## 4 Results and Discussion
### Experimental Validation of Current Injection Models
#### 4.2.1 Dirac Delta Function  
Case 1: Current Flows in and out of the Same Boundary  
As shown in Figure 7, if current \(I\) is injected from the right boundary point (\(x=a\), \(y=y_{1}\)) of the rectangular object to point \(x=a\), \(y=y_{2}\), the function of the boundary condition at \(x=a\) can be expressed by the Dirac delta function. The Dirac delta function is written as \(\delta(y-y_{k})\), which is \(\delta(y-y_{k})=\infty\) at \(y=y_{k}\) and \(\delta(y-y_{k})=0\) otherwise. Using the Dirac delta function, Equation (25) becomes \(g(y)=0\) and \(f(y)=(I/\sigma)\cdot\{\delta(y-y_{1})-\delta(y-y_{2})\}\), and \(A_{n}\) in Equation (17) is calculated as follows:  
\[ \begin{split} A_{n}&=\frac{2\cdot(I/\sigma)}{n\pi \sinh\bigl{(}\frac{\pi\alpha}{\beta}\bigr{)}}\cdot\Bigl{(}\int_{0}^{b}\left( \delta(y-y_{1})-\delta(y-y_{2})\right)\cdot\cos\bigl{(}\frac{\pi\pi y}{b} \bigr{)}dy\Bigr{)}\\ &=\frac{2\cdot(I/\sigma)}{n\pi\sinh\bigl{(}\frac{\pi\alpha}{b} \bigr{)}}\cdot\bigl{(}\cos\bigl{(}\frac{n\pi\alpha}{b}\bigr{)}-\cos\bigl{(} \frac{n\pi\alpha}{b}\bigr{)}\bigr{)}\end{split}  \tag{26}\]  
where, \(I\) is the injected current, \(\sigma\) is the electrical conductivity of concrete, and \(a\) and \(b\) are the lengths of the concrete sample along the \(x\)- and \(y\)-axes, respectively. As \(A_{n}\) is applied to Equation (27), the electric potential \(u(x,y)\) can be calculated, and thus the potential difference between two electrodes corresponds to the measured voltages. Figure 6: Sum of voltages V1 and V2 as a function of series \(n\). \[u(x,y)=\sum_{n=1}^{50}\Big{(}A_{n}\cdot\cosh\Big{(}\frac{n\pi x}{b}\Big{)}\cdot \cos\Big{(}\frac{n\pi y}{b}\Big{)}\Big{)} \tag{27}\]  
Using nonlinear regression [17], the voltages calculated by Equation (27) are fitted to the measured voltages. Table 3 shows the nonlinear regression results. The fitted electrical conductivity was 16.077 mS\(\cdot\)m\({}^{-1}\), the rate of relative absolute error (RAE) was 22.009%:  
\[\text{Rate of Relative Absolute Error }(\text{RAE rate})=\frac{\Sigma|\text{ Residuals}|}{\Sigma|\text{Caculation}|} \tag{28}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Experimental Configuration** & **Measured** & **Calculated** & **Residual (Mea.-Cal.)** & **I Res. /Cal. (\%)** \\ \hline \multirow{3}{*}{Set \#1} & V1 & \multirow{3}{*}{V2} & \multirow{3}{*}{\(\
u_{1}\)} & 0.384 & 0.950 & \(-\)0.566 & 59.579 \\ \cline{2-6}  & V2 & & & 0.964 & 1.058 & \(-\)0.094 & 8.885 \\ \cline{2-6}  & V3 & & & 0.964 & 0.765 & 0.199 & 26.013 \\ \hline \multirow{3}{*}{Set \#2} & V1 & \multirow{3}{*}{V2} & \multirow{3}{*}{\(\
u_{1}\)} & 1.269 & 2.008 & \(-\)0.739 & 36.803 \\ \cline{2-6}  & V2 & & & 2.433 & 2.373 & 0.059 & 2.486 \\ \cline{2-6}  & V3 & & & 2.561 & 1.823 & 0.738 & 40.483 \\ \hline \multirow{3}{*}{Set \#3} & V1 & \multirow{3}{*}{V2} & \multirow{3}{*}{\(\
u_{1}\)} & 2.011 & 2.773 & \(-\)0.762 & 27.479 \\ \cline{2-6}  & V2 & & & 3.738 & 3.431 & 0.307 & 8.948 \\ \cline{2-6}  & V3 & & & 3.127 & 2.773 & 0.354 & 12.766 \\ \hline \multirow{3}{*}{Set \#4} & V1 & \multirow{3}{*}{V2} & \multirow{3}{*}{\(\
u_{1}\)} & 0.872 & 1.058 & \(-\)0.186 & 17.580 \\ \cline{2-6}  & V2 & & 1.367 & 1.315 & 0.052 & 3.954 \\ \cline{1-1} \cline{2-6}  & V3 & & & 1.708 & 1.058 & 0.650 & 61.437 \\ \hline \end{tabular}
\end{table} 
Table 3: Comparisons between the measured and calculated voltages using the Dirac delta function for Case 1 (unit: 10\({}^{-3}\) Volt). Figure 7: Boundary illustration of the current injection model using the Dirac delta function for Case 1. As shown in Table 3, V1 and V3 resulted in higher residuals between the calculated and measured voltages than V2 in all experimental sets (Sets #1-#4). Regardless of the path of electric current flow, the inner areas of the cement-based samples provided more stable measurements than the corner areas. This was likely due to the higher volatility of the electric potential in the corner areas. To better understand the residuals, a histogram and Q-Q (quantile-quantile) plot were drawn.