Boosting nonlinear ultrasonic detection performance: a framework combining coded excitation and sequence filtering for weak nonlinear feature enhancement

Abstract

Ultrasonic techniques are widely employed in ensuring structural safety. Traditional ultrasonic techniques are effective for detecting macro-damage yet struggle with early-stage micro-damage. Nonlinear ultrasonic techniques offer promising technical solutions for early-stage micro-damage detection and imaging in structural components. However, the intrinsic weakness of nonlinear features may lead to measurement inaccuracies or imaging artifacts, hindering practical engineering applications. To combat this, a framework integrating pulse compression with coded excitation is proposed to enhance nonlinear acoustic field energy. Specifically, complementary Golay sequences are employed for phase-coded excitation, boosting average transmitting power without increasing peak power. While considering the need to retain the nonlinear features, a sequence filter is used to replace the conventional matched filter to preserve the second harmonic component during signal decoding. Simulations and experiments validate the method’s efficacy in material nonlinearity assessment and micro-damage imaging, demonstrating improved nonlinear feature extraction, artifact suppression, and imaging quality. This work applies coded excitation/decoding techniques to the field of nonlinear ultrasound, significantly enhancing micro-damage detection performance and paving a promising way for early damage assessment in practical engineering structures.

Keywords

Nonlinear ultrasound pulse compression coded excitation micro-damage detection nonlinear feature enhancement

Introduction

Structural health monitoring (SHM) technology plays a critically important role in safeguarding structural integrity and operational safety across high-stakes industries such as nuclear power, aviation, and chemical processing. The timely and reliable detection of potential defects, coupled with a scientific assessment of their impact on structural strength, provides an effective way to compensate for limitations arising during design and manufacturing processes. Among various SHM methods, ultrasonic techniques have gained significant prominence.^1,2 Traditional ultrasonic techniques generally assess damage status by analyzing the linear features, including velocity, phase, and amplitude attenuation, which are extracted from the reflected and transmitted ultrasonic waves. These techniques are well-suited for identifying wavelength-level macro-damage like macrocracks, inclusions, and pores. Nevertheless, the detection of early-stage micro-damage or material degradation poses a huge challenge based on traditional ultrasonic techniques due to the limited sensitivity of the linear features.

It has been reported that nonlinear ultrasonic techniques exhibit high sensitivity to invisible micro-damage, effectively overcoming the limitations of traditional linear ultrasonic techniques.^3,4 The representative nonlinear ultrasonic techniques include the second harmonic generation technique,⁵ the sideband technique,⁶ and the frequency mixing technique.⁷ By analyzing and establishing the correlation between micro-damage status, material degradation, and nonlinear features, the research on nonlinear ultrasonic techniques provides a novel way for early-stage micro-damage detection, attracting considerable attention in recent years. Sagar et al.⁸ introduced the nonlinear ultrasonic technique to assess the fatigue damage status of the heat-resistant alloy steel, and the results indicated that the nonlinear parameter exhibited a changing pattern of rising, falling, and then rising during the fatigue experiment. Xiang et al.⁹ conducted the nonlinear ultrasonic experiment for evaluating Ti-60 creep damage status, the nonlinear parameter revealed a trend of initial increase followed by a decrease with the progression of the high-temperature creep experiment. In addition to the application of the micro-damage assessment, such as fatigue, creep, and so on, nonlinear ultrasonic techniques have been widely studied for micro-damage imaging, represented by the fatigue crack. Potter et al.¹⁰ conducted contact acoustic nonlinearity (CAN) imaging research based on the nonlinear ultrasonic phased array technique, which has been one of the practical methods for fatigue crack imaging. On this basis, multiple micro-damage imaging methods are proposed, including fundamental amplitude difference, subharmonic detection, and acoustic diffusion field.^11–13 While nonlinear ultrasound has demonstrated effectiveness in micro-damage assessment and imaging, one critical issue remains: the nonlinear features are very weak, which may lead to detection misjudgments, imaging artifacts, and considerably restrict the development and engineering applications of micro-damage detection and imaging research. How to enhance the nonlinear features is one of the urgent priorities in the field of nonlinear ultrasonic research.

