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Stochastic Models of Uncertainties in Computational Structural Dynamics and Structural Acoustics

  • Christian Soize
Part of the CISM Courses and Lectures book series (CISM, volume 539)

Abstract

We present an overview concerning the main concepts, formulations and advances for the stochastic modeling of uncertainties in computational structural dynamics and structural acoustics. The parametric probabilistic approach, the nonparametric probabilistic approach and the generalized probabilistic approach of uncertainties are presented in the context of structural dynamics and in structural acoustics and vibration, including not only the construction of prior probability models but also the identification of posterior probability models.

Keywords

Random Matrice Random Matrix Theory Stochastic Partial Differential Equation Maximum Entropy Principle Polynomial Chaos 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. T. W. Anderson. Introduction to Multivariate Statistical Analysis. John Wiley & Sons, New York, 1958.zbMATHGoogle Scholar
  2. M. Arnst and R. Ghanem. Probabilistic equivalence and stochastic model reduction in multiscale analysis. Computer Methods in Applied Mechanics and Engineering, 197:3584–3592, 2008.MathSciNetzbMATHGoogle Scholar
  3. M. Arnst, D. Clouteau, H. Chebli, R. Othman, and G. Degrande. A nonparametric probabilistic model for ground-borne vibrations in buildings. Probabilistic Engineering Mechanics, 21(1):18–34, 2006.Google Scholar
  4. M. Arnst, D. Clouteau, and M. Bonnet. Inversion of probabilistic structural models using measured transfer functions. Computer Methods in Applied Mechanics and Engineering, 197(6–8):589–608, 2008.MathSciNetzbMATHGoogle Scholar
  5. M. Arnst, R. Ghanem, and C. Soize. Identification of bayesian posteriors for coefficients of chaos expansions. Journal of Computational Physics, 229(9):3134–3154, 2010.MathSciNetzbMATHGoogle Scholar
  6. S.K. Au and J.L. Beck. Subset simulation and its application to seismic risk based on dynamic analysis. Journal of Engineering Mechanics — ASCE, 129(8):901–917, 2003a.Google Scholar
  7. S.K. Au and J.L. Beck. Important sampling in high dimensions. Structural Safety, 25(2):139–163, 2003b.Google Scholar
  8. I. Babuska, R. Tempone, and G. E. Zouraris. Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation. Computer Methods in Applied Mechanics and Engineering, 194(12–16):1251–1294, 2005.MathSciNetzbMATHGoogle Scholar
  9. I. Babuska, F. Nobile, and R. Tempone. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Journal on Numerical Analysis, 45(3):1005–1034, 2007.MathSciNetzbMATHGoogle Scholar
  10. A. Batou and C. Soize. Identification of stochastic loads applied to a nonlinear dynamical system using an uncertain computational model and experimental responses. Computational Mechanics, 43(4):559–571, 2009a.MathSciNetGoogle Scholar
  11. A. Batou and C. Soize. Experimental identification of turbulent fluid forces applied to fuel assemblies using an uncertain model and fretting-wear estimation. Mechanical Systems and Signal Processing, 23(7):2141–2153, 2009b.Google Scholar
  12. A. Batou, C. Soize, and M. Corus. Experimental identification of an uncertain computational dynamical model representing a family of structures. Computer and Structures, pages In press,doi:10.1016/j.compstruc.2011.03.004, 2011.Google Scholar
  13. J. L. Beck and L. S. Katafygiotis. Updating models and their uncertainties. i: Bayesian statistical framework. Journal of Engineering Mechanics, 124(4):455–461, 1998.Google Scholar
  14. J.L. Beck. Bayesian system identification based on probability logic. Structural Control and Health Monitoring, 17(7):825–847, 2010.Google Scholar
  15. J.L. Beck and S.K. Au. Bayesian updating of structural models and reliability using markov chain monte carlo simulation. Journal of Engineering Mechanics — ASCE, 128(4):380–391, 2002.Google Scholar
  16. J.L. Beck, E. Chan, A. Irfanoglu, and et al. Multi-criteria optimal structural design under uncertainty. Earthquake Engineering and Structural Dynamics, 28(7):741–761, 1999.Google Scholar
  17. J. M. Bernardo and A. F. M. Smith. Bayesian Theory. John Wiley & Sons, Chichester, 2000.zbMATHGoogle Scholar
  18. M. Berveiller, B. Sudret, and M. Lemaire. Stochastic finite element: a nonintrusive approach by regression. European Journal of Computational Mechanics, 15:81–92, 2006.zbMATHGoogle Scholar
  19. G. Blatman and B. Sudret. Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. Comptes Rendus Mcanique, 336(6):518–523, 2007.Google Scholar
