Bridge sensor self-optimization arrangement and state evaluation method based on modal confidence criterion

Abstract

The application discloses a bridge sensor self-optimization arrangement and state evaluation method based on a modal confidence criterion, and relates to the technical field of big data analysis and processing; the method comprises the following steps: step one, collecting vertical acceleration time series at each sensor candidate installation coordinate through a preset vehicle excitation; step two, determining a bridge sensor self-optimization arrangement result by using a physical information neural network to drive a sparse damage reconstruction and sensor importance joint optimization algorithm; and step three, installing vertical acceleration sensors on a target bridge according to the bridge sensor self-optimization arrangement result, dividing bridge state grades according to finite element unit damage estimation results, and outputting bridge state evaluation results. The application can be continuously used during the operation period of the bridge, provides quantitative and traceable structure state evaluation results, helps to discover potential risks in advance, and significantly improves the reliability, practicality and intelligent level of bridge health monitoring.

Description

Technical Field

[0001] This invention relates to the field of big data analysis and processing technology, and in particular to a method for self-optimizing bridge sensor placement and state assessment based on modal confidence criteria. It is a robust method for self-optimizing sensor placement and state assessment. Background Technology

[0002] Bridges endure various factors throughout their service life, including vehicle loads, temperature variations, material aging, and environmental erosion. Inevitably, internal structural degradation occurs, leading to crack propagation, loosening of connections, and cross-sectional damage. To monitor bridge structural condition without disrupting normal traffic, conventional bridge health monitoring methods widely employ a monitoring network composed of accelerometers, displacement sensors, and strain sensors. Vibration response analysis is used to infer the bridge's stiffness distribution, damage location, and severity. Modal parameter-based structural health monitoring is currently a widely adopted approach in engineering, utilizing indicators such as mode shapes, frequencies, and damping ratios to reflect overall structural performance. However, in practical engineering applications, traditional modal analysis methods typically rely on a limited number of sensor points. Improper sensor placement can result in insufficient mode shape reconstruction accuracy, thus reducing the reliability of damage identification. Furthermore, due to the large spans of bridge structures and the significant environmental influences on vibration signals, obtaining vibration data that meets high stability requirements is often difficult, leading to large fluctuations in modal parameters and unstable damage identification results.

[0003] To address the sensor placement problem, existing technologies typically employ methods such as the effective independence method, genetic algorithms, and particle swarm optimization to optimize sensor placement and improve modal identification quality by enhancing modal observability. While these algorithms can optimize the measurement point layout to some extent, they still have significant limitations. For example, the effective independence method only optimizes the linear independence of the modal matrix and cannot simultaneously consider frequency identification stability and modal quality under noisy conditions. Although genetic algorithms and particle swarm optimization possess global search capabilities, their optimization objectives are usually based solely on finite element modal data. However, there are model errors between finite element analysis and actual bridge structures, leading to deviations in optimization results in practical applications. Furthermore, none of the above methods fully utilize actual monitoring data and physical constraints, thus failing to handle situations where sensor sampling is insufficient or measurement point locations are limited by construction conditions. In terms of damage identification, traditional methods often rely on single indicators such as frequency changes, modal curvature changes, and compliance matrix changes for judgment. These methods depend on high-quality modal parameters, but in operational bridges, traffic disturbances, temperature and humidity fluctuations, and measurement noise can all cause changes in modal parameters, obscuring damage characteristic signals. For example, frequency variations often fail to distinguish between structural damage and environmental influences, while modal curvature methods are sensitive to higher-order modes, which are often difficult to reliably identify in actual measurements. Furthermore, to address the problem of insufficient data, some studies have attempted to recover sparse damage distributions using compressed sensing methods; however, these methods require specific conditions of mutual incompatibility, which are difficult to fully satisfy in real-world bridges, resulting in limited reconstruction accuracy. Summary of the Invention

[0004] The purpose of this invention is to provide a method for self-optimizing bridge sensor placement and condition assessment based on modal confidence criteria. This method utilizes a physical information neural network trained with actual monitoring data and dynamic constraints to achieve high-precision reconstruction of bridge vibration response, and filters unreliable modes based on modal confidence criteria, making damage identification more stable. Combined with a sparse damage reconstruction model formed by unit sensitivity response sequences, it can accurately locate damage areas even with a limited number of sensors.

[0005] To address the aforementioned technical problems, this invention provides a method for self-optimizing bridge sensor placement and state assessment based on modal confidence criteria. This method includes: Step 1: Establish a finite element model of the target bridge in the computer, divide the candidate installation coordinates of the sensor according to the main beam structure of the bridge, install vertical acceleration sensors on the target bridge according to the candidate installation coordinates, and collect the vertical acceleration time series at each candidate installation coordinate by preset vehicle excitation. Step 2: Train a physical information neural network modal analysis model based on the bridge finite element model and vertical acceleration time series. Construct modal confidence criteria using the physical information neural network modal analysis model and the bridge finite element model. Establish a sparse damage reconstruction model based on the bridge finite element model. Use a physical information neural network-driven sparse damage reconstruction and sensor importance joint optimization algorithm to determine the self-optimization layout result of bridge sensors. Step 3: Install vertical acceleration sensors on the target bridge according to the bridge sensor self-optimization layout results. During the bridge operation, periodically collect vertical acceleration time series. Input the vertical acceleration time series and the bridge finite element model into the physical information neural network modal analysis model. Under the constraints of the modal confidence criterion, obtain the finite element damage estimation results through the sparse damage reconstruction model. Based on the finite element damage estimation results, classify the bridge state level and output the bridge state assessment results.

[0006] Furthermore, step one is performed as follows: The design information of the target bridge is input into the finite element analysis software to establish a three-dimensional finite element model of the bridge, including the main beam, piers and bearings. Multiple finite element elements are divided along the axis of the main beam at a preset equal interval. A candidate sensor installation coordinate is set at the neutral axis position of the top plate of the main beam in each finite element element. A vertical acceleration sensor is installed at the corresponding position. The vertical acceleration time series at each candidate sensor installation coordinate is collected at a uniform time sampling interval under the action of a preset load time history, and then uniformly stored in the data acquisition system.

[0007] Furthermore, the physical information neural network modal analysis model in step two has a feedforward fully connected structure, including at least an input layer, three or more hidden layers, and an output layer. The input quantities are the coordinates of the bridge main beam axis, the vertical position coordinates, and time, and the output quantities are the vertical displacement and vertical acceleration at the corresponding positions and times.

[0008] Furthermore, the physical information neural network modal analysis model is trained in the following way: Multiple internal segmentation points are set along the main beam axis in the bridge finite element model. These internal segmentation points are combined with the support positions to form a set of physical constraint sampling points. This set of physical constraint sampling points is combined with the time sampling point set to form a physical constraint sampling grid. On this grid, the difference between the inertial force, damping force, and elastic restoring force and the external load is calculated using the bridge finite element model to obtain the physical residual target. At the combined location of the sensor candidate installation coordinates and the time sampling points, the vertical acceleration time series is used as the measurement residual target. In each training round, the coordinates and time of the physical constraint sampling grid and the measurement residual target are used as inputs to calculate and output the vertical displacement and vertical acceleration. The vertical acceleration estimate is obtained from the vertical displacement time series through numerical differentiation. The sum of the absolute values ​​of the differences between the vertical acceleration estimate and the vertical acceleration time series, as well as the sum of the absolute values ​​of the physical residuals, are calculated. The sum of these two values ​​is used as the loss value. The adjustable coefficients within the physical information neural network modal analysis model are updated through backpropagation and gradient descent until the loss value meets the preset stopping condition.

[0009] Furthermore, in step two, the position coordinates and time sampling points at each candidate sensor installation coordinate are input into the trained physical information neural network modal analysis model to obtain the predicted vertical displacement time series. The Fast Fourier Transform is applied to the predicted vertical displacement time series to obtain the correspondence between frequency and amplitude, and multiple principal vibration frequencies corresponding to the amplitude peaks are identified. The complex amplitudes of each principal vibration frequency are arranged in the order of the candidate sensor installation coordinates at all candidate sensor installation coordinates to form the measured modal vector. The theoretical modal vectors corresponding one-to-one with each order of principal vibration frequency are calculated in the bridge finite element model.

[0010] Furthermore, in step two, the correlation index is calculated using the dot product of the measured modal vector and the theoretical modal vector, as well as the modal lengths of the measured and theoretical modal vectors. The frequency deviation is calculated using the absolute value of the difference between the measured principal vibration frequency and the theoretical principal vibration frequency. The physical consistency index is calculated using the average absolute value of the physical residuals at the physical constraint sampling grid points in the region where the principal vibration mode amplitude exceeds the preset amplitude threshold. When the correlation index reaches the preset correlation threshold and the frequency deviation does not exceed the preset frequency deviation threshold and the physical consistency index does not exceed the preset physical residual threshold, the current order mode is included in the modal set allowed by the modal confidence criterion. For subsequent sparse damage reconstruction and sensor importance assessment, only the modal set allowed by the modal confidence criterion is used.

