A structural damage system identification method based on physically consistent neural network

Abstract

The present invention relates to a structural damage system identification method based on a physically consistent neural network, which solves the dilemma of limited generalization performance and insensitivity to local losses in current physically based system identification methods. The steps include: 1. Collecting target building information and monitoring data; 2. Establishing a physically consistent neural network; 3. Constructing a network training data set; 4. Conducting physically consistent neural network training; 5. Obtaining inverted structural parameters. The present invention utilizes the similarity between Timoshenko beam theory and neural network principles to build a physically consistent neural network. Each parameter in the network has a strict physical meaning, which greatly enhances the interpretability and generalization performance of the network; at the same time, prior knowledge constraints are added to the loss function, which significantly reduces the uncertainty and difficulty of the inversion.

Description

Technical Field

[0001] The present invention relates to the field of earthquake engineering, and in particular to a structural damage system identification method based on a physical consistent neural network. Background Art

[0002] Structural health monitoring (SHM) typically involves deploying multiple sensors on a structure to measure its response in real time. By analyzing the recorded waveforms, the dynamic characteristics of the structure can be identified, enabling early detection of potential damage and rapid assessment of structural damage after natural or man-made disasters such as earthquakes and explosions. However, accurately identifying dynamic characteristics from the measured response data is a very challenging task. Missing identification may lead to an incorrect assessment of the structural health status, failure to detect and repair damage in a timely manner, and thus endangering the safety of the structure. Conversely, false positives about structural damage may lead to unnecessary repairs, increased costs, and construction risks. Therefore, it is crucial to develop a reliable, damage-sensitive and accurate system identification method.

[0003] Numerous system identification methods have been developed, which can be broadly categorized into two main categories: vibration-based methods and wave propagation methods. Vibration-based methods are the most widely studied and involve extracting modal properties of the structure, such as natural frequencies, damping ratios, and mode shapes, from measured acceleration records using various modal analysis techniques. By identifying changes in these modal properties due to damage, structural health can be monitored. While these methods are very effective in identifying damage that affects the overall stiffness of the structure, they may fail to detect localized damage, as such damage has no significant effect on global properties such as natural frequencies and mode shapes. Wave propagation methods address this issue because waves are highly sensitive to localized damage during propagation. Localized damage can significantly alter wave velocity, attenuation, reflection, and scattering characteristics. By carefully analyzing these changes in wave propagation properties, localized damage can be accurately located and assessed. Another significant advantage of wave propagation methods is their insensitivity to soil-structure interaction effects, even in the presence of more extensive foundation sway and coupled horizontal and rocking responses.

[0004] Wave propagation methods consider the seismic response of a structure as a superposition of waves propagating through the structure, which reflect from the structural boundaries and interfere with each other. These methods, also known as seismic interferometry, usually assume that the structure is a shear beam and use wave velocity to characterize the structure. Wave velocity can be identified by deconvolution, a key step of which is to monitor the peaks of the impulse response function (IRF) between the signals recorded by different sensors. However, some scholars have demonstrated that the accuracy of identification by monitoring these peaks alone is limited. Therefore, they proposed an identification algorithm based on least squares fitting of the IRF, which significantly improved the accuracy of identification. In addition, considering that the shear beam assumption ignores the contribution of bending deformation to the response, they first proposed a wave propagation method based on Timoshenko beam theory for the identification of high-rise buildings, which further improved the identification accuracy.

[0005] In recent years, deep learning has been widely used in the field of system identification due to its excellent nonlinear fitting and feature extraction capabilities. These deep learning methods can be divided into two categories: data-driven methods and physics-based methods. The former assumes that the mechanism model is unknown and aims to use neural network models to approximate the input-output relationship of the structural system without involving the actual physical parameters of the system. These methods avoid the high computational cost and numerical problems caused by building and verifying structural numerical models. In addition, they can automatically adapt to the uncertainty caused by measurement. However, data-driven methods face two major challenges: (1) a large amount of data is required for network training, and insufficient data increases the possibility of overfitting, thereby reducing the generalization performance of the model; (2) the trained network model is a "black box" and lacks sufficient physical interpretability. To address these problems, scholars have proposed various physics-based methods. These methods involve establishing a representative physical model of the structure and introducing a physics-guided loss function during the network training process. The loss function is in the form of the residual of the control equation corresponding to the physical model. By minimizing the difference between the network prediction and the measured data, the parameters of the physical model can be updated. However, the network model itself still lacks physical interpretability and still faces the challenge of limited generalization performance. Furthermore, these physics-based approaches typically consider the seismic response of the structure as a vibration problem rather than a wave propagation problem, and therefore face the same challenge of being insensitive to local losses. Summary of the Invention

[0006] The purpose of this invention is to solve the problems of limited generalization performance and insensitivity to local losses in current physics-based system identification methods, and to propose a physically consistent neural network to achieve system identification more efficiently and reliably.

