Polynomial Correlated Function Expansion for Nonlinear Stochastic Dynamic Analysis
Publication: Journal of Engineering Mechanics
Volume 141, Issue 3
Abstract
A new computational tool, referred to as the polynomial correlated function expansion (PCFE), has been developed for predicting response statistics of nonlinear stochastic dynamic problems. This method facilitates a systematic mapping between the input and output by expressing the output as a ranked order of component functions, with higher-order component functions representing higher-order cooperative effects. The component functions are expressed in terms of extended bases, and the unknown coefficients associated with the basis are determined by using a homotopy algorithm. This algorithm considers the hierarchical orthogonality of the component functions as a constraint and determines the coefficients associated with the basis. Implementation of the proposed approach for nonlinear stochastic dynamic problems has been demonstrated with three numerical problems. Results obtained certify the accuracy of the proposed method. Furthermore, the proposed approach significantly reduces the number of calls to the partial differential equation solver, as observed in the numerical problems.
Get full access to this article
View all available purchase options and get full access to this article.
References
Alibrandi, U. (2011). “A response surface method for nonlinear stochastic dynamic analysis.” Proc., 11th Int. Conf. on Applications of Statistics and Probability in Civil Engineering, ETH Zurich, Switzerland, 1–7.
Alibrandi, U., and Der Kiureghian, A. (2012). “A gradient-free method for determining the design point in nonlinear stochastic dynamic analysis.” Probab. Eng. Mech., 28(Apr), 2–10.
Alibrandi, U., Di Paola, M., and Ricciardi, G. (2007). “Path integral solution solved by the kernel density maximum entropy approach.” Proc., Int. Symp. on Recent Advances in Mechanics, Dynamical Systems and Probability Theory, Patron Editore, Granarolo, Italy.
Alış, Ö. F., and Rabitz, H. (2001). “Efficient implementation of high dimensional model representations.” J. Math. Chem., 29(2), 127–142.
Beer, M., and Spanos, P. D. (2009). “A neural network approach for simulating stationary stochastic processes.” Struct. Eng. Mech., 32(1), 71–94.
Bollig, E. F., Flyer, N., and Erlebacher, G. (2012). “Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs.” J. Comput. Phys., 231(21), 7133–7151.
Bratley, P., and Fox, B. L. (1988). “Algorithm 659: Implementing Sobol’s quasirandom sequence generator.” ACM Trans. Math. Softw., 14(1), 88–100.
Cai, G. Q., and Lin, Y. K. (1988). “A new approximate solution technique for randomly excited non-linear oscillators.” Int. J. Non Linear Mech., 23(5–6), 409–420.
Caughey, T. K. (1971). “Nonlinear theory of random vibrations.” Advances in applied mechanics, C.-S. Yih, ed., Vol. 11, Academic Press, New York, 209–253.
Caughey, T. K. (1986). “On the response of non-linear oscillators to stochastic excitation.” Probab. Eng. Mech., 1(1), 2–4.
Chu, C.-T. (1985). “Random vibration of non-linear building-foundation systems.” Ph.D. thesis, Univ. of Illinois at Urbana–Champaign, Urbana, IL.
De Marchi, S., and Santin, G. (2013). “A new stable basis for radial basis function interpolation.” J. Comput. Appl. Math., 253(Dec), 1–13.
Der Kiureghian, A. (2000). “The geometry of random vibrations and solutions by FORM and SORM.” Probab. Eng. Mech., 15(1), 81–90.
DiazDelaO, F. A., Adhikari, S., Flores, E. I. S., and Friswell, M. I. (2013). “Stochastic structural dynamic analysis using Bayesian emulators.” Comput. Struct., 120(Apr), 24–32.
Di Paola, M., and Vasta, M. (1997). “Stochastic integro-differential and differential equations of non-linear systems excited by parametric Poisson pulses.” Int. J. Non Linear Mech., 32(5), 855–862.
Echard, B., Gayton, N., Lemaire, M., and Relun, N. (2013). “A combined importance sampling and kriging reliability method for small failure probabilities with time-demanding numerical models.” Reliab. Eng. Syst. Saf., 111(Mar), 232–240.
Faure, H. (1992). “Good permutations for extreme discrepancy.” J. Number Theory, 42(1), 47–56.
Fujimura, K., and Der Kiureghian, A. (2007). “Tail-equivalent linearization method for nonlinear random vibration.” Probab. Eng. Mech., 22(1), 63–76.
Galanti, S., and Jung, A. (1997). “Low-discrepancy sequences: Monte Carlo simulation of option prices.” J. Derivatives, 5(1), 63–83.
Grigoriu, M. (1995). Applied non-Gaussian processes: Examples, theory, simulation, linear random vibration, and MATLAB solutions, Prentice Hall, Englewood Cliffs, NJ.
Halton, J. H. (1960). “On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals.” Numer. Math., 2(1), 84–90.
Jamshidi, A. A., and Kirby, M. J. (2010). “Skew-radial basis function expansions for empirical modeling.” SIAM J. Sci. Comput., 31(6), 4715–4743.
Kliemann, W., and Namachchivaya, N. S., eds. (1995). Nonlinear dynamics and stochastic mechanics, CRC Press, Boca Raton, FL.
Kougioumtzoglou, I. A., and Spanos, P. D. (2012). “An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillators.” Probab. Eng. Mech., 28(Apr), 125–131.
Kougioumtzoglou, I. A., and Spanos, P. D. (2014). “Nonstationary stochastic response determination of nonlinear systems: A Wiener path integral formalism.” J. Eng. Mech., 04014064.