Some potential schemes can be implemented to intensify the nonlinear features, all of which have the core idea of improving the energy of the acoustic field. The simplest method involves increasing the amplitude of the transducers’ excitation voltage. However, it is noted that an instantaneous high-voltage input can produce strong electromagnetic interference, affect the performance of the adjacent transducers, and even cause transducer damage. The second method employs array sensing techniques to adjust the transmission time of each transducer, thereby achieving physical focusing.¹⁴ The amplitude of the acoustic field in the target region is increased, and the nonlinear response can be further enhanced. Nevertheless, this scheme requires multiple transducers to excite simultaneously, which imposes significant requirements on equipment hardware and is limited in the data processing and imaging rate. The local defect resonance (LDR) is also an effective technique for enhancing nonlinear ultrasonic features, which achieves energy localization of nonlinear behavior through resonant excitation. When the external excitation frequency matches the natural frequency of the defective region, vibrational energy concentrates and is significantly amplified within the damage zone.¹⁵ This drives interactions such as collision, friction at contact interfaces, and the opening-closing of crack surfaces, thereby markedly enhancing nonlinear features that are difficult to detect under conventional excitation, which enables high signal-to-noise ratio (SNR) detection of damage. LDR has been applied in areas such as debonding detection in composite materials and micro-crack identification in metallic structures.^16–18 However, the resonant frequency is closely related to the damage morphology. Given that actual defects often involve complex and variable geometric and boundary conditions, accurately determining the resonant frequency remains a challenging task.

Pulse compression technique, generally combined with coded excitation technique, is widely applied for improving signal quality in different fields, including radar, ultrasonic non-destructive testing, and ultrasonic medical imaging.^19–21 It generally adopts a coded modulation excitation signal with a fixed excitation pulse amplitude to amplify the energy transmission and improve the average acoustic field power by extending the signal transmission duration. However, the introduction of long pulse waves leads to the wide echo, which can cause overlapping between echoes from closely spaced targets and sacrifice the range resolution.²² To deal with it, the pulse compression technique is employed to convert the long pulse echo waves into a single narrow pulse with high-intensity power using a designed filter, and the decoding waves obtained have a resolution equivalent to that of the echo under traditional single-pulse excitation. Pulse compression combined with coded excitation technique effectively synthesizes the advantages of high-resolution signal with short pulse excitation and high energy transmission under high amplitude excitation, which has become a practical and effective method in ultrasonic testing and imaging research dealing with coarse grains, high-attenuation materials, and large-scale structures.^23–25

The advantage of pulse compression combined with the coded excitation technique in balancing “high energy” and “high resolution” paves a potential avenue for tackling the problem of the weak nonlinear features existing in nonlinear ultrasonic research. However, existing pulse compression techniques in ultrasonic testing research mostly focus on macro-damage detection and imaging using linear features. It is doubtful whether the traditional pulse compression combined with the coded excitation technique, especially the traditional pulse compression technique, is suitable for micro-damage detection based on nonlinear ultrasonic techniques. The signal decoding effectiveness of the pulse compression technique is related to the designed filter. A matched filter is commonly employed, that is, the cross-correlation between the echo waves and the excitation waves is calculated.^22,26–29 The yielded decoded signal has a bandwidth identical to that of the excitation waves, resulting in some valuable frequency information being suppressed, such as higher-order harmonic components, sideband frequency, and other nonlinear features. Therefore, the traditional pulse compression technique using the matched filter is not suitable for micro-damage detection based on nonlinear ultrasonic techniques.

Aiming at the typical yet critical problem of weak nonlinear features in nonlinear ultrasonic research, a framework for nonlinear acoustic field energy enhancement guided by the pulse compression technique combined with the coded excitation technique is established in this work. In terms of signal excitation, the phase-coded excitation technique with the complementary Golay sequences (CGS) is introduced to augment the average transmitting power without increasing the peak transmitting power, thereby enhancing the acoustic field energy. Regarding signal processing, the pulse compression technique with a sequence filter, as a replacement for the conventional matched filter, is introduced for realizing echo wave decoding without distortion while fully retaining the nonlinear features of the echo waves. To confirm the validity of the proposed combined CGS-coded excitations and sequence filtering method in nonlinear ultrasonic research, the material nonlinearity assessment research is carried out and implemented through finite element (FE) simulation and experiment. Furthermore, the analysis of the CAN represented by the fatigue crack is employed as a demonstrative case to verify the framework’s effectiveness in enhancing nonlinear features, suppressing artifacts, and improving micro-damage imaging quality.

The remainder of the paper is organized as follows: the second section demonstrates the proposed methodology along with the detailed theory. The validity of the proposed framework is verified in the third and fourth sections through research on material nonlinearity assessment and CAN imaging, respectively. Conclusions are summarized in the fifth section.

