D 2008
%P 518-523
%V 336
%N 6
%I Elsevier
%R 10.1016/j.crme.2008.02.013
%G en
%F CRMECA_2008__336_6_518_0
Géraud Blatman; Bruno Sudret. Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. Comptes Rendus. Mécanique, Volume 336 (2008) no. 6, pp. 518-523. doi : 10.1016/j.crme.2008.02.013. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2008.02.013/

[1] R. Ghanem; P. Spanos Stochastic Finite Elements: A Spectral Approach, Courier Dover Publications, 2003

[2] C. Soize; R. Ghanem Physical systems with random uncertainties: chaos representations with arbitrary probability measure, SIAM J. Sci. Comput., Volume 26 (2004) no. 2, pp. 395-410

[3] D. Xiu; G.E. Karniadakis A new stochastic approach to transient heat conduction modeling with uncertainty, Int. J. Heat Mass Transfer, Volume 46 (2003), pp. 4681-4693

[4] O.P. Le Maître; O.M. Knio; H.N. Najm; R.G. Ghanem A stochastic projection method for fluid flow, J. Comput. Phys., Volume 173 (2001), pp. 481-511

[5] O.M. Knio; O.P. Le Maître Uncertainty propagation in CFD using polynomial chaos decomposition, Fluid Dyn. Res., Volume 38 (2006) no. 9, pp. 616-640

[6] B. Sudret; A. Der Kiureghian Comparison of finite element reliability methods, Prob. Eng. Mech., Volume 17 (2002), pp. 337-348

[7] G. Blatman; B. Sudret; M. Berveiller Quasi-random numbers in stochastic finite element analysis, Mécanique & Industries, Volume 8 (2007), pp. 289-297

[8] D.M. Ghiocel; R.G. Ghanem Stochastic finite element analysis of seismic soil–structure interaction, J. Eng. Mech. (ASCE), Volume 128 (2002), pp. 66-77

[9] M. Berveiller; B. Sudret; M. Lemaire Stochastic finite element: a non-intrusive approach by regression, Rev. Européenne Mécanique Numérique, Volume 15 (2006) no. 1–3, pp. 81-92

[10] A. Nouy A generalized spectral decomposition technique to solve stochastic partial differential equations, Comput. Methods Appl. Mech. Engrg., Volume 196 (2007) no. 45–48, pp. 4521-4537

[11] S.K. Choi; R.V. Grandhi; R.A. Canfield Structural reliability under non-Gaussian stochastic behavior, Computers & Structures, Volume 82 (2004), pp. 1113-1121

[12] B. Sudret, Uncertainty propagation and sensitivity analysis in mechanical models – Contributions to structural reliability and stochastic spectral methods, Habilitation à diriger des recherches, Université Blaise-Pascal, Clermont-Ferrand, France, 2007

[13] B. Sudret Global sensitivity analysis using polynomial chaos expansions, Reliab. Eng. Sys. Safety, Volume 93 (2008), pp. 964-979

[14] M.D. McKay; R.J. Beckman; W.J. Conover A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, Volume 2 (1979), pp. 239-245

Cited by Sources:

Comments - Policy