Structural health monitoring using ultrasonic guided-waves and the degree of health index

sensors Article Structural Health Monitoring Using Ultrasonic Guided-Waves and the Degree of Health Index Sergio Cantero-Chinchilla 1, Gerardo Aranguren 2,* , José Manuel Royo 3, Manuel Chiachío 4,5 , Josu Etxaniz 2and Andrea Calvo-Echenique 3   Citation: Cantero-Chinchilla, S.; Aranguren, G.; Royo, J.M.; Chiachío, M.; Etxaniz, J.; Calvo-Echenique, A. Structural Health Monitoring Using Ultrasonic Guided-Waves and the Degree of Health Index. Sensors 2021, 21, 993. https://doi.org/10.3390/ s21030993 Academic Editor: Theodore E. Matikas Received: 1 January 2021 Accepted: 27 January 2021 Published: 2 February 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Department of Mechanical Engineering, University of Bristol, Bristol BS8 1TR, UK; sergio.canter[email protected] 2Electronic Design Group, University of the Basque Country (UPV/EHU), 48013 Bilbao, Spain; [email protected] 3Techonological Institute of Aragón (ITAINNOVA), María de Luna 8, 50018 Zaragoza, Spain; [email protected] (J.M.R.); [email protected] (A.C.-E.) 4Department of Structural Mechanics & Hydraulics Engineering, University of Granada, 18001 Granada, Spain; [email protected] 5Andalusian Research Institute of Data Science & Computational Intelligence, University of Granada, 18001 Granada, Spain *Correspondence: gerardo.arangur[email protected] Abstract: This paper proposes a new damage index named degree of health (DoH) to efficiently tackle structural damage monitoring in real-time. As a key contribution, the proposed index relies on a pattern matching methodology that measures the time-of-flight mismatch of sequential ultrasonic guided-wave measurements using fuzzy logic fundamentals. The ultrasonic signals are generated using the transmission beamforming technique with a phased-array of piezoelectric transducers. The acquisition is carried out by two phased-arrays to compare the influence of pulse-echo and pitch-catch modes in the damage assessment. The proposed monitoring approach is illustrated in a fatigue test of an aluminum sheet with an initial notch. As an additional novelty, the proposed pattern matching methodology uses the data stemming from the transmission beamforming technique for structural health monitoring. The results demonstrate the efficiency and robustness of the proposed framework in providing a qualitative and quantitative assessment for fatigue crack damage. Keywords: structural health monitoring; ultrasonic guided-waves; fatigue damage detection; transmission beamforming; degree of health index 1. Introduction Crack initiation and propagation are the main driving forces of damage in many engineering materials [ 1 ]. In the absence of corrective actions, for example, decreasing the load levels or performing corrective maintenance actions, a crack may propagate up to the catastrophic failure of a structural component. A wide range of non-destructive testing (NDT) techniques have been proposed to support operation and maintenance decision making [ 2 – 4 ]. Visual inspection, radiography, or ultrasound [ 5 , 6 ] are some examples of these NDT techniques; these are regarded as highly reliable but also time consuming (requiring the interruption of service) and dependent on the structure [ 7 ]. Alternatively, structural health monitoring (SHM) techniques enable a continuous on-board monitoring of the structural health facilitating the conditionbased maintenance of the structure [ 8 ]. Some SHM techniques are especially suited for structures made of conductive materials such as metals (e.g., eddy currents [ 9 ]). Others, however, allow a more generic application to structures made of any kind of isotropic or anisotropic materials. For instance, strain gauges and fiber Bragg gratings [ 10 ] are typically used to monitor strain deformation in structures. However, their area of coverage is relatively low, thus potentially missing small defects unless a dense network of sensors Sensors 2021,21, 993. https://doi.org/10.3390/s21030993 https://www.mdpi.com/journal/sensors Sensors 2021,21, 993 2 of 17 is used. The acoustic emission technique has shown efficiency in fatigue damage detection and localization [ 11 ], although it is known as a passive method that requires capturing the instant of the damage growth in order to measure damage-related information. Alternatively, ultrasonic guided-waves [ 12 , 13 ] overcome most of the referred drawbacks and can be efficiently used as a non-intrusive and on-board SHM technique [ 14 ], due to their ability to actively explore large areas