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Time reversal damage localization method of ocean platform based on particle swarm optimization algorithm
Marine Structures 69 (2020) 102672
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Marine Structures journal homepage: www.elsevier.com/locate/marstruc
Time reversal damage localization method of ocean platform based on particle swarm optimization algorithm
T
Weilei Mub, Jiangang Suna, Ren Xina, Guijie Liub,∗, Shuqing Wangb a b
School of Ocean Engineering, Ocean University of China, Qingdao, 266100, China Shandong Provincial Key Laboratory of Ocean Engineering, Ocean University of China, Qingdao, 266100, China
ARTICLE INFO
ABSTRACT
Keywords: Time reversal Damage localization Particle swarm
The traditional time reversal is considered a promising approach for non-destructive testing and health monitoring of key region and structure, but it is considerably time consuming. This paper presents a time reversal damage localization method, based on particle swarm optimization algorithm, which is capable of improving the real-time performance of health monitoring in ocean platform. Firstly, the virtual focusing model of time reversal is constructed, and a succinct expression of virtual focusing for sensor pairs is proposed. Then, on the definition of the evaluation index, the PSO based time reversal algorithm is proposed, and the proper coefficients is given. Finally, the finite element simulation and experimental case validate that the proposed method is capable of find the damage location within limit iterative steps. Thus, the proposed method is a hopeful method for online monitoring and damage localization of large sized structure.
1. Introduction Ocean platform is an important tool for marine exploration. Given the complicated loads during the service period [1], the health monitoring of the offshore structure is of the essence for its safe and stable running [2,3]. Mode parameters identification is one of the most commonly used approaches for structural health monitoring, such as floating structures [4], offshore wind turbines [5] and offshore platform. However, it is rather difficult to detect the minor crack for mode identification method. Ultrasonic guided waves are characterized by small attenuation in long-distance propagation and large monitoring range [6]. Thus, structural health monitoring based on guided waves has received considerable attention and become a research hotspot [7]. Generally, the crack of the offshore platform is usually caused by the concentration of stress. When the stress reaches the fatigue limit, the fatigue crack will be generated and the stress will be released. Thus, the object of this study is open crack. The health monitoring method based on guided waves can be divided into passive and active modes [8,9]. The passive mode realizes damage positioning and analysis by monitoring the guided wave signals that are excited at the occurrence of damages. While the active mode needs excite the guided wave, and then detects the reflected wave caused by defects and localize defects [10]. The passive mode is only valid to active damages, whereas the active mode can detect existing damages. This study applied the active mode for minor damage localization in the key region of offshore structure. Time reversal damage localization method is a commonly used positive damage positioning method [11,12], which is based on the acoustic reciprocity. In the active mode, the emittor excites the ultrasonic guided wave, and the wave will reflect when it encounter the defect. Then, the sensors will collect the reflected wave. Affected by dispersion characteristics of guided wave, the guided wave packet will extend during propagation [13], thereby the damage Corresponding author. E-mail addresses: [email protected] (W. Mu), [email protected] (J. Sun), [email protected] (R. Xin), [email protected] (G. Liu), [email protected] (S. Wang). ∗
https://doi.org/10.1016/j.marstruc.2019.102672 Received 8 July 2018; Received in revised form 26 June 2019; Accepted 30 August 2019 0951-8339/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Schematic of time reversal focusing.
positioning resolution will be decreased. The time reversal method assumes that the wave propagetes from the sensor to the emittor reversely; thus, it can compensate the extended wave packet to some extent [14]. So that, it is benificial to improve the localization accuracy. However, in the traditional time reversal method, it is necessary to calculate the wave field of the entire region at each iterative time. Therefore, longer positioning time is required when the positioning area is large and the iterative step length is small. Particle swarm optimization (PSO) algorithm is a high-efficiency optimization algorithm and is often used to optimize traversal calculation [15]. Passive damage localization method based on PSO algorithm has been proved that POS is capable of improving the efficiency of the positioning algorithm [16]. Thus, PSO algorithm and time reversal damage positioning algorithm can be combined to realize the rapid damage positioning of large-sized components. In this study, a mathematical model of time reversal damage positioning process was initially constructed and the mathematical expression of the virtual focus field was then established. On this basis, the time reversal damage positioning method based on PSO algorithm was proposed. This method optimizes the calculation of focusing field by using the PSO algorithm, which reduces operation loads. At last, the proposed method was verified by both a case of finite element simulation and an experimental case. 2. Principle of time reversal damage localization Time reversal damage localization method mainly includes two processes, namely, acquisition process of damage signals and loading process of time reversal signals. In Fig. 1, the guided wave excited by the left exciter propagates to the damage position, thereby generating scattered waves that propagate to the right receiving sensor. This is the acquisition process of damage signals. Thus, signals collected by the right sensor contain the direct and scattered waves. Generally, the direct wave can be distinguished and eliminated easily, thereby leaving the pure scattered wave of damage. The received signals were processed by time reversal and then inputted into the right sensor for excitation. The generated guided wave would focus at the position of the damage. Thus, the scattered wave source was reconstructed. In this theoritical study, we suppose the wave guided plate is a piece of infinite plate, which means the infinite boundary would not generate the reflection echo in the specimen. Thus, the sensor will only collect the direct and scattered waves. The guided wave, which was generated by the exciter, would propagate to the receiving sensor and damage position separately. The transfer functions of these propagetion routes were denoted by HA1 (w ) and HA2 (w ) . In this study, only the A0 mode of the guided wave is considered, because the amplitude of S0 mode is too slight to be collected. The guided wave, which propagated to the damage position, scattered at that position. The transfer function of the propagetion route from damage to the receiving position was denoted by HA3 (w ) . The direct wave received at the receiving position Y1A (w ) and the scattered wave of damage Y2A (w ) can be expressed, if the excitation signla is S (w )
Y1A (w ) = S (w ) HA1 (w ) Y2A (w ) = S (w ) HA2 (w ) HA3 (w )
(1)
Usually, Y1A (w ) can generate disturbances to time reversal damage localization. Thus, certain measurements are often taken to eliminate the direct wave. In this study, sensor pair with short distance are adopted. In this way, Y1A (w ) can arrive at the receiving
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sensor rapidly, whereas the scattered wave of damage would arrive after a certain propagation period. In this way, the direct wave packet can be eliminated easily. When Y2A (w ) was zoomed in the time domain at the scale of −1, it is equivalent to the complex conjugation processing in the frequency domain. Suppose that the time length of the time reversal signal of the damage scattered wave was . To ensure the zero point of the initial moment, the signal should be moved rightward horizontally by the time length of . The time reversal signal for excitation Y2A (w ) can be expressed
Y2A (w )(w ) = Y2A (w ) e
iw
= S (w ) H A2 (w ) H A3 (w ) e
(2)
iw
Usually, the transfer function of the guided wave propagetion route is
HA (r , w ) = AA (r , w ) e
(3)
ikA (w ) r
where AA (r , w ) is the attenuation coefficient which is relative to the propagetion distance and frequency of the guided wave. Due to the capacity of long range detecting, the effect of the amplitude attenuation was neglected. When the guided wave excited by the time reversal signal propagated to r , the wave can be expressed as follows:
Y3A (r , w ) = Y2A (w ) HA (r , w ) = S (w ) eid1 kA (w) eid2 kA (w ) e
iw
e
(4)
irkA (w )
where d1 is the distance between the damage and the exciter, and d2 is the distance between the damage and the sensor. Equation (4) can be simplified as
Y3A (r , w ) = S (w ) e
(5)
i ( + r d1 d2) kA (w )
