Transient Response of Structural Dynamic Systems with Parametric Uncertainty
Publication: Journal of Engineering Mechanics
Volume 140, Issue 2
Abstract
The time-domain response of a randomly parameterized structural dynamic system is investigated with a polynomial chaos expansion approach and a stochastic Krylov subspace projection, which has been proposed here. The latter uses time-adaptive stochastic spectral functions as weighting functions of the deterministic orthogonal basis onto which the solution is projected. The spectral functions are rational functions of the input random variables and depend on the spectral properties of the unperturbed system. The stochastic system response can be accurately resolved even when using low-order spectral functions, which are computationally advantageous. The time integration required for the resolution of the transient stochastic response has been performed with the unconditionally stable single-step implicit Newmark scheme using a stochastic integration operator. A semistatistical hybrid analytical and simulation-based computational approach has been utilized to obtain the moments and probability density functions of the solution. The simulations have been performed for different degrees of variability of the input randomness and different dimensions of the input stochastic space and compared with the direct Monte-Carlo simulations for accuracy and computational efficiency.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
Abhishek Kundu acknowledges the support of Swansea University through the award of a Zienkiewicz Scholarship. Sondipon Adhikakri gratefully acknowledges the support of the Royal Society of London through the Wolfson Research Merit Award.
References
Adhikari, S. (2011). “Stochastic finite element analysis using a reduced orthonormal vector basis.” Comput. Methods Appl. Mech. Eng., 200(21–22), 1804–1821.
Babuska, I., Tempone, R., and Zouraris, G. E. (2005). “Solving elliptic boundary value problems with uncertain coefficients by the finite element method: The stochastic formulation.” Comput. Methods Appl. Mech. Eng., 194(12–16), 1251–1294.
Bathe, K. J. (1996). Finite element procedures, Prentice Hall, Englewood Cliffs, NJ.
Blatman, G., and Sudret, B. (2010). “An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis.” Probab. Eng. Mech., 25(2), 183–197.
Bobrowski, A. (2005). Functional analysis for probability and stochastic processes: An introduction, Cambridge University Press, Cambridge, U.K.
Bucher, C., and Bourgund, U. (1990). “A fast and efficient response surface approach for structural reliability problems.” Struct. Saf., 7(1), 57–66.
Caflisch, R. E. (1998). “Monte Carlo and quasi-Monte Carlo methods.” Acta Numer., 7, 1–49.
Deb, M. K., Babuska, I. M., and Oden, J. T. (2001). “Solution of stochastic partial differential equations using Galerkin finite element techniques.” Comput. Methods Appl. Mech. Eng., 190(48), 6359–6372.
Falsone, G., and Impollonia, N. (2002). “A new approach for the stochastic analysis of finite element modelled structures with uncertain parameters.” Comput. Methods Appl. Mech. Eng., 191(44), 5067–5085.
Gerritsma, M., van der Steen, J.-B., Vos, P., and Karniadakis, G. (2010). “Time-dependent generalized polynomial chaos.” J. Comput. Phys., 229(22), 8333–8363.
Ghanem, R., and Spanos, P. D. (1991). Stochastic finite elements: A spectral approach, Springer, New York.
Hahn, G. D. (1991). “A modified Euler method for dynamic analyses.” Int. J. Numer. Methods Eng., 32(5), 943–955.
Hurtado, J., and Barbat, A. (1998). “Monte Carlo techniques in computational stochastic mechanics.” Arch. Comput. Meth. Eng., 5(1), 3–29.
Ipsen, I. C. F., and Meyer, C. D. (1998). “The idea behind Krylov methods.” Am. Math. Mon., 105(10), 889–899.
Kiureghian, A. D., and Ditlevsen, O. (2009). “A leatory or epistemic? Does it matter?” Struct. Saf., 31(2), 105–112.
Kleiber, M., and Hien, T. D. (1992). The stochastic finite element method, Wiley, New York.
Kleijnen, J. (2009). “Kriging metamodeling in simulation: A review.” Eur. J. Oper. Res., 192(3), 707–716.
Kundu, A., and Adhikari, S. (2013). “A novel reduced spectral function approach for finite element analysis of stochastic dynamical systems.” Computational methods in stochastic dynamics, M. Papdrakakis, G. Stefanou, and V. Papadopoulos, eds., Vol. 2, Springer, Dordrecht, Netherlands.
Lei, Z., and Qiu, C. (2000). “Neumann dynamic stochastic finite element method of vibration for structures with stochastic parameters to random excitation.” Comp. Struct., 77(6), 651–657.
Li, C.-C., and Kiureghian, A. D. (1993). “Optimal discretization of random fields.” J. Eng. Mech., 1136–1154.
Lin, Y. K. (1967). Probabilistic theory of structural dynamics, McGraw Hill, New York.
Liu, W. K., Belytschko, T., and Mani, A. (1986). “Random field finite-elements.” Int. J. Numer. Methods Eng., 23(10), 1831–1845.
