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Multi-Mode Electromagnetic Ultrasonic Lamb Wave Tomography Imaging for Variable-Depth Defects in Metal Plates Songling Huang *, Yu Zhang, Shen Wang and Wei Zhao State Key Lab of Power Systems, Department of Electrical Engineering, Tsinghua University, Beijing 100084, China; zhang-yu13@mails.tsinghua.edu.cn (Y.Z.); wangshen@mail.tsinghua.edu.cn (S.W.); zhaowei@mail.tsinghua.edu.cn (W.Z.) * Correspondence: huangsling@tsinghua.edu.cn; Tel./Fax: +86-10-6277-2131 Academic Editor: Xiaoning Jiang Received: 24 March 2016; Accepted: 27 April 2016; Published: 2 May 2016

Abstract: This paper proposes a new cross-hole tomography imaging (CTI) method for variable-depth defects in metal plates based on multi-mode electromagnetic ultrasonic Lamb waves (LWs). The dispersion characteristics determine that different modes of LWs are sensitive to different thicknesses of metal plates. In this work, the sensitivities to thickness variation of A0- and S0-mode LWs are theoretically studied. The principles and procedures for the cooperation of A0- and S0-mode LW CTI are proposed. Moreover, the experimental LW imaging system on an aluminum plate with a variable-depth defect is set up, based on A0- and S0-mode EMAT (electromagnetic acoustic transducer) arrays. For comparison, the traditional single-mode LW CTI method is used in the same experimental platform. The imaging results show that the computed thickness distribution by the proposed multi-mode method more accurately reflects the actual thickness variation of the defect, while neither the S0 nor the A0 single-mode method was able to distinguish thickness variation in the defect region. Moreover, the quantification of the defect’s thickness variation is more accurate with the multi-mode method. Therefore, theoretical and practical results prove that the variable-depth defect in metal plates can be successfully quantified and visualized by the proposed multi-mode electromagnetic ultrasonic LW CTI method. Keywords: multi-mode Lamb waves (LWs); variable-depth defects; electromagnetic acoustic transducers (EMATs); cross-hole tomography imaging (CMI)

1. Introduction The dispersion and multi-mode characteristics of ultrasonic Lamb waves (LW) indicate that the group velocity of a particular mode of LW varies with the product of frequency and waveguide thickness (frequency–thickness product, FTP) [1,2]. Based on the dispersion characteristics, the cross-hole tomography imaging (CTI) method utilizes the time-of-flight (TOF) to reconstruct the thickness distribution of the defect area so as to obtain the image of the defect [3–5]. The concerned imaging area is divided into M ˆ N meshes, and the transmitters and receivers are respectively arranged along the two sides of the imaging area. When some LW rays pass through the meshes with defects, the group velocities will change with the area thickness, and this leads to the changes in the TOF of the LW rays. The distribution of the slowness (the reciprocal of the group velocity) can be calculated by solving: n ÿ Ti “ Lij ˚ S j pi “ 1, 2, ..., mq (1) j “1

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where LLijij is the length of the i-th LW LW ray ray in in the the j-th j-th mesh, mesh, TTii is the TOF of the the i-th i-th LW LW ray, ray, SSj j is is the the where slowness of of the the j-th j-th mesh, mesh, m m is is the the total total number number of ofLW LWrays, rays,and andnnisisthe thetotal totalnumber numberof ofmeshes. meshes. slowness Each particular mode of is Each of LW LW has has different differentthrough-thickness through-thicknessdisplacement displacementvalues, values,and andeach each sensitive to todifferent image is sensitive differentdepths depthsofofdefects defects[6]. [6]. The Themajority majorityofof CTI CTI research research is is focused on image reconstruction for for fixed-depth fixed-depth defects, defects, but but the the imaging imaging for for variable-depth variable-depth defects defects suffers suffers relatively relatively reconstruction poor quality, quality, and and the the depth depth variation variation can can hardly hardlybebereconstructed. reconstructed. The The relationship relationship between between poor detecting signals signals and and the the depths depths of of defects defects has has been been studied studied [7–10], [7–10], but but the the research research objects objects had had detecting different defects of different depths, as opposed to varying depths within a single defect. Endoh different defects of different depths, as opposed to varying depths within a single defect. Endoh demonstrated a photoacoustic a photoacoustic microscope detect metal titled or demonstrated microscope (PAM)(PAM) to detecttometal planes withplanes titled orwith variable-depth variable-depth subsurface defects [11], while the detecting accuracy wasthe limited the image subsurface defects [11], while the detecting accuracy was limited because imagebecause of the defect was of the defect was not constructed. not constructed. Related research researchabout about multi-mode is mostly confined to comparing the imaging Related multi-mode LWLW CTICTI is mostly confined to comparing the imaging quality quality of different experimental methods usingmodes different modes LWs, only one particular of different experimental methods using different of LWs, andofonly oneand particular mode of LW mode of LW is applied in each imagingAexperiment. continuous wavelet transform method was is applied in each imaging experiment. continuousA wavelet transform method was used in Ho’s used in work to analyzeLW multi-mode LW [12]. propagation [12]. However, at present, there is nearly work to Ho’s analyze multi-mode propagation However, at present, there is nearly no practical no practical LW multi-mode LW for CTIvariable-depth method for variable-depth defects. multi-mode CTI method defects. This work work attempts attempts to to define define the the sensitivity sensitivity of of multi-mode multi-mode LWs LWs in in terms terms of of working working frequency frequency This and waveguide waveguide thickness, thickness, to to explore explore the the variation variation pattern pattern of the defined sensitivity, sensitivity, and and to to propose propose aa and novel multi-mode electromagnetic LW CTI method for variable-depth defects in metal plates. novel multi-mode electromagnetic LW CTI method for variable-depth defects in metal plates.

