{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T10:14:54Z","timestamp":1782209694000,"version":"3.54.5"},"reference-count":32,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2019,3,15]],"date-time":"2019-03-15T00:00:00Z","timestamp":1552608000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>For the inverse finite element method (iFEM), an inappropriate scheme of strain senor distribution would cause severe degradation of the deformation reconstruction accuracy. The robustness of the strain\u2013displacement transfer relationship and the accuracy of reconstruction displacement are the two key factors of reconstruction accuracy. Previous research studies have been focused on single-objective optimization for the robustness of the strain\u2013displacement transfer relationship. However, researchers found that it was difficult to reach a mutual balance between robustness and accuracy using single-objective optimization. In order to solve this problem, a bi-objective optimal model for the scheme of sensor distribution was proposed for this paper, where multi-objective particle swarm optimization (MOPSO) was employed to optimize the robustness and the accuracy. Initially, a hollow circular beam subjected to various loads was used as a case to perform the static analysis. Next, the optimization model was established and two different schemes of strain sensor were obtained correspondingly. Finally, the proposed schemes were successfully implemented in both the simulation calculation and the experiment test. It was found that the results from the proposed optimization model in this paper proved to be a promising tool for the selection of the scheme of strain sensor distribution.<\/jats:p>","DOI":"10.3390\/s19061306","type":"journal-article","created":{"date-parts":[[2019,3,18]],"date-time":"2019-03-18T04:06:55Z","timestamp":1552882015000},"page":"1306","update-policy":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":43,"title":["Multi-Objective Particle Swarm Optimization of Sensor Distribution Scheme with Consideration of the Accuracy and the Robustness for Deformation Reconstruction"],"prefix":"10.3390","volume":"19","author":[{"given":"Feifei","family":"Zhao","sequence":"first","affiliation":[{"name":"Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, Xidian University, Xi\u2019an 710071, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hong","family":"Bao","sequence":"additional","affiliation":[{"name":"Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, Xidian University, Xi\u2019an 710071, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Song","family":"Xue","sequence":"additional","affiliation":[{"name":"Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, Xidian University, Xi\u2019an 710071, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qian","family":"Xu","sequence":"additional","affiliation":[{"name":"Xinjiang Observatory, National Astronomical Observatories, Chinese Academy of Science, Urumqi 830011, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3100","DOI":"10.1016\/j.ijsolstr.2012.06.009","article-title":"Shape sensing of 3D frame structures using an inverse Finite Element Method","volume":"49","author":"Gherlone","year":"2012","journal-title":"Int. J. Solids Struct."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2487","DOI":"10.1007\/s11012-015-0146-8","article-title":"Real-time displacement monitoring of a composite stiffened panel subjected to mechanical and thermal loads","volume":"50","author":"Cerracchio","year":"2015","journal-title":"Meccanica"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Yin, W., Fu, T., and Liu, J. (2009, January 3\u20137). Structural shape sensing for variable camber wing using FBG sensors. Proceedings of the SPIE 7292, Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems 2009, San Diego, CA, USA.","DOI":"10.1117\/12.812484"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1064","DOI":"10.2514\/1.J053986","article-title":"Wing shape sensing from measured strain","volume":"54","author":"Pak","year":"2016","journal-title":"AIAA J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1088\/0964-1726\/3\/2\/002","article-title":"Shape estimation from incomplete measurements: A neural-net approach","volume":"3","author":"Bruno","year":"1994","journal-title":"Smart Mater. Struct."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Mao, Z., and Todd, M. (2008, January 9\u201313). Comparison of shape reconstruction strategies in a complex flexible structure. Proceedings of the SPIE 6932, Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems 2008, San Diego, CA, USA.","DOI":"10.1117\/12.775931"},{"key":"ref_7","unstructured":"Ko, W.L., Richards, W., and Tran, V. (2007). Displacement Theories for In-Flight Deformed Shape Predictions of Aerospace Structures."},{"key":"ref_8","unstructured":"Ko, W.L., Richards, W.L., and Fleischer, V.T. (2009). Applications of the Ko Displacement Theory to the Deformed Shape Predictions of the Doubly-Tapered Ikhana Wing."},{"key":"ref_9","first-page":"035011","article-title":"Structural shape reconstruction with consideration of the reliability of distributed strain data from a Brillouin-scattering-based optical fiber sensor","volume":"19","author":"Nishio","year":"2010","journal-title":"Comput. Phys. Commun."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Bang, H., Ko, S.W., and Jang, M.S. (2012, January 13\u201316). Shape estimation and health monitoring of wind turbine tower using a FBG sensor array. Proceedings of the IEEE Instrumentation and Measurement Technology Conference, Graz, Austria.","DOI":"10.1109\/I2MTC.2012.6229407"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"035011","DOI":"10.1088\/0964-1726\/20\/3\/035011","article-title":"Shape estimation with distributed fiber Bragg grating sensors for rotating structures","volume":"20","author":"Kim","year":"2011","journal-title":"Smart Mater. Struct."