Minimum variance imaging based on correlation analysis of Lamb wave signals

Ultrasonics 70 (2016) 107–122

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Minimum variance imaging based on correlation analysis of Lamb wave signals Jiadong Hua a, Jing Lin a,b,⇑, Liang Zeng a, Zhi Luo a a b

State Key Laboratory of Manufacturing System Engineering, Xi’an Jiaotong University, Xi’an, Shannxi Province 710049, PR China Shaanxi Key Laboratory of Mechanical Product Quality Assurance and Diagnostics, Xi’an Jiaotong University, Xi’an, Shannxi Province 710049, PR China

a r t i c l e

i n f o

Article history: Received 11 November 2015 Received in revised form 12 March 2016 Accepted 24 April 2016 Available online 26 April 2016 Keywords: Lamb wave Minimum variance distortionless response Local signal correlation coefficient Damage detection

a b s t r a c t In Lamb wave imaging, MVDR (minimum variance distortionless response) is a promising approach for the detection and monitoring of large areas with sparse transducer network. Previous studies in MVDR use signal amplitude as the input damage feature, and the imaging performance is closely related to the evaluation accuracy of the scattering characteristic. However, scattering characteristic is highly dependent on damage parameters (e.g. type, orientation and size), which are unknown beforehand. The evaluation error can degrade imaging performance severely. In this study, a more reliable damage feature, LSCC (local signal correlation coefficient), is established to replace signal amplitude. In comparison with signal amplitude, one attractive feature of LSCC is its independence of damage parameters. Therefore, LSCC model in the transducer network could be accurately evaluated, the imaging performance is improved subsequently. Both theoretical analysis and experimental investigation are given to validate the effectiveness of the LSCC-based MVDR algorithm in improving imaging performance. Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction Lamb waves can propagate over a long distance with low attenuation and interact with small changes in structural property [1–3]. As a result, it is considered as a promising tool for damage detection and evaluation in plate- and pipe-like structures. In order to visualize the damage intuitively in a two- or three-dimensional image, several Lamb wave imaging techniques have been developed, such as tomography [4–7], elliptical imaging [8–10], hyperbolic imaging [8,11,12], correlation imaging [8,13–15] and multipath imaging [16]. Elliptical imaging, also known as delay-and-sum (DAS), is a representative Lamb wave imaging method. In this method, based on the assumption that scattering occurs when Lamb wave encounters damage, the image is generated by shifting and adding the damage features (i.e. the features extracted from the captured Lamb wave signals that can be linked with damage) according to an appropriate time shifting rule [9,10,17,18]. In practical application, DAS is capable of monitoring a large structure with sparse transducer network. A major disadvantage of this method is the poor suppression of the undesired imaging artifacts. ⇑ Corresponding author at: No. 28 Xianning West Road, Xi’an, Shaanxi 710049, PR China. E-mail address: [email protected] (J. Lin). http://dx.doi.org/10.1016/j.ultras.2016.04.020 0041-624X/Ó 2016 Elsevier B.V. All rights reserved.

Recently, minimum variance distortionless response method (MVDR) has been incorporated into elliptical imaging to successfully suppress the imaging artifacts [19–22]. The imaging artifacts suppression comes as additional information is incorporated into the algorithm. In particular, DAS generates each pixel value by simply summing all extracted damage features, while MVDR sets a steering angle (or look direction) to minimize all contributions other than those that satisfy the expected relationship. The relationship, theoretically, is satisfied only when the damage features correspond to the damage position. Previous studies in MVDR method use signal amplitude as the input damage feature. However, accurate evaluation of the amplitude relationship for scattered signals from different actuator– receiver pairs is difficult, because the scattering characteristic is highly dependent on damage parameters such as type, orientation and size [23–26]. Unfortunately, these damage parameters are unknown beforehand. The evaluation error of the amplitude relationship can severely degrade algorithm imaging performance. In this study, a more reliable damage feature, local signal correlation coefficient (LSCC), is established to replace signal amplitude. Firstly, both Barker code (BC) and Golay complementary code (GCC) are used as excitation signals to actuate Lamb waves. Subsequently, pulse compression and dispersion compensation are applied, which are aimed at compressing the Lamb wave responses into excitation auto-correlation shape. Finally, local signals are

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extracted by applying a rectangle window to the two responses from both BC and GCC excitations centered at the propagation time. A correlation coefficient between the two local signals (i.e. LSCC), as a measure of the shape matching extent, is computed as the damage feature. Functioning as a damage feature, LSCC is based on: (i) the phenomenon that the excitation (i.e. BC and GCC) auto-correlations exhibit the same main-lobe and different side-lobes; (ii) the assumption that the scattering by a damage does not change the shape of the propagating Lamb wave signal of limited bandwidth. In practice, this assumption is widely used in correlation imaging methods, damages with different types [13], different orientations [14] and different sizes [15] are successfully detected and localized. In comparison with the original damage feature (i.e. signal amplitude), LSCC is independent of damage parameters. As a result, the relationship of LSCCs from different actuator–receiver pairs could be accurately evaluated, so that the imaging performance is improved. The rest of this paper is organized as follows. In Section 2, the original damage feature (i.e. signal amplitude) is briefly reviewed, then, the calculation steps of LSCC are presented in detail. In Section 3, both amplitude-based and LSCC-based MVDR algorithms are presented. In Section 4, imaging performance of the two algorithms is theoretically analyzed and compared. In Section 5, an experiment is given to demonstrate the effectiveness of the LSCC-based MVDR algorithm in improving imaging performance. In Section 6, the robustness of the algorithm to different input parameters is demonstrated. In Section 7, further experiments are given to demonstrate the effectiveness of LSCC under different damage parameters. In Section 8, several conclusions are summarized. 2. Damage feature