When the excitation signal was a narrow-band signal, the guided wave packet approximately propagates at the group velocity of vg . At this moment, the Fourier inversion of Eq. (5) is
y3A (r , t ) = s ( (t
+ (d1 + d2
(6)
r )/ vg ))
For the damage position, r = d2 and the time domain of the guided wave can be expressed
y3A (d2 , t ) = s ( (t
(7)
+ d1/vg ))
If the maximum amplitude of the excitation signal arrived at the sensor at the time of t1, then the maximum guided wave d1 t1, which is called as the focusing time. amplitude at the damage position would be achieved at v At the focusing time, Eq. (6) can be simplified as
y = s (t1
g
(8)
d2/ vg + r / vg )
As many studies mentioned, n pairs of sensors are customarily applied for damage detection. Thereby, the virtual focusing d1 t1 is amplitude of any position in the searching region at the focusing time t = v g
n
y
=
si (t1i
d2i / vg + ri / vg ), i
n
(9)
i=1
where ri is distance between i-th sensor and the objective point, d2i / vg is the time for guided wave propagation from the damage position to sensor group i . If the distance between the exciter and sensor is reasonably close, which is called as a sensor pair, d2i / vg is approximately half of the time for guided wave propagation from the exciter to the sensor. Let t2i be the time for guided wave propagation to sensor group i . Therefore, Eq. (9) can be expressed as: n
y
=
si (t1i
0.5t2i + ri /vg ), i
n
(10)
i=1
The distance between one point in the searching region and the sensor was calculated and denoted by ri . The wave field in the region was obtained by using the virtual focusing calculation based on Eq. (10). The position with the highest virtual focusing amplitude in the region was the damage position. 3. Time reversal localization method based on PSO algorithm PSO algorithm is motivated from the simulation of social behavior. Through cooperation and competition among the potential solutions, PSO can find optimal solution more rapidly when applied to complex optimization problems. Therefore, the searching strategy of time reversal positioning was optimized by PSO algorithm to approximate optimization rapidly. The time reversal damage positioning process based on PSO algorithm is shown in Fig. 2. The time reversal damage positioning flowchart based on PSO algorithm is: 1) Acquisition of the scattered wave of damage
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Fig. 2. Time reversal damage positioning based on PSO algorithm.
2)
3)
4)
5)
6)
The scattered wave signal was collected by the receiving sensor. Then, the time reversal excitation signals were acquired by the time reversal processing of the scattered wave signal. Regional encoding A rectangular coordinate system was constructed on the specimen. The domains of definition on the x- and y-axes in the target region were used as the coding range. Generally, real number encoding does not need specific encoding and decoding processes, thereby simplifying the algorithm operation and increasing the execution efficiency of the algorithm. Therefore, real number encoding was applied here for regional encoding. Initialization of particle swarm The PSO algorithm has to arrange particles in the target region randomly. Then, these particles shall move to the optimal position. Finally, they shall approach to the optimal solution rapidly. If the particle number is extremely small, then the solving process is influenced by accidental factors, which make the optimal solution difficult to obtain. Therefore, the computation loads and solving accuracy should be considered [17]. In this work, the particle number was set to 30. Boundary judgment The position of particles continuously change in each iteration. Particles may exceed the boundaries of the search region when they move, which may result in wrong solutions and should be avoided. In this work, the closest points, which were located in the boundary, were selected as the updating position of the cross-border particles. Then, the cross-border particles were deleted. Particle evaluation Particles, which presented virtual focusing points in searching region, should be evaluated before the next updating. In this work, Eq. (10) was used as the evaluation index to assess the targeting points represented by particles in the particle swarm. High values of the evaluation index demonstrated the remarkable performance of the current particle. Updating of particle position During the updating process of the particle swarm, particle movement was realized by tracing the best position of individuals during the evolution and the best position of all particles in the current particle swarm. The amplitude of position change was mainly determined by the advancing direction and distance. The advancing direction, distance, and positional movement of the ith particle in the particle swarm can be expressed as:
Vi = wVi + c1 r1 (Pbest Xi = Xi + Vi
Xi ) + c2 r2 (Gbest
Xi ) (11)
where Vi and Vi are the displacement vectors of the previous and current iterations of the i-th particle, respectively; Xi and Xi are the current and target positions of the i-th particle, respectively; w is the inertia weight coefficient; c1 and c2 are acceleration constants; r1 and r2 are random coefficients with range of [0, 1]; Pbest is the best position of the i-th particle during evolution; and
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Table 1 Material parameters. Material
Density(kg/m^3)
Elastic modulus(Pa)
Poisson's ratio
Q235
7850
2.1 × 1011
0.3
Gbest is the best position of all particles in the particle swarm [18]. Perez's work demonstrated that when the acceleration constant and inertia weight coefficient satisfied Eq. (12), the algorithm obtains improved convergence characteristic [19]. 0 < (c1 + c2) < 4 ( c1 + c 2 ) 2
1
(12)
The inertia weight coefficient w presents dynamic changes during the searching process, which enables the algorithm to achieve high global searching capability in the early period and good local search capability after the searching. The calculation formula is:
w = wmax
(wmax
wmin ) n n max
(13)
where wmax is the maximum inertia coefficient and is often set as 0.8; wmin is the initial inertia coefficient, which is calculated from c1 and c2 . n is the number of particle generation, and n max is the maximum generations. 7) Terminal conditions Many terminal conditions of particle swarm are available, including termination based on number of operations, time of operation, and updating distance changes. To approach the real solutions, the number of iterations or iterative times can be set as a large one. In this work, the highest number of iteration and the highest iterative time were set at 100 and 100 s, respectively. Particles evolution would stop when any one of these two parameters exceeded the limits. 4. Finite element simulation of time reversal damage positioning 4.1. Finite element model The simulation model was constructed in ABAQUS CAE™ application, which was with the dimensions of 500 mm (length) × 500 mm (width) × 5 mm(thickness). The material was set as carbon steel, which is commonly used in the offshore platform, and the grade of the carbon steel is Q235. The material parameters were listed in Table 1. To obtain the signals of intact and damaged state, two simulation models was constructed, as shown in Fig. 3(a) and (b). The red marks represent the position of the excitator, and the dark marks represent that of the sensor. There are three pairs of sensors mounted on the surface of the plate-like model. Among the three pairs of sensors, namely, S1–S4, S2–S5, and S3–S6, the former is the exciter and the latter is the sensor, as listed in Table 2. The coordinates of the excita The blue mark represents the damage which is a blind hole with 3 mm diameter and 2 mm depth, located at the coordinate (200,240). There are two sorts of simulation solver for dynamic problem in ABAQUS CAE™ application, explicit and implicit. Although piezoelectric element are not available in ABAQUS CAE™ explicit, researchers could apply an equivalent load instead of a voltage as actuation loading. Furthermore, explicit solver is more economical and accurate for wave propagation simulation. Meanwhile, there is no convergence problem in explicit solution, because it does not need iteration. Thus, the explicit procedure is strongly recommended in the previous study [20]. To avoid the influence of reflected wave from the model boundaries, the elements of the model boundaries were set as infinite element type (CIN3D8), as shown in Fig. 3. In the numerical simulation, the size of the grid will affect the final calculation result. The smaller the grid size is, the more accurate the result will be. However, with the decrease of the grid size, the computational consumption will increase exponentially. Therefore, element size is usually limited into one-tenth of the wavelength. The grid size in this study is set as 1 mm. According to the Nyquist's theory, the acquisition frequency must be twice more than the maximum frequency component at least. The central Table 2 Coordinates of the sensors. Sensor
S1
S2
S3
S4
S5
S6
Coordinate
(150,150)
(350,350)
(350,150)
(165,150)
(350,335)
(365,150)
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Fig. 3. Simulation model.
