Generation of random field

A number of different techniques to generate random fields is available (see, e.g., [5]). In this following, the most simple method based on Cholesky decomposition of the correlation matrix.

First, vector $\bf X$ of statistically independent random numbers ${x_1, x_2, \dots, x_n}$ (where $n$ is number of elements in the FE mesh) with a standard normal distribution (i.e., with probability function of Eq. (18) with $\mu_C=0$ and $\sigma_C=1$) is generated.

A correlation matrix $\bf K$, which represents the correlation coefficient between each of the element used in the finite element analysis, is assembled. The correlation matrix $\bf K$ has the following form:

$${\bf K}= \left[ \begin{array}{llll} 1 & \rho_{12} & \dots & \rho_... ... & \ddots & \vdots \\ \rho_{n1} & \rho_{n2} & \dots & 1 \\ \end{array}\right]$$ (21)

where $\rho_{ij}$ is the correlation coefficient between elements $i$ and $j$, calculated using Markov function:

$$\rho_{ij}=\exp\left[-\frac{2x_{ij}}{\theta_C}\right]$$ (22)

where $x_{ij}$ is absolute distance between elements $i$ and $j$ (distance between centers of gravity of elements $i$ and $j$). For anisotropic case Eq. (22) reads

$$\rho_{ij}=\exp\left[-2\sqrt{\left(\frac{\tau_{xij}}{\theta_{Cx}}\... ...}{\theta_{Cy}}\right)^2 + \left(\frac{\tau_{zij}}{\theta_{Cz}}\right)^2}\right]$$ (23)

where $\theta_{Cx}$ is a correlation coefficient in direction of $x$-axis and $\tau_{xij}$ is a distance between two elements $i$ and $j$ in $x$ direction. The same notation applies for $y$ and $z$ directions.

The matrix $\bf K$ is positive definite and hence, the standard Cholesky decomposition algorithm can be used to factor the matrix into upper and lower triangular forms, $\bf S$ and ${\bf S}^T$, respectively:

$${\bf S}^T{\bf S}={\bf K}$$ (24)

The vector of correlated random variables ${\bf G}$ (i.e., ${G_1, G_2, \dots , G_n}$, where $G_i$ specifies the random component of variable $C$ in element $i$) is calculated by

$${\bf G}={\bf S}^T{\bf X}$$ (25)

Vector $\bf X$ is generated as described above.

Finally, value of the variable $C$ is assigned to each element ($C_i$) by the following transformation:

• for normally distributed variable $C$:

  $$C_i=\mu_C+\sigma_{CA} G_i$$ (26)

  
  
  where $\sigma_{CA}$ is calculated from $\sigma_C$ as described in the following section.
• for log-normally distributed variable $C$:

  $$C_i=\exp\left(\mu_{\ln C}+\sigma_{\ln CA} G_i\right)$$ (27)

  
  
  where $\mu_{\ln C}$ is calculated by Eq. (20)b using $\sigma_{\ln CA}$ instead of $\sigma_{\ln C}$; $\sigma_{\ln CA}$ is calculated from $\sigma_{\ln C}$ as described in the following section.