[0007] The embodiment of the present application provides a structural damage system identification method based on a physical consistent neural network. The structural damage system identification method based on a physical consistent neural network is characterized by comprising the following steps:

[0008] Step 1: Collect target building information and monitoring data:

[0009] Collect detailed geometric dimensions and spatial layout information of the target building, as well as historical acceleration responses recorded by acceleration monitors placed inside the target building;

[0010] Step 2: Establish a physically consistent neural network:

[0011] The physically consistent neural network is consistent with the cantilever layered Timoshenko beam theory and consists of an input layer, an output layer, and n hidden layers, where n corresponds to the number of layers in the cantilever layered Timoshenko beam model. The parameters of the physically consistent neural network are related to the parameters of the Timoshenko model, and its input and output are the Fourier spectra corresponding to the horizontal acceleration responses of the top and bottom floors of the building, respectively.

[0012] Step 3: Build a network training dataset:

[0013] The amplitude of each frequency component is obtained by using discrete Fourier transform on the historical acceleration response recorded by the acceleration monitor arranged in the target building collected in step 1;

[0014] Step 4: Conduct physically consistent neural network training:

[0015] The loss function used to guide network training consists of data loss and physical constraint loss; data loss is defined as follows:

[0016]

[0017] in, and denotes the sth element in the observed and predicted non-top acceleration response Fourier spectra, respectively; M denotes the total number of elements in these Fourier spectra; R(·) and Ι(·) denote the real and imaginary part operations, respectively;

[0018] The physical constraint loss is defined as follows:

[0019]

[0020] in, and are the shear wave velocity V S,j and modulus ratio R j The lower bound of the search space; ReLU(·) represents the rectified linear unit function;

[0021] The final loss function is:

[0022] L all =λ1×LPC +λ2×L data Formula (29)

[0023] Among them, λ1 and λ2 are weight factors;

[0024] Step 5: Obtain inverted structural parameters:

[0025] Since the parameter weights of the physical consistent neural network PCNN are related to the shear wave velocity V S Related to the modulus ratio R, when the PCNN training is completed, the shear wave velocity V of each layer of the Timoshenko beam model is obtained through the network parameters S and modulus ratio R.

[0026] 2. The structural damage system identification method based on a physically consistent neural network according to claim 1 is characterized in that the specific method of establishing the physically consistent neural network in step 2 is:

[0027] The building is modeled as a cantilevered layered Timoshenko beam with its base subjected to horizontal seismic motion u g The model contains n layers, the number of which is equal to the number of horizontal acceleration monitors arranged on the building minus one; each layer of the model is assumed to be uniform and isotropic, and adjacent layers are fully bonded; the layers of the model are numbered from top to bottom, and their geometric properties are determined by the cross-sectional area A j , second-order moment of inertia I j , height H j , width W j and shear correction factor k G,j Characterization, where j = 1, ..., n represents the index of each layer; the shear correction factor k G The non-uniform distribution of shear stress on the cross section is taken into account; the material properties of each layer are determined by the mass density ρ j , Young's modulus E j and shear modulus G j Characterized by mass density ρ j , longitudinal wave velocity V L,j and shear wave velocity V S,j Characterization; the relationship between modulus and wave velocity is defined as and The damping adopts the Kelvin-Voigt material model and assumes that the bending and shear deformations have the same viscosity constant μ j ;

[0028] The cross section of the beam is perpendicular to the neutral axis before deformation and remains planar after deformation. However, due to shear deformation, the cross section is no longer perpendicular to the deformation axis after deformation. The neutral axis of the beam is assumed to undergo only horizontal displacement, denoted by u(z,t). The total rotation of the beam consists of two components: the rotation θ(z,t) due to bending deformation and the rotation γ(z,t) due to shear deformation. M(z,t) and V(z,t) represent the bending moment and shear force, respectively.

[0029] Under the small deformation assumption, formula (1) applies to any layer in the above layered model:

[0030]

[0031] Substituting the Kelvin-Voigt material model into formula (1) yields formula (2):

[0032]

[0033] According to the force, the dynamic equation shown in formula (3) can be obtained:

[0034]

[0035] By combining formula (2) and formula (3), the following results can be derived, as shown in formula (4) to formula (7):

[0036]

[0037]

[0038]

[0039]

[0040] Assuming that the load is simple harmonic excitation, the displacement u(z, t), angle θ(z, t), bending moment M(z, t) and shear force V(z, t) can be expressed by formulas (8) and (9):

[0041] u(z,t)=U(z)e -iωt , θ(z,t)=Θ(z)e -iωt Formula (8)

[0042]

[0043] Where ω is the circular frequency. Substituting formula (8) and formula (9) into formula (4)-formula (7), we obtain formula (10):

[0044]

[0045] in, represents the state vector, and the matrix B is defined as:

[0046]

[0047] According to propagation matrix theory, if the matrix B is a continuous function of z (which is true in each layer), the state vector f(z) at any point in the same layer can be calculated using formula (11):

[0048] f(z)=P(z,z0)f(z0) Formula (11)

[0049] Where P(z,z0) is the propagation matrix from z0 to z. According to the general solution form of the ordinary differential equation given in formula (10), the form of the propagation matrix can be defined as shown in formula (12):

[0050]

[0051] Among them, e (·) represents the matrix exponential function; given that the state vector must be continuous between adjacent layers, any interface z starting from the top j The state vector at can be determined as formula (13):

[0052]

[0053] For a cantilever beam, the free end (top) must satisfy the condition of zero stress, which means that the shear force and bending moment at this point are both zero; the bottom of the beam is the fixed end, so the rotation angle must be zero; based on these conditions, the following formula can be derived:

[0054] f(z0)=[U0,Θ0,0,0] T Formula (14)

[0055]

[0056] Five of the physical quantities are non-zero; assume that U0 is a known quantity, which is applicable to the system identification task; then, the other four quantities can be calculated according to formula (13); the solution process is divided into two main steps. First, determine the rotation angle Θ0 of the top; let us assume that:

[0057]

[0058] It can be deduced that:

[0059]

[0060] in, Representation matrix The element in the p-th row and q-th column of ; therefore, Θ0 can be derived as:

[0061]

[0062] According to formula (18), the state vector f(z0) at the top can be completely determined; then according to formula (13), the state vector at any interface can be obtained starting from the top;

[0063] Let u(z ref ,t) represents the reference point z ref The movement at coordinate point z and the reference point z ref The transfer function between can be expressed as:

[0064]

[0065] To improve numerical stability and avoid division by very small numbers, the transfer function is often regularized as follows:

[0066]

[0067] Where, the horizontal bar represents the complex conjugate, and η represents the regularization parameter;

[0068] For a neural network with L+1 layers, the relationship between the lth layer and the (l+1)th layer can be expressed as formula (21):

[0069] z l =σ l (W l ·z l-1 +b l ),l=1,...,L Formula (21)

[0070] Among them, z 0 and z L Represent the input and output of the network respectively, W l and b l Represent the trainable weight matrix and bias vector of layer l respectively; function σ l (·) is the nonlinear activation function of layer l;

[0071] A physically consistent neural network is established, consisting of an input layer, an output layer, and n hidden layers, where n corresponds to the number of layers in the cantilevered layered Timoshenko beam model. Each hidden layer contains four neurons, which is equal to the number of elements in each state vector. The output of the lth hidden layer corresponds to the state vector at the top of the lth layer in the Timoshenko beam model. A post-processing layer is added after the output layer to concatenate the displacement output U(zj) of each network layer starting from the second hidden layer to calculate the subsequent loss function.

[0072] According to formula (18), the weight matrix of the first hidden layer of the physically consistent neural network can be derived as:

[0073]

[0074] The weight matrices of other hidden layers can be derived according to formula (16) as follows:

[0075] W j =P(z j-1 ,z j-2 ) Formula (23)

[0076] Among them, the value range of j is 2 to n; similarly, the weight matrix of the output layer can be derived as follows:

[0077] W n+1 =p1(z n ,z n-1 ) T Formula (24)

[0078] Among them, p1(z n ,z n-1 ) represents the propagation matrix P(z n ,z n-1 ), the superscript T indicates the transpose operation.

[0079] 3. The structural damage system identification method based on a physically consistent neural network according to claim 2, wherein the specific method for constructing the network training data set in step 3 is:

[0080] For a (digitized) acceleration record containing m equally spaced element values, the decomposition is as follows:

[0081]

[0082] Among them, ω s is the sth equidistant frequency, a s and b s Respectively represent the frequency corresponding to ω s and -ω s The input dataset is constructed based on the amplitude of each frequency component of the acceleration response of the top floor of the building, and the label dataset is constructed using the amplitude of the corresponding frequency components of the acceleration response of the remaining floors.

[0083] 4. The structural damage system identification method based on physical consistent neural network according to claim 3 is characterized in that a in formula (25) s and b s Calculated by formula (30) and formula (31) respectively:

[0084]

[0085]

[0086] 5. The structural damage system identification method based on physical consistent neural network according to claim 1 is characterized in that in step 4, in the actual physical scene, the shear wave velocity V of each layer is S,j and modulus ratio R j The following relationship must be satisfied:

[0087] V S,j >0,R j >0 Formula (27)

[0088] The value of j ranges from 1 to n.

[0089] 6. The structural damage system identification method based on physical consistent neural network according to claim 1 is characterized in that in step 4, the Adam optimizer and the learning rate decay strategy are used for training, and the shear wave velocity V S The initial learning rates of and modulus ratio R are set to 100 and 10-5 respectively. The learning rate decays to 0.1 times the original value every 50 training rounds, and the total number of training rounds is 150.

[0090] The beneficial effects of the present invention are as follows: the present invention utilizes the similarities between Timoshenko beam theory and neural network principles to build a physically consistent neural network, in which each parameter has a strict physical meaning, greatly enhancing the interpretability and generalization performance of the network; at the same time, prior knowledge constraints are added to the loss function, which significantly reduces the uncertainty and difficulty of inversion and provides strong technical support for system identification.

[0091] Compared with the current physics-based system identification method, the system identification method based on the physical consistent neural network of the present invention has clear physical meaning and strong generalization performance. It also has the advantage of being sensitive to local losses. Therefore, it is better to use and more convenient and quick in actual use. BRIEF DESCRIPTION OF THE DRAWINGS

[0092] Figure 1 Schematic diagrams related to Timoshenko beams: (A) Cantilever layered Timoshenko beam model representing a building; (B) The geometric shape of the deformed micro-beam element; (C) Schematic diagram of the force on the micro-beam element.

[0093] Figure 2Figure 1 shows the architecture of the proposed PCNN. Orange circles represent input neurons, green circles represent output neurons, and other blue circles represent hidden neurons. The symbol within each circle represents the output value of the corresponding neuron. The post-processing layer generates the final output by concatenating the displacement outputs U(zj) from each network layer, starting with the second hidden layer.

