Automated Operational Modal Analysis and its Applications in Structural Health Monitoring
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Lin Cheng, Fei Tong, Jie Yang, Dongjian Zheng, "Online Modal Identification of Concrete Dams Using the Subspace TrackingBased Method", Shock and Vibration, vol. 2019, Article ID 7513261, 18 pages, 2019. https://doi.org/10.1155/2019/7513261
Online Modal Identification of Concrete Dams Using the Subspace TrackingBased Method
Abstract
To investigate the timevarying dynamic characteristics of concrete dams under the excitation of large earthquakes for online structural health monitoring and damage evaluation, an online modal identification procedure based on strongmotion records is proposed. The online modal identification of concrete dams is expressed as a subspace tracking problem, and a newly developed recursive stochastic subspace identification (RSSI) method based on the generalized yet another subspace tracker (GYAST) algorithm, which exploits both the accuracy of the subspace identification and fast computational capability, is used to extract the timevarying modal parameters of concrete dams during earthquakes. With the simulated vibration response records, a numerical example is used to verify the accuracy, robustness, and efficiency of the proposed GYASTbased, timevarying modal identification method. Then, the realistic strongmotion records of the Pacoima arch dam are analysed using the proposed modal identification procedure, and the timevarying characteristics of the concrete arch dam during three different earthquakes are analysed.
1. Introduction
The modal parameters of concrete dams, including the natural frequencies, damping ratios, and modal shape vectors, extracted from earthquake response records can be used for structural antiearthquake capacity analysis, health monitoring, and postearthquake structural damage evaluation. These parameters have obvious physical meanings and represent the practical dynamic characteristics of structures. Some researchers have made efforts to address the modal identification problem of concrete dams based on earthquake response records [1–6]. However, the modal identification methods adopted in the past studies are restricted to linear systems and stationary processes, and using such methods would require the assumption that the damreservoirfoundation system is linear and time invariant for the duration of a vibration measurement. The damreservoirfoundation system under the excitation of large earthquakes may present some timevarying characteristics, resulting from the timevarying character of the material parameters, boundary conditions, damreservoirfoundation interactions, and structural damage [7]. Using the offline modal identification procedure to extract the constant modal parameters cannot track the timevarying dynamic properties of a structure during a large earthquake, which may be important information for online structural health monitoring. Therefore, the online modal identification method of concrete dams is an important research topic.
The structural online modal identification problem is a research topic with many challenges that still attract significant attention for its significance in aerospace engineering [8], mechanical engineering [9], and civil engineering [10], among other fields. For timevarying structures, the assumption of the classic modal identification methods that the structure is a timeinvariant system is no longer applicable. Thus, Liu [11] proposed the concept of “pseudomodal parameters,” which use the timefreezing concept. Three types of methods have been developed for timevarying modal identification, i.e., the time series model method and the recursive stochastic subspace identification (RSSI) method in the time domain and the timefrequency decomposition method in the timefrequency domain [12, 13]. For the time series model method, Gong [14] used adaptive forgetting through multiple models (AFMM) to track the timevarying modal parameters of tall buildings during a large earthquake and evaluated the structural health according to the modal identification results. Poulimenos and Fassois [15] performed an important literature review of different timevarying modal identification methods using different nonstationary time series models and point out their pros and cons. Klepka and Uhl [16] used the timefrequency domain decomposition technique to analyse the structural vibration response and identified the modal parameters. Goethals et al. [17] and Nezam Sarmadi and Venkatasubramanian [18] used the projection approximation subspace tracking (PAST) algorithmbased RSSI (PASTRSSI), and Loh et al. [19, 20] used the extended instrumental variable version of the PAST algorithmbased RSSI (EIVPASTRSSI) to carry out the online modal identification of a power system, an inflight flutter, a tall building, and a television tower. Among these methods, the RSSI method has attracted the most attention. The advantage of the RSSI method is mainly reflected in its strong numerical stability and strong robustness to noise interference. Therefore, it is suitable for the online modal identification of structures under random vibration excitation such as earthquakes. Although the online modal identification problem has been studied in many other fields, very few research works have been reported that use the online modal identification for concrete dams [21]. The damreservoirfoundation system is characterized by its large size, large number of degrees of freedom, and closely spaced modes, and the vibration measurement of concrete dams is usually performed with a low signaltonoiseratio (SNR) as a result of external disturbances. This brings great difficulties to the online modal identification of concrete dams.