Consider a transducer network where N actuator–receiver pairs are involved for damage identification. Focusing on an arbitrarily selected pair (e.g. kth) as shown in Fig. 1, assuming a single mode is propagating, let rk(t) denote the Lamb wave signal captured by the receiver. Generally, rk(t) is pre-processed by dispersion compensation to avoid wave packet overlapping. The compensation result could be either raw format or envelope format. At the location (x, y) within the inspection area, the time for the Lamb wave travelling from the actuator to (x, y) and then to the receiver is

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx  xk1 Þ2 þ ðy  yk1 Þ2 þ ðx  xk2 Þ2 þ ðy  yk2 Þ2 cg ; ð1Þ

where (x, y), (xk1, yk1) and (xk2, yk2) denote the coordinates of the inspecting point, actuator and receiver, respectively, cg is the group velocity of the mode of interest. If damage exists at (x, y), the damage-reflected wave packet will be present at t = tk(x, y), thus the corresponding amplitude rk[tk(x, y)] is highlighted. Comparatively, the amplitude rk[tk(x, y)] is much smaller if no damage exists at (x, y) due to free of the reflected wave. By changing the coordinates in Eq. (1), the signal amplitudes for each location (x, y) from the total N paths are calculated. These amplitudes are organized as a row vector,

~ rðx; yÞ ¼ ½r1 ðt1 ðx; yÞÞr2 ðt 2 ðx; yÞÞ    r N ðtN ðx; yÞÞ:

2.2. LSCC (local signal correlation coefficient) LSCC is established as a new damage feature. The calculation consists of four steps: binary code excitation, pulse compression, dispersion compensation and correlation analysis between the extracted local signals. Step 1: binary code excitation As two commonly used binary codes, BC and GCC are used as the excitation signals, respectively. A N-bit BC can be represented as

B½N ¼ ½b0 ; b1 ; . . . ; bN1  bi 2 f1; þ1g:

2.1. Review of signal amplitude

t k ðx;yÞ ¼

Fig. 1. A graphical description of damage identification with a particular actuator– receiver pair.

ð2Þ

Functioning as a damage feature, the elements in ~ rðx; yÞ are believed to be bigger for damage location than those for non-damage location.

ð3Þ

Generally, the number of bits N takes 2, 3, 4, 5, 7, 11 and 13. BCs of these bits are listed in Table 1. In this paper, the 13-bit BC is used due to high signal-to-noise ratio (SNR) after pulse compression, which is shown in Fig. 2. A N-bit GCC is composed of two binary sequences,

  A G A ½N ¼ g 0A ; g 1A ; . . . ; g N1 g iA 2 f1; þ1g;   GB ½N ¼ g B0 ; g B1 ; . . . ; g BN1 g Bi 2 f1; þ1g:

ð4Þ

Different from BC, the number of bits N for GCC is infinite because a longer GCC can be constructed by recursively operating on a shorter GCC with the ‘‘negate and concatenate” method [27]. In particular, if GA[N] and GB[N] are the representations of the N-bit GCC, then the 2N-bit GCC can be generated by concatenating GB[N] to GA[N] and concatenating GB[N] to GA[N] where GB[N] is the complement of GB[N]. Therefore, GA[2N] = {GA[N] GB[N]} and GB[2N] = {GA[N]  GB[N]}. For example, if GA[N] = [+1 +1] and GB[N] = [+1 1], then the 4-bit GCC can be generated as GA[2N] = [+1 +1 +1 1] and GB[2N] = [+1 +1 1 +1]. This procedure can be repeated recursively to produce GCC arbitrarily long. In this paper, the 8-bit GCC as shown in Fig. 3 is used. In practice, neither BC nor GCC of this type (Figs. 2 and 3) can be directly used as the excitation signal because the spectrum of the binary sequence is not matched to the desired frequency range for Lamb wave inspection. A binary phase shift keying (BPSK) modulation scheme is usually employed to adapt this binary sequence to the desired frequency range [28]. Instead of a positive value +1 or a negative value 1, each code bit is modulated with a predesigned signal. In this paper, a single-frequency sinusoidal signal is used as the modulation signal. Thus, BPSK modulation for BC and GCC can be represented as the convolution of a sinusoidal signal with a train of Dirac delta functions, as

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process is also known as matched filtering, which can be represented as

Table 1 Barker codes of different bits. bit (N)

Z

Barker code

2 3 4 5 7 11 13

1 1 1 1 1 1 1

bðtÞ ¼ sðtÞ 

N 1 X

1 or 1 1 1 1 1 1 1 or 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Z

g iA dðt  ikTÞ



0 6 t 6 ðN  1ÞkT

ð5Þ

i¼0

g B ðtÞ ¼ sðtÞ 

N1 X 

g Bi dðt  ikTÞ



0 6 t 6 ðN  1ÞkT;

where s(t) with 0 6 t 6 kT is the k-cycle sinusoidal signal, T = 1/f is the period, bi is the binary element of the BC in Eq. (3), b(t) is the modulated BC, g iA and g Bi are the binary elements of the GCC pair in Eq. (4), g A ðtÞ and g B ðtÞ are the modulated GCC pairs. An example is given for illustration. In this example, a sinusoidal signal with the frequency f = 80 kHz and the cycle k = 1 is used for modulation. The corresponding modulation results for 13-bit BC and 8-bit GCC are shown in Fig. 4. Step 2: Pulse compression

