{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T23:58:51Z","timestamp":1780444731411,"version":"3.54.1"},"reference-count":46,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2022,10,8]],"date-time":"2022-10-08T00:00:00Z","timestamp":1665187200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NUPTSF","award":["NY220074"],"award-info":[{"award-number":["NY220074"]}]},{"name":"NUPTSF","award":["62001245"],"award-info":[{"award-number":["62001245"]}]},{"name":"NUPTSF","award":["BK20200757"],"award-info":[{"award-number":["BK20200757"]}]},{"name":"NUPTSF","award":["K202108"],"award-info":[{"award-number":["K202108"]}]},{"name":"National Nature Science Foundation of China for Youth","award":["NY220074"],"award-info":[{"award-number":["NY220074"]}]},{"name":"National Nature Science Foundation of China for Youth","award":["62001245"],"award-info":[{"award-number":["62001245"]}]},{"name":"National Nature Science Foundation of China for Youth","award":["BK20200757"],"award-info":[{"award-number":["BK20200757"]}]},{"name":"National Nature Science Foundation of China for Youth","award":["K202108"],"award-info":[{"award-number":["K202108"]}]},{"name":"Natural Science Foundation of Jiangsu Province for Youth","award":["NY220074"],"award-info":[{"award-number":["NY220074"]}]},{"name":"Natural Science Foundation of Jiangsu Province for Youth","award":["62001245"],"award-info":[{"award-number":["62001245"]}]},{"name":"Natural Science Foundation of Jiangsu Province for Youth","award":["BK20200757"],"award-info":[{"award-number":["BK20200757"]}]},{"name":"Natural Science Foundation of Jiangsu Province for Youth","award":["K202108"],"award-info":[{"award-number":["K202108"]}]},{"name":"State Key Laboratory of Millimeter Waves","award":["NY220074"],"award-info":[{"award-number":["NY220074"]}]},{"name":"State Key Laboratory of Millimeter Waves","award":["62001245"],"award-info":[{"award-number":["62001245"]}]},{"name":"State Key Laboratory of Millimeter Waves","award":["BK20200757"],"award-info":[{"award-number":["BK20200757"]}]},{"name":"State Key Laboratory of Millimeter Waves","award":["K202108"],"award-info":[{"award-number":["K202108"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Model assisted probability of detection (MAPoD) is crucial for quantifying the inspection capability of a nondestructive testing (NDT) system which uses the coil or probe to sense the size and location of the cracks. Unfortunately, it may be computationally intensive for the simulation models. To improve the efficiency of the MAPoD, in this article, an efficient 3D eddy current nondestructive evaluation (ECNDE) forward solver is proposed to make estimations for PoD study. It is the first time that singular value decomposition (SVD) is used as the recompression technique to improve the overall performance of the adaptive cross approximation (ACA) algorithm-based boundary element method (BEM) ECNDE forward solver for implementation of PoD. Both the robustness and efficiency of the proposed solver are demonstrated and testified by comparing the predicted impedance variations of the coil with analytical, semi-analytical and experimental benchmarks. Calculation of PoD curves assisted by the proposed simulation model is performed on a finite thickness plate with a rectangular surface flaw. The features, which are the maximum impedance variations of the coil for various flaw lengths, are obtained entirely by the proposed model with selection of the liftoff distance as the uncertain parameter in a Gaussian distribution. The results show that the proposed ACA-SVD based BEM fast ECNDE forward solver is an excellent simulation model to make estimations for MAPoD study.<\/jats:p>","DOI":"10.3390\/s22197625","type":"journal-article","created":{"date-parts":[[2022,10,10]],"date-time":"2022-10-10T05:12:21Z","timestamp":1665378741000},"page":"7625","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Efficient Model Assisted Probability of Detection Estimations in Eddy Current NDT with ACA-SVD Based Forward Solver"],"prefix":"10.3390","volume":"22","author":[{"given":"Yang","family":"Bao","sequence":"first","affiliation":[{"name":"College of Electronic and Optical Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210023, China"},{"name":"State Key Laboratory of Millimeter Waves, Southeast University, Nanjing 210096, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Minxuan","family":"Xu","sequence":"additional","affiliation":[{"name":"College of Electronic and Optical Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210023, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiahao","family":"Qiu","sequence":"additional","affiliation":[{"name":"College of Electronic and Optical Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210023, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiming","family":"Song","sequence":"additional","affiliation":[{"name":"Center for Nondestructive Evaluation, Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011-2042, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4557","DOI":"10.1016\/S0020-7683(97)00038-3","article-title":"An integral equation for changes in the structural dynamics characteristics of damaged structures","volume":"34","author":"Luo","year":"1997","journal-title":"Int. J. Solids Struct."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1109\/MIM.2007.364985","article-title":"Microwave and millimeter wave nondestructive testing and evaluation-Overview and recent advances","volume":"10","author":"Kharkovsky","year":"2007","journal-title":"IEEE Instrum. Meas. Mag."