  20. A. W. Bowman and A. Azzalini. Applied Smoothing Techniques for Data Analysis. Oxford University Press, Oxford, 1997.zbMATHGoogle Scholar
  21. E. Capiez-Lernout and C. Soize. Nonparametric modeling of random uncertainties for dynamic response of mistuned bladed disks. Journal of Engineering for Gas Turbines and Power, 126(3):600–618, 2004.Google Scholar
  22. E. Capiez-Lernout and C. Soize. Robust design optimization in computational mechanics. Journal of Applied Mechanics — Transactions of the ASME, 75(2):021001-1–021001-11, 2008a.MathSciNetGoogle Scholar
  23. E. Capiez-Lernout and C. Soize. Design optimization with an uncertain vibroacoustic model. Journal of Vibration and Acoustics, 130(2):021001-1–021001-8, 2008b.MathSciNetGoogle Scholar
  24. E. Capiez-Lernout and C. Soize. Robust updating of uncertain damping models in structural dynamics for low-and medium-frequency ranges. Mechanical Systems and Signal Processing, 22(8):1774–1792, 2008c.Google Scholar
  25. E. Capiez-Lernout, C. Soize, J.-P. Lombard, C. Dupont, and E. Seinturier. Blade manufacturing tolerances definition for a mistuned industrial bladed disk. Journal of Engineering for Gas Turbines and Power, 127(3):621–628, 2005.Google Scholar
  26. E. Capiez-Lernout, M. Pellissetti, H. Pradlwarter, G. I. Schueller, and C. Soize. Data and model uncertainties in complex aerospace engineering systems. Journal of Sound and Vibration, 295(3–5):923–938, 2006.Google Scholar
  27. B. P. Carlin and T. A. Louis. Bayesian Methods for Data Analysis. Third Edition, Chapman & Hall / CRC Press, Boca Raton, 2009.Google Scholar
  28. G. Casella and E.I. George. Explaining the gibbs sampler. The American Statistician, 46(3):167–174, 1992.MathSciNetGoogle Scholar
  29. E. Cataldo, C. Soize, R. Sampaio, and C. Desceliers. Probabilistic modeling of a nonlinear dynamical system used for producing voice. Computational Mechanics, 43(2):265–275, 2009.zbMATHGoogle Scholar
  30. H. Chebli and C. Soize. Experimental validation of a nonparametric probabilistic model of non homogeneous uncertainties for dynamical systems. Journal of the Acoustical Society of America, 115(2):697–705, 2004.Google Scholar
  31. C. Chen, D. Duhamel, and C. Soize. Probabilistic approach for model and data uncertainties and its experimental identification in structural dynamics: Case of composite sandwich panels. Journal of Sound and Vibration, 294(1–2):64–81, 2006.Google Scholar
  32. S.H. Cheung and J.L. Beck. Bayesian model updating using hybrid monte carlo simulation with application to structural dynamic models with many uncertain parameters. Journal of Engineering Mechanics — ASCE, 135(4):243–255, 2009.Google Scholar
  33. S.H. Cheung and J.L. Beck. Calculation of posterior probabilities for bayesian model class assessment and averaging from posterior samples based on dynamic system data. Computer-Aided Civil and Infrastructure Engineering, 25(5):304–321, 2010.Google Scholar
  34. J.Y. Ching, J.L. Beck, and K.A. Porter. Bayesian state and parameter estimation of uncertain dynamical systems. Probabilistic Engineering Mechanics, 21(1):81–96, 2006.Google Scholar
  35. P. Congdon. Bayesian Statistical Modelling. Second Edition, John Wiley & Sons, Chichester, 2007.Google Scholar
  36. R. Cottereau, D. Clouteau, and C. Soize. Construction of a probabilistic model for impedance matrices. Computer Methods in Applied Mechanics and Engineering, 196(17–20):2252–2268, 2007.MathSciNetzbMATHGoogle Scholar
  37. R. Cottereau, D. Clouteau, and C. Soize. Probabilistic impedance of foundation, impact of the seismic design on uncertain soils. Earthquake Engineering and Structural Dynamics, 37(6):899–918, 2008.Google Scholar
  38. S. Das and R. Ghanem. A bounded random matrix approach for stochastic upscaling. Multiscale Model. Simul., 8(1):296325, 2009.MathSciNetGoogle Scholar
  39. S. Das, R. Ghanem, and J. C. Spall. Asymptotic sampling distribution for polynomial chaos representation from data: a maximum entropy and fisher information approach. SIAM Journal on Scientific Computing, 30 (5):2207–2234, 2008.MathSciNetzbMATHGoogle Scholar
  40. S. Das, R. Ghanem, and S. Finette. Polynomial chaos representation of spatio-temporal random field from experimental measurements. Journal of Computational Physics, 228:8726–8751, 2009.MathSciNetzbMATHGoogle Scholar
  41. M. Deb, I. Babuska, and J.T. Oden. Solution of stochastic partial differential equations using galerkin finite element techniques. Computer Methods in Applied Mechanics and Engineering, 190:6359–6372, 2001.MathSciNetzbMATHGoogle Scholar
