- Email: [email protected]
Shape reconstruction of three-dimensional flaw from backscattering data
Measurement 40 (2007) 854–859 www.elsevier.com/locate/measurement
Shape reconstruction of three-dimensional flaw from backscattering data Zheng Gangfeng a
a,b,*
, Wu Bin a, He Cunfu
a
College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, PingLe Yuan 100#, ChaoYang District, Beijing 100022, China b Department of Materials Science and Engineering, Anhui University of Science and Technology, Huainan 232001, China Received 22 April 2006; received in revised form 1 November 2006; accepted 12 January 2007 Available online 2 February 2007
Abstract Three dimensional Born and Kirchhoff inverse scattering methods are modified to convenient forms for a cylindrical specimen that includes three dimensional defect. One aluminum cylinder with flaw model is prepared and ultrasonic measurements are carried out. The measurement area in the modified methods is restricted in the plane perpendicular to the axis of cylindrical specimen. That is to say that the methods are modified to convenient form to use measured waveforms in the x1–x2 plane. The measured wave data are fed into the inversion method and cross-sectional images are obtained. Then, three dimensional shape reconstruction of flaw model in aluminum specimen is performed by piling up the cross-sectional images. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: NDE; Elastodynamics; Shape reconstruction; Born inversion; Kirchhoff inversion
1. Introduction Three-dimensional linearized elastodynamic inversion methods to reconstruct the shape of scatterer have been investigated in this paper. The volume and surface types of integral representations were formulated for the scattered field. Then Born and Kirchhoff approximations were introduced for *
Corresponding author. Address: College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, PingLe Yuan 100#, ChaoYang District, Beijing 100022, China. Tel.: +86 1067391720x11; fax: 86 1067391617. E-mail address: [email protected] (Z. Gangfeng).
the far-field expressions of scattering amplitudes in these integral representations. The shape reconstruction of the scatterer was performed by the inverse Fourier transform of the approximated scattering amplitudes. The experimental measurement is performed to collect the scattering data from defect. The processed data from the measurement are fed into the inversion method and the performance of the shape reconstruction is confirmed. It is very important to determine the geometrical features of defects in the structural component for the engineering judgement on an accepted or rejected decision. There are many cylindrical-shaped civil structures like a bridge pier. In this paper, the object is cylindrical aluminum specimen. When the
0263-2241/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2007.01.005
Z. Gangfeng et al. / Measurement 40 (2007) 854–859
specimen is inspected by ultrasonics, the access point of transducer is limited to the surface of the cylinder. The three-dimensional inverse scattering method [1] is modified to the convenient form for cylindrical structures. This method is based on the elastodynamic Born [2,3] and Kirchhoff [4,5] inverse scattering methods. These inverse scattering methods are applicable to the cement-based material[6], since the low frequency component of measured wave data plays key roles in these methods. In experimental measurement of this paper, one aluminum cylinder with flaw model is prepared. Moving the measurement plane along the axis of the cylinder, cross-sectional images of the cylinder are obtained by the modified method. Then, the threedimensional shape reconstruction of flaw model is demonstrated by piling up the obtained cross-sectional images. 2. Three-dimensional linearized inverse scattering methods
where Am(yˆ) and Bm(yˆ) are scattering amplitudes for longitudinal and transverse waves respectively and
DC
y
3
D \ DC 2 1
Fig. 1. Schematic of flaw’s far-field scattering.
2.1. Born inversion The Born approximation is to replace the displacement field u in the defect DC by the incident wave u0 and the incident wave is assumed to be a plane longitudinal wave _ _ u0 ðxÞ ¼ u0 d 0 exp ik 0 p0 x ð2Þ _
where u0is the amplitude, d 0 is the unit polarization 0 vector, _ k is the wave number of the incident wave, 0 and p is the unit propagation vector. For a cavity, the longitudinal scattering amplitude can be expressed by the following form Z _ _ ð3Þ Am k L ; y ¼ 2u0 y m k 2L CðxÞeiKx dV D
ð1Þ
sc m
D(r) = eir/(4pr). The longitudinal scattering amplitude is used in the following inversions.