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"112951","DOI":"10.1016\/j.compstruct.2020.112951","article-title":"Advances, limitations and prospects of nondestructive testing and evaluation of thick composites and sandwich structures: A state-of-the-art review","volume":"256","author":"Nsengiyumva","year":"2021","journal-title":"Compos. Struct."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"112030","DOI":"10.1016\/j.sna.2020.112030","article-title":"A novel magnetic force transmission eddy current array probe and its application for nondestructive testing of defects in pipeline structures","volume":"309","author":"Xie","year":"2020","journal-title":"Sens. Actuators A Phys."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ciampa, F., Mahmoodi, P., Pinto, F., and Meo, M. (2018). Recent advances in active infrared thermography for non-destructive testing of aerospace components. Sensors, 18.","DOI":"10.3390\/s18020609"},{"key":"ref_6","first-page":"911","article-title":"Nondestructive testing research and development efforts at Argonne National Laboratory: An overview","volume":"76","author":"Bakhtiari","year":"2018","journal-title":"Mater. Eval."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1109\/58.265834","article-title":"Ultrasonic flaw detection based on mathematical morphology","volume":"41","author":"Saniie","year":"1994","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1109\/58.585186","article-title":"Signal detection and noise suppression using a wavelet transform signal processor: Application to ultrasonic flaw detection","volume":"44","author":"Abbate","year":"1997","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1016\/j.engfailanal.2013.06.008","article-title":"Reliability considerations of NDT by probability of detection (POD) determination using ultrasound phased array","volume":"35","author":"Kurz","year":"2013","journal-title":"Eng. Fail. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.ultras.2018.02.012","article-title":"Issues in estimating probability of detection of NDT techniques\u2014A model assisted approach","volume":"87","author":"Rentala","year":"2018","journal-title":"Ultrasonics"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"102300","DOI":"10.1016\/j.ndteint.2020.102300","article-title":"Experimental and numerical methodology to obtain the probability of detection in eddy current NDT method","volume":"114","author":"Bato","year":"2020","journal-title":"NDT E Int."},{"key":"ref_12","first-page":"1733","article-title":"Demonstration of model-assisted probability of detection evaluation methodology for eddy current nondestructive evaluation","volume":"Volume 1430","author":"Aldrin","year":"2012","journal-title":"Proceedings of the Review of Progress in Quantitative Nondestructive Evaluation"},{"key":"ref_13","first-page":"1589","article-title":"Case studies for model-assisted probabilistic reliability assessment for structural health monitoring systems","volume":"Volume 1335","author":"Aldrin","year":"2011","journal-title":"Proceedings of the Review of Progress in Quantitative Nondestructive Evaluation"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.ndteint.2015.07.001","article-title":"Efficient finite element modelling of eddy current probability of detection with transmitter\u2013receiver sensors","volume":"75","author":"Rosell","year":"2015","journal-title":"NDT E Int."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1857","DOI":"10.1109\/20.250768","article-title":"Numerical model based approach for estimating probability of detection in NDE applications","volume":"29","author":"Rajesh","year":"1993","journal-title":"IEEE Trans. Magn."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2528\/PIERL20092701","article-title":"A model assisted probability of detection approach for ECNDT of hidden defect in aircraft structures","volume":"95","author":"Chebout","year":"2021","journal-title":"Prog. Electromagn. Res. Lett."},{"key":"ref_17","first-page":"1903","article-title":"Statistical study of ECT detection around fasteners using simulation based POD curves","volume":"Volume 1211","author":"Reboud","year":"2010","journal-title":"Proceedings of the Review of Progress in Quantitative Nondestructive Evaluation"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Hall, W.S. (1994). The Boundary Element Method, Springer.","DOI":"10.1007\/978-94-011-0784-6"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"8128","DOI":"10.1063\/1.356511","article-title":"Eddy-current interaction with an ideal crack. I. The forward problem","volume":"75","author":"Bowler","year":"1994","journal-title":"J. Appl. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.ndteint.2009.08.005","article-title":"Rapid computation of eddy current signals from narrow cracks","volume":"43","author":"Theodoulidis","year":"2009","journal-title":"NDT E Int."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"102401","DOI":"10.1016\/j.ndteint.2020.102401","article-title":"Integration of multiple response signals into the probability of detection modelling in eddy current NDE of flaws","volume":"118","author":"Baskaran","year":"2021","journal-title":"NDT E Int."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"102499","DOI":"10.1016\/j.ndteint.2021.102499","article-title":"Probability of detection modelling in eddy current NDE of flaws integrating multiple correlated variables","volume":"123","author":"Baskaran","year":"2021","journal-title":"NDT E Int."