Nonlinear feature enhancement framework

The proposed framework for nonlinear feature enhancement based on the coded excitation technique and the pulse compression technique includes three steps: (1) the phase-coded excitation technique is implemented to enhance the acoustic field energy, with the resultant effect of augmenting the absolute amplitude of the nonlinear features; (2) the sequence filter is used to decode system response waves without distortion, which preserves the nonlinear features generated by the micro-damage; (3) the nonlinear features existed in the decoded signals are extracted for nonlinear ultrasonic research, with typical applications include material damage state assessment and micro-damage imaging.

Coded excitation and pulse compression technique

The coded excitation technique employs the signal modulation method to boost the output energy of transducers by prolonging the signal transmission duration. Common signal modulation techniques encompass frequency modulation and phase modulation. The typically employed frequency modulation signal is the chirp excitation signal, which is not appropriate for nonlinear ultrasonic research due to its broadband nature. The phase-modulated signal $m (t)$ commonly introduces one fixed pulse signal $s (t)$ , referred to as the symbol waves, which is continuously sent in accordance with a varying binary sequence $p (t)$ :

m (t) = s (t) \otimes p (t),

(1)

where $\otimes$ represents the convolution operation. The common phase sequences include the Barker sequence, M-sequence, and CGS. CGS exhibits ideal autocorrelation, and the sidelobe can be eliminated after the complementary signals are superimposed, achieving the optimal pulse compression effect and making it more suitable for ultrasonic detection. The phase sequences employed in the following work are CGS unless stated otherwise.

With the introduction of the coded excitation method, the transducer emits long pulses, and the received signal should be compressed using a filter to obtain an echo signal comparable to that of narrow pulse excitation, which improves the system’s detection performance while preserving the signal resolution. The matched filtering, commonly employed in the pulse compression technique, is the operation of convolving the received signal with the time reverse of the modulated signal. Assuming that the ultrasonic measurement system is a linear time-invariant system, the impulse response function is $f (t)$ . The ideal modulated signal $m (t)$ is utilized as the excitation signal, with the ideal modulated signal being defined as a signal that can be compressed into a single value with a high-intensity peak via the matched filter, that is:

m (t) \otimes m (- t) = A \cdot δ (t),

(2)

where A is the amplitude gain, and $δ (t)$ is the Dirac function. The response signal of the ultrasonic measurement system $r (t)$ undergoes matched filtering, yielding the filtering signal $\overset{'}{r} (t)$ .

r (t) = f (t) \otimes m (t),

(3)

\overset{'}{r} (t) = r (t) \otimes m (- t) = f (t) \otimes m (t) \otimes m (- t) = A \cdot f (t) .

(4)

Assuming the presence of external interference factors such as micro-damage and noise, the new system response $n (t)$ is generated, and the matched filtering result is updated:

\begin{matrix} \overset{'}{r} (t) = r (t) \otimes m (- t) = [f (t) \otimes m (t) + n (t)] \otimes m (- t) \\ = A \cdot f (t) + n (t) \otimes m (- t) . \end{matrix}

(5)

The frequency band range of $n (t)$ typically differs from that of the modulated signal $m (t)$ , and the convolution of $n (t)$ and $m (- t)$ is equivalent to filtering $n (t)$ , where the used filter possesses a frequency band range consistent with $m (t)$ . Therefore, the introduction of the matched filter suppresses the frequency components outside the frequency band range of the modulated signal, rendering it unsuitable for nonlinear ultrasonic research that focuses on the analysis of harmonic components. To demonstrate that the matched filtering operation is inappropriate for nonlinear ultrasonic research, one numerical simulation is performed based on the propagation mechanism of the acoustic nonlinearity. The S0 mode Guided wave propagates 100 mm along the aluminum plate with a thickness of 2 mm (density: 2750 kg/m³, Young’s modulus: 69 GPa, and Poisson’s ratio: 0.331). Two types of excitation waves, the standard excitation $s (t)$ comprising a five-cycle Hanning-windowed sinusoidal tone bursts with a central frequency of 300 kHz, and the CGS-coded excitations (L = 8) $m (t)$ with the standard excitation mentioned above as the symbol waves, are introduced. Suppose there is one closed fatigue crack in the plate structure, and the second harmonic component can be generated. The response waves $b (t)$ and $r (t)$ under the two types of excitation waves are:

b (t) = s (t) \otimes f (t) + n_{s} (t),

(6)

r (t) = m (t) \otimes f (t) + n_{m} (t),

(7)

f (t) = F^{- 1} [e^{- ik (ω) l}],

(8)

n_{i} (t) = a \cdot ℱ^{- 1} {ℱ [x (t)] \cdot e^{- i k (ω) l}} \cdot e^{i ω_{f} t},