frequency of the exciting signal is 150 kHz, and the acquisition frequency is 10 MHz in the following experiment. Thus, the time step in the simulation is set to be 0.1 μs in accordance with the experiment. 4.2. Finite element simulation In the simulations, the excitation signal is a five-cycle sinusoidal signal modulated with Hanning window, as shown in Fig. 11(a).
y = Asin (2 fc ) × (1
(14)
cos (2 fc /N ))
where A is the amplitude of the sinusoidal signal, N denotes the cycle number of the modulated signal, and fc represents the time serial ranging from zero to N. In both the intact and damaged model, when the exciter S1 is actuated, only the sensor S4 is assigned to collect the wave motion. The sensor pairs S2–S5 and S3–S6 work in the same way with S1–S4. In the intact model, there is no reflected wave, as shown in Fig. 4, and the signals collected by S4, S5 and S6 are called as the reference signals. In the damaged model, the signals collected by S4, S5 and S6 are called as the reference signals. As shown in Fig. 5, there are obvious reflected wave in the damaged model, which will be collected by the sensors.
Fig. 4. The wave field of intact model. 6
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Fig. 5. The wave field of damaged model.
4.3. Analysis of simulation results From the signals collected by S4, S5 and S6, there is a direct wave packet with the same shape in both the intact and damaged state, which appears at the same time. While, there is a minor wave packet after the direct wave packet, which appears only in the damaged state, as shown in Fig. 6. The different part is caused by the reflected wave, which could be obtained by making a subtraction of the signal of intact state from that of damaged state. Usually, affected by the difference between the intact and damaged model, the direct waves in the intact and damaged state will differ with each other slightly. Thus, there is still a residual direct wave packet as shown in Fig. 7. Nevertheless, the residual wave packet could be easily eliminated by a limited width window. The subtraction calculation results after removing the residual wave packet were called as the scattered wave signals, as shown in Fig. 8. The damage positioning was investigated by using the scattered wave of damage after the elimination of the disturbance wave. The proper length of scattered wave signals was of the essence in localizing the potential damage. The duration of the signals must satisfy the following equation.
T
2L max +t vwave
(15)
where L max is the maximum distance between the sensor and the farthest point in the searching field, vwave is the group velocity of the guided wave, t is the duration of the excitation signal in Eq. (15). The duration of the scattered signals was set to be 300 µs according to the criterion above. Thus, the time reversal signals of the scattered wave of damage were conducted, as shown in Fig. 9. The enveloped curve of the time reversal signals were excited virtually at their corresponding sensors. The virtual wave fields at different moments are shown in Fig. 10. When the duration of excited signal is 230 μs, all the three sensors have excited waves. After that, the three waves begin to superpose with each other, and the peak of those waves superpose only at the momont of 270 μs, which is called as the focusing time (see Table 2). In the traditional time reversal method, the virtual wave field at each iterative moment ought to be calculated. The position with the highest amplitude is taken as the damage position. Thus, when the iterative step of time domainis rather small, the traditional method is considerably time-consuming. Moreover, when the detecting region is much large, the situation of time consumption is worse. As listed in Table 3, the time consumption of traditional time reversal rises with the increase of searching area. Whereas, the PSO based time reversal consumes the same smaller time, no matter how large the searching area is. This is because the PSO algorithm is capacity of finding the optimal solution within limited iterative step. The PSO algorithm might be affected by acceleration constants. In our previous study, the effect of acceleration constants has been analysed [15]. With the proper values, the PSO based time reversal offers a stable positioning results. Although both positioning methods achieve small positioning error, as listed in Table 4, the PSO based method could accomplish the damage positioning rapidly under a large searching area. No matter how large the searching area is, the localization error remains the same, which represent that changing the sensor location will not affect the accuracy of this method. 5. Time reversal damage positioning experiment 5.1. Experimental process The experimental specimen is with the same material and dimension with the simulation model. Six piezoelectric wafers were
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Fig. 6. The damage and reference waveform. 8
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Fig. 7. The subtraction calculation result.
mounted on the specimen surface. Among them, S1–S4, S2–S5, and S3–S6 formed the sensor pairs, the intact specimen is shown in Fig. 11. The experimental setup consisted of a Tektronix AFG1022 function/arbitrary waveform generator, a DS-2B(Beijing Softland Times Scientific &Technology Co.Ltd.) data collecting instrument and charge amplifiers. The piezoelectric wafers has the dimension of φ 10 mm. The dielectric constant of PZT sensor in the z direction is far larger than that in the x and y directions, and the relative dielectric constants in the x and y directions are approximately equal to 0. Therefore, only the displacement in z direction is extracted. The piezoelectric wafers S1, S2 and S3 are connected with the waveform generator, and the piezoelectric wafers S4, S5 and S6 are connected with the data collector by the charge amplifiers. Meanwhile, waveform generator is connected with the collector to record excitation signals and provide a start point of data collection. After the reference guided wave signals in the intact specimen are acquired, a through hole with φ 4 mm diameter is created by using an electric drill at (200, 240) of the specimen surface as shown in Fig. 12. 5.2. Experimental results The reference signals collected by piezoelectric sensor under the intact state was subtracted by that under the damaged state, then the reflected wave signal caused by damage is obtained. When sensors S1 and S4 are used as the exciter and receiving sensor, respectively, the waveform data were normalized after the subtraction operation (Fig. 13). In the subtraction result, there are commonly error packet caused by the direct wave, scattered wave caused by damage, reflected wave caused by the specimen boundary and the damage reflected wave caused by scattered wave and boundaries. The direct wave of reference and damaged signals may differ with each other due to sampling and system errors, thus, error packet could not be eliminated completely. However, the waveform and the frequency spectrum had remarkable differences with the other wave packets, thereby, the error packet could be eliminated easily. Scattered signals arrived at the sensor after the direct wave, and this wave packet was similar to the excitation signals. Specimen boundaries will reflect waves, and the reflected wave could be eliminated completely, because the nearest reflection routine of S1–S4 is known. The scattered wave of damage would be reflected upon the specimen boundaries, thereby reflected wave occurs after the reflected wave. The collected signal except the part of damage scattered wave was eliminated to avoid the influences of the other wave packets, as shown in Fig. 14. The wave field of the specimen region at different times are shown in Fig. 15. Only at the moment of focusing time, the amplitude of superimposed wave field will achieve the maximum value. In the study, the resolution of wave field in both the vertical and horizon direction is 1 mm. If the searching area are 0.25 m2, 1 m2, and 16 m2, the traditional time reversal will consume more time with the increase of the area. The same with the simulation, the POS based time reversal will consume the same time no matter how much the searching area is, as listed in Table 5. It means that the POS based time reversal will achieve the damage localization within some iterative step. The localization coordinate of traditional time reversal approach is (201,236), which is with a localization error of 4.1 mm to the real damage coordinate of (200,240). No matter how large the searching area is, the localization coordinate remain (201,236). This is because traversing search strategy of traditional approach is capable of finding out the global optimal solution. The PSO based time reversal approach localize the damage at the coordinate of (198,229), which is with a localization error of 11.2 mm. 5.3. Localization error analysis The localization accuracy of ultrasonic detection is affected by the half-wave length, thus, localizing the damage with no localization error is impossible. In this study, the excitation signal with the central frequency 150 kHz will generate a guided wave with the wave length of 20.4 mm in the 5 mm thick plate. Thus, the localization accuracy limit is 10.2 mm. If there is an extra positive
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Fig. 8. The scattered wave signals.