Lucor, D., Su, C.-H., and Karniadakis, G. E. (2004). “Generalized polynomial chaos and random oscillators.” Int. J. Numer. Methods Eng., 60(3), 571–596.
Matthies, H. (2007). “Uncertainty quantification with stochastic finite elements.” Encyclopedia of computational mechanics, John Wiley & Sons.
Matthies, H. G., and Keese, A. (2005). “Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations.” Comput. Methods Appl. Mech. Eng., 194(12–16), 1295–1331.
Nair, P. B. (2002). “Equivalence between the combined approximations technique and Krylov subspace methods.” AIAA J., 40(5), 1021–1023.
Nair, P. B., and Keane, A. J. (2002). “Stochastic reduced basis methods.” AIAA J., 40(8), 1653–1664.
Najm, H. N. (2009). “Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics.” Annu. Rev. Fluid Mech., 41(1), 35–52.
Newmark, N. M. (1959). “A method of computation for structural dynamics.” J. Engrg. Mech. Div., 85(7), 67–94.
Nickel, R. E. (1971). “On the stability of approximation operators in problems of structural dynamics.” Int. J. Solids Struct., 7(3), 301–319.
Nouy, A. (2007). “A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations.” Comput. Methods Appl. Mech. Eng., 196(45–48), 4521–4537.
Nouy, A. (2008). “Generalized spectral decomposition method for solving stochastic finite element equations: Invariant subspace problem and dedicated algorithms.” Comput. Methods Appl. Mech. Eng., 197(51–52), 4718–4736.
Papadrakakis, M., and Papadopoulos, V. (1996). “Robust and efficient methods for stochastic finite element analysis using Monte Carlo simulation.” Comput. Methods Appl. Mech. Eng., 134(3–4), 325–340.
Pettit, C., and Beran, P. (2006). “Spectral and multiresolution wiener expansions of oscillatory stochastic processes.” J. Sound Vibrat., 294(45), 752–779.
Powell, C., and Elman, H. (2008). “Block-diagonal preconditioning for spectral stochastic finite-element systems.” IMA J. Numer. Anal., 29(2), 350–375.
Pradlwarter, H. J., and Schuëller, G. I. (1997). “On advanced Monte Carlo simulation procedures in stochastic structural dynamics.” Int. J. Non-linear Mech., 32(4), 735–744.
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), 182–192.
Sarkar, A. and Ghanem, R. (2003). “A substructure approach for the midfrequency vibration of stochastic systems.” J. Acoust. Soc. Am., 113(4), 1922–1934.
Schuëller, G. I. (2001). “Computational stochastic mechanics—Recent advances.” Comp. Struct., 79(22–25), 2225–2234.
Schuëller, G. I., Pradlwarter, H. J., and Bucher, C. G. (1991). “Efficient computational procedures for reliability estimates of MDOF-systems.” Int. J. Non-linear Mech., 26(6), 961–974.
Shinozuka, M. (1972). “Monte Carlo solution of structural dynamics.” Comp. Struct., 2(5–6), 855–874.
Vanmarcke, E. H. (1983). Random fields, MIT Press, Cambridge, MA.
Wall, F., and Bucher, C. (1987). “Sensitivity of expected exceedance rate of SDOF-system response to statistical uncertainties of loading and system parameters.” Probab. Eng. Mech., 2(3), 138–146.
Wan, X. L. and Karniadakis, G. E. (2006). “Beyond Wiener-Askey expansions: Handling arbitrary PDFs.” J. Sci. Comput., 27(3), 455–464.
Xiu, D., and Karniadakis, G. E. (2002). “The Wiener-Askey polynomial chaos for stochastic differential equations.” SIAM J. Sci. Comput., 24(2), 619–644.
Xiu, D., and Karniadakis, G. E. (2003a). “A new stochastic approach to transient heat conduction modeling with uncertainty.” Int. J. Heat Mass Transfer, 46(24), 4681–4693.
Xiu, D., and Karniadakis, G. E. (2003b). “Modeling uncertainty in flow simulations via generalized polynomial chaos.” J. Comput. Phys., 187(1), 137–167.
Yamazaki, F., Member, A., Shinozuka, M., and Dasgupta, G. (1988). “Neumann expansion for stochastic finite element analysis.” J. Eng. Mech., 1335–1354.
Yamazaki, F., and Shinozuka, M. (1988). “Digital generation of non-Gaussian stochastic fields.” J. Eng. Mech., 1183–1197.
Zhu, W. Q., Ren, Y. J., and Wu, W. Q. (1992). “Stochastic FEM based on local averages of random vector fields.” J. Eng. Mech., 496–511.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 140 • Issue 2 • February 2014
Pages: 315 - 331
Copyright
© 2014 American Society of Civil Engineers.
History
Received: Aug 6, 2012
Accepted: Mar 22, 2013
Published online: Apr 1, 2013
Published in print: Feb 1, 2014
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.