Bridge full-field deflection reconstruction method based on inverse finite element and discrete deflection correction

Abstract

The application discloses a bridge deflection reconstruction method based on modal auxiliary inverse finite element and discrete deflection correction, and belongs to the technical field of bridge structure health monitoring. A finite element model is established according to the geometric size and boundary conditions of a bridge, and a strain modal matrix is extracted; sparse measured strain data are acquired, a strain modal submatrix is extracted, modal coordinates are solved, and an extended full-field strain distribution is obtained; the extended full-field strain is input into an inverse finite element model, and preliminary full-field deflection is solved and obtained; discrete deflection observation data are acquired, and an augmented observation model fusing discrete deflection observation and inverse finite element constraint information is constructed; the preliminary full-field deflection is corrected based on a prediction-correction mechanism, and final full-field deflection is obtained. The application fully utilizes discrete deflection observation information, improves the accuracy and stability of bridge full-field deflection reconstruction under a sparse sensing condition, and is suitable for in-service bridge structure health monitoring.

Description

Technical Field

[0001] This invention belongs to the field of bridge structural health monitoring and structural response identification technology, specifically involving a bridge full-field deflection reconstruction method based on modal-assisted inverse finite element method and discrete deflection correction. Background Technology

[0002] Bridges are critical nodes in transportation infrastructure networks. During long-term service, they are susceptible to performance degradation and even structural damage due to factors such as traffic loads, environmental erosion, and material aging. Deflection is an important indicator reflecting the overall stress state and load-bearing capacity of a bridge. Deflection monitoring results can effectively characterize the deformation response of a bridge under load, serving as a crucial basis for bridge safety assessment, health monitoring, condition early warning, and maintenance decisions. Currently, methods for obtaining bridge deflection mainly include direct measurement methods and indirect reconstruction methods.

[0003] Direct measurement methods typically employ static levels, laser displacement gauges, and visual measurement equipment to obtain bridge displacement information. These methods usually only yield deflection data at a small number of discrete measurement points, making it difficult to reflect the overall deflection distribution across the entire bridge.

[0004] Indirect reconstruction methods utilize structural responses such as strain, acceleration, and tilt angle for deflection inversion. Among these, the inverse finite element method (IFM), as a deformation reconstruction technique, can achieve full-field structural deformation reconstruction using structural strain information without relying on external load information or complex boundary condition assumptions, showing promising application prospects in bridge health monitoring. However, existing IFM methods still face the following problems in practical engineering applications: While traditional strain inversion methods can be used for full-field deflection estimation, they are sensitive to the number of measurement points, boundary conditions, and measurement noise, and reconstruction accuracy decreases significantly under sparse sensing conditions; furthermore, existing bridge monitoring systems can typically acquire discrete deflection observation data as well as bridge design parameters and structural geometric information, but existing deflection reconstruction methods have not fully utilized the prior information of bridge finite element modal analysis to enhance strain data, nor have they effectively integrated discrete deflection observations to correct preliminary reconstruction results, thus limiting the accuracy and engineering applicability of full-field bridge deflection reconstruction. Summary of the Invention

[0005] To address the shortcomings and deficiencies of existing technologies, this invention provides a method for reconstructing the full-field deflection of bridges based on modal-assisted inverse finite element method and discrete deflection correction. This method solves the problems of high dependence on strain sensor deployment density, insufficient reconstruction accuracy under sparse monitoring conditions, and inadequate utilization of discrete deflection observations in existing full-field bridge deflection reconstruction processes. By fusing prior modal information of the bridge finite element method to expand sparse strain data, and combining it with an inverse finite element model to achieve preliminary full-field deflection reconstruction, further prediction and correction are performed using discrete deflection observations. This achieves high-precision and stable reconstruction of the full-field bridge deflection, improving its engineering applicability.