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Fig. 9. The time reversal signals.
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Fig. 10. Virtual wave fields.
Fig. 11. Experimental specimen.
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Table 3 Comparison of time consumption. Searching area (Width x Length)
Time reversal (s)
PSO based time reversal (s)
0.5 × 0.5 m2 1 × 1 m2 4 × 4 m2
0.7 1.1 12.8
0.3 0.3 0.3
Table 4 The damage coordinate of different methods. Searching area (Width x Length)
Time reversal
Localization error of time reversal (mm)
PSO based time reversal
Localization error of PSO based time reversal (mm)
0.5 × 0.5 m2 1 × 1 m2 4 × 4 m2
(206, 238) (206, 238) (206, 238)
6.3 6.3 6.3
(205, 238) (205, 238) (205, 238)
5.4 5.4 5.4
Table 5 Consumption of calculation time. Searching area (Width x Length) 0.5 × 0.5 m 1 × 1 m2 4 × 4 m2
2
Time reversal (s)
PSO based time reversal (s)
1 1.6 13.3
1.9 1.9 1.9
Fig. 12. Artificial damages.
Fig. 13. Subtraction result of S4.
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Fig. 14. Waveform amplitude difference of S4.
Fig. 15. Wave fields at different moments.
error, the localization error will larger than the localization limit. On the contrary, an negative error will cause a little localization error. 6. Conclusions In this study, the PSO algorithm based time reversal positioning method was proposed, which was capable of localizing the damage online. At first, a mathematical model of time reversal damage positioning method was constructed and the mathematical expression of the virtual focus field was then established. The time reversal needs to calculate the wave field at each moment, which is highly time-consuming. Thus, PSO method is combined with the traditional time reversal, which is capable of finding out the optimal localization within limit iterative step. The finite simulation and experimental case prove that the proposed method can position the damage rapidly even in the large-sized structure. It is a solution for the offshore platform to monitor its health online. Moreover, the arrangement of sensor pairs can effectively avoid the influence of direct wave on the scattered wave of damage. The focusing moment is calculated by the group velocity of the central frequency component of guided wave, which could influence 14
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positioning to a certain extent. Therefore, future studies should pay attention to the acquisition of focusing moment to increase positioning accuracy. Funding The authors acknowledge the support by the National Science Fund for Distinguished Young Scholars (No. 51625902), the National Natural Science Foundation of China (No. 61501418), and Qingdao Source Innovation Program (No. 18-2-2-69-jch). Conflicts of interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgment The valuable comments from the anonymous reviewers are highly appreciated. References [1] Liu FS, Li HJ, Lu HC. Weak-mode identification and time-series reconstruction from high-level noisy measured data of offshore structures. Appl Ocean Res 2016;56:92–106. [2] Mieloszyk M, Ostachowicz W. An application of Structural Health Monitoring system based on FBG sensors to offshore wind turbine support structure model. Mar Struct 2017;51:65–86. [3] Hillis AJ, Courtney CRP. Structural health monitoring of fixed offshore structures using the bicoherence function of ambient vibration measurements. J Sound Vib 2011;330(6):1141–52. [4] Liu FS, Chen JF, Qin HD. Frequency response estimation of floating structures by representation of retardation functions with complex exponentials. Mar Struct 2017;54:144–66. [5] Liu FS, Gao SJ, Han HW, Tian Z, Liu P. Interference reduction of high-energy noise for modal parameter identification of offshore wind turbines based on iterative signal extraction [J]. Ocean Eng. [6] Wilcox P, Lowe M, Cawley P. The effect of dispersion on long-range inspection using ultrasonic guided waves. NDT E Int 2001;34(1):1–9. [7] Clarke T, Cawley P, Wilcox PD, et al. Evaluation of the damage detection capability of a sparse-array guided-wave SHM system applied to a complex structure under varying thermal conditions[J]. IEEE Trans Ultrason Ferroelectr Freq Control 2009;56(12):2666–78. [8] Staszewski WJ, Mahzan S, Traynor R. Health monitoring of aerospace composite structures Active and passive approach. Compos Sci Technol 2009;69(11):1678–85. [9] Yu LY, Giurgiutiu V. In situ 2-D piezoelectric wafer active sensors arrays for guided wave damage detection. Ultrasonics 2008;48(2):117–34. [10] Guan JF, Shen ZH, Ni XW, et al. Numerical simulation of the reflected acoustic wave components in the near field of surface defects. J Phys D Appl Phys 2006;39(6):1237. [11] Fink M. Time reversal of ultrasonic fields. I. Basic principles. IEEE Trans Ultrason Ferroelectr Freq Control 1992;39(5):555–66. [12] Zhu R, Huang GL, Yuan FG. Fast damage imaging using the time-reversal technique in the frequency–wavenumber domain. Smart Mater Struct 2013;22(7):075028. [13] Liu L, Yuan FG. A linear mapping technique for dispersion removal of Lamb waves. Struct Control Health Monit 2010;9:75–86. [14] Chen CL, Li YL, Yuan FG. Development of time-reversal method for impact source identification on plate structures. Shock Vib 2013;20(3):561–73. [15] Zhao CL, Sun XB, Sun SL, et al. Fault diagnosis of sensor by chaos particle swarm optimization algorithm and support vector machine. Expert Syst Appl 2011;38(8):9908–12. [16] Mu WL, Qu WS, Liu GJ, et al. Acoustic emission beamforming localisation approach based on particle swarm optimizations. Insight 2018;60:575–80. [17] Jordehi AR, Jasni J. Parameter selection in particle swarm optimizations: a survey. J Exp Theor Artif Intell 2013;25(4):527–42. [18] Kennedy J. The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Trans Evol Comput 2002;6(1):58–73. [19] Perez RE, Behdinan K. Particle swarm approach for structural design optimization. Comput Struct 2007;85(19–20):1579–88. [20] Mu WL, Sun JG, Liu GJ, Wang SQ. High-resolution crack localization approach based on diffraction wave. Sensors 2019;19(8):1951.
