JP5674038B2 - Defect evaluation method by leakage magnetic flux density measurement - Google Patents

Description

  The present invention relates to a nondestructive inspection method using leakage magnetic flux density, and more particularly to a method using a fluxgate sensor.

In order to correctly evaluate the safety and reliability of a structure, nondestructive inspection to identify damaged parts such as cracks and holes existing in the structure is important.
The material of the structure is often a magnetic material, and electromagnetic nondestructive inspection is effective. The leakage magnetic flux flaw detection method is an inspection method for detecting a defect by measuring a leakage magnetic flux density generated on a crack surface. However, it is impossible to quantitatively evaluate the defect size and shape only by measuring the leakage magnetic flux density. In order to quantitatively evaluate these defects, a response function indicating the geometrical correspondence between the measured magnetic flux density distribution and the magnetic charge existing on the defect is formulated, and the magnetic charge is obtained by inverse analysis. It is necessary to restore the distribution of.
The fluxgate sensor (hereinafter referred to as FG sensor) is a high-sensitivity magnetic sensor of about 10 ⁻⁷ T (see Non-Patent Document 1), and has a wide measurement range of ± 2 × 10 ⁻⁴ and a low lift-off of about 1 mm ( Since it can be used in a wide range from a specimen-observation surface distance) region to a high lift-off region of about 10 mm, it has been attracting attention in recent years (see Non-Patent Documents 2 and 3).

IEEJ Magnetics Technical Committee, Fundamentals and Applications of Magnetic Engineering, p. 171, Corona, 1999 T. T. et al. Suzuki, A .; Terasaki, A .; Sasamoto, Y. et al. Nishimura and T. Teramoto, Nondestructive evaluation of ferrous magnetic structural materials FG sensor, Electromagnetic Nondestructive Evaluation (XII) -2, ISP S. Takaya, T .; Suzuki, Y. et al. Matsumoto, K .; Demochi and M.M. Uesaka, Magnetic microstructure of the sensitized SUS304 stainless steel, Electromagnetic Nondestructive Evaluation (VII), IOSPless, 2006, pp 313-320

In the measurement using this FG sensor, high accuracy can be expected in a low lift-off region. However, in general, the spatial change rate of the magnetic flux density is very large in the low lift-off region, and if this is measured by an FG sensor having a length of about 3 mm, such a local change cannot be accurately captured. The high sensitivity of the sensor may not lead to a high-accuracy evaluation of the defect shape. However, as far as the present inventors know, there are no reports of researches that have been performed reverse analysis in consideration of the influence of the sensor length. On the other hand, in the high lift-off region, the magnetic flux density is remarkably attenuated, and errors due to measurement greatly affect the inverse analysis.
Therefore, in the present invention, it is assumed that this FG sensor is used for measuring the leakage magnetic flux density, and the response function is improved in consideration of the influence of the sensor length.

The present invention performs a reverse analysis using a response function indicating a geometric correspondence between a leakage magnetic flux density distribution measured by an FG sensor and a magnetic charge existing on a defect, thereby restoring the magnetic charge distribution. In the defect shape evaluation method for evaluating the defect shape of a body structure, the response function is evaluated using a corrected response function corrected in consideration of the fact that the FG sensor has a sensor length of 2L. .
According to the present invention, in the defect evaluation method described above, the correction response function has a z-axis as a normal direction of the structure and a y-axis as a direction perpendicular to the crack. Magnetized dipole due to magnetic charge is non-zero component only in the y-axis direction
The magnetic flux density component in the y-axis direction at this time is
When the FG sensor has a length 2L in a direction parallel to the y-axis,
Magnetic flux density observed by sensor
But
Where m is the number of trapezoids (number of divisions), and
Is the y-direction section
Is the y coordinate of the dividing point dividing m into
Is
Is given by (9)
The following equation (10) obtained using
The modified response function matrix is expressed as follows.
Further, the present invention provides the defect shape evaluation method, wherein the inverse analysis using the corrected response function matrix is performed by using the Tikhonov regularization method, α is a regularization parameter,
Is the identity matrix,
Is a magnetic flux density distribution measured by an FG sensor having a sensor length of 2 L, the magnetic charge distribution obtained by inverse analysis
Is the following equation (15)
It is obtained by.

  In the present invention, since the response function considering the sensor length of the FG sensor is used, it is possible to perform measurement with higher accuracy than the conventional sensor length that does not consider the sensor length (the sensor is regarded as a point).

