## 3 Numerical Experimentation and Analysis
### Grey Relational Analysis (GRA) of Concrete Mechanical Properties and Pore Structure Parameters  
When the volume of the sub-sample is not large enough to reflect all the information of the parent body, we say that the sub-sample has a certain gray scale. GRA is an effective analysis method for the analysis of the relationship among objects with incomplete run mechanism, score behavior data, and unclear, inherent connotations, which was proposed by [PERSON] [55] and can be regarded as an overall comparison with a reference system. The specific calculation of relational grade is shown below. 1. The evaluation index system is determine according to the evaluation purpose and collect evaluation data:  
Assuming that there are n data columns, the following original data matrix \((\mathrm{X}_{1}^{\prime},\mathrm{X}_{2}^{\prime},\ldots,\mathrm{X}_{n}^{ \prime})\) is formed. \[(\mathrm{X}_{1}^{\prime},\mathrm{X}_{2}^{\prime},\ldots,\mathrm{X}_{n}^{ \prime})=\left( \begin{array}{cccc}\mathrm{X}_{1}^{\prime}(1)&\mathrm{X}_{2}^ {\prime}(1)&\cdots&\mathrm{X}_{n}^{\prime}(1)\\ \mathrm{X}_{1}^{\prime}(2)&\mathrm{X}_{2}^{\prime}(2)&\cdots&\mathrm{X}_{n}^{ \prime}(2)\\ \vdots&\vdots&\vdots&\vdots\\ \mathrm{X}_{1}^{\prime}(\mathrm{m})&\mathrm{X}_{2}^{\prime}(\mathrm{m})& \cdots&\mathrm{X}_{n}^{\prime}(\mathrm{m})\end{array} \right) \tag{9}\]  
where m is the number of indicators, \(\mathrm{X}_{i}^{\prime}=(\mathrm{X}_{i}^{\prime}(1),\mathrm{X}_{i}^{\prime}(2 ),\cdots\mathrm{X}_{i}^{\prime}(\mathrm{m}))^{T},i=1,2,\cdots,\mathrm{n}\). 2. Determine the reference data column \(\mathrm{X}_{0}^{\prime}\):  
The reference column should be an ideal comparison standard, which can be constituted by the optimal value of each index or other reference values selected based on the evaluation purpose, denoted as \(\mathrm{X}_{0}^{\prime}=(\mathrm{X}_{0}^{\prime}(1),\mathrm{X}_{0}^{\prime}(2 ),\cdots,\mathrm{X}_{0}^{\prime}(\mathrm{m}))^{T}\). Other data columns are comparison data columns. 3. Dimensionless processing of index data columns:  
The commonly-used dimensionless methods are the mean method, as shown in Equation (10) and the initial value method as shown in Equation (11):  
\[\mathrm{X}_{i}(\mathrm{k})=\frac{\mathrm{X}_{i}^{\prime}(\mathrm{X})}{\frac{ \mathrm{1}}{\mathrm{m}}\sum\limits_{\mathrm{k}=1}^{\mathrm{m}}\mathrm{X}_{i}^{ \prime}(\mathrm{X})} \tag{10}\]  
\[\mathrm{X}_{i}(\mathrm{k})=\frac{\mathrm{X}_{i}^{\prime}(\mathrm{X})}{\mathrm{ X}_{1}^{\prime}(\mathrm{X})Calculate the relational grade.  
Equation (15) is used to calculate the relational grade of each evaluation object to reflect its relationship with the reference column. The greater the relational grade, the greater the relationship between the comparison column and the reference column, and vice versa.  
\[\text{r}_{\text{i}}=\frac{1}{\text{m}}\sum_{\text{k}=1}^{\text{m}}\xi_{\text{i} }(\text{k}) \tag{15}\]  
To study the influence of pore structure parameters on the macroscopic mechanical properties of concrete, the sub-porosity of each gradation segment of pores, total porosity, average pore radius, and PSSA are taken as the comparison columns for the GRA. Elastic modulus, tensile strength, and peak strain are taken as the reference columns for the GRA, respectively. Using the mean dimensionless method, the calculated relational grades are shown in Table 7.  
It can be seen from Table 7 that the descending order of the relational grades between elastic modulus and pore structure parameters is: total porosity \(>\text{P}_{\text{P}}\) [e\({}_{\text{1}}\), e\({}_{\text{2}}\)] \(>\text{P}_{\text{P}}\) [e\({}_{\text{2}}\), e\({}_{\text{3}}\)] \(>\text{P}_{\text{P}}\) [e\({}_{\text{3}}\), e\({}_{\text{4}}\)] \(>\text{P}_{\text{P}}\) [e\({}_{\text{4}}\), e\({}_{\text{5}}\)] \(>\) average pore size \(>\) PSSA. In the range of pore size studied in this paper, the smaller the pore size, the greater the influence of its content on the elastic modulus, and the total porosity has the greatest influence.