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Time reversal damage localization method of ocean platform based on particle swarm optimization algorithm
T
Weilei Mub, Jiangang Suna, Ren Xina, Guijie Liub,∗, Shuqing Wangb a b
School of Ocean Engineering, Ocean University of China, Qingdao, 266100, China Shandong Provincial Key Laboratory of Ocean Engineering, Ocean University of China, Qingdao, 266100, China
ARTICLE INFO
ABSTRACT
Keywords: Time reversal Damage localization Particle swarm
The traditional time reversal is considered a promising approach for non-destructive testing and health monitoring of key region and structure, but it is considerably time consuming. This paper presents a time reversal damage localization method, based on particle swarm optimization algorithm, which is capable of improving the real-time performance of health monitoring in ocean platform. Firstly, the virtual focusing model of time reversal is constructed, and a succinct expression of virtual focusing for sensor pairs is proposed. Then, on the definition of the evaluation index, the PSO based time reversal algorithm is proposed, and the proper coefficients is given. Finally, the finite element simulation and experimental case validate that the proposed method is capable of find the damage location within limit iterative steps. Thus, the proposed method is a hopeful method for online monitoring and damage localization of large sized structure.
1. Introduction Ocean platform is an important tool for marine exploration. Given the complicated loads during the service period [1], the health monitoring of the offshore structure is of the essence for its safe and stable running [2,3]. Mode parameters identification is one of the most commonly used approaches for structural health monitoring, such as floating structures [4], offshore wind turbines [5] and offshore platform. However, it is rather difficult to detect the minor crack for mode identification method. Ultrasonic guided waves are characterized by small attenuation in long-distance propagation and large monitoring range [6]. Thus, structural health monitoring based on guided waves has received considerable attention and become a research hotspot [7]. Generally, the crack of the offshore platform is usually caused by the concentration of stress. When the stress reaches the fatigue limit, the fatigue crack will be generated and the stress will be released. Thus, the object of this study is open crack. The health monitoring method based on guided waves can be divided into passive and active modes [8,9]. The passive mode realizes damage positioning and analysis by monitoring the guided wave signals that are excited at the occurrence of damages. While the active mode needs excite the guided wave, and then detects the reflected wave caused by defects and localize defects [10]. The passive mode is only valid to active damages, whereas the active mode can detect existing damages. This study applied the active mode for minor damage localization in the key region of offshore structure. Time reversal damage localization method is a commonly used positive damage positioning method [11,12], which is based on the acoustic reciprocity. In the active mode, the emittor excites the ultrasonic guided wave, and the wave will reflect when it encounter the defect. Then, the sensors will collect the reflected wave. Affected by dispersion characteristics of guided wave, the guided wave packet will extend during propagation [13], thereby the damage Corresponding author. E-mail addresses: [email protected] (W. Mu), [email protected] (J. Sun), [email protected] (R. Xin), [email protected] (G. Liu), [email protected] (S. Wang). ∗
https://doi.org/10.1016/j.marstruc.2019.102672 Received 8 July 2018; Received in revised form 26 June 2019; Accepted 30 August 2019 0951-8339/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Schematic of time reversal focusing.
positioning resolution will be decreased. The time reversal method assumes that the wave propagetes from the sensor to the emittor reversely; thus, it can compensate the extended wave packet to some extent [14]. So that, it is benificial to improve the localization accuracy. However, in the traditional time reversal method, it is necessary to calculate the wave field of the entire region at each iterative time. Therefore, longer positioning time is required when the positioning area is large and the iterative step length is small. Particle swarm optimization (PSO) algorithm is a high-efficiency optimization algorithm and is often used to optimize traversal calculation [15]. Passive damage localization method based on PSO algorithm has been proved that POS is capable of improving the efficiency of the positioning algorithm [16]. Thus, PSO algorithm and time reversal damage positioning algorithm can be combined to realize the rapid damage positioning of large-sized components. In this study, a mathematical model of time reversal damage positioning process was initially constructed and the mathematical expression of the virtual focus field was then established. On this basis, the time reversal damage positioning method based on PSO algorithm was proposed. This method optimizes the calculation of focusing field by using the PSO algorithm, which reduces operation loads. At last, the proposed method was verified by both a case of finite element simulation and an experimental case. 2. Principle of time reversal damage localization Time reversal damage localization method mainly includes two processes, namely, acquisition process of damage signals and loading process of time reversal signals. In Fig. 1, the guided wave excited by the left exciter propagates to the damage position, thereby generating scattered waves that propagate to the right receiving sensor. This is the acquisition process of damage signals. Thus, signals collected by the right sensor contain the direct and scattered waves. Generally, the direct wave can be distinguished and eliminated easily, thereby leaving the pure scattered wave of damage. The received signals were processed by time reversal and then inputted into the right sensor for excitation. The generated guided wave would focus at the position of the damage. Thus, the scattered wave source was reconstructed. In this theoritical study, we suppose the wave guided plate is a piece of infinite plate, which means the infinite boundary would not generate the reflection echo in the specimen. Thus, the sensor will only collect the direct and scattered waves. The guided wave, which was generated by the exciter, would propagate to the receiving sensor and damage position separately. The transfer functions of these propagetion routes were denoted by HA1 (w ) and HA2 (w ) . In this study, only the A0 mode of the guided wave is considered, because the amplitude of S0 mode is too slight to be collected. The guided wave, which propagated to the damage position, scattered at that position. The transfer function of the propagetion route from damage to the receiving position was denoted by HA3 (w ) . The direct wave received at the receiving position Y1A (w ) and the scattered wave of damage Y2A (w ) can be expressed, if the excitation signla is S (w )
Y1A (w ) = S (w ) HA1 (w ) Y2A (w ) = S (w ) HA2 (w ) HA3 (w )
(1)
Usually, Y1A (w ) can generate disturbances to time reversal damage localization. Thus, certain measurements are often taken to eliminate the direct wave. In this study, sensor pair with short distance are adopted. In this way, Y1A (w ) can arrive at the receiving
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sensor rapidly, whereas the scattered wave of damage would arrive after a certain propagation period. In this way, the direct wave packet can be eliminated easily. When Y2A (w ) was zoomed in the time domain at the scale of −1, it is equivalent to the complex conjugation processing in the frequency domain. Suppose that the time length of the time reversal signal of the damage scattered wave was . To ensure the zero point of the initial moment, the signal should be moved rightward horizontally by the time length of . The time reversal signal for excitation Y2A (w ) can be expressed
Y2A (w )(w ) = Y2A (w ) e
iw
= S (w ) H A2 (w ) H A3 (w ) e
(2)
iw
Usually, the transfer function of the guided wave propagetion route is
HA (r , w ) = AA (r , w ) e
(3)
ikA (w ) r
where AA (r , w ) is the attenuation coefficient which is relative to the propagetion distance and frequency of the guided wave. Due to the capacity of long range detecting, the effect of the amplitude attenuation was neglected. When the guided wave excited by the time reversal signal propagated to r , the wave can be expressed as follows:
Y3A (r , w ) = Y2A (w ) HA (r , w ) = S (w ) eid1 kA (w) eid2 kA (w ) e
iw
e
(4)
irkA (w )
where d1 is the distance between the damage and the exciter, and d2 is the distance between the damage and the sensor. Equation (4) can be simplified as
Y3A (r , w ) = S (w ) e
(5)
i ( + r d1 d2) kA (w )
When the excitation signal was a narrow-band signal, the guided wave packet approximately propagates at the group velocity of vg . At this moment, the Fourier inversion of Eq. (5) is