In this work, the RSSI method is used to extract the modal parameters of concrete dams based on the strongmotion records of the structure. The online version eigenvalue decomposition (EVD) is solved as a subspace tracking problem using the generalized yet another subspace tracker (GYAST) algorithm [22]. GYAST is a newly proposed advanced algorithm which makes several robust modifications of the original subspace tracking algorithm. The YAST algorithm relies on an interesting idea of optimally extracting the updated subspace weighting matrix in each step. In the GYAST algorithm, computation reduction of optimal subspace extraction is achieved by an approximation. It caused the GYAST has the advantages of a low computational complexity, high computational efficiency, and strong robustness. Combining GYAST with RSSI is expected to improve the accuracy and efficiency of the online modal identification. The online timevarying modal identification procedure based on the GYASTRSSI is verified using a numerical example to evaluate the algorithm identification accuracy, robustness, and computation efficiency. Finally, the timevarying modal identification of the Pacoima arch dam is carried out using the proposed timevarying modal identification procedure and the strongmotion records of three different earthquakes to track the timevarying characteristics of the arch dam during earthquakes.
2. Linear TimeVarying Modal Identification Using the GYASTBased RSSI
2.1. TimeVarying State Space Expression of the DamReservoirFoundation System
In the design stage of a concrete dam, the structure is generally assumed to be linear, and the observed vibration response of the structure is stationary. However, for some reasons, the damreservoirfoundation system may be timevarying. As shown in Figure 1, the dynamic constitutive model of the concrete and the base rock may be nonlinear. Additionally, the joints on the dam body may close or open with variations in the amplitude of the earthquake excitation, which will change the stiffness of the connections between different dam monoliths. The effects of the structurefoundation interaction may also introduce some timevarying factors into the dynamic features of concrete dams [23]. Moreover, a damage structure normally exhibits nonlinear dynamic behaviours and a timedependent stiffness, and the damping variations over time lead to the timevarying modal parameters of the system. In this study, the damreservoirfoundation system is assumed to be a linear timevarying system. The “frozentime” assumption that consists of modelling a linear timevarying system as a piecewise linear timeinvariant system is adopted. The parameters will be assumed to vary in a stepwise way; i.e., they will change abruptly at instants separated by small time intervals and will remain constant during these time intervals. If the time intervals are sufficiently small, this model can be considered an approximation of the continuously varying case [24]. The expression of this system will be discussed in this section.
It should be noted that the material characteristics, the boundary conditions, the damping, etc. may be affected by some environmental variables, such as the water level, temperature, and humidity. The impact of these variables will not be taken into account in this study since the observation time of an earthquake response record is not very long.
The equation of motion for an degreesoffreedom timevarying linear system with timevarying structural parameters can be expressed aswhere , , and are the time relevant mass matrix, damping matrix, and stiffness matrix, respectively. , , and are assumed to slowly change with time. For the damreservoirfoundation system, the change of mass is usually not taken into account, and thus, the mass matrix is constant. is the displacement response of the structure, and is the external excitation force.
Equation (1) is a system of differential equations with timevarying coefficients. The response can be found by solving the differential equations using initial conditions, but in general, there is no closedform solution set to these equations. Thus, an approximation is introduced to solve the vibration problem. In other words, the instantaneous modal parameters are obtained at each time instant under the slowvarying assumption of the structural system. According to this assumption, the continuous “frozen” timevarying structure is made up of the mass, damping, and stiffness of every instantaneous moment , which can be expressed as follows:where is the end time of the time series of the vibration response signal, represents the timeinvariant structure of , and is a set of the timefreezing descriptions of the linear timevarying structure. Analytically, during the time interval , the parameter matrices of equation (2) have entries with constant values. When , the matrices may suddenly change from , , and to , , and since the system during the time interval is time invariant. The state space expression for the timevarying system at discrete time intervals is as follows [26]:where and are the discrete state space matrix and observation matrix of the time interval , respectively. is a vector of state variables, and the observation vector is a vector of channels and can be any or some of the three types of structural vibration responses, i.e., the displacement, velocity, and acceleration. is the stochastic excitation, and is the observational noise.
