Crack Growth Signal Processing Approach Combining Wavelet Threshold Denoising and Variable Amplitude DCPD Technique

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Introduction
e direct current potential drop (DCPD) technique is commonly used to monitor the crack growth rate of metal specimens in laboratory environments [1][2][3][4].It is based on the principle that crack growth in a metal specimen increases its equivalent resistance [5,6].e method involves passing a constant DC to the tested specimens and then observing the potential drop between two points across the crack surface to monitor the crack growth [7].e relationship between crack growth and DCPD can be obtained from the electric field analysis of a specimen using the finite element method [8][9][10].
e DCPD signal is significantly small, approximately tens of microvolts to hundreds of microvolts [11].Because the connections between the measuring leads, specimens, and measuring instruments are often connections of dissimilar materials and because of the existence of temperature difference, the thermal electromotive force (TEMF) is superimposed on the signal.However, owing to the environmental electromagnetic interference and the influence of the instrument itself, the DCPD signal is also accompanied by complex random background noise interference signals.e amplitude of these interference signals accounts for a large part of the effective DCPD signal amplitude; thus, it is difficult to distinguish weak DCPD signals, which significantly affects the measurement accuracy of the signal and the resolution of the crack growth measurement [12][13][14].To suppress these interference signals, the reversing DCPD method is used to eliminate the TEMF [15,16]. is method requires the use of relay groups in the test circuit, thereby increasing the complexity of the circuit.To suppress the signal drift caused by the temperature drift of the instruments, the reference specimen method or the reference potential method is employed to provide an additional reference DCPD signal to achieve the correction of the main DCPD signal [17].To the best of our knowledge, no signal processing method has been reported for random noise interference signals on DCPD signals.If these random noises are not managed properly, they will inevitably affect the measurement accuracy and resolution.Wavelet analysis is a good time-frequency analysis method, which has the characteristics of multiresolution analysis and has good analysis characteristics in both the time and frequency domains [18].erefore, wavelet analysis is suitable for the analysis and processing of weak DCPD signals and other nonstationary signals [19].e wavelet threshold denoising algorithm is a signal denoising method based on wavelet analysis developed in recent years.It uses wavelet transform to decompose the signal into wavelet coefficients in different frequency bands that can represent the overview and details of the signal, and then processes these coefficients by thresholding methods to achieve denoising [20].In addition, this algorithm can perform multiresolution analysis of signals and has been applied to many fields, such as geophysics, biomedicine, and vibration analysis in recent years.Compared to other signal processing methods, it has the advantages of multiresolution features and localized features in the time and frequency domains [21].e wavelet threshold denoising algorithm can also be combined with other signal processing algorithms to achieve a better denoising effect [22,23].
A signal composition model was proposed in this study to improve the measurement accuracy of the DCPD signal.According to this model, a crack propagation signal processing approach using a wavelet threshold denoising algorithm and variable current amplitude DCPD signal synthesis technique is proposed.
e 316 L stainless steel compact tension (C(T)) specimens were selected to conduct DCPD signal monitoring experiments under static and dynamic crack growth conditions.e validity and practicability of the signal model and signal processing approaches were verified based on experimental data.e proposed approach helps to further improve the measurement accuracy of the crack growth monitoring DCPD signal and the measurement resolution of crack growth.