[0006] The technical solution adopted by this invention to solve its technical problem is as follows: A method for reconstructing the full-field deflection of a bridge based on inverse finite element method and discrete deflection correction includes the following steps: Step 1: Establish a finite element model of the bridge based on its geometric dimensions, cross-sectional parameters, and boundary conditions. Perform modal analysis on the finite element model to extract several strain mode shapes of the bridge and form a strain mode matrix. Step 2: Obtain sparse measured strain data of the bridge under test, extract the corresponding strain mode submatrix from the strain mode matrix according to the location of the strain measurement points, establish the mapping relationship between the sparse measured strain data and the modal coordinates, solve the modal coordinates, and obtain the extended full-field strain distribution based on the modal coordinates and the strain mode matrix. Step 3: Input the extended full-field strain distribution into the inverse finite element model and solve to obtain the preliminary full-field deflection; Step 4: Obtain discrete measured deflection data of the bridge under test, and construct an augmented observation model that integrates the discrete measured deflection data and the constraint information of the inverse finite element model; Step 5: Use the augmented observation model to correct the initial full-field deflection to obtain the final full-field deflection of the bridge.

[0007] Furthermore, the sparse measured strain data mentioned in step two are collected from the strain measuring points of the bridge health monitoring system under test, and arranged in the order of the preset strain measuring point numbers to form the first... k Sparse measured strain vector at each sampling time ; Based on the location of the strain measurement points, from the strain mode matrix Extract the corresponding strain mode submatrix Establish the sparse measured strain vector With modal coordinate vector Mapping relationship between them: ; The modal coordinate vector is solved using the least squares principle. And based on the modal coordinate vector and strain mode matrix The extended full-field strain vector at the k-th sampling time is obtained: .

[0008] Furthermore, step three specifically involves: expanding the full-field strain vector. As input to the inverse finite element model, the inverse finite element global equation is constructed and solved to obtain the preliminary global field nodal displacement vector at the k-th sampling time, and the vertical deflection component is extracted from it to obtain the preliminary global field deflection vector.

[0009] Furthermore, the inverse finite element model is established by dividing the bridge under test into n inverse beam elements along the longitudinal direction. Each inverse beam element includes two nodes, and each node includes three degrees of freedom: axial displacement, vertical deflection, and cross-sectional rotation angle. For any inverse beam element, the theoretical section strain vector is established based on the strain-displacement matrix of the inverse beam element, according to the... k Extended full-field strain vector at each sampling time point Obtain the corresponding extended section strain vector and construct the element equations for the inverse beam element; The element equations of each inverse beam element are assembled as a whole to obtain the global equation of the inverse finite element method.

[0010] Furthermore, the element equation of the inverse beam element is expressed as: ; in, Represents the unit coefficient vector; This represents the element node displacement vector of any inverse beam element; Represents the unit's equivalent vector; and: ; ; ; in, Let be the element length of the inverse beam element. This is the weight matrix; This is the strain-displacement matrix of the inverse beam element; For the extended cross section strain vector; These represent the axial displacement, vertical deflection, and section rotation angle of node one, respectively. These represent the axial displacement, vertical deflection, and section rotation angle of node two, respectively.

[0011] Furthermore, the discrete measured deflection data mentioned in step four are collected from the deflection measuring points of the bridge health monitoring system under test, and arranged into the first set according to the preset deflection measuring point numbering order. k Discrete measured deflection vector at each sampling time ; Establish the state vector at the k-th sampling time. The state vector This includes the total nodal deflection components and nodal rotation components of the bridge under test. Based on discrete measured deflection vector Constructing augmented observation vectors from inverse finite element global equations And establish an augmented observation model.

[0012] Furthermore, the augmented observation model is expressed as: ; in, To augment the observation vector, To augment the observation matrix, For the observed noise vector; Let be the state vector at the k-th sampling time.

[0013] Furthermore, the state vector Represented as: ; in, This represents the total field node deflection vector at the k-th sampling time. This represents the rotation vector of all nodes at the k-th sampling time. The augmented observation vector ; in, For the first k Discrete measured deflection vector at each sampling time; It is the inverse finite element global equivalent vector formed by the extended full-field strain vector at the k-th sampling time; The augmented observation matrix Represented as: ; Where M is the discrete deflection observation matrix; K is the global coefficient matrix in the inverse finite element global equation.

[0014] Furthermore, the correction described in step five employs a prediction-correction mechanism, including: Based on the posterior state estimate at the (k-1)th sampling time Perform state prediction to obtain the prior state estimate at the k-th sampling time, and calculate the prior error covariance matrix at the k-th sampling time; The corrected gain matrix is ​​calculated based on the prior error covariance matrix. The prior state estimate is corrected using the corrected gain matrix and the augmented observation vector to obtain the posterior state estimate at the k-th sampling time, and the posterior error covariance matrix is ​​updated. The first state is extracted from the posterior state estimate. k The final full-field deflection vector at each sampling time point .