The expression of the structural vibration response for a timevarying system during a time instant is discussed first. Since the system during the interval is assumed to be time invariant, a basic expression for the linear vibration system is applicable during the time interval.
To identify the complex modal shape vector, the doublesized mixing model is adopted [27]. It is well known that the shift of the phase angle between the displacement and velocity and the velocity and acceleration is 90°. The acceleration response is generally measured for the structural strongmotion response observation system. Since the physical parameters vary slowly with time, component is an asymptotic signal. To track the timevarying characteristics of a structure with degreesoffreedom, the Hilbert transformation is performed to obtain the pseudoresponse of the structure with a 90° transferred phase angle . Then, in the discrete time domain, the augmented response vector can be obtained as follows:
Equation (4) is equivalent to the representation of the original signal in the analytical form as follows:where and are the real and imaginary parts of the analytical signal, respectively, and is the imaginary unit; is the mathematical operator of the Hilbert transformation.
Based on the derivation by McNeill [28] and Masuda et al. [29], the free acceleration vibration of a structure can be expressed aswhere represents the kth time interval, i.e., . The complex modal acceleration response matrix is defined as and is composed of the modal acceleration responses of all the activated modes. The terms and are the real and imaginary parts of the modal acceleration response, respectively. is the real part of a number.
If the acceleration response is measured and the output selection matrix is , the analytical expression of the acceleration response signal is
Based on the coordinate transformation shown in equation (6), the augmented response vector shown in equation (4) is expressed aswhere and .
Assume the observational noise components are independently identically distributed with a variance . Then, the covariance matrix of can be expressed aswhere the superscript , denotes the conjugate transpose of a matrix and is the covariance matrix of the modal acceleration responses. For a structure system with weak damping,
After the discrete sampling of the physical acceleration of the channels at identical time intervals, a timedelayed vector , which is composed of and its timedelayed data, can be written as follows:in which the number of time delays should satisfy .
Then, the Hankel matrix can be defined as follows:
Based on the derivation of the free acceleration response shown in equation (9), the Hankel matrix can be decomposed into the following form:where . are the eigenvalues of the continuous system matrix.
For the free vibration response of a structure system without damping, the modal acceleration responses of the different modes are independent of each other. However, the damping effect and the disturbance of the noise result in the matrix being approximately a diagonal matrix. Since the damping of concrete structures is usually weak, the error caused by the approximation is acceptable.
At the time instant , new observational data are available, and the Hankel matrix should be updated online. One way to update the Hankel matrix is called the exponential forgetting method. Using this matrix, the Hankel matrix at the time is expressed asin which is the forgetting factor, with a suggested value between 0.995 and 0.999 [30].
2.2. Modal Subspace Tracking Using GYAST
For each time step, the EVD of (where is a square matrix) should be performed at each time instant as follows:
If contains the first dominant eigenvectors of , it can be taken as the dominant signal subspace, which represents the contributions of different modes. is a diagonal matrix that is composed of the first n dominant eigenvalues of . and correspond to the remaining eigenvectors and eigenvalues. However, the total computation time will then be very large and is not suitable for online applications. Thus, the online version of the EVD algorithm should be adopted. Since only the left part of the EVD is needed to retrieve an estimate of the observability matrix, an economical (socalled thin) EVD, which computes only that left part, can be used. Using the subspace tracking procedure, the GYAST, in each time step, a sufficient approximation of the dominant subspace, namely,can be obtained. It has been proven that, in a stationary situation, the desired steadystate weights can be obtained by maximizing the mean square value of the output of the combiners, given byin which the operator trace stands for the trace of a matrix.