  42. B.J. Debusschere, H.N. Najm, P.P. Pebay, and et al. Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM Journal on Scientific Computing, 26(2):698–719, 2004.MathSciNetzbMATHGoogle Scholar
  43. G. Deodatis and P. D. Spanos. 5th international conference on computational stochastic mechanics. Special issue of the Probabilistic Engineering Mechanics, 23(2–3):103–346, 2008.Google Scholar
  44. C. Desceliers, C. Soize, and S. Cambier. Non-parametric — parametric model for random uncertainties in nonlinear structural dynamics — application to earthquake engineering. Earthquake Engineering and Structural Dynamics, 33(3):315–327, 2004.Google Scholar
  45. C. Desceliers, R. Ghanem, and C. Soize. Maximum likelihood estimation of stochastic chaos representations from experimental data. International Journal for Numerical Methods in Engineering, 66(6):978–1001, 2006.MathSciNetzbMATHGoogle Scholar
  46. C. Desceliers, C. Soize, and R. Ghanem. Identification of chaos representations of elastic properties of random media using experimental vibration tests. Computational Mechanics, 39(6):831–838, 2007.zbMATHGoogle Scholar
  47. C. Desceliers, C. Soize, Q. Grimal, M. Talmant, and S. Naili. Determination of the random anisotropic elasticity layer using transient wave propagation in a fluid-solid multilayer: Model and experiments. Journal of the Acoustical Society of America, 125(4):2027–2034, 2009.Google Scholar
  48. A. Doostan and G. Iaccarino. A least-squares approximation of partial differential equations with highdimensional random inputs. Journal of Computational Physics, 228(12):4332–4345, 2009.MathSciNetzbMATHGoogle Scholar
  49. A. Doostan, R. Ghanem, and J. Red-Horse. Stochastic model reductions for chaos representations. Computer Methods in Applied Mechanics and Engineering, 196(37–40):3951–3966, 2007.MathSciNetzbMATHGoogle Scholar
  50. J. Duchereau and C. Soize. Transient dynamics in structures with nonhomogeneous uncertainties induced by complex joints. Mechanical Systems and Signal Processing, 20(4):854–867, 2006.Google Scholar
  51. J.-F. Durand, C. Soize, and L. Gagliardini. Structural-acoustic modeling of automotive vehicles in presence of uncertainties and experimental identification and validation. Journal of the Acoustical Society of America, 124(3):1513–1525, 2008.Google Scholar
  52. B. Faverjon and R. Ghanem. Stochastic inversion in acoustic scattering. Journal of the Acoustical Society of America, 119(6):3577–3588, 2006.Google Scholar
  53. C. Fernandez, C. Soize, and L. Gagliardini. Fuzzy structure theory modeling of sound-insulation layers in complex vibroacoustic uncertain sytems — theory and experimental validation. Journal of the Acoustical Society of America, 125(1):138–153, 2009.Google Scholar
  54. C. Fernandez, C. Soize, and L. Gagliardini. Fuzzy structure theory modeling of sound-insulation layers in complex vibroacoustic uncertain sytems — theory and experimental validation. Journal of the Acoustical Society of America, 125(1):138–153, 2009.Google Scholar
  55. C. Fernandez, C. Soize, and L. Gagliardini. Sound-insulation layer modelling in car computational vibroacoustics in the medium-frequency range. Acta Acustica united with Acustica (AAUWA), 96(3):437–444, 2010.Google Scholar
  56. G.S. Fishman. Monte Carlo: Concepts, algorithms, and applications. Springer-Verlag, New York, 1996.zbMATHGoogle Scholar
  57. C. Fougeaud and A. Fuchs. Statistique. Dunod, Paris, 1967.Google Scholar
  58. P. Frauenfelder, C. Schwab, and R.A. Todor. Finite elements for elliptic problems with stochastic coefficients. Computer Methods in Applied Mechanics and Engineering, 194(2–5):205–228, 2005.MathSciNetzbMATHGoogle Scholar
  59. B. Ganapathysubramanian and N. Zabaras. Sparse grid collocation schemes for stochastic natural convection problems. Journal of Computational Physics, 225(1):652–685, 2007.MathSciNetzbMATHGoogle Scholar
  60. S. Geman and D. Geman. Stochastic relaxation, gibbs distribution and the bayesian distribution of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol PAM I-6(6):721–741, 1984.Google Scholar
  61. R. Ghanem. Ingredients for a general purpose stochastic finite elements formulation. Computer Methods in Applied Mechanics and Engineering, 168(1–4):19–34, 1999.MathSciNetzbMATHGoogle Scholar
  62. R. Ghanem and S. Dham. Stochastic finite element analysis for multiphase flow in heterogeneous porous media. Transp. Porous Media, 32:239–262, 1998.MathSciNetGoogle Scholar