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"025006","DOI":"10.1088\/0964-1726\/18\/2\/025006","article-title":"Displacement field estimation for a two-dimension structure using fiber Bragg grating sensors","volume":"18","author":"Rapp","year":"2009","journal-title":"Smart Mater. Struct."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Bogert, P.B., Haugse, E.D., and Gehrki, R.E. (2003, January 7\u201310). Structural shape identification from experimental strains using a modal transformation technique. Proceedings of the 44th AIAA Structures Conference, Norfolk, VA, USA.","DOI":"10.2514\/6.2003-1626"},{"key":"ref_14","unstructured":"Foss, G., and Haugse, E. (1995). Using modal test results to develop strain to displacement transformations. Proceedings\u2014SPIE the International Society for Optical Engineering, SPIE."},{"key":"ref_15","unstructured":"Zhu, M., and Todd, M. (2008). Comparison of Shape Reconstruction Strategies in a Complex Flexible Structure. [Master\u2019s Thesis, University of California]."},{"key":"ref_16","unstructured":"Jutte, C.V., Ko, W.L., and Stephens, C.A. (2011). Deformed Shape Calculation of a Full-Scale Wing Using Fiber Optic Strain Data from a Ground Loads Test."},{"key":"ref_17","unstructured":"Shkarayev, S., Krashanitsa, R., and Tessler, A. (2001, January 12\u201314). An Inverse Interpolation Method Utilizing In-Flight Strain Measurements for Determining Loads and Structural Response of Aerospace Vehicles. Proceedings of the 3th International Workshop on Structural Health Monitoring, Stanford, CA, USA."},{"key":"ref_18","unstructured":"Alexander, T., and Spangler, A. (2003). Variational Principal for Reconstruction of Elastic Deformation of Shear Deformable Plates and Shells."},{"key":"ref_19","unstructured":"Tessler, A., and Spangler, J.L. (2004, January 7\u20139). Inverse FEM for full-field reconstruction of elastic deformations in shear deformable plates and shells. Proceedings of the 2nd European Work shop on Structural Health Monitoring, Munich, Germany."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/j.cma.2004.03.015","article-title":"A least-squares variational method for full-field reconstruction of elastic deformations in shear-deformable plates and shells","volume":"194","author":"Alexander","year":"2005","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Quach, C., Vazquez, S.L., and Tessler, A. (2005, January 15\u201318). Structural anomaly detection using fiber optic sensors and inverse Finite element method. Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, San Francisco, CA, USA.","DOI":"10.2514\/6.2005-6357"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.ymssp.2011.05.019","article-title":"The effect of prediction error correlation on optimal sensor placement in structural dynamics","volume":"28","author":"Papadimitriou","year":"2012","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Zhao, Y. (2018). Optimal Sensor Placement for Inverse Finite Element Reconstruction of Three Dimensional Frame Deformation. Int. J. Aerosp. Eng., 7.","DOI":"10.1155\/2018\/6121293"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Zhao, Y., Du, J., Bao, H., and Xu, Q. (2018). Optimal Sensor Placement Based on Eigenvalues Analysis for Sensing Deformation of Wing Frame Using iFEM. Sensors, 18.","DOI":"10.3390\/s18082424"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"045027","DOI":"10.1088\/0964-1726\/23\/4\/045027","article-title":"An inverse finite element method for beam shape sensing: Theoretical framework and experimental validation","volume":"23","author":"Mattone","year":"2014","journal-title":"Smart Mater. Struct."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Zhao, Y., Bao, H., and Duan, X. (2017). The application research of inverse finite element method for frame deformation Estimation. Int. J. Aerosp. Eng., 3.","DOI":"10.1155\/2017\/1326309"},{"key":"ref_27","unstructured":"Kunsoo, H., and Stein, J.L. (1993, January 2\u20134). Well-Conditioned Observer Design for Observer-Based Monitoring Systems. Proceedings of the 1993 American Control Conference, San Francisco, CA, USA."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1109\/TEVC.2004.826067","article-title":"Handling multiple objective with particle swarm optimization","volume":"8","author":"Coello","year":"2004","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"5580","DOI":"10.1109\/TAP.2014.2352361","article-title":"Multiobjective Particle Swarm Optimization of Boresight Error and Transmission Loss for Airborne Radomes","volume":"62","author":"Xu","year":"2014","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1656","DOI":"10.1109\/TSMCB.2012.2227469","article-title":"Particle Swarm optimization for Feature Selection in Classification: A Multi-Objective Approach","volume":"43","author":"Xue","year":"2017","journal-title":"IEEE Trans. Cybern."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1109\/TAP.2004.823969","article-title":"Particle Swarm Optimization in Electromagnetics","volume":"52","author":"Robinson","year":"2004","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1007\/s40031-018-0323-y","article-title":"A Review of Particle Swarm Optimization","volume":"99","author":"Jain","year":"2018","journal-title":"J. Inst. Eng. Ser. B"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/www.mdpi.com\/1424-8220\/19\/6\/1306\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:39:02Z","timestamp":1760186342000},"score":1,"resource":{"primary":{"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/www.mdpi.com\/1424-8220\/19\/6\/1306"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,15]]},"references-count":32,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2019,3]]}},"alternative-id":["s19061306"],"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/doi.org\/10.3390\/s19061306","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,15]]}}}