2. LW’s Sensitivity to Thickness 2. LW’s

Healthy area

Part 1

Part 2

Vertical view

Part 1 Part 2 Healthy area

d2

(a)

d1

d0

Side view

Group Velocity (km/s)

To To be be clear, clear,in inthe thenear nearand andlocal local range range of of aagiven given defect, defect, the the thickness thickness of of the the area area outside outside the the defect defect is is defined defined as as the the “fundamental “fundamental thickness” thickness” in in this this work. work. Figure Figure 1a 1a illustrates illustrates aa typical typical circular circular defect depth in aninaluminium plate. According to the above fundamental defect with withvariable variable depth an aluminium plate. According to definition, the abovethe definition, the thickness of Part 1 is d0 ,ofand the fundamental thickness of thickness Part 2 is dof fundamental thickness Part 1 is d0, and the fundamental 1 . Part 2 is d1.

FTP (Hz*m)

(b)

Figure 1. 1. Research Research object object in in an an aluminium aluminium plate. plate. (a) (a) A A typical typical circular circular defect defect with with variable variable depth depth and and Figure (b) the dispersion curves of a Lamb wave (LW)’s group velocity in the aluminium plate. (b) the dispersion curves of a Lamb wave (LW)’s group velocity in the aluminium plate.

Based on the dispersion characteristics of LWs, higher-order LWs can hardly propagate Based on the dispersion characteristics of LWs, higher-order LWs can hardly propagate through through the defect area where the FTP is relatively small. Therefore, only A0- and S0-mode LWs are the defect area where the FTP is relatively small. Therefore, only A0- and S0-mode LWs are applied in applied in the CTI in this work, and the working points are selected at the relatively small FTP area the CTI in this work, and the working points are selected at the relatively small FTP area in order to in order to avoid the interference of multi-mode LWs. avoid the interference of multi-mode LWs. According to the dispersion equations of LWs [13], the dispersion curves of a LW’s group According to the dispersion equations of LWs [13], the dispersion curves of a LW’s group velocity in the small FTP area (less than 3000 Hz*m) of an aluminium plate are calculated and shown velocity in the small FTP area (less than 3000 Hz*m) of an aluminium plate are calculated and shown in Figure 1b, where the longitudinal wave velocity and transverse wave velocity are respectively in Figure 1b, where the longitudinal wave velocity and transverse wave velocity are respectively evaluated as 6300 m/s and 3300 m/s. The working point Ac is the demarcation of an A0-mode LW’s evaluated as 6300 m/s and 3300 m/s. The working point Ac is the demarcation of an A0-mode LW’s group velocity monotonicity varying with the FTP. The group velocities of working points on the left group velocity monotonicity varying with the FTP. The group velocities of working points on the left side of Ac will get bigger as the FTP grows, and those on the right side of A c will get smaller as the FTP grows. Therefore, in this work, the working point A c is defined as the cut-off working point of

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side of2016, Ac will get bigger as the FTP grows, and those on the right side of Ac will get smaller as the Sensors 16, 628 3 of 10 FTP grows. Therefore, in this work, the working point Ac is defined as the cut-off working point of and the corresponding FTP xaa is defined as the cut-off an A0-mode LW, LW, and cut-off FTP of an an A0-mode A0-mode LW. LW. In a the working working point Scc in Figure 1b is defined as the cut-off working point of an S0-mode similar way, the LW, and and the the corresponding correspondingFTP FTPxxss is is defined defined as as the the cut-off cut-off FTP FTP of of an anS0-mode S0-modeLW. LW. LW, LW’s sensitivity sensitivity to to thickness thickness SSsen sen(f, In this work, the LW’s (f, d) d) at a given given working point is defined as the LW’s theoretical theoretical slowness slowness of of thickness thickness variation. variation. Therefore, the sensitivity of a rate of change of a LW’s particular mode modeofofLW LW sen(f, d) can be expressed the absolute partial derivative a LW’s particular SsenS(f, d) can be expressed as the as absolute partial derivative of a LW’s of theoretical theoreticalSjslowness Sj of its fundamental slowness of its fundamental thickness d:thickness d: ˇ S j _ the (f, d) ˇ Ssen (f, d) ˇˇ BS j_the pf, dq ˇˇ Ssen pf, dq “ ˇ d ˇ Bd