The conceptual diagram which detects the slit-like crack (Defect) which exists in the test piece surface of a ferromagnetic material by the leakage magnetic flux density measurement (Measurement of magnetic flux leak) using FG sensor (Flux gate sensor). The figure which showed the y-axis direction distribution of the magnetic flux density calculated | required on the line of x = 0 when lift-off (Liftoff) is 1 mm, 5 mm, and 10 mm with respect to the test piece shown in FIG. The conceptual diagram which showed replacing integral calculation with the numerical integration based on a composite trapezoidal rule. The magnetic flux density distribution in 1 mm of lift-off was shown to (a) with respect to the test piece shown in FIG. 1, and the magnetic flux density distribution in 10 mm of lift-off was shown to (b). The figure which showed the test piece of the semi-elliptical surface crack model used for evaluation of a reverse analysis. FIG. 6A shows the distribution of simulated magnetic flux density when the lift-off is 1 mm in the model of FIG. 5, and FIG. 6B shows the distribution of simulated magnetic flux density when the lift-off is 10 mm. The figure which showed the rank of the response function matrix when changing a lift-off to 1 mm-10 mm, and a correction response function matrix. The figure which showed the flow of forward analysis (Forward analysis) and reverse analysis (Inverse analysis). The figure which shows L curve in case lift-off is 1 mm (vertical axis | shaft: residual vector norm: Regularized term, horizontal axis = regularization term norm: Residuals term). It is the figure which showed the magnetic charge distribution obtained by the reverse analysis in case liftoff is 1 mm, (a) shows the case where a conventional response function matrix is used, (b) shows the case where a modified response function matrix is used. The figure which shows L curve in case liftoff is 10 mm. The figure which showed the magnetic charge distribution obtained by the reverse analysis in case liftoff is 10 mm.

FIG. 1 shows a state where a slit-like defect exists on the surface of a ferromagnetic test piece. The normal direction of the surface of the structure is taken as the z axis, and the direction perpendicular to the crack surface is taken as the y axis. When the positive and negative magnetic charges existing on the opposing crack surfaces are regarded as magnetic dipoles, the magnetic dipoles are non-zero components only in the y-axis direction.
have. At this time, the magnetic flux density is a non-zero component mainly in the y-axis direction and the z-axis direction.
When
In the present invention, the former is used, and the magnetic flux density in the text is
Shall be pointed to. The coordinates of the magnetic dipole
Is written as
Magnetic flux density observed at
Is given by equation (1). However,
Is the entire region where magnetic dipoles are distributed.

(1) Expression function
Is a function that takes the relative coordinate value of the observation point with respect to the magnetic charge as an input variable, and is specifically expressed by equation (2). However,
Is the vacuum permeability.

Note that equation (2) is an expression when a magnetic dipole exists at the origin. Replace the integral calculation of equation (1) with the sum of magnetic dipoles with discrete points.

here
When
The right superscript j represents the number of the magnetic dipole, and N is the total number of magnetic dipoles.
When the observation point is moved on a plane that maintains a certain lift-off, the equation (3) is established for each discretized observation point, and the simultaneous equations of the equation (4) are obtained when arranged.

here
Is the observation point number, and M is the total number. Also
Is a response function of the equation (2) that relates the observation point i and the magnetic dipole j. Formula (4) is expressed by Formula (5). (5) Formula matrix
Is hereinafter referred to as a response function matrix.

Equation (4) is an equation that gives the magnetic flux density observed at an arbitrary point from the arranged magnetic dipole, and this is positioned as forward analysis. In the following, it is verified how lift-off affects the magnetic flux density distribution obtained by forward analysis.
FIG. 2 shows a magnetic flux density distribution in the y-axis direction obtained on the x = 0 line when the lift-off is 1 mm, 5 mm, and 10 mm for the test piece of FIG. However, the graph corresponding to each lift-off is normalized so that all the maximum values become 1 by dividing by the respective maximum values.
The basic distribution tendency is confirmed by taking the case where the lift-off is 1 mm as an example. Although the magnetic flux density is zero at infinity, it decreases to a negative value when approaching a defect. However, after reaching a minimum value in the vicinity of y = ± 2 mm, it rapidly increases and becomes a peak distribution having a maximum value on the crack (y = 0 mm).
The FG sensor (Flux gate sensor) assumed in this study has a cylindrical shape with a length of about 3 mm and a diameter of about 1 mm (see FIG. 1). Although the magnetic flux penetrating the entire inside of the sensor is measured, the magnetic flux density showing such a rapid increase / decrease is accurately measured within a section of | y | <2 mm, which is almost the same as the sensor length. It may not be possible.
On the other hand, when the lift-off is 10 mm, the change is very slow compared to the lift-off of 1 mm, and the influence of the sensor length is considered to be small.

