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3D FEM analysis in magnetic flux leakage method
NDT&E International 39 (2006) 61–66 www.elsevier.com/locate/ndteint
3D FEM analysis in magnetic flux leakage method Huang Zuoying*, Que Peiwen, Chen Liang Department of Information Measurement Technology and Instruments, Institute of Automatic Detection, Shanghai Jiaotong University, Shanghai 200030, People’s Republic of China Received 8 May 2005; revised 7 June 2005; accepted 7 June 2005 Available online 2 August 2005
Abstract The magnetic flux leakage (MFL) method is currently the most commonly used pipeline inspection technique. In this paper, 3D FEM is used to analyze the MFL signals, a generalized potential formulation to the magnetostatic field MFL problem is discussed, typical 3D defects are accurately modeled and detail MFL signal in test surface are calculated by the method. The relation between defect parameters and MFL signals are also analyzed. q 2005 Elsevier Ltd. All rights reserved. PACS: 75.80; 07.55.G; 07.55 Keywords: MFL; FEM; Defect; Inspection; Pipeline
1. Introduction Magnetic flux leakage technique is generally considered to be the most cost-effective method for corrosion monitoring. In this technique, the wall of the pipeline is magnetized axially to near saturation flux density. If, at some point, the wall thickness is reduced by a defect, a higher fraction of the magnetic flux will ‘leak’ from the wall into the air inside and outside the pipe. The magnetic leakage field measured on the nearby of the pipe contains information about the pipe conditions. The MFL inspection tool, propelled by the oil pressure or driven equipment, magnetizes the pipe wall; MFL signals due to any irregularities are detected and stored within the tool. The inspection vehicle consists of a magnetizer which is made of permanent magnet or coil and circumferentially distributed sensor assembly which is hall sensor or signal pick up coil [1,2]. Analytical approaches to the modeling of MFL have largely been difficult due to both the awkward boundaries * Corresponding author. Tel.: C86 21 6293 2851; fax: C86 21 6293 2810. E-mail address: [email protected] (H. Zuoying).
0963-8695/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ndteint.2005.06.006
associated with realistic defect shapes and the lack of generality that ensures when making the necessary assumptions needed to obtain tractable analytical solutions [3,4]. Viable models are needed in order to understand the ways in which fields and defects interact to produce measurable indications, to help in the design of detection probes for diverse applications, to simulate those testing environments which are difficult and/or expensive to replicate in the laboratory, and perhaps most important in many of the critical testing situations facing industries today, to provide training data for automated defect characterization schemes. More recently, 2D finite element methods have been used to research MFL signals under different defects shapes, materials, magnetizing situation and so on and it is also proved to be an effective method [5]. However, in 2D FEM defects are also treated as 2D profile instead of actually 3D geometry, and the result MFL signal is single channel whereas the actual signals are multichannel. In this paper, 3D FEM is adopted to analyze the MFL method, accurate 3D defect are modeled and detail MFL signal in test surface are calculated by the method. The relation between defect geometry and MFL signals are discussed detailed, the influence of lift-off value is also studied.
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2. 2 Three dimensional finite element computation of MFL numerical models The MFL problem can be treated as magnetostatic problem by magnetic scalar potential method. It can be expressed by Maxwell’s equations below: V !fHg Z fJs g
(1)
V$fBg Z 0
(2)
where {H} is magnetic field intensity vector, {Js} is applied source current density vector and {B} is magnetic flux density vector. The field equations are supplemented by the constitutive relation that describes the behavior of electromagnetic materials. In permanent magnet region fBg Z ½mfHg C m0 fM0 g
(3)
In other region fBg Z ½mfHg
(4)
Fig. 3. Vector plot of calculated B.
In the domain of a magnetostatic field problem, a solution is sought which satisfies the Maxwell Eqs. (1) and (2) and the constitutive relation (3) in the following form: [6,7] fHg Z fHg gKVfg
where m is magnetic permeability matrix, {M0} is remanent intrinsic magnetization vector.
Fig. 1. Geometry of 3D MFL model.
Fig. 2. B–H curve of the material.
Fig. 4. Surface plot of axial magnetic flux density.