_
Three-dimensional isotropic elastic material DnDC with three-dimensional flaw DC is shown in Fig. 1. The flaw is located near the origin O and the measurement point y is far enough to introduce the far-field approximation. In the Fig. 1, yˆ = y/jyj is the unit vector which points to the measurement point y from the origin. The incident wave is assumed to be a plane longitudinal wave and its propagation direction is y, since the longitudinal–longitudinal pulse-echo configuration is used in the present measurement. The scattered wave field usc m ðyÞ at far-field measurement point can be separated into longitudinal and transverse waves as follows: usc y Þ þ Dðk T jyjÞBm ð^y Þ m ðyÞ ¼ Dðk L jyjÞAm ð^
855
where K ¼ 2k L ; y . The integral in the above equae tion is the Fourier transform, wðCðxÞÞ ¼ CðKÞ, of the characteristic function in the K space. From the backscattered longitudinal waves in the frequency domain, we can obtain the scattering amplie tude Am(kL, yˆ) and thus the function CðKÞ in the K space. The characteristic function C(x) is reconstructed from the inverse Fourier transform Z 2p Z p Z 1 ^y m 1 CðxÞ ¼ Am ðk L ; h; /Þ 3 2u0 k 2L ð2pÞ 0 0 0 e2ikL ðx1 sin h cos /þx2 sin h sin /þx3 cos /Þ 8k 2L sin h dk L dhd/
ð4Þ
The characteristic function C(x) reconstructs the inside of the defect in the three dimensional elastic body. 2.2. Kirchhoff inversion The Kirchhoff approximation is to replace the displacement field u on the defect surface by the incident wave u0 and reflected waves. Then we can obtain the following expression for the longitudinal scattering amplitude Z _ 0_ Am k L ; y ¼ 2iu y m k L cðxÞeiKx dV ð5Þ D
The singular function c(x) is obtained as the inverse Fourier transform of the scattering amplitude
856
cðxÞ ¼
Z. Gangfeng et al. / Measurement 40 (2007) 854–859
Z
1 3
ð2pÞ e
2p
0
Z
p
Z
0
0
1
^y m Am ðk L ; h; /Þ 2iu0 k L
2ik L ðx1 sin h cos /þx2 sin h sin /þx3 cos /Þ
8k 2L
sin h dk L dh d/
ð6Þ
The singular function c(x) reconstructs the defect surface. 2.3. Three-dimensional shape reconstruction from the side of cylinder If we can obtain the scattering amplitude Am(yˆ) from all directions around origin O, we can calculate the distribution of C(x) and cH(x), and reconstruct the three-dimensional shape of flaw from Eqs. (4) and (6). But in the field, it is difficult to measure from all directions. In this study, the object to be inspected is cylindrical structure. And in the measurement, the size of flaw is larger than the diameter of the incident beam. We modify the three dimensional inverse scattering method to convenient form for cylindrical structure and try to reconstruct the three-dimensional flaw shape. The process of three-dimensional shape reconstruction is shown in Fig. 2. Here, the access point of transducer is limited to the surface of the cylinder. In this configuration, it is possible to measure from all directions in a measurement section S in the x1–x2 plane. In this case, the scattering amplitude ALm ðk L ; h; /Þ, in Eqs. (4) and (6) can be represented as Z 2p Z p Z 1 ^y m 1 Am ðk L ; p=2; /Þ CðxÞ ¼ 3 2u0 k 2L ð2pÞ 0 0 0 e2ikL ðx1 sin h cos /þx2 sin h sin /þx3 cos /Þ 8k 2L sin hdk L dh d/
is the as ALm ðk L ; p=2; uÞ. Then, we find: where C characteristic function reconstructed by scattering amplitude in x1–x2 plane. The singular function can be written as Z Z 1 1 cH ðxÞ ¼ ð2pÞ3 0 ^y m Am ðk L ; p=2; uÞe2ikL ^y x 8k 2L dk L dXð^y Þ 2iu0 k L ð8Þ Moving the measurement section S along the x3axis, cross-sectional images are obtained in each measurement section. Then, these cross-sectional images are piled up along x3-axis, and the threedimensional shape reconstruction can be performed. 3. Experiment measurement Aluminum specimen with diameter 80 mm is prepared as shown in Fig. 3. It includes a cylindrical hole as flaw model. The experimental setup is shown in Fig. 4. The scattered wave from flaw model is measured by longitudinal–longitudinal pulse-echo method. A contact type transducer with diameter 12.7 mm moves on the surface of the specimen 20° a step in x1–x2 plane and 20 mm a step in x3 direction from x3 = 0 to 100 mm. Glutinous honey syrup is used as a couplant, and it brings good acoustic coupling and fixation of the transducer. The transducer is driven by 300 V square wave pulser-receiver, and digital oscilloscope and PC are used for data processing and shape reconstruction. The center frequency of transducer is 5 MHz. As an example, measured waveforms Osc(t) obtained from the aluminum specimen with the hole are shown in Fig. 5.
