## 3 Uncertainty Quantification of Macro Elastic Parameters Based on Non-Probabilistic Thought
Convex Polyhedral Modeling of Multi-Dimensional Elastic Parameters Based on Grey Mathematical Theory  
In Section 3, we have proposed three kinds of analysis methods to perform uncertainty quantification for macro elastic parameters of composites from the perspective of micro-macro scale on the basis of non-probabilistic thoughts, where uncertain micro input parameters are treated as independent interval variables. Although these methods can give an exact prediction for the intervals of elastic mechanical parameters or evaluate correlations between any two of them, they cannot give a clear boundary or outline for dependent macro elastic parameters of composites. Therefore, in this section one will carry out uncertainty quantification of dependent elastic mechanical parameters based on some finite sample points or experimental data by virtue of grey mathematical theory. Not only can make full use of the obtained available information to determine and evaluate uncertainty characteristic information, but it can also take the correlations between uncertain parameters into account and give an appropriate geometry description for macro elastic mechanical properties of composite laminates. Here, uncertain multi-dimensional elastic parameters are dealt with as uncertain-but-bounded variables or modeled as a closed and bounded convex polyhedral model. It can be described analytically by some linear inequalities constraints by virtue of some sample points or experimental points. According to grey mathematical theory, for the sake of carrying out uncertainty quantification of elastic properties for composites, the obtained sample points or experimental points should be validated and they can be quantified in a fixed interval. Then, for any two uncertain macro elastic mechanical parameters \(x_{i}\) and \(x_{j}\), the dependent uncertain-but-bounded parameters \(x\) or the combination sequence should satisfy the following linear inequalities constraints as:  
\[\left|(x_{i}-x_{i}^{c})\cdot\cos\theta_{k}+(x_{j}-x_{j}^{c})\cdot\sin\theta_{k }\right|\leq K\cdot G\cdot\frac{\Delta_{\max}^{ij}(\theta_{k})}{N}\ i=1,2, \cdots,n-1;j=i+1,\cdots,n \tag{19}\]  
where \(G\) is the grey identification coefficient, which is nearly unaffected by the distribution type of measurement values and can be calculated by the grey evaluation model [43]; \(n\) is the number of uncertainties. In addition, according to the Chebycheff's inequity [44] and some statistical analysis methods [45], the quantified parameter \(K\) can be calculated, which is practically set to 3. \(x_{i}^{c}\) and \(x_{j}^{c}\) are nominal values or mean values of uncertain macro elastic mechanical parameters, which can be calculated as:  
\[x_{i}^{c}=\frac{1}{N}\sum_{p=1}^{N}x_{ip}\ i=1,2,\cdots,n \tag{20}\]Further, the macro elastic mechanical parameter \(x_{i}\) of composites should be in the following estimated interval as:  
\[x_{i}\in x_{i}^{l}=[x_{i}^{c}-K\cdot s_{i},x_{i}^{c}+K\cdot s_{i}]\ i=1,2,\cdots,n \tag{21}\]  
\(s_{i}\) is the standard deviation of an uncertain parameter. \(\theta_{k}\) is the polar angle, which represents different quantification directions. \(N\) is the number of obtained sample points. \(\Delta_{\text{max}}^{ij}(\theta_{k})\) is the maximum difference between the mean-value accumulated sequence and the accumulated data sequence, namely:  
\[\Delta_{\text{max}}^{ij}(\theta_{k})=\text{max}(\left|l\cdot\frac{\sum\limits_ {p=1}^{N}\left(x_{ip}\cdot\cos\theta_{k}+x_{jp}\cdot\sin\theta_{k}\right)}{N} -\sum\limits_{p=1}^{l}\left(x_{ip}\cdot\cos\theta_{k}+x_{jp}\cdot\sin\theta_{ k}\right)\right|),l=1,2,\cdots,N \tag{22}\]  
Certainly, the Formula (19) can be rewritten as the following formula as:  
\[\left\{