Z BðxÞHðwÞ expðixtÞdx:

ð6Þ

If the actuator is excited with GCC pairs, g A ðtÞ and g B ðtÞ are both used as input, and the outputs are added to obtain GCC response as

r g ðtÞ ¼

Z A

G ðxÞHðwÞ expðixtÞdx þ  expðixtÞdx:

 expðixtÞdx Z ¼ fjG A ðxÞj2 þ jGB ðxÞj2 gHðxÞ expðixtÞdx:

ð9Þ

gate, the terms in the right-hand-side, jBðxÞj2 , jG A ðxÞj2 and jGB ðxÞj2 , represent the Fourier transform of the auto-correlation functions of the excitation signal b(t), g A ðtÞ and g B ðtÞ, respectively. In practice, Eqs. (8) and (9) indicate that pulse compression to original response is equivalent to applying this auto-correlation curve as the virtual excitation signal. It should be noted that GCC auto-correlation is the sum of g A ðtÞ auto-correlation and g B ðtÞ auto-correlation. The auto-correlation function of N-bit BC in Eq. (3) can be represented as



Consider that two transducers are positioned on the specimen under inspection, which act as the actuator and receiver, respectively. The entire system comprises the instrumentation, actuator, receiver and specimen, which may be considered as a linear system. If the actuator is excited with BC, b(t), the response captured by the receiver can be represented as

Z

G A ðxÞHðxÞG A ðxÞ expðixtÞdx Z þ GB ðxÞHðxÞGB ðxÞ expðixtÞdx Z Z ¼ jG A ðxÞj2 HðxÞ expðixtÞdx þ jGB ðxÞj2 HðxÞ

cgcc ðtÞ ¼

In Eqs. (8) and (9), the superscript ⁄ represents the complex conju-

i¼0

r b ðtÞ ¼

ð8Þ

Z

1

ðbi dðt  ikTÞÞ 0 6 t 6 ðN  1ÞkT

N1 X 

jBðxÞj2 HðxÞ expðixtÞdx:

¼

i¼0

g A ðtÞ ¼ sðtÞ 

BðxÞHðxÞBðxÞ expðixtÞdx

cbc ðtÞ ¼

WBB ½n ¼

ð7Þ

In Eqs. (6) and (7), B(x), GA(x) and GB(x) are the Fourier transform of b(t), g A ðtÞ and g B ðtÞ in Eq. (5), respectively. H(x) is the transfer function of the system. In this step, pulse compression is performed by crosscorrelating the response with the excitation signal [29–31]. This

n¼0

ð10Þ

0 or 1 n – 0:

The peak of the auto-correlation function equals N, and the side-lobe levels fall between 0 and 1. For illustration, the auto-correlation curves of 13-bit BC before and after BPSK modulation are shown in Fig. 5. For GCC in Eq. (4), the auto-correlation functions of GA[N] and GB[N] have side-lobes with equivalent magnitude however opposite sign. Therefore the sum of the auto-correlation functions has a main peak and zero side-lobes, which is represented as



WG A G A ½n þ WGB GB ½n ¼

B

G ðxÞHðwÞ

N

2N

n¼0

0

n – 0:

ð11Þ

In practice, GCC is the only binary code that theoretically exhibits a perfect pulse compression without any side-lobe [32]. For the modulated GCC as shown in Fig. 4(b), its auto-correlation curve is shown in Fig. 6, which clearly exhibits the property of side-lobe cancelling. As the virtual excitation signals, BC and GCC auto-correlation curves are plotted simultaneously in Fig. 7 for comparisons. Both curves are normalized. It can be seen that their main-lobes (enclosed by the dashed line rectangle) are with the same shape,

Fig. 2. 13-bit Barker code.

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Fig. 3. 8-bit Golay complementary code.

Fig. 4. Modulation result for: (a) 13-bit Barker code and (b) 8-bit Golay complementary code.

while the side-lobes are with different shapes. In practice, the main-lobes will be the same if the same sinusoidal signal s(t) is employed for BPSK modulation in Eq. (5) [33]. The side-lobes can never be the same because BC and GCC are with different properties in auto-correlation.

Step 3: Dispersion compensation After pulse compression, dispersion compensation based on the wavenumber linear Taylor expansion is applied, the frequency range covered in this expansion is consistent with excitation

J. Hua et al. / Ultrasonics 70 (2016) 107–122

111

Fig. 5. Auto-correlation curves of 13-bit BC: (a) before modulation and (b) after modulation.

Fig. 6. Auto-correlation curve of 8-bit GCC after modulation.

frequency range [34]. By transforming the dispersive signals cbc(t) and cgcc(t) to frequency domain and interpolating the signals at wavenumber values satisfying the linear relationship, the shapes of the excitation signals (i.e. BC and GCC auto-correlation) can be recovered. In particular, the pulse compression signals cbc(t) and cgcc(t) after dispersion compensation are with the same shape as the BC and GCC auto-correlation curves (Fig. 7), respectively [33–35]. Step 4: Correlation analysis between the extracted local signals Focusing on the kth actuator–receiver pair as shown in Fig. 1, let ckbc ðtÞ and ckgcc ðtÞ denote the received Lamb wave signals after binary code excitation, pulse compression and dispersion

compensation (i.e. Step 1, Step 2 and Step 3). Here ckbc ðtÞ corresponds to BC excitation, and ckgcc ðtÞ corresponds to GCC excitation. Containing enough information, global signals ckbc ðtÞ and ckgcc ðtÞ can depict the structural state of the whole area under inspection. However, local signals (i.e. a part of global signals) only depicting structural state at a single point are needed here. Define that ckbc ðt; x; yÞ and ckgcc ðt; x; yÞ represent such signals, exclusively related to the structural state at point (x, y). By applying a timeshift and a windowing function to global signals, local signals can be obtained as

ckbc ðt; x; yÞ ¼ ckbc ½t þ t k ðx; yÞwðtÞ; ckgcc ðt; x; yÞ ¼ ckgcc ½t þ tk ðx; yÞwðtÞ:

ð12Þ

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Fig. 7. Comparison of auto-correlation curves for BC and GCC.