ð7Þ
3
θ=
π 2
2a
S
2
1
φ
Fig. 2. The process of three-dimensional shape reconstruction from the side of a cylinder.
Fig. 3. Aluminum specimen with cylindrical hole.
Z. Gangfeng et al. / Measurement 40 (2007) 854–859
857
Fig. 4. Experimental setup and process of shape reconstruction.
Fig. 5. Measured waveform Osc(t) from aluminum specimen with cylindrical hole.
To extract the scattering amplitude ALm from measured waveforms Osc(t), the data processing with reference waveform is adopted: The longitudinal scattering amplitude ^y m ALm ðk L ; p=2; uÞ in the elastic solid is required for the shape reconstruction from Eqs. (7) and (8). To extract the waveform in the solid, the following data processing is adopted here. The received waveform Osc(f) in the experimental system is expressed in the frequency domain Osc ðf Þ ¼ Iðf ÞT ðf ÞCðf ÞH cs ðf ÞEsc ðf ÞH sc ðf ÞCðf ÞRðf Þ ð9Þ where I(f) is the input signal, T(f), C(f), Hcs(f), cHsc(f), R(f) are the effects of the transmitting transducer, couplant path, transmission from the couplant to solid, transmission from solid to the couplant, receiving transducer, respectively. The term Esc(f) represents the scattering effect by flaws in the solid. The reference specimen with the diameter 80 mm is prepared as shown in Fig. 6. The spec-
imen for reference measurement has a planar surface at the center, and it is made of the same material as the specimens with flaw model. The reference signal Osc(f) is measured waveform as shown in Fig. 6 and it can be written by Osc ðf Þ ¼ Iðf ÞT ðf ÞCðf ÞH cs ðf ÞEref ðf ÞH sc ðf ÞCðf ÞRðf Þ ð10Þ where Eref(f) is the reflection coefficient at a planar free surface. It is to be remarked that the reference signal in Eq. (10) is the same as the measured signal in Eq. (9) except for terms Eref(f) and Esc(f). Therefore, the scattered waveform in the elastic solid is obtained from Esc ðf Þ ¼ Eref ðf Þ
Osc ðf Þ Oref ðf Þ
ð11Þ
Where Osc(f) is the Fourier transform of the measured waveform and Oref(f) is the Fourier transform of the reference waveform. Moreover, Eref(f) is the reflection coefficient at the planar free surface and
858
Z. Gangfeng et al. / Measurement 40 (2007) 854–859
Fig. 6. Characteristics of reference waveform Osc(t) and Oref(f).
it has a constant value. The processed waveform is used as the scattering amplitude in Eqs. (7) and (8). Then, the distributions of CðxÞ and cH ðxÞ are calculated on six measurement sections. Obtained cross-sectional images of CðxÞ and cH ðxÞ for cylindrical hole are shown in Fig. 7 as an example. Total cross-sectional images (Fig. 8) are piled up, then three dimensional shape reconstructions are carried out by introducing the interpolation for
the piled up cross-sectional images. The results of three-dimensional shape reconstruction are shown in Fig. 8. From the results, we can see that Born approximation is more effective means than Kirchhoff approximation and the Born model is supposed to reconstruct the flaw volume and the Kirchhoff model the flaw surface. Here, the values of CðxÞ and cH ðxÞ greater than each threshold value are displayed. The shapes of flaw models are well reconstructed.
