Wishart Random Matrices in Probabilistic Structural Mechanics
Publication: Journal of Engineering Mechanics
Volume 134, Issue 12
Abstract
Uncertainties need to be taken into account for credible predictions of the dynamic response of complex structural systems in the high and medium frequency ranges of vibration. Such uncertainties should include uncertainties in the system parameters and those arising due to the modeling of a complex system. For most practical systems, the detailed and complete information regarding these two types of uncertainties is not available. In this paper, the Wishart random matrix model is proposed to quantify the total uncertainty in the mass, stiffness, and damping matrices when such detailed information regarding uncertainty is unavailable. Using two approaches, namely, (a) the maximum entropy approach; and (b) a matrix factorization approach, it is shown that the Wishart random matrix model is the simplest possible random matrix model for uncertainty quantification in discrete linear dynamical systems. Four possible approaches for identifying the parameters of the Wishart distribution are proposed and compared. It is shown that out of the four parameter choices, the best approach is when the mean of the inverse of the random matrices is same as the inverse of the mean of the corresponding matrix. A simple simulation algorithm is developed to implement the Wishart random matrix model in conjunction with the conventional finite-element method. The method is applied vibration of a cantilever plate with two different types of uncertainties across the frequency range. Statistics of dynamic responses obtained using the suggested Wishart random matrix model agree well with the results obtained from the direct Monte Carlo simulation.
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Acknowledgments
The writer acknowledges the financial support from the Royal Society, London and Engineering and Physical Sciences Research Council (U.K.) (EPSRC)EPSRC-GB through the award of an advanced research fellowship.
References
Adhikari, S. (2007a). “Matrix variate distributions for probabilistic structural mechanics.” AIAA J., 45(7), 1748–1762.
Adhikari, S. (2007b). “On the quantification of damping model uncertainty.” J. Sound Vib., 305(1–2), 153–171.
Adhikari, S., and Manohar, C. S. (1999). “Dynamic analysis of framed structures with statistical uncertainties.” Int. J. Numer. Methods Eng., 44(8), 1157–1178.
Adhikari, S., and Manohar, C. S. (2000). “Transient dynamics of stochastically parametered beams.” J. Eng. Mech., 126(11), 1131–1140.
Cotoni, V., Shorter, P., and Langley, R. S. (2007). “Numerical and experimental validation of a hybrid finite element-statistical energy analysis method.” J. Acoust. Soc. Am., 122(1), 259–270.
Eaton, M. L. (1983). Multivariate statistics: A vector space approach, Wiley, New York.
Elishakoff, I., and Ren, Y. J. (2003). Large variation finite element method for stochastic problems, Oxford Univ. Press, Oxford, U.K.
Ghanem, R., and Spanos, P. (1991). Stochastic finite elements: A spectral approach, Springer, New York.
Girko, V. L. (1990). Theory of random determinants, Kluwer, The Netherlands.
Graham, A. (1981). Kronecker products and matrix calculus with applications, mathematics and its applications, Ellis Horwood Limited, Chichester, U.K.
Gupta, A., and Nagar, D. (2000). Matrix variate distributions, monographs and surveys in pure and applied mathematics, Chapman and Hall/CRC, London.
Haldar, A., and Mahadevan, S. (2000). Reliability assessment using stochastic finite element analysis, Wiley, New York.
Horn, R. A., and Johnson, C. R. (1985). Matrix analysis, Cambridge Univ. Press, Cambridge, U.K.
Janik, R. A., and Nowak, M. A. (2003). “Wishart and anti-wishart random matrices.” J. Phys. A, 36(2), 3629–3637.
Johnson, N. L., Kotz, S., and Balakrishnan, N. (1994). Continuous univariate distributions, Vol. 1, Wiley Series in Probability and Mathematical Statistics, Wiley, New York.
Kleiber, M., and Hien, T. D. (1992). The stochastic finite element method, Wiley, Chichester, U.K.
Langley, R. S., and Bremner, P. (1999). “A hybrid method for the vibration analysis of complex structural-acoustic systems.” J. Acoust. Soc. Am., 105(3), 1657–1671.
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), 3445–3463.
