Bridge model correction method of frequency and deflection influence line dual-objective function

Abstract

The invention discloses a bridge model correction method of a frequency and deflection influence line dual-objective function, and belongs to the technical field of bridge finite element model correction. According to the method, the VMD is combined with the Tikhonov regularization influence line identification method, vehicle-induced bridge response power components and multi-axis effect interference are effectively eliminated, the actually-measured deflection time history response of the bridge is restored to the deflection influence line under the unit concentrated load, the effect of identifying the variable-cross-section steel-concrete combined continuous beam bridge influence line is good, and the applicability is wide; according to the model correction method based on combination of a PSO-BP agent model and NSGA-II dual-objective optimization, a complex implicit function relationship between a to-be-corrected parameter and an objective function can be effectively constructed, a Pareto coordinated optimal solution under the dual-objective optimization problem is obtained, then a corrected parameter is obtained, and finite element model correction is achieved.

Description

Technical Field

[0001] The present invention relates to the technical field of bridge finite element model updating, and particularly relates to a bridge model updating method with a dual-objective function of frequency and deflection influence line. Background Technique

[0002] In recent years, bridge health monitoring technology has developed vigorously, and the research on updating bridge finite element models based on measured data has gradually received extensive attention. However, most traditional model updating methods rely on a single objective function, such as dynamic index parameters or static index parameters. Although the model accuracy can be improved to a certain extent, due to the complexity of actual bridge structures, a single updating objective often fails to meet the analysis accuracy. Therefore, it is urgent to combine multiple performance indicators to update the bridge model and conduct comprehensive optimization methods and experimental research.

[0003] Qin Shiqiang et al. proposed a multi-objective model updating method for railway cable-stayed bridges based on test data and displacement confidence criteria, and used the Kriging surrogate model to update the initial finite element model of the bridge. Peng Tao et al. constructed an updating objective function using the measured dynamic responses of structures such as modal frequencies and vibration modes, and used the measured values to test the model updating results. Wei Jinhui et al. used the measured static and dynamic responses on site, such as static load displacements and vibration frequencies of the structure, to construct a joint static and dynamic structure finite element model updating objective function, and proposed a bridge static and dynamic finite element model updating method based on the response surface.

[0004] The above research mostly constructs an objective function based on response residual functions such as frequency, displacement, or vibration mode to update the finite element model. However, there are still certain limitations. For example, the responses of a small number of local measuring points are difficult to reflect the overall characteristic information of the structure, and the information of the unselected local measuring points is difficult to substitute into the updating process; the objective function constructed by a single performance index parameter is difficult to comprehensively reflect the static and dynamic performance of the structure.

[0005] The above problems need to be solved urgently. For this reason, a bridge model updating method with a dual-objective function of frequency and deflection influence line is proposed. Summary of the Invention

[0006] The technical problem to be solved by the present invention is: how to solve the limitations of the existing model updating research that uses static or dynamic index parameters alone to update the model, and provides a bridge model updating method with a dual-objective function of frequency and deflection influence line.

[0007] As Figure 20 shown, the present invention solves the above technical problems through the following technical solutions. The present invention includes the following steps:

[0008] S1: Conduct modal testing on the beam bridge based on ambient excitation, use the EFDD method for analysis, and obtain the first three modal frequencies;

[0009] S2: Use the idling passage of a single heavy vehicle for loading to obtain the measured deflection time-history response data. Use the VMD method to eliminate the dynamic components of the vehicle-induced response, and use the Tikhonov regularization method to eliminate the interference of the multi-axle effect to identify the bridge deflection influence line.

[0010] S3: Construct a combined objective function using the modal frequency and the deflection influence line and select the parameters to be corrected. Adopt the PSO-BP surrogate model and combine it with the NSGA-II algorithm for bi-objective function optimization to correct the finite element model.