Fig. 7. CðxÞ and cH ðxÞ reconstructed from aluminum specimen with cylindrical hole.
Z. Gangfeng et al. / Measurement 40 (2007) 854–859
859
Fig. 8. Reconstructed shape of cylindrical hole.
4. Conclusions
References
Three-dimensional linearized inverse scattering methods are modified to convenient form to use measured waveforms in the x1–x2 plane, and cross-sectional images are obtained with experimentally measured waveforms. Then, three-dimensional shape reconstructions of flaw models in aluminum specimen are performed by piling up the cross-sectional images.
[1] J.H. Rose, Elastic wave inverse scattering in nondestructive evaluation [J], J. Pure Appl.Geophys. 11 (1989) 715–739. [2] D.L. Jain, The Born approximation for the scattering theory of elastic waves by two dimensional flaws [J], J. Appl. Phys. 53 (6) (1982) 4208–4217. [3] J.H. Rose, T.M. Richardson, Time domain born approximation [J], J. Nondestruct. Eval. 3 (1982) 45. [4] J.K. Cohen, N. Bleistein, The singular function of a surface and physical optics inverse scattering [J], Wave Motion 1 (1979) 153–161. [5] A. Sedov, L.W. Schmerr, The time domain elastodynamic Kirchhoff approximation for cracks:the inverse problem [J], Wave Motion 8 (1986) 15. [6] M. Yamada, K. Murakami, K. Nakahata, M. Kitaharata M, Three dimensional Born and Kirchhoff inversions of shape reconstruction of defects [J], Rev. Quant. Nondestruct. Eval. 22 (2003) 734.
Acknowledgements This work was supported by the national natural science foundation (No. 10272007), and Beijing natural science foundation (No. 4052008).
Shape reconstruction of three-dimensional flaw from backscattering data Zheng Gangfeng a
a,b,*
, Wu Bin a, He Cunfu
a
College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, PingLe Yuan 100#, ChaoYang District, Beijing 100022, China b Department of Materials Science and Engineering, Anhui University of Science and Technology, Huainan 232001, China Received 22 April 2006; received in revised form 1 November 2006; accepted 12 January 2007 Available online 2 February 2007
Abstract Three dimensional Born and Kirchhoff inverse scattering methods are modified to convenient forms for a cylindrical specimen that includes three dimensional defect. One aluminum cylinder with flaw model is prepared and ultrasonic measurements are carried out. The measurement area in the modified methods is restricted in the plane perpendicular to the axis of cylindrical specimen. That is to say that the methods are modified to convenient form to use measured waveforms in the x1–x2 plane. The measured wave data are fed into the inversion method and cross-sectional images are obtained. Then, three dimensional shape reconstruction of flaw model in aluminum specimen is performed by piling up the cross-sectional images. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: NDE; Elastodynamics; Shape reconstruction; Born inversion; Kirchhoff inversion
1. Introduction Three-dimensional linearized elastodynamic inversion methods to reconstruct the shape of scatterer have been investigated in this paper. The volume and surface types of integral representations were formulated for the scattered field. Then Born and Kirchhoff approximations were introduced for *
Corresponding author. Address: College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, PingLe Yuan 100#, ChaoYang District, Beijing 100022, China. Tel.: +86 1067391720x11; fax: 86 1067391617. E-mail address: [email protected] (Z. Gangfeng).