[0094] Figure 3 54-story office building in downtown Los Angeles, California: (A) Photograph, (B) west-east elevation, (C) floor plan with speed monitoring equipment.

[0095] Figure 4 The location of the 54-story office building (marked by a triangle with coordinates 61.21528°N and 149.89296°W) and the epicenters of the five selected earthquakes (marked by circles).

[0096] Figure 5 is the shear wave velocity VS of each layer identified from the five earthquake events, and the corresponding bending stiffness EI and shear stiffness GA.

[0097] Figure 6 Figure 3. Mean and CV of the shear wave velocity (VS) profiles, and the corresponding bending stiffness (EI) and shear stiffness (GA) profiles identified from the five earthquake events. (a) Mean values ​​of the shear wave velocity (VS), bending stiffness (EI), and shear stiffness (GA) identified from the five earthquake events, and (b) distribution plot of the corresponding coefficient of variation (CV).

[0098] Figure 7 Comparison of the transfer function (between roof and ground response) of the identified Timoshenko beam model for five earthquake events with the observed transfer function. DETAILED DESCRIPTION

[0099] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0100] A structural damage system identification method based on physically consistent neural networks is implemented in the following steps:

[0101] Step 1: Collect target building information and monitoring data:

[0102] Collect detailed geometric dimensions and spatial layout information of the target building, as well as historical acceleration responses recorded by acceleration monitors arranged in the target building.

[0103] In this embodiment, the target building selected is a 54-story office building located in downtown Los Angeles, California. Figure 3 As shown, the building has 54 above-ground floors (210.2 meters) and 4 basement floors (14.0 meters). Its plan shape is rectangular with two rounded corners. The dimensions of the building are 196 feet × 121 feet (59.7 meters × 36.9 meters) before the 36th floor, 176 feet × 121 feet (53.6 meters × 36.9 meters) between the 36th and 46th floors, and 156 feet × 121 feet (47.5 meters × 36.9 meters) above the 46th floor to the top floor. The horizontal acceleration responses in the north-south direction of each floor of the building were collected from five earthquake events with magnitudes between 6.4 and 7.3 between 1992 and 2010. The epicenter locations are shown in Figure 1. Figure 4 shown.

[0104] Step 2: Establish a physically consistent neural network:

[0105] The physically consistent neural network is consistent with the theory of a cantilevered layered Timoshenko beam (hereinafter referred to as a Timoshenko beam). It consists of an input layer, an output layer, and n hidden layers, where n corresponds to the number of layers in the cantilevered layered Timoshenko beam model. The parameters of the physically consistent neural network are related to the parameters of the Timoshenko model. Its input and output are the Fourier spectra corresponding to the horizontal acceleration responses of the top and bottom floors of the building, respectively.

[0106] (1) Cantilever layered Timoshenko beam model

[0107] The building is modeled as a cantilevered layered Timoshenko beam with its base subjected to horizontal seismic motion u g ,like Figure 1 The model consists of n layers, where the number of layers is equal to the number of horizontal acceleration monitors arranged on the building minus one. Each layer of the model is assumed to be uniform and isotropic, and adjacent layers are fully bonded. The layers of the model are numbered from top to bottom, and their geometric properties are represented by the cross-sectional area A. j , second-order moment of inertia I j , height H j , width W j and shear correction factor k G,j Characterization, where j = 1, ..., n represents the index of each layer. Shear correction factor k G The non-uniform distribution of shear stress on the cross section is taken into account. The material properties of each layer are given by the mass density ρ j , Young's modulus E j and shear modulus G j Characterized by mass density ρ j , longitudinal wave velocity VL,j and shear wave velocity V S,j Characterization. The relationship between modulus and wave velocity is defined as and The damping adopts the Kelvin-Voigt material model and assumes that the bending and shear deformations have the same viscosity constant μ j .

[0108] Timoshenko beam theory takes into account not only bending deformation but also rotational inertia and shear deformation. Figure 1 Figure (B) shows the geometry of a deformed tiny beam element at coordinate z. The beam's cross section is perpendicular to its neutral axis before deformation and remains planar after deformation. However, due to shear deformation, the deformed cross section is no longer perpendicular to the deformation axis. The beam's neutral axis is assumed to be displaced only horizontally, denoted by u(z,t). The total rotation of the beam consists of two components: the rotation θ(z,t) due to bending deformation and the rotation γ(z,t) due to shear deformation. Figure 1 (C) in the figure depicts the force diagram of the micro-beam element at coordinate z, where the dotted line represents the inertial force. In this figure, M(z,t) and V(z,t) represent the bending moment and shear force, respectively.