  As described above, in the low lift-off region where the lift-off is about 1 mm, the FG sensor may not accurately capture a sudden change near the crack. In the present invention, the response function matrix of the equation (5) is corrected in consideration of this.

It is considered realistic that the magnetic flux density distribution inside the sensor has a weight near the center of the sensor, but in the present invention, a simple average is used to correct the observed magnetic flux density for simplicity. point
Magnetic flux density observed by sensor
Is given by:

Here, 2L = sensor length. By the same procedure as the expressions (1) to (3), the expression (6) can be expressed as the expressions (7) and (8), and the averaging process acts on the response function.

(8) on the right side of the expression
Is given by equation (2). this
Is difficult to analytically integrate in the range of real numbers, so replacing the integral calculation of equation (8) with numerical integration based on the composite trapezoidal rule (see FIG. 3) yields equation (9).

Where m is the number of trapezoids (number of divisions), and
Is the y-direction section
Is the y-coordinate of the division point dividing m. (9)
Modified response function matrix
Is obtained by the equation (10).

The effect of modifying the response function matrix is verified here. In the following, the expression (10)
To obtain the magnetic flux density obtained by forward analysis.
Is written. That is, equation (11) holds.

FIG. 4 shows the magnetic flux density distribution obtained by forward analysis when the lift-off is 1 mm and 10 mm with respect to the dimensions and magnetic dipole arrangement shown in FIG.
as well as
Are the magnetic flux densities obtained by the forward analysis shown in equations (5) and (11), respectively. Also
M is the number of divided trapezoids.
In FIG. 4 (a),
Is
Lost its steepness and became a rounded distribution when y = 0. Further, the position where the sign is reversed is moved about 1 mm outside, that is, in the direction in which | y | becomes larger. As a result of the author's trial, if the number of divisions m is increased, m = 5
The distribution shape of almost converged. On the other hand, in FIG.
When
Showed almost the same behavior.
From the above, it was found that the distribution of magnetic flux density observed in the low lift-off region is greatly affected by the modification of the response function, and the effect is small in the high lift-off region.

In the above, a response function considering the influence of sensor length was derived and the effect was confirmed by forward analysis. Here, we reverse-analyze a practical model and verify its effect. The model is a semi-elliptical surface crack on the test piece, and an external view is shown in FIG. As for the dimensions of the crack cross section, the major axis radius and minor axis radius are 5 mm and 2.5 mm, respectively.
The magnetic flux density which is input information in the inverse analysis is originally obtained by experimental measurement, but here, this is substituted by numerical calculation, that is, forward analysis. This magnetic flux density is called a simulated magnetic flux density, and the derivation process will be described below.
Magnetic dipoles are arranged in a lattice pattern on the cross section of y = 0 in FIG. The arrangement interval was 0.25 mm in both the x-axis direction and the z-axis direction. And the thing located in the inside of a semi-elliptical area | region gave the magnitude | size with -1. Further, the magnetic flux density is observed at both the x-axis and y-axis directions within the range of −15 ≦ x ≦ 15 [mm] and −20 ≦ y ≦ 20 [mm] on the observation surface (z = xy plane where lift-off occurs). They were placed on the grid at intervals of 0.5 mm. As a result, the number of observation points M = 4941 and the number of magnetic dipole points N = 1701.
The magnetic flux density used for the inverse analysis is obtained from equation (5).
But not from (11)
Should be. Furthermore, in accordance with reality, here we take into account errors associated with observations. The magnitude of the error is
Was set to 1% of the maximum value, and the distribution was a uniform random number with an average value of zero. From the above, the simulated magnetic flux density is
Given in.

(12)
Is a vector representing the error described above. Also
Is a vector representing the distribution of magnetic dipoles shown in FIG. Obtained by equation (12) when lift-off is 1 mm and 10 mm
The distribution of is shown in FIG.

The distribution of magnetic dipoles to be obtained by inverse analysis is
And (12)
Is used, the evaluation function of equation (13)
Minimize
It becomes.

(12)
Is the input value in forward analysis, whereas
Note that is the output value in the inverse analysis.
More appropriate by minimizing
Is a matrix
Dominated by the nature of. So in the following
And (4)
Are examined from the viewpoint of primary independence.
When the lift-off is changed from 1mm to 10mm
When
FIG. 7 shows the rank (number of linearly independent equations). Since the number of observation points is M = 4941, there are apparently M equations, but it is understood that the number of the substantially independent ones is 600 or less. This is less than the number N of dipoles to be obtained.
Also, as the lift-off increases, the rank decreases, and through all the lift-offs
But
It can be seen that the rank is lower than that. As for the former, as shown in FIG. 2, it is considered that the magnetic flux density distribution becomes sluggish by increasing the lift-off, and for the latter, the averaging process of the magnetic flux density contributes to the direction of increasing the same equation. .
When trying to obtain a solution from simultaneous equations with reduced primary independence in this way, the error is often overestimated and becomes a vibration solution. Therefore, as a method for obtaining an appropriate solution, the expression (13) is optimized by the Tikhonov regularization method. That is, the equation (13)
In place of (14)
Minimize.