In Eq. (12), the time-shift tk(x, y) is the time duration for the Lamb wave travelling from the actuator to (x, y) and then to the receiver, which is calculated in Eq. (1), w(t) is a unit amplitude rectangular window centered at t = 0 with a given width. In practice, the window width controls the length of the local signal, which is equivalent to the main-lobe width (Fig. 7) in this paper. If damage exists at (x, y), assuming that reflection from the damage does not change the shape of the propagating Lamb wave signal due to narrowband excitation [13–15], the extracted local signals ckbc ðt; x; yÞ and ckgcc ðt; x; yÞ will be exactly the main-lobes of damage-reflected wave packets corresponding to BC and GCC excitation, respectively. As mentioned previously, they are with the same shape. Comparatively, if no damage exists at (x, y), ckbc ðt; x; yÞ and ckgcc ðt; x; yÞ are free of the main-lobes of damagereflected wave packets. Actually, they will be the side-lobes of other wave components (e.g. direct wave, reflected wave from the boundary, or reflected wave from the damage at other place), thus exhibiting different shapes. As a result, a measure can be built to quantify the similarity between the local signals ckbc ðt; x; yÞ and ckgcc ðt; x; yÞ, and this measure is associated with the structural state at point (x, y). Correlation coefficient is employed for this purpose. The correlation coefficient, q, between the two local signals, X = ckbc ðt; x; yÞ and Y = ckgcc ðt; x; yÞ, is

q¼

Cov ðX; YÞ

rX rY

;

ð13Þ

K X ðX k  lX ÞðY k  lY Þ

ð14Þ

k¼1

and the standard deviations,

rX



! LSCCðx; yÞ ¼ ½LSCC1 ðx; yÞLSCC2 ðx; yÞ    LSCCM ðx; yÞ;

3. Imaging algorithm 3.1. MVDR In MVDR imaging algorithm, the pixel value at the point (x, y) is calculated in vector format as [19–22]

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XK XK 2 ¼ ðX k  lX Þ ; rY ¼ ðY k  lY Þ2 : k¼1 k¼1

ð15Þ

Then, the local signal correlation coefficient is, LSCCk(x,y) = q. It varies between 0, when two local signals have no linear relationship, and 1, when two local signals are exactly with the same shape. Therefore, LSCCk(x, y) = 1 if damage exists at (x, y); on the other hand, LSCCk(x, y) is much smaller and close to 0 if no damage exists at (x, y). Damage identification accuracy at point (x, y) can be improved with the incorporation of excitation information. As shown in Fig. 7, the excitation main-lobe surrounded by the dashed line rectangle window has a finite length, which can be regarded as another local signal. Based on the fact that damage-reflected main-lobe and

ð17Þ

where the superscript T represents the transpose. The middle term, Fxy , is a correlation matrix calculated as

f Hxy : Fxy ¼ ~ f xy~

rX and rY , are

ð16Þ

where M ¼ ð2Nþ1Þð2NÞ . 2 For damage point, the elements in Eq. (16) are the same and equal to 1. For non-damage point, the elements in Eq. (16) are different and believed to be very small.

~ Txy Fxy w ~ xy ; Pxy ¼ w

where the covariance, Cov, is

Cov ðX; YÞ ¼

excitation main-lobe are with the same shape after dispersion compensation (step 3), the similarity between excitation mainlobe with ckbc ðt; x; yÞ or ckgcc ðt; x; yÞ by Eq. (13) also generates LSCC to depict the structural state at point (x, y). Considering a transducer array consisting of N actuator– receiver pairs, for each point (x, y) within the inspection area, 2N local signals corresponding to both BC and GCC excitation can be obtained by Eq. (12). Taking into account the excitation mainlobe, we get 2N + 1 local signals. Two arbitrarily selected local signals are used to calculate the correlation coefficient, generating a total of (2N + 1)  (2N)/2 LSCCs from every possible combination. These LSCCs are organized as a row vector,

ð18Þ

where ~ f xy is a vector containing the input damage features, referred to as damage feature vector. The damage features used in this study include signal amplitude and LSCC, which are introduced in Sections 2.1 and 2.2, respectively. ~ xy , is chosen to satisfy the following The other term in Eq. (17), w constrained optimization problem,

~ Hxy Rxy w ~ Hxy~ ~ xy such that w min w exy ¼ 1;

ð19Þ

where ~ exy is a pre-designed unit column vector, referred to as the look direction, which describes the anticipated relationship between elements in ~ f xy when damage is present at (x, y). As with ~ f xy , vector ~ exy is specific to the point (x, y). The solution to Eq. (17) can be found using Lagrange multipliers, and is [36]

J. Hua et al. / Ultrasonics 70 (2016) 107–122

113

Fig. 8. Scattering characteristic analysis with different damage parameters: (a) case I, (b) case II, (c) case III and (d) case IV.