with a relatively small attenuation [ 15 ]. These beneficial properties of ultrasonic guided-waves have attracted the attention of safety-critical industries such the aerospace over the last few decades [ 16 , 17 ]. The aforementioned SHM technique uses piezoelectric wafer active sensors (PWAS) [ 18 ] permanently attached to the structure to generate and acquire ultrasonic guided-waves [ 19 , 20 ], which have to be further processed to infer damage-related information. Several signal processing and damage inference strategies can be applied to extract damage information out of the ultrasonic data. A number of them are based on some forms of inverse problems that infer relevant damage parameters from the comparison between experimental and computationally simulated ultrasonic data [ 21 ]. For example, the localization of damage using ultrasonic guided-waves has been addressed by several authors using uncertainty quantification frameworks and time-of-flight (ToF) models [ 22 , 23 ]. The quantification of damage has also been addressed using ultrasonic measurements in the context of a probabilistic Bayesian inverse problem in [ 24 , 25 ]. An alternative approach for ultrasound-based damage inference is the time-reversal method [ 26 ], which makes it possible to focus ultrasonic guided-waves towards a defect in the structure without the need for a baseline. This technique overcomes the limitations stemming from both the wave dispersion and the unknown material deformations by focusing the signal [ 26 ]. Most of these methods are baseline free, hence needing no comparison between undamaged and non-pristine states. However, the latter is at the cost of employing a significant amount of physically-grounded model evaluations, which bounds its applicability in real-life realtime engineering scenarios not only for its computational cost but also for its model and implementation complexity. Alternatively, the use of baseline-based methods for damage detection might render the required efficiency for complex structures. These are typically model-free, and a straightforward comparison between signal features obtained in the pristine state and the subsequent damage states is the only information required. In this sense, a number of methods have been proposed in the literature. One such example is the reconstruction algorithm for probabilistic inspection of defects (RAPID) [ 27 ] and further variations of the same [28]. This method is able to localize defects by using signal difference coefficients of sensor pairs. Also, in [ 27 ], the authors proposed the use of signal correlation coefficients to detect and monitor damage evolution in an aerospace panel. Other examples for the reconstruction of damage using a baseline from an undamaged state are the embeddedultrasonics structural radar [ 29 ] and the delay-and-sum imaging algorithm [ 30 ]. These techniques use the difference between signals (acquired using a receiver beamforming mode) in healthy and damaged conditions to obtain only the signatures stemming from the defect [ 31 – 33 ]. More recently, an unsupervised feature-extraction method for online damage detection was proposed in [ 34 ], whereby a subset of the ultrasonic signals carrying the majority of the energy content was used. The baseline approach was also used in [35], where the effectiveness of the shear-horizontal guided-wave mode in monitoring damage was investigated. Besides, the change in the ToF of the signal peaks due to the presence of debonding in a composite plate was demonstrated in [ 36 ]. This group of approaches enables the inference of a lower degree of damage-related information compared to the aforementioned inverse problems, and they approach real-world engineering scenarios with a higher efficiency and model simplicity. Nevertheless, they still require a significant computational effort, which poses an important limitation for on-board SHM applications. Therefore, there is still a need to provide computationally efficient post-processing methods of ultrasonic guided-waves for real-time damage assessment. Sensors 2021,21, 993 3 of 17 To overcome this limitation, this paper proposes the use of a novel pattern matching post-processing method for the detection and monitoring of fatigue damage in isotropic materials based on signal features stemming from each characteristic point (CP) of the signal. These points correspond to the peak amplitudes (maximum and minimum) of the acquired signal. The ToF of the peaks are chosen as the base of comparison to detect and monitor damage in a structure [ 37 ]. More specifically, a set of