the far-field expressions of scattering amplitudes in these integral representations. The shape reconstruction of the scatterer was performed by the inverse Fourier transform of the approximated scattering amplitudes. The experimental measurement is performed to collect the scattering data from defect. The processed data from the measurement are fed into the inversion method and the performance of the shape reconstruction is confirmed. It is very important to determine the geometrical features of defects in the structural component for the engineering judgement on an accepted or rejected decision. There are many cylindrical-shaped civil structures like a bridge pier. In this paper, the object is cylindrical aluminum specimen. When the
0263-2241/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2007.01.005
Z. Gangfeng et al. / Measurement 40 (2007) 854–859
specimen is inspected by ultrasonics, the access point of transducer is limited to the surface of the cylinder. The three-dimensional inverse scattering method [1] is modified to the convenient form for cylindrical structures. This method is based on the elastodynamic Born [2,3] and Kirchhoff [4,5] inverse scattering methods. These inverse scattering methods are applicable to the cement-based material[6], since the low frequency component of measured wave data plays key roles in these methods. In experimental measurement of this paper, one aluminum cylinder with flaw model is prepared. Moving the measurement plane along the axis of the cylinder, cross-sectional images of the cylinder are obtained by the modified method. Then, the threedimensional shape reconstruction of flaw model is demonstrated by piling up the obtained cross-sectional images. 2. Three-dimensional linearized inverse scattering methods
where Am(yˆ) and Bm(yˆ) are scattering amplitudes for longitudinal and transverse waves respectively and
DC
y
3
D \ DC 2 1
Fig. 1. Schematic of flaw’s far-field scattering.
2.1. Born inversion The Born approximation is to replace the displacement field u in the defect DC by the incident wave u0 and the incident wave is assumed to be a plane longitudinal wave _ _ u0 ðxÞ ¼ u0 d 0 exp ik 0 p0 x ð2Þ _
where u0is the amplitude, d 0 is the unit polarization 0 vector, _ k is the wave number of the incident wave, 0 and p is the unit propagation vector. For a cavity, the longitudinal scattering amplitude can be expressed by the following form Z _ _ ð3Þ Am k L ; y ¼ 2u0 y m k 2L CðxÞeiKx dV D
ð1Þ
sc m
D(r) = eir/(4pr). The longitudinal scattering amplitude is used in the following inversions.
_
Three-dimensional isotropic elastic material DnDC with three-dimensional flaw DC is shown in Fig. 1. The flaw is located near the origin O and the measurement point y is far enough to introduce the far-field approximation. In the Fig. 1, yˆ = y/jyj is the unit vector which points to the measurement point y from the origin. The incident wave is assumed to be a plane longitudinal wave and its propagation direction is y, since the longitudinal–longitudinal pulse-echo configuration is used in the present measurement. The scattered wave field usc m ðyÞ at far-field measurement point can be separated into longitudinal and transverse waves as follows: usc y Þ þ Dðk T jyjÞBm ð^y Þ m ðyÞ ¼ Dðk L jyjÞAm ð^
855
where K ¼ 2k L ; y . The integral in the above equae tion is the Fourier transform, wðCðxÞÞ ¼ CðKÞ, of the characteristic function in the K space. From the backscattered longitudinal waves in the frequency domain, we can obtain the scattering amplie tude Am(kL, yˆ) and thus the function CðKÞ in the K space. The characteristic function C(x) is reconstructed from the inverse Fourier transform Z 2p Z p Z 1 ^y m 1 CðxÞ ¼ Am ðk L ; h; /Þ 3 2u0 k 2L ð2pÞ 0 0 0 e2ikL ðx1 sin h cos /þx2 sin h sin /þx3 cos /Þ 8k 2L sin h dk L dhd/