Lyon, R. H., and Dejong, R. G. (1995). Theory and application of statistical energy analysis, 2nd Ed., Butterworth-Heinmann, Boston.
Manohar, C. S., and Adhikari, S. (1998a). “Dynamic stiffness of randomly parametered beams.” Probab. Eng. Mech., 13(1), 39–51.
Manohar, C. S., and Adhikari, S. (1998b). “Statistical analysis of vibration energy flow in randomly parametered trusses.” J. Sound Vib., 217(1), 43–74.
Mathai, A. M. (1997). Jocobians of matrix transformation and functions of matrix arguments, World Scientific, London.
Mathai, A. M., and Provost, S. B. (1992). Quadratic forms in random variables: Theory and applications, Marcel Dekker, New York.
Matthies, H. G., Brenner, C. E., Bucher, C. G., and Soares, C. G. (1997). “Uncertainties in probabilistic numerical analysis of structures and solids—Stochastic finite elements.” Struct. Safety, 19(3), 283–336.
Mehta, M. L. (1991). Random matrices, 2nd Ed., Academic Press, San Diego.
Mezzadri, F., and Snaith, N. C., eds. (2005). Recent perspectives in random matrix theory and number theory, London Mathematical Society Lecture Note, Cambridge Univ. Press, Cambridge, U.K.
Muirhead, R. J. (1982). Aspects of multivariate statistical theory, Wiley, New York.
Nair, P. B., and Keane, A. J. (2002). “Stochastic reduced basis methods.” AIAA J., 40(8), 1653–1664.
Sachdeva, S. K., Nair, P. B., and Keane, A. J. (2006a). “Comparative study of projection schemes for stochastic finite element analysis.” Comput. Methods Appl. Mech. Eng., 195(19–22), 2371–2392.
Sachdeva, S. K., Nair, P. B., and Keane, A. J. (2006b). “Hybridization of stochastic reduced basis methods with polynomial chaos expansions.” Probab. Eng. Mech., 21(2), 182–192.
Sarkar, A., and Ghanem, R. (2002). “Mid-frequency structural dynamics with parameter uncertainty.” Comput. Methods Appl. Mech. Eng., 191(47–48), 5499–5513.
Sarkar, A., and Ghanem, R. (2003a). “Reduced models for the medium-frequency dynamics of stochastic systems.” J. Acoust. Soc. Am., 113(2), 834–846.
Sarkar, A., and Ghanem, R. (2003b). “A substructure approach for the midfrequency vibration of stochastic systems.” J. Acoust. Soc. Am., 113(4), 1922–1934.
Shinozuka, M., and Yamazaki, F. (1998). “Stochastic finite element analysis: An introduction.” Stochastic structural dynamics: Progress in theory and applications, S. T. Ariaratnam, G. I. Schuëller, and I. Elishakoff, eds., Elsevier Applied Science, London.
Soize, C. (2000). “A nonparametric model of random uncertainties for reduced matrix models in structural dynamics.” Probab. Eng. Mech., 15(3), 277–294.
Soize, C. (2001). “Maximum entropy approach for modeling random uncertainties in transient elastodynamics.” J. Acoust. Soc. Am., 109(5), 1979–1996.
Sudret, B., and Der-Kiureghian, A. (2000). “Stochastic finite element methods and reliability,” Rep. No. UCB/SEMM-2000/08, Dept. of Civil and Environmental Engineering, Univ. Of California, Berkeley, Berkeley, Calif.
Tulino, A. M., and Verdú, S. (2004). Random matrix theory and wireless communications, Now Publishers Inc., Hanover, Mass.
Wigner, E. P. (1958). “On the distribution of the roots of certain symmetric matrices.” Ann. Math., 67(2), 325–327.
Wishart, J. (1928). “The generalized product moment distribution in samples from a normal multivariate population.” Biometrika, 20(A), 32–52.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 134 • Issue 12 • December 2008
Pages: 1029 - 1044
Copyright
© 2008 ASCE.
History
Received: Dec 18, 2006
Accepted: Nov 28, 2007
Published online: Dec 1, 2008
Published in print: Dec 2008
Notes
Note. Associate Editor: Arvid Naess
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