(9)

where $F$ and $F^{- 1}$ are the Fourier transform and its inverse, respectively. $k (ω)$ is the wavenumber. x(t) denotes the excitation waves referred to s(t) and m(t) in Equations (6) and (7), respectively. a is the second harmonic to fundamental component amplitude ratio. l is the wave propagation distance. $ω_{f}$ represents the center frequency.

The system responses r(t) are decoded with the matched filter, and the comparative results with the system response under the standard excitation b(t) are illustrated in Figure 1(g), and the signals are normalized for convenient comparison. The observed phase difference between the two waves is partially related to the inherent dispersion nature of guided waves. The alternations in frequency components of the CGS decoded signals are more noteworthy. As shown in Figure 1(h), the system response waves’ spectrum under standard excitation contains the prominent second harmonic component (600 kHz), which is a typical kind of nonlinear feature and is generated by the interaction between the incident waves and micro-damage. However, the system response waves’ spectrum under the CGS-coded excitations, following matched filtering, comprises solely the fundamental frequency component at 300 kHz, with a complete absence of harmonic components. This demonstrates that the frequency components outside the symbol waves s(t) bandwidth are suppressed completely, and provides a more compelling demonstration of the inapplicability of matched filtering operation in nonlinear ultrasonic research.

Sequence filtering for signal decoding without distortion

The decoding of system response waves containing harmonic components via matched filtering is demonstrated in the preceding section, in which the frequency components beyond the modulated signal bandwidth are completely suppressed, leading to a loss of valuable frequency information. It is worthwhile to investigate how pulse compression techniques can be employed to simultaneously enhance the acoustic field energy while preserving the decoded time-domain signal undistorted and retaining the second harmonic component of the decoded signal.

The phase-coded excitation signal is derived from the convolution of the symbol waves $s (t)$ and the varying binary sequence $p (t)$ , where the symbol waves determine the frequency component of the coded excitation, while the binary sequence, serving as the modulation component, is independent of the frequency component and is only responsible for influencing the transmitted energy and the main lobe and side lobe shapes. Therefore, the commonly used matched filtering operation can be divided into two steps: (1) the sequence filtering operation serves to realize temporal compression of long pulse echo waves; (2) the symbol filtering operation serves to suppress the frequency components outside the symbol waves. Given the frequency independence inherent in the sequence filtering operation, the introduction of the sequence filter paves the way for pulse compression. When the sequence filtering operation can achieve ideal side lobe cancellation, as described in Equation (2), the decoded signal using the sequence filtering signal $\overset{″}{r} (t)$ is:

\begin{matrix} \overset{″}{r} (t) = r (t) \otimes p (- t) \\ = f (t) \otimes [s (t) \otimes p (t)] \otimes p (- t) . \\ = A \cdot [f (t) \otimes s (t)] = A \cdot b (t) \end{matrix}

(10)

Equation (10) indicates that the sequence filtering results $\overset{″}{r} (t)$ of the system response r(t) under the coded excitation m(t) exhibit identical output characteristics to those under the standard excitation s(t), with the only difference being signal energy. This implies that the sequence filtering operation does not distort the signal spectral features. To validate it, the system response waves containing the harmonic component depicted in Figure 1(d) and (f) are subjected to sequence filtering, and the comparison results are presented in Figure 2. The output of the system response signal containing the second harmonic component, when processed with the sequence filter, aligns with the system response under standard excitation conditions, as illuminated in Figure 2(c). Meanwhile, spectrum analysis reveals that the sequence filtering operation preserves the second harmonic component, differing substantially from the out-of-band spectral suppression achieved by the matched filtering operation. This difference is of considerable importance for nonlinear ultrasonic research. The preceding analysis omitted the significant contribution of phased coded excitation to energy enhancement since the normalized signals are employed. Figure 2(e) and (f) demonstrate that the phased coded excitation technique substantially boosts signal energy relative to the standard excitation, and the SNR is improved. Consequently, when employing CGS-coded excitations, pulse compression techniques based on sequence filtering enable both time-domain compression and distortion-free decoding of the received signal, accompanied by enhanced output energy.

