# Shape Reconstruction of Eccentric Defect in Cylindrical Component by Modified Born Approximation Method

Authors: Gangfeng Zheng; Ze Li; Songfeng Liu; Hao Dong; Bin Wu; Cunfu He
Journal: Journal of Sensors
 (2022)

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## Abstract

In this paper, a cylindrical aluminum sample with an eccentric circular hole was prepared, and ultrasonic measurements were carried out by experimental means. The measurement area was limited to a plane edge perpendicular to the axis of the cylindrical component. The measured waveform data were input into the formula of approximate correction method, and the section image was obtained by using a modified Born approximation (MBA) method. Then, the three-dimensional (3D) shape of the defects in the aluminum sample was reconstructed by superimposing the cross-sectional images. Results showed that the defect reconstruction effect of the two-dimensional section and the 3D defect reconstruction effect were significantly improved by the MBA method.


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## Body

## 1. Introduction

Nondestructive testing (NDT) and characterization of defects, such as cracks and voids, are important in many branches of industries [1–4]. Examples are the new types of bridges, grid structures, high-rise buildings, and aerospace crafts. The quality of the component directly affects the equipment performance and service life. Thus, NDT technology is continuously refined [5–9]. NDT of cylindrical components is essential. Cylindrical components have wide applications and considerable market potential. For several aperture geometries, exact inversion formulas can be derived under the assumption that scattering is sufficiently weak for Born approximation (BA) [10–13] to hold and that only backscattering is observed. The solution to this problem does not require far-field approximations. This condition is significant because the classical Born inversion procedure, which is perhaps the simplest and best-known embodiment of the inverse scattering problem, relies crucially on a far-field assumption.Suitably dealing with the defect echo signal will greatly benefit the quantitative inversion of the defects of cylindrical components [14]. The task of deriving the structure of an object from scattered radiation is known as the inverse scattering problem. The inversion procedure acquires sufficient independent data to reconstruct the scattering parameter by varying the frequency and/or direction of illumination (or object orientation with respect to the source) while measuring the scattered waves over an aperture located beyond the scattering region [15]. If the measurements are performed on a far field with respect to the scattering region, and if the BA is applicable, then the scattering amplitude distribution and the measured time-domain data bear a simple Fourier transform relationship. However, the defect image is distorted when the defect inside the cylindrical component is inverted using the BA method. Therefore, based on the BA method for defect inversion, this paper proposes a quantitative inversion study of defects in cylindrical components using the backscatter echo amplitude correction method for the backscatter echo amplitudes corresponding to different integral wave numbers.