(2) (2)

Working Frequency (kHz)

Working Frequency (kHz)

where S Sj_the(f,(f,d)d)isisthe the sensitivity sensitivity of each where thetheoretical theoreticalslowness slownessof ofaa particular particularmode modeof of LW. LW. Since Since the of each j_the mode of of LW LW will will vary vary in in fundamental fundamental thickness, thickness, relatively relatively high high sensitivity sensitivity at at aa given given fundamental fundamental mode thickness can hardly be guaranteed if only one single-mode LW is applied in the CTI. Therefore, by thickness can hardly be guaranteed if only one single-mode LW is applied in the CTI. Therefore, by the the complementation of a multi-mode LW, there will almost always be at least one mode of LW that complementation of a multi-mode LW, there will almost always be at least one mode of LW that is is relatively highly sensitive to different fundamental thicknesses sogreatly as to improve greatly improve the relatively highly sensitive to different fundamental thicknesses so as to the imaging imaging quality of a variable-depth defect by the multi-mode LW CTI method. quality of a variable-depth defect by the multi-mode LW CTI method. On the of slowness slowness in On the basis basis of of dispersion dispersion curves curves in in Figure Figure 1b, 1b, the the definition definition of in Equation Equation (1) (1) and and the the definition of a LW’s sensitivity in Equation (2), the variation curves of A0and S0-mode LW definition of a LW’s sensitivity in Equation (2), the variation curves of A0- and S0-mode LW sensitivity sensitivity Ssen(f, d) varying in fundamental thickness at different working frequencies can be respectively S sen (f, d) varying in fundamental thickness at different working frequencies can be respectively calculated and in in thethe frequency–thickness plane, as calculated and illustrated illustratedas asthe thetwo-dimensional two-dimensionaldistribution distribution frequency–thickness plane, shown in Figure 2. In consideration of a LW’s high attenuation due to a higher frequency, as well as as shown in Figure 2. In consideration of a LW’s high attenuation due to a higher frequency, as well its poor recognition of defects due to a lower frequency, the working frequencies in this work are as its poor recognition of defects due to a lower frequency, the working frequencies in this work are selected as as 100 100 kHz kHz to to 1000 1000 kHz. kHz. Different Different values values of of sensitivity sensitivity SSsen(f, d) are expressed as different selected sen (f, d) are expressed as different contour lines in Figure 2, and the difference values between adjacent contour lines are are the the same. same. contour lines in Figure 2, and the difference values between adjacent contour lines

Fundamental Thickness (mm)

Fundamental Thickness (mm)

(a)

(b)

Figure 2. Two-dimensional distribution of sensitivity Ssen(f, d) on the frequency–thickness plane. Figure 2. Two-dimensional distribution of sensitivity Ssen (f, d) on the frequency–thickness plane. (a) A0-mode LW and (b) S0-mode LW. (a) A0-mode LW and (b) S0-mode LW.

As illustrated in Figure 2a, the sensitivity of an A0-mode LW decreases as the fundamental As illustrated in Figure 2a, increases. the sensitivity of an A0-mode LW decreases theis fundamental thickness or working frequency The dotted line represents the FTP as that equal to the thickness or working frequency increases. The dotted line represents the FTP that is of equal to the cut-off FTP of the A0-mode LW xa, and the working area should be at the lower left part the dotted cut-off FTP the A0-mode LW xa , andthe thesensitivity working area should behigh at the lower part of the line. For a of given working frequency, is relatively when theleft fundamental dotted line. For a given working frequency, the sensitivity is relatively high when the fundamental thickness is relatively small. Figure 2b illustrates that the sensitivity of an S0-mode LW increases as thickness is relatively small.or Figure 2b illustrates the sensitivity of an LW than increases as the the fundamental thickness working frequencythat increases and when theS0-mode FTP is less the cut-off FTP of an S0-mode LW. The dotted line represents the FTP that is equal to the cut-off FTP of an S0-mode LW xs, and the working area should be at the lower left part of the dotted line. For a given working frequency, the sensitivity is relatively high when the fundamental thickness is relatively large.

