# Predicting the [PERSON]'s Modulus of Concrete Using a Particle-Based Movable Cellular Automata Method
## 3 Modelling Concrete in Uniaxial Static Compression Test
### Parametric Sensitivity Analysis  
Grey Relational Analysis (GRA) is a method belonging to Grey System Theory (GST) that is used to assess and compare the influence of various input factors on the result. The first step in sensitivity analysis using the GRA method is to normalize the data matrix \(X\), where each row corresponds to one observation and each column contains the changing value of the parameter being studied and the result vector \(\vec{Y}\):  
\[X= \begin{bmatrix}x_{11}&x_{12}&\cdots&x_{1n}\\ x_{21}&x_{22}&\cdots&x_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ x_{m1}&x_{m2}&\cdots&x_{mn}\end{bmatrix} ,\,\vec{Y}= \begin{bmatrix}y_{1}\\ y_{2}\\ \vdots\\ y_{m}\end{bmatrix}  \tag{9}\]  
where \(m\) is the number of observations and \(n\) is the number of evaluated parameters. In this study, we examine the impact of the selected parameters on the [PERSON]'s modulus. Therefore, we assume that higher values are better and perform normalization according to the following formula:  
\[x^{\prime}_{ij} = \frac{x_{ij}-\min(x_{j})}{\max(x_{j})-\min(x_{j})} \tag{10}\] \[y^{\prime}_{i} = \frac{y_{i}-\min(\vec{Y})}{\max(\vec{Y})-\min(\vec{Y})}. \tag{11}\]  
The grey relational coefficient \(\xi\) is determined as follows:  
\[\xi_{ij}=\frac{\Delta_{ij}+\zeta\cdot\Delta_{\max}}{\Delta_{\min}+\zeta\cdot \Delta_{\max}} \tag{12}\]  
where \(\Delta_{ij}=|x^{\prime}_{ij}-y^{\prime}_{i}|\), \(\Delta_{\min}\) and \(\Delta_{\max}\) are the minimum and maximum values of this absolute difference, respectively, and \(\zeta\in[0,1]\) is the differentiation coefficient, typically taken as 0.5. For each input variable (i.e., column), the Grey Relational Grade (GRG) is calculated as the average relational coefficient:  
\[\gamma_{j}=\frac{1}{m}\sum_{i=1}^{m}\xi_{ij}. \tag{13}\]  
The GRG provides information about the global impact of a variable on the result, with higher values indicating stronger influence. To perform sensitivity analysis using the grey method, series of numerical simulations were carried out for a hypothetical concrete composed of cement paste, fine aggregate, and coarse aggregate. The numerical experiments were conducted according to the Taguchi L81 plan [59; 60]. The effects of the following parameters were tested, with three values used for each:  
* Viscous friction of cement paste, with values of 0.01, 0.1, and 1 kg/s. * Dry friction coefficient of coarse aggregate, with values of 0.1, 0.5, and 0.9. * Coarse aggregate compressive strength, with values of 120, 180, and 240 MPa. * Coarse aggregate Poisson's ratio, with values of 0.2, 0.25, and 0.3. * Coarse aggregate elastic modulus, with values of 60 GPa, 70 GPa, and 80 GPa. * Water/concrete ratio (expressed as the cement paste elastic modulus, with values of 15.24, 20, and 28.23 GPa), corresponding to w/c ratio values of 0.3, 0.45, and 0.6, respectively, according to the data from [52] presented in Table 1.
* Cement content (with values of 30%, 35%, and 40% by volume of the sample) and coarse aggregate content (with values of 20%, 30%, and 40% by volume of the sample), resulting in three different aggregate/cement ratios and six different coarse aggregate/total aggregate ratios. The values of the viscous friction coefficients for fine and coarse aggregates were not changed, being set at 0.1 kg/s, while the dry friction coefficients for cement paste and fine aggregate were set to 0.5 and their Poisson's ratios were equal to 0.2. Results are presented in Figure 4. The analysis shows that the coarse aggregate elastic modulus, water-to-cement (w/c) ratio, and coarse aggregate-to-total aggregate ratio have the greatest impact on the Young's modulus of concrete modeled using the MCA method. The weakest influence is observed for the friction coefficients, Poisson's ratio, and compressive strength of one of the components, specifically the coarse aggregate.