(5)
H. Zuoying et al. / NDT&E International 39 (2006) 61–66
63
The development of {Hg} varies depending on the problem and the formulation. Basically, {Hg} must satisfy Ampere’s law so that the remaining part of the field can be derived as the gradient of the generalized scalar potential fg. This ensures that fg is singly valued. Additionally, the absolute value of {Hg} must be greater than that of Dfg. In other words, {Hg} should be a good approximation of the total field. This avoids difficulties with cancellation errors. The finite element matrix equations can be derived by variational principles. The element matrices of scalar potential can be presented in the following form: ½K m Z ½K L C ½K N
Fig. 5. Relation between MFLpp and defect depth.
V$½mVfg KV$½mfHg gKV$m0 ðM0 Þ Z f0g
ð
½K Z ðVfNgT ÞT ½mðVfNgT ÞdV L
(8)
V
(6) N
where {Hg} is preliminary magnetic field, fg is generalized potential.
(7)
½K Z
ð
vmh dV ðfHgT VfNgT ÞT ðfHgT VfNgT Þ jHj vjHj
V
Fig. 6. Contour plot of magnetic flux density at different defect depth values (lZ10 and wZ10 mm).
(9)
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H. Zuoying et al. / NDT&E International 39 (2006) 61–66
3. Analysis procedure
Fig. 7. Relation between MFLpp and defect width.
where {N} is the element shape functions (fZ {N}T{fe}) and vmh =ðvjHjÞ is the derivative of permeability with respect to magnitude of the magnetic field intensity.
Calculations are made for a simplified MFL detector. Fig. 1 shows the geometry of the problem in a 3D model for the influence of the flaw sizes and lift-off value on the magnetic leakage flux density from a rectangular flaw in the specimen. The magnetic circuit is constituted by yoke, magnets, brushes and specimen. l, w and d denote the length, width and depth size of flaw, respectively. The flaw locates at the center of the specimen. We set the specimen length to be 500 mm breath to be 120 mm and thickness to be 14 mm. Two permanent magnets, made of NdFeB material, are used as the magnetic flux induction; this material has a coercive force of 872,000 A/m. The material of specimen is X52, and that of yoke is mild steel. Fig. 2 shows the B–H curve of them. We assume the same B–H curve for brush. The calculations are made using the ANSYS finite element software. The most fundamental element of 3D is a tetrahedron. In order to have a precise result, we refine the elements near the flaw.
Fig. 8. Contour plot of magnetic flux density at different defect width values (lZ10 mm and dZ8 mm).
H. Zuoying et al. / NDT&E International 39 (2006) 61–66
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Fig. 3 shows the vector plot of the calculated magnetic flux density.
4. Calculated results and consideration Fig. 4 shows a surface plot of the amplitude of the radial and axial component of magnetic flux density in the vicinity of a flaw whose sizes are lZ10 mm, wZ10 mm, and dZ 10 mm. The signal patterns are very similar to each other and to experimental measurements. The two peaks in the amplitude are due to the flux being diverted into and returning from the air around the flaw. 4.1. Relation between MFL peak–peak value and the depth of defect Fig. 9. Relation between MFLpp and defect length.
Fig. 5 shows the relation between the MFL peak–peak value (MFLpp) and the percentage of defect depth to wall thickness at different lift-off values with constant width (10 mm) and length (mm). It can be seen that MFLpp is
Fig. 10. Contour plot of magnetic flux density at different defect length values (wZ10 mm and dZ8 mm).
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H. Zuoying et al. / NDT&E International 39 (2006) 61–66
values. It can be seen that the defect length does not affect MFLpp much, the MFLpp decreases slightly when defect length increases from 10 to 80 mm. Fig. 10 shows the pattern of MFL at different length values, the patterns are similar and the magnitudes are close, the distance of MFL peaks varies with the length of defect. 4.4. The influence of lift-off value to MFL peak–peak value Fig. 11 shows the relation between the MFL peak–peak value (MFLpp) and the lift-off value in the case of three different defects. The MFL amplitude has a noticeable decrease when lift-off value increases from 1 to 2 mm. When the lift-off value becomes larger, its influence to MFL amplitude decreases. Fig. 11. Relation between MFLpp and lift-off value.
strongly related to defect depth, a small depth percentage causes a small MFLpp, the relation between defect depth and MFLpp are nearly linear if other parameters keeps constant. Fig. 6 shows the pattern of MFL signal at different depth values by contour plot. The patterns are very similar to each other and the different is the flux density magnitude.