## 2. Modified Born Approximation (MBA) Method

### 2.1. Amplitude Correction Technique

#### 2.1.1. Acoustic Pressure of the Probe Emission Sound Field

The application of the BA method is based on the condition that the incident wave is a plane wave. Therefore, as long as the vertical distance between the probe and defect is unchanged, the energy of the incident wave can be consistent, and the ideal result of defect reconstruction can be obtained. However, when the plane nonfocus probe is used to carry out the experimental research, the sound energy will attenuate with the propagation of sound waves in the measured medium due to the divergence of the sound beam emitted by the probe. At the same time, when the defect deviates from the center of the sound beam, its incident energy will change, affecting the amplitude of the backscattered echo signal (Figure1). The acoustic pressure variation of the probe emission sound field pr⇀,ω in space must therefore be studied to accurately correct the backscatter echo amplitude Am. Here, r is the distance between the source and field points, and ω is the angular frequency.Figure 1
Incident waves scattered from a finite volume DC in an infinite region.The probe crystal sheet is discretized into a series of point sources, and the response of the point sound source can be represented by Green’s formula function, which can be substituted into the Helmholtz equation.(1)∇2Gr,ω+k2Gr,ω=−4πδr,where δr denotes the Dirac pulse function and k is the wave number. In accordance with the Kirchhoff boundary integral equation, only the sound field of the point source is required, and the scattered sound field produced by any sound source can be written as follows:
(2)pr⇀,ω=∬SGr⇀,ω∂p0∂n−p0∂Gr⇀,ω∂ndS,where r denotes the distance between the source and field points, dSdenotes the discrete elements of the transducer surface, and n is the normal vector of the transducer surface. p0 denotes the acoustic pressure on the transducer surface. Green’s formula for the sound source at the point in free space can be expressed as follows:
(3)Gr⇀,ω=14πexp−ikLrr,where kL is the longitudinal wave number and i is the imaginary unit, i2=−1.
(4)p0=ρcLvn,where vn=eikr is the surface vibration velocity of the probe crystal sheet, cL is the longitudinal velocity, and ρ is the medium density.On the boundary, the partial derivative of the sound pressure is calculated in the normal direction:(5)∂p0∂n=ρcLikeikrcosr,n=iωρvncosr,n.Given that the surface vibration direction and normal angle of the probe crystal sheet are zero,cosr,n=1:
(6)∂p0∂n=iωρvn,∂Gr⇀,ω∂n=−ikLexp−ikLr⟶r⇀−exp−ikLr⇀r⇀2cosr,n.In the remote area,r>>λ, the normal derivative of Green’s function can be approximated as follows:
(7)∂Gr⇀,ω∂n=−ikLexp−ikLr⟶r⇀,pr⇀,ω=∫∫Gr⇀,ω∂p0∂n−p0∂Gr⇀,ω∂ndS=14π∫∫exp−ikLr⇀r⇀iwρvn−ρcLvn−ikLexp−ikLr⇀r⇀dS=12π∫∫iwρvnexp−ikLr⇀r⇀dS.


### 2.2. Amplitude Correction Method

The ultrasonic used in the experiment has a certain bandwidth, and the calculation modelpr⇀,ω of sound field distribution is based on a single frequency. Therefore, the sound field distribution values at different frequencies in a series of bandwidth ranges must be calculated based on the probe bandwidth. By contrast, the reconstruction model should be used for integration. According to the backscattered echo amplitude AmkL corresponding to different integral wave numbers kL, the correction coefficient composed of the sound field model pr⇀,ω of the corresponding frequency is used to correct the amplitude.The concrete implementation process assumes that the crossing time of the probe to the center of the measured component ist0 (which can be obtained by using the reference component measurement) and an eccentric circular hole defect exists in the measured component. As shown in Figure 2, when the probe is measured at position θ1, and the signal transit time is t1>t0, the defect incident signal sound pressure value p1kL is less than the component center position sound pressure value p0kL; that is, p1kL<p0kL. Given the weak incident energy, the amplitude AmkL,θ1 of the backscatter echo measured is small. Thus, the ratio of the incident sound pressure p1 to the sound pressure p0 at the center of the component should be used as the correction coefficient to correct the amplitude AmkL,θ1 of the backscatter echo, as shown in
(8)A~mkL,θ1=AmkL,θ1p1kL/p0kL⋯Figure 2
Relative position of probe and eccentric circular hole defect (t1>t0).At different frequenciesω (wave numberkL), the resulting sound fields vary, and the corresponding p0kL is different from p1kL. Therefore, the sound field distribution map at different frequencies must be obtained by calculation, followed by the construction of the sound field matrix. In this way, thus, the position of the defect in the sound field can be assessed by time t of defect signal transition, and the sound pressure value pnkL (similar to the look-up table method) at different frequencies ω (wave number kL) of the position can be obtained.In the same way, as shown in Figure3, when the probe is measured in position θ2, and the signal transit time is t2<t0, the defect incident signal sound pressure value p2 is larger than the sound pressure value p0 of the central position of the component; that is, p2>p0. The backscatter echo amplitude AmkL,θ2 is also measured. Given that the incident energy is relatively strong, and AmkL,θ2 is large, the ratio of the incident sound pressure p2 to the sound pressure p0 at the center position of the component must be used as the correction coefficient to correct the amplitude of the backscattering echo AmkL,θ2, as shown in
(9)A~mkL,θ2=AmkL,θ2p2kL/p0kL⋯Figure 3
Relative position of the probe and eccentric circular hole defect (t2<t0).