[0015] Furthermore, the modified gain matrix is ​​expressed as: ; in, To correct the gain matrix; Let be the prior error covariance matrix at the k-th sampling time. To augment the observation matrix; The observation noise covariance matrix is ​​expressed as: ; in, The measurement noise covariance matrix is ​​the discrete measured deflection vector. The equivalent noise covariance matrix corresponding to the initial reconstruction constraints of the inverse finite element method; The posterior state estimate at the k-th sampling time is: ; in, This is the prior state estimate for the k-th sampling time; To correct the gain matrix; To augment the observation vector; To augment the observation matrix; The posterior error covariance matrix is ​​updated as follows: in, Represents a unit matrix.

[0016] The beneficial effects of this invention include: 1. This invention extracts strain modal information based on the finite element model of a bridge and obtains the extended full-field strain distribution of the bridge by combining sparse measured strain data. It realizes the reconstruction of the continuous strain field of the bridge under the condition of a small number of strain measurement points, reduces the dependence of the full-field deflection reconstruction on the density of strain sensor layout, and has the advantages of low sensor requirements and strong engineering applicability.

[0017] 2. This invention introduces extended full-field strain into the inverse finite element framework to solve the full-field deflection of bridges. It can achieve preliminary full-field deflection reconstruction of bridges without the need to accurately obtain external load information and material parameters, and has the advantages of high computational efficiency and wide applicability.

[0018] 3. This invention integrates discrete deflection observation data with preliminary reconstruction results of inverse finite element method, and corrects the bridge's deflection across the entire field through a prediction-correction mechanism. This can effectively improve the accuracy and stability of the bridge's deflection reconstruction results, and is suitable for long-term health monitoring, condition assessment, and safety early warning of bridges. Attached Figure Description

[0019] Figure 1 This is a flowchart of the method framework of the present invention; Figure 2 This is an experimental case model and sensor layout diagram of the present invention; Figure 3 This is a diagram showing the results of the method of the present invention. Detailed Implementation

[0020] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. 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.

[0021] Furthermore, the technical features involved in the different embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0022] A method for reconstructing the full-field deflection of bridges based on modal-assisted inverse finite element method and discrete deflection correction is disclosed. This method utilizes strain modal information extracted from the bridge's finite element model as a spatial prior, expands the sparse measured strain data obtained from a health monitoring system into a full-field strain distribution, and introduces the expanded full-field strain distribution into the inverse finite element model to obtain the preliminary full-field deflection of the bridge. Based on this, the discrete deflection data from the health monitoring system is fused to construct an augmented observation model, and the preliminary deflection results are corrected through a prediction-correction mechanism to obtain the final full-field deflection of the bridge. This invention can fully utilize discrete deflection observation information, improve the accuracy and stability of full-field deflection reconstruction of bridges under sparse sensing conditions, and is applicable to the health monitoring of in-service bridge structures.

[0023] Example 1: As Figure 1 As shown, this embodiment provides a method for reconstructing the full-field deflection of a bridge based on modal-assisted inverse finite element method and discrete deflection correction. It includes four stages: finite element model establishment, full-field strain expansion, preliminary inverse finite element reconstruction, and discrete deflection correction. By fusing bridge finite element modal information, sparse strain monitoring data, and discrete deflection observation data, it achieves full-field deflection reconstruction of the bridge. Specifically, it includes: Step 1: Establish a finite element model of the bridge based on its geometric dimensions, cross-sectional parameters, and boundary conditions. Perform modal analysis on the model to extract several strain mode shapes of the bridge and form a strain mode matrix. .

[0024] Step 2: Acquire sparse measured strain data collected by the strain measurement points of the bridge health monitoring system under test, and arrange them into the first set according to the preset strain measurement point numbering order. k Sparse measured strain vector at each sampling time ; Based on the strain mode matrix Extract the strain mode submatrix corresponding to the location of the strain measurement point. Establish the sparse measured strain vector With modal coordinate vector Mapping relationship between them: ; The modal coordinate vector is solved using the least squares principle. And based on the modal coordinate vector and strain mode matrix The extended full-field strain vector at the k-th sampling time is obtained: .