y3A (r , t ) = s ( (t
+ (d1 + d2
(6)
r )/ vg ))
For the damage position, r = d2 and the time domain of the guided wave can be expressed
y3A (d2 , t ) = s ( (t
(7)
+ d1/vg ))
If the maximum amplitude of the excitation signal arrived at the sensor at the time of t1, then the maximum guided wave d1 t1, which is called as the focusing time. amplitude at the damage position would be achieved at v At the focusing time, Eq. (6) can be simplified as
y = s (t1
g
(8)
d2/ vg + r / vg )
As many studies mentioned, n pairs of sensors are customarily applied for damage detection. Thereby, the virtual focusing d1 t1 is amplitude of any position in the searching region at the focusing time t = v g
n
y
=
si (t1i
d2i / vg + ri / vg ), i
n
(9)
i=1
where ri is distance between i-th sensor and the objective point, d2i / vg is the time for guided wave propagation from the damage position to sensor group i . If the distance between the exciter and sensor is reasonably close, which is called as a sensor pair, d2i / vg is approximately half of the time for guided wave propagation from the exciter to the sensor. Let t2i be the time for guided wave propagation to sensor group i . Therefore, Eq. (9) can be expressed as: n
y
=
si (t1i
0.5t2i + ri /vg ), i
n
(10)
i=1
The distance between one point in the searching region and the sensor was calculated and denoted by ri . The wave field in the region was obtained by using the virtual focusing calculation based on Eq. (10). The position with the highest virtual focusing amplitude in the region was the damage position. 3. Time reversal localization method based on PSO algorithm PSO algorithm is motivated from the simulation of social behavior. Through cooperation and competition among the potential solutions, PSO can find optimal solution more rapidly when applied to complex optimization problems. Therefore, the searching strategy of time reversal positioning was optimized by PSO algorithm to approximate optimization rapidly. The time reversal damage positioning process based on PSO algorithm is shown in Fig. 2. The time reversal damage positioning flowchart based on PSO algorithm is: 1) Acquisition of the scattered wave of damage
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Fig. 2. Time reversal damage positioning based on PSO algorithm.
2)
3)
4)
5)
6)
The scattered wave signal was collected by the receiving sensor. Then, the time reversal excitation signals were acquired by the time reversal processing of the scattered wave signal. Regional encoding A rectangular coordinate system was constructed on the specimen. The domains of definition on the x- and y-axes in the target region were used as the coding range. Generally, real number encoding does not need specific encoding and decoding processes, thereby simplifying the algorithm operation and increasing the execution efficiency of the algorithm. Therefore, real number encoding was applied here for regional encoding. Initialization of particle swarm The PSO algorithm has to arrange particles in the target region randomly. Then, these particles shall move to the optimal position. Finally, they shall approach to the optimal solution rapidly. If the particle number is extremely small, then the solving process is influenced by accidental factors, which make the optimal solution difficult to obtain. Therefore, the computation loads and solving accuracy should be considered [17]. In this work, the particle number was set to 30. Boundary judgment The position of particles continuously change in each iteration. Particles may exceed the boundaries of the search region when they move, which may result in wrong solutions and should be avoided. In this work, the closest points, which were located in the boundary, were selected as the updating position of the cross-border particles. Then, the cross-border particles were deleted. Particle evaluation Particles, which presented virtual focusing points in searching region, should be evaluated before the next updating. In this work, Eq. (10) was used as the evaluation index to assess the targeting points represented by particles in the particle swarm. High values of the evaluation index demonstrated the remarkable performance of the current particle. Updating of particle position During the updating process of the particle swarm, particle movement was realized by tracing the best position of individuals during the evolution and the best position of all particles in the current particle swarm. The amplitude of position change was mainly determined by the advancing direction and distance. The advancing direction, distance, and positional movement of the ith particle in the particle swarm can be expressed as:
Vi = wVi + c1 r1 (Pbest Xi = Xi + Vi
Xi ) + c2 r2 (Gbest
Xi ) (11)
where Vi and Vi are the displacement vectors of the previous and current iterations of the i-th particle, respectively; Xi and Xi are the current and target positions of the i-th particle, respectively; w is the inertia weight coefficient; c1 and c2 are acceleration constants; r1 and r2 are random coefficients with range of [0, 1]; Pbest is the best position of the i-th particle during evolution; and
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Table 1 Material parameters. Material
Density(kg/m^3)
Elastic modulus(Pa)
Poisson's ratio
Q235
7850
2.1 × 1011
0.3
Gbest is the best position of all particles in the particle swarm [18]. Perez's work demonstrated that when the acceleration constant and inertia weight coefficient satisfied Eq. (12), the algorithm obtains improved convergence characteristic [19]. 0 < (c1 + c2) < 4 ( c1 + c 2 ) 2
1
(12)
The inertia weight coefficient w presents dynamic changes during the searching process, which enables the algorithm to achieve high global searching capability in the early period and good local search capability after the searching. The calculation formula is:
w = wmax
(wmax
wmin ) n n max
(13)
where wmax is the maximum inertia coefficient and is often set as 0.8; wmin is the initial inertia coefficient, which is calculated from c1 and c2 . n is the number of particle generation, and n max is the maximum generations. 7) Terminal conditions Many terminal conditions of particle swarm are available, including termination based on number of operations, time of operation, and updating distance changes. To approach the real solutions, the number of iterations or iterative times can be set as a large one. In this work, the highest number of iteration and the highest iterative time were set at 100 and 100 s, respectively. Particles evolution would stop when any one of these two parameters exceeded the limits. 4. Finite element simulation of time reversal damage positioning 4.1. Finite element model The simulation model was constructed in ABAQUS CAE™ application, which was with the dimensions of 500 mm (length) × 500 mm (width) × 5 mm(thickness). The material was set as carbon steel, which is commonly used in the offshore platform, and the grade of the carbon steel is Q235. The material parameters were listed in Table 1. To obtain the signals of intact and damaged state, two simulation models was constructed, as shown in Fig. 3(a) and (b). The red marks represent the position of the excitator, and the dark marks represent that of the sensor. There are three pairs of sensors mounted on the surface of the plate-like model. Among the three pairs of sensors, namely, S1–S4, S2–S5, and S3–S6, the former is the exciter and the latter is the sensor, as listed in Table 2. The coordinates of the excita The blue mark represents the damage which is a blind hole with 3 mm diameter and 2 mm depth, located at the coordinate (200,240). There are two sorts of simulation solver for dynamic problem in ABAQUS CAE™ application, explicit and implicit. Although piezoelectric element are not available in ABAQUS CAE™ explicit, researchers could apply an equivalent load instead of a voltage as actuation loading. Furthermore, explicit solver is more economical and accurate for wave propagation simulation. Meanwhile, there is no convergence problem in explicit solution, because it does not need iteration. Thus, the explicit procedure is strongly recommended in the previous study [20]. To avoid the influence of reflected wave from the model boundaries, the elements of the model boundaries were set as infinite element type (CIN3D8), as shown in Fig. 3. In the numerical simulation, the size of the grid will affect the final calculation result. The smaller the grid size is, the more accurate the result will be. However, with the decrease of the grid size, the computational consumption will increase exponentially. Therefore, element size is usually limited into one-tenth of the wavelength. The grid size in this study is set as 1 mm. According to the Nyquist's theory, the acquisition frequency must be twice more than the maximum frequency component at least. The central Table 2 Coordinates of the sensors. Sensor
S1
S2
S3
S4
S5
S6
Coordinate
(150,150)
(350,350)
(350,150)
(165,150)
(350,335)
(365,150)
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Fig. 3. Simulation model.