Material nonlinearity assessment verification

The acoustic nonlinearity parameter β increases with the wave propagation distance in nonlinear ultrasonic research. This classical phenomenon can be employed for material damage status assessment. FE simulation and experimental research on material nonlinearity assessment are carried out in this section. The proposed method is thereby verified as suitable for nonlienar ultrasonic research.

FE simulation

Nonlinear ultrasonic guided wave propagation of the mode pair S1–S2 in the 2 mm-thick aluminum alloy plate is simulated. A user-defined material subroutine VUMAT is developed and employed to incorporate the nonlinear constitutive relationship, accounting for both convective and inherent material nonlinearities, and the material parameters used are cited in the study by Zuo et al.³⁰ The two-dimensional plate structure is set as the plane strain model and is discretized by plane strain elements CPE4R. The maximum mesh size is 0.05 mm, and the time increment is 2e⁻⁹ s. To meet the requirement of the nonlinearity generation, including group velocity matching, phase velocity matching, as well as non-zero power flux from primary modes to the secondary modes, a 10-cycle Hanning windowed tone burst with a carrier frequency of 1.8 MHz and an amplitude of 1e⁻⁴ mm is selected as the excitation signal, which is subsequently called the standard excitation, and is parallelly loaded onto the left surface of the model. Analogous to the standard excitation setup, the CGS-coded excitations (L = 8) with the 1.8 MHz/10-cycle Hanning windowed weighted sine wave symbol are introduced as the excitation signals. Fifteen nodes, distributed across the upper surface of the model, are designated to record the in-plane displacement waves, and the specific coordinates of these nodes are displayed in Figure 3.

The normalized system response waves are displayed in Figure 4(a) and (b), corresponding to the standard excitation and the CGS-coded excitations (L = 8), respectively. The sequence filter and matched filter are introduced to decode the system response waves, respectively, and the decoded waveforms are illuminated in Figure 4(c). The waveform decoded using the sequence filter aligns with the system response waves obtained under the standard excitation, while the waveform decoded using the matched filter exhibits differences in both phase and amplitude. Furthermore, spectrum analysis, as depicted in Figure 4(d), reveals that the spectrum of the waveform decoded using the matched filter contains only the fundamental component, with complete suppression of the harmonic components. In contrast, the sequence filter-decoded waveform retains a prominent second harmonic component, facilitating the calculation of the acoustic nonlinearity parameter β for material nonlinearity assessment with short-time Fourier transform method. Moreover, as shown in Figure 4(e) and (f), the acoustic nonlinearity parameters, calculated from the system response under the standard excitation and from the sequence filter-decoded waveform, grow cumulatively with the propagation distance and exhibit a similar trend, which further indicates that the sequence filtering operation effectively preserve the features representing the material state, and is well-applicable in nonlinear ultrasonic research.

The influence of noise on nonlinear ultrasonic measurement results is also significant. In FE simulation, Gaussian white noise at SNR levels of 10, 20, and 30 dB is added to compare the outcomes of standard excitation and the CGS-coded excitations with the sequence filtering under varying noise levels.

The spectrograms of the response signals under different noise levels are shown in Figure 5. At the SNR of 30 dB, the spectrogram of the response signals from both excitation methods exhibits some oscillations, but the second harmonic components all remain discernible. The calculated nonlinear parameters are presented in Figure 6(a) and (b). As the propagation distance increases, the nonlinear parameter generally rises, though it shows fluctuations compared to the noise-free condition. At an SNR of 10 dB, the spectrogram from both excitation methods is completely overwhelmed by noise, making the second harmonic components undetectable. Under an SNR of 20 dB, the second-harmonic component in the response spectrum under standard excitation is largely submerged in noise, whereas under the CGS-coded excitations with the sequence filtering, the amplitude of the second harmonic components is visible. The corresponding nonlinear parameters, shown in Figure 6(c) and (d), reveal that the standard excitation yields result with pronounced oscillations heavily influenced by noise. In contrast, the nonlinear parameter from the CGS-coded excitations with the sequence filtering, while still affected by noise, maintains a clear increasing trend with propagation distance. These comparative results indicate that the proposed method in this work can enhance the nonlinear ultrasonic features under certain noise interference.

