Simulation of multivariate stationary stochastic processes using dimension-reduction representation methods

Abstract In view of the Fourier-Stieltjes integral formula of multivariate stationary stochastic processes, a unified formulation accommodating spectral representation method (SRM) and proper orthogonal decomposition (POD) is deduced. By introducing random functions as constraints correlating the orthogonal random variables involved in the unified formulation, the dimension-reduction spectral representation method (DR-SRM) and the dimension-reduction proper orthogonal decomposition (DR-POD) are addressed. The proposed schemes are capable of representing the multivariate stationary stochastic process with a few elementary random variables, bypassing the challenges of high-dimensional random variables inherent in the conventional Monte Carlo methods. In order to accelerate the numerical simulation, the technique of Fast Fourier Transform (FFT) is integrated with the proposed schemes. For illustrative purposes, the simulation of horizontal wind velocity field along the deck of a large-span bridge is proceeded using the proposed methods containing 2 and 3 elementary random variables. Numerical simulation reveals the usefulness of the dimension-reduction representation methods.

[1]  Jianbing Chen,et al.  The Number Theoretical Method in Response Analysis of Nonlinear Stochastic Structures , 2007 .

[2]  A. Pisano,et al.  Multivariate stochastic wave generation , 1996 .

[3]  H. Niederreiter.  Pseudo-random numbers and optimal coefficients☆ , 1977 .

[4]  Y. Peng,et al.  Random function based spectral representation of stationary and non-stationary stochastic processes , 2016 .

[5]  Yongbo Peng,et al.  A stochastic harmonic function representation for non-stationary stochastic processes , 2017 .

[6]  Ahsan Kareem,et al.  Simulation of Multivariate Nonstationary Random Processes by FFT , 1991 .

[7]  Jianbing Chen,et al.  Orthogonal expansion of Gaussian wind velocity field and PDEM-based vibration analysis of wind-excited structures , 2011 .

[8]  S. Rice.  Mathematical analysis of random noise , 1944 .

[9]  Hong Hao,et al.  Modelling and simulation of spatially varying earthquake ground motions at sites with varying conditions , 2012 .

[10]  Weiling Sun,et al.  Stochastic Harmonic Function Representation of Stochastic Processes , 2013 .

[11]  K. Fang,et al.  Number-theoretic methods in statistics , 1993 .

[12]  Ahsan Kareem,et al.  Proper Orthogonal Decomposition-Based Modeling, Analysis, and Simulation of Dynamic Wind Load Effects on Structures , 2005 .

[13]  Yongbo Peng,et al.  Modeling and simulation of fluctuating wind speeds using evolutionary phasespectrum , 2013 .

[14]  Giovanni Solari,et al.  Wind modes for structural dynamics: a continuous approach , 2002 .

[15]  Paul Glasserman,et al.  Monte Carlo Methods in Financial Engineering , 2003 .

[16]  Yongbo Peng,et al.  Dimension reduction of Karhunen-Loeve expansion for simulation of stochastic processes , 2017 .

[17]  Dayong Li,et al.  Simulation of spatially correlated earthquake ground motions for engineering purposes , 2011 .

[18]  Yongbo Peng,et al.  Nonlinear Response of Structures Subjected to Stochastic Excitations via Probability Density Evolution Method , 2014 .

[19]  Guoqing Huang,et al.  New formulation of Cholesky decomposition and applications in stochastic simulation , 2013 .

[20]  M. D. Paola,et al.  Digital simulation of wind field velocity , 1998 .

[21]  Masanobu Shinozuka,et al.  Simulation of Multivariate and Multidimensional Random Processes , 1971 .

[22]  Haifan Xiang,et al.  An efficient ergodic simulation of multivariate stochastic processes with spectral representation , 2011 .

[23]  Takuji Nishimura,et al.  Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator , 1998, TOMC.

[24]  Haifan Xiang,et al.  SIMULATION OF STOCHASTIC WIND VELOCITY FIELD ON LONG-SPAN BRIDGES , 2001 .

[25]  Emil Simiu,et al.  Wind Effects on Structures , 2019 .

[26]  J.-N. Yang,et al.  Simulation of random envelope processes. , 1972 .

[27]  L. Carassale,et al.  Monte Carlo simulation of wind velocity fields on complex structures , 2006 .

[28]  P. Caines.  Linear Stochastic Systems , 1988 .

[29]  Masanobu Shinozuka,et al.  Simulation of Multi-Dimensional Gaussian Stochastic Fields by Spectral Representation , 1996 .

[30]  M. Shinozuka,et al.  Simulation of Stochastic Processes by Spectral Representation , 1991 .

[31]  M. Di Paola,et al.  Digital generation of multivariate wind field processes , 2001 .

[32]  P. Spanos,et al.  Monte Carlo Treatment of Random Fields: A Broad Perspective , 1998 .

[33]  Jianbing Chen,et al.  Strategy for selecting representative points via tangent spheres in the probability density evolution method , 2008 .

[34]  Yongbo Peng,et al.  Field measurement and investigation of spatial coherence for near-surface strong winds in Southeast China , 2018 .

[35]  A. Davenport.  The spectrum of horizontal gustiness near the ground in high winds , 1961 .

[36]  Svend Ole Hansen,et al.  Wind Loads on Structures , 1997 .