### 2.3. MBA Method

The BA method is a low-frequency approximation method sensitive to bulk defects and insensitive to crack defects. Based on the BA method, the backscatter echo amplitude in the theoretical formula of the BA method is corrected. The corrected method is known as the MBA method.The formula for reconstructing defects is given in [16] by BA methods.
(10)Γx=12π3∫02π∫0π∫0∞y^m2u0kL2AmkL,θ,φe2ikLx1sinθcosφ+x2sinθsinφ+x3cosφ8kL2sinθdkLdθdφ.On this basis, the backscattering echo amplitude in the formula is corrected. After correction, the backscattering echo amplitude becomesA~mkL,θ. Thus, the theoretical formula about the MBA method is obtained as follows:
(11)Γ~x=12π3∫02π∫0π∫0∞y^m2u0kL2A~mkL,θ,φe2ikLx1sinθcosφ+x2sinθsinφ+x3cosφ8kL2sinθdkLdθdφ,where
(12)A~mkL,θ=AmkL,θpkL/p0kL.


### 2.4. Experimental Measurements

An aluminum specimen with a diameter of 80 mm was prepared (Figure4). The specimen includes a cylindrical eccentric circular hole (diameter: 8 mm) as a defect model. Figure 5 shows the experimental setup. The scattered wave from the defect model was measured by the longitudinal–longitudinal pulse-echo method. A contact type transducer with a diameter of 0.125 is moved to the surface of the specimen 10°, a step in the x1–x2 plane, and 10 mm, a step in the x3 direction from x3 = 0 mm to 100 mm. An industrial-specific ultrasonic coupling agent was used as the couplant, resulting in good acoustic coupling and fixation of the transducer. The transducer was driven by a 300 V square wave pulser–receiver. A digital oscilloscope and PC were used for data processing and shape reconstruction. The center frequency of the transducer was 10 MHz, and frequencies ranging from 7.5 MHz to 12.5 MHz were used for defect inversion.Figure 4
Cylindrical aluminum specimen with eccentric circular hole.Figure 5
Ultrasonic testing equipment.Figures6 and 7 display the cross-sectional defect images obtained from Γx and Γ~x for the eccentric circular hole, respectively. Given that we collected the data of the scattering amplitudes 10° a step on the surface of the specimen in the x1–x2 plane, each cross-sectional defect image was reconstructed by looping 36 times around a cross section. The type of data we processed is a Microsoft Office Excel Worksheet (.xlsx). Figures 6 and 7 show the results of shape reconstruction of BA and MBA, respectively. Total cross-sectional images were piled up, and 3D shape reconstruction was performed by introducing the interpolation for the piled-up cross-sectional images. Figures 8 and 9 reveal the results of 3D shape reconstructions of BA and MBA, respectively.Figure 6
Reconstruction of two-dimensional (2D) section defect before amplitude correction.Figure 7
Reconstruction of 2D section defect after amplitude correction.Figure 8
Reconstruction of three-dimensional (3D) defect before amplitude correction (unit: mm).Figure 9
Reconstruction of 3D defect after amplitude correction (unit: mm).Figures7 and 9 show the results after introducing the defect echo amplitude correction coefficient. After the defect amplitude correction, the reconstruction effects of 2D and 3D defects were significantly improved.