these points acquired in the undamaged state are further used to build a trapezoidal function similarly to a membership function or fuzzy set [ 38 , 39 ]. Additional measurements in non-pristine states are evaluated in the proposed function, whereby a degree of health (DoH) of the structure is provided as output. The feasibility and efficiency of the proposed algorithm (based on [ 40 ]) are demonstrated in a real fatigue test of an aluminum plate. The ultrasonic transmission beamforming technique with a linear phased-array of six PWAS is adopted to monitor (i.e., excite and receive ultrasonic guided-waves) the structure with an enhanced power every 1000 loading cycles at the same load level. Another symmetric phased-array (which only receives ultrasonic signals) is used to support the results obtained from the former one. The experimental results show that the proposed methodology is able to detect fatigue damage at an early stage (i.e., the crack onset) and to monitor the crack growth in an effective and efficient manner. Furthermore, the ease of implementation of this methodology makes it potentially applicable to industrial environments in order to perform real-time SHM. The remainder of the paper is organized as follows—Section 2describes the proposed post-processing method based on the comparison of the ToF of the signal peaks; Section 3 shows experimental setup as well as the results obtained from this method in the fatigue experiment; Section 4discusses the potential impact of the proposed methodology on ultrasonic guided-wave based SHM; finally, Section 5provides concluding remarks and suggests future works. 2. Methodology The proposed detection and monitoring methodology based on a novel damage index is presented in this section. The index, referred to here as DoH, relies on ultrasonic guided-wave data taken from the SHM of structural panels. More specifically, the DoH uses baseline ultrasonic data acquired when the structure is in pristine state as a basis of comparison for further measurements when the structure is potentially damaged. To enhance the computational efficiency of this technique, only the ToF of the signal peaks above a user-specified amplitude threshold (named as At ) are used as representative information of the raw ultrasonic data, as shown in Figure 1a. The peaks are obtained applying a sliding window to the ultrasonic signal. Note that the width of this window is chosen to be proportional to the period of the signal, that is, the inverse of the central frequency. The maximum and minimum points within these moving windows are selected, hence obtaining the signal peaks. Furthermore, to partially address the irreducible uncertainty of these measurements and add robustness to the damage index, an arbitrary amount of repeated signals are acquired leading to a set of peaks with small differences in both time and amplitude (see Figure 1b). These variations are assumed to be an indicator of the measurement (or aleatory) uncertainty [ 41 ], which supports the previous robustness claim of the proposed damage index. Similarly to a fuzzy set, a trapezoidal function is created around the set of ToF points acquired in pristine state (see Figure 1c). This function makes it possible to assess the structural health by analyzing the ToF mismatch between the CPs (see Figure 1b for reference) acquired in both the degraded state and the pristine state. Note that as this function gives degree of membership values within the interval [ 0, 1 ] , the proposed damage index will carry qualitative information about the structural degradation. Sensors 2021,21, 993 4 of 17 2At Time Amplitude (a) Selection of maximum and minimum peaks Time Amplitude ( b ) Dispersion of consecutive measurements Aj iBj iCj iDj i µ=1 µ=0 bj icj i `j i={Cj i,Dj i} Lj i={Bj i,Cj i} `j i={Aj i,Bj i} Time Degree of membership µ Time Amplitude (c) Trapezoidal function Figure 1. Panel ( a ): Selection of characteristic points (CPs) above the threshold value At . Panel ( b ): Illustration of ToF dispersion due to repeated measurements. Panel ( c ): Trapezoidal function used to evaluate the ToF mismatch based on the repeated measurements of one CP (blue circles). Mathematically, the evaluation within the previously established trapezoidal function (or set) of CP j i , namely the i -th CP of the j -th signal, i= 1, . . . , N , j= 1, . . . , m , will provide the degree of membership µj i∈[ 0, 1 ] of such a point to the set, which is defined as follows: µj i=                      0 ToFj i≤Aj i ToFj i−Aj i `j i Aj i<ToFj i<Bj i 1Bj i≤ToFj i≤Cj i Dj i−ToFj i `j i Cj i<ToFj i<Dj