{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T22:41:34Z","timestamp":1777761694222,"version":"3.51.4"},"reference-count":22,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2019,4,16]],"date-time":"2019-04-16T00:00:00Z","timestamp":1555372800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["41674144, 41727804, 41464007"],"award-info":[{"award-number":["41674144, 41727804, 41464007"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Open Projects Funding of Lunar and Planetary Science Laboratory at MUST","award":["119\/2017\/A3"],"award-info":[{"award-number":["119\/2017\/A3"]}]},{"name":"Educational Foundation of Jiangxi Province","award":["GJJ160075"],"award-info":[{"award-number":["GJJ160075"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>How to effectively deploy all wireless sensors and save a system\u2019s energy consumption is a key issue in current wireless sensor network (WSN) applications. Theoretical analysis has proven that a hexagonal structure is the best topology in the two-dimensional network, which can provide the maximum coverage area with the minimum number of sensor nodes and minimum energy consumption. Recently, many scientists presented their self-deployment strategies based on different virtual forces and discussed the corresponding efficiency via several case studies. However, according to our statistical analysis, some virtual force algorithms, e.g., virtual spring force, can still cause holes or twisted structure in a small region of the final network distribution, which cannot achieve the ideal network topology and will waste the system energy in real applications. In this paper, we first statistically analyzed the convergence and deployment effect of the virtual spring force algorithm to derive our question. Then we presented an optimized strategy that sensor deployment begins from the center of the target region by adding an external central force. At the early stage, the external force will be added to the most peripheral nodes to promote the formation of hexagonal topology and avoid covering holes or unusual structure. Finally, a series of independent simulation experiments and corresponding statistical results proved that our optimized deployment solution is very stable and effective, which can improve the energy consumption of the whole sensor network and be used in the application of a large scale WSN.<\/jats:p>","DOI":"10.3390\/s19081817","type":"journal-article","created":{"date-parts":[[2019,4,17]],"date-time":"2019-04-17T03:02:01Z","timestamp":1555470121000},"page":"1817","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":36,"title":["An Optimized Node Deployment Solution Based on a Virtual Spring Force Algorithm for Wireless Sensor Network Applications"],"prefix":"10.3390","volume":"19","author":[{"given":"Xiaohua","family":"Deng","sequence":"first","affiliation":[{"name":"Institute of Space Science and Technology, Nanchang University, Nanchang 330031, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhiyong","family":"Yu","sequence":"additional","affiliation":[{"name":"Institute of Space Science and Technology, Nanchang University, Nanchang 330031, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0154-3456","authenticated-orcid":false,"given":"Rongxin","family":"Tang","sequence":"additional","affiliation":[{"name":"Institute of Space Science and Technology, Nanchang University, Nanchang 330031, China"},{"name":"Lunar and Planetary Science Laboratory, Macau University of Science and Technology, Macau 999078, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xin","family":"Qian","sequence":"additional","affiliation":[{"name":"One Microsoft Way, Redmond, WA 98052-6399, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kai","family":"Yuan","sequence":"additional","affiliation":[{"name":"Institute of Space Science and Technology, Nanchang University, Nanchang 330031, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shiyun","family":"Liu","sequence":"additional","affiliation":[{"name":"Usun Microelectronics, Nanchang 330072, Jiangxi, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5487","DOI":"10.1109\/TII.2018.2803758","article-title":"3D Multi-Objective Deployment of an Industrial Wireless Sensor Network for Maritime Applications Utilizing a Distributed Parallel Algorithm","volume":"12","author":"Cao","year":"2018","journal-title":"IEEE Trans. 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