{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T03:30:05Z","timestamp":1761708605959,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2010,9,20]],"date-time":"2010-09-20T00:00:00Z","timestamp":1284940800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>A fast crack profile reconstitution model in nondestructive testing is developed using an arrayed eddy current sensor. The inverse problem is based on an iterative solving of the direct problem using genetic algorithms. In the direct problem, assuming a current excitation, the incident field produced by all the coils of the arrayed sensor is obtained by the translation and superposition of the 2D axisymmetric finite element results obtained for one coil; the impedance variation of each coil, due to the crack, is obtained by the reciprocity principle involving the dyadic Green\u2019s function. For the inverse problem, the surface of the crack is subdivided into rectangular cells, and the objective function is expressed only in terms of the depth of each cell. The evaluation of the dyadic Green\u2019s function matrix is made independently of the iterative procedure, making the inversion very fast.<\/jats:p>","DOI":"10.3390\/s100908696","type":"journal-article","created":{"date-parts":[[2010,9,27]],"date-time":"2010-09-27T11:23:33Z","timestamp":1285586613000},"page":"8696-8704","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Inverse Problem in Nondestructive Testing Using Arrayed Eddy Current Sensors"],"prefix":"10.3390","volume":"10","author":[{"given":"Abdelhalim","family":"Zaoui","sequence":"first","affiliation":[{"name":"Electromagnetic Systems Laboratory, EMP, BP-17 Bordj El Bahri, 16111 Algiers, Algeria"},{"name":"IREENA, 37, Boulevard de l\u2019universit\u00e9, BP-406, 44602 Saint-Nazaire Cedex, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hocine","family":"Menana","sequence":"additional","affiliation":[{"name":"IREENA, 37, Boulevard de l\u2019universit\u00e9, BP-406, 44602 Saint-Nazaire Cedex, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mouloud","family":"Feliachi","sequence":"additional","affiliation":[{"name":"IREENA, 37, Boulevard de l\u2019universit\u00e9, BP-406, 44602 Saint-Nazaire Cedex, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"G\u00e9rard","family":"Berthiau","sequence":"additional","affiliation":[{"name":"IREENA, 37, Boulevard de l\u2019universit\u00e9, BP-406, 44602 Saint-Nazaire Cedex, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2010,9,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1016\/S0963-8695(03)00050-1","article-title":"Design of an eddy-current array probe for crack sizing in steam generator tubes","volume":"36","author":"Huang","year":"2003","journal-title":"NDT&E Int"},{"key":"ref_2","unstructured":"Takagi, T, Uchimoto, T, Nagaya, Y, Huang, H, and Endo, H (2003, January 4\u20136). Design of eddy current camera for non-destructive testing. Fukui, Japan."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1016\/j.ndteint.2005.08.004","article-title":"2D Eddy current sensor array","volume":"39","author":"Grimberg","year":"2006","journal-title":"NDT&E Int"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3407","DOI":"10.1109\/20.312670","article-title":"Reconstruction of crack shape by optimization using eddy current field measurement","volume":"30","author":"Pavo","year":"1994","journal-title":"IEEE Trans. Mag"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1419","DOI":"10.1109\/TMAG.2006.872490","article-title":"Calculation of eddy current testing probe signal with global approximation","volume":"42","author":"Pavo","year":"2006","journal-title":"IEEE Trans. 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Int. de G\u00e9nie \u00c9lectrique"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1745","DOI":"10.1109\/20.877781","article-title":"Eddy current interaction with a thin-opening crack in a plate conductor","volume":"36","author":"Dezhi","year":"2000","journal-title":"IEEE Trans. Magn"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3145","DOI":"10.1109\/20.34388","article-title":"On the use of the magnetic vector potential in the finite element analysis of three-dimensional eddy currents","volume":"25","author":"Biro","year":"1989","journal-title":"IEEE Trans. Magn"},{"key":"ref_11","unstructured":"Geuzaine, C, and Remacle, JF Available online: www.geuz.org\/gmsh (accessed on 17 September 2010)."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1052","DOI":"10.1109\/20.877622","article-title":"Genetic algorithms with assistant chromosomes for inverse shape optimization of electromagnetic devices","volume":"36","author":"Yokose","year":"2000","journal-title":"IEEE Trans. Magn"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1686","DOI":"10.1109\/20.767341","article-title":"Shape optimization of magnetic devices using genetic algorithms with dynamically adjustable parameters","volume":"35","author":"Yokose","year":"1999","journal-title":"IEEE Trans. Magn"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Haupt, RL, and Werner, DH (2007). Genetic Algorithms in Electromagnetics, Wiley and Sons Ltd.-IEEE Press. [1st ed].","DOI":"10.1002\/047010628X"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/10\/9\/8696\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:03:24Z","timestamp":1760220204000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/10\/9\/8696"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,9,20]]},"references-count":14,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2010,9]]}},"alternative-id":["s100908696"],"URL":"https:\/\/doi.org\/10.3390\/s100908696","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2010,9,20]]}}}