[0011] Furthermore, in the step S1, the specific processing process is as follows:

[0012] S11: Arrange the measuring points of the continuous beam bridge test section according to the quarter points of the bridge span. A total of multiple measuring points are arranged in two measuring lines, upstream and downstream, and the velocity signals of each measuring point are collected simultaneously.

[0013] S12: Use the EFDD method to analyze the velocity signals of each measuring point to obtain the first three modal frequencies of the bridge structure.

[0014] Furthermore, in the step S12, the relationship between the input and output signals in the EFDD method is as follows:

[0015] G yy (jw) = H(jw)G xx (jw)H H (jw)

[0016] Among them, G yy (jw) is the power spectral density matrix of the output signal, H(jw) is the frequency response function matrix, and G xx (jw) is the power spectral density matrix of the input signal, and the superscript H represents the complex conjugate transpose.

[0017] Furthermore, in the step S2, the processing process of the VMD method is as follows:

[0018] S201: Under the constraint conditions, the sum of all modes is equal to the original signal, and the bandwidth of each mode is minimized. The constraint condition is the variational constraint, which is expressed as follows:

[0019]

[0020] Among them, u k is the k-th IMF component, ω k is the center frequency of the k-th IMF component, is the partial derivative with respect to time, δ(t) is the Dirac function, * is the convolution operator, and f(t) is the original input signal, that is, the measured deflection time-history response data obtained;

[0021] S202: Introduce the alternating multiplication operator λ(t) and the quadratic penalty factor α, transform the constrained variational problem in step S201 into an unconstrained variational problem, and the expanded unconstrained Lagrange function expression is as follows:

[0022]

[0023] Solve it by iteratively updating {u k 1}, {ω k 1}, λ 1 until the iterative stop when the allowable error ε is satisfied. The allowable error discriminant is as follows:

[0024]

[0025] S203: Gradually take values starting from K = 2. When the dominant frequency of the IMF (K-1) is lower than the structural fundamental frequency, reconstruct the signals of the IMF (K-1 ) and the IMF (K) . The obtained signal is the measured quasi-static time history response data of the bridge.

[0026] Furthermore, in the said step S2, the process of identifying the bridge deflection influence line is as follows:

[0027] S211: Use the Tikhonov regularization method, introduce the error e, and establish a mathematical model for influence line identification:

[0028] R S = Lφ + e

[0029] where Lφ is the measured quasi-static time history response of the bridge, L is the vehicle information matrix, and φ is the influence line coefficient;

[0030] S212: Tikhonov regularization restricts the least squares expression by using the L2 norm as the penalty function. The regularization equation is as follows:

[0031]

[0032] where λ is the regularization coefficient and T is the regularization matrix:

[0033]

[0034] S213: Establish the vehicle information matrix as follows:

[0035]

[0036] where A k is the axle weight;

[0037] S214: Substitute the regularization matrix T into the regularization equation and set the derivative function to 0, then the influence line R s The solution expression is as follows:

[0038] φ=(L T L + λ 2 T T T) -1 L T R s .

[0039] Furthermore, in the step S3, the objective function constructed by using the modal frequency and the deflection influence line is as follows:

[0040]

[0041] Among them, F1 and F2 are the deflection influence line and the modal frequency objective function; α s , β s are the static and dynamic parameter weight coefficients, with values of 1 / n and 1 / k respectively, u mi is the measured value of the influence line, u ti is the theoretically calculated value of the influence line; f mi is the measured value of the frequency, f ti is the theoretically calculated value of the frequency; n is the number of influence line coefficients; k is the frequency order.

