AbstractPrompt identification of structural damage is essential for effective postdisaster responses. To this end, this paper proposes a deep neural network (DNN)–based framework to identify seismic damage based on structural response data recorded during an earthquake event. The DNN in the proposed framework is constructed by Variational Autoencoder, which is one of the self-supervised DNNs that can construct the continuous latent space of the input data by learning probabilistic characteristics. The DNN is trained using the flexibility matrices obtained by operational modal analysis (OMA) of simulated structural responses of the target structure under the undamaged state. To consider the load-dependency of OMA results, the undamaged state of the structure is represented by the flexibility matrix, which is closest to that obtained from the measured seismic response in the latent space. The seismic damage of each member is then estimated based on the difference between the two matrices using the flexibility disassembly method. As a numerical example, the proposed method is applied to a 5-story, 5-bay steel frame structure for which structural analyses are first performed under artificial ground motions to create train and test datasets. The proposed framework is verified with the near-real-time simulation using ground motions of El Centro and Kobe earthquakes. The example demonstrates that the proposed DNN-based method can identify seismic damage accurately in near-real-time.IntroductionExtreme loadings caused by natural or man-made disasters can introduce structural damage, which may lead to disproportionate outcomes. A failure to identify such damage may shorten the service life of the structural system or lead to the violation of the requirements for safety and functionality. Therefore, it is essential to implement a proper postdisaster inspection process that can assess the integrity of the structural system and detect potential damage.Generally, nondestructive inspections (NDI), such as acoustic, electromagnetic, and visual inspections, are often used to inspect infrastructures. NDI methods are considered more cost-effective than destructive testing and can detect both surface and embedded damages, such as cracks and corrosions, using a simple device. However, NDI methods are known to be labor-intensive and time-consuming because they are conducted manually by human inspectors. Relying on the inspector’s experience and subjective judgment, the NDI may produce inaccurate results (Liu et al. 2017) and hamper a prompt inspection of infrastructure systems, which is essential for effective postdisaster responses. The inefficiency of the inspection process, combined with the aging of infrastructures, such as deterioration, makes it difficult to allocate limited resources to the maintenance practice. Consequently, it becomes increasingly difficult to satisfy the short and long-term demand for integrity, safety, and resiliency of the structural system by using conventional NDI methods (An et al. 2019).To overcome these challenges, an inspection strategy exploiting numerous sensor data and measurements has been replacing conventional inspection methods. This strategy, generally called structural health monitoring (SHM), can deal with many important tasks related to the infrastructure system, including postdisaster damage identification. SHM-based damage identification practice generally includes four levels: (1) detecting the existence of damage; (2) localizing the damage; (3) quantifying the severity of the damage; and (4) predicting the remaining service life of the structure (Rytter 1993). In general, SHM processes are categorized into long- and short-term SHM (Dawson 1976). A long-term SHM aims to monitor the ability to fulfill the intended function based on information periodically updated to reflect the aging and degradation of the structural system caused by operational environments. On the other hand, a short-term SHM aims at quick identification of changes in the structural characteristics and providing information about the integrity of the structure in near-real-time.With rapid developments of sensing technologies and data analysis, vibration-based damage identification methods have been gaining much attention as an application of machine learning to SHM. These methods are built upon the fact that vibration characteristics, such as mode shape and natural frequency, can be affected by a change in structural characteristics, including mass and stiffness. Such methods have a great potential in detecting local damage or failures in the structural system immediately after a disastrous event in efforts to protect human lives and maintain or recover the serviceability of infrastructures. Recently, pattern recognition methods are being combined with the vibration-based damage identification approach to facilitate the short-term SHM. This is because such methods can utilize a pretrained model, such as a deep neural network (DNN) using features extracted from the vibration signals (Pathirage et al. 2018) and, thus, require low computational time to recognize the changes in vibration characteristics. This paper focuses on developing a new vibration-based near-real-time damage identification method for structures under earthquake excitations using DNN-based pattern recognition.During the development, we noted the fact that it is generally challenging to identify damage from the raw vibration signal, which is insensitive to the changes in structural characteristics. By contrast, modal properties, such as mode shape and modal strain energy, are relatively sensitive to structural damage. This is the reason why modal-based methods are widely used for damage identification. Because it is challenging to perform a forced-vibration test to obtain modal properties during the operation of an infrastructure system, the operational modal analysis (OMA), which aims to identify modal properties only with structural responses, is commonly used. There are several variations of OMA methods, such as the Ibrahim time domain (ITD; Ibraham and Mikulcik 1977), frequency domain decomposition (FDD; Brincker et al. 2000), time domain decomposition (TDD; Kim et al. 2005), and automated covariance-driven stochastic subspace identification (SSI-COV; Magalhaes et al. 2009). In this paper, the TDD method is used to obtain modal properties. In the TDD method, the size of the singular value decomposition (SVD) matrix is determined by the number of sensors, whereas, in most other OMA methods, the matrix size depends on the number of time or frequency samples. Therefore, the TDD method requires a low computational cost to obtain modal properties and, thus, is more suitable for the near-real-time damage identification than other OMA methods.It