To ensure the orthogonality of , a maximization optimization problem with constraints is expressed as follows:
To reduce the computational cost of updating online, the ideal subspace projection (SP) type algorithm limits the search space to the range space of plus one additional search direction, i.e., . Then, the dimensional range space of is found as a subspace of the dimensional space spanned by :
If is an orthonormal basis of the dimensional range space , then can be expressed asin which .
Then, replacing equation (20) with the optimization problem shown in equation (18), the following new optimization problem is obtained:in which the matrix .
For the GYAST algorithm, an approximation is introduced that calculates the modal acceleration responses as
Then, the orthonormal basis of the augmented dimensional range space is given bywhere is the unitnorm variant of , is the complement of the orthonormal projection of , and is the L_{2}norm of .
Define the matrix . Based on equation (14) and equations (22) and (23), can be deduced asin which , , and .
For the subspace tracking algorithm, GYAST, to track the signal subspace of , four basic steps must be performed, which are (i) calculating the orthonormal basis of the range space , (ii) constructing the matrix , (iii) conducting the online updating of , and (iv) conducting the online updating of . Using equations (14) and (22)–(24) and the following equations (25a)–(25k), these calculations can be realized online in a recursive way, and can be updated online for each time instant [23, 31]:
2.3. Flowchart of the Online Modal Identification Method Using the GYASTRSSI
The proposed online modal identification procedure based on the GYASTRSSI is shown in Figure 2. Some basic steps of this procedure are summarized as follows:(1)Initialize some parameters of the procedure, such as the number of time delays p, the system order n, the forgetting factor β, and the initial data length N_{1}. The system order n can be determined using the SVD spectrum method. The number of time delays should satisfy 2pl > n. The forgetting factor β is set within the range of [0.995–0.999]. Through some numerical experiments, the choice of N_{1} will have an effect on the calculation result of the initial Hankel matrix, but considering that GYAST is an iterative algorithm, the result of H(N_{1}) only affects the results of the previous steps. Here, for convenience, set and k = N_{1} and use the initial strongmotion records of l channels with data length N_{1} to calculate the initial Hankel matrix as .(2)GYAST is a recursive subspace tracking algorithm, and it should first be initialized. Perform EVD on the Hankel matrix and initialize the orthonormal matrix as follows:(3)Set k = k + 1 and update using the GYAST algorithm to calculate the orthonormal basis , , , etc., using equation (14) and equations (22)–(24).(4)Since , the two matrices have the same eigenvalues and eigenvectors, and so we set . From the above recursive procedures, the orthonormal matrix can be updated. Then, at the kth time interval, the discrete state space matrix A_{k} and the observation matrix G_{k} are realized using the following expressions:where and .(5)After we obtain the modal identification results at time instant t_{k}, some meaningless mathematical poles should first be discarded. If the identified natural frequencies and damping ratios at time instant t_{k} satisfy the following criteria, then mode i is deemed to be a real structural mode:in which f_{min} and f_{max} are the minimum and maximum natural frequencies of the active modes of the structure and ξ_{min} and ξ_{max} are the minimum and maximum damping ratios.(6)Continue with step 3∼step 5 of the above calculation procedure until reaching the end of the strongmotion records.
3. Examples
3.1. Numerical Example
Using the massspringdamper system with 4 degreesoffreedom shown in Figure 3, the timevarying modal identification method proposed in this work is validated for its identification accuracy, robustness, and computation efficiency. Some parameters of the system are set as follows: m_{1} = m_{2} = m_{3} = 1.0 kg, m_{4} = 0.9 kg, k_{3} = 7000 N/m, k_{2} = k_{4} = 8000 N/m, c_{1} = c_{2} = 0.6 N·s^{2}/m, and c_{3} = c_{4} = 0.55 N·s^{2}/m. The timevarying stiffness coefficients k_{1} are simulated as follows. The time intervals are in seconds. Case 1: abrupt change of k_{1}: Case 2: continuous linear change of k_{1}:
The initial state of the system is zero, and the support excitation is simulated using a white noise signal. The structural vibration response is obtained by adding some white noise to the calculated responses using a Runge–Kutta algorithm (using MATLAB code ODE45). The sampling frequency of the simulated structural vibration response is 1000 Hz. Figures 4 and 5 show the support excitation and the acceleration response of m_{1} for two simulation cases of k_{1}.
