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Maximum entropy approach to the identification of stochastic reduced-order models of nonlinear dynamical systems

Published online by Cambridge University Press:  03 February 2016

M. Arnst ,
R. Ghanem  and
S. Masri
M. Arnst
Affiliation:
arnst@usc.edu
R. Ghanem
Affiliation:
University of Southern California, Los Angeles, USA
S. Masri
Affiliation:
University of Southern California, Los Angeles, USA
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Abstract

Data-driven methodologies based on the restoring force method have been developed over the past few decades for building predictive reduced-order models (ROMs) of nonlinear dynamical systems. These methodologies involve fitting a polynomial expansion of the restoring force in the dominant state variables to observed states of the system. ROMs obtained in this way are usually prone to errors and uncertainties due to the approximate nature of the polynomial expansion and experimental limitations. We develop in this article a stochastic methodology that endows these errors and uncertainties with a probabilistic structure in order to obtain a quantitative description of the proximity between the ROM and the system that it purports to represent. Specifically, we propose an entropy maximization procedure for constructing a multi-variate probability distribution for the coefficients of power-series expansions of restoring forces. An illustration in stochastic aeroelastic stability analysis is provided to demonstrate the proposed framework.

Type
Research Article
Information
The Aeronautical Journal , Volume 114 , Issue 1160 , October 2010 , pp. 637 - 650
Copyright
Copyright © Royal Aeronautical Society 2010 

