G. Q. Cai¹, R. H. Huan² and W. Q. Zhu^2
1Department of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, Florida, USA
²Department of Mechanics, State Key Laboratory of Fluid Power and Mechatronic and Control, Zhejiang University, Hangzhou, Zhejiang, China

Part of the book: Stochastic Processes: Fundamentals and Emerging Applications

Chapter DOI: https://doi.org/10.52305/KMDP8644

Abstract

For the purposes of analysis, numerical calculation and/or simulation, stochastic processes and their samples are required to be properly modeled and generated mathematically. Various methods have been developed to model stochastic processes according to their probabilistic and statistical properties. The present chapter is a review of these methods mostly based on the authors’ work in recent years. For a stochastic process, the two most important properties are its probability density function and power spectral density function. For a Gaussian process, four methods have been developed, the method of linear filters, the series expansions with random amplitudes, random phases, and both random amplitudes and random phases. To generate a non-Gaussian process defined in infinite range, semi-infinite range, or finite interval, two models are proposed, the nonlinear filter model and the randomized harmonic model. In some problems, two or more stochastic processes are involved, and they are correlated. Thus, feasible methods to model and generate correlated processes are needed. Two correlated stationary Gaussian processes are generated using the three methods, the method of linear filters, the method of series expansion with random amplitudes, and the method of series expansion with random phases. All three methods intend to match the power spectral density for each process, but use information of different levels of correlation. In the chapter, the systematic procedures are presented for these methods, and their application scopes and relative advantages are pointed out. Illustration examples are given to show the feasibility and accuracy of these methods.

Keywords: stochastic processes, Gaussian and non-Gaussian distributions, correlated processes, modeling, simulation

