# Development of a Vibroacoustic Stochastic Finite Element Prediction Tool for a CLT Floor
## 4 Stochastic Process
### Decomposition of the Random Elastic Tensor  
Construction of PDF of a random value in a high-dimension approach can be applied to any arbitrary material symmetry class [15; 18; 19], such as isotropic symmetry, cubic symmetry, and transversely isotropic symmetry. In this work, this stochastic approach aims to seek a reasonable probability distribution of the random elastic tensor of the target material (CLT). Therefore, only the orthotropic symmetry case is considered in this section. The dimension of the random variable \(\mathbf{N}\) is limited to 9. Let \(\mathbf{C}\) be a fourth-order random elastic tensor, which could be decomposed by using the equation below. \[\mathbf{C}=\sum_{i=1}^{N}c_{i}\mathbf{E}_{i}, \tag{3}\]  
where \(c_{i}\) is a set of random coefficients that can be described by its PDFs and \(\mathbf{E}_{i}\) is the tensor basis of the random elastic tensor \(\mathbf{C}\), based on [PERSON]'s derivation [29]. The CLT slab is modelled as orthotropic material in Abaqus so that the tensor basis \(\mathbf{E}_{i}\) of the orthotropic symmetric elastic tensor are defined in the following form [18]. Figure 10: Flow chart of the application of the stochastic process. \[\left\{ \begin{array}{l}\mathbf{E^{11}}=\mathbf{a}\otimes\mathbf{a}\otimes\mathbf{a},\mathbf{E^{12}}= \mathbf{a}\otimes\mathbf{a}\otimes\mathbf{b}\otimes\mathbf{b},\mathbf{E^{13}}=\mathbf{a}\otimes\mathbf{a} \otimes\mathbf{c}\otimes\mathbf{c},\\ \mathbf{E^{21}}=\mathbf{b}\otimes\mathbf{b}\otimes\mathbf{a}\otimes\mathbf{a},\mathbf{E^{22}}=\mathbf{b} \otimes\mathbf{b}\otimes\mathbf{b}\otimes\mathbf{b},\mathbf{E^{23}}=\mathbf{b}\otimes\mathbf{b} \otimes\mathbf{c}\otimes\mathbf{c},\\ \mathbf{E^{31}}=\mathbf{c}\otimes\mathbf{c}\otimes\mathbf{a}\otimes\mathbf{a},\mathbf{E^{32}}=\mathbf{c} \otimes\mathbf{c}\otimes\mathbf{b}\otimes\mathbf{b},\mathbf{E^{33}}=\mathbf{c}\otimes\mathbf{c} \otimes\mathbf{c},\\ \hskip 28.452756 ptE^{i}_{jkl}=\left(a_{i}b_{j}+b_{i}a_{j}\right)\left(a_{k}b_ {l}+b_{k}a_{l}\right)/2,\\ \hskip 28.452756 ptE^{i}_{jkl}=\left(b_{i}c_{j}+c_{i}b_{j}\right)\left(b_{k}c_ {l}+c_{k}b_{l}\right)/2,\\ \hskip 28.452756 ptE^{6}_{ijkl}=\left(c_{i}a_{j}+a_{i}c_{j}\right)\left(c_{k}a _{l}+a_{k}c_{l}\right)/2,\end{array} \right. \tag{4}\]  
where \(\mathbf{a},\mathbf{b},\) and \(\mathbf{c}\) are the unit orthogonal vectors, \(\otimes\) is the Kronecker product. The fourth-order elastic tensor \(\mathbf{C}\) is decomposed as:  
\[\mathbf{C}=c_{1}\mathbf{E^{11}}+c_{2}\mathbf{E^{22}}+c_{3}\mathbf{E^{33}}+c_{4} \big{(}\mathbf{E^{12}}+\mathbf{E^{21}}\big{)}+c_{5}\big{(}\mathbf{E^{23}}+\mathbf{E^{32}} \big{)}+c_{6}\big{(}\mathbf{E^{31}}+\mathbf{E^{13}}\big{)} \tag{5}\] \[+c_{7}\mathbf{E^{4}}+c_{8}\mathbf{E^{5}}+c_{9}\mathbf{E^{6}}.\]  
Construction of Probability Distribution Function in High-Dimension Using the Maximum Entropy Principle  
The objective of this section is to establish the PDFs of the random coefficients \(\mathbf{c}_{i}\), which control the statistical fluctuation of the fourth-order random elastic tensor. Let \(c=\left(c_{1},\ldots,c_{N}\right)\) be a vector in \(\mathbb{R}^{N}\)-valued second order random variable, which obeys certain unknown probability distribution \(P_{\mathbf{c}}(dc)\) with the Lebesgue measure \(d\mathbf{c}=dc_{1}\ldots dc_{N}\). The element in vector \(\mathbf{c}=\left(c_{1},\ldots,c_{N}\right)\) represent the random coefficient of the random elasticity matrix in the previous section (Equation (3)). The unknown probability distribution \(P_{\mathbf{c}}(d\mathbf{c})\) of the vector \(\mathbf{c}\) can be presented by a probability density function \(p_{\mathbf{c}}(\mathbf{c})\), which satisfies the following normalization condition. \[\int p_{\mathbf{c}}(\mathbf{c})d\mathbf{c}=1. \tag{6}\]  
The Maximum Entropy Principle applied here aims to construct the unknown probability distribution \(P_{\mathbf{c}}(d\mathbf{c})\) with the help of the available information. In this case, the probability density function (PDF) \(p_{\mathbf{c}}\) could be written using the equation below.