ð4Þ
The characteristic function C(x) reconstructs the inside of the defect in the three dimensional elastic body. 2.2. Kirchhoff inversion The Kirchhoff approximation is to replace the displacement field u on the defect surface by the incident wave u0 and reflected waves. Then we can obtain the following expression for the longitudinal scattering amplitude Z _ 0_ Am k L ; y ¼ 2iu y m k L cðxÞeiKx dV ð5Þ D
The singular function c(x) is obtained as the inverse Fourier transform of the scattering amplitude
856
cðxÞ ¼
Z. Gangfeng et al. / Measurement 40 (2007) 854–859
Z
1 3
ð2pÞ e
2p
0
Z
p
Z
0
0
1
^y m Am ðk L ; h; /Þ 2iu0 k L
2ik L ðx1 sin h cos /þx2 sin h sin /þx3 cos /Þ
8k 2L
sin h dk L dh d/
ð6Þ
The singular function c(x) reconstructs the defect surface. 2.3. Three-dimensional shape reconstruction from the side of cylinder If we can obtain the scattering amplitude Am(yˆ) from all directions around origin O, we can calculate the distribution of C(x) and cH(x), and reconstruct the three-dimensional shape of flaw from Eqs. (4) and (6). But in the field, it is difficult to measure from all directions. In this study, the object to be inspected is cylindrical structure. And in the measurement, the size of flaw is larger than the diameter of the incident beam. We modify the three dimensional inverse scattering method to convenient form for cylindrical structure and try to reconstruct the three-dimensional flaw shape. The process of three-dimensional shape reconstruction is shown in Fig. 2. Here, the access point of transducer is limited to the surface of the cylinder. In this configuration, it is possible to measure from all directions in a measurement section S in the x1–x2 plane. In this case, the scattering amplitude ALm ðk L ; h; /Þ, in Eqs. (4) and (6) can be represented as Z 2p Z p Z 1 ^y m 1 Am ðk L ; p=2; /Þ CðxÞ ¼ 3 2u0 k 2L ð2pÞ 0 0 0 e2ikL ðx1 sin h cos /þx2 sin h sin /þx3 cos /Þ 8k 2L sin hdk L dh d/
is the as ALm ðk L ; p=2; uÞ. Then, we find: where C characteristic function reconstructed by scattering amplitude in x1–x2 plane. The singular function can be written as Z Z 1 1 cH ðxÞ ¼ ð2pÞ3 0 ^y m Am ðk L ; p=2; uÞe2ikL ^y x 8k 2L dk L dXð^y Þ 2iu0 k L ð8Þ Moving the measurement section S along the x3axis, cross-sectional images are obtained in each measurement section. Then, these cross-sectional images are piled up along x3-axis, and the threedimensional shape reconstruction can be performed. 3. Experiment measurement Aluminum specimen with diameter 80 mm is prepared as shown in Fig. 3. It includes a cylindrical hole as flaw model. The experimental setup is shown in Fig. 4. The scattered wave from flaw model is measured by longitudinal–longitudinal pulse-echo method. A contact type transducer with diameter 12.7 mm moves on the surface of the specimen 20° a step in x1–x2 plane and 20 mm a step in x3 direction from x3 = 0 to 100 mm. Glutinous honey syrup is used as a couplant, and it brings good acoustic coupling and fixation of the transducer. The transducer is driven by 300 V square wave pulser-receiver, and digital oscilloscope and PC are used for data processing and shape reconstruction. The center frequency of transducer is 5 MHz. As an example, measured waveforms Osc(t) obtained from the aluminum specimen with the hole are shown in Fig. 5.
ð7Þ
3
θ=
π 2
2a
S
2
1
φ
Fig. 2. The process of three-dimensional shape reconstruction from the side of a cylinder.
Fig. 3. Aluminum specimen with cylindrical hole.
Z. Gangfeng et al. / Measurement 40 (2007) 854–859
857
Fig. 4. Experimental setup and process of shape reconstruction.
Fig. 5. Measured waveform Osc(t) from aluminum specimen with cylindrical hole.