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fundamental thickness or working frequency increases and when the FTP is less than the cut-off FTP of an S0-mode LW. The dotted line represents the FTP that is equal to the cut-off FTP of an S0-mode LW xs , and the working area should be at the lower left part of the dotted line. For a given working frequency, the sensitivity is relatively high when the fundamental thickness is relatively large. The sensitivity of an S0-mode LW decreases near the point Sc (cut-off FTP) because the derivative of group velocity curve almost reaches zero. Furthermore, the cut-off curve fd = xs is defined according to the cut-off FTP. In theoretical research, the cut-off FTP can be decreased to less than the FTP of point Sc so as to prevent the working area of the S0-mode LW from falling into the near neighbor of Sc . In practical detection and imaging for defects, the working area of an S0-mode LW nearly never reaches close to the near neighbor of Sc , because the thickness of the defect area is always smaller than the thickness of the healthy area. Therefore, the potential poor sensitivity of an S0-mode LW near the neighbor of Sc can be theoretically and practically avoided. The sensitivity value of a corresponding selected contour line is defined as the sensitivity threshold STH in this work, for both A0-mode and S0-mode LWs. Further, the area in which Ssen (f, d) is greater than STH is defined as the sensitive area. Therefore, the A0-mode LW is more sensitive to thickness variation when the thickness is relatively small, and the S0-mode LW is more sensitive when the thickness is relatively large. This lays the theoretical foundation for the cooperation principle and method of multi-mode LW CTI for variable-depth defects. 3. Principles and Procedures of Multi-Mode LW CTI On the basis of the A0-mode and S0-mode LWs’ sensitivity to FTP, the cooperation principles for multi-mode LW CTI are provided as: (1)

(2) (3)

(4)

The FTP of an A0-mode LW and an S0-mode LW should be respectively less than xa and xs , which are respectively the cut-off FTP of an A0-mode LW and an S0-mode LW, in order to preserve the monotonicity of the LW group velocity. Based on the sensitivity threshold STH , the working areas of an A0-mode LW and S0-mode LW should be respectively located in their own sensitive areas to increase the imaging sensitivity. An A0-mode LW should be mainly applied in a small-thickness situation, and an S0-mode LW should be mainly applied in a large-thickness situation, according to the investigation results of LW’s sensitivity to thickness. The working areas of an A0-mode LW and an S0-mode LW at their own working frequencies should partially overlap each other in order to guarantee that, for a given thickness, at least one mode of LW is sensitive. Specifically, the upper limit thickness of an A0-mode LW working area should be greater than the lower limit thickness of an S0-mode LW working area.

The determination of working parameters and operating points of an A0-mode LW and S0-mode LW is significantly important for their cooperation and thus multi-mode LW CTI. Figure 3 illustrates the principles and methods for determining the working parameters of an A0-mode LW and an S0-mode LW. The procedures for working parameter and operating point determination are as follows: Set the sensitivity thresholds STH respectively for the A0-mode and S0-mode LWs. Locate the curve fd = xs in Figure 3a, which is the sensitivity distribution of the S0-mode LW. The working area of the S0-mode LW should be at the lower left part of the curve. (3) Locate the straight line d = d0 in Figure 3a, where d0 is the thickness of the healthy area. The working area of the S0-mode LW should be confined to the left part of the straight line. Because the S0-mode LW is more sensitive when the thickness is relatively large, the thickness of the healthy area (known quantity) should be figured on the sensitivity distribution plane of the S0-mode LW, and the working areas of the S0-mode LW should firstly be determined. (1) (2)

