## 3 Generation of Cross-Correlated Random Fields
### Methodology
#### 3.1.3 Cross-Correlated Random Fields  
In the context of nonlinear material modeling, the correlation complex in nature, because the individual parameters are approximated best by different correlation functions and the parameters are strongly cross-correlated [24; 51]. Therefore, to achieve a proper representation of the material properties, the generated random fields must include the information about the cross-correlation. [PERSON] et al. introduced two different algorithms for the expansion of multi-correlated processes, namely the multiple uncorrelated Karhunen-Loeve expansion (muKL) and the multiple correlated Karhunen-Loeve expansion (mcKL) [46]. The main difference between these two procedures lies in the generation of the used random variables. In the context of the muKL, all correlated processes are described by a single set of uncorrelated random variables [46]. In contrast to this, the mcKL provides multiple sets of mutually correlated random variables for the discretization of correlated processes. Since the modeling of nonlinear material behavior requires a set of random variables for each parameter, the mcKL is applicable in this context.  
Within this algorithm, the same equations as for uncorrelated random fields are used and solved. However, in an additional step, the uncorrelated random variables and eigenfunctions are transformed to obtain correlated random variables and eigenfunctions, respectively, which meet the given correlation structure. Here, first, the matrix \(\mathbf{K}\) is introduced. The matrix is defined as \[K_{km}^{ij}=\mathbb{E}[Z_{k}^{i}\;Z_{m}^{j}], \tag{11}\]  
where \(Z_{k}^{i}\) and \(Z_{m}^{j}\) are a set of zero-mean uncorrelated random variables for processes \(i\) and \(j\), respectively. Using Equation (2), the cross-covariance between the processes \(i\) and \(j\) is written as [52]  
\[C_{ij} =\mathbb{E}[f_{i}(\omega,\mathbf{x})\;f_{j}(\omega,\mathbf{y})]\] \[=\sum_{k,m=1}^{\infty}K_{km}^{ij}\sqrt{\lambda_{k}^{i}\lambda_{m} ^{i}}\phi_{k}^{i}(\mathbf{x})\phi_{m}^{j}(\mathbf{y}). \tag{12}\]  
The result is an integral equation similar to the one for the auto-correlation shown in Section 3.1.2. Following Equation (3), the elements of \(\mathbf{K}\) are obtained by solving  
\[K_{km}^{ij}=\frac{1}{\sqrt{\lambda_{k}^{i}\lambda_{m}^{j}}}\;\int_{\Omega_{1}} \int_{\Omega_{2}}C_{ij}(\mathbf{x},\mathbf{y})\phi_{k}^{i}(\mathbf{x})\phi_{m }^{j}(\mathbf{y})\;\mathrm{d}\,\mathbf{y}\;\mathrm{d}\,\mathbf{x}. \tag{13}\]  
In a final step, the matrix \(\mathbf{K}\) is used to determine sets of correlated random variables for each process. It is assumed that \(\mathbf{K}\) is positive definite, which allows a Cholesky decomposition given by [52]  
\[\mathbf{K}=\mathbf{R}\mathbf{R}^{T}. \tag{14}\]  
Applying \(\mathbf{R}\) on a set of random variables leads to  
\[\hat{\mathbf{Z}}=R^{-1}\mathbf{Z}, \tag{15}\]  
where  
\[\mathbb{E}[\hat{\mathbf{Z}}\hat{\mathbf{Z}}^{T}]=\mathbf{I}. \tag{16}\]  
This means that \(\hat{\mathbf{Z}}\) contains only uncorrelated random variables. In reverse, applying \(\mathbf{R}\) on a set of uncorrelated random variables \(\hat{\mathbf{Z}}\) gives a set of correlated random variables that represent cross-correlated random fields. The same procedure is also applicable to obtain cross-correlated eigenfunctions.