[0025] Step 3: Expand the full-field strain vector As input to the inverse finite element model, construct and solve the global equations of the inverse finite element model: ; Where K is the inverse finite element global coefficient matrix; This is the initial global node displacement vector at the k-th sampling time; It is the inverse finite element global equivalent vector formed by the extended global strain vector; The preliminary global nodal displacement vector at the k-th sampling time is obtained. The vertical deflection component is extracted from it to obtain the preliminary global deflection vector. .

[0026] The inverse finite element model is established by dividing the bridge under test into n iBeam3 inverse beam elements along the longitudinal direction. Each inverse beam element includes two nodes, and each node includes three degrees of freedom: axial displacement, vertical deflection, and cross-sectional rotation. For any inverse beam element, its nodal displacement vector is expressed as: ; in, These represent the axial displacement, vertical deflection, and section rotation angle of node one, respectively. These represent the axial displacement, vertical deflection, and section rotation angle of node two, respectively. Based on the strain-displacement matrix of the inverse beam element Establish the theoretical cross-sectional strain vector: ; According to the k Extended full-field strain vector at each sampling time point Obtain the corresponding strain vector of the extended section. And construct the element equations for the inverse beam element: ; in, Represents the unit coefficient vector; This represents the element node displacement vector of any inverse beam element; Represents the unit's equivalent vector; and: ; ; in, Let be the element length of the inverse beam element. This is the weight matrix; The element equations of each inverse beam element are assembled as a whole to obtain the global equation of the inverse finite element method.

[0027] Step 4: Obtain discrete measured deflection data collected by the deflection measurement points of the bridge health monitoring system, and assemble them into a series according to the preset deflection measurement point numbering order. k Discrete measured deflection vector at each sampling time ; Establish the state vector at the k-th sampling time. The state vector This includes the total nodal deflection components and nodal rotation components of the bridge under test; based on the discrete measured deflection vector. Constructing augmented observation vectors from inverse finite element global equations And establish an augmented observation model: ; in, To augment the observation vector, To augment the observation matrix, For the observed noise vector; Let be the state vector at the k-th sampling time.

[0028] The state vector Represented as: ; in, This represents the total field node deflection vector at the k-th sampling time. This represents the total field node deflection vector at the k-th sampling time. The augmented observation vector ; in, For the first k Discrete measured deflection vector at each sampling time; It is the inverse finite element global equivalent vector formed by the extended full-field strain vector at the k-th sampling time; The augmented observation matrix Represented as: ; Where M is the discrete deflection observation matrix; K is the global coefficient matrix in the inverse finite element global equation.

[0029] Step 5: Adjust the preliminary full-field deflection vector using a prediction-correction mechanism. After making corrections, we obtain the first... k The final full-field deflection vector at each sampling time point .

[0030] The prediction-correction mechanism includes: According to the k Posterior state estimation at -1 sampling time point Perform state prediction to obtain the first k Prior state estimation at each sampling time: ; And calculate the first k The prior error covariance matrix at each sampling time point: ; in, Here is the state transition matrix. For the first k The posterior error covariance matrix at -1 sampling time point The process noise covariance matrix; Calculate the corrected gain matrix: ; in, Let be the prior error covariance matrix at the k-th sampling time. To augment the observation matrix; The observation noise covariance matrix is ​​expressed as: ; in, The measurement noise covariance matrix is ​​the discrete measured deflection vector. The equivalent noise covariance matrix corresponding to the initial reconstruction constraints of the inverse finite element method; Using the modified gain matrix and the augmented observation vector, the prior state estimate is modified to obtain the first... k Posterior state estimation at each sampling time: ; in, This is the prior state estimate for the k-th sampling time; To correct the gain matrix; To augment the observation vector; To augment the observation matrix; And update the posterior error covariance matrix: ; in, Represents a unit matrix.

[0031] The first state is extracted from the posterior state estimate. k The final full-field deflection vector at each sampling time point .

