To reduce the computational effort using polynomial chaos expansion to predict the dynamic characteristics of structures with several uncertain parameters, hybrid techniques combining stochastic finite element analysis with either deterministic or stochastic model order reduction (MOR) are developed. For the deterministic MOR, the Arnoldi-based Krylov subspace technique is implemented to reduce the system matrices of the finite element model. For the stochastic MOR, a stochastic reduced basis method is implemented in which the structural modal and frequency responses are approximated by a small number of basis vectors using stochastic Krylov subspace. To demonstrate the computational efficiency of each reduced stochastic finite element model, variability in the natural frequencies and frequency responses of a simply supported flexible plate randomized by uncertain geometrical and material parameters is examined. Results are compared with both Monte Carlo (MC) simulations and nonreduced stochastic models. Using the reduced models, the effects of the individual uncertain parameters as well as the combined uncertainties on the dynamic characteristics of the plate are examined.
Issue Section:
Research Papers
References
1.
Fishman
, G. S.
, 1995
, MC: Concepts, Algorithms, and Applications
, Springer
, New York
.2.
Zhou
, K.
, and Tang
, J.
, 2015
, “Reducing Dynamic Response Variation Using NURBS Finite Element-Based Geometry Perturbation
,” ASME J. Vib. Acoust.
, 137
(6
), p. 061008
.3.
Chen
, S. H.
, Qiu
, Z. P.
, and Liu
, Z. S.
, 1994
, “A Method for Computing Eigenvalue Bounds in Structural Vibration Systems With Interval Parameters
,” Comput. Struct.
, 51
(3
), pp. 309
–313
.4.
Gao
, W.
, and Kessissoglou
, N. J.
, 2007
, “Dynamic Response Analysis of Stochastic Truss Structures Under Non-Stationary Random Excitation Using the Random Factor Method
,” Comput. Methods Appl. Mech. Eng.
, 196
, pp. 2765
–2773
.5.
Weaver
, R. L.
, 1996
, “The Effect of an Undamped Finite Degree of Freedom ‘Fuzzy’ Substructure: Numerical Solutions and Theoretical Discussion
,” J. Acoust. Soc. Am.
, 100
(5
), pp. 3159
–3164
.6.
Soize
, C.
, 1993
, “A Model and Numerical Method in the Medium Frequency Range for Vibroacoustic Predictions Using the Theory of Structural Fuzzy
,” J. Acoust. Soc. Am.
, 94
(2
), pp. 849
–865
.7.
Wiener
, N.
, 1938
, “The Homogeneous Chaos
,” Am. J. Math.
, 60
(4
), pp. 897
–936
.8.
Soize
, C.
, 2000
, “A Nonparametric Model of Random Uncertainties for Reduced Matrix Models in Structural Dynamics
,” Probab. Eng. Mech.
, 15
(3
), pp. 277
–294
.9.
Mehta
, M. L.
, 1991
, Random Matrices
, Academic Press
, New York
.10.
Kessissoglou
, N. J.
, and Lucas
, G. I.
, 2009
, “Gaussian Orthogonal Ensemble Spacing Statistics and the Statistical Overlap Factor Applied to Dynamic Systems
,” J. Sound Vib.
, 324
, pp. 1039
–1066
.11.
Cicirello
, A.
, and Langley
, R. S.
, 2013
, “The Vibro-Acoustic Analysis of Built-Up Systems Using a Hybrid Method With Parametric and Non-Parametric Uncertainties
,” J. Sound Vib.
, 332
(9
), pp. 2165
–2178
.12.
Langley
, R. S.
, and Cotoni
, V.
, 2007
, “Response Variance Prediction for Uncertain Vibro-Acoustic Systems Using a Hybrid Deterministic-Statistical Method
,” J. Acoust. Soc. Am.
, 122
(6
), pp. 3445
–3463
.13.
Ghanem
, R. G.
, and Spanos
, P. D.
, 1991
, Stochastic Finite Elements: A Spectral Approach
, Springer-Verlag
, New York
.14.
Sarsri
, D.
, Azrar
, L.
, Jebbouri
, A.
, and El Hami
, A.