[0011] Furthermore, in step two, a set of finite element elements corresponding one-to-one with the axis of the main beam is determined in the bridge finite element model. For each finite element, the bending stiffness of the corresponding finite element is reduced to a preset ratio while keeping other finite element elements in an undamaged state. Under the load time history corresponding to the preset vehicle excitation in step one, the vertical displacement time series at all candidate sensor installation coordinates is calculated. The vertical displacement time series in the undamaged state is subtracted from the vertical displacement time series after the stiffness of the corresponding finite element is reduced to obtain the element sensitivity response value. The element sensitivity response values ​​at each candidate sensor installation coordinate and each time sampling point are arranged in a unified order to form an element sensitivity response sequence. The operation of reducing the bending stiffness and calculating the element sensitivity response sequence is repeated for all finite element elements to form a set of element sensitivity response sequences.

[0012] Furthermore, in step two, the trained physical information neural network modal analysis model is used to calculate the predicted vertical displacement time series of the non-destructive bridge finite element model under load time history at all candidate sensor installation coordinates. The predicted vertical displacement time series is subtracted from the vertical displacement time series obtained by directly solving the non-destructive bridge finite element model to obtain the measurement difference time series. The measurement difference time series is used as the observation vector, the set of element sensitivity response sequences is used as the basis vector set, and the sparse damage reconstruction model is constructed with the finite element damage coefficient as the unknown.

[0013] Furthermore, step three includes the bridge state estimation process during the operation phase. The bridge state estimation process includes the following steps: installing vertical acceleration sensors at the neutral axis position of the top plate of the main beam of the target bridge according to the bridge sensor self-optimization arrangement results; periodically collecting the vertical acceleration time series at the installation coordinates of each sensor at the same time sampling interval as in step one; inputting the vertical acceleration time series and the bridge finite element model into the physical information neural network modal analysis model to obtain the predicted vertical displacement time series under the current non-destructive reference state; constructing the current measurement difference time series using the predicted vertical displacement time series and the vertical displacement time series obtained by the bridge finite element model under the current load; constructing the current element sensitivity response sequence set under the constraints of the modal confidence criterion and executing the sparse damage reconstruction iteration process to obtain the current finite element damage estimation result.

[0014] The bridge sensor self-optimization layout and state assessment method based on modal confidence criteria of this invention has the following beneficial effects: By simultaneously introducing measurement constraints and dynamic balance constraints during the training phase, the physical information neural network modal analysis model can still reconstruct vertical displacement and vertical acceleration that conform to the laws of structural dynamics even when the data is relatively sparse, thereby effectively reducing the distortion of modal extraction caused by noise, environmental changes, and finite element model errors. The modal confidence criterion constructed on this basis not only relies on frequency and morphological similarity but also incorporates the dynamic balance error in high-response regions, making the modal screening process more reliable and actively eliminating pseudo-modalities caused by interference, thus improving the stability of damage identification. By constructing a sparse damage reconstruction model using the set of element sensitivity response sequences based on the bridge finite element model and the measurement difference time series, the damage degree and distribution location of finite element elements can be accurately identified even with limited observation data. A joint optimization algorithm for sparse damage reconstruction and sensor importance driven by a physical information neural network can automatically select the sensor installation coordinates with the greatest impact on the damage reconstruction results from a complete pool of candidate monitoring points. This eliminates reliance on experience-based judgments or single optimization metrics in the sensor placement process, instead focusing on damage identification capabilities as the core objective, significantly improving the overall efficiency of the monitoring system. The bridge condition assessment method developed in this invention can process monitoring data in real time during operation and quantify and classify the safety status of bridge structures. This allows maintenance personnel to obtain reliable structural condition information without relying on manual experience, facilitating the early detection of potential risks and the implementation of targeted measures. Furthermore, the entire process is based on unified physical constraints and data-driven principles, enabling the model to continuously adapt to the actual structural condition. It is applicable to various bridge structure types and possesses good scalability and engineering application value. Attached Figure Description

[0015] Figure 1 This is a schematic diagram showing the comparison of sensor arrangement before and after optimization in an embodiment of the present invention; Figure 2 This is a schematic diagram of vertical displacement time series FFT frequency domain analysis and modal frequency identification provided in an embodiment of the present invention; Figure 3 A schematic diagram comparing the measured and theoretical vibration modes of the first-order vertical bending mode provided in an embodiment of the present invention; Figure 4 This is a schematic diagram of the element sensitivity response time series of a typical finite element element provided in an embodiment of the present invention. Detailed Implementation

[0016] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0017] A bridge sensor self-optimization arrangement and state assessment method based on modal confidence criteria, comprising: Step 1: Establish a finite element model of the target bridge in the computer, divide the candidate installation coordinates of the sensor according to the main beam structure of the bridge, install vertical acceleration sensors on the target bridge according to the candidate installation coordinates, and collect the vertical acceleration time series at each candidate installation coordinate by preset vehicle excitation. Step 2: Train a physical information neural network modal analysis model based on the bridge finite element model and vertical acceleration time series. Construct modal confidence criteria using the physical information neural network modal analysis model and the bridge finite element model. Establish a sparse damage reconstruction model based on the bridge finite element model. Use a physical information neural network-driven sparse damage reconstruction and sensor importance joint optimization algorithm to determine the self-optimization layout result of bridge sensors. Step 3: Install vertical acceleration sensors on the target bridge according to the bridge sensor self-optimization layout results. During the bridge operation, periodically collect vertical acceleration time series. Input the vertical acceleration time series and the bridge finite element model into the physical information neural network modal analysis model. Under the constraints of the modal confidence criterion, obtain the finite element damage estimation results through the sparse damage reconstruction model. Based on the finite element damage estimation results, classify the bridge state level and output the bridge state assessment results.

[0018] In one implementation, the original design data of the target bridge is first obtained, including the bridge's total length, span arrangement, main girder type, bridge deck width, transverse diaphragm arrangement, pier and abutment arrangement, support type, and design load level. Based on this data, a finite element model of the target bridge is established in finite element analysis software. During modeling, the main girder is constructed as beam element or solid element model according to its actual cross-sectional form, and the piers and abutments are constructed as corresponding column or pile cap elements. The connection conditions between the main girder and the supports are set through hinged, fixed, or elastic connections, so that the bridge finite element model can reflect the actual stress and deformation behavior of the target bridge in terms of vertical stiffness and vertical vibration characteristics.

[0019] In the aforementioned bridge finite element model, the model is discretized along the main beam axis, starting from the center of the main beam support on one abutment. For example, in a specific implementation, the main beam axis can be divided into several segments at 2-meter intervals. The midpoint of each segment serves as a candidate sensor installation coordinate. Correspondingly, in the bridge finite element model, a vertical displacement degree of freedom and a vertical acceleration output are set at the neutral axis of the top plate of the main beam where the midpoint of each segment is located. This way, the established candidate sensor installation coordinates correspond one-to-one with the degree of freedom positions in the bridge finite element model, facilitating the direct correlation between the measured vertical acceleration time series and the finite element calculation results. Furthermore, the uniform 2-meter intervals cover the entire bridge length without making the number of candidate sensor installation coordinates too large, thus avoiding excessive complexity in subsequent optimization calculations.

[0020] In another implementation approach, the density of candidate sensor installation coordinates in the mid-span region can be increased for long bridges. For example, candidate sensor installation coordinates can be divided at 1-meter intervals within the mid-span third of the span, and at 3-meter intervals within the third of the span closer to the supports. With this unequal spacing, the mid-span region, due to its larger bending moment and deflection, is more sensitive to overall vertical vibration and damage. Increasing the density of candidate sensor installation coordinates in the mid-span region can improve modal identification accuracy and damage location accuracy. Conversely, the region near the supports, with relatively smaller responses, can have a smaller spacing to reduce the number of candidate sensor installation coordinates, thus achieving a balance between monitoring accuracy and engineering implementation costs.