## 2.1. Amplitude Correction Technique

### 2.1.1. Acoustic Pressure of the Probe Emission Sound Field

The application of the BA method is based on the condition that the incident wave is a plane wave. Therefore, as long as the vertical distance between the probe and defect is unchanged, the energy of the incident wave can be consistent, and the ideal result of defect reconstruction can be obtained. However, when the plane nonfocus probe is used to carry out the experimental research, the sound energy will attenuate with the propagation of sound waves in the measured medium due to the divergence of the sound beam emitted by the probe. At the same time, when the defect deviates from the center of the sound beam, its incident energy will change, affecting the amplitude of the backscattered echo signal (Figure1). The acoustic pressure variation of the probe emission sound field pr⇀,ω in space must therefore be studied to accurately correct the backscatter echo amplitude Am. Here, r is the distance between the source and field points, and ω is the angular frequency.Figure 1
Incident waves scattered from a finite volume DC in an infinite region.The probe crystal sheet is discretized into a series of point sources, and the response of the point sound source can be represented by Green’s formula function, which can be substituted into the Helmholtz equation.(1)∇2Gr,ω+k2Gr,ω=−4πδr,where δr denotes the Dirac pulse function and k is the wave number. In accordance with the Kirchhoff boundary integral equation, only the sound field of the point source is required, and the scattered sound field produced by any sound source can be written as follows:
(2)pr⇀,ω=∬SGr⇀,ω∂p0∂n−p0∂Gr⇀,ω∂ndS,where r denotes the distance between the source and field points, dSdenotes the discrete elements of the transducer surface, and n is the normal vector of the transducer surface. p0 denotes the acoustic pressure on the transducer surface. Green’s formula for the sound source at the point in free space can be expressed as follows:
(3)Gr⇀,ω=14πexp−ikLrr,where kL is the longitudinal wave number and i is the imaginary unit, i2=−1.
(4)p0=ρcLvn,where vn=eikr is the surface vibration velocity of the probe crystal sheet, cL is the longitudinal velocity, and ρ is the medium density.On the boundary, the partial derivative of the sound pressure is calculated in the normal direction:(5)∂p0∂n=ρcLikeikrcosr,n=iωρvncosr,n.Given that the surface vibration direction and normal angle of the probe crystal sheet are zero,cosr,n=1:
(6)∂p0∂n=iωρvn,∂Gr⇀,ω∂n=−ikLexp−ikLr⟶r⇀−exp−ikLr⇀r⇀2cosr,n.In the remote area,r>>λ, the normal derivative of Green’s function can be approximated as follows:
(7)∂Gr⇀,ω∂n=−ikLexp−ikLr⟶r⇀,pr⇀,ω=∫∫Gr⇀,ω∂p0∂n−p0∂Gr⇀,ω∂ndS=14π∫∫exp−ikLr⇀r⇀iwρvn−ρcLvn−ikLexp−ikLr⇀r⇀dS=12π∫∫iwρvnexp−ikLr⇀r⇀dS.


## 2.1.1. Acoustic Pressure of the Probe Emission Sound Field

The application of the BA method is based on the condition that the incident wave is a plane wave. Therefore, as long as the vertical distance between the probe and defect is unchanged, the energy of the incident wave can be consistent, and the ideal result of defect reconstruction can be obtained. However, when the plane nonfocus probe is used to carry out the experimental research, the sound energy will attenuate with the propagation of sound waves in the measured medium due to the divergence of the sound beam emitted by the probe. At the same time, when the defect deviates from the center of the sound beam, its incident energy will change, affecting the amplitude of the backscattered echo signal (Figure1). The acoustic pressure variation of the probe emission sound field pr⇀,ω in space must therefore be studied to accurately correct the backscatter echo amplitude Am. Here, r is the distance between the source and field points, and ω is the angular frequency.Figure 1
Incident waves scattered from a finite volume DC in an infinite region.The probe crystal sheet is discretized into a series of point sources, and the response of the point sound source can be represented by Green’s formula function, which can be substituted into the Helmholtz equation.(1)∇2Gr,ω+k2Gr,ω=−4πδr,where δr denotes the Dirac pulse function and k is the wave number. In accordance with the Kirchhoff boundary integral equation, only the sound field of the point source is required, and the scattered sound field produced by any sound source can be written as follows:
(2)pr⇀,ω=∬SGr⇀,ω∂p0∂n−p0∂Gr⇀,ω∂ndS,where r denotes the distance between the source and field points, dSdenotes the discrete elements of the transducer surface, and n is the normal vector of the transducer surface. p0 denotes the acoustic pressure on the transducer surface. Green’s formula for the sound source at the point in free space can be expressed as follows:
(3)Gr⇀,ω=14πexp−ikLrr,where kL is the longitudinal wave number and i is the imaginary unit, i2=−1.
(4)p0=ρcLvn,where vn=eikr is the surface vibration velocity of the probe crystal sheet, cL is the longitudinal velocity, and ρ is the medium density.On the boundary, the partial derivative of the sound pressure is calculated in the normal direction:(5)∂p0∂n=ρcLikeikrcosr,n=iωρvncosr,n.Given that the surface vibration direction and normal angle of the probe crystal sheet are zero,cosr,n=1:
(6)∂p0∂n=iωρvn,∂Gr⇀,ω∂n=−ikLexp−ikLr⟶r⇀−exp−ikLr⇀r⇀2cosr,n.In the remote area,r>>λ, the normal derivative of Green’s function can be approximated as follows:
(7)∂Gr⇀,ω∂n=−ikLexp−ikLr⟶r⇀,pr⇀,ω=∫∫Gr⇀,ω∂p0∂n−p0∂Gr⇀,ω∂ndS=14π∫∫exp−ikLr⇀r⇀iwρvn−ρcLvn−ikLexp−ikLr⇀r⇀dS=12π∫∫iwρvnexp−ikLr⇀r⇀dS.


