# Numerical Infeasibilities of Nanofibrous Mats Process Design
## 7 Discussion  
The target issue is the distinction between failed and successful simulations. The simulation is considered to have failed when the examined set of points do not satisfy at least one of the imposed constraints or if they do not belong to the function domain. Let us reduce Equation (3) to the form  
\[\text{F}_{\text{D}}=\text{A}\ \sqrt{\frac{1}{2\text{ln}\chi-3}}, \tag{14}\]  
where \(\text{A}=\text{C}^{0.5}\left(\gamma\epsilon\frac{\text{Q}^{2}}{1^{2}}\frac{2} {\pi}\right)^{1/3}\in\mathcal{R}\), for a given set of process parameters. Figure 5 depicts the detailed value of the \(\chi\) parameter (\(\chi*\approx 4.48\)) that leads to the feasible region. More accurate and time-consuming numerical computations resulted with \(\chi^{*}\approx 4481.69\times 10^{-3}\). The infeasible region was marked in grey; these results are beyond expectations. Therefore, in order to obtain successful simulations, we should take \(\chi\geq\chi*\). Otherwise, our simulations will have failed. The \(\chi<\chi*\) causes a denominator (Equation (14)) of less than zero. As a result we have a cube root of a negative number. This leads to  
1. Negative values of the \(\text{F}_{\text{D}}\) (real cube root), where \(\chi<\chi*\) violates the Equation (4);
2. To the situation, where the \(\text{F}_{\text{D}}\) value is expressed as the complex number (complex cube root), where \(\chi<\chi*\) violates the Equation (7). In both cases the obtained result is undesirable. In practice, the obtained solutions were expressed in the form of complex numbers due to the power function used in all calculations (see [46; 47]). Figure 6 shows the model results when the fiber diameter approaches 500 nm and \(\chi\in\left[\chi*,\,7.00\times 10^{5}\right]\). Figure 7 depicts the area where \(\chi\in\left[\chi*,50\right]\). Here, one can observe the decreasing function \(\text{F}_{\text{D}}(\chi)\). Let us find  
\[\lim_{\chi\rightarrow+\infty}\text{F}_{\text{D}}(\chi)=0 \tag{15}\]  
Figure 9: Fiber diameter as a function of \(\chi\); model results when the fiber diameter approaches 3.25 mm and the \(\chi\) range is very tight. The infeasible region is marked in grey. Thus, we can see that as the \(\chi\) increases the \(\text{F}_{\text{D}}(\chi)\) goes to zero. This situation is more clearly illustrated in Figure 8. Here, we can see the initial significant change in the \(\text{F}_{\text{D}}\) value (sharp slope area). \(\text{F}_{\text{D}}\) varies at a slower rate. It is worth noting that in the sharp slope area even a small change in the \(\chi\) value may lead to incorrect, distorted results, whereas approaching the desired solution may be achieved by inconsiderable change of the \(\chi\) parameter, especially for the smaller \(\text{F}_{\text{D}}\) (i.e., when \(\text{F}_{\text{D}}=100\) nm or less). It should be also added that  
\[\underset{\chi\rightarrow\chi^{+}}{\text{lim}}\text{F}_{\text{D}}(\chi)=+\infty \tag{16}\]  
This is particularly well demonstrated by Figure 9, where \(\text{F}_{\text{D}}\) approaches 3.25 mm (\(3.25\times 10^{6}\)) nm. According to Equation (16), the \(\text{F}_{\text{D}}\) goes to \(+\infty\), hence \(\text{F}_{\text{D}}\) may reach high values. However, \(\text{F}_{\text{D}}=3.25\) mm is against the process nature. Moreover, we can also write, that \(\text{F}_{\text{D}}(\chi)=\text{A}\sqrt[3]{\frac{1}{2\text{m}}\chi^{-3}}\) and  
\[\widetilde{\text{F}_{\text{D}}}(\chi)=\frac{a}{\chi-\chi*} \tag{17}\]  
with \(A,a\in\mathcal{R}\), characterized by similar asymptotical behavior. One should also emphasize that the sampling frequency affects the results. The sampling should be appropriately adjusted to the process nature and to the model that describes it. The sampling frequency for the simulations depicted above was: 1.00 (Figure 6); \(1.00\times 10^{-5}\) (Figures 5, 7 and 8); \(1.00\times 10^{-11}\) (Figure 9). Dense sampling needs to be carried out for uncertain, suspicious regions.