## 4 Parameter Effects on VFA-SF and VFA-SF-OPT
### Effect of Different Obstacles  
In order to present our statistical analysis for the VFA-SF and the optimized method VFA-SF-OPT for the first time, optimal network deployment was only considered to cover the whole target area without obstacles. However, in physical WSN applications, obstacles exist. It is very important to discuss the effect based on the different obstacle types. Experiments with some typical obstacles are shown in the figures, implemented by using both the VFA-SF and the VFA-SF-OPT. Figure 7 presents the final node deployments from the VFA-SF (left panels) and the VFA-SF-OPT (right panels) with a circle obstacle. Top panels are the cases when the obstacle was at the center, and bottom panels are cases when the obstacle was off-center. It is clear that the VFA-SF-OPT had a better deployment result than the VFA-SF referring to network topology. The right panels had more perfect hexagonal cells than the left panels. However, it should be noted that network structures with the off-center obstacle were worse than the cases with the obstacle at the center. The reason is that both the VFA-SF and the VFA-SF-OPT calculate the total virtual force and the corresponding physical velocity and acceleration based on the hexagonal network assumption. The six neighbor nodes were distributed around a given node like a circle. The external central force added in the VFA-SF-OPT pointed to the center of the target region, and the corresponding optimization began node redeployment from the central region. Therefore, the position of the obstacle affects final deployment to some extent. The off-center obstacle caused worse node deployment, i.e., more twisted structures, as shown in the bottom panels of Figure 7. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(v_{max}\) & \(F_{extern}\) & **Node Distance** & **Moving Distance** & \(v_{max}\) & \(F_{extern}\) & **Node Distance** & **Moving Distance** \\ \hline
0.3 & 0 & 1.7315 & 1.1759 & 1.0 & 0 & 1.7310 & 1.1759 \\  & 1 & 1.6942 & 1.6919 & & 1 & 1.695 & 2.0404 \\  & 2 & 1.6819 & 2.7282 & & 2 & 1.6796 & 2.6884 \\  & 3.5 & 1.6718 & 3.9669 & & 3.5 & 1.6641 & 3.1902 \\  & 5.5 & 1.6692 & 4.2612 & & 5.5 & 1.6587 & 3.4111 \\  & 8 & 1.6561 & 4.1812 & & 8 & 1.6544 & 3.4891 \\  & 10 & 1.6557 & 4.1766 & & 10 & 1.6529 & 3.4779 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average equilibrium distance and average moving distance within radius = 10 for different values of external central force from the VFA-SF-OPT, when \(v_{max}=0.3\), and 1.0, respectively. Similarly, Figure 8 shows the final node deployments for the square obstacle both at the center (top panels) and off-center (bottom panels) of the target region from the VFA-SF (left panels) and the VFA-SF-OPT (right panels). The square obstacle is also a central symmetry obstacle like a circle. Network deployment is significantly affected by four straight sides, however. This is because these straight sides break the normal balance of the hexagonal topology. By comparing with Figure 7, it is clear that the final deployments for the square obstacle have more twisted structures than the cases of the circle obstacle from both the VFA-SF and the VFA-SF-OPT. In such a case, the VFA-SF-OPT only optimizes the network topology in some extent but cannot obtain a perfect hexagonal network. We then considered a rectangular obstacle that had no central symmetry. Figure 9 presents the final node deployments for the rectangular obstacle, as shown in Figures 7 and 8. It is obvious that the rectangular obstacle led to worse final node deployment than the square obstacle since noncentral-symmetry obstacles brought more edge effects to virtual-force calculations. In such a case, the VFA-SF-OPT cannot further improve network topology because the preferred central optimization by adding an external central force is invalid to some extent. Moreover, Table 5 lists the corresponding computed PCD values for Figures 7-9 from these obstacle tests. The regions for PCD calculations are indicated by the red dashed circles in the panels.