~ xy ¼ w

~ R1 xy exy ~ ~ eHxy R1 xy exy

;

ð20Þ

where the superscript 1 represents a matrix inverse. In Eq. (20), R1 xy may become ill posed and require regularization. Generally, the regularization is achieved through diagonal loading [37], as 1

R1 xy ¼ ðR xy þ f k1 IÞ :

ð21Þ

Here, k1 is the largest eigenvalue of Rxy , f is a fraction used to describe the degree of diagonal loading. ~ xy into Eq. (17), the pixel value is minSubstituting the solved w exy (i.e. ~ f xy is proimized unless ~ f xy has the relationship described in ~ ~ xy in Eq. (19) ~ Hxy Rxy w portional to ~ exy ). In particular, the term min w minimizes the pixel value, but the constraint of the inner product ~ Hxy~ exy ¼ 1 preserves the pixel value when the relationship w exy describes the described in ~ exy is satisfied. Because the vector ~ relationship of the damage feature elements only if damage is present at (x, y), the pixel value is preserved only at damage point. On this basis, MVDR algorithm is effective to reduce imaging artifacts at non-damage points. A big challenge associated with MVDR imaging algorithm is the sensitivity to the pre-designed look direction, ~ exy . Error in ~ exy can severely degrade algorithm imaging performance. Details about ~ exy design for both amplitude-based and LSCC-based MVDR are discussed in Sections 3.2 and 3.3, respectively.

3.2. MVDR based on amplitude If amplitude is used as the input damage feature, ~ f xy is equivalent to the vector expressed in Eq. (2),

~ f xy ¼ ½r 1 ðt 1 ðx; yÞÞr 2 ðt2 ðx; yÞÞ    r N ðt N ðx; yÞÞ:

ð22Þ

As discussed in Section 2.1, when damage is present at (x, y), each element in the ~ f xy vector is the amplitude of the damage-reflected wave packet from the corresponding actuator–receiver pair. Referring to [19,20], wave packet amplitude is expected to be a function of the scattering characteristic and the propagation distance. As mentioned in Section 3.1, a reasonable look direction is proportional to ~ f xy at damage position, thus ~ exy based on this information is designed as

~ exy 

 w1xy w2xy wixy wNxy ;   d1xy d2xy dixy dNxy

ð23Þ

where dixy is the product of the distances from the actuator to point (x, y) and from (x, y) to the receiver for the ith actuator–receiver pair, wixy is a scattering coefficient that characterizes the amount of energy scattered by the damage at (x, y) for the ith actuator–receiver pair. Here, the multiplication of distances is appropriate to account for the geometric spreading due to the reflector. The scatter inforexy mation wixy is incorporated into look direction to improve the ~ accuracy. Note that, ~ exy should be scaled to be a unit-norm vector.

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Fig. 9. Aluminum plate with three introduced defects and the clock-like sensor array: (a) photograph; (b) schematic diagram.

3.3. MVDR based on LSCC If LSCC is used as the input damage feature, ~ f xy is equivalent to the vector expressed in Eq. (16),

~ f xy ¼ ½LSCC1 ðx; yÞLSCC2 ðx; yÞ    LSCCM ðx; yÞ;

ð24Þ

where M ¼ ð2Nþ1Þð2NÞ . 2 As mentioned in Section 2.2, when damage is present at (x, y), the elements in the ~ f xy vector are equivalent. A reasonable look direction is proportional to ~ f xy at damage position, thus ~ exy based on this information is designed as

~ exy  ½11    1T :

ð25Þ

Note that ~ exy here is an M dimensional vector because ~ f xy has M elements. As before, ~ exy should be scaled to be a unit-norm vector. 4. Performance analysis Two imaging methods are employed for comparisons: (1) amplitude-based MVDR and (2) LSCC-based MVDR. Pixel values at both damage and non-damage locations are compared for different imaging methods. In practice, in a perfect image, pixel value at damage location would be preserved and all other pixel values would be minimized. 4.1. Pixel value at damage location The pixel value at damage location could be preserved if the look direction ~ exy accurately describes the relationship of the damage feature elements in ~ f xy . In amplitude-based MVDR, however, accurate design of ~ exy model is difficult, and the primary difficulty is the evaluation of the scattering characteristics wixy in Eq. (23). The reason lies that accurate evaluation of the scattering characteristic depends on damage parameters [23–26]. To investigate the influence of damage parameters (i.e. type, orientation and size) on scattering characteristic, four cases (I, II, III and IV) of two actuator–receiver pairs with different damage parameters are considered, as shown in Fig. 8. Using both actuator–receiver pairs, the damage-reflected Lamb waves could be captured. The corresponding reflection coefficients (i.e. the ratio of the magnitude of the damage-reflected wave to that of incident wave) are denoted as R1 and R2, respectively. In case I, it is obvious that R1 = R2 because the symmetric damage (i.e. a hole) has no

orientation dependence [26]. In case II, R1 > R2 because the reflection coefficient under normal incidence to a crake is much higher than that under oblique incidence [23–25]. In case III, the crack orientation is varied and the incident Lamb wave of the 2nd actuator– receiver pair is normal to the crack, thus R1 < R2. In case IV, the crack orientation is the same as that in case II. However, the two cracks are with different lengths, thus exhibit different scattering characteristics [25]. That is to say, the ratio R1:R2 is different in the two cases. Based on previous analysis, it could be concluded that the scattering characteristic correlates highly with damage type, orientation and size. With these damage parameters unknown in most practical situation, it is difficult or even impossible to give the accurate scattering model in Eq. (23). In published research, the scattering model is generally assumed to be uniformly distributed for simplicity [21,22]. The same assumption is used in this study, then the ~ exy model in Eq. (23) reduces to