[0032] Explanation: The technical essence of "correcting the preliminary full-field deflection" in step 5 lies in the following: The preliminary full-field nodal displacement vector obtained by solving the inverse finite element global equation in step 3 contains the estimated information of the bridge's full-field nodal deflection components. However, this estimation is only based on extended full-field strain data and does not incorporate the measured constraints of discrete deflection observations. In step 4, when constructing the augmented observation model, the inverse finite element global equation is incorporated into the augmented observation vector in the form of equivalent observation constraints, so that the preliminary full-field deflection information participates in the prediction-correction process as a priori state constraint. In step 5, the prediction stage recursively obtains the priori state estimate at the k-th sampling time from the posterior state estimate at the (k-1)-th sampling time. The full-field nodal deflection components in this priori state estimate are the information of the preliminary full-field deflection in step 3. In the correction stage, the priori state estimate is weighted and corrected using the discrete measured deflection data in the augmented observation vector and the equivalent constraints of the inverse finite element, thereby realizing the correction and optimization of the preliminary full-field deflection. Finally, a more accurate final full-field deflection vector is extracted from the corrected posterior state estimate.

[0033] Example 2: As Figure 2 As shown, the test bridge is a three-span continuous beam bridge with a span arrangement of 3.2m + 5.6m + 3.2m. Three mid-span sections were used as monitoring sections, and strain sensors were placed at the upper and lower edges of each section to collect the bridge's dynamic strain response. Simultaneously, deflection sensors were placed at the three mid-span positions to acquire the mid-span deflection response. Additionally, a deflection sensor was placed at the left side span's quarter-span position to compare and verify the reconstruction results of the proposed method. During the experiment, a trolley was used as a moving load, traveling at a constant speed along the bridge's longitudinal centerline from left to right, continuously collecting dynamic response data for 30 seconds. The bridge's full-field deflection reconstruction method based on modal-assisted inverse finite element method and discrete deflection correction, proposed in this invention, was used to reconstruct the bridge's full-field deflection time history, and the reconstructed deflection time history at the left side span's quarter-span position was extracted.

[0034] Figure 3 The comparison between the reconstructed deflection and the measured deflection at the quarter-span location of the left side span is presented. Figure 3It can be seen that the deflection time history reconstructed by the method of the present invention is basically consistent with the trend of the measured results, and can accurately reflect the dynamic deflection response of the bridge under moving load.

[0035] Obviously, the above embodiments are merely illustrative examples for clear explanation and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations here. However, obvious variations or modifications derived therefrom are still within the scope of protection of this invention.

Claims

1. A method for reconstructing the full-field deflection of a bridge based on inverse finite element method and discrete deflection correction, characterized in that, Includes the following steps: Step 1: Establish a finite element model of the bridge based on its geometric dimensions, cross-sectional parameters, and boundary conditions. Perform modal analysis on the finite element model to extract several strain mode shapes of the bridge and form a strain mode matrix. Step 2: Obtain sparse measured strain data of the bridge under test, extract the corresponding strain mode submatrix from the strain mode matrix according to the location of the strain measurement points, establish the mapping relationship between the sparse measured strain data and the modal coordinates, solve the modal coordinates, and obtain the extended full-field strain distribution based on the modal coordinates and the strain mode matrix. Step 3: Input the extended full-field strain distribution into the inverse finite element model and solve to obtain the preliminary full-field deflection; Step 4: Obtain discrete measured deflection data of the bridge under test, and construct an augmented observation model that integrates the discrete measured deflection data and the constraint information of the inverse finite element model; Step 5: Use the augmented observation model to correct the initial full-field deflection to obtain the final full-field deflection of the bridge.

2. The bridge full-field deflection reconstruction method based on inverse finite element method and discrete deflection correction as described in claim 1, characterized in that, The sparse measured strain data mentioned in step two are collected from the strain measuring points of the bridge health monitoring system under test, and arranged in the order of the preset strain measuring point numbers to form the first... k Sparse measured strain vector at each sampling time ; Based on the location of the strain measurement points, from the strain mode matrix Extract the corresponding strain mode submatrix Establish the sparse measured strain vector With modal coordinate vector Mapping relationship between them: ； The modal coordinate vector is solved using the least squares principle. And based on the modal coordinate vector and strain mode matrix The extended full-field strain vector at the k-th sampling time is obtained: 。 3. The bridge full-field deflection reconstruction method based on inverse finite element method and discrete deflection correction as described in claim 2, characterized in that, Step three specifically involves: expanding the full-field strain vector. As input to the inverse finite element model, the inverse finite element global equation is constructed and solved to obtain the preliminary global field nodal displacement vector at the k-th sampling time, and the vertical deflection component is extracted from it to obtain the preliminary global field deflection vector.