[0042] Furthermore, in the step S3, the specific process of selecting the parameters to be corrected is as follows:

[0043] S301: Initially select the parameters to be corrected as the elastic modulus E1, Poisson's ratio v1 and unit weight r1 of the concrete roof slab; the elastic modulus E2, Poisson's ratio v2, unit weight r2, roof slab thickness tf1, web thickness tw and bottom slab thickness tf2 of the rigid box girder;

[0044] S302: Use the difference method to calculate the relative sensitivity of the modal frequency and the deflection influence line after regularization for the parameters to be corrected. The calculation formula is as follows:

[0045]

[0046] Among them, S ij is the sensitivity of the modal frequency and the deflection influence line index with respect to each parameter to be corrected, p j is the initial value of the parameter to be corrected, f i represents the structural response, Δp j Take p j / 100, i represents the index of the modal frequency or the deflection influence line index, and j represents the index of the parameter to be corrected;

[0047] S303: After sensitivity analysis, the final parameters to be corrected are determined as the elastic modulus E2 of the steel box girder, the top plate thickness tf1, the web thickness tw, and the bottom plate thickness tf2, and the correction ranges of each parameter to be corrected are determined.

[0048] Furthermore, in the PSO-BP surrogate model of step S3, the PSO algorithm is used to optimize the initial weights and biases of the BP neural network, accelerating convergence and improving the model accuracy simultaneously.

[0049] Among them, in each iteration of the PSO algorithm, the particle velocity and position are updated, and the particle moves towards the individual historical optimal center, making the population approach the optimal solution. The specific update formulas are as follows:

[0050] v ij = wv ij + c1r1(p bij - x ij ) + c2r2(g bij - x ij )

[0051] x ij = x ij + v ij

[0052] Among them, v ij represents the velocity of particle i in the j-th dimension, x ij represents the position of particle i in the j-th dimension, w represents the inertia weight, p bij represents the individual optimal solution of particle i in the j-th dimension, g bij represents the global optimal solution of the entire population in the j-th dimension, c1 and c2 respectively represent the individual learning factor and the social learning factor, and r1 and r2 are random numbers in (0, 1).

[0053] Furthermore, in step S3, the specific processing process of the NSGA-II algorithm is as follows:

[0054] S311: Randomly generate a group of initial solutions and calculate the fitness values of each solution on the objective function.

[0055] S312: Perform non-dominated sorting on the solutions in the population, divide them into different levels, and assign better solutions to higher levels; calculate the crowding distance of the solutions within each level to measure the distribution density of the solutions in the objective function space.

[0056] S313: According to the non-dominated solution level and the crowding distance, select the high-level solutions as the parents, and then generate higher-level solutions.

[0057] S314: Perform crossover and mutation operations on the selected parent solutions to generate new solutions.

[0058] S315: Combine the parental solutions and the newly generated solutions into a new population and truncate it to the size of the original population;

[0059] S316: Repeat the above steps until the predetermined number of iterations or the termination condition is met;

[0060] S317: Output the Pareto front.

[0061] The present invention has the following advantages compared with the prior art:

[0062] 1. The influence line identification method using VMD combined with Tikhonov regularization can effectively eliminate the interference of the dynamic components of vehicle-induced bridge responses and the multi-axle effect, restore the time-history response of bridge deflection to the deflection influence line under a unit concentrated load, and has good effects in identifying the influence line of variable-section steel-concrete composite continuous beam bridges, with wide applicability;

[0063] 2. The model correction method based on the PSO-BP surrogate model combined with NSGA-II dual-objective optimization can effectively construct the complex implicit function relationship between the parameters to be corrected and the objective function, obtain the Pareto coordinated optimal solution under the dual-objective optimization problem, and then obtain the corrected parameters;

[0064] 3. It can significantly improve the description accuracy of the bridge finite element model for the actual operation performance, and has the coordination and applicability of static and dynamic performances, and can provide a theoretical basis and method basis for subsequent performance prediction, health diagnosis and safety assessment of existing bridges. Brief Description of the Drawings

[0065] Figure 1 is a schematic diagram of the research path of the bridge model correction method with the dual-objective functions of frequency and deflection influence line in the embodiment of the present invention;