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The sensitivity of an S0-mode LW decreases near the point Sc (cut-off FTP) because the 5 of 10 derivative of group velocity curve almost reaches zero. Furthermore, the cut-off curve fd = xs is defined according to the cut-off FTP. In theoretical research, the cut-off FTP can be decreased to less than FTP ofthe point Sc so as to prevent the working of curve the S0-mode LWthe from falling into (4) the Calculate intersection point coordinates (d0 , f s )area of the fd = xs and straight line d =the d0 , near neighbor S cthe . In practical detection and imaging for defects, the working area of an S0-mode and take fof as working frequency of the S0-mode LW. If the working frequency of the S0-mode s LW nearly near neighbor Sc,cover because thickness of healthy the defect area is LW is never biggerreaches than f s ,close then to thethe working area willof not the the thickness of the area, and, always smaller than the thickness of the healthy area. Therefore, the potential poor sensitivity of an if the working frequency is smaller than f s , the sensitive thickness range that the S0-mode LW S0-mode LWgreatly near the neighbor Therefore, of Sc can bef theoretically and practically avoided. covers decreases. s at the intersection point is almost the perfect working The sensitivity value of a corresponding selected contour line is defined as the sensitivity frequency of the S0-mode LW. threshold STHthe in contour this work, forS both A0-mode and S0-mode LWs. Further, the area in which Ssen(f, d) (5) Locate line sen (f, d) = STH in Figure 3a, and the working area of the S0-mode LW is greater than S TH is defined as the sensitive area. Therefore, the A0-mode LW is more sensitive to should be confined to the upper right part of the contour line, which is the sensitive area of the thickness variationLW. when the thickness is relatively small, and the S0-mode LW is more sensitive the S0-mode when the thickness is relatively large. This lays(dthe theoretical foundation for the cooperation (6) Calculate the intersection point coordinates L , f s ) of the straight line f = f s and the contour principle and method of multi-mode LW CTI for variable-depth defects. line Ssen (f, d) = STH to obtain the lower limit dL , and the working area of the thickness of the S0-mode LW at working frequency f s is [dL , d0 ]. The sensitive-thickness range of the S0-mode LW 3. Principles and Procedures of Multi-Mode LW CTI is maximized by the method proposed above. On basis of the width A0-mode S0-mode LWs’ sensitivity to FTP, cooperation (7) Set the overlapping dZ (dand of the A0-mode and S0-mode LWsthe working areas inprinciples Figure 3b Z > 0) for multi-mode LWworking CTI are provided to obtain the area of theas:thickness of the A0-mode LW [0, dL + dZ ]. The overlapping width is to that theLW sensitive by the thethan A0-mode (1) The FTP of ensure an A0-mode and anworking S0-modethickness LW should be cooperation respectively of less xa andand xs, S0-mode LWs can completely cover the thickness of the variable-depth defect area. which are respectively the cut-off FTP of an A0-mode LW and an S0-mode LW, in order to Locate the curve fd = xa in Figure 3b. The working area of the A0-mode LW should be confined to (8) preserve the monotonicity of the LW group velocity. the lower leftsensitivity part of thethreshold curve, which theworking cut-off FTP (2) Based on the STH, isthe areasofofthe anA0-mode A0-modeLW LW and S0-mode LW (9) should Calculate the intersection point coordinates (d + d , f ) of the curve = ximaging straight line a and thesensitivity. L Z areas a1 be respectively located in their own sensitive to increasefdthe d = A0-mode dL + dZ . LW should be mainly applied in a small-thickness situation, and an S0-mode LW (3) An (10) should Locate be themainly contour line Ssen d) = STH in Figure 3b, and the working area of the A0-mode applied in (f, a large-thickness situation, according to the investigation resultsLW of should be confined to the lower left part of the contour line, which is the sensitive area of the the LW’s sensitivity to thickness. A0-mode LW.areas of an A0-mode LW and an S0-mode LW at their own working frequencies (4) The working Calculate the intersection point coordinates (dLguarantee + dZ , f a2 ) that, of thefor contour Ssen (f, d)at=least STH one and (11) should partially overlap each other in order to a givenline thickness, the straight d = dL + Specifically, dZ . mode of LWline is sensitive. the upper limit thickness of an A0-mode LW working area be smallest greater than thickness of an S0-mode working area. (12) should Select the of f a1the andlower f a2 aslimit the working frequency f a of theLW A0-mode LW. If the contour line Ssen (f, d) = STH is under the curve fd = xa , f a2 is selected as the working frequency f a . Otherwise, if The determination of working parameters and operating points of an A0-mode LW and the contour line Ssen (f, d) = STH is above the curve fd = xa , f a1 is selected as the working frequency S0-mode LW is significantly important for their cooperation and thus multi-mode LW CTI. Figure 3 f a . This is to maximize the sensitive working thickness of the A0-mode LW to be [0, dL + dZ ]. illustrates the principles and methods for determining the working parameters of an A0-mode LW and an S0-mode LW. Sensors 2016, 16, 628

2

fd=xs

Frequency(kHz) Working工作频率 (kHz)

f = fs

Ssen(f，d)=STH fd=xa

d = d0

1000

Ssen (s/m ) >0.2

900

0.18

800

0.16

700

0.14

600

0.12

（dL，fs）

0.06

300

0.04

200

0.02 100

0

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1

1.5 2 2.5 Fundamental Thickness 基础板厚 (mm) (mm)

3

3.5

Ssen (s/m ) >0.2

900

0.18 0.16

800

0.08

400

2

fd=xa

0.1

500

d = d L + dZ

1000

Frequency Working 工作频率 (kHz)(kHz)

fd=xs

Ssen(f，d)=STH

（dL+ dZ，fa1）

700

0.14

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0.1

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（dL+ dZ，fa2）

f = fa 200 100

0.04 0

0.5

1

1.5

2

2.5

3

3.5

0.02

Fundamental Thickness 基础板厚 (mm) (mm)

(a)

(b)

Figure 3. Principles for determining the working parameters. (a) S0-mode LW working parameters Figure 3. Principles for determining the working parameters. (a) S0-mode LW working parameters and (b) A0-mode LW working parameters. and (b) A0-mode LW working parameters.

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After the determination of working parameters and operating points of the A0-mode LW and S0-mode LW, the procedures of multi-mode LW CTI can be concluded: (1) (2) (3) (4)

Calculate the working frequencies and working areas of the A0-mode and S0-mode LWs based on the proposed cooperation principles. Respectively implement the single A0-mode and single S0-mode LWs to detect the defect area and obtain the TOF. Conduct the CTI respectively based on the A0-mode and S0-mode LWs using the TOF. Conduct data fusion of CTI results from the A0-mode and S0-mode LWs.