~ exy 



1 1 1 1 :   d1xy d2xy dixy dNxy

ð26Þ

The error in this ~ exy model will severely reduce the pixel value at damage location, thus the damage may not be distinguished from imaging artifacts, which will result in false identification result. In LSCC-based MVDR, the design of ~ exy model in Eq. (25) is independent of the damage parameters. This is attributed to the fact that LSCC is a measure of the shape matching extent, which is independent of scattered characteristic. In practice, the accuracy of ~ exy in describing the relationship in LSCC vector is based on the following assumptions: (1) the main-lobes of binary code responses after pulse compression are with the same shape; (2) scattering by a damage does not change the signal shape of limited bandwidth [13–15]; (3) the shape of excitation signal is recovered by dispersion compensation. The former assumption is reasonable because BC and GCC auto-correlations are with the same main-lobe, which can be clearly seen in Fig. 7. The latter two assumptions also could be satisfied by choosing narrowband excitation and proper dispersion compensation method [34,35]. As discussed in step 3 in Section 2.2, the linear wavenumber mapping technique is chosen in this study, due to its advantage in excitation waveform reconstruction [35]. In this study, Lamb wave signals are used in envelope format to reduce the influence of weak/small difference and achieve improved signal matching extent, so that the accuracy of the ~ exy model in LSCC-based MVDR could be further improved. With

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Fig. 10. Residual responses captured by the sensor array corresponding to different excitation signals: (a) tone burst excitation, (b) BC and GCC excitation.

artifacts is minimized. In amplitude-based MVDR, however, the amplitude relationship at non-damage location is difficult to control. It may agree with the corresponding look direction ~ exy by coincidence, particularly when only few actuator–receiver pairs are used. Under this circumstance, the imaging artifacts is preserved rather than minimized. Such coincidence in LSCC-based MVDR could be avoided because the used binary codes (i.e. BC and GCC) show considerably different side-lobes, which can be seen in Fig. 7. As discussed in step 4 in Section 2.2, most of the extracted local signals at nondamage location belong to side-lobes, thus they are with different shapes. Arbitrary combination of two local signals will not generate the same correlation coefficient. This means no likelihood that the generated LSCC vector will coincidently agree with the designed model in Eq. (25), ~ exy  ½11    1T .

5. Experimental investigation 5.1. Specimen and experimental setup

Fig. 11. MVDR images based on different input damage features: (a) amplitude, (b) LSCC.

accurate ~ exy model, the pixel value at damage location could be perfectly preserved. 4.2. Pixel value at non-damage location The pixel value at non-damage location could be minimized f xy , so that the imaging through the disagreement between ~ exy and ~

The experiment was carried out on an aluminum plate with dimensions 1000 mm  1000 mm  2 mm. Three artificial defects (r, s and t) were introduced in the form of through-thickness rectangular slots. Bonded with the plate, 7 piezoelectric ceramic discs with a diameter of 8 mm and 0.5 mm in thickness were networked as a clock-like sensor array where the circle has the dimensions of U60 mm. Fig. 9 shows the photograph and the schematic diagram of the aluminum specimen with defects and the PZT array. In the array, the PZT element located at the center (the red1 point in Fig. 9(b)) serves as the actuator, while others (the blue points in Fig. 9(b)) serve as receivers. A coordinate system was employed with the plane of the monitoring area spanned by the horizontal, x, and vertical, y, axes, where the origin of coordinate was set to be the center of the array. Three artificial defects are 255 mm, 297 mm and 348 mm away from the actuator, the coordinates of the center of which are (212, 142), (172, 241), (64, 342) in the form of (x, y), respectively. In the experiment, the actuator is driven with three types of excitation signal: (1) 3-period Hanning-windowed sinusoidal tone burst signal centered at 80 kHz, (2) BC and (3) GCC. Tone burst excitation is used for amplitude-MVDR imaging, and binary code excitation is used for LSCC-MVDR imaging. Note that both BC and GCC need BPSK modulation. For fair comparison, a 1-period sinusoidal signal with the frequency f = 80 kHz is used for modulation, so that 1 For interpretation of color in Fig. 9, the reader is referred to the web version of this article.

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Fig. 12. Comparisons between the look direction ~ exy and the damage feature vector in different methods at different defect locations: (a) amplitude-based MVDR, defect r location; (b) amplitude-based MVDR, defect s location; (c) amplitude-based MVDR, defect t location; (d) LSCC-based MVDR, defect r location; (e) LSCC-based MVDR, defect s location; (f) LSCC-based MVDR, defect t location.

the frequency range is approximately identical with that of the tone burst. Every excitation signal is generated by an Agilent 33220A function/arbitrary waveform generator. After being amplified with the peak-to-peak voltage of 50 V (by a Piezo Systems EPA-104 voltage amplifier), it is applied to the actuator of the sensor array. The Lamb wave signals pass through the monitoring area and are captured by the receivers. After being amplified by the AVANT NI-2000 conditioning amplifiers, they are acquired at a sampling rate of 2 MHz by a NI PXIe-1082 data acquisition. With the excitation signals centered at 80 kHz, only the fundamental Lamb modes A0 and S0 exist, and A0 is dominant. Thus, A0 mode is selected. 5.2. Result Fig. 10(a) shows the residual tone burst responses after dispersion compensation captured by the sensor array. The residual signal is obtained by differencing the signal from the defected plate and that from the same perfect plate (before introduction of the defects). In Fig. 10(a), the three wave packets enclosed by the dashed line rectangle are the ones scattered by the defects. To utilize the amplitude relationship of the scattered wave packets, the signals are scaled appropriately to ensure that the amplitudes of the direct wave packets are consistent for different actuator–receiver pairs, so that the differences in both sensors and bonding are eliminated.