[0109] (2) Control equations and their analytical solutions

[0110] Under the small deformation assumption, formula (1) applies to any layer in the above layered model:

[0111]

[0112] The layer index j is omitted here for simplicity. Substituting the Kelvin-Voigt material model into formula (1) yields formula (2):

[0113]

[0114] according to Figure 1 From the force diagram in (C), we can get the dynamic equation shown in formula (3):

[0115]

[0116] By combining formula (2) and formula (3), the following results can be derived, as shown in formula (4) to formula (7):

[0117]

[0118]

[0119]

[0120]

[0121] Assuming that the load is simple harmonic excitation, the displacement u(z, t), angle θ(z, t), bending moment M(z, t) and shear force V(z, t) can be expressed by formulas (8) and (9):

[0122] u(z,t)=U(z)e -iωt ,θ(z,t)=Θ(z)e -iωt Formula (8)

[0123]

[0124] Where ω is the circular frequency. Substituting formula (8) and formula (9) into formula (4)-formula (7), we get formula (10):

[0125]

[0126] in, represents the state vector, and the matrix B is defined as:

[0127]

[0128] According to propagation matrix theory, if the matrix B is a continuous function of z (which is true in each layer), the state vector f(z) at any point in the same layer can be calculated using formula (11):

[0129] f(z)=P(z,z0)f(z0) Formula (11)

[0130] Where P(z,z0) is the propagation matrix from z0 to z. According to the general solution form of the ordinary differential equation given in formula (10), the form of the propagation matrix can be defined as shown in formula (12):

[0131]

[0132] Among them, e (·) represents the matrix exponential function. Given that the state vector must be continuous between adjacent layers, any interface z starting from the top j The state vector at can be determined as formula (13):

[0133]

[0134] For a cantilever beam, the free end (top) must satisfy the condition of zero stress, which means that the shear force and bending moment at this point are both zero. The bottom of the beam is the fixed end, so the rotation angle must be zero. Based on these conditions, the following formula can be derived:

[0135] f(z0)=[U0,Θ0,0,0] T Formula (14)

[0136]

[0137] Five of the physical quantities are non-zero. Assume that U0 is a known quantity, which is suitable for system identification tasks. Then, the other four quantities can be calculated according to formula (13). The solution process is divided into two main steps. First, the rotation angle Θ0 of the top is determined. Let's assume that:

[0138]

[0139] It can be deduced that:

[0140]

[0141] in, Representation matrix The element in the p-th row and q-th column of . Therefore, Θ0 can be derived as:

[0142]

[0143] According to formula (18), the state vector f(z0) at the top can be completely determined. Then, according to formula (13), the state vector at any interface can be obtained starting from the top.

[0144] (3) Transfer function

[0145] The transfer function (TF) is a mathematical representation of the relationship between the input and output of a linear time-invariant system in the frequency domain. Let u(z ref ,t) represents the reference point z ref The movement at coordinate point z is different from the movement at reference point z ref The transfer function between can be expressed as:

[0146]

[0147] To improve numerical stability and avoid division by very small numbers, the transfer function is often regularized as follows:

[0148]

[0149] Where, the horizontal bar represents the complex conjugate, and η represents the regularization parameter. The regularization parameter η is taken as U(z ref ,ω) is 0.1% of the average power spectral density.

[0150] (4) Construction of physically consistent neural networks

[0151] For a neural network with L+1 layers, the relationship between the lth layer and the (l+1)th layer can be expressed as formula (21):

[0152] z l =σ l (W l ·z l-1 +b l ),l=1,...,L Formula (21)

[0153] Among them, z 0 and z L Represent the input and output of the network respectively, W l and b l Represent the trainable weight matrix and bias vector of layer l respectively. Function σ l (·) is the nonlinear activation function of layer l, which is crucial for the network to learn complex mappings from input to output.

[0154] Based on the traditional neural network, a physical consistent neural network (PCNN) consistent with Timoshenko beam theory is proposed. Its structure is as follows: Figure 2 As shown in Figure 1. The network consists of an input layer, an output layer, and n hidden layers, where n corresponds to the number of layers in the cantilever layered Timoshenko beam model. Each hidden layer contains 4 neurons, which is equal to the number of elements in each state vector. The output of the lth hidden layer corresponds to the state vector at the top of the lth layer in the Timoshenko beam model. A post-processing layer is added after the output layer to connect the displacement output U(zj) of each network layer starting from the second hidden layer in order to calculate the subsequent loss function. It should be noted that the activation function of all layers is set to a linear function, and the bias vector is fixed to a non-trainable zero vector.

[0155] According to formula (18), the weight matrix of the first hidden layer of the PCNN can be derived as:

[0156]

[0157] The weight matrices of other hidden layers can be derived according to formula (16) as follows:

[0158] W j =P(z j-1 ,z j-2 ) Formula (23)

[0159] Among them, the value of j ranges from 2 to n. Similarly, the weight matrix of the output layer can be derived as follows:

[0160] W n+1 =p1(z n ,zn-1 ) T Formula (24)

[0161] Among them, p1(z n ,z n-1 ) represents the propagation matrix P(z n ,z n-1 ), where the superscript T indicates a transpose operation. As described above, each parameter in the PCNN has a clear physical interpretation and is related to the structural parameters of the Timoshenko beam model. Therefore, parameter optimization during network training effectively corresponds to the process of inverting the structural parameters. By minimizing the difference between the network output and the target value, the PCNN is able to identify the optimal structural parameters.