[0021] refer to Figure 1 , Figure 1 This diagram illustrates the comparison between the sensor arrangement before and after optimization. The diagram uses a two-part comparison to show the advantages of the self-optimized sensor arrangement method of this invention compared to the traditional equidistant arrangement method. Figure 1The upper part of the diagram shows the sensor layout before optimization. In this layout, the main girder of the bridge is drawn as a horizontal straight line along the longitudinal direction. The bridge is 200 meters long, and piers or abutments are located at 0 meters, 60 meters, 120 meters, and 200 meters, respectively. These support positions are connected to the axis of the main girder by short vertical lines, and solid square marks are marked at the connection points between the main girder axis and the supports to indicate the support positions. In the original, evenly spaced layout, sensors were arranged at fixed intervals of 10 meters, starting from the 10-meter position and evenly distributed at 20 meters, 30 meters, 40 meters, and up to 190 meters, for a total of 20 sensor installation positions. These sensor installation positions are represented by hollow circles in the diagram. All circles are the same size and are evenly spaced along the axis of the main girder, reflecting the uniformity and regularity of sensor position selection in the traditional layout method. The advantage of this equidistant arrangement method is that the arrangement rules are simple and easy to implement. However, its disadvantage is that it does not take into account the difference in contribution of different locations on the bridge to damage identification, which may result in insufficient sensor density in damage-sensitive areas and redundant sensor arrangement in non-critical areas. Figure 1 The lower half of the diagram shows the sensor arrangement scheme optimized by the method of this invention. In this scheme, the basic structure of the bridge remains consistent with the upper half, and the main beam axis, support positions, and total bridge length are also marked. The optimized sensor arrangement scheme also includes 20 sensors, but the distribution of these sensors exhibits a significant non-uniformity. As can be observed from the figure, the sensor density increases significantly in the mid-span region, around 100 meters, forming relatively dense sensor clusters at positions such as 90 meters, 100 meters, and 110 meters. Simultaneously, the sensor density is also relatively high in the areas near each support, such as at 5 meters and 12 meters near the 0-meter support, at 50 meters, 58 meters, 62 meters, and 75 meters near the 60-meter support, at 142 meters and 150 meters near the 120-meter support, and at 188 meters and 195 meters near the 200-meter support. In contrast, the sensor spacing is relatively large in the transition area between the mid-span and the supports. This non-uniform arrangement pattern is calculated based on the importance index of sensors. Under the condition of limited number of sensors, it can prioritize the allocation of limited sensor resources to the positions that contribute the most to modal identification and damage reconstruction, thereby obtaining higher damage identification accuracy and more reliable bridge condition assessment results under the condition of the same number of sensors.

[0022] After determining the candidate sensor installation coordinates in the bridge finite element model, these coordinates are projected onto the actual structure of the target bridge. Specifically, in the construction drawings of the target bridge, based on the mileage marker of the main beam axis or the measured length of the main beam on site, the distance from the inner edge of the expansion joint on one abutment towards the center of the span is measured, for example, 3 meters, and this coordinate is marked at the neutral axis position of the top plate of the main beam. Subsequently, the positions of subsequent candidate sensor installation coordinates are measured and marked according to the intervals, until the mid-span and near the abutment on the other side, ensuring that the marked positions on site correspond one-to-one with the positions of the candidate sensor installation coordinates in the bridge finite element model along the bridge direction. At the same time, in the transverse direction, the candidate sensor installation coordinates are fixed on the longitudinal centerline of the neutral axis position of the top plate of the main beam to avoid the influence of torsional response caused by transverse asymmetry, so that the vertical acceleration sensor mainly reflects the vertical bending vibration of the main beam.

[0023] In one specific implementation, a vertical accelerometer is installed at each candidate sensor installation coordinate. During installation, the concrete or steel plate surface at the neutral axis position of the main beam top plate is first ground and cleaned to remove laitance and rust, ensuring a tight fit between the vertical accelerometer base and the structural surface. Then, the vertical accelerometer base is secured using epoxy-based high-strength structural adhesive or mechanical bolts, ensuring that the sensitive axis of the vertical accelerometer is strictly perpendicular to the bridge deck, thereby guaranteeing that the acquired signal is primarily vertical acceleration. To reduce environmental interference with the acquisition results, a waterproof and dustproof cover can be installed outside the vertical accelerometer, and the sensor signal cable can be routed through a sealed joint into a nearby cable tray or bridge deck conduit.

[0024] The vertical acceleration sensor is connected to the data acquisition device via a shielded signal cable. The data acquisition device is set to a uniform sampling frequency, such as 200 Hz or 500 Hz. The sampling frequency is chosen to be more than twice the first to several higher-order modal frequencies of the target bridge, which can fully meet the needs of signal spectrum reconstruction. According to the Nyquist sampling theory, the sampling frequency should be at least twice the highest vibration frequency of interest. In this embodiment, 200 Hz or 500 Hz can cover most of the vertical natural frequency range of highway bridges and retain sufficient high-frequency content for noise analysis and modal identification. All data acquisition channels are synchronized through a unified clock source to ensure the alignment of the vertical acceleration time series at each candidate sensor installation coordinate on the time axis, enabling subsequent modal analysis and sparse damage reconstruction to utilize the phase relationship and relative amplitude information between the measurement points.

[0025] refer to Figure 2 , Figure 2This diagram illustrates the FFT frequency domain analysis and modal frequency identification of vertical displacement time series. It shows the process of converting the vertical displacement time series collected by the sensor to the frequency domain and then identifying the main vibration modal frequencies of the bridge through spectrum analysis. Figure 2 The horizontal axis represents frequency, measured in Hertz, with a frequency range from 0 Hertz to 20 Hertz. The horizontal axis is marked in 2 Hertz increments, covering the main frequency range of vertical vibrations common in highway bridges. Figure 2 The vertical axis represents the amplitude of the FFT spectrum, measured in meters. The axis scale extends from 0 to approximately 1.2 meters, marked in 0.2-meter intervals. This amplitude reflects the contribution of different frequency components to the vertical displacement response. The continuous black solid line curve in the figure represents the spectral amplitude distribution obtained after performing a Fast Fourier Transform on the vertical displacement time series. From this curve, it can be observed that the spectral amplitude remains at a low level over most frequency ranges, with significant peaks appearing only at a few specific frequency positions. These peaks correspond to the natural vibration mode frequencies of the bridge structure. Specifically, the first significant spectral peak appears at 2.5 Hz, with an amplitude of approximately 0.8 meters, corresponding to the bridge's first vertical bending mode. The second spectral peak appears at 5.8 Hz, with an amplitude of approximately 0.5 meters, corresponding to the bridge's second vertical bending mode. The third spectral peak appears at 9.2 Hz, with an amplitude of approximately 0.3 meters, corresponding to the bridge's third vertical bending mode. A fourth spectral peak appears at 13.5 Hz, with an amplitude of approximately 0.2 m, corresponding to the fourth vertical bending mode of the bridge. To clearly identify these modal frequencies, a vertical dashed line extending upwards from the horizontal axis to the peak point is drawn at each peak location, and a hollow circular marker is added at the peak point. Above each marker is a textual description, labeled as first-order mode 2.5 Hz, second-order mode 5.8 Hz, third-order mode 9.2 Hz, and fourth-order mode 13.5 Hz. The decreasing trend of the spectral amplitude indicates that lower-order modes dominate the bridge's vertical vibration response, while the contribution of higher-order modes gradually decreases. This frequency domain analysis plot allows for the intuitive identification of the bridge's dominant vibration frequencies, providing fundamental data for subsequent construction of modal confidence criteria and modal matching.

[0026] The pre-set vehicle excitation is generated by the driving of test vehicles on the bridge deck. Test vehicles can be single or in pairs, such as a three-axle truck with a total mass of approximately 30 tons or a four-axle vehicle with a total mass of approximately 40 tons. The total mass and wheelbase arrangement of the vehicles meet the design load requirements of the target bridge. The test vehicles travel across the bridge according to a pre-set path and speed, for example, at different constant speeds such as 40 km / h and 60 km / h in the outer lane. Using a constant speed approach allows for better repeatability and comparability of the vehicle excitation, facilitating comparisons between multiple test results. Furthermore, when vehicles travel at a constant speed, the main frequency components of the bridge response are more stable, which helps extract clear modal characteristics from the vertical acceleration time series.

[0027] In one implementation, a test vehicle enters the target bridge from the approach section in front of one abutment at a constant speed, maintaining this constant speed in the mid-span region until it exits the approach section behind the other abutment. Throughout the journey, data acquisition equipment begins collecting data several seconds before the test vehicle enters the target bridge and ends several seconds after the vehicle has completely left the target bridge, thus obtaining a vertical acceleration time series covering the entire process of the vehicle entering the bridge, traveling in the mid-span, and exiting the bridge. To improve statistical stability, the test vehicle can be repeatedly tested in one direction or in both directions, for example, five times in one direction and five times in both directions, with the vertical acceleration time series collected from each test stored independently.

[0028] In another optional implementation, the preset vehicle excitation can also include convoy driving excitation, where two or three test vehicles pass the target bridge sequentially at a fixed distance. Multi-vehicle combinations can generate larger bending moments and shear forces under certain span and vehicle arrangement conditions, thereby stimulating higher-order vertical vibration modes of the bridge. By controlling the vehicle spacing to 10 or 20 meters and maintaining consistent speeds among the test vehicles, a load application sequence with a specific rhythm can be formed in the vertical acceleration time series, which is beneficial for distinguishing the response components of different modes. This method is suitable for scenarios with high requirements for high-order mode recognition.

