{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:27:19Z","timestamp":1760171239954,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,3,17]],"date-time":"2022-03-17T00:00:00Z","timestamp":1647475200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministerio de Ciencia e Innovaci\u00f3n, Espa\u00f1a","award":["PID2019-104347RB-I00"],"award-info":[{"award-number":["PID2019-104347RB-I00"]}]},{"DOI":"10.13039\/501100003451","name":"University of the Basque Country","doi-asserted-by":"publisher","award":["GIU19\/058"],"award-info":[{"award-number":["GIU19\/058"]}],"id":[{"id":"10.13039\/501100003451","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Lock-in vibrothermography has proven to be very useful to characterizing kissing cracks producing ideal, homogeneous, and compact heat sources. Here, we approach real situations by addressing the characterization of non-compact (strip-shaped) heat sources produced by open cracks and inhomogeneous fluxes. We propose combining lock-in vibrothermography data at several modulation frequencies in order to gather penetration and precision data. The approach consists in inverting surface temperature amplitude and phase data by means of a least-squares minimization algorithm without previous knowledge of the geometry of the heat source, only assuming knowledge of the vertical plane where it is confined. We propose a methodology to solve this ill-posed inverse problem by including in the objective function penalty terms based on the expected properties of the solution. These terms are described in a comprehensive and intuitive manner. Inversions of synthetic data show that the geometry of non-compact heat sources is identified correctly and that the contours are rounded due to the penalization. Inhomogeneous smoothly varying fluxes are also qualitatively retrieved, but steep variations of the flux are hard to recover. These findings are confirmed by inversions of experimental data taken on calibrated samples. The proposed methodology is capable of identifying heat sources generated in lock-in vibrothermography experiments.<\/jats:p>","DOI":"10.3390\/s22062336","type":"journal-article","created":{"date-parts":[[2022,3,20]],"date-time":"2022-03-20T21:37:17Z","timestamp":1647812237000},"page":"2336","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Vertical Cracks Excited in Lock-in Vibrothermography Experiments: Identification of Open and Inhomogeneous Heat Fluxes"],"prefix":"10.3390","volume":"22","author":[{"given":"Arantza","family":"Mendioroz","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica Aplicada, Escuela de Ingenier\u00eda de Bilbao, Universidad del Pa\u00eds Vasco UPV\/EHU, Plaza Ingeniero Torres Quevedo 1, 48013 Bilbao, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alazne","family":"Castelo","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica Aplicada, Escuela de Ingenier\u00eda de Bilbao, Universidad del Pa\u00eds Vasco UPV\/EHU, Plaza Ingeniero Torres Quevedo 1, 48013 Bilbao, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2183-2159","authenticated-orcid":false,"given":"Ricardo","family":"Celorrio","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada, EINA\/IUMA, Universidad de Zaragoza, Campus R\u00edo Ebro, Edificio Torres Quevedo, 50018 Zaragoza, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Agust\u00edn","family":"Salazar","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica Aplicada, Escuela de Ingenier\u00eda de Bilbao, Universidad del Pa\u00eds Vasco UPV\/EHU, Plaza Ingeniero Torres Quevedo 1, 48013 Bilbao, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,17]]},"reference":[{"key":"ref_1","unstructured":"Maldague, X.P.V. 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