5. Conclusions This paper presents a 3D FEM for MFL analysis in pipeline NDE. Comparing with previous 2D method, it gives an accurate problem description and gets an ideal result. A typical defect is analyzed by this method, the influence of defect parameters and lift-off value to MFL are discussed. The results show it is an effective way to MFL analysis.
4.2. Relation between MFL peak–peak value and the width of defect Acknowledgements Fig. 7 shows the relation between the MFL peak–peak value (MFLpp) and the defect width at different lift-off values. It can be seen that MFLpp is also strongly related to defect depth, the MFLpp increases with increasing defect width. In the case of a narrow defect the coupling of the two flanks is stronger and therefore less magnetic flux is leaking in the air. Fig. 8 shows the pattern of MFL at different width values by contour plot, the patterns are also similar while the magnitudes differ from each other and a wider defect lead to a larger MFLpp and wider MFL signals. 4.3. Relation between MFL peak–peak value and the length of defect Fig. 9 shows the relation between the MFL peak–peak value (MFLpp) and the defect length at different lift-off
This project was supported by ‘863’ of the High Technology Research and Development Program of China (No. 2001 AA602021).
References [1] Crouch A.E., Gas Research Institute Topical Report GRI-91/0365, 1993; 12–16 [2] Mandal K, Dufour D, Atherton DL. IEEE Trans Magn 1999;35:2007. [3] Edwards C, Palmer SB. Appl Phys 1986;19:657. [4] Catalin M, Lynann C. Appl Phys 2003;36:2427. [5] Mitsuaki K. NDT E Int 2003;36:479. [6] Gyimesi M, Lavers D, Pawlak T, Ostergaard D. IEEE Trans Magn 1993;29:1345. [7] Gyimesi M, Lavers JD. IEEE Trans Magn 1992;28:1924.
3D FEM analysis in magnetic flux leakage method Huang Zuoying*, Que Peiwen, Chen Liang Department of Information Measurement Technology and Instruments, Institute of Automatic Detection, Shanghai Jiaotong University, Shanghai 200030, People’s Republic of China Received 8 May 2005; revised 7 June 2005; accepted 7 June 2005 Available online 2 August 2005
Abstract The magnetic flux leakage (MFL) method is currently the most commonly used pipeline inspection technique. In this paper, 3D FEM is used to analyze the MFL signals, a generalized potential formulation to the magnetostatic field MFL problem is discussed, typical 3D defects are accurately modeled and detail MFL signal in test surface are calculated by the method. The relation between defect parameters and MFL signals are also analyzed. q 2005 Elsevier Ltd. All rights reserved. PACS: 75.80; 07.55.G; 07.55 Keywords: MFL; FEM; Defect; Inspection; Pipeline
1. Introduction Magnetic flux leakage technique is generally considered to be the most cost-effective method for corrosion monitoring. In this technique, the wall of the pipeline is magnetized axially to near saturation flux density. If, at some point, the wall thickness is reduced by a defect, a higher fraction of the magnetic flux will ‘leak’ from the wall into the air inside and outside the pipe. The magnetic leakage field measured on the nearby of the pipe contains information about the pipe conditions. The MFL inspection tool, propelled by the oil pressure or driven equipment, magnetizes the pipe wall; MFL signals due to any irregularities are detected and stored within the tool. The inspection vehicle consists of a magnetizer which is made of permanent magnet or coil and circumferentially distributed sensor assembly which is hall sensor or signal pick up coil [1,2]. Analytical approaches to the modeling of MFL have largely been difficult due to both the awkward boundaries * Corresponding author. Tel.: C86 21 6293 2851; fax: C86 21 6293 2810. E-mail address: [email protected] (H. Zuoying).