  63. R. Ghanem and R. Doostan. Characterization of stochastic system parameters from experimental data: A bayesian inference approach. Journal of Computational Physics, 217(1):63–81, 2006.MathSciNetzbMATHGoogle Scholar
  64. R. Ghanem and D. Ghosh. Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition. International Journal for Numerical Methods in Engineering, 72(4):486–504, 2007.MathSciNetzbMATHGoogle Scholar
  65. R. Ghanem and R. M. Kruger. Numerical solution of spectral stochastic finite element systems. Computer Methods in Applied Mechanics and Engineering, 129:289–303, 1996.zbMATHGoogle Scholar
  66. R. Ghanem and M. Pellissetti. Adaptive data refinement in the spectral stochastic finite element method. Comm. Numer. Methods Engrg., 18: 141–151, 2002.MathSciNetzbMATHGoogle Scholar
  67. R. Ghanem and J. Red-Horse. Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach. Physica D, 133(1–4):137–144, 1999.MathSciNetzbMATHGoogle Scholar
  68. R. Ghanem and A. Sarkar. Reduced models for the medium-frequency dynamics of stochastic systems. Journal of the Acoustical Society of America, 113(2):834–846, 2003.Google Scholar
  69. R. Ghanem and P. D. Spanos. Stochastic finite elements: a spectral approach. Springer-Verlag, New York, 1991.zbMATHGoogle Scholar
  70. R. Ghanem and P.D. Spanos. Polynomial chaos in stochastic finite elements. Journal of Applied Mechanics — Transactions of the ASME, 57(1):197–202, 1990.zbMATHGoogle Scholar
  71. R. Ghanem and P.D. Spanos. Stochastic Finite Elements: A spectral Approach. (revised edition) Dover Publications, New York, 2003.Google Scholar
  72. R. Ghanem, S. Masri, M. Pellissetti, and R. Wolfe. Identification and prediction of stochastic dynamical systems in a polynomial chaos basis. Computer Methods in Applied Mechanics and Engineering, 194(12-16): 1641–1654, 2005.zbMATHGoogle Scholar
  73. R. Ghanem, R. Doostan, and J. Red-Horse. A probability construction of model validation. Computer Methods in Applied Mechanics and Engineering, 197(29–32):2585–2595, 2008.zbMATHGoogle Scholar
  74. D. Ghosh and R. Ghanem. Stochastic convergence acceleration through basis enrichment of polynomial chaos expansions. International Journal for Numerical Methods in Engineering, 73(2):162–184, 2008.MathSciNetzbMATHGoogle Scholar
  75. B. Goller, H.J. Pradlwarter, and G.I. Schueller. Robust model updating with insufficient data. Computer Methods in Applied Mechanics and Engineering, 198(37–40):3096–3104, 2009.zbMATHGoogle Scholar
  76. J. Guilleminot and C. Soize. Non-gaussian positive-definite matrix-valued random fields with constrained eigenvalues: application to random elasticity tensors with uncertain material symmetries. International Journal for Numerical Methods in Engineering, page (To appear), 2011.Google Scholar
  77. J. Guilleminot, C. Soize, D. Kondo, and C. Benetruy. Theoretical framework and experimental procedure for modelling volume fraction stochastic fluctuations in fiber reinforced composites. International Journal of Solid and Structures, 45(21):5567–5583, 2008.zbMATHGoogle Scholar
  78. J. Guilleminot, C. Soize, and D. Kondo. Mesoscale probabilistic models for the elasticity tensor of fiber reinforced composites: experimental identification and numerical aspects. Mechanics of Materials, 41(12):1309–1322, 2009.Google Scholar
  79. J. Guilleminot, A. Noshadravanb, C. Soize, and R. Ghanem. A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures. Computer Methods in Applied Mechanics and Engineering, 200(17–20):1637–1648, 2011.MathSciNetzbMATHGoogle Scholar
  80. W. K. Hastings. Monte carlo sampling methods using markov chains and their applications. Biometrika, 109:57–97, 1970.Google Scholar
  81. E. T. Jaynes. Information theory and statistical mechanics. Physical Review, 108(2):171–190, 1957.MathSciNetGoogle Scholar
  82. J. Kaipio and E. Somersalo. Statistical and Computational Inverse Problems. Springer-Verlag, New York, 2005.zbMATHGoogle Scholar
  83. M. Kassem, C. Soize, and L. Gagliardini. Energy density field approach for low-and medium-frequency vibroacoustic analysis of complex structures using a stochastic computational model. Journal of Sound and Vibration, 323(3–5):849–863, 2009.Google Scholar
  84. M. Kassem, C. Soize, and L. Gagliardini. Structural partitioning of complex structures in the medium-frequency range. an application to an automotive vehicle. Journal of Sound and Vibration, 330(5):937–946, 2011.Google Scholar