is also noted that structural responses under seismic ground motion may not fulfill traditional OMA assumptions because they are nonstationary processes with short durations. Therefore, the modal properties identified by OMA methods may tend to show more variabilities under seismic loading conditions. In addition, this variability becomes larger if one aims to identify higher mode properties (Pioldi and Rizzi 2018). This makes it challenging to identify structural damage by comparing the modal properties before and after damage. To handle this, a variational autoencoder (VAE; Kingma and Welling 2013), an autoencoder (AE) with probabilistic learning, is used in this study. AE is one of the well-known DNN methods used to learn latent variables of input data, i.e., hidden patterns of the system, in a self-supervised manner (Kramer 1991). However, the latent variable learned by AE cannot construct a latent space because AE learns the latent variable in a one-to-one manner between the input and latent variable. In contrast, VAE learns how to encode an input as a distribution over the continuous latent space by learning probabilistic characteristics. Because of its continuity, the proximity of any two points in the latent space can be measured by their distance. Therefore, if modal properties at the undamaged state are used to train VAE, latent variables should be clustered in terms of the load conditions because uncertain characteristics are significantly affected by those. Because modal properties do not change dramatically by local damage, modal properties at a damaged state shall be located around a latent cluster that has a similar load condition. Therefore, by using VAE, it is possible to find the undamaged modal property (preestablished) that is the most similar to the damaged one (input data), i.e., being subjected to similar load conditions, based on the distance in the latent space. This makes it possible to identify structural damage by comparing modal properties under various load conditions.Based on these considerations, this paper proposes a DNN-based framework for near-real-time damage identification using the mode shapes obtained by the TDD method from the raw acceleration data of a structure from an earthquake event. The proposed DNN hinges on convolutional VAE (CVAE) composed of convolutional layers for the pattern recognition of highly nonlinear two-dimensional data. The proposed framework finds the modal properties of the undamaged condition (train data) that is the most similar to the damaged condition (input data) in terms of the distance in the latent space of CVAE. Next, structural damage of the target structure is quantified by the flexibility disassembly method (Yang 2011). After providing theoretical background, the paper presents the following four steps of the proposed framework: (1) flexibility matrices of an undamaged state under artificial ground motions are prepared as input data; (2) the network is trained to learn the latent space and reconstruct the training input data and is then verified using the test data; (3) real-time computational simulations are performed with real ground motions for various damage condition scenarios to verify the pretrained network; and (4) the structural damage is then calculated in near-real-time using the aforementioned damage identification method. The proposed framework is demonstrated and tested by a numerical example of the two-dimensional steel frame structure, followed by concluding remarks.Theoretical Background for Deep Neural NetworkAutoencoderAutoencoder is one of the DNN methods, which is widely used for dimensionality reduction or anomaly detection (An and Cho 2015). The aim of AE is to learn patterns hidden in a set of data by finding a DNN structure that can reconstruct the input data by going through the mapping functions called encoder and decoder sequentially, as illustrated in Fig. 1. Once the DNN is successfully trained by an AE algorithm, one can describe the data in the latent variable space whose dimension is generally lower than that of the original space. Because the AE model trained by data from a normal condition would not be able to reconstruct the data from an abnormal condition effectively, the reconstruction error can be used as an indicator of anomaly, such as structural damage. A classical AE consists of an encoder and decoder with a single hidden layer (Vincent et al. 2010), but one can construct a deep AE by introducing multiple hidden layers, as shown in Fig. 1. For simplicity, the following mathematical description considers the AE with one hidden layer, i.e., the traditional AE, without losing applicability to deep AEs.EncoderThe mapping function f(x), which transforms a d-dimensional input vector x∈Rd into an r-dimensional latent variable z∈Rr in which d>r, is called an encoder; f(x) is usually described as a nonlinear transformation (1) where W∈Rr×d denotes the mapping weight matrix of the encoder; b∈Rr is the bias vector; and σ is the activation function, which is usually a nonlinear function, such as sigmoid, tangent hyperbolic, rectified linear unit (ReLU), and exponential linear unit (ELU) function. In this paper, the hidden layers employ the ELU function ELUα(x), with the hyperparameter α (=1.0) (2) ELUα(x)={xx≥0α(ex−1)x<0DecoderThe mapping function g(z), which transforms the latent variable z back into a reconstructed vector x′∈Rd, is called a decoder. Note that the dimension of x and x′ are the same. Usually, g(z) is described (3) where W^∈Rd×r denotes the mapping weight matrix of the decoder; b^∈Rd is the bias vector; and σ is the activation function described previously.To obtain an optimal estimation of the parameters θ=[W,b,W^,b^] based on the training data, AE algorithms aim to minimize the loss function often defined as the mean squared error (MSE) (4) LAE(x)=1m∑i=1m12∥x(i)−g(f(x(i)))∥2where m = number of training samples; and x(i) = i-th input sample. To minimize the nonlinear function LAE(x), gradient descent-based optimizers, such as Adam (Kingma and Ba 2014), are commonly used. One also can utilize the regularized AE, such as AE with the L1 or L2 regularization terms, sparse AE (SAE; Ng 2011), and denoising AE (DAE; Vincent et al. 2010), to prevent the overfitting during the training process.Variational Autoencoder and β-VAEA variational autoencoder (VAE) is also composed of both an encoder and a decoder, which are trained to minimize the reconstruction error between the decoded and input data. The main difference between VAE and AE is that the VAE encodes the input data as a probabilistic distribution over the latent space to introduce regularization of the latent space instead of encoding an input as a single point. Due to this characteristic, VAE can generate synthetic samples, which