## 2.2. Amplitude Correction Method

The ultrasonic used in the experiment has a certain bandwidth, and the calculation modelpr⇀,ω of sound field distribution is based on a single frequency. Therefore, the sound field distribution values at different frequencies in a series of bandwidth ranges must be calculated based on the probe bandwidth. By contrast, the reconstruction model should be used for integration. According to the backscattered echo amplitude AmkL corresponding to different integral wave numbers kL, the correction coefficient composed of the sound field model pr⇀,ω of the corresponding frequency is used to correct the amplitude.The concrete implementation process assumes that the crossing time of the probe to the center of the measured component ist0 (which can be obtained by using the reference component measurement) and an eccentric circular hole defect exists in the measured component. As shown in Figure 2, when the probe is measured at position θ1, and the signal transit time is t1>t0, the defect incident signal sound pressure value p1kL is less than the component center position sound pressure value p0kL; that is, p1kL<p0kL. Given the weak incident energy, the amplitude AmkL,θ1 of the backscatter echo measured is small. Thus, the ratio of the incident sound pressure p1 to the sound pressure p0 at the center of the component should be used as the correction coefficient to correct the amplitude AmkL,θ1 of the backscatter echo, as shown in
(8)A~mkL,θ1=AmkL,θ1p1kL/p0kL⋯Figure 2
Relative position of probe and eccentric circular hole defect (t1>t0).At different frequenciesω (wave numberkL), the resulting sound fields vary, and the corresponding p0kL is different from p1kL. Therefore, the sound field distribution map at different frequencies must be obtained by calculation, followed by the construction of the sound field matrix. In this way, thus, the position of the defect in the sound field can be assessed by time t of defect signal transition, and the sound pressure value pnkL (similar to the look-up table method) at different frequencies ω (wave number kL) of the position can be obtained.In the same way, as shown in Figure3, when the probe is measured in position θ2, and the signal transit time is t2<t0, the defect incident signal sound pressure value p2 is larger than the sound pressure value p0 of the central position of the component; that is, p2>p0. The backscatter echo amplitude AmkL,θ2 is also measured. Given that the incident energy is relatively strong, and AmkL,θ2 is large, the ratio of the incident sound pressure p2 to the sound pressure p0 at the center position of the component must be used as the correction coefficient to correct the amplitude of the backscattering echo AmkL,θ2, as shown in
(9)A~mkL,θ2=AmkL,θ2p2kL/p0kL⋯Figure 3
Relative position of the probe and eccentric circular hole defect (t2<t0).