  85. L.S. Katafygiotis and J.L. Beck. Updating models and their uncertainties. ii: Model identifiability. Journal of Engineering Mechanics — ASCE, 124 (4):463–467, 1998.Google Scholar
  86. O.M. Knio and O.P. Le Maitre. Uncertainty propagation in cfd using polynomial chaos decomposition. Fluid Dynamics Research, 38(9):616–640, 2006.MathSciNetzbMATHGoogle Scholar
  87. T. Leissing, C. Soize, P. Jean, and J. Defrance. Computational model for long-range non-linear propagation over urban cities. Acta Acustica united with Acustica (AAUWA), 96(5):884–898, 2010.Google Scholar
  88. O. P. LeMaitre, O. M. Knio, H. N. Najm, and R. Ghanem. A stochastic projection method for fluid flow. ii. random process. Journal of Computational Physics, 181:9–44, 2002.MathSciNetGoogle Scholar
  89. O. P. LeMaitre, O. M. Knio, H. N. Najm, and R. Ghanem. Uncertainty propagation using wiener-haar expansions. Journal of Computational Physics, 197(1):28–57, 2004a.MathSciNetGoogle Scholar
  90. O. P. LeMaitre, H. N. Najm, R. Ghanem, and O. Knio. Multi-resolution analysis of wiener-type uncertainty propagation schemes. Journal of Computational Physics, 197(2):502–531, 2004b.MathSciNetGoogle Scholar
  91. O. P. LeMaitre, H. N. Najm, P. P. Pebay, R. Ghanem, and O. Knio. Multiresolutionanalysis scheme for uncertainty quantification in chemical systems. SIAM Journal on Scientific Computing, 29(2):864–889, 2007.MathSciNetGoogle Scholar
  92. O.P. LeMaitre and O.M. Knio. Spectral Methods for Uncerainty Quantification with Applications to Computational Fluid Dynamics. Springer, Heidelberg, 2010.Google Scholar
  93. D. Lucor, C.H. Su, and G.E. Karniadakis. Generalized polynomial chaos and random oscillators. International Journal for Numerical Methods in Engineering, 60(3):571–596, 2004.MathSciNetzbMATHGoogle Scholar
  94. D. Lucor, J. Meyers, and P. Sagaut. Sensitivity analysis of large-eddy simulations to subgrid-scale-model parametric uncertainty using polynomial chaos. Journal of Fluid Mechanics, 585:255–279, 2007.MathSciNetzbMATHGoogle Scholar
  95. X. Ma and N. Zabaras. An efficient bayesian inference approach to inverse problems based on an adaptive sparse grid collocation method. Inverse Problems, 25(3):Article Number: 035013, 2009.Google Scholar
  96. R. Mace, W. Worden, and G. Manson. Uncertainty in structural dynamics. Special issue of the Journal of Sound and Vibration, 288(3):431–790, 2005.Google Scholar
  97. Y.M. Marzouk and H.N. Najm. Dimensionality reduction and polynomial chaos acceleration of bayesian inference in inverse problems. Journal of Computational Physics, 228(6):1862–1902, 2009.MathSciNetzbMATHGoogle Scholar
  98. Y.M. Marzouk, H.N. Najm, and L.A. Rahn. Stochastic spectral methods for efficient bayesian solution of inverse problems. Journal of Computational Physics, 224(2):560–586, 2007.MathSciNetzbMATHGoogle Scholar
  99. L. Mathelin and O. LeMaitre. Dual based a posteriori estimation for stochastic finite element method. Comm. App. Math. Comp. Sci., 2(1):83–115, 2007.MathSciNetzbMATHGoogle Scholar
  100. H. G. Matthies. Stochastic finite elements: Computational approaches to stochastic partial differential equations. Zamm-Zeitschrift Fur Angewandte Mathematik Und Mechanik, 88(11):849–873, 2008.MathSciNetzbMATHGoogle Scholar
  101. H.G. Matthies and A. Keese. Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Computer Methods in Applied Mechanics and Engineering, 194(12–16):1295–1331, 2005.MathSciNetzbMATHGoogle Scholar
  102. M. L. Mehta. Random Matrices, Revised and Enlarged Second Edition. Academic Press, New York, 1991.Google Scholar
  103. N. Metropolis and S. Ulam. The monte carlo method. Journal of American Statistical Association, 49:335–341, 1949.MathSciNetGoogle Scholar
  104. M. P. Mignolet and C. Soize. Nonparametric stochastic modeling of linear systems with prescribed variance of several natural frequencies. Probabilistic Engineering Mechanics, 23(2–3):267–278, 2008a.Google Scholar
  105. M. P. Mignolet and C. Soize. Stochastic reduced order models for uncertain nonlinear dynamical systems. Computer Methods in Applied Mechanics and Engineering, 197(45–48):3951–3963, 2008b.MathSciNetzbMATHGoogle Scholar
  106. H.N. Najm. Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics. Journal Review of Fluid Mechanics, pages 35–52, 2009.Google Scholar
  107. A. Nouy. A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations. Computer Methods in Applied Mechanics and Engineering, 196(45–48):4521–4537, 2007.MathSciNetzbMATHGoogle Scholar
