{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T06:06:49Z","timestamp":1778738809444,"version":"3.51.4"},"reference-count":31,"publisher":"MDPI AG","issue":"13","license":[{"start":{"date-parts":[[2023,7,7]],"date-time":"2023-07-07T00:00:00Z","timestamp":1688688000000},"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":["62101356"],"award-info":[{"award-number":["62101356"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022YFC3004802"],"award-info":[{"award-number":["2022YFC3004802"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key R&amp;D Program of China","doi-asserted-by":"publisher","award":["62101356"],"award-info":[{"award-number":["62101356"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key R&amp;D Program of China","doi-asserted-by":"publisher","award":["2022YFC3004802"],"award-info":[{"award-number":["2022YFC3004802"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Pipeline magnetic flux leakage inspection is widely used in the evaluation of material defect detection due to its advantages of having no coupling agent and easy implementation. The quantification of defect size is an important part of magnetic flux leakage testing. Defects of different geometrical dimensions produce signal waveforms with different characteristics after excitation. The key to achieving defect quantification is an accurate description of the relationship between the magnetic leakage signal and the size. In this paper, a calculation model for solving the defect leakage field based on the non-uniform magnetic charge distribution of magnetic dipoles is developed. Based on the traditional uniformly distributed magnetic charge model, the magnetic charge density distribution model is improved. Considering the variation of magnetic charge density with different depth positions, the triaxial signal characteristics of the defect are obtained by vector synthesis calculation. Simultaneous design of excitation pulling experiment. The leakage field distribution of rectangular defects with different geometries is analyzed. The experimental results show that the change in defect size will have an impact on the area of the defect leakage field distribution, and the larger the length and wider the width of the defect, the more sensitive the impact on the leakage field distribution. The solution model is consistent with the experimentally obtained leakage signal distribution law, and the model is a practical guide by which to improve the quality of defect evaluation.<\/jats:p>","DOI":"10.3390\/s23136221","type":"journal-article","created":{"date-parts":[[2023,7,10]],"date-time":"2023-07-10T01:02:50Z","timestamp":1688950970000},"page":"6221","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Research on the Forward Solving Method of Defect Leakage Signal Based on the Non-Uniform Magnetic Charge Model"],"prefix":"10.3390","volume":"23","author":[{"given":"Pengfei","family":"Gao","sequence":"first","affiliation":[{"name":"College of Information Science and Engineering, Shenyang University of Technology, Shenyang 110870, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hao","family":"Geng","sequence":"additional","affiliation":[{"name":"College of Information Science and Engineering, Shenyang University of Technology, Shenyang 110870, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lijian","family":"Yang","sequence":"additional","affiliation":[{"name":"College of Information Science and Engineering, Shenyang University of Technology, Shenyang 110870, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuming","family":"Su","sequence":"additional","affiliation":[{"name":"College of Information Science and Engineering, Shenyang University of Technology, Shenyang 110870, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,7]]},"reference":[{"key":"ref_1","unstructured":"Nestleroth, B., Burgoon, D., and Haines, H. (1996, January 24\u201329). Determining Corrosion Defect Geometry from Magnetic Flux Leakage Pig Data. Proceedings of the Corrosion 96, Denver, CO, USA."},{"key":"ref_2","first-page":"38","article-title":"Magnetic flux leakage signal analysis for metal-loss defect characterization","volume":"1","author":"Nestleroth","year":"1997","journal-title":"NDT E Int."},{"key":"ref_3","unstructured":"Reber, K. (2006, January 25\u201329). Reliability of Flaw Size Calculation based on Magnetic Flux Leakage Inspection of Pipelines. Proceedings of the 9th European Conference on NDT, (ECNDT 2006), Berlin, Germany."},{"key":"ref_4","unstructured":"Reber, K., Boenisch, A., and Asher, S. (2023, March 01). Magnetic Eddy Current (MEC\u2122) As a Novel Technique for the Internal Inspection of CRA-Lined Pipe. Available online: https:\/\/ppsa-online.com\/papers\/15-Aberdeen\/2015-08-Innospection-paper.pdf."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1573","DOI":"10.1109\/20.497552","article-title":"A comparative study of 3D and axisymmetric magnetizer assemblies used in magnetic flux leakage inspection of pipelines","volume":"32","author":"Katragadda","year":"1996","journal-title":"IEEE Trans. Magn."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1581","DOI":"10.1109\/20.497554","article-title":"Alternative magnetic flux leakage modalities for pipeline inspection","volume":"32","author":"Katragadda","year":"1996","journal-title":"IEEE Trans. Magn."},{"key":"ref_7","first-page":"247","article-title":"Research on quantification technology of magnetic flux leakage detection for oil and gas pipeline defects","volume":"25","author":"Wang","year":"2004","journal-title":"Statistics"},{"key":"ref_8","first-page":"304","article-title":"Effect of groove depth on leakage magnetic field","volume":"19","author":"Zhong","year":"1997","journal-title":"Nondestruct. Test."},{"key":"ref_9","first-page":"551","article-title":"Progress of magnetic dipole theory in China in the past 20 years","volume":"12","author":"Zhong","year":"2000","journal-title":"Nondestruct. Test."