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MSolve.Stochastic

Facilitates Stochastic sampling of input data and statistical response analysis of systems with uncertain parameters.

Spectral Representation of Random Processes

The Fourier transform of a stochastic process x(t) is a stochastic process X(ω) given by

(1)

The integral is interpreted as an MS limit. Reasoning as in Eq.3, Ch.2, we can show that (inversion formula)

(2)

in the MS sense.

Let f0(t) be a ID-IV stationary stochastic process with mean value equal to zero, autocorrelation func-tion Rf0f0(τ) and two-sided power spectral density function Sf0f0(ω). Then the following relations hold:

(3)

(4)

(5)

(6)

where the last two equations constitute the well-known Wiener-Khintchine transform pair. The following theorem is fundamental in the theory of 1DIV stationary stochastic processes with mean value equal to zero (e.g. Yaglom 1962, Cramer and Leadbetter 1967).

To every real-valued ID-IV stationary stochastic process f0t) with mean value equal to zero and two-sided power spectral density function SF0F0(ω), two mutually orthogonal real processes u(ω) and v(ω) with orthogonal increments du(ω) and dv(ω) can be assigned such that:

(7)

Under certain requirements for the processes u(ω) and v(ω) Eq.7 can be rewritten in the following form:

(8)

where with sufficiently small but finite Δω, and if are independent random variables with mean value equal to zero and standard deviation (2Sf0f0(ωk)Δω)1/2 , then

(9)

where

(10)

(11)

(12)

and the ’s are independent random phase angles uniformly distributed ine the range [0,2π]. For further reading the reader is referred to Shinozuka and Deodatis 1991, Simulation of Stochastic Processes with Spectral Representation.

For a 2D-1V homogeneous stochastic field case, the following series as can be used for its simulation:

(13)

where:

(14)

(15)

For further reading: Shinozuka and Deodatis 1996, Simulation of multi-dimensional Gaussian stochastic fields by spectral representation.