[0066] Figure 2 is a schematic diagram of the Pareto front of the dual-objective optimization problem in the embodiment of the present invention;

[0067] Figure 3 is a schematic diagram of the calculation process of the PSO-BP surrogate model in the embodiment of the present invention;

[0068] Figure 4 is a schematic diagram of the initial model and cross-sectional dimensions of the bridge in the embodiment of the present invention, where (a) is the initial finite element model and the layout of measurement points, and (b) is the cross-sectional dimensions of the bridge;

[0069] Figure 5 is the D2 of the bridge modal test in the embodiment of the present invention # Power spectral density diagram of the measurement point, where (a) is the time-domain waveform diagram and (b) is the frequency-domain waveform diagram;

[0070] Figure 6It is a comparison diagram of the first three vertical bending vibration modes in the embodiment of the present invention. Among them, (a) is the measured first-order vertical bending vibration mode, with a frequency of 1.375 Hz; (b) is the measured second-order vertical bending vibration mode, with a frequency of 2.125 Hz; (c) is the measured third-order vertical bending vibration mode, with a frequency of 3.125 Hz; (a1) is the theoretical first-order vertical bending vibration mode, with a frequency of 1.133 Hz; (a2) is the theoretical second-order vertical bending vibration mode, with a frequency of 1.904 Hz; (a3) is the theoretical third-order vertical bending vibration mode, with a frequency of 2.688 Hz;

[0071] Figure 7 It is a schematic diagram of the test loading process in the embodiment of the present invention;

[0072] Figure 8 It is a comparison diagram of the deflection time-history response before and after VMD processing in the embodiment of the present invention

[0073] Figure 9 It is the measured deflection influence line after Tikhonov regularization in the embodiment of the present invention;

[0074] Figure 10 It is a comparison diagram of the deflection influence line before correction and the measured deflection influence line in the embodiment of the present invention;

[0075] Figure 11 It is a result diagram of the sensitivity analysis of the frequency index in the embodiment of the present invention;

[0076] Figure 12 It is a result diagram of the sensitivity analysis of the influence line index in the embodiment of the present invention;

[0077] Figure 13 It is the neural network topology diagram in the embodiment of the present invention;

[0078] Figure 14 It is the neural network training effect diagram in the embodiment of the present invention;

[0079] Figure 15 It is the fitness curve of the PSO-BP neural network in the embodiment of the present invention;

[0080] Figure 16 It is a comparison diagram of the relative error between the predicted value and the sample value in the embodiment of the present invention;

[0081] Figure 17 It is the Pareto optimal solution set diagram in the embodiment of the present invention;

[0082] Figure 18 It is a comparison diagram of the deflection influence line after correction and the measured deflection influence line in the embodiment of the present invention;

[0083] Figure 19 It is a schematic comparison diagram of the absolute error of the deflection influence line before and after correction in the embodiment of the present invention;

[0084] Figure 20It is a schematic flow chart of the bridge model correction method for the dual-objective function of the frequency and deflection influence line of the present invention. Detailed implementation manners

[0085] The following makes a detailed description of the embodiments of the present invention. The embodiments are implemented on the premise of the technical solution of the present invention, and detailed implementation manners and specific operation processes are given. However, the protection scope of the present invention is not limited to the following embodiments.

[0086] Embodiment 1

[0087] As Figure 1 shown, this embodiment provides a technical solution: a bridge model correction method for the dual-objective function of frequency and deflection influence line, including the following steps:

[0088] Step 1: Perform modal testing on the beam bridge based on ambient excitation, and use the Enhanced Frequency Domain Decomposition (EFDD) method to obtain the first three natural frequencies.

[0089] Step 2: Use a single heavy vehicle to idle through for loading, obtain the measured deflection time history response data, and use the Variational Mode Decomposition (VMD) method and the Tikhonov regularization method to remove the vehicle-induced response dynamic components and multi-axial effects respectively, and identify the bridge deflection influence line.