Experimental verification

On the basis of FE simulation research, experimental investigations are conducted to further validate the suitability of the framework combining CGS-coded excitations and sequence filtering in nonlinear ultrasonic research. The experimental testing system, including a gated radio frequency (RF) amplifier (RITEC® GA-2500A); Warwick, Rhode Island, USA), an arbitrary function generator with dual channels (Textronix® AFG-31000); Beaverton, Oregon, USA), and a mixed domain oscilloscope (Textronix® MDO34); Beaverton, Oregon, USA). The schematic of the experimental setup is illustrated in Figure 7.

One 7075-T6 aluminum alloy plate with 2 mm thickness is chosen as the sample. Considering the requirement of the nonlinearity generation and the specific characteristics of the sample, a 10-cycle Hanning windowed sine tone burst with a central frequency of 2 MHz is set as the excitation signal. A narrow-band transducer with a center frequency of 2.25 MHz is selected as the actuator, and one broadband transducer with a center frequency of 5 MHz serves as the sensor. The transducers, coupled with organic glass wedges at an angle of 23.5°, are used to generate the S1–S2 mode pair propagating within the plate. Based on the setting of standard excitation parameters, the CGS-coded excitation experiment is also carried out. The initial distance between the actuator and the sensor is set as 100 mm. The sensor is then moved in 10 mm increments, and response waves are captured under both standard and CGS-coded excitations at each interval, yielding a total of 10 sampling points.

The experimental results of material nonlinearity assessment, employing both the standard excitation and the CGS-coded excitations, are displayed in Figure 8. The comparison of the matched filtering and the sequence filtering operation in the decoding system response waves is shown in Figure 8(c). The decoded waves obtained using the sequence filter are virtually identical to the system response under the standard excitation. In contrast, the signal decoded with the matched filter exhibits significant differences in both phase and amplitude. The spectral comparison, as illuminated in Figure 8(d), highlights the distinctions between the two kinds of filtering operation.

The matched filtering operation inherently suppresses the second harmonic feature indicative of material nonlinearity. This limitation is not present in the sequence filtering operation, which effectively preserves these features, thereby rendering it a more appropriate method for the evaluation of material nonlinearity. The observed shift in harmonic frequency is directly related to the center frequency of the used transducers. A comparative analysis of the calculated acoustic nonlinearity parameters for system response waves obtained under the standard excitation and the sequence filtering all reveals a consistent cumulative increase with the propagation distance. This convergence in trends provides compelling evidence verifying the effectiveness of the sequence filter algorithm within the nonlinear ultrasonic research. It is worth noting that the system response waves in both the FE simulation and experimental research are normalized to facilitate comparison. Compared to the standard excitation, the phased coded excitation technique enhances signal amplitude, coupled with a sequence filtering operation, which reduces interference from electronic and scattered noise sources while preserving the second harmonic component indicative of material state, ultimately resulting in more accurate and reliable acoustic nonlinearity parameter calculations. This perspective will be subjected to further validation in subsequent investigations focusing on CAN enhancement and micro-damage imaging.

CAN enhancement and experimental validation

Imaging local micro-damage represented by the fatigue crack is a representative research content of nonlinear ultrasound. However, the weak CAN and susceptibility to external interference obscure the second harmonic features generated by the fatigue crack easily, leading to inaccurate nonlinear feature extraction and artifact generation. The enhancement of CAN is vital to improving the efficacy of nonlinear ultrasonic guided waves for fatigue crack imaging. Two kinds of experiments are conducted to demonstrate that the combined CGS-coded excitations and sequence filtering method enhances the CAN and suppresses artifacts.

Amplitude boosting of CAN

The comparative experiments are conducted to evaluate the influence of the standard excitation versus the combined CGS-coded excitations and sequence filtering method on the amplitude boosting of the second harmonic component. The 7075-T6 aluminum alloy plate specimen is subjected to the high-cycle fatigue test with sinusoidal tensile loading, and the applied tensile loading from 1.5 to 15 kN at a frequency of 15 Hz. After about 156,000 cycles, a barely visible fatigue crack originating from the notch tip, measuring about 6.9 mm in length, is produced. As illuminated in Figure 9, three sets of Lead Zirconate Titanate (PZT) wafers are bonded on the aluminum alloy plate specimen. Ultrasonic wave propagation is evaluated along three distinct paths. Waves from path 1 (PZT-1/2) are directly incident upon the fatigue crack, while waves from path 2 (PZT-3/4) and path 3 (PZT-5/6) propagate away from it. The distance between the PZT wafers and the fatigue crack is maintained at 100 mm, and the PZT wafer with a diameter of 10 mm and a thickness of 0.5 mm only serves as the actuator, and the PZT wafer with a diameter of 6 mm and a thickness of 0.5 mm only serves as the sensor. The experimental apparatus used in this study is the same as that shown in Figure 7. Two distinct excitation signal types are utilized: the standard excitation and the CGS-coded excitations (L = 8). The standard excitation signal is composed of a five-cycle Hanning windowed sine tone burst with a central frequency of 300 kHz, and the 300 kHz/five-cycle Hanning-windowed weighted sine wave symbol is employed as the CGS-coded excitations. The excitation signals are enhanced using a gated RF amplifier, and the system response waves are captured by the oscilloscope. The sampling rate is set as 25 MHz, and 128 averages are acquired.