[0029] After data acquisition, the data acquisition equipment stores the vertical acceleration time series corresponding to the candidate sensor installation coordinates in a unified data format, such as one file per channel or multiple channels combined, and adds information such as the timestamp of each test, test vehicle type, vehicle speed, and driving direction. These vertical acceleration time series correspond one-to-one with the candidate sensor installation coordinates in the bridge finite element model, providing basic data for subsequent modal analysis using physical information neural networks and damage identification using sparse damage reconstruction models. Through the above process of establishing the bridge finite element model, dividing the candidate sensor installation coordinates, installing vertical acceleration sensors on the target bridge, and collecting vertical acceleration time series using preset vehicle excitation, the vertical dynamic response region of the target bridge's main beam can be fully covered while ensuring test repeatability and data quality. This facilitates subsequent sensor self-optimization layout and state assessment based on modal confidence criteria.

[0030] In one implementation, the computer first trains the physical information neural network modal analysis model based on the bridge finite element model and the vertical acceleration time series obtained in step one. This enables the physical information neural network modal analysis model to simultaneously reconstruct the bridge's vertical displacement and vertical acceleration in both space and time. Based on this, modal confidence criteria are constructed, a sparse damage reconstruction model is established, and a physical information neural network-driven sparse damage reconstruction and sensor importance joint optimization algorithm is executed to obtain the bridge sensor self-optimization layout result.

[0031] In this embodiment, the physical information neural network modal analysis model adopts a feedforward fully connected structure. The inputs include the coordinates of the main beam's axis, the vertical coordinates, and time. The outputs include the vertical displacement and vertical acceleration at the corresponding positions and times. Specifically, the inputs to the physical information neural network modal analysis model can be represented by three real numbers: one representing the distance along the main beam axis (e.g., in meters), one representing the vertical position of the cross-section (e.g., the distance relative to the neutral axis of the main beam's top plate), and one representing time (e.g., in seconds). The outputs of the physical information neural network modal analysis model can be represented by two real numbers: one representing the vertical displacement at the input position and time, and the other representing the vertical acceleration at the input position and time. To balance expressive power and computational efficiency in practical engineering, three to five hidden layers can be set, each containing, for example, 64 or 128 neurons, using continuously differentiable activation functions, such as the hyperbolic tangent activation function, enabling the physical information neural network modal analysis model to learn the smooth variation law of the bridge's vertical dynamic response in space and time.

[0032] To ensure that the physical information neural network modal analysis model not only fits the vertical acceleration time series but also satisfies the bridge's dynamic equilibrium, both measurement and physical constraints are introduced during training. Measurement constraints are constructed based on the vertical acceleration time series acquired in step one. At each candidate sensor installation coordinate and each time sampling point, the vertical acceleration time series is used as the target measurement value. The candidate sensor installation coordinates and time are combined into an input sample, which is then fed into the physical information neural network modal analysis model to obtain the corresponding vertical displacement and vertical acceleration outputs. To ensure that the vertical acceleration output reflects the second-order trend of the vertical displacement over time, two methods can be used for constraint: one method is to directly calculate the difference between the vertical acceleration output of the physical information neural network modal analysis model and the vertical acceleration time series; the other method is to apply numerical differentiation methods, such as the central difference method, to the vertical displacement time series output by the physical information neural network modal analysis model to calculate the corresponding estimated vertical acceleration value, and then calculate the difference with the vertical acceleration time series. In practical implementation, both can be used simultaneously. This means constraining both the vertical acceleration output and the estimated vertical acceleration derived from the vertical displacement time series, ensuring that the physical information neural network modal analysis model satisfies both the consistency of acceleration amplitude and the consistency of the time relationship between displacement and acceleration at the measurement locations. The differences at all measurement points and time sampling points can be summed by taking their absolute values ​​or squares to obtain the measurement residual metric. The existence of the measurement residual metric ensures that the physical information neural network modal analysis model closely matches the actual measurement data at the candidate sensor installation coordinates, thereby guaranteeing that the subsequent modal identification results are consistent with the actual bridge response.

[0033] Physical constraints are constructed based on a bridge finite element model. In the bridge finite element model, several internal partition points are set within each finite element element along the main beam axis, for example, two or three internal partition points are set along the length of each finite element element. These internal partition points and the positions of each support are then combined to form a set of physical constraint sampling points. A physical constraint sampling mesh is formed on the Cartesian product of the physical constraint sampling point set and the time sampling point set. For each point in the physical constraint sampling mesh, using the bridge finite element model and the load time history corresponding to the preset vehicle excitation, the inertial force, damping force, and elastic restoring force at each time sampling point are calculated and compared with the force generated by the external load. The physical residual target is obtained by accumulating the absolute value or square of the difference between the sum of the inertial force, damping force, and elastic restoring force and the external load. At this point, the physical constraint sampling points and time are fed into the physical information neural network modal analysis model as input to obtain the predicted vertical displacement and vertical acceleration. Substituting the predicted vertical displacement and vertical acceleration back into the bridge's dynamic equilibrium relationship, the inertial force, damping force, and elastic restoring force caused by the predicted response are recalculated. This is then compared with the external load to obtain the physical residuals output by the physical information neural network modal analysis model. By measuring the difference between the physical residuals directly calculated from the bridge finite element model and those output by the physical information neural network modal analysis model, for example using absolute values ​​or sums of squares, it is possible to evaluate whether the physical information neural network modal analysis model follows the same dynamic behavior as the bridge finite element model in non-measurement point regions. The advantage of setting physical constraints in this way is that even in data-sparse regions where no sensors are installed, the physical information neural network modal analysis model is still constrained to meet the bridge's dynamic equilibrium conditions, avoiding unreasonable vibration modes and displacement fields in other regions due to only locally fitting measurement points.

[0034] During training, both measurement residuals and physical residuals are considered and summed to form the training objective function of the physical information neural network modal analysis model. In each training round, the value of this objective function is calculated, and the gradient of the objective function with respect to adjustable coefficients is calculated within each layer of neurons using the backpropagation algorithm. Gradient descent algorithms or adaptive learning rate algorithms, such as gradient descent with momentum or adaptive moment estimation, are then used to update the adjustable coefficients. The training process can employ mini-batch random sampling, randomly selecting a certain number of samples from the measurement constraint samples and physical constraint samples in each batch (e.g., 1024 to 4096 samples per batch) to improve training efficiency and enhance the model's generalization ability. Training stops when the objective function reaches a pre-set error threshold (e.g., the sum of the measurement and physical residuals no longer decreases significantly), or when the number of training rounds reaches a preset upper limit (e.g., 2000 rounds), resulting in a stable and converged physical information neural network modal analysis model.

[0035] After obtaining the trained physical information neural network modal analysis model, modal confidence criteria are constructed based on the physical information neural network modal analysis model and the bridge finite element model. The construction process includes frequency domain analysis of the predicted vertical displacement time series, extracting the principal vibration frequencies and measured modal vectors, and comparing them with the theoretical modal vectors in the bridge finite element model. At each candidate sensor installation coordinate, the location and time sampling point are input into the physical information neural network modal analysis model to obtain the predicted vertical displacement time series under non-destructive conditions. A fast Fourier transform is performed on each predicted vertical displacement time series to convert the time domain signal to the frequency domain, obtaining a series of correspondences between frequencies and complex amplitudes. Several peak values ​​with significantly higher amplitudes than neighboring frequency points are selected as principal vibration frequencies, for example, frequencies with amplitudes exceeding the average amplitude across the entire frequency band and within the common natural frequency range of the target bridge. These principal vibration frequencies represent the significant vertical vibration modes of the bridge under the current operating conditions. For each principal vibration frequency, the complex amplitudes of the corresponding frequency points at all candidate sensor installation coordinates are arranged in order of the candidate sensor installation coordinates to form the measured modal vector. The measured modal vector contains both the relative displacement amplitude information of different measuring points and the phase information between different measuring points, which can completely describe the spatial vibration mode characteristics of this mode.

[0036] In the finite element model of a bridge, the theoretical modal frequencies and vectors of each order can be obtained by solving the natural vibration problem under the lossless state. To establish a one-to-one correspondence with the measured modal vectors, the measured principal vibration frequencies and theoretical modal frequencies can be matched in frequency order. For example, for each measured principal vibration frequency, one or more candidate frequencies with the closest frequency are found in the theoretical modal frequencies, and the most suitable corresponding candidate theoretical mode is selected based on the subsequent correlation index. To this end, a correlation index is calculated between each pair of candidate measured modal vectors and theoretical modal vectors. The correlation index can be obtained as follows: First, the measured modal vector and the theoretical modal vector are multiplied one by one at corresponding positions and summed to obtain a value representing the overall consistency; second, the square root of the sum of squares of the elements of the measured modal vector and the sum of squares of the elements of the theoretical modal vector are calculated respectively, representing the overall amplitude level of the two vectors; finally, the ratio of the sum of the aforementioned point-by-point multiplications to the product of the two square roots is calculated, and the ratio is made to fall between zero and one. The closer the ratio is to one, the closer the measured modal morphology is to the theoretical modal morphology. This correlation index allows for a quantitative measurement of the consistency between the modal morphology predicted by the physical information neural network modal analysis model and the theoretical modes of the bridge finite element model.

