Reconstruction of the shape of a flaw in ferromagnetic plate by solving inverse problem of magnetostatics and series of direct problems

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Abstract

The article presents a verification technique for solving the inverse geometric problem of magnetostatics in a soft magnetic ferromagnet plate. The technique involves solving a number of direct problems, in which the shape of the defect obtained by solving the inverse geometric problem of magnetostatics is used as a first approximation, and then increasing or decreasing the depth of the defect without changing the shape of the boundary surface — comparing the topographies of the magnetic field components obtained during measurements above the plate surface and calculated (as a result of solving the direct problem) at the same points of the components of the magnetic stray field from the reconstructed three-dimensional defect. As a result of applying the technique, the geometric parameters of the defect under study can also be refined. Obtaining the initial conditions for solving the inverse problem and solving direct problems of magnetostatics is carried out using the finite element method in the ELMER program. The technique works with one-sided access to any surface of the plate (a defect-free surface or a surface with a defect).

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About the authors

A. V. Nikitin

M.N. Mikheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences

Author for correspondence.
Email: an@imp.uran.ru
Russian Federation, 620108 Yekaterinburg, S. Kovalevskaya Str., 18

L. V. Mikhaylov

M.N. Mikheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences

Email: mikhaylov_lv@imp.uran.ru
Russian Federation, 620108 Yekaterinburg, S. Kovalevskaya Str., 18

A. V. Mikhaylov

M.N. Mikheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences

Email: mikhaylov_lv@imp.uran.ru
Russian Federation, 620108 Yekaterinburg, S. Kovalevskaya Str., 18

Yu. L. Gobov

M.N. Mikheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences

Email: mikhaylov_lv@imp.uran.ru
Russian Federation, 620108 Yekaterinburg, S. Kovalevskaya Str., 18

V. N. Kostin

M.N. Mikheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences

Email: mikhaylov_lv@imp.uran.ru
Russian Federation, 620108 Yekaterinburg, S. Kovalevskaya Str., 18

Ya. G. Smorodinskii

M.N. Mikheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences

Email: mikhaylov_lv@imp.uran.ru
Russian Federation, 620108 Yekaterinburg, S. Kovalevskaya Str., 18

References

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  11. Nikitin A.V., Gobov Yu.L., Mikhailov A.V., Mikhailov L.V. Methodology for solving the inverse geometric problem of magnetostatics for surface defects of a soft magnetic ferromagnet // Defectoscopiya. 2022. No. 58. P. 24—34.
  12. Nikitin A.V., Mikhailov A.V., Mikhailov L.V., Gobov Yu.L., Kostin V.N., Smorodinskii Ya.G. Scope of applicability of the technique for constructing magnetic induction lines for flaw detection of extended objects // Defectoscopiya. 2023. No. 59. P. 51—59.
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Supplementary files

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2. Figure 1. Example of the function f(x, y) represented as a segment of the double Fourier series in the investigated area.

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3. Figure 2. Cross section of the studied ferromagnetic plate in the transverse direction. The magnetizing field H0 is directed along the OX axis.

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4. Figure 3. Cross-section of the plate with reconstruction of the metal boundary in the defect region obtained by solving the inverse geometric problem.

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5. Figure 4. Boundary of a dome-shaped defect with dimensions 25×25×3 mm obtained as a result of solving the inverse geometric problem of magnetostatics for a 10 mm thick plate of steel 20.

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6. Figure 5. Cross section of the plate. Reconstruction of the defect shape was performed taking into account its real dimensions on the plate surface.

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7. Figure 6. Graph of values of x-components of the magnetic field strength on the measurement surface, on the line passing through the center of the defect along the OX axis. 1 - Hx0 values of the investigated defect; 2 - Hxr values of the reconstructed defect.

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8. Figure 7. Dependence of the average deviation D on the vertical displacement of the defect surface. The positive sign of S corresponds to the increase of the defect depth.

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