[0162] Step 3: Build a network training dataset:

[0163] The historical acceleration response recorded by the acceleration monitors placed in the target building collected in step 1 is used to obtain the amplitude of each frequency component using discrete Fourier transform. For a (digitized) acceleration record containing m equally spaced element values, it is decomposed into the following form:

[0164]

[0165] Among them, ω s is the sth equidistant frequency, a s and b s Respectively represent the frequency corresponding to ω s and -ω s The complex Fourier amplitude of a. s and b s Calculated by formula (30) and formula (31) respectively:

[0166]

[0167]

[0168] The input dataset is constructed based on the amplitude of each frequency component of the acceleration response of the top floor of the building, and the label dataset is constructed using the amplitude of the corresponding frequency components of the acceleration response of the remaining floors.

[0169] Step 4: Conduct physically consistent neural network training:

[0170] The loss function used to guide network training consists of data loss and physical constraint loss. Data loss is defined as follows:

[0171]

[0172] in, and denotes the sth element in the observed and predicted non-top acceleration response Fourier spectra, respectively. M denotes the total number of elements in these Fourier spectra. R(·) and I(·) denote the real and imaginary part operations, respectively.

[0173] In actual physical scenarios, the shear wave velocity V of each layer S,j and modulus ratio R j The following relationship must be satisfied:

[0174] V S,j >0,R j >0 Formula (27)

[0175] The value of j ranges from 1 to n. In order to ensure that the identified parameters meet these constraints, in addition to the data loss, a physical constraint loss is introduced to enhance the reliability of the recognition results. The formula is as follows:

[0176]

[0177] in, and R min are the shear wave velocity V S,j and modulus ratio R j The lower limit of the search space is set to 0 in this embodiment. ReLU(·) represents the rectified linear unit function.

[0178] The final loss function of the PCNN is:

[0179] L all =λ1×L PC +λ2×L data Formula (29)

[0180] Among them, λ1 and λ2 are weight factors, which are set to 1 and 0.001 respectively. By minimizing the final loss function, PCNN can effectively identify structural parameters. The shear wave velocity V of each layer of the Timoshenko beam model during training S The initial values ​​of the velocity and modulus ratio R are set to 100 m / s and 0.01 respectively, and the remaining parameters including density ρ and viscosity constant μ are assumed to be 300 kg / m 3 Only data in the 0.1-1.7 Hz frequency band is used for training, and the Adam optimizer and learning rate decay strategy are used for training. The shear wave velocity V S The initial learning rates of and modulus ratio R are set to 100 and 10-5 respectively. The learning rate decays to 0.1 times the original value every 50 training rounds, and the total number of training rounds is 150.

[0181] Step 5: Obtain inverted structural parameters:

[0182] Since the PCNN parameter weights are related to the shear wave velocity V S Related to the modulus ratio R, when the PCNN training is completed, the shear wave velocity V of each layer of the Timoshenko beam model is obtained through the network parameters S and modulus ratio R.

[0183] Table 1 summarizes the identification results, showing a high degree of consistency between different earthquake events. S Observation of the cross section shows that V S Instead of increasing monotonically from the top, there is an inversion between the second and third layers, that is, V S The first layer has a significantly higher modulus ratio than the third layer. Furthermore, analysis of the modulus ratio profile shows that the first layer has a significantly higher modulus ratio than the other layers. The last two columns of Table 1 present the statistical characteristics of the recognition parameters. It can be seen that the coefficient of variation (CV) for all recognition parameters is less than 6%, indicating low variability in the recognition results. This confirms the excellent robustness of the proposed PCNN.

[0184] Table 1. Parameters identified using the North-South (NS) acceleration waveforms (frequency range 0.1-1.7 Hz) of five earthquakes recorded in a 54-story office building.

[0185]

[0186] Figure 5 The shear wave velocity V of each layer is graphically shown S , bending stiffness (EI), and shear stiffness (GA), where the histograms are arranged from bottom to top in chronological order of the earthquake events. It can be observed that the VS profile identified from the Big Bear earthquake shows a significant decrease compared to the Landers earthquake, indicating that the profiles are arranged from bottom to top in chronological order of the earthquake events. It can be observed that the Big Bear earthquake may have caused a permanent decrease in stiffness compared to the Landers earthquake, as reflected in both the EI and GA profiles. A similar decrease in VS was also observed after the Northridge earthquake, suggesting that this event may have also caused a permanent decrease in stiffness. In contrast, no significant decrease in VS was observed during the last two earthquake events. Further analysis of the EI and GA profiles revealed similar trends. Furthermore, during the Hectormine earthquake, the VS of the third floor was found to be lower than that of the second floor, while the GA of the third floor was higher than that of the second floor. This difference is due to the building's gradually decreasing cross-sectional area from the base upwards, resulting in a larger cross-sectional area on the third floor than on the second floor.

[0187] Figure 6The mean and CV values ​​of the shear wave velocity (VS) profiles identified during the five earthquake events are presented, along with the corresponding bending stiffness (EI) and shear stiffness (GA) profiles. Notably, the previously discussed VS inversion phenomenon between the second and third layers also appears in the EI and GA profiles, despite the smaller cross-sectional area of ​​the second layer. A possible explanation for this phenomenon is the unique structural configuration between the 36th and 46th floors, which corresponds to the second layer in the Timoshenko beam model. Furthermore, a comparison of the CV values ​​of the three profiles reveals that the CV values ​​of the EI and GA profiles are higher than those of the VS profile, with overall variability decreasing from top to bottom. Furthermore, the CV values ​​of both stiffness profiles are below 10%, further validating the robustness of the proposed PCNN model.