References

[1] Cai, G. Q. (2018). Generation of correlated random variables and stochastic
processes. Probabilistic Engineering Mechanics, 52, 40-46.
[2] Cai, G. Q., Huan, R., and Zhu, W. Q. (2021). Generation of two correlated stationary
Gaussian Processes. Mathematics, 9, 2687, https://doi.org/10.3390/ math9212687.
[3] Cai, G. Q. and Lin, Y. K. (1996). Generation of non-Gaussian stationary stochastic
processes. Physical Review E, 54, 299-303.
[4] Cai, G. Q. and Lin, Y. K. (1997a). Response spectral densities of strongly nonlinear
systems under random excitation. Probabilistic Engineering Mechanics, 12, 41-47.
[5] Cai, G. Q. and Lin, Y. K. (1997b). Reliability of dynamical systems under non-Gaussian random excitations. In Shirarishi N. et al. edItôrs. Structural Safety and
Reliability. Proceedings of the 7th International Conference on Structural Safety and
Reliability, Japan: Kyoto, 819-826.
[6] Cai, G. Q., Lin, Y. K., and Xu, W. (1998). Response and reliability of nonlinear
systems under stationary non-Gaussian excitations. In: Spencer BF, Johnson EA,
edItôrs. Structural Stochastic Dynamics. Proceedings of the 4th International
Conference on Structural Stochastic Dynamics, Indiana: Notre Dame, 17-22.
[7] Cai, G. Q. and Wu, C. (2004). Modeling of bounded stochastic processes.
Probabilistic Engineering Mechanics, 19, 197-203.
[8] Cai, G. Q., Yu, J. S., and Lin, Y. K. (1995). Toppling of a rigid block under
evolutionary random base excitations. Journal of Engineering Mechanics, 121, 924-
929.
[9] Cai, G. Q. and Zhu, W. Q. (2016). Elements of Stochastic Dynamics. World
Scientific Publisher, Singapore.
[10] Cai, G. Q. and Zhu, W. Q. (2019). Generation of two correlated stationary Gaussian
processes and application to ship rolling motion. Probabilistic Engineering
Mechanics, 57, 26-31.
[11] Chai, W., Naess, A., and Leira, B. J. (2015). Filter models for prediction of stochastic
ship roll response. Probabilistic Engineering Mechanics, 41, 104-114.
[12] Chai, W., Naess, A., and Leira, B. J. (2016). Stochastic nonlinear ship rolling in
random beam seas by path integration method. Probabilistic Engineering Mechanics,
44, 43-52.
[13] Dalzell, J. F. (1971). A Study of the distribution of maxima of non-Linear ship rolling
in a seaway. Report DL-71-1562, Stevens Institute of Technology, Hoboken, NJ.
[14] Dalzell, J. F. (1973). A Note on the distribution of maxima of ship rolling. Journal
of Ship Research, 17, 217-226.
[15] Deodatis, G. (1996). Simulation of ergodic multivariate stochastic processes, 122,
778-787.
[16] Deodatis, G. and Micaletti, R. C. (2001). Simulation of highly skewed non-Gaussian
stochastic processes. Journal of Engineering Mechanics, 127, 1284-1295.
[17] Dimentberg, M. F. (1988). Statistical Dynamics of Nonlinear and Time-Varying
Systems. Wiley, New York.
[18] Dimentberg, M. F. (1991). A stochastic model of parametric excitation of a straight
pipe due to slug flow of a two-phase fluid. Proceedings of the 5th International
Conference on Flow-Induced Vibrations. Brighton, UK, 207-209.
[19] Dimentberg, M. F. (1992). Stability and subcritical dynamics of structures with
spatially disordered parametric excitation. Probabilistic Engineering Mechanics, 7,
131-134.
[20] Faltinsen, O. M. (1993). Sea Loads on Ships and Offshore Structures. Cambridge
University Press, Oxford.
[21] Grigoriu, M. (1998). Simulation of stationary non-Gaussian translation processes.
Journal of Engineering Mechanics, 124, 121-126.
[22] Hsieh, S. R., Troesch, A. W., and Shaw, S. W. (1994). A nonlinear probabilistic
method for predicting vessel capsizing in random beam seas. Proceedings of Royal
Society A of London, 446, 195-211.
[23] Itô, K. (1951a). On stochastic differential equations. Memoirs American
Mathematical Society, 4, 289-302.
[24] Itô, K. (1951b). A formula concerning stochastic differentials. Nagoya Mathematical
Journal, 3, 55-65.
[25] Kougioumtzoglou, I. A. and Spanos, P. D. (2014). Spanos, Stochastic response
analysis of the softening Duffing oscillator and ship capsizing probability
determination via a numerical path integral approach. Probabilistic Engineering
Mechanics, 35, 67-74.
[26] Li, Q. C. and Lin, Y. K. (1995). New stochastic theory for bridge stability in turbulent
flow, II. Journal of Engineering Mechanics, 121, 102-116.
[27] Li, Y. and Kareem, A. (1993). Simulation of multi-variate random processes, hybrid
DFT and digital filtering approach. Journal of Engineering Mechanics, 119, 1078-
1098.
[28] Lin, Y. K. and Cai, G. Q. (1995). Probabilistic Structural Dynamics, Advanced
Theory and Applications, McGraw-Hill, New York.
[29] Mignolet, M. P. and Spanos, P. D. (1877a). Recursive simulation of stationary multivariate random processes, Part I. Journal of Applied Mechanics, 54, 674-680.
[30] Mignolet, M. P. and Spanos, P. D. (1877b). Recursive simulation of stationary multivariate random processes, Part II. Journal of Applied Mechanics, 54, 681-687.
[31] Muhuri, P. K. (1980). A study of the stability of the rolling motion of a ship in an
irregular seaway. International Shipbuilding Progress, 27, 139-142.
[32] Paroka, D., Ohkura, Y., and Umeda, N. (2006). Analytical prediction of capsizing
probability of a ship in beam wind and waves. Journal of Ship Research, 50, 187-
195.
[33] Pierson, W. J. and Moskowitz, L. A. (1964). A proposal spectral form for fully
developed wind seas based on the similarity theory of S. A. Kitaigorodskii. Journal
of Geophysical Research, 69, 5181-5190.
[34] Ramadan, O. and Novak, M. (1993). Simulation of spatially incoherent random
ground motions. Journal of Engineering Mechanics, 119, 997-1016.
[35] Roberts, J. B. (1982). A stochastic theory for nonlinear ship rolling in irregular seas.
Journal of Ship Research, 26, 229-245.
[36] Shinozuka, M. and Deodatis, G. (1991). Simulation of stochastic processes by
spectral representation. Applied Mechanics Review, 44, 191-204.
[37] Shinozuka, M. and Jan, C. M. (1972). Digital simulation of random processes and its
applications. Journal of Sound and Vibration, 25, 111-128.
[38] Wedig, W. V. (1989). Analysis and simulation of nonlinear stochastic systems. In:
Schiehlen W, edItôr. Nonlinear dynamics in engineering systems. Berlin: SpringerVerlag; 337-344.
[39] Winterstein, S. R. (1988). Nonlinear vibration models for extremes and fatigue.
Journal of Engineering Mechanics, 114, 1772-1790.
[40] Wong, E. and Zakai, M. (1965). On the relation between ordinary and stochastic
equations. International Journal of Engineering Sciences, 3, 213-229.
[41] Wu, C. and Cai, G. Q. (2004). Effects of excitation probability distribution on system
responses. International Journal of Non-Linear Mechanics, 39, 1463-1472.
[42] Zhu, W. Q. and Cai, G. Q. (2013). On bounded stochastic processes. In Bounded
Noises in Physics, Biology and Engineering, Series: Modeling and Simulation in
Science, Engineering and Technology, ed. A. d’Onofrio, Springer Science+Business
Media, New York, 3-24.
[43] Zhu, W. Q. and Cai, G. Q. (2014). Generation of non-Gaussian stochastic processes
using nonlinear filters. Probabilistic Engineering Mechanics, 36, 56-62