},{"key":"ref_10","first-page":"2","article-title":"Magnetic dipole chains and magnetic particle flaw detection","volume":"25","author":"Zhong","year":"2003","journal-title":"Nondestruct. Test."},{"key":"ref_11","first-page":"154","article-title":"Measurement of magnetic flux leakage due to oblique surface cracks and estimation methods of its oblique angle","volume":"31","author":"Uetake","year":"1989","journal-title":"Trans. Natl. Res. Inst. Met."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/S0963-8695(97)00002-9","article-title":"Magnetic flux leakage by adjacent parallel surface slots","volume":"30","author":"Uetake","year":"1997","journal-title":"NDT E Int."},{"key":"ref_13","first-page":"164","article-title":"Reconstruction method of complex defects in uniform sampling magnetic flux leakage detection based on space mapping","volume":"39","author":"Wu","year":"2018","journal-title":"J. Instrum. Meter"},{"key":"ref_14","first-page":"15","article-title":"Magnetic flux leakage defect contour reconstruction based on improved artificial bee colony algorithm","volume":"41","author":"Han","year":"2016","journal-title":"Fire Command. Control."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"862","DOI":"10.1134\/S1061830917120075","article-title":"Magnetic Flux Leakage Signal Inversion Based on Improved Efficient Population Utilization Strategy for Particle Swarm Optimization","volume":"53","author":"Han","year":"2017","journal-title":"Russ. J. Nondestruct. Test."},{"key":"ref_16","first-page":"592","article-title":"3D contour reconstruction of magnetic flux leakage defects based on sparse LS-SVM","volume":"29","author":"Ji","year":"2008","journal-title":"Army J."},{"key":"ref_17","first-page":"876","article-title":"Analysis of leakage field based on three-dimensional finite element method","volume":"28","author":"Ji","year":"2007","journal-title":"J. Mil. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1049\/iet-smt.2009.0054","article-title":"Sizing of multiple cracks using magnetic flux leakage measurements","volume":"4","author":"Ravan","year":"2010","journal-title":"IET Sci. Meas. Technol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2058","DOI":"10.1109\/TMAG.2008.923228","article-title":"A Space Mapping Methodology for Defect Characterization from Magnetic Flux Leakage Measurements","volume":"44","author":"Amineh","year":"2008","journal-title":"IEEE Trans. Magn."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"418","DOI":"10.1049\/iet-smt.2014.0173","article-title":"Three-dimensional defect inversion from magnetic flux leakage signals using iterative neural network","volume":"9","author":"Chen","year":"2015","journal-title":"IET Sci. Meas. Technol."},{"key":"ref_21","first-page":"1","article-title":"Some analytical solutions for the magnetic dipole model of defect leakage magnetic field","volume":"37","author":"Shi","year":"2015","journal-title":"Nondestruct. Test."},{"key":"ref_22","first-page":"097501","article-title":"An analytical model for magnetic Barkhausen stress detection in ferromagnetic plates","volume":"71","author":"Zhang","year":"2022","journal-title":"J. Physics."},{"key":"ref_23","first-page":"266","article-title":"The influence of the direction of surface defects on the distribution of leakage magnetic field","volume":"66","author":"Wu","year":"2017","journal-title":"Accord. Phys."},{"key":"ref_24","first-page":"5","article-title":"Transient simulation analysis of oil and gas long-distance pipeline crack leakage detection","volume":"30","author":"Wu","year":"2009","journal-title":"J. Pet."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1520\/JTE20130133","article-title":"3-D FEM Simulation and Analysis on the Best Range of Lift-off Values in MFL Testing","volume":"3","author":"Wu","year":"2015","journal-title":"J. Test. Eval."},{"key":"ref_26","first-page":"97","article-title":"Magnetic Flux Leakage Defect Reconstruction Based on Gravitational Search Algorithm","volume":"41","author":"Han","year":"2016","journal-title":"Fire Command. Control."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2427","DOI":"10.1088\/0022-3727\/36\/20\/001","article-title":"A model for magnetic flux leakage signal predictions","volume":"36","author":"Mandache","year":"2003","journal-title":"J. Phys. D Appl. Phys."},{"key":"ref_28","first-page":"269","article-title":"Defect separation considerations in magnetic flux leakage inspection","volume":"47","author":"Mandache","year":"2005","journal-title":"Insight-Non-Destr. Test. Cond. Monit."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1109\/TMAG.2008.2006246","article-title":"Study of Lift-Off Invariance for Pulsed Eddy-Current Signals","volume":"45","author":"Tian","year":"2009","journal-title":"IEEE Trans. Magn."},{"key":"ref_30","first-page":"2740","article-title":"Inspection of three-dimensional geographic coordinate in pipeline","volume":"22","author":"Yang","year":"2014","journal-title":"Guangxue Jingmi Gongcheng\/Opt. Precis. Eng."},{"key":"ref_31","first-page":"20200362","article-title":"Research on the magnetic flux leakage field distribution characteristics of defect in low-frequency electromagnetic detection technique","volume":"18","author":"Yang","year":"2021","journal-title":"Inst. Electron. Inf. Commun. Eng. (IEICE)"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/13\/6221\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:07:58Z","timestamp":1760126878000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/13\/6221"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,7]]},"references-count":31,"journal-issue":{"issue":"13","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["s23136221"],"URL":"https:\/\/doi.org\/10.3390\/s23136221","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,7]]}}}