Simulation of Multivariate Nonstationary Random Processes: Hybrid Stochastic Wave and Proper Orthogonal Decomposition Approach
Publication: Journal of Engineering Mechanics
Volume 143, Issue 9
Abstract
The classical spectral representation method for simulation of multivariate nonstationary Gaussian random processes may be less efficient due to difficulty in a straightforward application of the fast Fourier transform (FFT). Although several attempts have been made to invoke the FFT, there remains the need to further improve the efficiency for cases where a large number of points need to be simulated. In this paper, a stochastic wave-based simulation scheme for the multivariate nonstationary random process along a straight line is introduced in conjunction with either a direct summation of cosine functions or the application of a two-dimensional (2D) FFT. Central to the proposed schemes is the transformation of the simulation of a multivariate nonstationary random process to that of a nonstationary one-dimensional stochastic wave. The stochastic wave can be simulated based on a direct summation of cosine functions or by invocation of a 2D FFT. The latter requires that the points to be simulated are evenly distributed. To implement FFT, proper orthogonal decomposition (POD) is employed to factorize the time-dependent and space-dependent 2D decomposed evolutionary power spectral density of the converted stochastic wave. The proposed hybrid approach of a stochastic wave and POD is quite general, and can be used to simulate nonstationary multivariate random processes with complex coherence functions. Numerical examples show that the proposed hybrid approach is very efficient in comparison with existing approaches when the number of simulation locations is large, and it offers a desired level of simulation accuracy when appropriate discretization parameters are selected.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The support by the Doctoral Innovation Fund of Southwest Jiaotong University and the National Natural Science Foundation of China (Grant Nos. 51578471 and 51478401) are greatly acknowledged. The fourth author acknowledges the support by NSF (CMMI 1301008).
References
Benowitz, B. A., and Deodatis, G. (2015). “Simulation of wind velocities on long span structures: A novel stochastic wave based model.” J. Wind. Eng. Ind. Aerodyn., 147, 154–163.
Benowitz, B. A., Shields, M. D., and Deodatis, G. (2015). “Determining evolutionary spectra from non-stationary autocorrelation functions.” Prob. Eng. Mech., 41, 73–88.
Cacciola, P., and Deodatis, G. (2011). “A method for generating fully non-stationary and spectrum-compatible ground motion vector processes.” Soil. Dyn. Earthq. Eng., 31(3), 351–360.
Chen, J., Hui, M. C. H, and Xu, Y. L. (2007). “A comparative study of stationary and non-stationary wind models using field measurements.” Bound. Lay Meteor., 122(1), 105–121.
Chen, X. (2008). “Analysis of alongwind tall building response to transient nonstationary winds.” J. Struct. Eng., 782–791.
Chen, X., and Kareem, A. (2005). “Proper orthogonal decomposition-based modeling, analysis, and simulation of dynamic wind load effects on structures.” J. Eng. Mech., 325–339.
Conte, J. P., and Peng, B. F. (1997). “Fully nonstationary analytical earthquake ground-motion model.” J. Eng. Mech., 15–24.
Davenport, A. G. (1961). “The spectrum of horizontal gustiness near the ground in high winds.” Q. J. R. Meteorol. Soc., 87(372), 194–211.
Deodatis, G. (1996). “Non-stationary stochastic vector processes: Seismic ground motion applications.” Prob. Eng. Mech., 11(3), 149–167.
Deodatis, G., and Shinozuka, M (1988). “Auto-regressive models for non-stationary stochastic processes.” J. Eng. Mech., 1995–2012.
Deodatis, G., and Shinozuka, M. (1989). “Simulation of seismic ground motion using stochastic waves.” J. Eng. Mech., 2723–2737.
Di Paola, M., and Gullo, I. (2001). “Digital generation of multivariate wind field processes.” Prob. Eng. Mech., 16(1), 1–10.
Grigoriu, M. (1993). “Simulation of nonstationary Gaussian processes by random trigonometric polynomials.” J. Eng. Mech., 328–343.
Gurley, K., and Kareem, A. (1999). “Applications of wavelet transforms in earthquake, wind and ocean engineering.” Eng. Struct., 21(2), 149–167.
Harichandran, R., and Vanmarcke, E. (1986). “Stochastic variation of earthquake ground motion in space and time.” J. Eng. Mech., 154–174.