[0090] Step 3: Use the frequency and deflection influence line to construct a combined objective function and select the parameters to be corrected, and use the PSO-BP surrogate model and combine it with the fast non-dominated sorting genetic algorithm with elitist strategy (Non-dominated Sorting Genetic Algorithm-II, NSGA-II) to optimize the dual-objective function and correct the finite element model.

[0091] The following makes a more specific description of the core methods in the above steps:

[0092] 1. Modal frequency and influence line identification

[0093] 1.1 Modal frequency identification method

[0094] The Enhanced Frequency Domain Decomposition (EFDD) is an improved method of the frequency domain decomposition method. Its basic principle is to perform the inverse Fourier transform on the single-degree-of-freedom power spectral density function after singular value decomposition to obtain the corresponding autocorrelation function, and then use the logarithmic decrement method to identify the modal frequency and modal shape. The relationship between its input and output signals is shown in Equation (1):

[0095] G yy S(jw)=H(jw)G xx H(jw) H S(jw)(1)

[0096] where G yy S(jw) is the power spectral density matrix of the output signal, H(jw) is the frequency response function matrix, and G xx S(jw) is the power spectral density matrix of the input signal, and the superscript H represents the complex conjugate transpose.

[0097] 1.2. Deflection Influence Line Identification Method

[0098] 1.2.1. Pretreatment of Deflection Time History Response

[0099] After VMD processing, the time-domain signal can be decomposed into several finite-bandwidth IMFs from high frequency to low frequency. Assuming that each finite-bandwidth IMF fluctuates around its own center frequency, the optimal solution of the variational mode is searched iteratively. Under the constraint conditions, the sum of all modes is equal to the original signal, and the bandwidth of each mode is minimized. The variational constraint can be described as shown in (2):

[0100]

[0101] where u k is the k-th IMF component, ω k is the center frequency of the k-th IMF component, is the partial derivative with respect to time, δ(t) is the Dirac function, * is the convolution operator, and f(t) is the original input signal.

[0102] To find the optimal solution of the constrained variational problem, the alternating multiplicative operator λ(t) and the quadratic penalty factor α are introduced, and the constrained variational problem in equation (2) is transformed into an unconstrained variational problem. The extended unconstrained Lagrange function expression is shown in equation (3):

[0103]

[0104] Equation (3) is solved by iteratively updating λ 1 until the allowable error ε is satisfied and the iteration stops. The allowable error discriminant is shown in the following equation (4):

[0105]

[0106] To avoid over-decomposition or under-decomposition of the measured deflection time history response, the value is gradually taken starting from K = 2. When the dominant frequency of IMF (K-1) is lower than the fundamental frequency of the structure, for IMF (K-1 ) and IMF (K)Perform signal reconstruction, and the obtained signal is the measured quasi-static time history response of the bridge.

[0107] 1.2.2 Identification of deflection influence line

[0108] Based on the vehicle multi-axle effect information contained in the measured quasi-static time history response of the bridge processed by VMD, in order to eliminate the interference of the multi-axle effect, the Tikhonov regularization method is used, introducing the error e, and a mathematical model for influence line identification is established as shown in the following formula (5):

[0109] R S = Lφ + e (5)

[0110] Among them, Lφ is the quasi-static response of the bridge, L is the vehicle information matrix, and φ is the influence line coefficient.

[0111] Tikhonov regularization uses the L2 norm as a penalty function to limit the least squares expression, thus solving the ill-conditioning of the influence line identification model equation caused by the error term. The regularization equation is shown in formula (6):

[0112]

[0113] Among them, λ is the regularization coefficient, and the regularization matrix T is shown in formula (7):

[0114]

[0115] Establish a vehicle information matrix, which is determined by parameters such as sampling frequency, vehicle axle weight, number of axles, axle distance, and vehicle speed. Taking the moment when the front axle of the vehicle gets on the bridge and the moment when the rear axle gets off the bridge as the starting and ending times, the vehicle information matrix is shown in formula (8):

[0116]

[0117] Among them, A k is the axle weight.