Kumar, P., and Narayanan, S. (2006). “Solution of Fokker–Planck equation by finite element and finite difference methods for nonlinear systems.” Sadhana, 31(4), 445–461.
Kumar, P., and Narayanan, S. (2009). “Numerical solution of multidimensional Fokker–Planck equation for nonlinear stochastic dynamical systems.” Adv. Vib. Eng., 8(2), 153–163.
Kumar, P., and Narayanan, S. (2010). “Modified path integral solution of Fokker–Planck equation: Response and bifurcation of nonlinear systems.” J. Comput. Nonlinear Dynam., 5(1), 011004.
Li, G., Hu, J., Wang, S.-W., Georgopoulos, P. G., Schoendorf, J., and Rabitz, H. (2006). “Random sampling-high dimensional model representation (RS-HDMR) and orthogonality of its different order component functions.” J. Phys. Chem. A, 110(7), 2474–2485.
Li, G., and Rabitz, H. (2010). “D-MORPH regression: Application to modeling with unknown parameters more than observation data.” J. Math. Chem., 48(4), 1010–1035.
Li, G., Rey-de-Castro, R., and Rabitz, H. (2012). “D-MORPH regression for modeling with fewer unknown parameters than observation data.” J. Math. Chem., 50(7), 1747–1764.
Lutes, L. D. (1986). “Cumulants of stochastic response for linear systems.” J. Eng. Mech., 1062–1075.
Lutes, L. D., and Sarkani, S. (2004). Random vibrations: Analysis of structural and mechanical systems, Elsevier, Burlington, MA.
Muscolino, G., Ricciardi, G., and Cacciola, P. (2003). “Monte Carlo simulation in the stochastic analysis of non-linear systems under external stationary Poisson white noise input.” Int. J. Non Linear Mech., 38(8), 1269–1283.
Muscolino, G., Ricciardi, G., and Vasta, M. (1997). “Stationary and non-stationary probability density function for non-linear oscillators.” Int. J. Non Linear Mech., 32(6), 1051–1064.
Narayanan, S., and Kumar, P. (2012). “Numerical solutions of Fokker–Planck equation of nonlinear systems subjected to random and harmonic excitations.” Probab. Eng. Mech., 27(1), 35–46.
Newland, D. E. (1996). An introduction to random vibrations, spectral and wavelet analysis, 3rd Ed., Prentice Hall, London.
Ng, S. H., and Yin, J. (2012). “Bayesian kriging analysis and design for stochastic simulations.” ACM Trans. Model. Comput. Simul., 22(3), 17:1–17:26.
Nigam, N. C. (1983). Introduction to random vibrations, MIT Press, Cambridge, MA.
Papadimitriou, C. (1995). “Stochastic response cumulants of MDOF linear systems.” J. Eng. Mech., 1181–1192.
Rabitz, H., and Aliş, Ö. (1999). “General foundations of high-dimensional model representations.” J. Math. Chem., 25(2–3), 197–233.
Rao, C. R., and Mitra, S. K. (1971). “Generalized inverse of matrix and its applications.” Proc., 6th Berkeley Symp. on Mathematical Statistics and Probability, University of California Press, Berkeley, CA.
Roberts, J. B., and Spanos, P. D. (1990). Random vibration and statistical linearization, Wiley, New York.
Rothman, A., Ho, T.-S., and Rabitz, H. (2005a). “Observable-preserving control of quantum dynamics over a family of related systems.” Phys. Rev. A, 72(2), 023416.
Rothman, A., Ho, T.-S., and Rabitz, H. (2005b). “Quantum observable homotopy tracking control.” J. Chem. Phys., 123(13), 134104.
Rubinstein, R. Y. (1981). Simulation and the Monte Carlo method, Wiley, New York.
Sobezyk, K., and Trȩbicki, J. (1990). “Maximum entropy principle in stochastic dynamics.” Probab. Eng. Mech., 5(3), 102–110.
Sobol, I. (1993). “Sensitivity estimates for nonlinear mathematical models.” Math. Model. Comput. Exp., 1(4), 407–414.
Sobol, I. M. (1976). “Uniformly distributed sequences with an additional uniform property.” USSR Comput. Math. Math. Phys., 16(5), 236–242.
To, C. W. S. (2013). Stochastic structural dynamics: Application of finite element methods, Wiley, New York.
Wang, S.-W., Georgopoulos, P. G., Li, G., and Rabitz, H. (2003). “Random sampling-high dimensional model representation (RS-HDMR) with nonuniformly distributed variables: Application to an integrated multimedia/multipathway exposure and dose model for trichloroethylene.” J. Phys. Chem. A, 107(23), 4707–4718.
Yong, Y., and Lin, Y. K. (1987). “Exact stationary-response solution for second order nonlinear systems under random parametric and external white-noise excitation.” J. Appl. Mech., 54(2), 414–418.
Zeng, Y., and Li, G. (2013). “Stationary response of bilinear hysteretic system driven by Poisson white noise.” Probab. Eng. Mech., 33(Jul), 135–143.
Zhao, W., Liu, J. K., Li, X. Y., Yang, Q. W., and Chen, Y. Y. (2013). “A moving kriging interpolation response surface method for structural reliability analysis.” Comput. Model. Eng. Sci., 93(6), 469–488.
Zhu, W.-Q., and Yu, J.-S. (1989). “The equivalent non-linear system method.” J. Sound Vib., 129(3), 385–395.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 141 • Issue 3 • March 2015
Copyright
© 2014 American Society of Civil Engineers.
History
Received: Mar 15, 2014
Accepted: Jul 28, 2014
Published online: Aug 29, 2014
Published in print: Mar 1, 2015
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.