are similar to the trained data, by sampling from the constructed latent space.VAE uses a variational inference approach for latent representation learning, which results in the loss function with a regularization term, called the stochastic gradient variational Bayes (SGVB) estimator (Kingma and Welling 2013). In detail, VAE assumes that the data is generated from the decoder pθ(x|z), and the encoder learns an approximation of the true posterior distribution pθ(z|x), denoted by qϕ(z|x) where ϕ and θ denote the parameters of the encoder and decoder, respectively. The loss function of VAE is given (5) LVAE(x;ϕ,θ)=DKL(qϕ(z|x)∥pθ(z))−Eqϕ(z|x)(logpθ(x|z))where DKL = the Kullback–Leibler divergence (KL divergence) between the two distribution models; pθ(z) = true prior distribution over the latent variables; and qϕ(z|x) = variational distribution, which is restricted to a family of distributions, such as the Gaussian or Bernoulli distribution, which are simpler than pθ(z). A proper type of distribution should be selected to make qϕ(z|x) similar to pθ(z). Note that the first term in the loss function forces qϕ(z|x) to be similar to pθ(z), i.e., the works as a regularization term, and the expectation in the second term helps reduce the reconstruction error. Introducing the regularization term to the loss function makes the latent space have the following two main properties: (1) continuity; and (2) completeness, which respectively mean that close points in the latent space should give similar output once decoded, and a point sampled from the latent space should give meaningful output (Carraro et al. 2020).Fig. 2 provides a conceptual illustration of VAE. The training process of the VAE model consists of the following four steps: 1.The input is encoded as a probabilistic distribution over the latent space (characterized by μz and logσz2 if the latent variables are assumed to follow a multivariate Gaussian distribution);2.A point is sampled in the latent space according to the encoded distribution;3.The sampled point is decoded through pθ(x|z); and4.The reconstruction error is calculated and back-propagated through the network.If the latent space is expected to be highly non-Gaussian, one can utilize other variational distributions, such as the von Mises-Fisher (vMF) distribution (Davidson et al. 2018) or Gaussian mixture (Dilokthanakul et al. 2016; Kingma et al. 2014).It is known that the original VAE constructs a relatively entangled latent space, which means that the data with different characteristics can be located closely in the latent space. In such cases, it becomes difficult to generate meaningful samples from the latent space and to cluster the data in terms of the latent features. To overcome this issue, a disentangled latent space can be defined such that a single latent variable vector z is sensitive to changes in only a single feature of data x, while being relatively invariant to changes in the other features (Bengio et al. 2013). An example of this approach is β-VAE (Higgins et al. 2016), which modifies VAE to construct the disentangled latent space. The loss function of β-VAE is given (6) Lβ−VAE(x;ϕ,θ)=β·DKL(qϕ(z|x)∥pθ(z))−Eqϕ(z|x)(logpθ(x|))where β = hyperparameter that controls the constraint on the latent space and the representation capacity of the latent variable z. When β=1, the loss function is the same as that of the original VAE. The larger β becomes, the more disentangled latent space is constructed by limiting the capacity of z. This is because limiting the representation capacity makes the network learn a more efficient representation of the data. Meanwhile, there is a trade-off between the degree of disentanglement and the reconstruction quality because the reconstruction error, the second term of Eq. (6), can be neglected during the training if β becomes too large. Therefore, the value of β should be selected appropriately based on the dimensions of the input and latent variable (Bengio et al. 2013).Convolutional Variational AutoencoderIt is known that VAEs consisting only of fully connected layers cannot learn well from two-dimensional input data in the matrix form with local features, such as images and video (Goodfellow et al. 2016). To handle this, the convolutional VAE (CVAE) combines the standard VAE with the convolutional layer to recognize local features in the input data. For this reason, CVAE is widely used in SHM with various kinds of input data. Ma et al. (2020) used one-dimensional CVAE to extract the essential features from structural responses in the matrix form directly, and San Martin et al. (2019) utilized two-dimensional CVAE with the spectrogram image of the vibration data for dimensionality reduction to perform fault diagnosis of ball bearing elements.The convolutional layer employs the convolution operation instead of the matrix multiplication generally used in fully connected layers. In other words, the mapping matrix of f(x), i.e., W, consists of multiple small filters, called kernels, instead of one large weight matrix. The conceptual illustration of the convolutional operation is given in Fig. 3. In the convolutional operation, the dimension of input is x∈Rn×l×l where n is the number of features, and l is the dimension of one feature map. The convolutional layer has kernels W∈Rm×k×k. Then, the dimension of the output layer is h∈Rm×p×p where p≤l. In the deconvolutional operation, so-called the inverse convolutional operation, by contrast, the dimension of the output is larger than that of the input, i.e., p>l. The procedure to obtain the hidden representation from the input through the convolutional operation is called convolutional encoder f(x). In reverse, the procedure to obtain the reconstructed input from hidden representation through the deconvolutional operation is called convolutional decoder g(f(x)).Theoretical Background for Structural Health MonitoringTime Domain DecompositionTo extract modal properties from sensor data by vibration-based damage identification methods, it is required to have a sufficient level of spatial resolution. Furthermore, for an accurate estimation of higher mode shape, a high resolution of sensor spacing is also required. Therefore, most OMA techniques, such as FDD or SSI-COV, may require a great computational effort as the size of the matrix to be solved during the procedure relies on not only time or frequency samples but also the number of sensors. To handle this, the time domain decomposition (TDD; Kim et al. 2005) method was introduced to extract modal properties with a low computational cost when many sensors are installed. The computational complexity of TDD is related to the number of sensors only, i.e., not to those of time or frequency samples. Therefore, the TDD is well-suited for the online