By comprehensively analyzing and comparing the results of signals transmitted through the path 1, path 2, and path 3, as shown in Figures 10 to 12, the efficacy of the proposed method in nonlinear feature enhancement can be validated. The system response of path 1 under the standard excitation and the CGS-coded excitations, as illuminated in Figure 10(a) and (b). The encoded excitation response waves are decoded using the sequence filter, and the output comparison of the standard excitation and the sequence filtering is presented in Figure 10(c). The pulse inversion technique is implemented to suppress the fundamental component and augment the second harmonic component. The direct waves, which contain the information of the acoustic nonlinearity features, are intercepted using the rectangular window and analyzed. The spectral comparison of direct waves subjected to the standard excitation and the CGS-coded excitations with the sequence filtering reveals that the latter significantly boosts the amplitude of the second harmonic component.

The previous experiment confirms the efficacy of the combined CGS-coded excitations and sequence filtering method in boosting the amplitude of the second harmonic component. However, one key question remains: Considering that the second harmonic component originates from various sources, including material nonlinearity, CAN, and machine nonlinearity, is the proposed methodology suitable for enhancing the CAN attributed to the fatigue crack?

Initially, a comparative analysis of the system responses under different kinds of excitation signals is performed for both path 1 and path 2. The certain distance between path 2 and the fatigue crack, in conjunction with the presence of the rivet hole, is theoretically anticipated to attenuate the amplitude of the second harmonic component generated by the fatigue crack under the standard excitation. This prediction is corroborated by the spectral analysis presented in Figure 11, in which the amplitude of the second harmonic component observed in path 1 is marginally higher than that observed in path 2 under the standard excitation (Figure 11(a)), while the amplitude difference of the second harmonic component is significantly more pronounced under the combined CGS-coded excitations and sequence filtering method (Figure 11(b)). Following the preceding analysis, a comparative study of the system response in path 1 and path 3 is conducted with a reduced excitation amplitude. Given the significant distance separating path 3 from the fatigue crack, it is theoretically that the observed CAN effects are very weak. As displayed in Figure 12(a), notwithstanding the presence of the fatigue crack in path 1, the reduced amplitude excitation signals also result in a considerably attenuated CAN. The spectral analysis depicted in Figure 12(b) provides compelling evidence for the efficacy of the proposed method in enhancing the CAN associated with the fatigue crack. Under conditions of equivalent excitation amplitude, the application of the combined CGS-coded excitations and sequence filtering technique resulted in a significantly larger second harmonic component in the response waves of path 1 compared to that of path 3. Two comparative experiments are conducted to confirm the applicability of the proposed combined CGS-coded excitations and sequence filtering method to augment the CAN generated by the fatigue crack.

Artifact suppression in fatigue crack imaging

Within the field of nonlinear ultrasound research, fatigue crack imaging constitutes a significant area of investigation. However, the successful realization of this goal is hampered by the inherent weakness of nonlinear features, which present significant difficulties in the plate structures. A framework for imaging fatigue cracks, based on the reconstruction of the second harmonic component emanating from fatigue cracks, leveraging the sparse representation theory, is proposed in previous work.³¹ A key limitation is the presence of artifacts in the imaging results. These artifacts are associated with a degree of deviation in the reconstructed second harmonic component, a phenomenon attributable to the weak CAN. This work employs a nonlinear ultrasonic guided wave phased array technique combined CGS-coded excitations and the sequence filtering method to conduct fatigue crack imaging experiments, which verifies the effectiveness of the proposed method in both enhancing the CAN and mitigating the presence of imaging artifacts.