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Setting β = 0.995, , and and using the simulated structural vibration responses (SNR = 40 dB) of case 1 and case 2, the timevarying modal identification results for the frequencies and damping ratios are shown in Figures 6 and 7. The identification accuracy of the frequencies is sufficient, but the identification accuracy of the damping ratios needs to be further improved.
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The forgetting factor is an important parameter in the GYASTRSSI method proposed in this paper. To verify its influence on the recognition result, different values of β are selected for a parameter sensitivity analysis. Under simulation case 1, the analysis results are shown in Figure 8. It can be seen from Figure 8 that when β is in the interval of [0.995 0.999], the result does not change much. In this example, the result of β = 0.995 is better.
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Based on the simulated vibration response (SNR = 40 dB), the identification results for frequencies f_{1} and f_{2} of mode 1 and mode 2 using the EIVPASTRSSI and GYASTRSSI methods are plotted in Figures 9 and 10 to make a comparison. Figures 9 and 10 show that the difference between the identification results of the EIVPASTRSSI and GYASTRSSI methods is small, but the identification results of GYASTRSSI are more stable.
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To verify the robustness of the proposed timevarying modal identification method, simulated vibration response signals with different SNRs are used to perform the timevarying modal identification. The modal identification results of the first two frequency orders f_{1} and f_{2} for simulation cases 1 and 2 are shown in Figures 11 and 12, respectively. Figures 11 and 12 show that, with the increase in the noise level (decrease in the SNR), the identification accuracy of the frequencies is still very high.
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The modal assurance criterion (MAC) indexes between the identified modal shape vectors and the theoretical modal shape vectors are calculated and shown in Figure 13. In addition to some sudden changes in the MAC value at 4 s, 8 s, and 12 s, which is caused by the instability of the algorithm, the identification results of the modal shape vectors indicate the sufficient accuracy of the GYASTRSSI method of tracking the timevarying structural modal shape vectors.
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To evaluate the identification accuracy of the proposed timevarying modal identification method, the average relative error (ARE) of the frequencies is calculated and is shown in Table 1. The ARE of the certain order frequency is defined asin which and are the identification results and the theoretical value of the certain order frequency at time instant m, respectively.

From Table 1, it can be seen that the identification accuracy is good for the two timevarying modal identification methods since the maximum ARE is 2.11%. With the decrease in the SNR, the identification accuracy for the frequencies has become low. Moreover, the GYASTRSSI has better identification than the EIVPASTRSSI method in most cases.
3.2. TimeVarying Modal Identification of the Pacoima Arch Dam
The Pacoima dam is a 113 m high arch dam located near Los Angeles, California. Since the Pacoima arch dam is near the famous San Fernando earthquakeprone area, it has experienced several earthquakes of different magnitudes. The dam experienced two severe earthquakes: the San Fernando earthquake (1971) and the Northridge earthquake (1994). Due to the damage caused by the San Fernando earthquake and the opening of the joints near the thrust block at the left abutment, rehabilitation operations were performed afterward, and an array of 17 accelerometers were installed on the dam in 1977 [32]. The arrangement of the 10 acceleration sensors with 17 measurement channels is shown in Figure 14. Among the 17 measurement channels, channels 1∼8 are located on the dam body and channels 9∼17 are located on the base rock. The measurement directions of channels 1, 2, 5, 6, 7, 8, 9, 12, and 15 are in the radial direction; channels 4, 11, 14, and 17 are in the tangential direction; and channels 3, 10, 13, and 16 are in the vertical direction. After the Northridge earthquake, the strongmotion observation system was updated, and the strongmotion responses of the three different earthquakes were recorded. The information of the three earthquakes is shown in Table 2.

The strongmotion records of the Pacoima arch dam for channels 1, 2, 5, 6, 7, and 8 in the radial direction during the Newhall earthquake are shown in Figure 15. The sampling frequency of the strongmotion monitoring data is 200 Hz. Thus, the cutoff frequency of the data is 50 Hz.