application, such as the near-real-time SHM proposed in this paper.For a target structure with p sensors, the output acceleration vector y¨(t)=[y1(t)⋯yp(t)]T, caused by an arbitrary load at continuous time t, can be spanned (7) y¨(t)=∑i=1∞c¨i(t)ϕiwhere c¨i(t) = i-th modal contribution factor of acceleration; and ϕi = i-th mode shape. Assuming that n dominant and well-separated poles are resolved in the discrete response, the acceleration at the discrete-time index k can be written (8) y¨(k)=∑i=1nc¨i(k)ϕi+ϵ¨t(k)where ϵ¨t(k) = truncation error at the discrete-time k.Using the mode isolation technique with the band-pass filter, the i-th filtered single-degree-of-freedom (SDoF) acceleration y¨i(k)∈Rp×1 is given (9) y¨i(t)=c¨i(k)ϕi+ϵ¨f(k)where ϵ¨f(k) = noise vector at time k obtained by the band-pass filtering and the residual of ϵ¨t(k). Suppose y¨i(k) contains the modal space and orthogonal noise space. Then, ϵ¨f(k) can be spanned, and y¨i(k) is written (10) y¨i(k)=c¨i(k)ϕi+∑j=1p−1d¨j(k)ψjwhere d¨j(k) = contribution factor of j-th noise mode; and ψj = j-th orthogonal noise base.When N response samples are measured, Eq. (10) is given in a matrix form, in other words (11) Y¨i=ϕic¨iT+∑j=1p−1ψjd¨jTwhere Y¨i∈Rp×N is the matrix form of the mode-isolated acceleration time history; ci¨=[c¨i(1)⋯c¨i(N)]T; and dj¨=[d¨j(1)⋯d¨j(N)]T.Then, a cross-correlation Ei∈Rp×p of the i-th mode-isolated acceleration time history signals can be represented as the energy correlation of the i-th mode (12) By substituting Eq. (11) into Eq. (12) and using the orthogonality of the basis vectors, Ei can be written (13) Ei=ϕiqiϕiT+∑j=1p−1ψjσjψjTwhere qi and σj are respectively termed as the level of energy at the modes i and j.Note that Eq. (13) can be represented alternatively using the form of singular value decomposition (SVD), in other words (14) where Ui=[ϕiψ1⋯ψp−1] is the singular vector matrix; and Ωi=diag([qiσ1⋯σp−1]) is the singular value matrix of Y¨i. The underlying assumption of the singular values is that qi>σ1>⋯>σp−1. Therefore, the i-th undamped mode shape can be extracted by taking the first singular vector of Ei.By premultiplying ϕiT in Eq. (11) and using the orthogonality, c¨iT can be written (15) Because Y¨i contains the i-th filtered modal response, the autospectrum of c¨iT has only one peak. The TDD method estimates the natural frequency of the i-th mode as the frequency at the peak of the autospectrum.Flexibility Disassembly MethodThe flexibility disassembly method (Yang 2011) is an effective damage identification method that can accurately identify structural parameters from the global flexibility matrix estimated by the identified modes without requiring any higher-order sensitivity analysis or iteration. Because of the low computational cost, the flexibility disassembly method is suitable for the near-real-time SHM proposed in this paper. The main idea is to decompose a flexibility matrix into a connectivity matrix between degrees-of-freedom (DoFs) and diagonal matrix-containing element stiffness parameters and quantify damage-induced changes in the latter.For a target structure with N elements and n DoFs, the method starts from disassembling the global stiffness matrix of the undamaged structure K∈Rn×n, using the Eigen-parameter decomposition of the i-th element stiffness matrix in the global coordinate Ki∈Rn×n (i=1,…,N) (Yang and Liu 2009) as follows (16) where K=∑i=1NKi; C∈Rn×N is the structural connectivity matrix; and P∈RN×N is a diagonal matrix containing element stiffness parameters pi representing the material and sectional properties. The C stays unchanged even if damage occurs because C contains only the structural contribution of the element stiffness to K, and C is thus independent of P. A full description of the elemental Eigen-parameter decomposition can be found in the study by Yang and Liu (2009). Introducing αi (0≤αi≤1) as the damage extent parameter of the i-th element, the global stiffness matrix of the damaged structure Kd∈Rn×n can be disassembled (17) where Pd=diag([pi(1−α1),…,pN(1−αN)]). In this paper, the subscript d stands for damaged.Because the flexibility matrix is the inverse of the stiffness matrix, the global flexibility matrices of the undamaged and damaged structure, i.e., F and Fd, can be respectively disassembled (18) (19) where C+ = pseudo inverse of C; E=(C+)T; and B=P−1 is the diagonal matrix containing the i-th flexibility parameter bi=1/pi. Then, the flexibility difference ΔF can be written (20) ΔF=Fd−F=E(Bd−B)ET=EΔBETwhere ΔB=diag([b1β1,…,bNβN]); and βi=[1/(1−αi)−1]. The damage extent parameter αi can be described in terms of βi, in other words (21) If exact F and Fd are available for the damage identification problem, the damage extent α=[α1,…,αN]T can be obtained using Eq. (20) and Eq. (21). On the other hand, when only the first few low-frequency modal properties can be obtained by the modal analysis method, such as TDD, one can instead use the global flexibility matrix approximated by the first few low-frequency mode shapes (Pandey and Biswas 1994) (22) F=∑j=1m1ωj2ϕ¯jϕ¯jTwhere ωj and ϕ¯j respectively denote the frequency and the mass-normalized mode shape of the j-th mode.Proposed DNN-Based Damage Identification FrameworkThe flexibility matrix of the target structure can be approximated by Eq. (22) with a limited number of modal properties. However, the approximated flexibility matrix is greatly affected by the external load condition when the modal properties are extracted by an OMA method (Ciloglu et al. 2012). This is because the dominant structural frequency varies according to the condition of the loading, e.g., earthquake or wind. Consequently, the performance of damage identification using the flexibility disassembly method is also affected by the load condition. (This issue will be subsequently demonstrated in the numerical example.)To deal with this load-dependency of OMA results, one can perform structural analyses of the undamaged numerical model under various load conditions to obtain flexibility matrices of the undamaged structure and then use the matrix, which is the most compatible with that of the damaged structure in terms of the load condition. Suppose the flexibility matrix at the unknown state Fu (damaged or undamaged) is obtained at current time step k by using Eq. (22), and the database (DB) of undamaged flexibility matrices of F has been built. In this paper, the subscript u stands for unknown. Then, the damage can be identified by utilizing the flexibility disassembly method using F from the DB and Fu at the current time step. However, it is challenging to extract F from the DB, which was under a load condition similar to that of the current Fu in a near-real-time sense.To overcome this challenge, a novel DNN-based damage identification framework is