Simulation of narrowband non-Gaussian processes using envelope distribution

An envelope distribution approach is developed for generation of a class of narrowband non-Gaussian processes. The process is prescribed by the probability density and the specified power spectrum with the non-zero dominant frequency. The proposed approach is based on the method using stochastic differential equations. In order to construct the stochastic differential equations, it is required to find a joint probability distribution, which takes the form derived from the corresponding FokkerPlanck equation and has the target non-Gaussian distribution as the marginal distribution. In the present method, the joint distribution is obtained from the envelope distribution corresponding to the target non-Gaussian probability density. The proposed approach is applicable to generation of narrowband processes with a variety of non-Gaussian probability densities. Two numerical examples are presented to illustrate the usefulness of the approach. Monte Carlo simulation is widely used to investigate the response and the reliability of mechanical and structural systems subjected to random excitation. For the purpose of obtaining the accurate results by the simulation, the excitation should be properly modeled as a stochastic process according to the information available. Therefore, a numerical algorithm for generation of stochastic processes with given statistical characteristics is widely studied (Shinozuka and Deodatis, 1991; Bocchini and Deodatis, 2008). One of the most well-known method for Gaussian processes is the spectral representation (Shinozuka and Jan, 1972; Shinozuka and Deodatis, 1991). In practical engineering problems, the random excitation often exhibits non-Gaussianity. The examples include road roughness (Grigoriu, 1995), shallow water waves (Bitner, 1980) and wind pressure acting on low-rise buildings (Kwon and Kareem, 2011). The probability distributions of these stochastic processes are far from Gaussian distribution. It is therefore required to generate non-Gaussian processes with the proper probability distributions to accurately estimate the reliability of the nonGaussian randomly excited systems. Another important characteristic of stochastic processes is the power spectral density. Especially, the narrowband power spectrum is observed in many types of random excitation. Furthermore, the narrowband spectral density may be particularly important in view of the resonance phenomenon of the system. The model of a non-Gaussian process is often constructed to match the prescribed first-order probability distribution and power spectrum. Grigoriu (1998) proposed the approach using the translation process defined by the memoryless nonlinear transform of the underlying Gaussian process. Yamazaki and Shinozuka (1988) developed the method combined with the spectral representation and the translation process to generate the prescribed non-Gaussian 12 International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 2 process. Many studies for the extensions of these methods also have been conducted (Masters and Gurley, 2003; Bocchini and Deodatis, 2008; Shields and Deodatis, 2011). These methods require iteration in order to match the target distribution and/or spectral density, and therefore are sometimes computationally expensive. In this paper, a simulation method using an envelope distribution is proposed to generate a class of narrowband non-Gaussian stochastic processes with the prescribed probability distribution and spectral density. This method is based on the scheme developed by Cai and Lin (1996). Since no iteration is required in the procedure, the approach can be easily implemented for fast and efficient simulation. To generate the nonGaussian process, a two-dimensional Ito stochastic differential equation is used. Then, in order to determine the diffusion coefficients, it is needed to find an appropriate joint probability density, which has the target probability density as the marginal distribution and satisfies a condition derived from the corresponding Fokker -Planck equation. However, it is difficult to obtain the joint probability density. In the proposed procedure, the joint probability density is obtained from the envelope distribution corresponding to the target nonGaussian probability density. To demonstrate the validity of the method, this method is applied to generate non-Gaussian processes with two different types of probability distributions. The proposed approach is applicable to generation of processes with various non-Gaussian probability densities observed in engineering and scientific fields. This model may also be useful for analytical studies since the process is expressed by stochastic differential equations. 1. NARROWBAND NON-GAUSSIAN PROCESS Let U(t) be a zero-mean stationary non-Gaussian stochastic process having the non-Gaussian probFigure 1: Power spectra of process U(t) ( ] [ E 2 U = 1) ability distribution pU(u) and the power spectrum ) (ω U S given by