{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T17:35:11Z","timestamp":1777743311836,"version":"3.51.4"},"reference-count":26,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2020,8,16]],"date-time":"2020-08-16T00:00:00Z","timestamp":1597536000000},"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":["51675398"],"award-info":[{"award-number":["51675398"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the joint Research Fund in Astronomy","award":["U1731135"],"award-info":[{"award-number":["U1731135"]}]},{"name":"CAS \u201cLight of West China\u201d Program","award":["No.2016-QNXZ-A-7"],"award-info":[{"award-number":["No.2016-QNXZ-A-7"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>When the inverse finite element method (inverse FEM) is used to reconstruct the deformation field of a multi-element structure with strain measurements, strain measurement errors can lower the reconstruction accuracy of the deformation field. Furthermore, the calibration ability of a self-structuring fuzzy network (SSFN) is weak when few strain samples are used to train the SSFN. To solve this problem, a novel two-step calibration method for improving the reconstruction accuracy of the inverse FEM method is proposed in this paper. Initially, the errors derived from measured displacements and reconstructed displacements are distributed to the degrees of freedom (DOFs) of nodes. Then, the DOFs of nodes are used as knots, in order to produce non-uniform rational B-spline (NURBS) curves, such that the sample size employed to train the SSFN can be enriched. Next, the SSFN model is used to determine the relationship between the measured strain and the DOFs of the end nodes. A loading deformation experiment using a three-element structure demonstrates that the proposed algorithm can significantly improve the accuracy of reconstruction displacement.<\/jats:p>","DOI":"10.3390\/s20164602","type":"journal-article","created":{"date-parts":[[2020,8,17]],"date-time":"2020-08-17T04:35:51Z","timestamp":1597638951000},"page":"4602","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Two-Step Calibration Method for Inverse Finite Element with Small Sample Features"],"prefix":"10.3390","volume":"20","author":[{"given":"Libo","family":"Xu","sequence":"first","affiliation":[{"name":"Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7251-6802","authenticated-orcid":false,"given":"Feifei","family":"Zhao","sequence":"additional","affiliation":[{"name":"Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9292-7633","authenticated-orcid":false,"given":"Jingli","family":"Du","sequence":"additional","affiliation":[{"name":"Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,16]]},"reference":[{"key":"ref_1","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_2","unstructured":"Tessler, A., and Spangler, J.L. (2003). A Variational Principle for Reconstruction of Elastic Deformations in Shear Deformable Plates and Shells."},{"key":"ref_3","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":"Tessler","year":"2005","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_4","unstructured":"Foss, G., and Haugse, E. (1995, January 13\u201319). Using modal test results to develop strain to displacement transformations. Proceedings of the 13th International Conference on Modal Analysis, Nashville, TN, USA."},{"key":"ref_5","unstructured":"Ko, W.L., and Fleischer, V.T. (2011). Extension of ko Straight-Beam Displacement Theory to Deformed Shape Predictions of Slender Curved Structures."},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/0045-7825(85)90114-8","article-title":"A three-node Mindlin plate element with improved transverse shear","volume":"50","author":"Tessler","year":"1985","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.oceaneng.2016.04.025","article-title":"Displacement and stress monitoring of a Panamax containership using inverse finite element method","volume":"119","author":"Kefal","year":"2016","journal-title":"Ocean Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.oceaneng.2015.11.032","article-title":"Displacement and stress monitoring of a chemical tanker based on inverse finite element method","volume":"112","author":"Kefal","year":"2016","journal-title":"Ocean Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.compstruct.2015.02.081","article-title":"A novel approach for displacement and stress monitoring of sandwich structures based on the inverse Finite Element Method","volume":"127","author":"Cerracchio","year":"2015","journal-title":"Compos. Struct."},{"key":"ref_11","unstructured":"Gherlone, M., Cerracchio, P., Mattone, M., Di Sciuva, M., and Tessler, A. (2011, January 13\u201315). Beam shape sensing using inverse finite element method: Theory and experimental validation. Proceedings of the 8th International Workshop on Structural Health Monitoring, Stanford, CA, USA."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.paerosci.2018.04.001","article-title":"Shape sensing methods: Review and experimental comparison on a wing-shaped plate","volume":"99","author":"Gherlone","year":"2018","journal-title":"Prog. Aerosp. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Cho, K., Park, Y.H., and Cho, J. (2019). Model Updating Using Measurements from Sensors Installed in Arbitrary Positions and Directions. Appl. Sci., 9.","DOI":"10.3390\/app9204309"},{"key":"ref_14","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_15","first-page":"1","article-title":"Optimal Sensor Placement for Inverse Finite Element Reconstruction of Three Dimensional Frame Deformation","volume":"2018","author":"Zhao","year":"2018","journal-title":"Int. J. Aerosp. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Islam, M., Hu, G., and Liu, Q. (2018). Online Model Updating and Dynamic Learning Rate-Based Robust Object Tracking. Sensors, 18.","DOI":"10.3390\/s18072046"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Zhao, F., Bao, H., Xue, S., and Xu, Q. (2019). Multi Objective Particle Swarm Optimization of Sensor Distribution Scheme with Consideration of the Accuracy and the Robustness for Deformation Reconstruction. Sensors, 19.","DOI":"10.3390\/s19061306"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Kim, S.H., Kim, N., Park, Y., and Jin, S.S. (2019). A Sequential Framework for Improving Identifiability of FE Model Updating Using Static and Dynamic Data. Sensors, 19.","DOI":"10.3390\/s19235099"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"102554","DOI":"10.1016\/j.tafmec.2020.102554","article-title":"Improved ANN technique combined with Jaya algorithm for crack identification in plates using XIGA and experimental analysis","volume":"107","author":"Khatir","year":"2020","journal-title":"Theor. Appl. Fract. Mech."},{"key":"ref_20","first-page":"360","article-title":"The in situ strain measurements modification based on Fuzzy nets for frame deformation reconstruction","volume":"38","author":"Pan","year":"2018","journal-title":"J. Vib. Meas. Diagn."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Fu, Z., Zhao, Y., Bao, H., and Zhao, F. (2019). Dynamic Deformation Reconstruction of Variable Section WING with Fiber Bragg Grating Sensors. Sensors, 19.","DOI":"10.3390\/s19153350"},{"key":"ref_22","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_23","doi-asserted-by":"crossref","unstructured":"Gherlone, M., Cerracchio, P., Mattone, M., Di Sciuva, M., and Tessler, A. (2011, January 26\u201328). Dynamic shape reconstruction of three-dimensional frame structures using the inverse finite element method. Proceedings of the 3rd ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Corfu, Greece.","DOI":"10.1016\/j.ijsolstr.2012.06.009"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","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":"Gherlone","year":"2014","journal-title":"Smart Mater. Struct."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1007\/s10483-013-1724-x","article-title":"2D mumerical manifold method based on quartic uniform B-spline interpolation and its application in thin plate bending","volume":"34","author":"Wen","year":"2013","journal-title":"Appl. Math. Mech. Engl. Ed."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1109\/TFUZZ.2011.2112369","article-title":"A New Methodology for the Online Adaptation of Fuzzy Self-Structuring Controllers","volume":"19","author":"Cara","year":"2011","journal-title":"IEEE Trans. Fuzzy Syst."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/16\/4602\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:01:48Z","timestamp":1760176908000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/16\/4602"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,16]]},"references-count":26,"journal-issue":{"issue":"16","published-online":{"date-parts":[[2020,8]]}},"alternative-id":["s20164602"],"URL":"https:\/\/doi.org\/10.3390\/s20164602","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,16]]}}}