proposed as follows. As illustrated in Fig. 4, the proposed framework is divided into two main parts: (1) offline; and (2) online processes.Offline ProcessTo get ready for the near-real-time damage identification by the online process, the data processing and network training should be performed in the offline process first. The offline process has two steps: (1) data collection from numerical analyses and modal analysis to obtain the undamaged flexibility matrix F; and (2) network training using the data from the first step. The conceptual illustration of the offline process is shown in the upper part of Fig. 4.Data Collection and Modal AnalysisTo obtain training data of structural responses under seismic ground motions, structural analyses are performed using the model of the undamaged target structure. To cover a wide range of ground motion properties that can occur in a future earthquake event, a variety of seismic ground motions are artificially generated. Furthermore, live loads are applied before and after the earthquake event to simulate the normal operational circumstance. After the data collection, the TDD is performed to collect modal properties from responses under each seismic ground motion. Using these modal properties, approximated flexibility matrices F are calculated by Eq. (22).Network TrainingAfter data collection, the DNN model is trained using the flexibility matrices from the undamaged structure as input data. Convolutional β-VAE (β-CVAE) is selected as the DNN model to construct the disentangled latent space of F, which is extracted under various load conditions. The network is trained to minimize the reconstruction error between the input and decoded data, as well as to establish the meaningful latent space of given flexibility matrices. To utilize the preestablished latent space in the online damage identification process, the decoder is removed from the network after training, and every point of the entire dataset in the latent space is stored in the DB. Note that the preestablished latent space represents flexibility matrices of the target structure at an undamaged state.Online ProcessThe online process for the near-real-time damage identification is performed using the trained DNN and the DB of the latent space obtained from the offline process. The online damage identification process employing the flexibility disassembly method has two steps: (1) searching the DB for the flexibility matrix representing the undamaged state, which is closest to the current data in the latent space; and (2) calculating damage extents using the flexibility disassembly method. These two steps are performed at every time step. The conceptual illustration of the online process is shown in the lower part of Fig. 4.Searching DB for Flexibility Matrix Representing Undamaged StateIn the online process, structural responses are obtained from sensors in real-time while the actual state of the structure is unknown, i.e., whether undamaged or damaged. The flexibility matrix at the unknown state Fu can be obtained in near-real-time by applying TDD to the recorded responses with a certain length of the time window. Note that the time step Δt of the structural response does not have to be the same as that of the training simulation when the OMA time window is long enough. This is because the input data to the network is the flexibility matrix, which is nonsequential data, and the effect of choosing different Δt decreases as more response data points are used to perform TDD.Then, the corresponding point in the latent space can be obtained through the encoder of the preestablished DNN. The similarity can be quantified by measuring the distance between the current (unknown) and prestored (undamaged) data. Because the network training or any computationally intensive calculations are not included in this step, the entire process can be finished in a few seconds. As a result, the pair of the matrices, i.e., the flexibility matrix representing the undamaged state, F, and the current matrix Fu, is obtained in near-real-time for the next step.Quantification of Element Damage ExtentThe element damage extent αk at time step k can be calculated by the flexibility disassembly method, i.e., Eqs. (20) and (21), using F and Fu, which were obtained from the previous step. Because this step includes just a few matrix calculations, αk can also be obtained in a short time. As a result, the near-real-time damage identification can be performed by repeating the online process at every time step.Numerical InvestigationsStructural Properties of Example Target StructureAs a target structure, a two-dimensional 5-story, 5-bay steel frame in Fig. 5 is considered. The uniform column heights and beam lengths are set to 14 ft (4.2672 m) and 24 ft (7.3152 m), respectively. The columns and beams have the section properties of wide flange beam W27×114 a36 and W24×94 a36 carbon steel, respectively. For simplicity, the elastic section is used for each element. The modal damping ratio ζ is set to 5%. At the undamaged state, the frequencies of the first 5 modes (shown in Fig. 6) are 1.01, 3.51, 6.94, 11.11, and 14.93 Hz, respectively.Data Generation and PreprocessingStructural analysis is performed by the object-oriented finite-element software, OpenSees, using artificial ground motion time histories generated for the earthquake magnitude from 7 to 9 with the time step Δt=0.02s (Rezaeian 2010). In addition, random excitations simulating live loads are applied to the first column before and after an earthquake event. To obtain structural responses under seismic ground motions with various characteristics, a total of 20,000 artificial ground motions were generated.To capture the modal properties effectively, the time window used in OMA should be set long enough. Accordingly, each simulation should be much longer than the OMA time window to extract the modal properties from each windowed response under various load conditions (random excitation or ground motion). To this end, the total time length of each simulation is set to 215×Δt=655.36s, and the length of the OMA time window is set as half of the total length, i.e., 214×Δt=327.68s, as illustrated in Fig. 7. The time interval for the beginning points of the OMA time window is set as 212×Δt=81.92s.As the structural response, only horizontal accelerations of all nodes are recorded to simulate the lack of sensors, often observed in real SHM practice. Because there are 30 horizontal DoFs in the target structure, the dimension of the flexibility matrix F is 30×30. The F is calculated using the first three mode shapes obtained by the TDD method from signal segments at each time step. The dataset is divided into the training and test set with a ratio of 9∶1. Because F is a nonnegative matrix, the