[0037] In addition to correlation indices, frequency deviation and physical consistency indices also need to be considered. Frequency deviation can be represented by the absolute value of the difference between the measured principal vibration frequency and the corresponding theoretical modal frequency. A small frequency deviation indicates that the vibration frequency derived by the physical information neural network modal analysis model based on the vertical acceleration time series is close to the theoretical frequency in the bridge finite element model, which helps to confirm that this mode does indeed reflect the inherent vibration characteristics of the bridge structure. The physical consistency index is used to evaluate the degree of dynamic balance of this mode on the physical constraint sampling grid. To this end, the principal vibration mode region of this theoretical mode can be determined in the bridge finite element model, for example, selecting those positions in the theoretical mode vector whose amplitude accounts for more than half of the maximum amplitude of the vector as high response regions. Then, at the physical constraint sampling grid points corresponding to the high response regions, the inertial force, damping force, and elastic restoring force are recalculated using the vertical displacement and vertical acceleration output by the physical information neural network modal analysis model. After combining with external loads, the dynamic balance error based on the physical information neural network modal analysis model is obtained. The absolute value or the average of the squares of these errors is then used to obtain the physical consistency index. The smaller the physical consistency index, the more likely that the displacement field corresponding to this mode not only matches the vertical acceleration time series at the measuring point, but also satisfies the dynamic equilibrium relationship of the bridge in the high response region, which is conducive to improving the confidence level of this mode.

[0038] refer to Figure 3 , Figure 3This diagram shows a comparison between the measured and theoretical modes of the first-order vertical bending mode. It is used to verify the consistency between the measured mode output by the physical information neural network modal analysis model and the theoretical mode calculated by the bridge finite element model. Figure 3 The horizontal axis represents the longitudinal position along the bridge, in meters. The horizontal axis scale extends from 0 meters to 200 meters, marked at 20-meter intervals, covering the entire length of the bridge. Figure 3 The vertical axis represents the normalized amplitude, with the scale extending from -0.2 to 1.2, marked in 0.2 increments. The normalized amplitude represents the relative displacement amplitude of the mode shape at different locations, with the maximum displacement amplitude normalized to 1. The continuous black solid line curve plotted in the figure represents the theoretical mode shape of the first-order vertical bending mode calculated based on the bridge finite element model. This theoretical mode shape curve exhibits a typical sinusoidal half-wave shape. The amplitude is zero at the supports at both ends of the bridge (0 m and 200 m), indicating that the vertical displacement at the supports is constrained. The amplitude reaches its maximum value of 1 at the mid-span (100 m), indicating that the mid-span is the point of maximum displacement of the first-order mode shape. The theoretical mode shape curve rises smoothly from the 0 m position, with an amplitude of approximately 0.7 at 50 m, reaching a peak at 100 m and then symmetrically decreasing to approximately 0.7 at 150 m, finally returning to zero at 200 m. The figure also marks the measured modal data points extracted from the measured vertical displacement time series using a physical information neural network modal analysis model. These measured data points are represented by hollow circles, with a representative measuring point selected approximately every 10 meters along the longitudinal direction of the bridge. The figure shows that the measured modal data points generally match the theoretical mode shape curves closely. The amplitude values ​​at each measuring point basically fall on the theoretical curve or fluctuate within a very small range near it, with a fluctuation range of approximately 0.02 to 0.03. This slight deviation mainly originates from measurement noise and model errors. To aid in determining the characteristics of the mode shapes, a horizontal dashed line is drawn at the zero position on the vertical axis to represent the baseline of the mode shapes. Simultaneously, a vertical dashed line is drawn at the 100-meter position on the horizontal axis, labeled with the text "spanning the middle," emphasizing that this position is a critical location for the modal shapes. The figure also shows a rectangular text box near the 150-meter and 0.7 amplitude, indicating that the modal confidence criterion (MAC) is 0.95. This value means that the correlation index between the measured modal vector and the theoretical modal vector reaches 0.95, indicating that the two have a high degree of consistency and meet the requirements of the modal confidence criterion.

[0039] A modal confidence criterion can be formed by combining correlation, frequency deviation, and physical consistency indices. For example, in one implementation, the correlation index can be set to be no lower than 0.9, the frequency deviation no more than 0.1 Hz or no more than 5% of the corresponding theoretical modal frequency, and the physical consistency index lower than a threshold determined by engineering experience. When a modality simultaneously meets all three conditions, that modality is included in the set of permissible modalities for the modal confidence criterion. Compared to the traditional method of judging modal reliability solely through a single similarity index, introducing a physical consistency index can prevent pseudo-modals, which, although similar in morphology to theoretical modalities, from entering subsequent sparse damage reconstruction models due to significant deviations in dynamic equilibrium, thereby improving the reliability of damage identification and sensor importance assessment.

[0040] After constructing the modal confidence criterion, a sparse damage reconstruction model is established based on the bridge finite element model. First, a set of finite element elements corresponding to the discrete division of the main beam axis is selected in the bridge finite element model, and each finite element is considered a potential damaged element. While keeping all finite element elements in an undamaged state, the vertical displacement time series at each candidate sensor installation coordinate under a preset vehicle excitation load is solved through finite element analysis to obtain the undamaged state reference displacement response. Then, for each finite element, a hypothetical damage condition is constructed, and the bending stiffness of the finite element is reduced by a preset ratio, for example, to half or one-third of the original stiffness, while keeping other finite element elements in an undamaged state. The vertical displacement time series at all candidate sensor installation coordinates is calculated again under the preset vehicle excitation load. The undamaged state reference displacement response is subtracted one-to-one from the displacement response under the hypothetical damage condition of the finite element, obtaining the element sensitivity response value at each candidate sensor installation coordinate and each time sampling point. These values ​​are then arranged in the order of the candidate sensor installation coordinates and time sampling points to form the element sensitivity response sequence of the finite element. Repeating the stiffness reduction and displacement calculation process for all finite element elements yields a set of element sensitivity response sequences. This set describes the influence pattern of local damage on the vertical displacement response at different locations for each finite element, serving as the basis vectors for the subsequent sparse damage reconstruction model.

[0041] In practical structural monitoring, physical information neural network modal analysis models can be used to improve the consistency between the non-destructive baseline response and the measured response. For example, under bridge operating conditions, the actual collected vertical acceleration time series is input into the physical information neural network modal analysis model to calculate the predicted vertical displacement time series at each candidate sensor installation coordinate. This is then compared with the vertical displacement time series calculated by the bridge finite element model under non-destructive conditions, and the difference between the two is constructed as a measurement difference time series. The measurement difference time series can be understood as the response deviation between the non-destructive finite element model and the physical information neural network modal analysis model that incorporates real monitoring data. Within the frequency range allowed by the modal confidence criterion for using modal sets, this deviation mainly originates from the difference between the actual stiffness distribution of the bridge and the non-destructive finite element model, and therefore can be used as the observation vector for the sparse damage reconstruction model. In this way, by combining the structural prior information of the bridge finite element model with the measurement data fitting capability of the physical information neural network modal analysis model, the damage effect can be more clearly separated from the measurement difference time series while reducing model error.

[0042] After establishing the measurement difference time series and the element sensitivity response series sets, the physical information neural network-driven sparse damage reconstruction and sensor importance joint optimization algorithm is formally executed. First, an iterative sparse damage reconstruction process is performed using all candidate sensor installation coordinates to obtain the reference damage estimation result and the reference residual time series. The sparse damage reconstruction iterative process achieves sparse estimation of the finite element damage coefficients by progressively selecting the finite element that contributes the most to the current residual and estimating its damage degree. Specifically, the current residual time series is initialized as the measurement difference time series, and the set of damaged elements is initialized as an empty set. In each iteration, for each finite element not yet selected into the damaged element set, the element sensitivity response series of that finite element is multiplied point-by-point with the current residual time series at all candidate sensor installation coordinates and all time sampling points, and all products are summed to obtain the correlation value of that finite element. The larger the absolute value of the correlation value, the better the response mode corresponding to the assumed damage of that finite element can explain the change in the current residual. Therefore, in this iteration, the finite element with the largest absolute value of correlation value is selected from the finite element elements that have not been selected into the damage element set and added to the damage element set.