Experiment
Two C(T) specimens were used for the fatigue tensile and DCPD monitoring experiments in an ambient laboratory air environment.e ambient temperature range was 24-30 °C.
e geometry and dimensions of the C(T) specimens are shown in Figure 1. e material used as the specimens was 316 L stainless steel.e initial crack lengths a 0 of specimens #1 and #2 were 10 mm and 12 mm, respectively.For specimen #2, it was cut with a width of 0.2 mm and a length of 2 mm slit at the notch root along the ligament plane by wire cutting to simulate cracks.
In the DCPD crack growth monitoring test, the specimen was injected with a constant current, and the DCPD signals between specific positions were monitored.e measured DCPD signals include the main DCPD signal V and the reference DCPD signal V r .e current lead positions (I+, I−) and the DCPD signal (V+, V−, V r +, V r −) lead positions of the specimens are shown in Figure 2.
e crack growth monitoring experimental system based on the DCPD technique is shown in Figure 3. e experimental equipment included a tensile testing machine and a set of DCPD crack growth monitoring system.e tensile testing machine can apply static or dynamic tensile loads to the specimen and can conduct fatigue precracking and crack growth experiments on specimens.Fatigue tensile tests with different frequencies and loads were performed using a tensile testing machine.During the test, the DCPD of the specimen was monitored using a DCPD crack growth monitoring system.e composition and connections of the DCPD crack growth monitoring system are shown in Figure 4.It consists of a computer, high-performance power supply, nanovoltmeter, data-acquisition switch instrument, and an ambient temperature measurement module.e ambient temperature measurement module was connected to a computer through an RS232 serial interface.Other instruments were interconnected through a general-purpose interface bus (GPIB) that was connected to a computer through a physical cell identifier-general-purpose interface bus (PCI-GPIB) interface card.
ese instruments work together under the control of software running on a computer.During the tensile test of the specimens, a power supply was used to provide a constant current for the specimens.It has low current ripple and noise as well as a high output current programming resolution.e output current range is 0-5 A, which can be dynamically set by the control software.A nanovoltmeter was used to measure the signal of a small potential drop.
e main and reference DCPD signals of the specimen were connected to the input channel of the data-acquisition switch instrument, and the output channel was connected to the measurement channel of the nanovoltmeter.During the test, the main control software controls the data-acquisition switch to switch the measured signals.Another software running on the computer records the ambient temperature during the test through the ambient temperature measurement module.
e recorded ambient temperature data were used to analyze the temperaturerelated interference signals.
e wiring and installation of the C(T) specimen on the testing machine are shown in Figure 5.To insulate the specimen from the tensile testing machine, zirconia ceramic tubes were used to separate the specimen loading hole and 2 Mathematical Problems in Engineering the pin, and zirconia ceramic sheets were used to separate the specimen side from the fixture.Both the current and DCPD signal leads used were platinum wires with diameters of 0.75 mm and 0.5 mm, respectively.ey were connected to the specimen via spot welding.e leads were covered with a polytetrafluoroethylene (PTFE) tube to ensure insulation.In addition, they were connected to the test instrument through twisted wire pairs after being drawn from the specimen.(1) TEMF interference signal: TEMF is generated when two junctions of dissimilar metals are maintained at different temperatures.In particular, the wire materials used in the DCPD measurement are not of the same material as the specimen or connectors on the measuring instruments.Any difference in temperature between the junctions generates the TEMF [24]. is type of interference signal has an additive relationship with real signals.Because it is related to temperature, this interference signal is not static and will change with the change in the complex temperature field distribution in the environment.

DCPD Signal Model and Anti-Interference Signal Processing Approach
During the test, the temperature fluctuation of the specimen and the fluctuation of the ambient temperature caused the fluctuation of the TEMF.(2) Multiplicative interference signals: is type of signal originates from the temperature drift of the semiconductor device of the measuring instruments, which is difficult to eliminate.is is caused by the change in the characteristics of the measurement channel with temperature.It is characterized by a temperature coefficient and has a multiplicative relationship with the measured signal.According to the experimental data of different current amplitudes under static crack conditions, it can be assumed that the amplitude is approximately linear with the real signal amplitude.(3) Random noise signal: e sources of random noise signals are complex, including interference from the mains and loading system, as well as the conversion noise of the A/D device of the instrument.For this type of noise signal with complex sources, it appears as flat broadband characteristics; thus, it can be approximately regarded as additive white Gaussian noise [25][26][27].

Mathematical Problems in Engineering 3
According to the above characteristics of the DCPD signal, the DCPD signal model can be expressed as follows: where t is time, a is the crack length, V is the measured DCPD signal, V * is the true value of the DCPD signal, which is a function of the crack length a, C is the TEMF interference signal, D is the multiplicative interference signal, and n is the random noise signal.

