Computation of Eigenvalues and Eigenfunctions in the Solution of Eddy Current Problems
The solution of the eigenvalue problem in bounded domains with planar and cylindrical stratification is a necessary preliminary task for the construction of modal solutions to canonical problems with discontinuities. The computation of the complex eigenvalue spectrum must be very accurate since losi...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-03-01
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| Series: | Sensors |
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| Online Access: | https://www.mdpi.com/1424-8220/23/6/3055 |
| _version_ | 1797609039822061568 |
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| author | Theodoros Theodoulidis Anastassios Skarlatos Grzegorz Tytko |
| author_facet | Theodoros Theodoulidis Anastassios Skarlatos Grzegorz Tytko |
| author_sort | Theodoros Theodoulidis |
| collection | DOAJ |
| description | The solution of the eigenvalue problem in bounded domains with planar and cylindrical stratification is a necessary preliminary task for the construction of modal solutions to canonical problems with discontinuities. The computation of the complex eigenvalue spectrum must be very accurate since losing or misplacing one of the thereto linked modes will have an important impact on the field solution. The approach followed in a number of previous works is to construct the corresponding transcendental equation and locate its roots in the complex plane using the Newton–Raphson method or Cauchy-integral-based techniques. Nevertheless, this approach is cumbersome, and its numerical stability decreases dramatically with the number of layers. An alternative, approach consists in the numerical evaluation of the matrix eigenvalues for the weak formulation for the respective 1D Sturm–Liouville problem using linear algebra tools. An arbitrary number of layers can thus be easily and robustly treated, with continuous material gradients being a limiting case. Although this approach is often used in high frequency studies involving wave propagation, this is the first time that has been used for the induction problem arising in an eddy current inspection situation. The developed method is implemented in Matlab and is used to deal with the following problems: magnetic material with a hole, a magnetic cylinder, and a magnetic ring. In all the conducted tests, the results are obtained in a very short time, without missing a single eigenvalue. |
| first_indexed | 2024-03-11T05:56:01Z |
| format | Article |
| id | doaj.art-29411af1facc400290f99b37ac885fea |
| institution | Directory Open Access Journal |
| issn | 1424-8220 |
| language | English |
| last_indexed | 2024-03-11T05:56:01Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Sensors |
| spelling | doaj.art-29411af1facc400290f99b37ac885fea2023-11-17T13:45:18ZengMDPI AGSensors1424-82202023-03-01236305510.3390/s23063055Computation of Eigenvalues and Eigenfunctions in the Solution of Eddy Current ProblemsTheodoros Theodoulidis0Anastassios Skarlatos1Grzegorz Tytko2Department of Mechanical Engineering, Faculty of Engineering, University of Western Macedonia, ZEP Campus, 50150 Kozani, GreeceCommissariat à l’Énergie Atomique et aux Énergies Alternatives (CEA), Laboratory for Integration of Systems and Technology (LIST), Université Paris-Saclay, F-91120 Palaiseau, FranceFaculty of Automatic Control, Electronics and Computer Science, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, PolandThe solution of the eigenvalue problem in bounded domains with planar and cylindrical stratification is a necessary preliminary task for the construction of modal solutions to canonical problems with discontinuities. The computation of the complex eigenvalue spectrum must be very accurate since losing or misplacing one of the thereto linked modes will have an important impact on the field solution. The approach followed in a number of previous works is to construct the corresponding transcendental equation and locate its roots in the complex plane using the Newton–Raphson method or Cauchy-integral-based techniques. Nevertheless, this approach is cumbersome, and its numerical stability decreases dramatically with the number of layers. An alternative, approach consists in the numerical evaluation of the matrix eigenvalues for the weak formulation for the respective 1D Sturm–Liouville problem using linear algebra tools. An arbitrary number of layers can thus be easily and robustly treated, with continuous material gradients being a limiting case. Although this approach is often used in high frequency studies involving wave propagation, this is the first time that has been used for the induction problem arising in an eddy current inspection situation. The developed method is implemented in Matlab and is used to deal with the following problems: magnetic material with a hole, a magnetic cylinder, and a magnetic ring. In all the conducted tests, the results are obtained in a very short time, without missing a single eigenvalue.https://www.mdpi.com/1424-8220/23/6/3055nondestructive testingeddy current testingeigenvalues and eigenfunctionscomplex roots |
| spellingShingle | Theodoros Theodoulidis Anastassios Skarlatos Grzegorz Tytko Computation of Eigenvalues and Eigenfunctions in the Solution of Eddy Current Problems Sensors nondestructive testing eddy current testing eigenvalues and eigenfunctions complex roots |
| title | Computation of Eigenvalues and Eigenfunctions in the Solution of Eddy Current Problems |
| title_full | Computation of Eigenvalues and Eigenfunctions in the Solution of Eddy Current Problems |
| title_fullStr | Computation of Eigenvalues and Eigenfunctions in the Solution of Eddy Current Problems |
| title_full_unstemmed | Computation of Eigenvalues and Eigenfunctions in the Solution of Eddy Current Problems |
| title_short | Computation of Eigenvalues and Eigenfunctions in the Solution of Eddy Current Problems |
| title_sort | computation of eigenvalues and eigenfunctions in the solution of eddy current problems |
| topic | nondestructive testing eddy current testing eigenvalues and eigenfunctions complex roots |
| url | https://www.mdpi.com/1424-8220/23/6/3055 |
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