# Development of a Vibroacoustic Stochastic Finite Element Prediction Tool for a CLT Floor
## 4 Stochastic Process
### Numerical Application of the Orthotropic Symmetric Material (CLT)  
The material properties of CLT gathered from the literature (c.f. Table 2) are regarded as an initial starting point (mean value) to proceed with the stochastic approach presented in the previous sections. In Abaqus, the orthotropic materials compliance matrix can be defined by the engineering constants:  
\[\left\{ \begin{array}{c}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{33}\\ \gamma_{12}\\ \gamma_{13}\\ \gamma_{23}\end{array} \right\}=\left[ \begin{array}{cccccc}\frac{1}{E_{1}}&- \frac{\
u_{12}}{E_{2}}&-\frac{\
u_{31}}{E_{3}}&0&0&0\\ -\frac{\
u_{12}}{E_{1}}&\frac{1}{E_{2}}&-\frac{\
u_{32}}{E_{3}}&0&0&0\\ -\frac{\
u_{13}}{E_{1}}&-\frac{\
u_{23}}{E_{2}}&\frac{1}{E_{3}}&0&0&0\\ 0&0&0&\frac{1}{C_{12}}&0&0\\ 0&0&0&0&\frac{1}{C_{13}}&0\\ 0&0&0&0&0&\frac{1}{C_{23}}\end{array} \right]\left\{ \begin{array}{c}\sigma_{ 11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{12}\\ \sigma_{13}\\ \sigma_{23}\end{array} \right\}. \tag{31}\]  
The compliance matrix of the CLT panel can be derived depending on the elastic constants reported in Table 2 and Equation (31). The stiffness matrix is the inverse of the compliance matrix. \[\mathbf{C}=\left[ \begin{array}{cccccc}10.58&2.3&2.3&0&0&0\\ 2.3&5.2619&2.4028&0&0&0\\ 2.3&2.4028&5.2619&0&0&0\\ 0&0&0&0.9&0&0\\ 0&0&0&0&0.09&0\\ 0&0&0&0&0&0.063\end{array} \right]\times 10^{9}. \tag{32}\]  
From Equation (32) and the corresponding orthotropic symmetric matrix basis (Equation (4)), the mean value of \(c_{i}\) defined in Equation (5) can be deduced by using the formulas below. \[(\overline{c}_{1},\ldots,\overline{c}_{9})=(10.58,\,5.2619,\,5.2619,\,2.3,\,2. 4048,\,2.3,\,0.9,\,0.09,\,0.063)\times 10^{9}. \tag{33}\]  
Let \(f^{target}=(\overline{c},\
u_{\mathcal{C}})\) be the target vector to compute the Lagrange multipliers. \[f^{target}=(10.58,\,5.2619,\,5.2619,\,2.3,\,2.4048,\,2.3,\,0.9,\,0.09,\,0.063,\, 5.3059). \tag{34}\]And let \(f^{est}(\lambda)=(\overline{c}(\lambda),\
u_{C}(\lambda))\) be the estimated vector to compare with the target vector so that the optimal solution of \(\lambda\) can be obtained by solving the following optimization function. \[J\big{(}\lambda^{opt}\big{)}=argmin\Big{(}(1-\alpha)(\overline{c}-\overline{c }(\lambda))^{2}+\alpha(\
u_{C}-\
u_{C}(\lambda))^{2}\Big{)}, \tag{35}\]  
where \(\alpha\in[0,1]\) is a free parameter. In this scenario, \(\alpha\) is set to be 0.5 to give a robust estimation. Lagrange multipliers associated with \(\lambda^{(1)}\) can be further expressed by means of \(\lambda^{(2)}\)[15]. \[\left\{ \begin{array}{c}\lambda_{1}^{(1)}=-\lambda^{2}\frac{c_{2}c_{3}-c_{4 }^{2}}{\triangle},\lambda_{2}^{(1)}=-\lambda^{(2)}\frac{c_{1}c_{3}-c_{6}^{2}}{ \triangle},\lambda_{3}^{(1)}=-\lambda^{(2)}\frac{c_{1}c_{2}-c_{6}^{2}}{ \triangle},\\ \lambda_{4}^{(1)}=-\lambda^{(2)}\frac{(c_{5}c_{6}+c_{4})}{\triangle},\lambda_ {5}^{(1)}=-\lambda^{(2)}\frac{(c_{4}c_{6}-c_{1}c_{5})}{\triangle},\lambda_{6} ^{(1)}=-\lambda^{(2)}\frac{(c_{4}c_{5}-c_{5}c_{6})}{\triangle},\\ \lambda_{7}^{(1)}=-\lambda^{(2)}\frac{1}{c^{\prime}},\lambda_{8}^{(1)}=- \lambda^{(2)}\frac{1}{c^{\prime}},\lambda_{9}^{(1)}=-\lambda^{(2)}\frac{1}{c ^{\prime}},\end{array} \right. \tag{36}\]  
with  
\[\triangle=c_{1}c_{2}c_{3}+2c_{4}c_{5}c_{6}-c_{1}c_{5}^{2}-c_{2}c_{6}^{2}-c_{3 }c_{4}^{2}. \tag{37}\]  
From the sensitivity analysis, the initial guess of \(\lambda^{(2)}\) is set to be \(-2\). The target optimization function (Equation (35)) is evaluated by using the interior-point method (_fmincon_ function) in Matlab [33]. From Figure 11, it could be observed that the optimization algorithm converges fast to a small value and the corresponding optimal values of the Lagrange multipliers are found to be:  
\[\lambda^{(1)}=(0.1263,\,0.2904,\,0.2905,\,\,-0.0758,-0.2324,-0.0758,\,12.9089, \,18.4413,\,1.2909), \tag{38}\] \[\lambda^{(2)}=-1.1618.\]  
The estimated vector \(f^{est}(\lambda)\) is evaluated by the Monte Carlo method.