frequency of the exciting signal is 150 kHz, and the acquisition frequency is 10 MHz in the following experiment. Thus, the time step in the simulation is set to be 0.1 μs in accordance with the experiment. 4.2. Finite element simulation In the simulations, the excitation signal is a five-cycle sinusoidal signal modulated with Hanning window, as shown in Fig. 11(a).
y = Asin (2 fc ) × (1
(14)
cos (2 fc /N ))
where A is the amplitude of the sinusoidal signal, N denotes the cycle number of the modulated signal, and fc represents the time serial ranging from zero to N. In both the intact and damaged model, when the exciter S1 is actuated, only the sensor S4 is assigned to collect the wave motion. The sensor pairs S2–S5 and S3–S6 work in the same way with S1–S4. In the intact model, there is no reflected wave, as shown in Fig. 4, and the signals collected by S4, S5 and S6 are called as the reference signals. In the damaged model, the signals collected by S4, S5 and S6 are called as the reference signals. As shown in Fig. 5, there are obvious reflected wave in the damaged model, which will be collected by the sensors.
Fig. 4. The wave field of intact model. 6
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Fig. 5. The wave field of damaged model.
4.3. Analysis of simulation results From the signals collected by S4, S5 and S6, there is a direct wave packet with the same shape in both the intact and damaged state, which appears at the same time. While, there is a minor wave packet after the direct wave packet, which appears only in the damaged state, as shown in Fig. 6. The different part is caused by the reflected wave, which could be obtained by making a subtraction of the signal of intact state from that of damaged state. Usually, affected by the difference between the intact and damaged model, the direct waves in the intact and damaged state will differ with each other slightly. Thus, there is still a residual direct wave packet as shown in Fig. 7. Nevertheless, the residual wave packet could be easily eliminated by a limited width window. The subtraction calculation results after removing the residual wave packet were called as the scattered wave signals, as shown in Fig. 8. The damage positioning was investigated by using the scattered wave of damage after the elimination of the disturbance wave. The proper length of scattered wave signals was of the essence in localizing the potential damage. The duration of the signals must satisfy the following equation.
T
2L max +t vwave
(15)
where L max is the maximum distance between the sensor and the farthest point in the searching field, vwave is the group velocity of the guided wave, t is the duration of the excitation signal in Eq. (15). The duration of the scattered signals was set to be 300 µs according to the criterion above. Thus, the time reversal signals of the scattered wave of damage were conducted, as shown in Fig. 9. The enveloped curve of the time reversal signals were excited virtually at their corresponding sensors. The virtual wave fields at different moments are shown in Fig. 10. When the duration of excited signal is 230 μs, all the three sensors have excited waves. After that, the three waves begin to superpose with each other, and the peak of those waves superpose only at the momont of 270 μs, which is called as the focusing time (see Table 2). In the traditional time reversal method, the virtual wave field at each iterative moment ought to be calculated. The position with the highest amplitude is taken as the damage position. Thus, when the iterative step of time domainis rather small, the traditional method is considerably time-consuming. Moreover, when the detecting region is much large, the situation of time consumption is worse. As listed in Table 3, the time consumption of traditional time reversal rises with the increase of searching area. Whereas, the PSO based time reversal consumes the same smaller time, no matter how large the searching area is. This is because the PSO algorithm is capacity of finding the optimal solution within limited iterative step. The PSO algorithm might be affected by acceleration constants. In our previous study, the effect of acceleration constants has been analysed [15]. With the proper values, the PSO based time reversal offers a stable positioning results. Although both positioning methods achieve small positioning error, as listed in Table 4, the PSO based method could accomplish the damage positioning rapidly under a large searching area. No matter how large the searching area is, the localization error remains the same, which represent that changing the sensor location will not affect the accuracy of this method. 5. Time reversal damage positioning experiment 5.1. Experimental process The experimental specimen is with the same material and dimension with the simulation model. Six piezoelectric wafers were
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Fig. 6. The damage and reference waveform. 8
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Fig. 7. The subtraction calculation result.
mounted on the specimen surface. Among them, S1–S4, S2–S5, and S3–S6 formed the sensor pairs, the intact specimen is shown in Fig. 11. The experimental setup consisted of a Tektronix AFG1022 function/arbitrary waveform generator, a DS-2B(Beijing Softland Times Scientific &Technology Co.Ltd.) data collecting instrument and charge amplifiers. The piezoelectric wafers has the dimension of φ 10 mm. The dielectric constant of PZT sensor in the z direction is far larger than that in the x and y directions, and the relative dielectric constants in the x and y directions are approximately equal to 0. Therefore, only the displacement in z direction is extracted. The piezoelectric wafers S1, S2 and S3 are connected with the waveform generator, and the piezoelectric wafers S4, S5 and S6 are connected with the data collector by the charge amplifiers. Meanwhile, waveform generator is connected with the collector to record excitation signals and provide a start point of data collection. After the reference guided wave signals in the intact specimen are acquired, a through hole with φ 4 mm diameter is created by using an electric drill at (200, 240) of the specimen surface as shown in Fig. 12. 5.2. Experimental results The reference signals collected by piezoelectric sensor under the intact state was subtracted by that under the damaged state, then the reflected wave signal caused by damage is obtained. When sensors S1 and S4 are used as the exciter and receiving sensor, respectively, the waveform data were normalized after the subtraction operation (Fig. 13). In the subtraction result, there are commonly error packet caused by the direct wave, scattered wave caused by damage, reflected wave caused by the specimen boundary and the damage reflected wave caused by scattered wave and boundaries. The direct wave of reference and damaged signals may differ with each other due to sampling and system errors, thus, error packet could not be eliminated completely. However, the waveform and the frequency spectrum had remarkable differences with the other wave packets, thereby, the error packet could be eliminated easily. Scattered signals arrived at the sensor after the direct wave, and this wave packet was similar to the excitation signals. Specimen boundaries will reflect waves, and the reflected wave could be eliminated completely, because the nearest reflection routine of S1–S4 is known. The scattered wave of damage would be reflected upon the specimen boundaries, thereby reflected wave occurs after the reflected wave. The collected signal except the part of damage scattered wave was eliminated to avoid the influences of the other wave packets, as shown in Fig. 14. The wave field of the specimen region at different times are shown in Fig. 15. Only at the moment of focusing time, the amplitude of superimposed wave field will achieve the maximum value. In the study, the resolution of wave field in both the vertical and horizon direction is 1 mm. If the searching area are 0.25 m2, 1 m2, and 16 m2, the traditional time reversal will consume more time with the increase of the area. The same with the simulation, the POS based time reversal will consume the same time no matter how much the searching area is, as listed in Table 5. It means that the POS based time reversal will achieve the damage localization within some iterative step. The localization coordinate of traditional time reversal approach is (201,236), which is with a localization error of 4.1 mm to the real damage coordinate of (200,240). No matter how large the searching area is, the localization coordinate remain (201,236). This is because traversing search strategy of traditional approach is capable of finding out the global optimal solution. The PSO based time reversal approach localize the damage at the coordinate of (198,229), which is with a localization error of 11.2 mm. 5.3. Localization error analysis The localization accuracy of ultrasonic detection is affected by the half-wave length, thus, localizing the damage with no localization error is impossible. In this study, the excitation signal with the central frequency 150 kHz will generate a guided wave with the wave length of 20.4 mm in the 5 mm thick plate. Thus, the localization accuracy limit is 10.2 mm. If there is an extra positive
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Fig. 8. The scattered wave signals.