[0118] Substitute the regularization matrix T into the regularization equation and set the derivative function to 0, then the influence line R s Solve the expression as shown in formula (9):

[0119] φ = (L T L + λ 2 T T T) -1 L T R s (9)

[0120] Among them, λ is the regularization coefficient, which is determined by the L-curve method.

[0121] 2 Establishment of multi-objective model correction method

[0122] 2.1 Multi-objective Optimization Method

[0123] The multi-objective optimization problem consists of an n-dimensional decision variable, m objective variables, and several constraint conditions, and can be described as in Equation (10):

[0124]

[0125] where x is the n-dimensional decision variable, which forms the decision space; F(x) is the objective function, and m is the number of objective functions; g i (x) is the inequality constraint; h j is the equality constraint; i and j are the numbers of constraints; X k-max and X k-min are the upper and lower limits of the k-th dimensional vector search.

[0126] In the multi-objective optimization problem, there are often mutual restraint relationships among the objectives. Therefore, it is impossible to make all objectives reach the optimal at the same time. Therefore, on the basis of comprehensively weighing each objective, a set of optimal solution sets need to be determined. These solutions are called Pareto optimal solutions, and the boundary formed by them is called the Pareto front. Taking the common two-objective optimization problem in engineering as an example, as Figure 2 shown. The area surrounded by the Pareto front in the figure represents the feasible solution domain of the problem, and the arrow represents the search direction. Through the optimization algorithm, all Pareto optimal solutions are searched in the feasible solution domain to form the Pareto front as in Figure 2 . On this front, points X L , X, and X R are all Pareto optimal solutions. Among them, point X is the point with the largest bending angle on the Pareto front curve, which is the convex point of the curve. This point is the coordinated optimal solution in the Pareto solution set.

[0127] 2.2 Model Updating Method Based on NSGA-II Two-Objective Optimization

[0128] Model updating is a reverse non-linear problem. Since it is impossible to directly establish the complex implicit function relationship between the parameters to be corrected of the structure and the optimized objective function. Therefore, first, a PSO-BP surrogate model is constructed. Because of its good learning ability and non-linear mapping ability, it can approximate any continuous function within a closed interval, complete the mapping from m dimensions to n dimensions, and avoid directly constructing complex implicit function expressions. Subsequently, based on the NSGA-II two-objective optimization algorithm, within the range of parameters to be corrected, the PSO-BP surrogate model is called to iteratively optimize, search for the coordinated optimal solution of the two-objective optimization problem, and thus obtain the exact values of the target parameters for finite element model updating.

[0129] 2.2.1 PSO-BP Surrogate Model

[0130] Considering that the traditional BP neural network is sensitive to the initial weights, has a slow convergence speed, and is prone to overfitting. The present invention uses the PSO algorithm to optimize the initial weights and biases of the BP neural network, accelerate convergence, and improve the model accuracy. The calculation process is shown in Figure 3 。

[0131] In each iteration of the PSO algorithm, the particle velocity and position are updated. The particles move towards the individual historical optimal center, enabling the population to approach the optimal solution and achieving the optimization effect. The specific update formulas are as follows:

[0132] v ij =wv ij +c1r1(p bij -x ij )+c2r2(g bij -x ij ) (11)

[0133] x ij =x ij +v ij (12)

[0134] Among them, v ij represents the velocity of particle i in the j-th dimension, x ij represents the position of particle i in the j-th dimension, w represents the inertia weight, p bij represents the individual optimal solution of particle i in the j-th dimension, g bij represents the global optimal solution of the entire population in the j-th dimension, c1 and c2 respectively represent the individual learning factor and the social learning factor, and r1 and r2 are random numbers in (0, 1).