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Chew, W.C., Tong, M.S., and Hu, B. (2008). Integral Equation Methods for Electromagnetic and Elastic Waves, Morgan & Claypool. [1st ed.].","DOI":"10.1007\/978-3-031-01707-0"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1109\/8.558669","article-title":"Fast solution methods in electromagnetics","volume":"45","author":"Chew","year":"1997","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1488","DOI":"10.1109\/8.633855","article-title":"Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects","volume":"45","author":"Song","year":"1997","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1002\/mop.4650100107","article-title":"Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering","volume":"10","author":"Song","year":"1995","journal-title":"Microw. Opt. Technol. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1007\/PL00005410","article-title":"Approximation of boundary element matrices","volume":"86","author":"Bebendorf","year":"2000","journal-title":"Numer. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4600","DOI":"10.1109\/TAP.2011.2165476","article-title":"Multilevel adaptive cross approximation (MLACA)","volume":"59","author":"Tamayo","year":"2011","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ndteint.2019.03.005","article-title":"Multilevel adaptive cross approximation for efficient modeling of 3D arbitrary shaped eddy current NDE problems","volume":"104","author":"Bao","year":"2019","journal-title":"NDT E Int."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1007\/s10921-018-0521-1","article-title":"Adaptive cross approximation algorithm for accelerating BEM in eddy current nondestructive evaluation","volume":"37","author":"Bao","year":"2018","journal-title":"J. Nondestruct. Eval."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/s006070050015","article-title":"A sparse matrix arithmetic based on H-matrices. Part I: Introduction to H-matrices","volume":"62","author":"Hackbusch","year":"1999","journal-title":"Computing"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3147","DOI":"10.1109\/TAP.2009.2028665","article-title":"An H2-matrix-based integral-equation solver of reduced complexity and controlled accuracy for solving electrodynamic problems","volume":"57","author":"Chai","year":"2009","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s10921-022-00849-2","article-title":"Multilevel kernel degeneration\u2013adaptive cross approximation method to model eddy current NDE problems","volume":"41","author":"Bao","year":"2022","journal-title":"J. Nondestruct. Eval."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"102544","DOI":"10.1016\/j.ndteint.2021.102544","article-title":"Integral equation fast solver with truncated and degenerated kernel for computing flaw signals in eddy current non-destructive testing","volume":"124","author":"Bao","year":"2021","journal-title":"NDT E Int."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1216\/JIE-2009-21-3-331","article-title":"Recompression techniques for adaptive cross approximation","volume":"21","author":"Bebendorf","year":"2009","journal-title":"J. Integral Equ. Appl."},{"key":"ref_36","unstructured":"Stratton, J.A. (1941). Electromagnetic Theory, McGraw-Hill."},{"key":"ref_37","unstructured":"Yang, M. (2010). Efficient Method for Solving Boundary Integral Equation in Diffusive Scalar Problem and Eddy Current Nondestructive Evaluation. [Ph.D. Dissertation, Iowa State University]."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"20190583","DOI":"10.1098\/rsta.2019.0583","article-title":"Analysis of electromagnetic non-destructive evaluation modelling using Stratton-Chu formulation-based fast algorithm","volume":"378","author":"Bao","year":"2020","journal-title":"Philos. Trans. R. Soc. A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1109\/TAP.1982.1142818","article-title":"Electromagnetic scattering by surfaces of arbitrary shape","volume":"30","author":"Rao","year":"1982","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"856","DOI":"10.1137\/0907058","article-title":"GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems","volume":"7","author":"Saad","year":"1986","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/s00211-005-0618-1","article-title":"Hybrid cross approximation of integral operators","volume":"101","author":"Grasedyck","year":"2005","journal-title":"Numer. Math."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"3123","DOI":"10.1098\/rspa.2005.1509","article-title":"Eddy current coil interaction with a right-angled conductive wedge","volume":"461","author":"Theodoulidis","year":"2005","journal-title":"Proc. R. Soc."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2829","DOI":"10.1063\/1.1656680","article-title":"Analytical solutions to eddy-current probe-coil problems","volume":"39","author":"Dodd","year":"1968","journal-title":"J. Appl. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/978-1-4615-3344-3_27","article-title":"Benchmark problems in eddy-current NDE","volume":"Volume 28","author":"Sabbagh","year":"1992","journal-title":"Proceedings of the Review of Progress in Quantitative Nondestructive Evaluation"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/s10921-019-0628-z","article-title":"Comparison of \u00e2 versus a and Hit\/Miss POD-Estimation methods: A European viewpoint","volume":"38","author":"Virkkunen","year":"2019","journal-title":"J. Nondestruct. Eval."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"592","DOI":"10.1214\/07-STS249","article-title":"The epic story of maximum likelihood","volume":"22","author":"Stigler","year":"2007","journal-title":"Stat. Sci."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/19\/7625\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:48:13Z","timestamp":1760143693000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/19\/7625"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,8]]},"references-count":46,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["s22197625"],"URL":"https:\/\/doi.org\/10.3390\/s22197625","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,8]]}}}