4. The bridge full-field deflection reconstruction method based on inverse finite element method and discrete deflection correction as described in claim 3, characterized in that, The inverse finite element model is established by dividing the bridge under test into n inverse beam elements along the longitudinal direction. Each inverse beam element includes two nodes, and each node includes three degrees of freedom: axial displacement, vertical deflection, and cross-sectional rotation angle. For any inverse beam element, the theoretical section strain vector is established based on the strain-displacement matrix of the inverse beam element, according to the... k Extended full-field strain vector at each sampling time point Obtain the corresponding extended section strain vector and construct the element equations for the inverse beam element; The element equations of each inverse beam element are assembled as a whole to obtain the global equation of the inverse finite element method.

5. The bridge full-field deflection reconstruction method based on inverse finite element method and discrete deflection correction as described in claim 4, characterized in that, The element equation of the inverse beam element is expressed as follows: ； in, Represents the unit coefficient vector; This represents the element node displacement vector of any inverse beam element; Represents the unit's equivalent vector; and: ； ； ； in, Let be the element length of the inverse beam element. This is the weight matrix; This is the strain-displacement matrix of the inverse beam element; For the extended cross section strain vector; These represent the axial displacement, vertical deflection, and section rotation angle of node one, respectively. These represent the axial displacement, vertical deflection, and section rotation angle of node two, respectively.

6. The bridge full-field deflection reconstruction method based on inverse finite element method and discrete deflection correction as described in claim 4 or 5, characterized in that, The discrete measured deflection data mentioned in step four are collected from the deflection measuring points of the bridge health monitoring system under test, and arranged in the order of the preset deflection measuring point numbers to form the [number of data]. k Discrete measured deflection vector at each sampling time ; Establish the state vector at the k-th sampling time. The state vector This includes the total nodal deflection components and nodal rotation components of the bridge under test. Based on discrete measured deflection vector Constructing augmented observation vectors from inverse finite element global equations And establish an augmented observation model.

7. The bridge full-field deflection reconstruction method based on inverse finite element method and discrete deflection correction as described in claim 6, characterized in that, The augmented observation model is expressed as follows: ； in, To augment the observation vector, To augment the observation matrix, For the observed noise vector; Let be the state vector at the k-th sampling time.

8. The bridge full-field deflection reconstruction method based on inverse finite element method and discrete deflection correction as described in claim 7, characterized in that, The state vector Represented as: ； in, This represents the total field node deflection vector at the k-th sampling time. This represents the rotation vector of all nodes at the k-th sampling time. The augmented observation vector ； in, For the first k Discrete measured deflection vector at each sampling time; It is the inverse finite element global equivalent vector formed by the extended full-field strain vector at the k-th sampling time; The augmented observation matrix Represented as: ； Where M is the discrete deflection observation matrix; K is the global coefficient matrix in the inverse finite element global equation.

9. The bridge full-field deflection reconstruction method based on inverse finite element method and discrete deflection correction as described in claim 8, characterized in that, The correction described in step five employs a prediction-correction mechanism, including: Based on the posterior state estimate at the (k-1)th sampling time Perform state prediction to obtain the prior state estimate at the k-th sampling time, and calculate the prior error covariance matrix at the k-th sampling time; The corrected gain matrix is ​​calculated based on the prior error covariance matrix. The prior state estimate is corrected using the corrected gain matrix and the augmented observation vector to obtain the posterior state estimate at the k-th sampling time, and the posterior error covariance matrix is ​​updated. The first state is extracted from the posterior state estimate. k The final full-field deflection vector at each sampling time point .

10. The bridge full-field deflection reconstruction method based on inverse finite element method and discrete deflection correction as described in claim 9, characterized in that, The modified gain matrix is ​​expressed as follows: ； in, To correct the gain matrix; Let be the prior error covariance matrix at the k-th sampling time. To augment the observation matrix; The observation noise covariance matrix is ​​expressed as: ； in, The measurement noise covariance matrix is ​​the discrete measured deflection vector. The equivalent noise covariance matrix corresponding to the initial reconstruction constraints of the inverse finite element method; The posterior state estimate at the k-th sampling time is: ； in, This is the prior state estimate for the k-th sampling time; To correct the gain matrix; To augment the observation vector; To augment the observation matrix; The posterior error covariance matrix is ​​updated as follows: in, Represents a unit matrix.

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