, 2011
, “Component Mode Synthesis and Polynomial Chaos Expansions for Stochastic Frequency Functions of Large Linear FE Models
,” Comput. Struct.
, 89
(3–4
), pp. 346
–356
.15.
Rupert
, C. P.
, and Miller
, C. T.
, 2007
, “An Analysis of Polynomial Chaos Approximations for Modeling Single-Fluid-Phase Flow in Porous Medium Systems
,” J. Comput. Phys.
, 226
(2
), pp. 2175
–2205
.16.
Witteveen
, J. A. S.
, Sarkar
, S.
, and Bijl
, H.
, 2007
, “Modeling Physical Uncertainties in Dynamic Stall Induced Fluid–Structure Interaction of Turbine Blades Using Arbitrary Polynomial Chaos
,” Comput. Struct.
, 85
, pp. 866
–878
.17.
Sepahvand
, K.
, Marburg
, S.
, and Hardtke
, H. J.
, 2012
, “Stochastic Free Vibration of Orthotropic Plates Using Generalized Polynomial Chaos Expansion
,” J. Sound Vib.
, 331
(1
), pp. 167
–179
.18.
Didier
, J.
, Sinou
, J. J.
, and Faverjon
, B.
, 2013
, “Nonlinear Vibrations of a Mechanical System With Non-Regular Nonlinearities and Uncertainties
,” Commun. Nonlinear Sci. Numer. Simul.
, 18
(11
), pp. 3250
–3270
.19.
Faverjon
, B.
, Ladevèze
, P.
, and Louf
, F.
, 2009
, “Validation of Stochastic Linear Structural Dynamics Models
,” Comput. Struct.
, 87
(13–14
), pp. 829
–837
.20.
Sinou
, J. J.
, and Faverjon
, B.
, 2012
, “The Vibration Signature of Chordal Cracks in a Rotor System Including Uncertainties
,” J. Sound Vib.
, 331
(1
), pp. 138
–154
.21.
Field
, R. V.
, and Grigoriu
, M.
, 2004
, “On the Accuracy of the Polynomial Chaos Approximation
,” Probab. Eng. Mech.
, 19
(1–2
), pp. 65
–80
.22.
Xiu
, D. B.
, and Hesthaven
, J. S.
, 2005
, “High-Order Collocation Methods for Differential Equations With Random Inputs
,” SIAM J. Sci. Comput.
, 27
(3
), pp. 1118
–1139
.23.
Stavroulakis
, G.
, Giovanis
, D. G.
, Papadrakakis
, M.
, and Papadopoulos
, V.
, 2014
, “A New Perspective on the Solution of Uncertainty Quantification and Reliability Analysis of Large-Scale Problems
,” Comput. Methods Appl. Mech. Eng.
, 276
, pp. 627
–658
.24.
Doostan
, A.
, Ghanem
, R. G.
, and Red-Horse
, J.
, 2007
, “Stochastic Model Reduction for Chaos Representations
,” Comput. Methods Appl. Mech. Eng.
, 196
(37–40
), pp. 3951
–3966
.25.
Arnst
, M.
, and Ghanem
, R. G.
, 2008
, “Probabilistic Equivalence and Stochastic Model Reduction in Multiscale Analysis
,” Comput. Methods Appl. Mech. Eng.
, 197
(43–44
), pp. 3584
–3592
.26.
Puri
, R. S.
, and Morrey
, D.
, 2011
, “An Arnoldi Based Projection Formulation for Fully Coupled Low Frequency Structural-Acoustic Analysis
,” Acta Acust. Acust.
, 97
(3
), pp. 509
–518
.27.
Volkov
, K. N.
, 2013
, “Implementation and Comparison of Various Approaches to Solving Coupled Thermal Analysis Problems on Unstructured Meshes
,” Comput. Math. Math. Phys.
, 53
(4
), pp. 503
–519
.28.
Puri
, R. S.
, and Morrey
, D.
, 2011
, “A Krylov–Arnoldi Reduced Order Modelling Framework for Efficient, Fully Coupled, Structural-Acoustic Optimization
,” Struct. Multidiscip. Optim.
, 43
(4
), pp. 495
–517
.29.
Bai
, Z. J.