  108. A. Nouy. Generalized spectral decomposition method for solving stochastic finite element equations: Invariant subspace problem and dedicated algorithms. Computer Methods in Applied Mechanics and Engineering, 197(51–52):4718–4736, 2008.MathSciNetzbMATHGoogle Scholar
  109. A. Nouy. Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations. Archives of Computational Methods in Engineering, 16(3):251–285, 2009.MathSciNetGoogle Scholar
  110. A. Nouy. Proper generalized decomposition and separated representations for the numerical solution of high dimensional stochastic problems. Archives of Computational Methods in Engineering, 17(4):403–434, 2010.MathSciNetGoogle Scholar
  111. A. Nouy and O. P. Le Maitre. Generalized spectral decomposition for stochastic nonlinear problems. Journal of Computational Physics, 228 (1):202–235, 2009.MathSciNetzbMATHGoogle Scholar
  112. R. Ohayon and C. Soize. Structural Acoustics and Vibration. Academic Press, San Diego, London, 1998.Google Scholar
  113. C. Papadimitriou, J.L. Beck, and S.K. Au. Entropy-based optimal sensor location for structural model updating. Journal of Vibration and Control, 6(5):781–800, 2000.Google Scholar
  114. C. Papadimitriou, J.L. Beck, and L.S. Katafygiotis. Updating robust reliability using structural test data. Probabilistic Engineering Mechanics, 16(2):103–113, 2001.Google Scholar
  115. M. Papadrakakis and A. Kotsopulos. Parallel solution methods for stochastic finite element analysis using monte carlo simulation. Computer Methods in Applied Mechanics and Engineering, 168(1–4):305–320, 1999.zbMATHGoogle Scholar
  116. M. Papadrakakis and N.D. Lagaros. Reliability-based structural optimization using neural networks and monte carlo simulation. Computer Methods in Applied Mechanics and Engineering, 191(32):3491–3507, 2002.zbMATHGoogle Scholar
  117. M. Papadrakakis and V. Papadopoulos. Robust and efficient methods for stochastic finite element analysis using monte carlo simulation. Computer Methods in Applied Mechanics and Engineering, 134(134):325–340, 1996.MathSciNetzbMATHGoogle Scholar
  118. M. Pellissetti, E. Capiez-Lernout, H. Pradlwarter, C. Soize, and G. I. Schueller. Reliability analysis of a satellite structure with a parametric and a non-parametric probabilistic model. Computer Methods in Applied Mechanics and Engineering, 198(2):344–357, 2008.MathSciNetzbMATHGoogle Scholar
  119. B. Peters and G. De Roeck. Stochastic system identification for operational modal analysis: A review. Journal of Dynamic Systems Measurement and Control-Transactions of The Asme, 123(4):659–667, 2001.Google Scholar
  120. H. J. Pradlwarter, G. I. Schueller, and G. S. Szekely. Random eigenvalue problems for large systems. Computer and Structures, 80:2415–2424, 2002.MathSciNetGoogle Scholar
  121. H.J. Pradlwarter and G.I. Schueller. On advanced monte carlo simulation procedures in stochastic structural dynamics. International Journal of Non-Linear Mechanics, 32(4):735–744, 1997.zbMATHGoogle Scholar
  122. H.J. Pradlwarter and G.I. Schueller. Local domain monte carlo simulation. Structural Safety, 32(5):275–280, 2010.Google Scholar
  123. T.G. Ritto, C. Soize, and R. Sampaio. Nonlinear dynamics of a drill-string with uncertainty model of the bit-rock interaction. International Journal of Non-Linear Mechanics, 44(8):865–876, 2009.Google Scholar
  124. T.G. Ritto, C. Soize, and R. Sampaio. Robust optimization of the rate of penetration of a drill-string using a stochastic nonlinear dynamical model. Computational Mechanics, 45(5):415–427, 2010.zbMATHGoogle Scholar
  125. R. Y. Rubinstein and D. P. Kroese. Simulation and the Monte Carlo Method. Second Edition, John Wiley & Sons, New York, 2008.zbMATHGoogle Scholar
  126. C.P. Rupert and C.T. Miller. An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems. Journal of Computational Physics, 226(2):2175–2205, 2007.MathSciNetzbMATHGoogle Scholar
  127. S. Sakamoto and R. Ghanem. Polynomial chaos decomposition for the simulation of non-gaussian nonstationary stochastic processes. Journal of Engineering Mechanics-ASCE, 128(2):190–201, 2002.Google Scholar
  128. R. Sampaio and C. Soize. On measures of non-linearity effects for uncertain dynamical systems — application to a vibro-impact system. Journal of Sound and Vibration, 303(3–5):659–674, 2007a.Google Scholar
  129. R. Sampaio and C. Soize. Remarks on the efficiency of pod for model reduction in nonlinear dynamics of continuous elastic systems. International Journal for Numerical Methods in Engineering, 72(1):22–45, 2007b.MathSciNetzbMATHGoogle Scholar