[0043] After determining the current set of damaged elements, a search is performed on the combinations of damage coefficients for the finite element elements within the set. For ease of implementation, a set of discrete candidate damage coefficient values ​​can be set for each finite element, for example, ranging from 0 (no damage) to 1 (complete loss of stiffness), discretized into 11 candidate values ​​with a step size of 0.1 within this range. By enumerating all combinations of damage coefficients for all finite element elements within the set, for each combination, the corresponding element sensitivity response sequence is linearly superimposed according to the damage coefficient combination to obtain the linear combined response time series for that combination. Then, the linear combined response time series is subtracted point-by-point from the measurement difference time series, and the squared differences at all candidate sensor installation coordinates and all time sampling points are calculated and summed to obtain the residual sum of squares. The smaller the residual sum of squares, the better the damage coefficient combination can explain the measurement difference time series. Therefore, in the current iteration, the damage coefficient combination with the smallest residual sum of squares is selected as the damage estimation result for the current iteration, and the linear combined response time series corresponding to this damage estimation result is subtracted from the measurement difference time series to obtain the new current residual time series. When the change in the sum of squared residuals of the current residual time series between two adjacent iterations is less than the preset error threshold, or when the number of iterations reaches the preset upper limit, such as 50 or 100 times, the sparse damage reconstruction iteration process is stopped, and the reference damage estimation result and reference residual time series are obtained with the participation of all sensor candidate installation coordinates.

[0044] In the iterative process of sparse damage reconstruction, the modal confidence criterion can be prioritized to allow the use of frequency components corresponding to the modal set, thereby improving the stability of damage identification. For example, before calculating the measurement difference time series and the element sensitivity response series, the vertical displacement time series can be bandpass filtered to retain only the frequency components near the principal vibration frequency of the modal set allowed by the modal confidence criterion. Then, the measurement difference time series and the element sensitivity response series can be constructed according to the above steps. The advantage of this approach is that it can filter out traffic noise, non-structural vibrations, and high-frequency interference unrelated to the bridge structure, allowing the sparse damage reconstruction model to focus primarily on the frequency range sensitive to the modal characteristics of the bridge structure, thus improving the robustness of damage localization and quantification.

[0045] After obtaining the reference damage estimation results and reference residual time series, the physical information neural network-driven sparse damage reconstruction and sensor importance joint optimization algorithm further executes the sensor importance assessment process. The sensor importance assessment process evaluates the impact of each candidate sensor installation coordinate on the overall sparse damage reconstruction result. When assessing a candidate sensor installation coordinate, the vertical displacement time series corresponding to that candidate coordinate is first removed from the measurement difference time series, and the element sensitivity response sequence component corresponding to that candidate coordinate is removed from the element sensitivity response sequence set, resulting in the measurement difference time series and element sensitivity response sequence set after removing the candidate sensor installation coordinate. Then, based on the new sequences, the sparse damage reconstruction iteration is re-executed following the same steps as the aforementioned sparse damage reconstruction iteration process to obtain the damage estimation results and new residual time series under the condition of removing the candidate sensor installation coordinate. By calculating the difference between the sum of squared residuals of the reference residual time series and the sum of squared residuals of the residual time series after removing the candidate sensor installation coordinate, the sensor importance index of that candidate sensor installation coordinate can be obtained. If the sum of squared residuals increases significantly after deleting a candidate sensor installation coordinate, it indicates that the candidate sensor installation coordinate provides crucial information for damage reconstruction, and its sensor importance index is high. Conversely, if the sum of squared residuals changes very little, it indicates that the information provided by the candidate sensor installation coordinate is already highly redundant among the information of other candidate sensor installation coordinates, and its sensor importance index is low.

[0046] By ranking all candidate sensor installation coordinates according to their sensor importance indices, a sequence of candidate sensor installation coordinates arranged from highest to lowest importance can be obtained. Considering the limitations on the number of sensors that can be installed in actual engineering projects, such as a maximum of 20 vertical acceleration sensors on a 200-meter-long bridge, the top 20 candidate sensor installation coordinates from the importance ranking sequence can be selected as the bridge sensor self-optimization layout results. The bridge sensor self-optimization layout results can ensure that, under the condition of a preset number of sensors, the interpretability loss of the sparse damage reconstruction model is minimized. That is, under the constraint of feasible sensor layout cost, the measurement difference time series between the bridge finite element model and the physical information neural network modal analysis model can be fitted as fully as possible by a linear combination of the unit sensitivity response sequence set. Since the entire bridge sensor self-optimization layout process adopts the modal confidence criterion to allow the use of modal sets when constructing the measurement difference time series and the unit sensitivity response sequence, and is driven by the physical information neural network modal analysis model, the bridge sensor self-optimization layout results can balance modal recognition quality, damage recognition accuracy, and sensor layout economy, providing a reliable sensor network foundation for subsequent bridge condition assessment.

[0047] refer to Figure 4 , Figure 4 This is a schematic diagram of the element sensitivity response time series of a typical finite element model. The diagram shows the change in vertical displacement response at each sensor installation location relative to the undamaged state when different elements in the bridge finite element model suffer assumed damage. Figure 4 The horizontal axis represents time, in seconds, ranging from 0 seconds to 10 seconds. The horizontal axis scale is marked at 1-second intervals. This time period covers the complete dynamic response process of the bridge under the preset vehicle excitation, from the initial static state, through vehicle load excitation, to free vibration decay. Figure 4 The vertical axis represents the element sensitivity response, measured in meters. The vertical axis scale extends from -0.4 meters to 0.4 meters, marked in 0.1-meter intervals. The physical meaning of element sensitivity response is the change in vertical displacement caused by local damage at various sensor locations when the bending stiffness of a finite element is reduced by a preset ratio. The figure shows three representative curves, corresponding to three typical finite element elements located at different positions on the bridge. The first curve, a solid black line, represents element 25, located at the 50-meter mark on the bridge. This element is located near the mid-span of the first span. Its element sensitivity response time series shows obvious oscillating characteristics. In the initial stage of vehicle load application (0 to 3 seconds), the response amplitude gradually increases, reaching a peak of approximately 0.25 meters at about 2 seconds. Subsequently, the response amplitude decays exponentially with time while maintaining periodic oscillations at a frequency of approximately 2.5 Hz, consistent with the bridge's first-order modal frequency. The second curve, a dashed black line, represents element 50, located at the 100-meter mark on the bridge. Located at the mid-span of the bridge, this element represents the region with the greatest vertical bending deformation. Therefore, damage to this element has the most significant impact on the overall response. As shown in the figure, the sensitivity response amplitude of element 50 is significantly higher than other elements, with a peak value of approximately 0.35 meters and a relatively slow decay rate, indicating that damage to the mid-span element will have a sustained impact on the bridge's dynamic response over a longer period. The third curve, represented by the black dotted line, represents element 75, located at the 150-meter mark of the bridge. This element is located near the mid-span of the third span. Its sensitivity response time series amplitude and oscillation characteristics are similar to those of element 25, with a peak value of approximately 0.2 meters and a similar decay trend. Comparing the three curves reveals that elements located in the mid-span region have larger sensitivity response amplitudes, indicating that damage to these elements has a more significant impact on the overall vertical displacement response of the bridge. Therefore, damage estimation for these elements is more sensitive during damage identification.

[0048] In one implementation, after obtaining the self-optimized layout results of the bridge sensors, vertical acceleration sensors are installed on the target bridge according to these results. During installation, positions that correspond one-to-one with the candidate sensor installation coordinates in the bridge finite element model are preferentially used. Similar to step one, surface treatment is performed at the neutral axis position of the main beam top plate, and the vertical acceleration sensors are fixed using structural adhesive or mechanical anchoring, ensuring that the sensitive axis of each vertical acceleration sensor is strictly arranged vertically. This ensures that the vertical acceleration time series collected during bridge operation matches the degrees of freedom in the bridge finite element model in spatial location. This facilitates direct correlation between the measured values ​​and the outputs of the bridge finite element model and the physical information neural network modal analysis model in subsequent analysis, eliminating the need for additional interpolation or coordinate transformation and reducing additional sources of error.

[0049] After the bridge enters normal operation, the data acquisition equipment periodically collects vertical acceleration time series according to a preset monitoring plan. One implementation method is continuous daily monitoring, where vertical acceleration time series are continuously collected at the sensor installation coordinates throughout the day the bridge is open to traffic. The sampling frequency is set to 200 Hz or 500 Hz to ensure coverage of the bridge's first and multiple vertical natural frequencies. Another implementation method is intermittent monitoring, for example, starting monitoring every hour for 10 minutes, continuously collecting vertical acceleration time series at all sensor installation coordinates during these 10 minutes. Intermittent monitoring reduces the resource consumption on storage and data transmission during long-term operation. Furthermore, by rationally selecting monitoring time periods, such as morning peak hours, evening peak hours, or periods with a high proportion of heavy traffic, a more sensitive response to the bridge's vertical dynamic characteristics can be obtained with a limited amount of data.