The CTI results of the single A0-mode and single S0-mode LWs are respectively obtained based on the above working parameters, and the data fusion of CTI results from the A0-mode and S0-mode LWs is to be conducted. For a certain imaging mesh, its thickness, calculated from single A0-mode LW CTI, is displayed as da , and its thickness, calculated from single S0-mode LW CTI, is displayed as ds . According to the working parameters and operating points of the A0-mode and S0-mode LWs, the sensitive thickness range of the A0-mode LW is [0, dL + dZ ], and the sensitive thickness range of the S0-mode LW is [dL , d0 ]. (1)

(2)

(3)

In the condition of da P[0, dL + dZ ] and ds P[dL , d0 ], or in the condition of da R[0, dL + dZ ] and ds R[dL , d0 ], the mesh thickness after fusion is dadd = 0.5da + 0.5ds . In this situation, da is in the sensitive thickness range of the A0-mode LW, and ds is in the sensitive thickness range of the S0-mode LW (or neither da nor ds is in the corresponding sensitive thickness range), which means the calculated thicknesses are both expected ones, so the weighted average value is selected as the final thickness of the mesh. In the condition of da P[0, dL + dZ ] and ds R[dL , d0 ], the mesh thickness after fusion is dadd = da . In this situation, only da is in the sensitive thickness range of the A0-mode LW, but ds is not in the sensitive thickness range of the S0-mode LW, so the calculated thickness ds should not be considered. That means the mesh thickness is in the sensitive thickness range of the A0-mode LW [0, dL + dZ ]; therefore, da is selected as the final thickness of the mesh. In the condition of da R[0, dL + dZ ] and ds P[dL , d0 ], the mesh thickness after fusion is dadd = ds . In this situation, only ds is in the sensitive thickness range of the S0-mode LW, but da is not in the sensitive thickness range of the A0-mode LW, so the calculated thickness da should not be considered. That means the mesh thickness is in the sensitive thickness range of the S0-mode LW [dL , d0 ]; therefore, ds is selected as the final thickness of the mesh.

4. Experiments and Results The multi-mode LW CTI method was implemented on a 3-mm thick aluminum plate with a variable-depth defect. Figure 4a illustrates the schematic diagram of the experimental system. The contra-flexure coils (CFC) were applied as the transmitting and receiving coils for the transmitters and receivers to separately stimulate and receive near-pure single A0-mode and S0-mode LWs [14]. The pure single-mode EMAT was realized by the configuration of the distance l between the two adjacent annular wires along the radial direction, which is supposed to be half of the wavelength of the desired mode of LW. The transmitters were excited by the power amplifier, which was controlled by the principal computer. The LW were converted into electrical signals by the receivers. Then, the detected signals were amplified and filtered by the signal conditioner and subsequently collected by the DAQ card. The collected data were finally sent to the principal computer to be analyzed for LW CTI. The 600 mm ˆ 600 mm imaging area in the aluminum plate was divided into 120 ˆ 120 meshes, as shown in Figure 4b. The variable-depth defect was composed of two parts: Part 1 is located at (310 mm, 350 mm), the diameter D1 is 120 mm, and the depth d01 is 1 mm; Part 2 is located at (300 mm, 350 mm), the diameter D2 is 60 mm, and the depth d02 is 2 mm. It can be seen that the fundamental thickness of Part 1 is 3 mm, and the fundamental thickness of Part 2 is 2 mm.

the desired mode of LW. The transmitters were excited by the power amplifier, which was controlled by the principal computer. The LW were converted into electrical signals by the receivers. Then, the detected signals were amplified and filtered by the signal conditioner and subsequently collected by the DAQ card. The collected data were finally sent to the principal computer to be Sensors 2016,for 16, LW 628 CTI. 7 of 10 analyzed

Principal Computer

DAQ Card

Power Amplifier

Signal Conditioner

N

N

Transmitter

Receivers Transmitters Part 2 Part 1

Receiver S

S

Aluminium Plate

(a)

(b)

Experimental platform platform for for multi-mode multi-mode LW LW cross-hole cross-hole tomography tomography imaging imaging (CTI). (CTI). Figure 4. 4. Experimental (a) Schematic diagram of the experimental system and (b) (b) location location parameters parameters of of the the experiment. experiment.

Based on the proposed principles and procedures of LW CTI, the calculated working frequencies of the A0-mode and S0-mode LWs were respectively 270 kHz and 700 kHz. Table 1 illustrates the structure parameters of the single A0-mode EMAT and the single S0-mode EMAT. For each single-mode LW detection, 14 transmitters and 14 receivers were respectively located at the two sides of the imaging area. Table 1. Structure parameters of single A0-mode EMAT and single S0-mode EMAT. Structure Parameters

A0 Mode EMAT

S0 Mode EMAT

Working frequency (kHz) l (mm) Outer diameter of the coil (mm) Inner diameter of the coil (mm) Number of windings