  130. G. I. Schueller. Uncertainties in structural mechanics and analysis-computational methods. Special issue of Computer and Structures, 83(14):1031–1150, 2005.Google Scholar
  131. G. I. Schueller. On the treatment of uncertainties in structural mechanics and analysis. Computer and Structures, 85(5–6):235–243, 2007.Google Scholar
  132. G. I. Schueller and H. A. Jensen. Computational methods in optimization considering uncertainties — an overview. Computer Methods in Applied Mechanics and Engineering, 198(1):2–13, 2008.MathSciNetzbMATHGoogle Scholar
  133. G.I. Schueller. Efficient monte carlo simulation procedures in structural uncertainty and reliability analysis — recent advances. Structural Engineering and Mechanics, 32(1):1–20, 2009.Google Scholar
  134. G.I. Schueller and H.J. Pradlwarter. Uncertain linear systems in dynamics: Retrospective and recent developments by stochastic approaches. Engineering Structures, 31(11):2507–2517, 2009.Google Scholar
  135. R. J. Serfling. Approximation Theorems of Mathematical Statistics. John Wiley & Sons, 1980.Google Scholar
  136. C. E. Shannon. A mathematical theory of communication. Bell System Technology Journal, 27(14):379–423 & 623–659, 1948.MathSciNetzbMATHGoogle Scholar
  137. C. Soize. The Fokker-Planck Equation for Stochastic Dynamical Systems and its Explicit Steady State Solutions. World Scientific Publishing Co Pte Ltd, Singapore, 1994.zbMATHGoogle Scholar
  138. C. Soize. A nonparametric model of random uncertainties on reduced matrix model in structural dynamics. Probabilistic Engineering Mechanics, 15 (3):277–294, 2000.Google Scholar
  139. C. Soize. Maximum entropy approach for modeling random uncertainties in transient elastodynamics. Journal of the Acoustical Society of America, 109(5):1979–1996, 2001.Google Scholar
  140. C. Soize. Random matrix theory and non-parametric model of random uncertainties. Journal of Sound and Vibration, 263(4):893–916, 2003a.MathSciNetzbMATHGoogle Scholar
  141. C. Soize. Uncertain dynamical systems in the medium-frequency range. Journal of Engineering Mechanics, 129(9):1017–1027, 2003b.Google Scholar
  142. C. Soize. A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics. Journal of Sound and Vibration, 288(3):623–652, 2005a.MathSciNetzbMATHGoogle Scholar
  143. C. Soize. Random matrix theory for modeling uncertainties in computational mechanics. Computer Methods in Applied Mechanics and Engineering, 194(12-16):1333–1366, 2005b.MathSciNetzbMATHGoogle Scholar
  144. C. Soize. Non gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators. Computer Methods in Applied Mechanics and Engineering, 195(1–3):26–64, 2006.MathSciNetzbMATHGoogle Scholar
  145. C. Soize. Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size. Probabilistic Engineering Mechanics, 23(2–3):307–323, 2008a.Google Scholar
  146. C. Soize. Construction of probability distributions in high dimension using the maximum entropy principle. applications to stochastic processes, random fields and random matrices. International Journal for Numerical Methods in Engineering, 76(10):1583–1611, 2008b.MathSciNetzbMATHGoogle Scholar
  147. C. Soize. Generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions. International Journal for Numerical Methods in Engineering, 81 (8):939–970, 2010a.MathSciNetzbMATHGoogle Scholar
  148. C. Soize. Random matrices in structural acoustics. In R. Weaver and M. Wright, editors, New Directions in Linear Acoustics: Random Matrix Theory, Quantum Chaos and Complexity, pages 206–230. Cambridge University Press, Cambridge, 2010b.Google Scholar
  149. C. Soize. Identification of high-dimension polynomial chaos expansions with random coefficients for non-gaussian tensor-valued random fields using partial and limited experimental data. Computer Methods in Applied Mechanics and Engineering, 199(33-36):2150–2164, 2010c.MathSciNetzbMATHGoogle Scholar
  150. C. Soize. A computational inverse method for identification of non-gaussian random fields using the bayesian approach in very high dimension. Computer Methods in Applied Mechanics and Engineering, 200(45-46):3083–3099, 2011.MathSciNetzbMATHGoogle Scholar
  151. C. Soize and H. Chebli. Random uncertainties model in dynamic substructuring using a nonparametric probabilistic model. Journal of Engineering Mechanics, 129(4):449–457, 2003.Google Scholar
  152. C. Soize and C. Desceliers. Computational aspects for constructing realizations of polynomial chaos in high dimension. SIAM Journal On Scientific Computing, 32(5):2820–2831, 2010.MathSciNetzbMATHGoogle Scholar