## 2.3. MBA Method

The BA method is a low-frequency approximation method sensitive to bulk defects and insensitive to crack defects. Based on the BA method, the backscatter echo amplitude in the theoretical formula of the BA method is corrected. The corrected method is known as the MBA method.The formula for reconstructing defects is given in [16] by BA methods.
(10)Γx=12π3∫02π∫0π∫0∞y^m2u0kL2AmkL,θ,φe2ikLx1sinθcosφ+x2sinθsinφ+x3cosφ8kL2sinθdkLdθdφ.On this basis, the backscattering echo amplitude in the formula is corrected. After correction, the backscattering echo amplitude becomesA~mkL,θ. Thus, the theoretical formula about the MBA method is obtained as follows:
(11)Γ~x=12π3∫02π∫0π∫0∞y^m2u0kL2A~mkL,θ,φe2ikLx1sinθcosφ+x2sinθsinφ+x3cosφ8kL2sinθdkLdθdφ,where
(12)A~mkL,θ=AmkL,θpkL/p0kL.


## 2.4. Experimental Measurements

An aluminum specimen with a diameter of 80 mm was prepared (Figure4). The specimen includes a cylindrical eccentric circular hole (diameter: 8 mm) as a defect model. Figure 5 shows the experimental setup. The scattered wave from the defect model was measured by the longitudinal–longitudinal pulse-echo method. A contact type transducer with a diameter of 0.125 is moved to the surface of the specimen 10°, a step in the x1–x2 plane, and 10 mm, a step in the x3 direction from x3 = 0 mm to 100 mm. An industrial-specific ultrasonic coupling agent was used as the couplant, resulting in good acoustic coupling and fixation of the transducer. The transducer was driven by a 300 V square wave pulser–receiver. A digital oscilloscope and PC were used for data processing and shape reconstruction. The center frequency of the transducer was 10 MHz, and frequencies ranging from 7.5 MHz to 12.5 MHz were used for defect inversion.Figure 4
Cylindrical aluminum specimen with eccentric circular hole.Figure 5
Ultrasonic testing equipment.Figures6 and 7 display the cross-sectional defect images obtained from Γx and Γ~x for the eccentric circular hole, respectively. Given that we collected the data of the scattering amplitudes 10° a step on the surface of the specimen in the x1–x2 plane, each cross-sectional defect image was reconstructed by looping 36 times around a cross section. The type of data we processed is a Microsoft Office Excel Worksheet (.xlsx). Figures 6 and 7 show the results of shape reconstruction of BA and MBA, respectively. Total cross-sectional images were piled up, and 3D shape reconstruction was performed by introducing the interpolation for the piled-up cross-sectional images. Figures 8 and 9 reveal the results of 3D shape reconstructions of BA and MBA, respectively.Figure 6
Reconstruction of two-dimensional (2D) section defect before amplitude correction.Figure 7
Reconstruction of 2D section defect after amplitude correction.Figure 8
Reconstruction of three-dimensional (3D) defect before amplitude correction (unit: mm).Figure 9
Reconstruction of 3D defect after amplitude correction (unit: mm).Figures7 and 9 show the results after introducing the defect echo amplitude correction coefficient. After the defect amplitude correction, the reconstruction effects of 2D and 3D defects were significantly improved.


## 3. Conclusions

An MBA method for an eccentric defect in a cylindrical component was presented, and it can relate the backscattering signal distributed on the frequency domain to the geometrical information of defects. Therefore, shape reconstruction can be treated as an inverse problem. The ultrasonic field was calculated in a homogeneous and isotropic aluminum medium after its passage through a curved interface, which was used to correct the amplitude of pulse echoes. The reconstruction qualities before and after compensation were compared, and the results showed that the reconstruction effect can be improved effectively after compensation. We find the following:(1)
The MBA method was proposed.(2)
The shape of the eccentric void defect was reconstructed by the MBA method.(3)
The results showed that the defect reconstruction effect of the 2D section and the 3D defect reconstruction effect were significantly improved by the MBA method.



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