training dataset is scaled to the range of [0, 1], normalized by the largest value.Network TrainingThe β-CVAE is then trained to construct the disentangled latent space of the undamaged flexibility matrix (two-dimensional data). Inspired by Na et al. (2018), the architecture of β-CVAE is proposed as illustrated in Fig. 8. The β-CVAE generally consists of three main parts: (1) the convolutional encoder, (2) latent space, and (3) convolutional decoder (deconvolutional layers). For a more efficient network representation of the data while preventing overfitting, this paper proposes to use β-CVAE consisting of small convolutional layers for limiting the representation capacity. In this paper, the convolutional encoder is composed of three convolutional layers with 4, 8, and 16 kernels, respectively. Stride 2 is used, rather than the Maxpooling2D layer, at every convolutional layer for effective dimension reduction (He et al. 2016). After convolutional operations, the output of the last convolutional layer is reshaped into the one-dimensional vector by the Flatten layer, which is followed by two Dense layers with 3 nodes representing μz and logσz2 of the latent distribution, respectively. Note that the latent space has much lower dimension than the input data in order to limit the representation capacity. The decoder has the inverse architecture of the encoder and uses the same hyperparameters as the encoder. The ELU activation function is used in all layers except the output layer, which uses the sigmoid activation function because the input data is scaled to the range of [0, 1]. The value of the loss function Lβ−VAE(·) is calculated using the SGVB estimator with β at the last step of the forward propagation, and the value is backpropagated through the network to optimize network parameters. The value of β is set to 10% of the ratio of dimensions of the input data and latent variable, i.e., β=0.1×[(30×30)/3]=30 (Higgins et al. 2016).The β-CVAE was constructed using the Python deep learning library Keras with the Tensorflow backend and trained on a server with 2x Intel(R) Xeon(R) Gold 6126 2.60 GHz, two NVIDIA TITAN RTX graphics cards, and 128GB RAM. The numbers of epochs and batch size were set to 200 and 32, respectively. The rectified Adam (Radam; Liu et al. 2019) optimizer with a learning rate of 0.001 was used for optimizing the loss function. Furthermore, to prevent the posterior collapse, in which the VAE model ignores the latent variable z and the posterior qϕ(z|x) mimics the prior pθ(z), the Kullback–Leibler annealing (KL annealing) method is introduced (Bowman et al. 2015; He et al. 2019). This method is used to first train the VAE using the reconstruction loss only for a few epochs and then slowly increase the KL divergence. In this numerical example, the β-CVAE network is trained only with the reconstruction loss for the first 10 epochs, and then the KL divergence is linearly increased over the next 100 epochs. The training process took about 12 h while the loss function converged fast and stably without overfitting or explosion of the validation loss, as shown in Fig. 9.Near-Real-Time Damage IdentificationTo verify the performance of the pretrained β-CVAE as a damage detection tool, real-time test simulations are performed with real ground motions: the El Centro Earthquake in 1940 with Δt=0.02s and the Kobe Earthquake in 1995 with Δt=0.01s, shown in Fig. 10. Note that Δt of the El Centro Earthquake record is the same as that of the training data, but Δt of the Kobe Earthquake record is different. As mentioned, choosing different Δt has little effect if the length of the OMA time window is long enough. To verify this, ground motions with different Δt are selected. It is assumed that the earthquake event occurs at the middle of the simulation, and random excitation is also applied to the first column before and after the earthquake event. The flexibility matrix is obtained at every time step, and the element damage extent αk is calculated simultaneously through the pretrained network.In both experiments, two assumptions are introduced: (1) the structural damage can be simulated by the degradation of the elastic modulus; and (2) the damage already exists at the initial time step. First, even though the degradation of the elastic modulus may not simulate the realistic damage well enough, it is reasonably appropriate for this numerical example because the elastic section is used for all structural elements. Next, the initial damage is introduced to verify the performance of the proposed method under various load conditions. Alternatively, one could assume that the damage occurs during the middle of the earthquake event. However, the damage occurrence time should be determined manually if the structure with the elastic section is considered.To verify the identification performance, two damage cases are investigated for each ground motion: (1) Case 1: only Column 3-1 is damaged; and (2) Case 2: Columns 1-1, 3-3, and 5-5 are damaged simultaneously. The damage of a column is simulated by a 50% degree of degradation in both cases.In the figures reporting the identification results in this paper, the results of the columns connected with each other vertically are shown in the same subplots, while the line colors indicate the floors of the frame structure. For example, the first subplot in Fig. 11 shows the damage extent parameters of Columns 1-1, 1-2, …,1-5, which are the columns located at the left end of the frame, as shown in Fig. 5. The transparent and solid lines in each subplot indicate the identification result at every time step and its 9-point moving average, respectively. In addition, a thicker line indicates the damaged column.Test Ground Motion 1: El Centro EarthquakeThe total duration of the simulation is set as 217×Δt=2,621.44  s to verify the stable performance over a long period of time. The length of the OMA time window is set as 214×Δt=327.68  s, i.e., the same length as that of the training. A shorter time interval of each window is set as 210×Δt=20.48  s to simulate the near-real-time damage identification. The near-real-time identification results for the two cases are shown in Figs. 11 and 12, and the temporal averages of each result are also described in Tables 1 and 2, respectively. In both tables, the bold and underlined numbers indicate the damaged columns in the corresponding scenarios. As shown by the figures and tables, the proposed method successfully identifies structural damage under both random excitation and earthquake event in near-real-time, i.e., it localizes and quantifies the damage in a stable manner.Table 1. Temporal average of identified damage extents (%) under El Centro earthquake (single member subject to damage)Table 1. Temporal average of identified damage