[0188] The dominant frequency of a building can be determined by analyzing the transfer function between the roof and ground responses, as this function will have a peak at the dominant frequency of the building. Figure 7 The transfer functions of the Timoshenko beam model identified from five earthquake events were compared with the transfer functions in the 0.1–1.7 Hz frequency band obtained from the observed responses. It can be seen that due to the influence of noise, the observed transfer functions also show significant peaks at non-dominant frequencies. For example, the transfer function of the Calexico earthquake exhibits a large peak in the 0.25–0.45 Hz frequency band. However, the identified transfer functions still match the observed values ​​overall, and the identified dominant frequencies are very close to the observed values, which is particularly evident in the Northridge earthquake event.

[0189] Table 2 presents a quantitative comparative analysis of the first two main frequencies identified in the five earthquake events. Since the third to fifth main frequencies are difficult to distinguish in most earthquake events, the analysis was not extended to these frequencies. From the analysis, it can be observed that the first two main frequencies identified in the Northridge earthquake show a significant decrease compared to the main frequencies in the Lander-Big Bear earthquake sequence, indicating that the Northridge earthquake may have caused damage to buildings. In contrast, the first two main frequencies identified in the Calexico earthquake show a slight increase compared to the main frequencies in the Hector Mine earthquake, which may be attributed to measurement errors or repair work performed before the Calexico event. Table 2 also shows the relative errors between the observed and identified first two main frequencies in the five earthquakes. The maximum relative error does not exceed 7%, further validating the reliability of the proposed PCNN model.

[0190] Table 2. Observed and identified dominant frequencies in the five earthquake events.

[0191]

[0192] The relative error is defined as the absolute difference between the observed dominant frequency and the identified dominant frequency divided by the observed dominant frequency.

Claims

1. A structural damage system identification method based on a physically consistent neural network, characterized in that: The following steps are involved: Step 1: Collect target building information and monitoring data: Collect detailed geometric dimensions and spatial layout information of the target building, as well as historical acceleration responses recorded by acceleration monitors placed inside the target building; Step 2: Establish a physically consistent neural network: The physically consistent neural network is consistent with the cantilever layered Timoshenko beam theory and consists of an input layer, an output layer, and n hidden layers, where n corresponds to the number of layers in the cantilever layered Timoshenko beam model. The parameters of the physically consistent neural network are related to the parameters of the Timoshenko model, and its input and output are the Fourier spectra corresponding to the horizontal acceleration responses of the top and bottom floors of the building, respectively. Step 3: Build a network training dataset: The amplitude of each frequency component is obtained by using discrete Fourier transform on the historical acceleration response recorded by the acceleration monitor arranged in the target building collected in step 1; Step 4: Conduct physically consistent neural network training: The loss function used to guide network training consists of data loss and physical constraint loss; data loss is defined as follows: in, and denotes the sth element in the observed and predicted non-top acceleration response Fourier spectra, respectively; M denotes the total number of elements in these Fourier spectra; R(·) and Ι(·) denote the real and imaginary part operations, respectively; The physical constraint loss is defined as follows: in, and R min are the shear wave velocity V S,j and modulus ratio R j The lower bound of the search space; ReLU(·) represents the rectified linear unit function; The final loss function is: L all =λ1×L PC +λ2×L data Formula (29) Among them, λ1 and λ2 are weight factors; Step 5: Obtain inverted structural parameters: Since the parameter weights of the physical consistent neural network PCNN are related to the shear wave velocity V S Related to the modulus ratio R, when the PCNN training is completed, the shear wave velocity V of each layer of the Timoshenko beam model is obtained through the network parameters S and modulus ratio R.