0963-8695/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ndteint.2005.06.006
associated with realistic defect shapes and the lack of generality that ensures when making the necessary assumptions needed to obtain tractable analytical solutions [3,4]. Viable models are needed in order to understand the ways in which fields and defects interact to produce measurable indications, to help in the design of detection probes for diverse applications, to simulate those testing environments which are difficult and/or expensive to replicate in the laboratory, and perhaps most important in many of the critical testing situations facing industries today, to provide training data for automated defect characterization schemes. More recently, 2D finite element methods have been used to research MFL signals under different defects shapes, materials, magnetizing situation and so on and it is also proved to be an effective method [5]. However, in 2D FEM defects are also treated as 2D profile instead of actually 3D geometry, and the result MFL signal is single channel whereas the actual signals are multichannel. In this paper, 3D FEM is adopted to analyze the MFL method, accurate 3D defect are modeled and detail MFL signal in test surface are calculated by the method. The relation between defect geometry and MFL signals are discussed detailed, the influence of lift-off value is also studied.
62
H. Zuoying et al. / NDT&E International 39 (2006) 61–66
2. 2 Three dimensional finite element computation of MFL numerical models The MFL problem can be treated as magnetostatic problem by magnetic scalar potential method. It can be expressed by Maxwell’s equations below: V !fHg Z fJs g
(1)
V$fBg Z 0
(2)
where {H} is magnetic field intensity vector, {Js} is applied source current density vector and {B} is magnetic flux density vector. The field equations are supplemented by the constitutive relation that describes the behavior of electromagnetic materials. In permanent magnet region fBg Z ½mfHg C m0 fM0 g
(3)
In other region fBg Z ½mfHg
(4)
Fig. 3. Vector plot of calculated B.
In the domain of a magnetostatic field problem, a solution is sought which satisfies the Maxwell Eqs. (1) and (2) and the constitutive relation (3) in the following form: [6,7] fHg Z fHg gKVfg
where m is magnetic permeability matrix, {M0} is remanent intrinsic magnetization vector.
Fig. 1. Geometry of 3D MFL model.
Fig. 2. B–H curve of the material.
Fig. 4. Surface plot of axial magnetic flux density.
(5)
H. Zuoying et al. / NDT&E International 39 (2006) 61–66
63
The development of {Hg} varies depending on the problem and the formulation. Basically, {Hg} must satisfy Ampere’s law so that the remaining part of the field can be derived as the gradient of the generalized scalar potential fg. This ensures that fg is singly valued. Additionally, the absolute value of {Hg} must be greater than that of Dfg. In other words, {Hg} should be a good approximation of the total field. This avoids difficulties with cancellation errors. The finite element matrix equations can be derived by variational principles. The element matrices of scalar potential can be presented in the following form: ½K m Z ½K L C ½K N
Fig. 5. Relation between MFLpp and defect depth.
V$½mVfg KV$½mfHg gKV$m0 ðM0 Þ Z f0g
ð
½K Z ðVfNgT ÞT ½mðVfNgT ÞdV L
(8)
V
(6) N
where {Hg} is preliminary magnetic field, fg is generalized potential.
(7)
½K Z
ð
vmh dV ðfHgT VfNgT ÞT ðfHgT VfNgT Þ jHj vjHj
V
Fig. 6. Contour plot of magnetic flux density at different defect depth values (lZ10 and wZ10 mm).
(9)
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H. Zuoying et al. / NDT&E International 39 (2006) 61–66
3. Analysis procedure
Fig. 7. Relation between MFLpp and defect width.
where {N} is the element shape functions (fZ {N}T{fe}) and vmh =ðvjHjÞ is the derivative of permeability with respect to magnitude of the magnetic field intensity.
Calculations are made for a simplified MFL detector. Fig. 1 shows the geometry of the problem in a 3D model for the influence of the flaw sizes and lift-off value on the magnetic leakage flux density from a rectangular flaw in the specimen. The magnetic circuit is constituted by yoke, magnets, brushes and specimen. l, w and d denote the length, width and depth size of flaw, respectively. The flaw locates at the center of the specimen. We set the specimen length to be 500 mm breath to be 120 mm and thickness to be 14 mm. Two permanent magnets, made of NdFeB material, are used as the magnetic flux induction; this material has a coercive force of 872,000 A/m. The material of specimen is X52, and that of yoke is mild steel. Fig. 2 shows the B–H curve of them. We assume the same B–H curve for brush. The calculations are made using the ANSYS finite element software. The most fundamental element of 3D is a tetrahedron. In order to have a precise result, we refine the elements near the flaw.