extents (%) under El Centro earthquake (single member subject to damage)Floor1st2nd3rd4th5th6th1st50.953.060.754.250.248.22nd6.956.695.836.216.085.853rd1.351.952.282.522.682.724th2.283.334.556.829.3910.25th12.015.219.623.225.125.7Table 2. Temporal average of identified damage extents (%) under El Centro earthquake (multiple members subject to damage)Table 2. Temporal average of identified damage extents (%) under El Centro earthquake (multiple members subject to damage)Floor1st2nd3rd4th5th6th1st35.719.09.966.935.615.262nd3.063.793.915.196.066.403rd2.758.1826.012.97.566.614th1.011.391.341.811.292.655th9.6126.046.461.169.469.1In Case 1 (single member subject to damage; Fig. 11 and Table 1), the damage extent αi of Column 3-1 is estimated at around 60%, which is close enough with the true value, throughout the duration. Furthermore, it only takes less than 5 s to obtain αk for every time step, which is much shorter than the preset time interval of 20.48 s. It is noted that only αi’s of elements, which are on the same floor as the damaged column, e.g., Columns 1-1 to 6-1, are identified incorrectly as damaged, i.e., false-positive cases. In addition, some elements on the 5th floor are estimated as damaged but with low degrees. Nevertheless, the damage can be readily located because the closer it is to the damaged column, the greater the damage extent is.In Case 2 (multiple members subject to damage; Fig. 12 and Table 2), the damage extents αi of Columns 1-1, 3-3, and 5-5 are estimated at around 35%, 25%, and 70%, respectively. The detection performance is slightly worse than in Case 1 because structural damage occurs at multiple locations in the structure. Nonetheless, the result is still helpful for the localization and quantification for all three damaged members while elements located at the same floor as damaged columns are estimated damaged but with much lower degrees of αi. Note that, as shown in the figure, the localization performance is much better than in Case 1. As getting far away from the damaged column, αi’s of undamaged elements decrease much faster than in Case 1.In addition, a simple experiment was performed using El Centro Earthquake to check the load-dependency of the flexibility matrix identified by the OMA method. Suppose the structure in Fig. 5 is damaged at the 25th element (Column 5-5), and its damage extent α25 is set to 0.5. Two cases are investigated: (1) flexibility matrices before and after damage, i.e., F and Fd, are extracted under the same load condition (earthquake); and (2) different load conditions (earthquake and random excitation). As shown in the identification results for the columns in Fig. 13, the performance is significantly worse when F and Fd are extracted under different load conditions. The estimation error of damage extent α25 is small, and only columns on the same floor as the damaged element (15th, 20th, and 30th elements) are estimated as damaged when F and Fd are extracted under the same load condition. In contrast, the estimation error of α25 is larger than the first case, and more undamaged columns are estimated as damaged when F and Fd are extracted under different load conditions. This result justifies that searching the most similar undamaged flexibility matrix in the latent space helps achieve more accurate results of damage identification.Test Ground Motion 2: Kobe EarthquakeBecause Δt of the Kobe Earthquake is set to 0.01 s, the lengths of total simulation time, OMA time window, and time interval in this simulation are set to half of the previous simulation, i.e., 217×Δt=1,310.72  s, 214×Δt=163.84  s, and 210×Δt=10.24  s, respectively. The near-real-time identification results for the two cases are shown in Figs. 14 and 15, and the temporal averages of each result are also described in Tables 3 and 4, respectively. In both tables, the bold and underlined numbers indicate the damaged columns in the corresponding scenarios. As shown in the figures and tables, the identification performance is less stable in both cases than the previous simulation using El Centro Earthquake, but the proposed method still successfully identifies damage in near-real-time even though a different Δt is used.Table 3. Temporal average of identified damage extents (%) under Kobe earthquake (single member subject to damage)Table 3. Temporal average of identified damage extents (%) under Kobe earthquake (single member subject to damage)Floor1st2nd3rd4th5th6th1st55.755.961.756.953.752.22nd11.48.195.845.575.385.253rd11.03.751.851.561.241.134th2.512.201.581.301.571.795th57.544.526.310.36.405.77Table 4. Temporal average of identified damage extents (%) under Kobe earthquake (multiple members subject to damage)Table 4. Temporal average of identified damage extents (%) under Kobe earthquake (multiple members subject to damage)Floor1st2nd3rd4th5th6th1st44.724.515.312.912.111.92nd3.413.362.593.143.253.223rd21.925.735.516.38.587.174th1.411.531. Case 1 (single member subject to damage; Fig. 14 and Table 3), the damage extent αi of Column 3-1 is estimated at around 60%, which is close enough to the true value, during the entire duration of the simulation. Just as in the previous simulation, αi’s of undamaged elements, which are on the same floor as the damaged column, are also estimated as damaged. On the other hand, the elements on the 5th floor are estimated as damaged with higher degrees, especially Column 1-5. Nonetheless, except for some elements on the 5th floor, the damage identification can still be well-performed in near-real-time.In Case 2 (multiple members subject to damage; Fig. 15 and Table 4), the estimation performance is less stable than in Case 1, but the method can localize and quantify the damage of multiple members successfully. Damage extents αi of Columns 1-1, 3-3, and 5-5 are respectively estimated at around 45%, 35%, and 60%, which are much closer to the true values than the previous simulation. Note that, except for the columns on the 5th floor, undamaged columns on the same floor as the damaged columns are estimated damaged with much lower degrees, and αi’s decrease much faster than in Case 1 as getting far away from the damaged column.Performance of the Proposed Damage Identification MethodDespite the remarkable performance, the proposed method still has room for improvements. First, even though damage extents are closely estimated, there are some estimation errors of up to 25%. This is the error propagated from the approximated flexibility matrix, which is calculated using only horizontal responses and limited mode shapes. This makes ΔB in Eq. (20), which was assumed as a diagonal matrix, a nondiagonal matrix. Especially, the damage extents of the columns on the top floor are identified consistently higher than the other column because the top floor has the free