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Fig. 9. The time reversal signals.
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Fig. 10. Virtual wave fields.
Fig. 11. Experimental specimen.
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Table 3 Comparison of time consumption. Searching area (Width x Length)
Time reversal (s)
PSO based time reversal (s)
0.5 × 0.5 m2 1 × 1 m2 4 × 4 m2
0.7 1.1 12.8
0.3 0.3 0.3
Table 4 The damage coordinate of different methods. Searching area (Width x Length)
Time reversal
Localization error of time reversal (mm)
PSO based time reversal
Localization error of PSO based time reversal (mm)
0.5 × 0.5 m2 1 × 1 m2 4 × 4 m2
(206, 238) (206, 238) (206, 238)
6.3 6.3 6.3
(205, 238) (205, 238) (205, 238)
5.4 5.4 5.4
Table 5 Consumption of calculation time. Searching area (Width x Length) 0.5 × 0.5 m 1 × 1 m2 4 × 4 m2
2
Time reversal (s)
PSO based time reversal (s)
1 1.6 13.3
1.9 1.9 1.9
Fig. 12. Artificial damages.
Fig. 13. Subtraction result of S4.
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Fig. 14. Waveform amplitude difference of S4.
Fig. 15. Wave fields at different moments.
error, the localization error will larger than the localization limit. On the contrary, an negative error will cause a little localization error. 6. Conclusions In this study, the PSO algorithm based time reversal positioning method was proposed, which was capable of localizing the damage online. At first, a mathematical model of time reversal damage positioning method was constructed and the mathematical expression of the virtual focus field was then established. The time reversal needs to calculate the wave field at each moment, which is highly time-consuming. Thus, PSO method is combined with the traditional time reversal, which is capable of finding out the optimal localization within limit iterative step. The finite simulation and experimental case prove that the proposed method can position the damage rapidly even in the large-sized structure. It is a solution for the offshore platform to monitor its health online. Moreover, the arrangement of sensor pairs can effectively avoid the influence of direct wave on the scattered wave of damage. The focusing moment is calculated by the group velocity of the central frequency component of guided wave, which could influence 14
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positioning to a certain extent. Therefore, future studies should pay attention to the acquisition of focusing moment to increase positioning accuracy. Funding The authors acknowledge the support by the National Science Fund for Distinguished Young Scholars (No. 51625902), the National Natural Science Foundation of China (No. 61501418), and Qingdao Source Innovation Program (No. 18-2-2-69-jch). Conflicts of interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgment The valuable comments from the anonymous reviewers are highly appreciated. References [1] Liu FS, Li HJ, Lu HC. Weak-mode identification and time-series reconstruction from high-level noisy measured data of offshore structures. Appl Ocean Res 2016;56:92–106. [2] Mieloszyk M, Ostachowicz W. An application of Structural Health Monitoring system based on FBG sensors to offshore wind turbine support structure model. Mar Struct 2017;51:65–86. [3] Hillis AJ, Courtney CRP. Structural health monitoring of fixed offshore structures using the bicoherence function of ambient vibration measurements. J Sound Vib 2011;330(6):1141–52. [4] Liu FS, Chen JF, Qin HD. Frequency response estimation of floating structures by representation of retardation functions with complex exponentials. Mar Struct 2017;54:144–66. [5] Liu FS, Gao SJ, Han HW, Tian Z, Liu P. Interference reduction of high-energy noise for modal parameter identification of offshore wind turbines based on iterative signal extraction [J]. Ocean Eng. [6] Wilcox P, Lowe M, Cawley P. The effect of dispersion on long-range inspection using ultrasonic guided waves. NDT E Int 2001;34(1):1–9. [7] Clarke T, Cawley P, Wilcox PD, et al. Evaluation of the damage detection capability of a sparse-array guided-wave SHM system applied to a complex structure under varying thermal conditions[J]. IEEE Trans Ultrason Ferroelectr Freq Control 2009;56(12):2666–78. [8] Staszewski WJ, Mahzan S, Traynor R. Health monitoring of aerospace composite structures Active and passive approach. Compos Sci Technol 2009;69(11):1678–85. [9] Yu LY, Giurgiutiu V. In situ 2-D piezoelectric wafer active sensors arrays for guided wave damage detection. Ultrasonics 2008;48(2):117–34. [10] Guan JF, Shen ZH, Ni XW, et al. Numerical simulation of the reflected acoustic wave components in the near field of surface defects. J Phys D Appl Phys 2006;39(6):1237. [11] Fink M. Time reversal of ultrasonic fields. I. Basic principles. IEEE Trans Ultrason Ferroelectr Freq Control 1992;39(5):555–66. [12] Zhu R, Huang GL, Yuan FG. Fast damage imaging using the time-reversal technique in the frequency–wavenumber domain. Smart Mater Struct 2013;22(7):075028. [13] Liu L, Yuan FG. A linear mapping technique for dispersion removal of Lamb waves. Struct Control Health Monit 2010;9:75–86. [14] Chen CL, Li YL, Yuan FG. Development of time-reversal method for impact source identification on plate structures. Shock Vib 2013;20(3):561–73. [15] Zhao CL, Sun XB, Sun SL, et al. Fault diagnosis of sensor by chaos particle swarm optimization algorithm and support vector machine. Expert Syst Appl 2011;38(8):9908–12. [16] Mu WL, Qu WS, Liu GJ, et al. Acoustic emission beamforming localisation approach based on particle swarm optimizations. Insight 2018;60:575–80. [17] Jordehi AR, Jasni J. Parameter selection in particle swarm optimizations: a survey. J Exp Theor Artif Intell 2013;25(4):527–42. [18] Kennedy J. The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Trans Evol Comput 2002;6(1):58–73. [19] Perez RE, Behdinan K. Particle swarm approach for structural design optimization. Comput Struct 2007;85(19–20):1579–88. [20] Mu WL, Sun JG, Liu GJ, Wang SQ. High-resolution crack localization approach based on diffraction wave. Sensors 2019;19(8):1951.
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