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References

1. Woon, S. and Marshall, S.. Design of multivariable control systems using reduced-order models, Electronics Letters, 1975, 11.Google Scholar
2. Legresley, P. and Alonso, J.. Airfoil design optimization using reduced order models based on proper orthogonal decomposition, 2000, American Institute of Aeronautics and Astronautics Paper 2000-2545, Fluids 2000 Conference and Exhibition.Google Scholar
3. Lall, S., Mardsen, J. and Glavaski, S.. Subspace approach to balanced truncation for model reduction of nonlinear control systems, Int J Robust and Nonlinear Control, 2002, 12, (519–535).Google Scholar
4. Yun, H. and Masri, S.. Stochastic change detection in uncertain nonlinear systems using reduced-order models: classication, Smart Materials and Structures, 2009, 18, 015004.Google Scholar
5. Ohayon, R. and Soize, C., Structural Acoustics and Vibration, 1998, Academic Press.Google Scholar
6. Grimme, E., Sorensen, D. and Vandooren, P.. Model reduction of state space systems via an implicitly restarted Lanczos method, Numerical Algorithms, 1996, 12, pp 1–31.Google Scholar
7. Willcox, K. and Peraire, J.. Balanced model reduction via the proper orthogonal decomposition, American Institute of Aeronautics and Astronautics J, 2002, 40, pp 2323–2330.Google Scholar
8. Masri, S. and Caughey, T.. A nonparametric identication technique for nonlinear dynamic problems, J Applied Mech, 1979, 46, pp 433–445.Google Scholar
9. Masri, S., Miller, R., Saud, A. and Caughey, T.. Identication of nonlinear vibrating structures: part I — formulation, J Applied Mech, 1987, 54, pp 918–929.Google Scholar
10. Masri, S., Carey, J., Caughey, T., Smyth, A. and Chassiakos, A.. A general databased approach for developing reduced-order models of nonlinear MDOF systems, Nonlinear Dynamics, 2005, 39, pp 95–112.Google Scholar
11. Tasbihgoo, F., Carey, J. and Masri, S.. Development of data-based model-free representations of non-conservative dissipative systems, Int J of Non-Linear Mechanics, 2007, 42, pp 99–117.Google Scholar
12. Ghanem, R. and Spanos, P., Stochastic Finite Elements: A Spectral Approach, 1991, Springer.Google Scholar
13. Ibrahim, R.. Structural dynamics with parameter uncertainties, ASME Applied Mechanics Reviews, 1987, 40, pp 309–328.Google Scholar
14. Manohar, C. and Ibrahim, R.. Progress in structural dynamics with stochastic parameter variations 1987-1998, ASME Applied Mechanics Reviews, 1999, 52, pp 177–197.Google Scholar
15. Schueller, G., Bergman, L., Bucher, C., Dasgupta, G., Deotdatis, G., Ghanem, R., Grigoriu, M., Hoshiya, M., Johnson, E., Naess, N., Pradlwarter, H., Shinozuka, M., Sobszyck, K., Spanos, P., Spencer, B., Sutoh, A., Takada, T., Wedig, W., Wojtkiewicz, S., Yoshida, I., Zeldin, B. and Zhang, R.. A state-of-the-art report on computational stochastic mechanics, Probabilistic Engineering Mechanics, 1997, 12, pp 197–321.Google Scholar
16. Schueller, G.. Computational stochastic mechanics — recent advances, Computers and Structures, 2001, 79, pp 2225–2234.Google Scholar
17. Soize, C.. Maximum entropy approach for modeling random uncertainties in transient elastodynamics, J Acoustical Society of America, 2001, 109, pp 1979–1996.Google Scholar
18. Soize, C.. Nonlinear dynamical systems with nonparametric model of random uncertainties, Uncertainties in Engineering Mechanics, 2001, 1, pp 1–38.Google Scholar
19. Ghanem, R., Masri, S., Pellissetti, M. and Wolfe, R.. Identication and prediction of stochastic dynamic systems in a polynomial chaos basis, Computer Methods in Applied Mechanics and Engineering, 2005, 194, pp 1641–1654.Google Scholar
20. Desceliers, C., Ghanem, R. and Soize, C.. Maximum likelihood estimation of stochastic chaos representations from experimental data, Int J for Numerical Methods in Engineering, 2006, 66, pp 978–1001.Google Scholar
21. Arnst, M., Ghanem, R. and Soize, C.. Identication of Bayesian posteriors for coecients of chaos expansions, J Computational Physics, 2010, 229, pp 3134–3154.Google Scholar
22. Soize, C.. Construction of probability distributions in high dimension using the maximum entropy principle: applications to stochastic processes, random elds and random matrices, Int J for Numerical Methods in Engineering, 2008, 76, pp 1583–1611.Google Scholar
23. Soize, C.. Reduced models in the medium frequency range for general dissipative structural-dynamics systems, European J of Mechanics, 1998, A: Solids, 17, pp 657–685.Google Scholar
24. Sarkar, A. and Ghanem, R.. Mid-Frequency structural dynamics with parameter uncertainty, Computer Methods in Applied Mechanics and Engineering, 2002, 191, pp 5499–5513.Google Scholar
25. Shannon, C.. A mathematical theory of communication, Bell System Technical J, 1948, 27, pp 379–423.Google Scholar
26. Jaynes, E.. Information theory and statistical mechanics, Physical Review, 1957, 106, pp 620–630.Google Scholar
27. Chebli, H. and Soize, C.. Experimental validation of a nonparametric probabilistic model of nonhomogeneous uncertainties for dynamical systems, J Acoustical Society of America, 2004, 115, pp 697–705.Google Scholar
28. Arnst, M., Clouteau, D., Chebli, H., Othman, R. and Degrande, G.. A non-parametric probabilistic model for ground-borne vibrations in buildings, Probabilistic Engineering Mechanics, 2006, 21, pp 18–34.Google Scholar
29. Ciarlet, P., Introduction to Numerical Linear Algebra and Optimisation, 1989, Cambridge University Press.Google Scholar
30. Levenberg, K.. A method for the solution of certain problems in least-squares, Quarterly Applied Mathematics, 1944, 2, pp 164–168.Google Scholar
31. Powell, M.. A Fortran subroutine for solving systems of nonlinear algebraic equations, Numerical Methods for Nonlinear Algebraic Equations, 1970, Rabinowitz, P. (Ed), Chapter 7, pp 115–161, Gordon and Breach.Google Scholar
32. Coleman, T. and Li, Y.. An interior, trust region approach for nonlinear minimization subject to bounds, SIAM J on Optimization, 1996, 6, pp 418–445.Google Scholar
33. Robert, C. and Casella, G., Monte Carlo Statistical Methods, 2004, Springer.Google Scholar
34. Belytschko, T., Liu, W. and Moran, B., Nonlinear Nite Elements for Continua and Structures, 2000, Wiley.Google Scholar
35. Casella, G. and Berger, R., Statistical Inference, 2001, Duxbury Press.Google Scholar
36. Parzen, E.. On estimation of probability density function and mode, Annals of Mathematical Statistics, 1962, 33, pp 1065–1076.Google Scholar
37. Rosenblatt, M.. Remarks on some nonparametric estimates of a density function, Annals of Mathematical Statistics, 1956, 27, pp 832–837.Google Scholar
38. Dowell, E., Aeroelasticity of Plates and Shells, 1974, Noordho International.Google Scholar
39. Lindsley, N., Pettit, C. and Beran, P.. Nonlinear plate aeroelastic response with uncertainstiness and boundary conditions, Structure and Infrastructure Engineering, 2006, 2, pp 201–220.Google Scholar
40. Hajj, M. and Beran, P.. Higher-order spectral analysis of limit cycle oscillations of fighter aircraft, J Aircr, 2008, 45, pp 1917–1923.Google Scholar
41. Heroux, M., Bartlett, R., Howle, V., Hoekstra, R., Hu, J., Kolda, T., Lehoucq, R., Long, K., Pawlowski, R., Phipps, E., Salinger, A., Thornquist, H., Tuminaro, R., Willenbring, J., Williams, A. and Stanley, K.. An overview of the Trilinos project, ACM Transactions on Mathematical Software, 2005, 31, (3), pp 397–423.Google Scholar