2. The structural damage system identification method based on physically consistent neural network according to claim 1 is characterized in that: The specific method for establishing a physically consistent neural network in step 2 is: The building is modeled as a cantilevered layered Timoshenko beam with its base subjected to horizontal seismic motion u g , the model contains n layers, the number of which is equal to the number of horizontal acceleration monitors arranged on the building minus one; each layer of the model is assumed to be uniform and isotropic, and adjacent layers are fully bonded; The layers of the model are numbered from top to bottom, and their geometric properties are determined by the cross-sectional area A. j , second-order moment of inertia I j , height H j , width W j and shear correction factor k G,j Characterization, where j = 1, ..., n represents the index of each layer; the shear correction factor k G The non-uniform distribution of shear stress on the cross section is taken into account; the material properties of each layer are determined by the mass density ρ j , Young's modulus E j and shear modulus G j Characterized by mass density ρ j , longitudinal wave velocity V L,j and shear wave velocity V S,j Characterization; the relationship between modulus and wave velocity is defined as and The damping adopts the Kelvin-Voigt material model and assumes that the bending and shear deformations have the same viscosity constant μ j ; The cross section of the beam is perpendicular to the neutral axis before deformation and remains planar after deformation. However, due to shear deformation, the cross section is no longer perpendicular to the deformation axis after deformation. The neutral axis of the beam is assumed to undergo only horizontal displacement, denoted by u(z,t). The total rotation of the beam consists of two components: the rotation θ(z,t) due to bending deformation and the rotation γ(z,t) due to shear deformation; M(z,t) and V(z,t) represent the bending moment and shear force, respectively. Under the small deformation assumption, formula (1) applies to any layer in the above layered model: Substituting the Kelvin-Voigt material model into formula (1) yields formula (2): According to the force, the dynamic equation shown in formula (3) can be obtained: By combining formula (2) and formula (3), the following results can be derived, as shown in formula (4) to formula (7): Assuming that the load is simple harmonic excitation, the displacement u(z, t), angle θ(z, t), bending moment M(z, t) and shear force V(z, t) can be expressed by formulas (8) and (9): u(z,t) = U(z)e -iωt , θ(z,t) = Θ(z)e -iωt Equation (8) Where ω is the circular frequency. Substituting formula (8) and formula (9) into formula (4)-formula (7), we obtain formula (10): in, represents the state vector, and the matrix B is defined as: According to propagation matrix theory, if the matrix B is a continuous function of z (which is true in each layer), the state vector f(z) at any point in the same layer can be calculated using formula (11): f(z)=P(z,z0)f(z0) Formula (11) Where P(z,z0) is the propagation matrix from z0 to z. According to the general solution form of the ordinary differential equation given in formula (10), the form of the propagation matrix can be defined as shown in formula (12): Among them, e (·) represents the matrix exponential function; given that the state vector must be continuous between adjacent layers, any interface z starting from the top j The state vector at can be determined as formula (13): For a cantilever beam, the free end (top) must satisfy the condition of zero stress, which means that the shear force and bending moment at this point are both zero; the bottom of the beam is the fixed end, so the rotation angle must be zero; based on these conditions, the following formula can be derived: f(z0)=[U0,Θ0,0,0] T Official(14) Five of the physical quantities are non-zero; assume that U0 is a known quantity, which is applicable to the system identification task; then, the other four quantities can be calculated according to formula (13); the solution process is divided into two main steps. First, determine the rotation angle Θ0 of the top; let us assume that: It can be deduced that: in, Representation matrix The element in the p-th row and q-th column of ; therefore, Θ0 can be derived as: According to formula (18), the state vector f(z0) at the top can be completely determined; then according to formula (13), the state vector at any interface can be obtained starting from the top; Let u(z ref ,t) represents the reference point z ref The movement at coordinate point z and the reference point z ref The transfer function between can be expressed as: To improve numerical stability and avoid division by very small numbers, the transfer function is often regularized as follows: Wherein, the horizontal bar represents the complex conjugate, and η represents the regularization parameter; For a neural network with L+1 layers, the relationship between the lth layer and the (l+1)th layer can be expressed as formula (21): z l = σ l (W l · z l-1 + b l ), l = 1, ..., L Equation (21) Among them, z 0 and z L Represent the input and output of the network respectively, W l and b l Represent the trainable weight matrix and bias vector of layer l respectively; function σ l (·) is the nonlinear activation function of layer l; A physically consistent neural network is established, consisting of an input layer, an output layer, and n hidden layers, where n corresponds to the number of layers in the cantilevered layered Timoshenko beam model. Each hidden layer contains four neurons, which is equal to the number of elements in each state vector. The output of the lth hidden layer corresponds to the state vector at the top of the lth layer in the Timoshenko beam model. A post-processing layer is added after the output layer to concatenate the displacement output U(zj) of each network layer starting from the second hidden layer to calculate the subsequent loss function. According to formula (18), the weight matrix of the first hidden layer of the physically consistent neural network can be derived as: The weight matrices of other hidden layers can be derived according to formula (16) as follows: W j =P(z j-1 ,z j-2 ) Formula (23) Among them, the value range of j is 2 to n; similarly, the weight matrix of the output layer can be derived as follows: W n+1 =p1(z n ,z n-1 ) T Formula (24) Among them, p1(z n ,z n-1 ) represents the propagation matrix P(z n ,z n-1 ), the superscript T indicates the transpose operation.

3. The structural damage system identification method based on physically consistent neural network according to claim 2 is characterized in that: The specific method for constructing the network training data set in step three is: For a (digitized) acceleration record containing m equally spaced element values, the decomposition is as follows: Among them, ω s is the sth equidistant frequency, a s and b s Respectively represent the frequency corresponding to ω s and -ω s The input dataset is constructed based on the amplitude of each frequency component of the acceleration response of the top floor of the building, and the label dataset is constructed using the amplitude of the corresponding frequency components of the acceleration response of the remaining floors.

4. The structural damage system identification method based on physically consistent neural network according to claim 3 is characterized in that: In formula (25), a s and b s Calculated by formula (30) and formula (31) respectively:

5. The structural damage system identification method based on physically consistent neural network according to claim 1 is characterized in that: In step 4, in the actual physical scene, the shear wave velocity V of each layer S,j and modulus ratio R j The following relationship must be satisfied: V S,j >0,R j >0 Formula (27) The value of j ranges from 1 to n.

6. The structural damage system identification method based on physically consistent neural network according to claim 1 is characterized in that: In step 4, the Adam optimizer and learning rate decay strategy are used for training, and the shear wave velocity V S The initial learning rates of and modulus ratio R are set to 100 and 10-5 respectively. The learning rate decays to 0.1 times the original value every 50 training rounds, and the total number of training rounds is 150.

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