boundary condition. That is, responses of the top floor have more vertical components, which are neglected, but only horizontal responses are used in this numerical example. Second, the proposed method showed false-positive errors during the damage identification. In particular, the damage extents of undamaged columns located on the same floor as the damaged column are identified as damaged ones with a similar degree. This is because of peaks in the difference between F and Fu are merged, which makes the damage localization difficult (Lu et al. 2002). Furthermore, the structural behavior of elements in the frame structure is significantly affected by other adjacent elements because of its complex structural typography (Brasiliano et al. 2004; Xia et al. 2002). Finally, the identification performance for every time step, i.e., the transparent lines in the results, shows large fluctuations over time. It is assumed that the undamaged flexibility matrix F extracted from the DB by the pretrained network and the current Fu are under the same load condition, i.e., F and Fu are in the same cluster in the latent space. Even so, each response data used to calculate F and Fu is observed under external loads that are not exactly the same. This can be handled by performing more structural analyses to obtain plenty of structural response data. Accordingly, the DB can be established with more undamaged F, and the fluctuation would be reduced because the extracted F would be more similar.ConclusionsIn this paper, a new unsupervised deep neural network (DNN)–based framework was proposed for near-real-time damage identification of a structural system that just underwent an earthquake excitation. A network structure of the convolutional β-variational autoencoder was proposed for constructing a disentangled latent space of input data. The β-CVAE was trained in an unsupervised manner to construct the latent space of the flexibility matrix identified from the undamaged structure using an efficient operational modal analysis (OMA) method, called time domain decomposition (TDD). Unlike the common usage of the variational autoencoder, the proposed framework only used the encoder part to utilize the preestablished latent space. In near-real-time identification, the undamaged flexibility matrix, which is the most similar to the seismic response data, could be extracted without any information of structural damage by measuring the distance in the latent space. The damage was identified by the flexibility disassembly method using the two flexibility matrices. Due to the computational efficiency of the proposed method, the near-real-time damage identification was possible using high-dimensional data without any complicated computation. A numerical example of the real-time simulation with real ground acceleration records was provided to test and demonstrate the proposed framework. To test the applicability of the proposed framework, we selected a steel frame structure as the target structure. Without any information of target structure or damage case, the framework showed a successful unsupervised damage identification performance in cases of single and multiple damages under various load conditions in near-real-time. Furthermore, even when earthquake records with different time steps were used, the framework showed stable identification performance over a long period of time. The proposed method showed a moderate level of false-positive errors, which are due to the inherent problem of the flexibility approximation and flexibility disassembly method, and the lack of observation data in real practice (simulated in the examples by using floor accelerations in the horizontal direction only).Although the proposed framework has been successfully verified through the numerical example of the two-dimensional frame structures, there may be the following challenges and limitations in its applications to real-world structures. First, because it is challenging to extract an accurate flexibility matrix from the complex structure, the proposed method may have limited applicability to real-world structures. Furthermore, because the dimension of input data increases exponentially as the structure becomes more complex, the inference time of the DNN model would also increase accordingly, even if the exact flexibility matrix is extracted. Second, due to the external effect, such as thermal effect or ground settlement, the numerical model of the structure may not describe the real structure properly. Although this can be calibrated through the model updating techniques, the difference between the real-world structure and numerical model makes it difficult for the DNN model to represent the undamaged actual structure. Third, in the case of complex structures, such as cable-stayed bridges, the use of OMA methods may be limited under the seismic load, which is the nonstationary process with a short duration. This is because its natural frequencies are closely distributed in the low-frequency band. Especially, applying TDD to the complex structure with high nonlinearity may be challenging because TDD assumes that the structure has linear characteristics (Kim et al. 2005).To improve the performance of the proposed framework and promote its applications, a damage identification method for real complex infrastructures based on the raw vibration signal is currently under development to achieve more stable and accurate performance. Recent research outcomes have proved that unsupervised DNN can extract meaningful features from the raw signal data (Geiger et al. 2020; Kwon et al. 2019). Moreover, the proposed method can be further developed to monitor the sudden change in structural parameters due to various types of the extreme event caused by natural or man-made disasters. Robust performance of the proposed method under various load conditions is expected to help reduce the time required for the postdisaster decision-making process by providing near-real-time damage assessment. Eventually, the proposed framework will be utilized to prepare an effective postdisaster operational and maintenance strategy as well.Data Availability StatementSome or all data, models, or code that support the findings of this study are available from the first author upon reasonable request.AcknowledgmentsThis work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (NRF-2021R1A2C2003553). The corresponding author is supported by the Institute of Construction and Environmental Engineering at Seoul National University.References An, J., and S. Cho. 2015. “Variational autoencoder based anomaly detection using reconstruction probability.” Spec. Lect. IE 2 (1): 1–18. An, Y., E. Chatzi, S. H. Sim, S. Laflamme, B. Blachowski, and J. 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