Hu, L., Li, L., and Gu, M. (2010). “Error assessment for spectral representation method in wind velocity field simulation.” J. Eng. Mech., 1090–1104.
Huang, G. (2014). “An efficient simulation approach for multivariate random process: Hybrid of wavelet and spectral representation method.” Prob. Eng. Mech., 37(10), 74–83.
Huang, G. (2015). “Application of proper orthogonal decomposition in fast Fourier transform-assisted multivariate nonstationary process simulation.” J. Eng. Mech., .
Huang, G., and Chen, X. (2009). “Wavelets-based estimation of multivariate evolutionary spectra and its application to nonstationary downburst winds.” Eng. Struct., 31(4), 976–989.
Huang, G., Liao, H., and Li, M. (2013). “New formulation of Cholesky decomposition and applications in stochastic simulation.” Prob. Eng. Mech., 34, 40–47.
Huang, G., Zheng, H., Xu, Y. L., and Li, Y. (2015). “Spectrum models for nonstationary extreme winds.” J. Struct Eng., .
International Electrotechnical Commission. (2005). “Wind Turbines. Part 1: Design Requirement.” IEC 61400-1, HIS, Geneva.
Kareem, A. (1987). “Wind effects on structures: A probabilistic viewpoint.” Prob. Eng. Mech., 2(4), 166–200.
Li, Y., and Kareem, A. (1991). “Simulation of multivariate nonstationary random processes by FFT.” J. Eng. Mech., 1037–1058.
Li, Y., and Kareem, A. (1993). “Simulation of multivariate random processes: Hybrid DFT and digital filtering approach.” J. Eng. Mech., 1078–1098.
Li, Y., and Kareem, A. (1995). “Stochastic decomposition and application to probabilistic dynamics.” J. Eng. Mech., 162–174.
Li, Y., and Kareem, A. (1997). “Simulation of multivariate non-stationary random processes: Hybrid DFT and digital filtering approach.” J. Eng. Mech., 1302–1310.
Liu, Z., Liu, W., and Peng, Y. (2016). “Random function based spectral representation of stationary and non-stationary stochastic processes.” Prob. Eng. Mech., 45, 115–126.
MATLAB 7.14.0.739 [Computer software]. MathWorks, Natick, MA.
Peng, L., Huang, G., Kareem, A., and Li, Y. (2016). “An efficient space-time based simulation approach of wind velocity field with embedded conditional interpolation for unevenly spaced locations.” Prob. Eng. Mech., 43, 156–168.
Priestley, M. B. (1965). “Evolutionary spectra and non-stationary processes.” J. R. Stat. Soc., 27(2), 204–237.
Shinozuka, M., and Deodatis, G. (1991a). “Simulation of stochastic processes by spectral representation.” Appl. Mech. Rev., 44(4), 191–204.
Shinozuka, M., and Deodatis, G. (1991b). “Stochastic wave models for stationary and homogeneous seismic ground motion.” Struct. Safety., 10(1), 235–246.
Shinozuka, M., Yun, C. B., and Seya, H. (1990). “Stochastic methods in wind engineering.” J. Wind. Eng. Ind. Aerodyn., 36(90), 829–843.
Solari, G., and Tubino, F. (2002). “A turbulence model based on principal components.” Prob. Eng. Mech., 17(4), 327–335.
Spanos, P. D., and Failla, G. (2004). “Evolutionary spectra estimation using wavelets.” J. Eng. Mech., 952–960.
Spanos, P. D., and Kougioumtzoglou, I. A. (2012). “Harmonic wavelets based statistical linearization for response evolutionary power spectrum determination.” Prob. Eng. Mech., 27(1), 57–68.
Tubino, F., and Solari, G. (2005). “Double proper orthogonal decomposition for representing and simulating turbulence fields.” J. Eng. Mech., 1302–1312.
Xu, Y. L., Hu, L., and Kareem, A. (2014). “Conditional simulation of non-stationary fluctuating wind speeds for long-span bridges.” J. Eng. Mech., 61–73.
Zerva, A. (1992). “Seismic ground motion simulations from a class of spatial variability models.” Earthquake Eng. Struct. Dyn., 21(4), 351–361.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 143 • Issue 9 • September 2017
Copyright
©2017 American Society of Civil Engineers.
History
Received: Mar 9, 2016
Accepted: Jan 27, 2017
Published online: Apr 25, 2017
Published in print: Sep 1, 2017
Discussion open until: Sep 25, 2017
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.