Where α is the regularization parameter,
Is the identity matrix. The first term on the right side is called the residual norm term, and the second term is called the regularization term.
If M> N
Is obtained by equation (15).

(15) is the modified response function matrix
Is the solution when is used for inverse analysis. Traditional response function to compare, as described below
Is used in inverse analysis, it is given by equation (16)
Is the solution.

The flow of forward analysis and reverse analysis described above is shown in FIG.

Curve with a series of plot points when the horizontal axis and the vertical axis are the residual vector norm and the regularization term norm, respectively, and the regularization coefficient α is changed in the range of 10 ⁻⁷ to 10 ³ when the lift-off is 1 mm. Is shown in FIG. This is also called an L curve, and α near the bent part of the curve is considered to give the most appropriate magnetic charge distribution. As a result of examining appropriate regularization parameters located in the vicinity based on the inverse analysis image,
Then α = 10 ⁻¹ ,
Then, α = 10 ⁻² was obtained.
With respect to, the optimum magnetic charge distribution tends to be obtained when α is somewhat larger than the bending point of the L curve.
The magnetic charge distribution obtained by inverse analysis at this time is shown in FIG. That is, (a) and (b) in FIG.
And (15)
The same applies to FIG. 12, which will be described later. The solid black line in the figure represents the boundary of the actual crack surface. Distribution of actually arranged magnetic dipoles
Is a negative value, but in FIG.
Clearly shows a positive magnetic charge near the surface (z = 0)
And the result was different. This tendency was observed even when α was slightly changed. As seen in FIG. 2, this is considered to be caused by the position where the magnetic flux density is reversed in the vicinity of the defect due to the difference in the response function. On the other hand, in FIG.
No reversal of the magnetic charge sign occurred, and the magnetic charge was accurately reproduced in the vicinity of the surface, and a profile almost coincident with the actual crack boundary was obtained.
Next, FIG. 11 shows a graph of the L curve when the lift-off is 10 mm, and FIG. 12 shows the magnetic charge distribution. In FIG.
When

Shows almost the same behavior. This is shown in FIG. 11 as compared to FIG. 9 from the viewpoint of regularization.
When
This means that the difference between is reduced. Similarly, in comparison with the case of FIG. 10, FIG.
And (b)
It can be said that the difference between them has also been reduced. However, if the two are compared in detail, even in this case, FIG. 12B shows a more accurate defect shape than FIG. 12A, and it is considered that the effect of correcting the response function appeared.
As described above, by correcting the response function matrix used for the inverse analysis, the effect of accurately reproducing the defect shape from the low lift-off region to the high lift-off region has been confirmed, particularly in the low lift-off region. I understand that.

  When the integration of equation (8) is performed by numerical integration, equation (9) is derived using numerical integration based on the composite trapezoidal rule, but other numerical integration methods can also be used.

Claims (1)

The structure of the magnetic body by restoring the magnetic charge distribution by performing a reverse analysis using a response function indicating the geometric correspondence between the leakage magnetic flux density distribution measured by the FG sensor and the magnetic charge existing on the defect In the defect shape evaluation method for evaluating the defect shape of
An evaluation method of a defect shape, characterized in that evaluation is performed using a corrected response function corrected in consideration of the fact that the FG sensor has a sensor length 2L as the response function,
The modified response function is such that the normal direction of the structure is the z-axis, the direction perpendicular to the crack is the y-axis, and the magnetization dipole due to the positive and negative magnetic charges existing on the opposing crack surface is in the y-axis direction. Only non-zero components
The magnetic flux density component in the y-axis direction at this time is
When the FG sensor has a length 2L in a direction parallel to the y-axis,
Magnetic flux density observed by sensor
But
Where m is the number of trapezoids (number of divisions), and
Is the y-direction section
Is the y coordinate of the dividing point dividing m into
Is
Is given by (9)
The following equation (10) obtained using
In that it has a modified response function matrix with a one represented,
In addition, the inverse analysis using the modified response function matrix is performed by using the Tikhonov regularization method, α is a regularization parameter,
Is the identity matrix,
Is a magnetic flux density distribution measured by an FG sensor having a sensor length of 2 L, the magnetic charge distribution obtained by inverse analysis
Is the following equation (15)
A method for evaluating a defect shape, which is obtained by: