# A Novel Method of Calibrating Micro-Scale Parameters of PFC Model and Experimental Validation
## 3 BB-DFO Experimental Design Method  
### Box-Behmken Response Surface Approach  
One of the important purposes of data analysis is to study the relationship between things and discover the potential laws. Generally, these objective regularities are often affected by various factors. Therefore, to reduce the error of analysis results and improve the efficiency at the same time, carrying out reasonable experimental design is indispensable. Otherwise, it will not only increase the number of experiments, prolong the experiment cycle, and cause waste of manpower, materials, and time but also give rise to unexpected results or even result in the entire study to fail. Response surface methodology, proposed by [PERSON] and [PERSON] [38], is a mathematical statistical method to solve nonlinear optimization problems. By using this method, it is able to obtain not only the functional relationship between multiple influencing factors and each response variable but also the interaction between different influencing factors. The basic principle of this method is to find a response surface function \(\hat{y}=f(x)\) that can approximate the mapping function \(y=f(x)\) between the independent variable \(x\) and the dependent variable \(y\) through experimental design, so that the response variable \(\hat{y}\to y\). Afterwards, the response surface function is optimized according to the constraints, and then the independent variable can be solved.  
Currently, response surface function has evolved from first-order linear functions to higher order polynomial functions that take into account the interaction between independent variables. The commonly used second-order response surface function can be expressed as  
\[\hat{y}=g(x_{1},x_{2},\cdots,x_{n})=\sum_{i=1}^{n}\beta_{ii}x_{i}^{2}+\sum_{i= 1}^{n-1}\sum_{j=1}^{n}\beta_{ij}x_{i}x_{j}+\sum_{i=1}^{n}\beta_{i}x_{i}+\beta _{0} \tag{1}\]  
where \(\hat{y}\) represents the predicted response variable, \(x_{i}\) denotes the independent variable, and \(n\) means the number of independent variables. \(\beta_{0}\), \(\beta_{i}\), \(\beta_{ij}\), and \(\beta_{ii}\) (\(i,j=1\), 2,\(\cdots\), \(n\)) are unknown constants which can be determined iteratively from test sample points, and the total number is \((n+1)(n+2)/2\).  
Provided that \(m\) trials were performed, the expression of the response surface variable can be written as  
\[\mathbf{Y}=\mathbf{\beta}\mathbf{X}+\mathbf{\varepsilon} \tag{2}\]  
where \(\mathbf{Y},\mathbf{X},\mathbf{\beta}\), and \(\mathbf{\varepsilon}\) denote vectors of dependent variables, independent variable, constant, and error, respectively, namely,  
\[\mathbf{Y}=\left[y^{(1)},y^{(2)}\cdots,y^{(m)}\right]^{T},\mathbf{\varepsilon}  
Based upon the test sample data, the constant \(\beta\) can be solved according to Equation (5), and then the response surface function \(\hat{y}\) can be easily found by substituting them into Equation (1). From the above, the design of experiment is the premise of applying the response surface methodology to find the response surface function and analyze the significance of the independent variables. At present, the commonly used experimental design methods include completely random design, paired block design, Latin square design, orthogonal experimental design, uniform experimental design, and factorial experimental design. As there is no axial point, Box-Behnken design that is particularly suitable for three-factor situation requires less experiments than other design methods. Thus, it was employed to design the experimental scheme in this study.