i 0Dj i≤ToFj i, (1) where ToF j i is the ToF of CP j i and `j i={Aj i , Bj i}={Cj i , Dj i} is the ToF interval between Aj i and Bj i , which results to be the same as the interval between Cj i and Dj i . The proposed membership function as well as their related parameters are illustrated in Figure 1c. Note that the interval Lj i={Bj i , Cj i} is larger than the one obtained by measuring the dispersion of the CPs (i.e., {bj i , cj i} ) as shown in Figure 1c. This is to account for further measurement uncertainty that is not captured by the initial repeated measurements. Note also the intervals `j iand Lj ineed to be suitably defined by the modeler in the proposed approach. Sensors 2021,21, 993 5 of 17 Thus, when assessing damage through an acquired guided-wave, this method evaluates if CP j i falls within Lj i , then µj i= 1 meaning that the structure is unaltered. Alternatively, if the structure suffers a permanent damage, the ultrasonic guided-waves will be affected by this damage through slight ToF mismatches [ 42 ]. In this case, the CP j i is likely to fall within the interval `j i due to an advance or delay of the acquired signal, which in turn leads to µj i∈[ 0, 1 ] . Lastly, if CP j i falls out of the greater interval {Aj i , Dj i} , then µj i= 0 and it is assumed that the structure may have suffered a severe damage or significant modification. The assessment of a degree of membership µj i∈[ 0, 1 ) may be an indication of structural damage, but it can also occur due to environmental or numerical noise. To overcome this issue, every CP j i is assessed and their degrees of membership µj i are obtained, the global degree of membership of the j -th ultrasonic signal (i.e., M j , with M being the capital letter of µ) is obtained as follows: Mj=1−g{µj i}N i=1=1−min{µj i}N i=1,j=1, . . . , m. (2) Note that the function g(·) , conservatively chosen in this paper as the minimum of the degrees of membership, might be adopted differently such as the weighting and segmentation of the ToF or the amplitude [ 43 ]. Note also that in Equation (2) the function g(µj i) has been subtracted from the unity, consequently, M j can be viewed as a damage index which increases as the defect becomes more severe, and vice versa. Moreover, when a set of ultrasonic signals are available and post-processed by this method, an array (or matrix) of M j values rather than a single value can be obtained. The use of additional M j values ultimately leads to a more reliable monitoring since the influence of sensor malfunctioning in the monitoring decision greatly decreases. In this case, the resulting array or matrix is referred here to as the DoH matrix of a structure, with values from 0 (maximum health) to 1 (maximum degradation). A schematic workflow of the proposed methodology is shown in Figure 2. Ultrasonic data PWAS sensors PWAS sensors Signal Peaks Figure 1a Trapezoidal function Figure 1b Structure in pristine state Signal Peaks Eval μij - Eq. (1) Eval Mj - Eq. (2) Structure in operation Damage index (DoH) SHM Decision Measurements Figure 2. Schematic workflow of the methodology divided between the data acquisition of ultrasonic data and its post-processing depending on the actual structural state (i.e., pristine or in operation). 3. Case Study The proposed DoH damage index (Section 2) for damage detection and monitoring is illustrated in this section. To this end, a fatigue test on an aluminum plate with a central notch has been carried out and its structural health is monitored by means of the DoH Sensors 2021,21, 993 6 of 17 using ultrasonic guided-waves excited and received by PWAS and a SHM ultrasound system (SHMUS). 3.1. Fatigue Testing Configuration The fatigue test has been performed using a middle tension [M(T)] specimen with a centered crack and loaded in tension using a positive loading ratio. The test specimen of dimensions 245 mm × 500 mm × 1 mm has been extracted from a 1003x503 QQA250/5 ‘O’ 2024 aeronautic grade aluminum sheet using a water jet cutting procedure. The notch has been centered with respect to the test sample centerline and machined using a laser cutting procedure, with the final dimensions specified in Figure 3a. The total length of the machined notch is 22.5 mm approximately. (a) Aluminum specimen and notch geometry ( b ) Actual fatigue testing setup Figure 3. Panel ( a ): Schematic of the specimen and notch geometry, along with the position of the piezoelectric wafer active sensors (PWAS) arrays. Panel ( b ): Picture of M(T) aluminum specimen with two permanently attached phased-arrays mounted on the fatigue testing machine. An