[0050] During each monitoring cycle, the data acquisition equipment synchronously samples all vertical accelerometers using a unified clock, ensuring that the vertical acceleration time series at each sensor's installation coordinates are aligned on the time axis. Synchronous sampling aims to guarantee accurate phase relationships between different measurement points, which is crucial for subsequent modal identification and sparse damage reconstruction. If there is a time shift between different sensors, errors will occur when constructing the measured modal vectors and measuring the difference time series, thus reducing the reliability of the modal confidence criteria and the sparse damage reconstruction model. Therefore, the data acquisition equipment can use a unified hardware clock or utilize the global positioning system's time signal for time alignment.

[0051] After completing a monitoring data acquisition, the raw vertical acceleration time series is preprocessed. Preprocessing may include DC offset removal, bandpass filtering, and segmentation. DC offset removal is achieved by averaging each vertical acceleration time series over the entire monitoring period and subtracting the average from each sampling point, thus eliminating constant offsets introduced by sensor zero-point drift or installation tilt. Bandpass filtering can be set according to the known or estimated natural frequency range of the target bridge; for example, it can retain frequency components between 0.2 Hz and 20 Hz while filtering out non-vibrational components such as temperature deformation and slow bridge deck deformation at excessively low frequencies, as well as localized impacts and noise at excessively high frequencies. Bandpass filtering highlights frequency components related to structural modes, allowing subsequent analysis based on modal confidence criteria to focus more on the structural response.

[0052] After preprocessing, the continuously monitored vertical acceleration time series is divided into multiple analysis time periods. For example, analysis time periods can be divided into 60-second or 120-second intervals. Each analysis time period needs to be long enough to include several principal vibration periods, thus displaying clear spectral peaks in the frequency domain analysis. However, it should not be too long to avoid significant changes in bridge condition and traffic flow characteristics within a single analysis time period, leading to unstable results. This segmentation method allows for the extraction of multiple representative analysis time periods from long-term monitoring data. Subsequent modal analysis and sparse damage reconstruction can be performed independently on each analysis time period to promptly capture trends in bridge condition changes.

[0053] Within each analysis time period, the vertical acceleration time series at each sensor installation coordinate is input into the physical information neural network modal analysis model along with the bridge finite element model. Specifically, for each combination of sensor installation coordinates and each time sampling point, the position coordinates and time are passed to the physical information neural network modal analysis model to obtain the predicted vertical displacement and predicted vertical acceleration. In one embodiment, the physical information neural network modal analysis model, which has already been trained in step two, is used, and incremental training or correction can be performed as needed based on new monitoring data to ensure that the physical information neural network modal analysis model continuously maintains consistency with the actual bridge response. Through the physical information neural network modal analysis model, predicted vertical displacement time series under a non-destructive reference state can be constructed at all sensor installation coordinates. These prediction results have been corrected through measurement constraints and physical constraints during the training phase, thus conforming to both the dynamic characteristics of the bridge finite element model and closely matching the statistical characteristics of the monitoring data.

[0054] After obtaining the predicted vertical displacement time series, the vertical displacement time series under the assumed undamaged state is calculated using the bridge finite element model, following the same calculation method as when establishing the sparse damage reconstruction model in step two. Then, the predicted vertical displacement time series output by the physical information neural network modal analysis model is subtracted one-to-one from the vertical displacement time series calculated by the bridge finite element model to obtain the measurement difference time series for the current analysis period. This measurement difference time series can be understood as the deviation between the actual monitoring data and the response of the reference structure when the bridge finite element model is considered as an undamaged reference structure. This deviation mainly includes response differences caused by possible stiffness degradation, loose connections, and other structural damage, and may also include a certain degree of model error and environmental influence. By introducing the trained physical information neural network modal analysis model, the systematic errors between the bridge finite element model and the actual structure can be partially offset, making the measurement difference time series more focused on reflecting damage factors.

[0055] After constructing the measurement difference time series, a sparse damage reconstruction model for the current analysis period is built using the set of element sensitivity response sequences established in step two. To avoid the influence of unstructured noise on the damage identification results, when constructing the measurement difference time series and element sensitivity response sequences, only the frequency components corresponding to the modal sets allowed by the modal confidence criterion are retained, according to the modal confidence criterion. Specifically, bandpass filtering can be applied to the predicted vertical displacement time series and the undamaged vertical displacement time series, retaining only the frequency components near the principal vibration frequencies of the modal sets allowed by the modal confidence criterion, and then differentially constructing the measurement difference time series. In this way, the measurement difference time series mainly reflects the modal behaviors that have been verified by the modal confidence criterion, thereby reducing the risk of misjudgment caused by abnormal noise.

[0056] Within the current analysis period, the measurement difference time series is used as the observation vector, and the set of element sensitivity response sequences is used as the basis vector set. The sparse damage reconstruction iterative process described in step two is used to solve for the finite element damage estimation result. The execution method of the sparse damage reconstruction iterative process in the running phase is basically the same as that in the training phase: First, the current residual time series is initialized as the measurement difference time series, and the damaged element set is initialized as an empty set; then, in each iteration, the finite element elements that have not yet been added to the damaged element set are traversed, and the element sensitivity response sequence of each finite element is multiplied point by point with the current residual time series and summed to calculate the correlation value. The finite element element with the largest absolute value of the correlation value is selected to be added to the damaged element set. Next, the damage coefficient combinations of each finite element are enumerated in the damaged element set, and a linear combination response time series is formed for each combination. The quality of the combination is evaluated by the sum of squared differences between the combination and the measurement difference time series. The damage coefficient combination with the smallest sum of squared differences is selected as the damage estimation result of the current iteration, and the current residual time series is updated with the linear combination response time series corresponding to this damage estimation result. The iterative process continues until the change in the sum of squared residuals between two adjacent iterations of the current residual time series is lower than a preset threshold, or the number of iterations reaches a preset upper limit, such as 50 or 100 times. At this point, the finite element damage estimation result is considered to be stable, and the finite element damage estimation result for the current analysis period is output.

[0057] In one implementation, the sparse damage reconstruction process described above can be performed on multiple analysis time periods within the same day or monitoring cycle to obtain multiple finite element damage estimation results. These results are then statistically averaged or the median is taken to reduce the impact of random noise or single traffic events on the damage estimation. For example, for the same finite element, damage coefficient estimates can be statistically analyzed over 10 analysis time periods, and the median can be used as the damage estimate for that finite element within the monitoring cycle. By statistically analyzing multiple analysis time periods, the stability of the damage estimation results can be improved, making the bridge condition assessment results less sensitive to single abnormal events and more suitable for long-term safety assessments.

[0058] After obtaining the finite element damage estimation results, the computer converts these results into bridge condition levels according to pre-defined bridge condition classification rules. In one implementation, the finite element damage estimation results can be converted into dimensionless damage indices, and the estimated damage coefficient of each finite element can be normalized to a range of 0 to 1, where 0 indicates no significant stiffness degradation and 1 indicates complete loss of bending stiffness. Then, the maximum value among all finite element damage indices, the number of finite element elements exceeding a certain damage level threshold, and the number of finite element elements continuously exceeding that threshold along the main beam axis are calculated. For example, the damage level threshold can be set to 0.3, meaning that when the damage index of a finite element exceeds 0.3, the element is considered to have significant stiffness degradation. If the maximum damage index is below 0.3, and no finite element damage index exceeds the damage level threshold, the bridge condition can be classified as normal. If the maximum damage index is between 0.3 and 0.5 and the number of finite element elements continuously exceeding the damage level threshold is small, for example, less than 3, the bridge condition can be classified as mild damage. If the maximum damage index is between 0.5 and 0.7, or the number of finite element elements continuously exceeding the damage level threshold along the main beam axis reaches 3 to 5, the bridge condition can be classified as moderate damage. If the maximum damage index exceeds 0.7, or the number of finite element elements continuously exceeding the damage level threshold exceeds 5, and these elements are concentrated in a certain mid-span or near a certain support area, the bridge condition can be classified as severe damage. By simultaneously considering the degree of damage and the continuous spatial distribution of damage, the overall safety status of the bridge structure can be more comprehensively reflected. On the one hand, a larger damage index reflects a significant reduction in local stiffness; on the other hand, a large number of continuously damaged elements indicates that there is extensive damage in a certain area, which requires more attention.

