# A Multifeatured Data-Driven Homogenization for Heterogeneous Elastic Solids
## 3 Design of Experiment (DOE)
### Extracting Microstructure's Descriptors
#### 3.2.1 N-Point Probability Function  
Consider \(x\) as a fixed point, where the domain of our interest consists of two phases: solid and void spaces. An indicator function \(\mathcal{L}^{(i)}(x)\) that has two possible values 0 (phase a) and 1 (phase b) for some realizations \(\theta\) can be defined. In other words, the indicator function, \(\mathcal{L}^{(i)}(x)\), can be considered a binary random variable (cf. [34]). Therefore, the probability of \(\mathcal{L}^{(i)}(x)=1\) may be simply expressed as,  
\[\mathcal{P}\Big{\{}\mathcal{L}^{(i)}(x)=1\Big{\}}. \tag{20}\]  
Accordingly, the probability of \(\mathcal{L}^{(i)}(x)=0\) can be expressed as,  
\[\mathcal{P}\Big{\{}\mathcal{L}^{(i)}(x)=1\Big{\}}=1-\mathcal{P}\Big{\{} \mathcal{L}^{(i)}(x)=1\Big{\}}. \tag{21}\]  
Therefore, the expectation value of any function \(\mathbb{F}[\mathcal{L}^{(i)}(x)]\) can be given as,  
\[\langle\mathbb{F}[\mathcal{L}^{(i)}(x)]\rangle=\mathcal{P}\Big{\{}\mathcal{L }^{(i)}(x)=1\Big{\}}\;\mathbb{F}(1)\;+\;\mathcal{P}\Big{\{}\mathcal{L}^{(i)}( x)=0\Big{\}}\;\mathbb{F}(0). \tag{22}\]  
It is noted that angular brackets indicate an ensemble average, that is, an average over all realizations \(\theta\) of the ensemble [34]. Based on this definition, the expectation value of a specific case is further considered when \(\mathbb{F}[\mathcal{L}^{(i)}(x)]=\mathcal{L}^{(i)}(x)\), that is,  
\[S_{1}^{(i)}(x)\equiv\langle\mathcal{L}^{(i)}(x)\rangle=\mathcal{P}\Big{\{} \mathcal{L}^{(i)}(x)=1\Big{\}}. \tag{23}\]  
In other words, the expectation of the indicator function \(\mathcal{L}^{(i)}(x)\) can be obtained directly from the probability \(\mathcal{P}\Big{\{}\mathcal{L}^{(i)}(x)=1\Big{\}}\). Followed by [PERSON] and [PERSON] [34], the \(S_{1}^{(i)}(x)\) is called the first order (one-point) probability function. To be specific, this one-point probability function, \(S_{1}^{(i)}(x)\), gives the probability of the phase \(i\) at position \(x\).  
Now, all \(x\) in the domain of our microscopic RVE \(\mathcal{V}\) is considered. According to the definition of the indicator function, the realization of \(\theta\) in \(\mathcal{V}\), that is \(\mathcal{V}_{i}(\theta)  
which can de defined as the n-point probability function for phase \(i\)[34]. In other words, this function yields the probability that \(n\) points at position \(x_{1},x_{2},\cdots,x_{n}\) are found in phase \(i\).  
The N-point probability function is implemented to represent the geometric information of microstructures within a succinct format. Firstly, the 1-point probability function (Figure 3) can be obtained by randomly drawing multiple points on the RVE domain and calculating a fraction of the points that are positioned inside voids. This 1-point probability function yields the porosity of the microstructure. Similarly, the distribution of voids can be represented using the 2-point probability function (Figure 3). For the 2-point probability function, two points are randomly drawn in one trial. Out of multiple casts, we calculate the probability when both two points are outside of any voids. The 3-point probability function (Figure 3) is further utilized to have more in-depth information about the distribution of voids. Again, the probability can be determined when all 3 points fall outside the voids [34]. Table 1 presents the max, min, and average of the N-point probability function for 100 randomly generated microstructures. The total number of trials to extract the N-point probability functions (1-point, 2-point, 3-point, respectively) is 1000, which is accepted as enough based on our preliminary study.