Instron 8850 servo-hydraulic fatigue testing machine has been used to conduct the fatigue experiment. A fatigue pre-cracking procedure has been applied to develop a fresh and straight crack front to mitigate the effect of the machined started notch [ 44 ]. The crack onset and growth is recorded through a digital still camera. A tension-controlled fatigue test has been performed with a stress ratio ∆R= 0.1 and maximum load amplitude Pmax = 10 kN, which corresponds to the 60% of the aluminum yield strength. The test was performed up to 100,000 cycles with a cycling frequency of 20 Hz. The reader is referred to Figure 3for further information about the experimental set-up. Note that the upper and lower 50 mm bands of the specimen are used to clamp the plate into the fatigue testing machine. 3.2. Ultrasonic Guided-Wave Based Tests The detection and monitoring of the crack onset and growth has been carried out using two linear phased-arrays of six PWAS, consisting of six evenly spaced PWAS that are linearly placed in the structure [ 19 ]. Piezoelectric ceramic disc transducers of 7 mm diameter and 0.5 mm thickness have been used in this fatigue test. These are used as transmitter-receiver (T i ) and receiver (S i ) arrays for the generation and reception of the ultrasonic guided-waves, respectively. Note that the transmitter-receiver array (T i ) works in pulse-echo mode by simultaneously generating and acquiring ultrasonic signals, while Sensors 2021,21, 993 7 of 17 the receiver array (S i ) functions as a sensor only, also known as pitch-catch mode. It is important to note that the PWAS T 5 malfunctioned during the fatigue test and therefore its associated data have not been considered for SHM purposes. These PWAS have a radial mode of vibration and a resonant frequency centered at 300 kHz. They have been evenly spaced with a separation of 10 mm and symmetrically placed with regards to the sample centerline, as observed in Figure 3, and bonded to the aluminum surface using 3M ™ Scotch-Weld ™ DP490 epoxy adhesive. The piezoelectric transducers are managed by a SHMUS, which is a custom-built compact electronic device for ultrasonic guided-wave based SHM. It includes 12 arbitrary waveform generators and 12 acquisition systems (refer to [ 45 ] for further details of the SHMUS). The SHMUS is controlled by USB using a tailor-made control and processing software installed in a laptop. The schematic of the experimental set-up for damage monitoring is shown in Figure 4. The excitation signals are sinusoids of 4 cycles, with 300 kHz frequency (in accordance to the resonant frequency of the chosen PWAS) and 45 volts peak to peak amplitude. The ultrasonic signals are acquired using a sampling frequency of 60 MHz and 12 bits of resolution. SHMUS 12 Signal generators 12 Signal acquirers Laptop Control & processing software Configuration parameters Ultrasonic data Fatigue testing machine PWAS Aluminum specimen Figure 4. Schematic of the ultrasonic guided-wave based tests. Furthermore, the transmission beamforming technique [ 46 ] has been adopted for the monitoring of the specimen during the fatigue test. Using this technique, synchronously delayed signals were applied to the T i PWAS (refer to Figure 3a), so that the wave fronts can be constructively summed at a pre-established direction creating a main wave beam [ 47 ]. The ultrasonic tests have been carried out by steering the main beam at different directions, that is, from ξ1= 0 ◦ to ξ37 = 180 ◦ with an increment of ∆ξ= 5 ◦ , which sweeps the entire monitoring area. Note that the ultrasonic signals are acquired by both the Tiand Si phased-arrays of PWAS, hence receiving two sets of 37 × 6 signals (i.e., 37 angles and 6 PWAS for each phased-array). Programmed inspections during the fatigue experiment have been carried out every 1000 cycles. For each individual inspection, the fatigue test has been stopped at the minimum load level and an ultrasonic inspection has been performed using the transmission beamforming technique. The received ultrasonic signals (available in [ 48 ]) by the SHMUS have been de-noised using a bandpass filter centered at the frequency of excitation (i.e., 300 kHz). This allows a range of frequencies around the frequency of excitation to pass, while attenuating frequencies out of the scope. In this case the passband is defined in the interval [250 kHz, 350 kHz], while the stopbands, that is, attenuated frequency ranges, are defined within [0 kHz, 200 kHz] and [400 kHz, ∞ ) with a minimum attenuation of − 60 dB. Figure 5shows the representation of this bandpass filter in both time and frequency domains. Sensors 2021,21, 993 8 of 17 0 10 20 30 −4 −2 0 2 4 Time [µs] Amplitude [×10−3] (a) Filter in time domain 0 200 400 600 −100 −50 0 Stopband Stopband Passband Frequency [kHz] Magnitude [dB] (b) Filter in frequency domain Figure 5. Time (panel (a)) and frequency (panel (b)) domain representations of the bandpass filter. 