{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:59:46Z","timestamp":1760151586892,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T00:00:00Z","timestamp":1648771200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Nature Science Foundation of China 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":"NUPTSF","award":["NY220074"],"award-info":[{"award-number":["NY220074"]}]},{"DOI":"10.13039\/501100011421","name":"State Key Laboratory of Millimeter Waves","doi-asserted-by":"publisher","award":["K202108"],"award-info":[{"award-number":["K202108"]}],"id":[{"id":"10.13039\/501100011421","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Jiangxi Provincial Outstanding Youth Talent Project of Science and Technology Innovation","award":["20192BCBL23003"],"award-info":[{"award-number":["20192BCBL23003"]}]},{"DOI":"10.13039\/501100004479","name":"Natural Science Foundation of Jiangxi Province","doi-asserted-by":"publisher","award":["20202BAB202004"],"award-info":[{"award-number":["20202BAB202004"]}],"id":[{"id":"10.13039\/501100004479","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, a novel hybrid method of multilevel kernel degeneration and adaptive cross approximation (MLKD-ACA) algorithm with integral kernel truncations is proposed to accelerate solving integral equations using method of moments (MoM), and to simulate the 3D eddy current nondestructive evaluation (NDE) problems efficiently. The MLKD-ACA algorithm with an integral kernel-truncations-based fast solver is symmetrical in the sense that: (1) the impedance matrix, which is generated by the MoM representing the interactions among the field and source basis functions, is symmetrical; (2) the factorized form of the integral kernel (Green\u2019s function) resulted from degenerating it by the Lagrange polynomial interpolation is symmetrical; (3) the structure of the truncated integral kernel for the interactions among the blocks, which ignores the trivial ones of the far block pairs, is symmetrical using the integral kernel truncations technique. The impedance variations predicted by the proposed symmetrical eddy current NDE solver are compared with other methods in benchmarks to show the remarkable accuracy and efficiency.<\/jats:p>","DOI":"10.3390\/sym14040712","type":"journal-article","created":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T21:23:55Z","timestamp":1648848235000},"page":"712","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Rapid Electromagnetic Modeling and Simulation of Eddy Current NDE by MLKD-ACA Algorithm with Integral Kernel Truncations"],"prefix":"10.3390","volume":"14","author":[{"given":"Yang","family":"Bao","sequence":"first","affiliation":[{"name":"College of Electronic and Optical Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210042, China"},{"name":"State Key Laboratory of Millimeter Waves, Southeast University, Nanjing 210096, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhiwei","family":"Liu","sequence":"additional","affiliation":[{"name":"Department of Information Engineering, East China Jiaotong University, Nanchang 330013, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiming","family":"Song","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, Center for Nondestructive Evaluation, Iowa State University, Ames, IA 50010, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TIM.2021.3118092","article-title":"Determination of surface crack orientation based on thin-skin regime using triple-coil drive\u2013pickup eddy-current sensor","volume":"70","author":"Lu","year":"2020","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_2","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_3","first-page":"1","article-title":"Eddy current modelling for nondestructive testing","volume":"8","author":"Pichenot","year":"2003","journal-title":"J. Nondestruct. Test."},{"key":"ref_4","unstructured":"Gilles-Pascaud, C., Pichenot, G., Premel, D., Reboud, C., and Skarlatos, A. (2008, January 25). Modelling of eddy current inspections with CIVA. Proceedings of the 17th World Conference on Nondestructive Testing, Shanghai, China."},{"key":"ref_5","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_6","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_7","doi-asserted-by":"crossref","first-page":"1290","DOI":"10.1109\/TMAG.2004.824585","article-title":"A fast 3-D multipole method for eddy-current computation","volume":"40","author":"Rubinacci","year":"2004","journal-title":"IEEE Trans. Mag."},{"key":"ref_8","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_9","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_10","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_11","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":"NDTE Int."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"141","DOI":"10.3233\/JAE-2000-178","article-title":"A FFT integral formulation using edge-elements for eddy current testing","volume":"11","author":"Tamburrino","year":"2000","journal-title":"Int. J. Appl. Electromag. Mech."},{"key":"ref_13","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_14","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_15","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_16","unstructured":"Stratton, J.A. (1941). Electromagnetic Theory, McGraw-Hill."},{"key":"ref_17","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. Comp."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1490","DOI":"10.1109\/8.43571","article-title":"A conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields","volume":"37","author":"Smith","year":"1989","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_19","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_20","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_21","doi-asserted-by":"crossref","first-page":"3994","DOI":"10.1109\/TAP.2015.2447033","article-title":"Multilevel fast adaptive cross approximation algorithm with characteristic basis functions","volume":"63","author":"Chen","year":"2015","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3815","DOI":"10.1109\/TAP.2019.2963619","article-title":"Efficient solution of electromagnetic scattering problems using multilevel adaptive cross approximation and LU factorization","volume":"68","author":"Gibson","year":"2020","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_23","first-page":"1303301","article-title":"An efficient MLACA-SVD solver for superconducting integrated circuit analysis","volume":"29","author":"Nel","year":"2019","journal-title":"IEEE Trans. Appl. Supercond."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1109\/TAP.2020.3010857","article-title":"Improving the accuracy of the adaptive cross approximation with a convergence criterion based on random sampling","volume":"69","author":"Heldring","year":"2021","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1177\/1094342020918305","article-title":"A parallel hierarchical blocked adaptive cross approximation algorithm","volume":"34","author":"Liu","year":"2022","journal-title":"Int. J. High Perform. Comput. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1023\/A:1021898520626","article-title":"Review of advances in quantitative eddy current nondestructive evaluation","volume":"18","author":"Auld","year":"1999","journal-title":"J. Nondestruct. Eval."},{"key":"ref_27","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_28","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_29","unstructured":"Antimirov, M.Y., Kolyshkin, A.A., and Vaillancourt, R. (1997). Mathematical Models for Eddy Current Testing, Les Publications CRM."},{"key":"ref_30","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_31","doi-asserted-by":"crossref","first-page":"102633","DOI":"10.1016\/j.ndteint.2022.102633","article-title":"Nested kernel degeneration-based boundary element method solver for rapid computation of eddy current signals","volume":"128","author":"Bao","year":"2022","journal-title":"NDTE Int."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Sabbagh, H.A., and Burke, S.K. (1992, January 2). Benchmark problems in eddy-current NDE. Proceedings of the Review of Progress in Quantitative Nondestructive Evaluation, Brunswick, ME, USA.","DOI":"10.1007\/978-1-4615-3344-3_27"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/4\/712\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:48:05Z","timestamp":1760136485000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/4\/712"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,1]]},"references-count":32,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["sym14040712"],"URL":"https:\/\/doi.org\/10.3390\/sym14040712","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,4,1]]}}}