## 5 Conclusions  
A virtual-force algorithm inspired by spring force (VFA-SF) was introduced in which sensors moved by the spring force are determined from the relative position of neighboring nodes. Corresponding optimization by adding an external central force during force calculation was proved to be valid and effective. It helps to avoid coverage holes ans twisted structures, and increases the possibility of a final hexagonal topology in node deployment [13]. In this paper, based on the VFA-SF and the VFA-SF-OPT, the simulation results of different parameter effects during the deploying process were investigated in detail. 1. As for the VFA-SF, statistical analysis indicates that, for maximum node velocity, a smaller \(v_{max}\) causes worse node deployment in WSN applications. When \(0.2<v_{max}<2.0\), the possibility of a perfect hexagonal network in the final deployment is stable at 42-50%. A larger \(v_{max}\) helps to improve the final network deployment but does not increase the possibility of perfect hexagonal topology. Most importantly, it should be noted that the whole algorithm converges quicker as the \(v_{max}\) limit increases. 2. In the VFA-SF-OPT optimization, it is very clear that it is harder for smaller external virtual force to eliminate a coverage hole or twisted structure in the final node distribution. When external central force increases, it causes the compression of the central area during the process of optimization deployment. The nodes newly involved in the VFA-SF-OPT need to move for a longer distance and consume more energy. Therefore, experimental \(F_{extern}\) should be slightly greater than the optimal value, which is determined based on the requirements of environment scenarios or location accuracy for some real WSN applications. 3. Experiments of three typical obstacles were simulated by using both the VFA-SF and the VFA-SF-OPT. For the circle obstacle, two algorithms could still obtain good final distributions, and the VFA-SF-OPT had a better effect. However, for the square and rectangular obstacles, final node deployment was worse. This means that the VFA-SF and the VFA-SF-OPT are more useful in a central-symmetry obstacle event based on virtual-spring-force theory. In summary, parameters have positive or negative effects on the performance and accuracy of the proposed VFA-SF and VFA-SF-OPT algorithms. In order to effectively achieve better hexagonal network topology and have better coverage performance for the target area, suitable parameters have to be selected in accordance with the experiment environments and application requirements, while a high overall coverage ratio is a goal. [PERSON] gave the basic idea of the paper. [PERSON] and [PERSON] designed the algorithms, experiments, and analysis tools; [PERSON] performed the experiments and analyzed the data; [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON] participated in discussing and interpretating of the data. [PERSON] wrote the paper. This research was supported by the National Natural Science Foundation of China under grants 41674144, 61861031, 41727804, and 41464007; the Open Projects Funding of Lunar and Planetary Science Laboratory at MUST under grant 119/2017/A3; the Electrostatic Research Foundation of Liu Shanghe Academicians and Experts workstation from Beijing Orient Institute of Measurement and Test under grant BOIMTLSHJD20181002; and the Educational Foundation of Jiangxi Province under grant GJJ160075. The authors declare no conflict of interest.