To extract the scattering amplitude ALm from measured waveforms Osc(t), the data processing with reference waveform is adopted: The longitudinal scattering amplitude ^y m ALm ðk L ; p=2; uÞ in the elastic solid is required for the shape reconstruction from Eqs. (7) and (8). To extract the waveform in the solid, the following data processing is adopted here. The received waveform Osc(f) in the experimental system is expressed in the frequency domain Osc ðf Þ ¼ Iðf ÞT ðf ÞCðf ÞH cs ðf ÞEsc ðf ÞH sc ðf ÞCðf ÞRðf Þ ð9Þ where I(f) is the input signal, T(f), C(f), Hcs(f), cHsc(f), R(f) are the effects of the transmitting transducer, couplant path, transmission from the couplant to solid, transmission from solid to the couplant, receiving transducer, respectively. The term Esc(f) represents the scattering effect by flaws in the solid. The reference specimen with the diameter 80 mm is prepared as shown in Fig. 6. The spec-
imen for reference measurement has a planar surface at the center, and it is made of the same material as the specimens with flaw model. The reference signal Osc(f) is measured waveform as shown in Fig. 6 and it can be written by Osc ðf Þ ¼ Iðf ÞT ðf ÞCðf ÞH cs ðf ÞEref ðf ÞH sc ðf ÞCðf ÞRðf Þ ð10Þ where Eref(f) is the reflection coefficient at a planar free surface. It is to be remarked that the reference signal in Eq. (10) is the same as the measured signal in Eq. (9) except for terms Eref(f) and Esc(f). Therefore, the scattered waveform in the elastic solid is obtained from Esc ðf Þ ¼ Eref ðf Þ
Osc ðf Þ Oref ðf Þ
ð11Þ
Where Osc(f) is the Fourier transform of the measured waveform and Oref(f) is the Fourier transform of the reference waveform. Moreover, Eref(f) is the reflection coefficient at the planar free surface and
858
Z. Gangfeng et al. / Measurement 40 (2007) 854–859
Fig. 6. Characteristics of reference waveform Osc(t) and Oref(f).
it has a constant value. The processed waveform is used as the scattering amplitude in Eqs. (7) and (8). Then, the distributions of CðxÞ and cH ðxÞ are calculated on six measurement sections. Obtained cross-sectional images of CðxÞ and cH ðxÞ for cylindrical hole are shown in Fig. 7 as an example. Total cross-sectional images (Fig. 8) are piled up, then three dimensional shape reconstructions are carried out by introducing the interpolation for
the piled up cross-sectional images. The results of three-dimensional shape reconstruction are shown in Fig. 8. From the results, we can see that Born approximation is more effective means than Kirchhoff approximation and the Born model is supposed to reconstruct the flaw volume and the Kirchhoff model the flaw surface. Here, the values of CðxÞ and cH ðxÞ greater than each threshold value are displayed. The shapes of flaw models are well reconstructed.
Fig. 7. CðxÞ and cH ðxÞ reconstructed from aluminum specimen with cylindrical hole.
Z. Gangfeng et al. / Measurement 40 (2007) 854–859
859
Fig. 8. Reconstructed shape of cylindrical hole.
4. Conclusions
References
Three-dimensional linearized inverse scattering methods are modified to convenient form to use measured waveforms in the x1–x2 plane, and cross-sectional images are obtained with experimentally measured waveforms. Then, three-dimensional shape reconstructions of flaw models in aluminum specimen are performed by piling up the cross-sectional images.
[1] J.H. Rose, Elastic wave inverse scattering in nondestructive evaluation [J], J. Pure Appl.Geophys. 11 (1989) 715–739. [2] D.L. Jain, The Born approximation for the scattering theory of elastic waves by two dimensional flaws [J], J. Appl. Phys. 53 (6) (1982) 4208–4217. [3] J.H. Rose, T.M. Richardson, Time domain born approximation [J], J. Nondestruct. Eval. 3 (1982) 45. [4] J.K. Cohen, N. Bleistein, The singular function of a surface and physical optics inverse scattering [J], Wave Motion 1 (1979) 153–161. [5] A. Sedov, L.W. Schmerr, The time domain elastodynamic Kirchhoff approximation for cracks:the inverse problem [J], Wave Motion 8 (1986) 15. [6] M. Yamada, K. Murakami, K. Nakahata, M. Kitaharata M, Three dimensional Born and Kirchhoff inversions of shape reconstruction of defects [J], Rev. Quant. Nondestruct. Eval. 22 (2003) 734.
Acknowledgements This work was supported by the national natural science foundation (No. 10272007), and Beijing natural science foundation (No. 4052008).