Fig. 8. Contour plot of magnetic flux density at different defect width values (lZ10 mm and dZ8 mm).
H. Zuoying et al. / NDT&E International 39 (2006) 61–66
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Fig. 3 shows the vector plot of the calculated magnetic flux density.
4. Calculated results and consideration Fig. 4 shows a surface plot of the amplitude of the radial and axial component of magnetic flux density in the vicinity of a flaw whose sizes are lZ10 mm, wZ10 mm, and dZ 10 mm. The signal patterns are very similar to each other and to experimental measurements. The two peaks in the amplitude are due to the flux being diverted into and returning from the air around the flaw. 4.1. Relation between MFL peak–peak value and the depth of defect Fig. 9. Relation between MFLpp and defect length.
Fig. 5 shows the relation between the MFL peak–peak value (MFLpp) and the percentage of defect depth to wall thickness at different lift-off values with constant width (10 mm) and length (mm). It can be seen that MFLpp is
Fig. 10. Contour plot of magnetic flux density at different defect length values (wZ10 mm and dZ8 mm).
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H. Zuoying et al. / NDT&E International 39 (2006) 61–66
values. It can be seen that the defect length does not affect MFLpp much, the MFLpp decreases slightly when defect length increases from 10 to 80 mm. Fig. 10 shows the pattern of MFL at different length values, the patterns are similar and the magnitudes are close, the distance of MFL peaks varies with the length of defect. 4.4. The influence of lift-off value to MFL peak–peak value Fig. 11 shows the relation between the MFL peak–peak value (MFLpp) and the lift-off value in the case of three different defects. The MFL amplitude has a noticeable decrease when lift-off value increases from 1 to 2 mm. When the lift-off value becomes larger, its influence to MFL amplitude decreases. Fig. 11. Relation between MFLpp and lift-off value.
strongly related to defect depth, a small depth percentage causes a small MFLpp, the relation between defect depth and MFLpp are nearly linear if other parameters keeps constant. Fig. 6 shows the pattern of MFL signal at different depth values by contour plot. The patterns are very similar to each other and the different is the flux density magnitude.
5. Conclusions This paper presents a 3D FEM for MFL analysis in pipeline NDE. Comparing with previous 2D method, it gives an accurate problem description and gets an ideal result. A typical defect is analyzed by this method, the influence of defect parameters and lift-off value to MFL are discussed. The results show it is an effective way to MFL analysis.
4.2. Relation between MFL peak–peak value and the width of defect Acknowledgements Fig. 7 shows the relation between the MFL peak–peak value (MFLpp) and the defect width at different lift-off values. It can be seen that MFLpp is also strongly related to defect depth, the MFLpp increases with increasing defect width. In the case of a narrow defect the coupling of the two flanks is stronger and therefore less magnetic flux is leaking in the air. Fig. 8 shows the pattern of MFL at different width values by contour plot, the patterns are also similar while the magnitudes differ from each other and a wider defect lead to a larger MFLpp and wider MFL signals. 4.3. Relation between MFL peak–peak value and the length of defect Fig. 9 shows the relation between the MFL peak–peak value (MFLpp) and the defect length at different lift-off
This project was supported by ‘863’ of the High Technology Research and Development Program of China (No. 2001 AA602021).
References [1] Crouch A.E., Gas Research Institute Topical Report GRI-91/0365, 1993; 12–16 [2] Mandal K, Dufour D, Atherton DL. IEEE Trans Magn 1999;35:2007. [3] Edwards C, Palmer SB. Appl Phys 1986;19:657. [4] Catalin M, Lynann C. Appl Phys 2003;36:2427. [5] Mitsuaki K. NDT E Int 2003;36:479. [6] Gyimesi M, Lavers D, Pawlak T, Ostergaard D. IEEE Trans Magn 1993;29:1345. [7] Gyimesi M, Lavers JD. IEEE Trans Magn 1992;28:1924.