Fig. 10(b) shows the residual BC and GCC responses after pulse compression and dispersion compensation captured by the sensor array. The responses are in envelope format, the amplitude of each envelope is normalized by its maximum. Here, the amplitude scaling in Fig. 10(a) is not applied because the shape information rather than the amplitude information is needed in the calculation of LSCC. As mentioned, amplitude vector and LSCC vector are calculated from the responses in Fig. 10(a) and (b), respectively. Both damage feature vectors are employed for MVDR imaging. In imaging, the diagonal loading coefficient f in Eq. (21) is first set as 0.1, the corresponding amplitude- and LSCC-based images after normalization are shown in Fig. 11(a) and (b), respectively. For both images, the ‘‘+” symbols indicate the central locations of actual defects. The image in Fig. 11(a) illustrates the typical characteristics of the image generated from amplitude-based MVDR. First, due to the disagreement between the look direction ~ exy and the amplitude vector, pixel value for some defect is reduced, which makes it difficult to distinguish the defect from imaging artifacts. In this case, the pixel value for defect r is relatively small, in practice, it is much smaller than those for defects s and t. The reason lies that ~ exy and the amplitude vector at defect r location are in worst agreement. For clear illustration, the comparisons between ~ exy and the amplitude vector at three defect locations are shown in Fig. 12(a)–(c), respectively. Note that the amplitude vectors in these figures are scaled to be unit-norm vectors for fair

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Fig. 13. Comparisons between amplitude- and LSCC-based MVDR images at different excitation time durations. The amplitude-based images are shown on the left hand side when the cycle number of the tone burst takes: (a) 2, (c) 3, (e) 4 and (g) 5; The LSCC-based images are shown on the right hand side when the cycle number of the sinusoidal modulation takes: (b) 0.75, (d) 1, (f) 1.5 and (h) 2.

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Fig. 14. Comparisons between amplitude- and LSCC-based MVDR images at different diagonal loading coefficients. The amplitude-based images are shown on the left hand side when the diagonal loading coefficient takes: (a) 0.02, (c) 0.05, (e) 0.1 and (g) 0.2. The LSCC-based MVDR images are shown on the right hand side when the diagonal loading coefficient takes: (b) 0.02, (d) 0.05, (f) 0.1 and (h) 0.2.

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Fig. 15. Schematic diagrams of the experiments: (a) case I, (b) case II and (c) case III. Several photographs of the experiments are in (d), (e) and (f).

Fig. 16. Damage reflection wave in case I corresponding to different crack sizes: (a) length is 10 mm, depth is 1 mm; (b) length is 12 mm, depth is 2 mm; (c) length is 16 mm, depth is 3 mm; (d) length is 20 mm, depth is 3.74 mm.

comparisons. Inner product is used to quantitatively describe the agreement degree between two unit-norm vectors. As can be seen, the inner product in Fig. 12(a) is the smallest, corresponding to the worst agreement. Second, due to coincidental and undesired agreement between the look direction ~ exy and the amplitude vector, pixel value at some non-defect location is preserved, resulting in imaging artifacts. In this case, obvious artifacts appear at the circle loci with a radius of defect occurrence.

The LSCC-based image in Fig. 11(b) clearly has improved image quality compared to the amplitude-based one in Fig. 11(a). First, pixel value for each defect is perfectly preserved, due to the perfect agreement between the look direction ~ exy and the LSCC vector at each defect location. For clear illustration, the comparisons between them at three defect locations are shown in Fig. 12(d)–(f), respectively. As can be seen, every inner product is very close to 1, even the smallest one (0.9976) shows much closer agreement

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Fig. 17. Damage reflection wave in case II captured by different receivers in respect to different crack orientations: (a) receiver 1, (b) receiver 2, (c) receiver 3 and (d) receiver 4.

Fig. 18. Damage reflection wave in case III corresponding to different through-hole diameters: (a) 5 mm, (b) 6 mm, (c) 8 mm and (d) 10 mm.

than those in Fig. 12(a)–(c). Second, imaging artifacts are considerably suppressed with negligible pixel values at all non-defect locations. 6. Discussions 6.1. Results with different excitation durations Excitation signals with different time durations are used and their effects on imaging performance are investigated. In amplitude-based MVDR, the tone burst signals are all centered at 80 kHz, but their cycle number changes from 2 to 5 with a step of 1. The corresponding MVDR images are shown on the left hand side of Fig. 13. When the cycle number is small, the image is clear. As the increase of cycle number, the images get fuzzier. The

reasons behind this phenomenon could be explained as follows. First, increasing the cycle number (i.e. time duration) causes overlap of the damage-reflected wave packets, which changes the amplitude relationship in their envelope curves and thus reduces the pixel value at damage location. Meanwhile, imaging artifacts are amplified during normalization. Second, increasing the cycle number increases the opportunity for undesired agreement between the look direction ~ exy and the amplitude vector at non-damage locations. In LSCC-based MVDR, the cycle number of the sinusoidal modulation for BC and GCC takes 0.75, 1, 1.5 and 2, respectively. The frequency ranges of the signals after modulation are approximately identical with that of the tone bursts. The corresponding MVDR images are shown on the right hand side of Fig. 13. It can be seen that these images are much more explicit

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defects are perfectly preserved. Also, even when f is 0.2, imaging artifacts are effectively suppressed. In summary, Fig. 14 clearly demonstrates the improved imaging performance of the LSCCbased algorithm as compared to the amplitude-based algorithm at different diagonal loading coefficients.