3.3. Damage Monitoring Results The ultrasonic data acquired in pristine state at the minimum load level (i.e., 1.0 kN) have been used to build the trapezoidal functions (refer to Equation (1) ). The parameters used to create these sets are: (1) amplitude threshold At= 15% of the maximum amplitude of the signal in the time window considered for the analysis; (2) `j i= 10% of the period of the excitation signal; and (3) Lj i= 10% of the same period in addition to the random dispersion measured from the 10 measurements. These values are selected so that measurements taken during the pristine state do not show any false damage indications. Moreover, `j i and Lj i are set so that the measurement uncertainty is considered while avoiding that the trapezoidal functions of two adjacent peaks overlap. Note that the evaluation of a new signal in the previous functions has been carried out using a smaller amplitude threshold of At= 10%, so that a control loop is created for the amplitude in an analogous manner to a hysteresis controller [49]. Figure 6a–d show the DoH matrix of the signals acquired at different fatigue cycles, namely 1000, 20,000, 50,000, and 100,000. Note that two DoH matrices are provided for each dataset, the one on the top is for the pulse-echo array (using T i PWAS) and the one on the bottom is for the pitch-catch array (using S i PWAS). As is evident from the results, a clear degradation of the structure is appreciated by the increase of the DoH damage index when increasing the number of cycles. Notwithstanding, working with the DoH matrices may be limited in practice due to their complex interpretation. Hence the mean value of M j , j= 1, . . . , m (i.e., all the values of the DoH matrices), is used here as a simplified statistic of the SHM data evolution, as shown in Figure 7a. Results also show that a higher damage index value is obtained using the data acquired in the S i sensors (grey line) compared to the T i sensors (black line). Furthermore, a comparison of the evolution of both the DoH index and fatigue crack length is also depicted in Figure 7a. The measurements of the crack length are obtained by digitizing different points in the crack path, as shown in Figure 7b. A remarkable agreement in the trend of both damage indicators is observed throughout the duration of the fatigue test. Note also that the early damage is detected using the two phased-arrays around 10,000 cycles, although the S i sensors are able to provide an earlier indication of damage (i.e., even before it is visible by optical means). Note that other statistics that illustrate the damage information obtained from the beamforming tests can be used. In particular, the adoption of the mean of the DoH matrix has shown accuracy in monitoring of the degradation of the structure. However, this statistic shows no indication of the dispersion of the cells of the DoH matrices and works as a filter of the evolution data. To provide such dispersion information, the evolution of the individual damage indices M j for the array of sensors T i when focusing at 60 ◦ and 120 ◦ are shown in Figure 8a,b, respectively. The equivalent data for the upper phased-array with S i Sensors 2021,21, 993 9 of 17 sensors are shown in Figure 8c,d. These curves depict the evolution of the 60 ◦ column of the DoH matrices, equivalent to the ones shown in Figure 6. ξ1ξ2ξ3ξ4ξ5ξ6ξ7ξ8ξ9ξ10 ξ11 ξ12 ξ13 ξ14 ξ15 ξ16 ξ17 ξ18 ξ19 ξ20 ξ21 ξ22 ξ23 ξ24 ξ25 ξ26 ξ27 ξ28 ξ29 ξ30 ξ31 ξ32 ξ33 ξ34 ξ35 ξ36 ξ37 T1 T2 T3 T4 T6 S1 S2 S3 S4 S5 S6 0 0.2 0.40.60.8 1 (a) DoH matrix at 1000 cycles ξ1ξ2ξ3ξ4ξ5ξ6ξ7ξ8ξ9ξ10 ξ11 ξ12 ξ13 ξ14 ξ15 ξ16 ξ17 ξ18 ξ19 ξ20 ξ21 ξ22 ξ23 ξ24 ξ25 ξ26 ξ27 ξ28 ξ29 ξ30 ξ31 ξ32 ξ33 ξ34 ξ35 ξ36 ξ37 T1 T2 T3 T4 T6 S1 S2 S3 S4 S5 S6 0 0.2 0.40.60.8 1 (b) DoH matrix at 20,000 cycles ξ1ξ2ξ3ξ4ξ5ξ6ξ7ξ8ξ9ξ10 ξ11 ξ12 ξ13 ξ14 ξ15 ξ16 ξ17 ξ18 ξ19 ξ20 ξ21 ξ22 ξ23 ξ24 ξ25 ξ26 ξ27 ξ28 ξ29 ξ30 ξ31 ξ32 ξ33 ξ34 ξ35 ξ36 ξ37 T1 T2 T3 T4 T6 S1 S2 S3 S4 S5 S6 0 0.2 0.40.60.8 1 (c) DoH matrix at 50,000 cycles ξ1ξ2ξ3ξ4ξ5ξ6ξ7ξ8ξ9ξ10 ξ11 ξ12 ξ13 ξ14 ξ15 ξ16 ξ17 ξ18 ξ19 ξ20 ξ21 ξ22 ξ23 ξ24 ξ25 ξ26 ξ27 ξ28 ξ29 ξ30 ξ31 ξ32 ξ33 ξ34 ξ35 ξ36 ξ37 T1 T2 T3 T4 T6 S1 S2 S3 S4 S5 S6 0 0.2 0.40.60.8 1 (d) DoH matrix at 100,000 cycles Figure 6. 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