Gaussian correlated random field

Thus the Aq are independent Gaussian random variables with zero mean. Accordingly u(, t) has become a random field, i.e., a random function of the four variables, t rather than of t alone. One is interested in its stochastic properties, for instance, the two-point correlation function... 

In contrast to the approach of simulating the physical synthesis of the material, the computational cost involved in the Gaussian random field method is minimal. Statistically independent realizations with the desired porosity and spatial correlations can be repeatedly generated simply by changing the random number seed of the Gaussian random field. 

The example structure is one quarter of a 1 in. thick, 32x32 in square composite plate with a 4-inch diameter circular core region under distributed edge loads, as shown in Fig. 5. The bulk and shear moduli K and G, respectively, of the outside material are modeled as homogeneous Gaussian random fields. The moduli and of the core material and the load intensity V/ are considered to be Gaussian random variables. The assumed means and coefficients of variation are listed in Table 2. The shear and bulk moduli for each material are assumed to be statistically independent, but correlation between the same moduli for the two materials, i.e, pG Gi is considered. 

This effective Q,t-range overlaps with that of DLS. DLS measures the dynamics of density or concentration fluctuations by autocorrelation of the scattered laser light intensity in time. The intensity fluctuations result from a change of the random interference pattern (speckle) from a small observation volume. The size of the observation volume and the width of the detector opening determine the contrast factor C of the fluctuations (coherence factor). The normalized intensity autocorrelation function g Q,t) relates to the field amplitude correlation function g (Q,t) in a simple way g t)=l+C g t) if Gaussian statistics holds [30]. g Q,t) represents the correlation function of the fluctuat-... 

This equation can be obtained as follows. Since the order parameter is conserved, it obeys a conservation law, 90/9/ - - V J = 0. The diffusion current J is given by J = — where fx is the chemical potential difference between oil and water. With the relation / = which follows from standard thermodynamics, we arrive at the first part of Eq. (49). The second part, a random (Gaussian) noise source, is necessary to describe thermal fluctuations. It can be shown that the order parameter field has the correct Boltzmann statistics if the noise has the correlations... 

The process of reconstruction of a three-dimensional homogeneous and isotropic random medium of a given porosity and a given correlation function is equivalent to generating a random discrete function Z(x) that satisfies the properties specified in Eqs. (90). It can be shown that such Z(x) can be devised from a Gaussian field Z(x) when the latter is passed successively through a linear and a nonlinear filter. 

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