[0135] 2.2.2, NSGA-II Two-Objective Optimization Algorithm

[0136] In the two-objective optimization problem, the fast non-dominated sorting genetic algorithm with elitist strategy (Non-dominated Sorting Genetic Algorithm-Ⅱ, NSGA-Ⅱ) is used to iteratively optimize within the search range. Due to the characteristics of this algorithm such as fast non-dominated sorting, elitist retention strategy, and reduced computational complexity, it has been initially applied in engineering.

[0137] The NSGA-II algorithm process is as follows:

[0138] (1) Initialize the population: Randomly generate a group of initial solutions and calculate the fitness values of each solution on the objective function.

[0139] (2) Non-dominated solution sorting and calculation of crowding distance: Sort the solutions in the population into non-dominated solutions, divide them into different levels, and assign better solutions to higher levels. Calculate the crowding distance of the solutions within each level to measure the distribution density of the solutions in the objective function space.

[0140] (3) Selection operation: According to the non-inferior solution level and congestion distance, a high-level solution is selected as the parent to generate a higher-level solution.

[0141] (4) Crossover and mutation operations: Perform crossover and mutation operations on the selected parent solution to generate a new solution.

[0142] (5) Update the population: combine the parent solution and the newly generated solution into a new population and truncate it to the size of the original population.

[0143] (6) Repeat iteration: Repeat the above steps until the predetermined number of iterations is reached or the termination condition is met.

[0144] (7) Output results: Output Pareto frontier.

[0145] Embodiment 2

[0146] 3. Bridge frequency and influence line testing and identification

[0147] 3.1 Project Overview

[0148] A three-span variable-section continuous steel-concrete composite beam bridge was used as the test object to carry out a model modification test based on frequency and influence line dual-objective optimization. The span of the bridge is (57+72+48)m, the bridge width is 12.75m, the main beam is a three-box single-chamber variable-height continuous composite beam, the steel structure is made of Q345 steel; the lower structure uses column piers and pile foundations; rib abutments are used, and the piers are arranged radially. The initial model, measurement point arrangement and section size are as follows Figure 4 shown.

[0149] 3.2 Modal frequency identification

[0150] The bridge modal was tested under environmental excitation using a 941B vibration pickup. According to the "Highway Bridge Load Test Code" (JTG T J21-01-2015), the test section measurement points of the continuous beam bridge were arranged according to the quarter points of the bridge span, and a total of 18 measurement points were arranged in two upstream and downstream measurement lines. The velocity signal of each measurement point was collected at the same time. The waveform measured on site is as follows Figure 5 As shown. The first three frequencies and vibration modes are analyzed by EFDD method as shown Figure 6 The first three modal frequencies are shown in Table 1. The test found that the first three natural frequencies measured were greater than the theoretical calculated values, indicating that the overall stiffness of the test bridge is good. The measured fundamental frequency of the bridge is 1.375Hz.

[0151] Table 1 Measured modal frequencies and finite element calculated frequencies before correction

[0152]

[0153] 3.3 Deflection influence line identification

[0154] To obtain the measured time - history response data of the bridge deflection, an optoelectronic dynamic deflection meter was used to test the deflection measuring points at the control section of the bridge mid - span. The loading path was selected as the center line of the bridge, and a three - axle loading vehicle with a full load of 340 kN was made to travel at idle speed along the loading route. The axle weights of the vehicle were 136 kN, 136 kN, and 68 kN respectively, and the axle spacings were 1.4 m and 3.8 m. The test loading process is as Figure 7 . The time - history response of the bridge deflection under the influence line loading was measured. The dynamic components of the vehicle - induced bridge response were removed by the VMD method, and the multi - axle effect was separated by combining with Tikhonov regularization, so as to restore the time - history response of the bridge deflection to the deflection influence line under the action of a unit concentrated load. The VMD pre - processing process is as Figure 8 , and the measured deflection influence line after using Tikhonov regularization is as Figure 9 .