270 4.2 37.8 12.6 24

700 3.6 32.4 10.8 24

Firstly, one of the A0-mode transmitters stimulated the pure A0-mode LW in the aluminum plate, all 14 A0-mode receivers received the pure A0-mode LW, and the 14 TOF were extracted by the short-time Fourier transform (STFT) method. Similar procedures were conducted for all 14 A0-mode transmitters, so there were a total of 196 (14 ˆ 14) groups of TOF extracted from the single A0-mode LW detection experiment. Secondly, the S0-mode transmitters and receivers were implemented in the detection experiment, and the procedures were similar. Fourteen S0-mode transmitters respectively stimulated the pure S0-mode LW in the aluminum plate, and the 14 S0-mode receivers received the pure S0-mode LW. Thus, there were also a total of 196 (14 ˆ 14) groups of TOF extracted from the single S0-mode LW detection experiment. The 196 groups of TOF extracted from the A0-mode LW detection, and the 196 groups of TOF extracted from the S0-mode LW detection were respectively implemented in the simultaneous iterative reconstruction technique to reconstruct the defect image, in which the solution threshold was set as 0.05%, and the relaxation parameter was 0.1. The thickness distribution results of single A0-mode and single S0-mode LW CTI are respectively shown in Figure 5a,b. As illustrated in Figure 5a, the reconstruction image and profile of Part 2 of the variable-depth defect are relatively clear when the single A0-mode LW is applied. However, many artifacts appear in the image, and the profile of Part 1 is nearly submerged by the artifacts. When the single S0-mode LW is applied, the reconstruction image and profile of Part 1 of the variable-depth defect are relatively clear. However, the image of Part 2 is seriously distorted, its profile severely blurred. Therefore, the

stimulated the pure S0-mode LW in the aluminum plate, and the 14 S0-mode receivers received the pure S0-mode LW. Thus, there were also a total of 196 (14 × 14) groups of TOF extracted from the single S0-mode LW detection experiment. The 196 groups of TOF extracted from the A0-mode LW detection, and the 196 groups of TOF extracted from the S0-mode LW detection were respectively implemented in the simultaneous Sensors 2016, 16, 628 8 of 10 iterative reconstruction technique to reconstruct the defect image, in which the solution threshold was set as 0.05%, and the relaxation parameter was 0.1. The thickness distribution results of single A0-mode and single LWofCTI respectively defect shownsuffer in Figure 5a,b. low quality when only reconstruction imageS0-mode and profile theare variable-depth relatively

the single-mode LW is implemented in the CTI. Thickness (mm)

Thickness (mm)

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(a)

(b) Thickness (mm)

Figure 5. Cont.

(c) Figure 5. Thickness Thickness distribution distributionresults resultsofofLW LWCTI. CTI. Single A0-mode CTI; single S0-mode Figure 5. (a)(a) Single A0-mode LWLW CTI; (b) (b) single S0-mode LW LW CTI; and (c) multi-mode LW CTI. CTI; and (c) multi-mode LW CTI.

ss (mm)

ess (mm)

As illustrated in Figure 5a, the reconstruction image and profile of Part 2 of the variable-depth On the basis of the proposed data fusion principles and methods, the thickness distributions defect are relatively clear when the single A0-mode LW is applied. However, many artifacts appear from the single A0-mode and single S0-mode LW CTI were fused, as illustrated in Figure 5c. The in the image, and the profile of Part 1 is nearly submerged by the artifacts. When the single reconstruction images and profiles of both Part 1 and Part 2 of the variable-depth defect are relatively S0-mode LW is applied, the reconstruction image and profile of Part 1 of the variable-depth defect clear, and the artifacts are greatly reduced. Additionally, the fused thickness distribution from the are relatively clear. However, the image of Part 2 is seriously distorted, its profile severely blurred. multi-mode LW CTI is closer to the actual shape and profile of the variable-depth defect. Therefore, the reconstruction image and profile of the variable-depth defect suffer relatively low The thickness distribution curves at y = 350 mm were respectively extracted from Figure 5a–c, quality when only the single-mode LW is implemented in the CTI. and the results are respectively shown in Figure 6a–c. On the basis of the proposed data fusion principles and methods, the thickness distributions It can be seen that, when the single A0-mode LW is applied, the thickness distribution curve from the single A0-mode and single S0-mode LW CTI were fused, as illustrated in Figure 5c. The of Part 2 fits the actual thickness curve quite well, but that of Part 1 suffers poor fitting. Figure 6b reconstruction images and profiles of both Part 1 and Part 2 of the variable-depth defect are indicates that when the single S0-mode LW is applied, the thickness distribution curve of Part 1 fits the relatively clear, and the artifacts are greatly reduced. Additionally, the fused thickness distribution actual thickness curve well, but that of Part 2 can hardly reflect the actual thickness curve. from the multi-mode LW CTI is closer to the actual shape and profile of the variable-depth defect. The thickness distribution curve extracted from the multi-mode LW CTI shows an obvious ladder The thickness distribution curves at y = 350 mm were respectively extracted from Figures 5a–c, shape, as illustrated in Figure 6c. The calculated thickness distribution curve displays a relatively and the results are respectively shown in Figures 6a–c. higher degree of fitting for the actual thickness curves of both Part 1 and Part 2.