[0059] In another implementation approach, the bridge condition level can be dynamically adjusted by incorporating the temporal trend of finite element damage estimation results. For example, for the same finite element, if the damage index shows a significant upward trend over multiple consecutive monitoring periods, even if the current damage index has not yet exceeded a certain high threshold, the bridge condition can be pre-classified as a higher damage level, and the bridge condition assessment results can indicate a damage development trend, recommending a detailed on-site inspection. By introducing the time dimension, the bridge condition assessment results can be expanded from a single static judgment to a long-term evolution analysis, which is beneficial for the early detection of potential safety hazards.

[0060] Finally, the bridge condition level and corresponding quantitative indicators are combined to form the bridge condition assessment result output. The bridge condition assessment result can include the current bridge condition level, maximum damage index, number of finite element elements exceeding the damage level threshold, location and range of continuous damage areas, and recommended management measures, such as continued monitoring, on-site inspection, or restriction of traffic on some lanes. The bridge condition assessment result can be provided to maintenance personnel in the form of text, charts, or visualization interfaces. For example, the damage degree of different finite element elements can be marked with colors on the main beam schematic diagram, allowing maintenance personnel to intuitively understand the distribution of structural damage. In this way, vertical acceleration sensors are installed on the target bridge according to the bridge sensor self-optimization layout results and long-term monitoring is carried out. The vertical acceleration time series and the bridge finite element model are input into the physical information neural network modal analysis model. Under the constraints of the modal confidence criterion, the finite element damage estimation results are obtained through the sparse damage reconstruction model. Based on the finite element damage estimation results, the bridge condition level is classified, thus forming a complete condition assessment process that can be used for online health monitoring and risk warning during the bridge operation period.

[0061] The present invention has been described in detail above. Specific examples have been used to illustrate the principles and implementation methods of the invention. The descriptions of the embodiments above are merely for the purpose of helping to understand the method and core ideas of the present invention. It should be noted that those skilled in the art can make various improvements and modifications to the present invention without departing from its principles, and these improvements and modifications also fall within the protection scope of the claims of the present invention.

Claims

1. A method for bridge sensor self-optimization placement and condition assessment based on modal confidence criteria, characterized in that, The method includes: Step 1: Establish a finite element model of the target bridge in the computer, divide the candidate installation coordinates of the sensor according to the main beam structure of the bridge, install vertical acceleration sensors on the target bridge according to the candidate installation coordinates, and collect the vertical acceleration time series at each candidate installation coordinate by preset vehicle excitation. Step 2: Train a physical information neural network modal analysis model based on the bridge finite element model and vertical acceleration time series. Construct modal confidence criteria using the physical information neural network modal analysis model and the bridge finite element model. Establish a sparse damage reconstruction model based on the bridge finite element model. Use a physical information neural network-driven sparse damage reconstruction and sensor importance joint optimization algorithm to determine the self-optimization layout result of bridge sensors. Step 3: Install vertical acceleration sensors on the target bridge according to the bridge sensor self-optimization layout results. During the bridge operation, periodically collect vertical acceleration time series. Input the vertical acceleration time series and the bridge finite element model into the physical information neural network modal analysis model. Under the constraints of the modal confidence criterion, obtain the finite element damage estimation results through the sparse damage reconstruction model. Based on the finite element damage estimation results, classify the bridge state level and output the bridge state assessment results. In step two, the correlation index is calculated by the dot product of the measured modal vector and the theoretical modal vector, as well as the modulus of the measured modal vector and the theoretical modal vector. The frequency deviation is calculated by the absolute value of the difference between the measured principal vibration frequency and the theoretical principal vibration frequency. The physical consistency index is calculated by the average absolute value of the physical residuals on the physical constraint sampling grid points in the region where the amplitude of the principal vibration mode exceeds the preset amplitude threshold. When the correlation index reaches the preset correlation threshold and the frequency deviation does not exceed the preset frequency deviation threshold and the physical consistency index does not exceed the preset physical residual threshold, the current order mode is included in the modal set allowed by the modal confidence criterion. For subsequent sparse damage reconstruction and sensor importance assessment, only the modal set allowed by the modal confidence criterion is used.

2. The method of claim 1, wherein, Step one is performed as follows: The design information of the target bridge is input into the finite element analysis software to establish a three-dimensional finite element model of the bridge, including the main beam, piers and bearings. Multiple finite element elements are divided along the axis of the main beam at a preset equal interval. A candidate sensor installation coordinate is set at the neutral axis position of the top plate of the main beam in each finite element element. A vertical acceleration sensor is installed at the corresponding position. The vertical acceleration time series at each candidate sensor installation coordinate is collected at a uniform time sampling interval under the action of a preset load time history, and then stored in the data acquisition system.

3. The method of claim 2, wherein, The physical information neural network modal analysis model in step two has a feedforward fully connected structure, including at least an input layer, three or more hidden layers, and an output layer. The input quantities are the position coordinates of the bridge main beam axis, the vertical position coordinates, and time, and the output quantities are the vertical displacement and vertical acceleration at the corresponding positions and times.

4. The method of claim 3, wherein, The physical information neural network modal analysis model is trained as follows: Multiple internal segmentation points are set along the main beam axis in the bridge finite element model. These internal segmentation points are combined with the support positions to form a set of physical constraint sampling points. This set of physical constraint sampling points is combined with the time sampling point set to form a physical constraint sampling grid. On this grid, the difference between the inertial force, damping force, and elastic restoring force and the external load is calculated using the bridge finite element model to obtain the physical residual target. At the combined location of the sensor candidate installation coordinates and the time sampling points, the vertical acceleration time series is used as the measurement residual target. In each training round, the coordinates and time of the physical constraint sampling grid and the measurement residual target are used as inputs to calculate and output the vertical displacement and vertical acceleration. The vertical acceleration estimate is obtained from the vertical displacement time series through numerical differentiation. The sum of the absolute values ​​of the differences between the vertical acceleration estimate and the vertical acceleration time series, as well as the sum of the absolute values ​​of the physical residuals, are calculated. The sum of these two values ​​is used as the loss value. The adjustable coefficients within the physical information neural network modal analysis model are updated through backpropagation and gradient descent until the loss value meets the preset stopping condition.

5. The method as described in claim 4, characterized in that, In step two, the position coordinates and time sampling points at each candidate sensor installation coordinate are input into the trained physical information neural network modal analysis model to obtain the predicted vertical displacement time series. The Fast Fourier Transform is applied to the predicted vertical displacement time series to obtain the correspondence between frequency and amplitude. Multiple principal vibration frequencies corresponding to the amplitude peaks are identified. The complex amplitudes of each principal vibration frequency are arranged in the order of the candidate sensor installation coordinates at all candidate sensor installation coordinates to form the measured modal vector. The theoretical modal vectors corresponding one-to-one with each order of principal vibration frequency are calculated in the bridge finite element model.

6. The method as described in claim 5, characterized in that, In step two, a set of finite element elements corresponding one-to-one with the axis of the main beam is determined in the finite element model of the bridge. For each finite element, the bending stiffness of the corresponding finite element is reduced to a preset ratio while keeping other finite element elements in an undamaged state. Under the load time history corresponding to the preset vehicle excitation in step one, the vertical displacement time series at all candidate sensor installation coordinates is calculated. The vertical displacement time series in the undamaged state is subtracted from the vertical displacement time series after the stiffness of the corresponding finite element is reduced to obtain the element sensitivity response value. The element sensitivity response values ​​at each candidate sensor installation coordinate and each time sampling point are arranged in a unified order to form an element sensitivity response sequence. The operation of reducing the bending stiffness and calculating the element sensitivity response sequence is repeated for all finite element elements to form a set of element sensitivity response sequences.

7. The method as described in claim 6, characterized in that, In step two, the trained physical information neural network modal analysis model is used to calculate the predicted vertical displacement time series of the non-destructive bridge finite element model under load time history at all candidate sensor installation coordinates. The predicted vertical displacement time series is subtracted from the vertical displacement time series obtained by directly solving the non-destructive bridge finite element model to obtain the measurement difference time series. The measurement difference time series is used as the observation vector, the set of element sensitivity response sequences is used as the basis vector set, and the sparse damage reconstruction model is constructed with the finite element damage coefficient as the unknown.

8. The method as described in claim 7, characterized in that, Step 3 includes the bridge state estimation process during the operation phase. The bridge state estimation process includes the following steps: Install vertical acceleration sensors at the neutral axis position of the top plate of the main beam of the target bridge according to the bridge sensor self-optimization layout results; periodically collect the vertical acceleration time series at the installation coordinates of each sensor at the same time sampling interval as in Step 1; input the vertical acceleration time series and the bridge finite element model into the physical information neural network modal analysis model to obtain the predicted vertical displacement time series under the current non-destructive reference state; construct the current measurement difference time series using the predicted vertical displacement time series and the vertical displacement time series obtained by the bridge finite element model under the current load; construct the current element sensitivity response sequence set under the modal confidence criterion constraint and execute the sparse damage reconstruction iteration process to obtain the current finite element damage estimation result.

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