  153. C. Soize and R. Ghanem. Physical systems with random uncertainties: Chaos representation with arbitrary probability measure. SIAM Journal On Scientific Computing, 26(2):395–410, 2004.MathSciNetzbMATHGoogle Scholar
  154. C. Soize and R. Ghanem. Reduced chaos decomposition with random coefficients of vector-valued random variables and random fields. Computer Methods in Applied Mechanics and Engineering, 198(21-26):1926–1934, 2009.MathSciNetzbMATHGoogle Scholar
  155. C. Soize, E. Capiez-Lernout, J.-F. Durand, C. Fernandez, and L. Gagliardini. Probabilistic model identification of uncertainties in computational models for dynamical systems and experimental validation. Computer Methods in Applied Mechanics and Engineering, 198(1):150–163, 2008a.zbMATHGoogle Scholar
  156. C. Soize, E. Capiez-Lernout, and R. Ohayon. Robust updating of uncertain computational models using experimental modal analysis. AIAA Journal, 46(11):2955–2965, 2008b.Google Scholar
  157. J. C. Spall. Introduction to Stochastic Search and Optimization. JohnWiley, 2003.Google Scholar
  158. G. Stefanou, A. Nouy, and A. Clément. Identification of random shapes from images through polynomial chaos expansion of random level set functions. International Journal for Numerical Methods in Engineering, 79(2):127–155, 2009.MathSciNetzbMATHGoogle Scholar
  159. G.S. Szekely and G.I. Schuller. Computational procedure for a fast calculation of eigenvectors and eigenvalues of structures with random properties. Computer Methods in Applied Mechanics and Engineering, 191(8–10):799–816, 2001.Google Scholar
  160. A.A. Taflanidis and J.L. Beck. An efficient framework for optimal robust stochastic system design using stochastic simulation. Computer Methods in Applied Mechanics and Engineering, 198(1):88–101, 2008.zbMATHGoogle Scholar
  161. M. T. Tan, G.-L. Tian, and K. W. Ng. Bayesian Missing Data Problems, EM, Data Augmentation and Noniterative Computation. Chapman & Hall / CRC Press, Boca Raton, 2010.zbMATHGoogle Scholar
  162. E. Walter and L. Pronzato. Identification of Parametric Models from Experimental Data. Springer-Verlag, Berlin, 1997.zbMATHGoogle Scholar
  163. X.L. Wan and G.E. Karniadakis. An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. Journal of Computational Physics, 209(2):617–642, 2005.MathSciNetzbMATHGoogle Scholar
  164. X.L. Wan and G.E. Karniadakis. Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM Journal on Scientific Computing, 28(3):901–928, 2006.MathSciNetzbMATHGoogle Scholar
  165. X.L. Wan and G.E. Karniadakis. Error control in multielement generalized polynomial chaos method for elliptic problems with random coefficients. Comm. Comput. Phys., 5(2–4):793–820, 2009.MathSciNetGoogle Scholar
  166. X.Q. Wang, M.P Mignolet, C. Soize, and V. Khannav. Stochastic reduced order models for uncertain infinite-dimensional geometrically nonlinear dynamical system — stochastic excitation cases. In IUTAM Symposium on Nonlinear Stochastic Dynamics and Control, Hangzhou, China, May 10–14 2010.Google Scholar
  167. C.G. Webster, F. Nobile, and R. Tempone. A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM Journal on Numerical Analysis, 46(5):2309–2345, 2007.MathSciNetGoogle Scholar
  168. N. Wiener. The homogeneous chaos. American Journal of Mathematics, 60 (1):897–936, 1938.MathSciNetGoogle Scholar
  169. M. Wright and R. Weaver. New Directions in Linear Acoustics: Random Matrix Theory, Quantum Chaos and Complexity. Cambridge University Press, Cambridge, 2010.Google Scholar
  170. D.B. Xiu and G.E. Karniadakis. Wiener-askey polynomial chaos for stochastic differential equations. SIAM Journal on Scientific Computing, 24(2):619–644, 2002a.MathSciNetzbMATHGoogle Scholar
  171. D.B. Xiu and G.E. Karniadakis. Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Computer Methods in Applied Mechanics and Engineering, 191(43):4927–4948, 2002b.MathSciNetzbMATHGoogle Scholar
  172. D.B. Xiu and G.E. Karniadakis. Modeling uncertainty in flow simulations via generalized polynomial chaos. Journal of Computational Physics, 187 (1):137–167, 2003.MathSciNetzbMATHGoogle Scholar
  173. N. Zabaras and B. Ganapathysubramanian. A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach. Journal of Computational Physics, 227(9):4697–4735, 2008.MathSciNetzbMATHGoogle Scholar

Copyright information

© CISM, Udine 2012

Authors and Affiliations

  • Christian Soize
    • 1
  1. 1.Laboratoire Modélisation et Simulation Multi-Echelle (MSME UMR 8208 CNRS)Université Paris-EstMarne-la-ValléeFrance

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