The specimen consists of a 7075-T6 aluminum alloy plate (300 × 200 × 2 mm³), with one barely invisible fatigue crack, measuring about 6.9 mm, and one 5 mm-diameter rivet hole (as illustrated in Figure 13(a)). A nonlinear ultrasonic guided wave phased array, composed of two types of PZT wafers, is employed for signal generation and reception. Six PZT wafers with a diameter of 10 mm and a thickness of 0.5 mm are configured as actuators, and five PZT wafers with a diameter of 6 mm and a thickness of 0.5 mm are configured as sensors. The nonlinear ultrasonic guided wave phased array is designed to ensure optimal performance across both the fundamental and second harmonic frequency components of the guided waves, and the selection of PZT wafers is detailed in previous work.³² Two distinct excitation waves, the standard excitation and the CGS-coded excitations (L = 8), are implemented in the experimental protocol, respectively. The standard excitation signal is comprised of a five-cycle, Hanning-windowed sine tone burst with a central frequency of 300 kHz. For CGS-coded excitations (L = 8), the 300 kHz/five-cycle Hanning-windowed weighted sine waves are used as the symbol. And the sampling rate is set as 25 MHz and 128 averages are acquired.

The system response waves under two kinds of excitation signals are recorded, and the pulse inversion technique is introduced to enhance the second harmonic component. The experimental system response waves elicited by standard excitation are shown in Figure 13(b), and the experimental output comparison of the standard excitation and the sequence filtering is shown in Figure 13(c). The sparse representation theory is employed to analyze the system response waves from two kinds of excitation signals and reconstruct the second harmonic waves generated by the fatigue crack. The reconstructed second harmonic waves are then processed using the total focusing method for fatigue crack imaging. The details of the specific methodology, including the establishment of the acoustic nonlinearity-aware dictionary and the calculation of sparse coefficients, are detailed in previous work.³¹

The outcomes of the fatigue crack imaging experiments, conducted using the two excitation signals, are depicted in Figure 14(a) and (c), and the nonlinear imaging details are presented in Figure 14(b) and (d). The appearance of bright spots in the nonlinear imaging results, corresponding to the fatigue crack tip, is evident under two types of excitation wave protocols. However, the nonlinear imaging results under the standard excitation waves exhibited a degree of artifacts. This suggests the presence of deviations within the reconstructed second harmonic component, a phenomenon attributable to the weak features inherent within the harmonic components themselves. In contrast to the nonlinear imaging results obtained under the standard excitation, the application of the combined CGS-coded excitations and sequence filtering method proposed in this work demonstrates the enhancement of the second harmonic component and further for marked suppression in nonlinear imaging artifacts, which provides compelling evidence of the applicability and efficacy of the proposed method within the nonlinear ultrasound research.

Conclusions

A framework integrating the coded excitation and sequence filtering techniques is proposed in this work for enhancing nonlinear ultrasonic features, including the material nonlinearity and the CAN. The proposed framework can effectively address the critical challenge of weak nonlinear feature extraction in micro-damage assessment and imaging by employing CGS-coded excitations to boost acoustic field energy without increasing peak excitation power, and introducing the sequence filter that preserves the second harmonic component while achieving effective pulse compression.

This work investigates the performance of standard excitation and CGS-coded excitation in evaluating material nonlinearity through comprehensive simulation and experiment. The results demonstrate that the sequence filtering method can effectively preserve the second harmonic component while revealing a progressive increase in nonlinear parameters with propagation distance. In contrast, the conventional matched filtering method significantly suppresses the nonlinear features. Furthermore, taking fatigue cracks as the research subject, this work validates that the proposed CGS-coded excitation combined with a sequence filter effectively enhances the CAN associated with fatigue cracks while simultaneously mitigating the presence of imaging artifacts.

Supplemental Material

sj-docx-1-shm-10.1177_14759217251410970 – Supplemental material for Boosting nonlinear ultrasonic detection performance: a framework combining coded excitation and sequence filtering for weak nonlinear feature enhancement

Supplemental material, sj-docx-1-shm-10.1177_14759217251410970 for Boosting nonlinear ultrasonic detection performance: a framework combining coded excitation and sequence filtering for weak nonlinear feature enhancement by Haiming Xu, Lishuai Liu, Hetang Wang, Siyuan Peng and Yanxun Xiang in Structural Health Monitoring

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant numbers 12327807, 12422415, 12374434, 52321002, and 52322404) and the Shanghai Science and Technology Innovation Action Plan (numbers 24TS1412200 and 24ZR1490900).

ORCID iDs

Haiming Xu

Lishuai Liu

Yanxun Xiang

Data availability

Data will be made available on request.

Supplemental material

Supplemental material for this article is available online.

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