, 2002
, “Krylov Subspace Techniques for Reduced-Order Modeling of Large-Scale Dynamical Systems
,” Appl. Numer. Math.
, 43
(1–2
), pp. 9
–44
.30.
Han
, J. S.
, Rudnyi
, E. B.
, and Korvink
, J. G.
, 2005
, “Efficient Optimization of Transient Dynamic Problems in MEMS Devices Using Model Order Reduction
,” J. Micromech. Microeng.
, 15
(4
), pp. 822
–832
.31.
Mi
, N.
, Tan
, S. X. D.
, Cai
, Y. C.
, and Hong
, X. L.
, 2008
, “Fast Variational Analysis of On-Chip Power Grids by Stochastic Extended Krylov Subspace Method
,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.
, 27
(11
), pp. 1996
–2006
.32.
Nair
, P. B.
, and Keane
, A. J.
, 2002
, “Stochastic Reduced Basis Methods
,” AIAA J.
, 40
(8
), pp. 1653
–1664
.33.
Sachdeva
, S. K.
, Nair
, P. B.
, and Keane
, A. J.
, 2006
, “Hybridization of Stochastic Reduced Basis Methods With Polynomial Chaos Expansions
,” Probab. Eng. Mech.
, 21
(2
), pp. 182
–192
.34.
Sachdeva
, S. K.
, Nair
, P. B.
, and Keane
, A. J.
, 2006
, “Comparative Study of Projection Schemes for Stochastic Finite Element Analysis
,” Comput. Methods Appl. Mech. Eng.
, 195
(19–22
), pp. 2371
–2392
.35.
Dohnal
, F.
, Mace
, B. R.
, and Ferguson
, N. S.
, 2009
, “Joint Uncertainty Propagation in Linear Structural Dynamics Using Stochastic Reduced Basis Methods
,” AIAA J.
, 47
(4
), pp. 961
–969
.36.
Bah
, M. T.
, Nair
, P. B.
, Bhaskar
, A.
, and Keane
, A. J.
, 2003
, “Forced Response Statistics of Mistuned Bladed Disks: A Stochastic Reduced Basis Approach
,” J. Sound Vib.
, 263
(2
), pp. 377
–397
.37.
Mohan
, P. S.
, Nair
, P. B.
, and Keane
, A. J.
, 2011
, “Stochastic Projection Schemes for Deterministic Linear Elliptic Partial Differential Equations on Random Domains
,” Int. J. Numer. Methods Eng.
, 85
(7
), pp. 874
–895
.38.
Xiu
, D.
, and Karniadakis
, G.
, 2002
, “The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations
,” J. Sci. Comput.
, 24
(2
), pp. 619
–644
.39.
Sepahvand
, K.
, Marburg
, S.
, and Hardtke
, H. J.
, 2010
, “Uncertainty Quantification in Stochastic Systems Using Polynomial Chaos Expansion
,” Int. J. Appl. Mech.
, 2
(2
), pp. 305
–353
.40.
Ghanem
, R.
, and Ghosh
, D.
, 2007
, “Efficient Characterization of the Random Eigenvalue Problem in a Polynomial Chaos Decomposition
,” Int. J. Numer. Methods Eng.
, 72
(4
), pp. 486
–504
.41.
Seguí
, B.
, Faverjon
, B.
, and Jacquet-Richardet
, G.
, 2013
, “Effects of Random Stiffness Variations in Multistage Rotors Using the Polynomial Chaos Expansion
,” J. Sound Vib.
, 332
(18
), pp. 4178
–4192
.42.
Debusschere
, B. J.
, Najm
, H. N.
, Pebay
, P. P.
, Knio
, O. M.
, Ghanem
, R. G.
, and Le Maitre
, O. P.
, 2004
, “Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes
,” SIAM J. Sci. Comput.
, 26
(2
), pp. 698
–719
.43.
Jacquelin
, E.
, Adhikari
, S.
, Sinou
, J. J.
, and Friswell
, M. I.
, 2015
, “Polynomial Chaos Expansion in Structural Dynamics: Accelerating the Convergence of the First Two Statistical Moment Sequences
,” J. Sound Vib.
, 356
, pp. 144
–154
.Copyright © 2017 by ASME
You do not currently have access to this content.