Adaptive Wavelet reshold Denoising of DCPD Signals.
e purpose of the adaptive wavelet threshold denoising of the DCPD signal is to reduce the random noise signal n(t), which has a wider frequency band compared with other wavelet transformation was performed on the signal, in the wavelet domain, the coefficient corresponding to the effective signal became large, whereas the coefficient corresponding to the noise became significantly small.e statistical properties of the coefficients corresponding to the noise in the wavelet domain were consistent with the original noise.e coefficient amplitude of the useful signal was greater than the coefficient amplitude of the noise.erefore, the wavelet coefficients at each scale can be obtained through wavelet decomposition.ereafter, the signal and noise can be distinguished by setting a reasonable threshold to achieve signal denoising [28].Although wavelet threshold denoising can be highly regarded as a low-pass filter, it is superior to the traditional low-pass filter.Wavelet threshold denoising can better retain the characteristics of the real signal itself while denoising, such as some local relatively fast change characteristics of the real signal itself.
erefore, the use of wavelet threshold denoising allows better retention of the characteristics of the real signal itself and other types of interference signals while denoising.us, the targeted processing of other types of interference signals cannot be affected.e wavelet threshold denoising process is conducted according to the following three steps [29]: (1) Perform a wavelet transform on a noisy signal.
Choose a wavelet basis and determine the level N of the wavelet decomposition, and then perform Nlevel wavelet decomposition on the signal.(2) Obtain threshold wavelet coefficients.To maintain the overall shape of the signal unchanged and retain the effective signal, the coefficients of each layer after decomposition are quantized by a hard threshold, soft threshold, or other threshold methods.(3) Perform an inverse wavelet transform to reconstruct the signal.
In this study, the db4 wavelet was selected as the wavelet base, and the signal to be processed was decomposed into four layers to obtain the wavelet coefficients.ereafter, the threshold of wavelet coefficients was obtained, and the signal was reconstructed.e selection of the threshold directly affects the denoising effect.Stein's unbiased risk estimate (SURE) was selected.It is an adaptive threshold selection method with minimum risk as an indicator.
e specific algorithm is as follows: (1) After wavelet decomposition, the absolute value of the wavelet coefficients of each layer ω k,j is sorted from small to large, and then the absolute value of the wavelet coefficients after sorting is squared to obtain the k-th layer vector: where N is the number of wavelet coefficients in each layer.(2) Calculate the threshold vector λ k of the k-th layer as (3) e risk estimation coefficient vector R k generated when λ k,j is selected on the kth layer is (4) By comparing the vector of the risk estimation coefficients, the unbiased risk estimation threshold of the kth layer decomposition is where j min is the j corresponding to the minimum risk assessment coefficient.e threshold function is a rule for modifying wavelet coefficients, and different threshold functions reflect different strategies for processing the wavelet coefficients.e threshold processing function of wavelet threshold denoising is divided into a soft threshold function and a hard threshold function.In this study, the soft threshold function was used to reduce the noise of the DCPD signal.e soft threshold function is

Mathematical Problems in Engineering
where ω k,j are the wavelet coefficients obtained after threshold processing, sign is the sign function, and λ k is the threshold of the kth layer.

Variable Current Amplitude DCPD Signal Synthesis
Technique.e reversing DCPD method was used for the TEMF interference signal.However, owing to the relay group introduced in the circuit, the complexity of the circuit increased, and additional interference signals were introduced.After the relay was switched, it was necessary to wait for the signal to stabilize before measuring; thus, the measurement speed was also affected.A new method of changing the current amplitude was adopted to eliminate the TEMF interference signal.In this method, two sets of DCPD signals were obtained by controlling the program-controlled constant current source to continuously output two different amplitude currents and then combining them to eliminate the TEMF.Consider the case in which the signal contains only multiplicative interference and TEMF.When the input currents are I1 and I2, the measured signal can be expressed as where V I1 and V I2 are the measured values of the signal when the input currents are I1 and I2, respectively; V * I1 and V * I2 are the true values of the signal; C I1 and C I2 are the TEMF; D I1 and D I2 are the multiplicative interference signals.Owing to the short interval between the two currents, erefore, the TEMF interference signal is eliminated after the difference between V I1 and V I2 as follows: Because the DCPD signal is proportional to the magnitude of the current, the multiplicative interference signal is proportional to the magnitude of the DCPD signal: Express the ratio of the two currents as e signal after removing TEMF is erefore, according to the signal measured under two consecutive different current signals and the ratio k c of the two currents, the DCPD signal without the TEMF signal can be obtained, and the multiplicative interference signal is retained.