Thickness (mm)

Thickness (mm)

reconstruction images and profiles of both Part 1 and Part 2 of the variable-depth defect are relatively clear, and the artifacts are greatly reduced. Additionally, the fused thickness distribution from the multi-mode LW CTI is closer to the actual shape and profile of the variable-depth defect. The thickness distribution curves at y = 350 mm were respectively extracted from Figures 5a–c, Sensors 16, 628 9 of 10 and the2016, results are respectively shown in Figures 6a–c.

Theoretical Extracted


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(a)

(b)

Thickness (mm)

Figure 6. Cont.


(c) Figure Figure 6. 6. Thickness Thickness distribution distributioncurves curvesat atyy ==350 350mm mmfrom fromLW LW CTI. CTI. (a) (a) Single Single A0-mode A0-mode LW LW CTI; CTI; (b) single S0-mode LW CTI; and (c) multi-mode LW CTI. (b) single S0-mode LW CTI; and (c) multi-mode LW CTI.

It can be seen that, when the single A0-mode LW is applied, the thickness distribution curve of Therefore, the proposed multi-mode LW CTI method is proven to be able to accurately quantify Part 2 fits the actual thickness curve quite well, but that of Part 1 suffers poor fitting. Figure 6b and visualize variable-depth defects in metal plates. indicates that when the single S0-mode LW is applied, the thickness distribution curve of Part 1 fits the actual thickness curve well, but that of Part 2 can hardly reflect the actual thickness curve. 5. Conclusions The thickness distribution curve extracted from the multi-mode LW CTI shows an obvious This workas proposes novel multi-mode electromagnetic ultrasonic LW curve CTI method ladder shape, illustrateda in Figure 6c. The calculated thickness distribution displaysfor a variable-depth defects in metal plates. The dispersion characteristics of LWs indicate that relatively higher degree of fitting for the actual thickness curves of both Part 1 and Part 2. different modes of LWs are different thicknesses metal plates, and to thebe sensitivity of a given Therefore, the sensitive proposedto multi-mode LW CTI of method is proven able to accurately mode of LWs varies with the thickness. Moreover, the single-mode LW CTI method has here been quantify and visualize variable-depth defects in metal plates. shown to suffer low resolution when it is used in variable-depth situations. The sensitivities of A0-mode and S0-mode LWs on thickness variation were theoretically studied and demonstrated on a 5. Conclusions frequency–thickness plane. Related results show that, for certain working frequencies, A0-mode LW This work proposes a novel multi-mode electromagnetic ultrasonic LW CTI method for is more sensitive to thickness variation on small thickness areas, and S0-mode LW is more sensitive variable-depth defects in metal plates. The dispersion characteristics of LWs indicate that different on large thickness areas. Based on the above theoretical results, the principles and procedures for the modes of LWs are sensitive to different thicknesses of metal plates, and the sensitivity of a given cooperation of A0- and S0-mode LW CTI are proposed. Additionally, the experimental platform for mode of LWs varies with the thickness. Moreover, the single-mode LW CTI method has here been multi-mode LW CTI on an aluminum plate with a variable-depth defect was set up in this work to shown to suffer low resolution when it is used in variable-depth situations. The sensitivities of verify the effectiveness and accuracy of the proposed imaging method. Results show that the defect A0-mode and S0-mode LWs on thickness variation were theoretically studied and demonstrated on imaging obtained by the multi-mode LW CTI method more accurately reflects the actual thickness a frequency–thickness plane. Related results show that, for certain working frequencies, A0-mode variation, while neither S0 nor A0 single-mode method was able to distinguish thickness variation LW is more sensitive to thickness variation on small thickness areas, and S0-mode LW is more in the defect region. Theoretical and practical results prove that the proposed multi-mode LW CTI sensitive on large thickness areas. Based on the above theoretical results, the principles and method can more accurately quantify and visualize variable-depth defects in metal plates. procedures for the cooperation of A0- and S0-mode LW CTI are proposed. Additionally, the experimental platform for multi-mode LW CTI on an aluminum plate with a variable-depth defect was set up in this work to verify the effectiveness and accuracy of the proposed imaging method. Results show that the defect imaging obtained by the multi-mode LW CTI method more accurately reflects the actual thickness variation, while neither S0 nor A0 single-mode method was able to

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Acknowledgments: This research was supported by the National Natural Science Foundation of China (NSFC) (No. 51277101), the Royal Society-NSFC International Exchanges Award (No. IE150600), the Royal Academy of Engineering Newton Research Collaboration Program (No. NRCP/1415/91) and the Royal Academy of Engineering Distinguished Visiting Fellowship program (No. DVF1415/2/17). Author Contributions: Wei Zhao contributed to the conception of the reported research and helped revise the manuscript. Songling Huang contributed significantly to the procedure and the design of the proposed multi-mode LW CTI method and helped revise the manuscript. Yu Zhang contributed significantly to the designing, the performing as well as the result analysis of the experiments, and contributed to the writing of the manuscript. Shen Wang helped design the experiments and perform the analysis with constructive discussions. Conflicts of Interest: The authors declare no conflict of interest.

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