7. Performance of LSCC under different damage parameters

Fig. 19. The calculated LSCCs in three cases: (a) case I, the evolution of LSCC with the change of crack sizes; (b) case II, the evolution of LSCC with the change of crack orientations; (c) case III, the evolution of LSCC with the change of through-hole sizes.

than the amplitude-based images. In practice, Fig. 13 clearly demonstrates the improved imaging performance of the LSCCbased algorithm as compared to the amplitude-based algorithm at different excitation time durations.

6.2. Results with different diagonal loading coefficients The diagonal loading coefficient f is set as 0.1 in previous experiments. For further illustration, considering the case in Section 5, f is set as 0.02, 0.05, 0.1 and 0.2 to investigate its effect on MVDR imaging performance. The amplitude-based MVDR images with different diagonal loading coefficients are shown on the left hand side of Fig. 14. When f is small, it is observed that pixel value for defect r is reduced due to the inaccuracy of the look direction ~ exy . As the increase of f, tolerance for inaccuracy in the look direction is also increased. This is the reason why the pixel value for defect r is much higher when f is 0.1 or 0.2 than that when f is 0.02. However, the tolerance comes at the cost of larger imaging artifacts, which can be seen from the evolution of the imaging artifacts associated with the changes of f. For comparison, LSCC-based MVDR images with different diagonal loading coefficients are shown on the right hand side of Fig. 14. As can be seen, even when f is 0.02, pixel values for three

The experiment in Sections 5 and 6 demonstrates the effectiveness of LSCC to detect cracks with different orientations. In this section, more experiments are investigated, in which damage is with varying type, orientation and size, to analyze the sensitivity of LSCC to these varying damage parameters. Three further experiments are carried out on three aluminum plates of dimensions 1000 mm  1000 mm  3.74 mm. The schematic diagrams for these cases are shown in Fig. 15(a)–(c), respectively. Several photographs of the aluminum specimen with the PZT distribution and the artificial damage are shown in Fig. 15(d)–(f). In Fig. 15(a) (case I), the length of introduced crack increases from 10 mm to 12 mm to 16 mm to 20 mm, meanwhile, the depth of the crack increases from 1 mm to 2 mm to 3 mm to 3.74 mm. In Fig. 15(b) (case II), the crack size is fixed at 20 mm (length) and 3.74 mm (depth), four receivers are used to capture the wave reflection in respect to different crack orientations. In Fig. 15(c) (case III), through-thickness hole is introduced, and the diameter of the through-hole increases from 5 mm to 6 mm to 8 mm to 10 mm. In these experiments, the geometric configuration of the aluminum plate and the location of PZT elements are properly arranged so that the damage reflection wave packet can be completely extracted for analysis without the interference of boundary reflections. As with the previous experiment, BC and GCC are used as the excitation signals. A 1-period sinusoidal signal with the frequency f = 80 kHz is used for BC and GCC modulation. With the excitation signals centered at 80 kHz, A0 mode is dominant, thus A0 mode is selected for analysis. In particular, the damage reflection wave packets propagating as the A0 mode after pulse compression and dispersion compensation are extracted for analysis. Figs. 16–18 show the captured damage reflection wave packets in case I, II and III, respectively. The reflections are in envelope format, the amplitude of each envelope is normalized by its maximum so that the shape comparison can be better highlighted. The three cases are representative because different damage parameters (including type, orientation and size) are taken into consideration. From each figure, it can be seen that the two main-lobes of the reflection waves (enclosed by the dashed line rectangle) are with similar shape. In addition, the two mainlobes also has similar shape to the excitation one in Fig. 7. For clearer illustration, LSCCs are calculated to quantitatively describe the shape matching extent in Figs. 16–18. At each damage parameter, two reflection main-lobes are extracted, considering also the excitation main-lobe, a total of three local signals are obtained. Combination of arbitrary two local signals generates a LSCC, thus there are three LSCCs at each damage parameter. The calculated LSCCs in case I, II and III are shown in Fig. 19. In particular, Fig. 19(a) shows the evolution of LSCC with the change of crack size in case I, Fig. 19(b) shows the evolution of LSCC with the change of crack orientation in case II, Fig. 19(c) shows the evolution of LSCC with the change of through-hole size in case III. From Fig. 19, it can be seen that most LSCCs are closely equivalent to 1, and even the smallest value is larger than 0.93, which quantitatively demonstrate the main-lobe matching phenomenon in Figs. 16–18. In summary, these experiments demonstrate that the proposed damage feature LSCC is effective to detect damage of different parameters.

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8. Conclusions LSCC is a new damage feature that can be employed in MVDR for damage identification and localization. Compared to the original amplitude-based MVDR, LSCC-based MVDR shows improved imaging performance. Several conclusions are obtained as follows. (1) In LSCC-based MVDR, the look direction ~ exy at damage location could be accurately designed, which is not available in amplitude-based MVDR. On this basis, pixel value at damage location is perfectly preserved. (2) In LSCC-based MVDR, the look direction ~ exy at non-damage location will never coincidently agree with the damage feature vector. On this bases, pixel value at non-damage location is minimized, thus the imaging artifacts could be further reduced.

Acknowledgement The work is supported by the National Natural Science Foundation of China (Grant Nos. 51421004, 51505365) and the China Postdoctoral Science Foundation (Grant No. 2015M572552), which are highly appreciated by the authors.

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