Elimination Method of Multiplicative Interference Signal.
As shown in Figure 2, to eliminate the multiplicative interference signal, the main DCPD signal V was measured, and the DCPD between two points on the same side of the specimen crack surface was also measured as a reference DCPD signal V r .e dimensionless parameter V norm obtained by the normalization of the two signals was used as the final parameter for calculating the crack length.e expression for V norm is where V 0 and V r0 are the initial main DCPD and the initial reference DCPD, respectively, which are determined by the initial crack length and the lead positions, and are measured before crack propagation; V * and V * r are the true values; and A is a constant determined by V 0 and V r0 .e calibration function can be obtained using a three-dimensional electrical finite element analysis.
Considering that there is only a multiplicative interference signal in the signal, the main DCPD signal and the reference DCPD signal can be expressed as where D and D r are the multiplicative interference signals of the two signals, respectively.Because the multiplicative interference signal is proportional to the real signal, the multiplicative interference signal can be expressed as where α is a coefficient related to the parameters of the measuring instrument and the ambient temperature.us, It can be observed that using V norm as the parameter for calculating the crack length can eliminate the multiplicative interference signal.

Signals of Static Crack.
e proposed signal processing approach was used to process the main DCPD signal and the reference DCPD signal when the specimen was unloaded (i.e.,  12).Moreover, it can be observed that the time-varying TEMF signal is eliminated, and the signal becomes more stable.
To further observe the effect of the signal processing approach, the signal-to-noise ratio (SNR) and root mean square error (RMSE) were used to quantitatively evaluate the performance of the algorithm.e calculation formulas are as follows: where N is the number of signal sampling points, s is the signal to be evaluated, s d is the signal processed by the proposed signal processing approach, and mean is the mean value function.Considering that the signal is obtained under static crack conditions, the mean value of s d is used as an estimate of the true signal.e larger the SNR and the smaller the RMSE, the more effective is the signal processing algorithm.Tables 1 and 2 list the RMSE and SNR of the main DCPD signal and the reference DCPD signal before and after the signal processing approach, respectively.It can be observed that the RMSE is reduced by approximately 10 times, and the SNR is increased by approximately 30 dB.
To estimate the crack length RMSE corresponding to the RMSE of the DCPD signal (RMSE_a), using the commercial FE software ABAQUS v6.12, the finite element model of the specimens used in the experiment was established.Finite Potential drop (µV) where Δa is the increment of the crack length relative to the initial crack length and ΔV is the increment of the DCPD.e RMSE values in Table 1 were substituted into equation (18), and the calculated Δa was used as the estimated value of RMSE_a, which is also listed in Table 1.It can be observed that the signal processing approach can increase the resolution of the crack growth measurement approximately by an order of magnitude.

Signals after the Specimen Is Loaded.
Owing to the effect of the resistance strain, the DCPD signal changes with the size of the crack and slightly changes with the deformation of the specimen under load.To evaluate the effect of the signal processing approach on changing signals, the testing machine was set to load control mode and a slow sine wave load was applied to the specimen to generate a slowly and periodically changing DCPD signal.ree sine wave loads with peaks of 2000 N, 5000 N, and 8000 N were used, and the loading frequency was 0.01 Hz.After each load slowly increased to its peak value, it changed according to the sinusoidal rule based on the set frequency.e loading time for each load was approximately five loading cycles.After each loading program segment, the next loading program segment did not start immediately.
e force was maintained at the minimum value for the same time to generate a no-load time period, which separated the loading program segments.
When the specimen was subjected to a sinusoidal load, the elastoplastic strain in the crack tip area of the specimen caused a small change, which was approximately linear with the load, in the DCPD signal.is caused the DCPD signal to produce regular sine wave fluctuations.Figure 9 shows the DCPD signals during loading (specimen #2, current I � 4.0 A, sampling frequency f s � 1.295 Hz).It can be observed that before the signal is processed, the sine wave law cannot be distinguished from the measured signal owing to interference signals, such as noise and TEMF.After signal processing, the signal shows an obvious sinusoidal law, which shows that the signal processing approach still has a good anti-interference effect on dynamic weakly changing signals.