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Timeinvariant modal identification results of the Pacoima dam using different methods are shown in Table 3. These identification results are used to make a comparison with the following timevarying modal identification results.
Channels 9∼17 are located on the base rock. The strongmotion records of channel 9 in the radial direction during the three major California earthquakes and the corresponding power spectrums are shown in Figure 15. As seen from Figure 15, for the San Fernando earthquake and the Chino Hills earthquake, the dominant frequencies of the earthquake excitations are less than 3.32 Hz. It can be seen from Table 3 that the dominant frequencies of the earthquake excitation are less than the firstorder modal frequency of the dam. However, for the Newhall earthquake, the range of the dominant frequencies of earthquake excitation is much wider than those of the other two earthquakes. Therefore, for the Newhall earthquake, the final modal identification results of the dam should not include these dominant frequencies of excitation shown in Figure 16(f).
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By setting β = 0.995, , and N_{1} = 200, the structural frequencies and damping ratios are identified using the proposed timevarying modal identification method based on GYASTRSSI, and the identification results are shown in Figures 17–19. The vibration response of channel 1 is plotted to show the variation of the modal parameters with the acceleration amplitude. From Figures 17–19, the timevarying characteristics of the arch dam during earthquakes are clearly observed, especially when the amplitude of the acceleration response is relatively large. According to Figures 17–19, the frequency after the earthquake tends to be stable, indicating that none of the three earthquakes may cause any damage to the dam.
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For the three earthquakes, the frequencies of the dam show a decreasing trend after the earthquake, which may be caused by the solidwater interaction and the reduction of the structural stiffness. As was shown in some previous works, the solidwater interaction can be simulated by the simple addingmass method. The additional mass that is added to the structural system will reduce the frequencies of the damreservoirfoundation system. However, the decrease in the frequencies can also be caused by the reduction of the structural stiffness during earthquakes, which needs to be studied in the future. When the amplitude of the acceleration is very low, the modal identification results tend to be time variant. These timevarying modal identification results demonstrate that the dynamic characteristics of concrete dams during large earthquakes are timevarying indeed.
For four typical time instants, t = 20 s, 25 s, 30 s, and 35 s, the identification results of the modal shape vectors using the strongmotion records of the Newhall earthquake are shown in Figure 20. Figure 20 shows the timevarying characteristics of the arch dam’s modal shape vectors during the earthquake.
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4. Conclusions
The online modal identification method of concrete dams proposed in this work has been verified using a numerical example and an engineering example. The GYAST is a good tool to track the signal subspaces of different modes to reduce the disturbance of the observation noise and stochastic excitation since it shows good identification accuracy, robustness, and computation efficiency. The identification accuracy of the frequencies and modal shape vectors is sufficient, while for the damping ratios, the performance of the proposed online modal identification method still needs to be improved. Some possible ways to obtain a better identification result of the damping include improving the robustness of the subspace tracking algorithms, signal denoising, and adopting inputoutput modal identification methods. The timevarying modal parameters identified using the strongmotion records of concrete dams can be used as important features to study the real dynamic characteristics of structures during large earthquakes, which is our research direction in the future.
Data Availability
The strong earthquake records data used to support the findings of this study may be released upon application to the Center for Engineering Strong Motion Data at https://strongmotioncenter.org. The MATLAB code for the GYAST algorithm online to support the findings of this study is available from the Professor Mostafa ArjomandiLari [22]. The numerical simulation and other data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant nos. 51409205, 51279052, 51409018, and 51309189), the National Natural Science Foundation for Excellent Young Scientists of China (Grant no. 51722907), a project funded by the China Postdoctoral Science Foundation (Grant no. 2015M572656XB), and the Open Foundation of the State Key Laboratory of HydrologyWater Resources and Hydraulic Engineering (Grant no. 2014491011). The authors appreciate Professor Mostafa ArjomandiLari for providing the MATLAB code for the GYAST algorithm online. The authors also appreciate the Center for Engineering Strong Motion Data; all strong earthquake records used in this article are derived from this centre.
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Copyright © 2019 Lin Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.