Signals in the Process of Crack Growth.
To observe the effect of the proposed signal processing approach during the crack growth process, the specimen was loaded according to the sine wave law at a frequency of 0.1 Hz to ensure slow growth of the crack.Figure 10 shows a section of the crack propagation signals when the peak force is 8000 N and the valley force is 6400 N (specimen #2, current I � 4.0 A, sampling frequency f s � 3.547 Hz).Comparing the DCPD signal measured during the crack growth process with the processed signal, it can be observed that the processed signal reflects both the process of slow crack growth and the small signal fluctuations caused by load changes.
Longer crack propagation signals at a loading frequency of 0.1 Hz with a peak force of 8000 N and valley force of 3000 N are shown in Figure 11 (specimen #2, current I � 4.0 A, sampling frequency f s � 0.385 Hz). e relative change in signal V I was compared with the dimensionless parameter V norm obtained according to equation (16).It can be observed from Figure 11 that a noise signal is superimposed on the signal V I1 /V I1_0 , and there is also a slow signal drift, which is the aforementioned multiplicative interference signal.
e noise on the signal V norm is significantly reduced, and the multiplicative interference signal is eliminated.e final processing effect also indirectly proves that the DCPD signal model proposed in this study is reasonable.Mathematical Problems in Engineering (1) e interference signal superimposed on the weak DCPD signal is composed of an additive TEMF, multiplicative interference signal, and random noise.e signal processing approach designed according to the three interference signals in this model can significantly reduce the amplitude of the interference signal, which shows that the signal composition model is reasonable.
(2) e proposed wavelet threshold denoising method based on Stein's unbiased risk estimate and adopting a soft threshold strategy can significantly reduce random interference signals and improve the SNR.
Combined with the proposed variable current signal synthesis technique, the time-varying TEMF interference signal superimposed on the signal is significantly reduced.e multiplicative interference signal can be effectively eliminated using the reference potential method.erefore, the proposed signal processing approach can effectively improve the accuracy and resolution of the crack growth measurements.(3) e crack growth monitoring experiment based on the DCPD technique has a wide range of applications (including fatigue crack growth tests, stress corrosion crack growth tests, and creep crack growth tests), and the test environment is complex.e proposed signal processing approach applies wavelet threshold denoising to weak DCPD signals and proposes new ideas for signal processing based on the DCPD crack growth monitoring technique.After adjusting the parameters of this method according to the specific experimental environment, the proposed method is expected to be applied to various crack growth monitoring tests to improve the crack growth accuracy.

6 Figure 1 :
Figure 1: Geometry and dimensions of the C(T) specimens.

3. 1 .
DCPD Signal Model.Owing to the good conductivity of the metal specimens and the limitation of the input current amplitude, the DCPD signal is only at the microvolt level.Several factors influence the DCPD signal, including environmental electromagnetic interference, instrumentation drift, TEMF, noise from the mains, crack surface contact, and specimen deformation under load[1,7,13].e influence of these factors on the DCPD signal is that different types of interference signals are superimposed on the measured signal.Figure6shows an example of the measured DCPD signals (specimen #1, current I � 0.28 A, sampling frequency f s � 0.385 Hz) when the specimens were not loaded (i.e., static crack).ese interference signals should be removed before they are converted into crack growth signals.According to our experimental data of the C(T) specimens and the reports in the literature[1,7,13], the DCPD signal contains the following types of interference signals:

Figure 5 :Figure 4 :
Figure 5: Installation and wiring of the C(T) specimen.

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Mathematical Problems in Engineering static crack), and four sets of signals were randomly selected to illustrate the effect of the signal processing approach (specimen #1, current I � 0.1 A, sampling frequency f s � 0.385 Hz). e signals shown in Figure 7 are the main DCPD signals, and the signals shown in Figure 8 are the reference DCPD signals.In Figures 7 and 8, V I1-d and V I2-d are the signals of V I1 and V I2 , respectively, after wavelet threshold denoising.To facilitate comparison, V I2 and V I2-d were divided by k c .It can be observed that after wavelet threshold denoising, the noise on signals V I1 and V I2 is significantly reduced while retaining the change characteristics of the lower frequency signals.e signal V SYN is the signal obtained by the signal synthesis algorithm in equation (

Figure 7 :
Figure 7: Signal processing results of the main DCPD signal under static crack conditions (specimen #1, current I � 0.1 A sampling frequency f s � 0.385 Hz).

Table 1 :
RMSE, SNR, and RMSE_a of the main DCPD signal before and after signal processing.

Table 2 :
RMSE and SNR of the reference DCPD signal before and after signal processing.According to the proposed DCPD signal composition model, the principle and feasibility of the proposed method were discussed theoretically.Using the static crack test and crack growth test data of 316 L stainless steel C(T) specimens, the anti-interference effect of the signal processing approach was analyzed.e following conclusions can be drawn: In this study, a signal processing approach for weak DCPD signals that combines wavelet threshold denoising and a variable current signal synthesis technique was proposed.