[0155] The deflection influence line of the bridge mid - span measured through the loading test conforms to the general law of the theoretical influence line of a three - span continuous beam bridge. The overall stiffness of the bridge is uniform, and there is no obvious mutation in the curve, indicating that the health state of the bridge structure is good [22-23] . The measured deflection influence line and the deflection influence line of the model before correction are shown in Figure 10 . The peak value of the deflection influence line of the mid - span of the bridge under vehicle - induced excitation is 2.8671×10 -6 mm / N. The peak value of the deflection influence line of the initial finite - element model established only based on the design information under the same loading condition is 2.0318×10 -6 mm / N. The relative error of the peak value of the mid - span control section is 41.1%. It is not difficult to know that the actual bridge is safer than the design model. However, due to the large deviation between the two, it is necessary to further carry out finite - element model correction research on the initial model.

[0156] 4. Finite - element model correction

[0157] 4.1. Selection of parameters to be corrected

[0158] When selecting the parameters to be corrected in the model, possible error parameters need to be considered, including structural geometric parameters, material properties parameters, and boundary conditions, etc. However, in the process of parameter selection, problems such as the parameters being insensitive to the characteristic information or being selected or misselected by mistake will be faced. Therefore, it is necessary to conduct a sensitivity analysis on the parameters to be corrected and screen out the parameters with greater sensitivity. Nine parameters are selected in this invention for sensitivity analysis: the elastic modulus E1, Poisson's ratio v1, and unit weight r1 of the concrete top plate of the steel - concrete composite beam bridge; the elastic modulus E2, Poisson's ratio v2, unit weight r2, top - plate thickness tf1, web - plate thickness tw, and bottom - plate thickness tf2 of the rigid steel box girder. The relative sensitivities of the frequency and the influence line to the regularized parameters to be corrected are calculated by the difference method, as shown in Equation (13):

[0159]

[0160] Among them, S ij is the sensitivity of the frequency and the deflection influence line index with respect to the parameter to be corrected, and Δp j can be taken as p j / 100, where p j is the initial value of the parameter to be corrected, f i represents the structural response, i represents the index of the modal frequency or the deflection influence line index, and j represents the index of the parameter to be corrected.

[0161] The frequency sensitivity analysis is as Figure 11 , and the peak sensitivity analysis of the influence line is as Figure 12 . According to the analysis results, parameters that have a greater impact on the influence line and frequency sensitivity are comprehensively selected. Specifically, four parameters, namely the elastic modulus E2 of the steel box girder, the top plate thickness tf1, the web thickness tw, and the bottom plate thickness tf2, are selected as the parameters to be corrected. For the elastic modulus E2 of the steel box girder, the correction range is (0.8 - 1.5)E2 based on empirical values; the correction range of the top plate thickness tf1 is 15 - 25 mm; since the web and bottom plate of the steel box girder have variable thicknesses of 16 - 24 mm and 24 - 36 mm respectively, the upper and lower limits of the variable thickness are used as the upper and lower limits of the correction values. To sum up, the selected parameters to be corrected and their value ranges are shown in Table 2.

[0162] Table 2 Value Ranges of Parameters to be Corrected in the Finite Element Model

[0163]

[0164]

[0165] 4.2. Construct the Objective Function

[0166] To overcome the limitations of using only static or dynamic index parameters for model correction, a model correction method that combines the dual-objective optimization of frequency and deflection influence line is adopted, so that the corrected model can more accurately reflect the dynamic and static performance of the structure. The construction of the objective function combining modal frequency and deflection influence line is shown in Equation (14):

[0167]

[0168] Among them, F1 and F2 are the objective functions of the deflection influence line and the modal frequency; α s , β s are the weight coefficients of static and dynamic parameters, with values of 1 / n and 1 / k respectively, u mi is the measured value of the influence line, u ti is the theoretically calculated value of the influence line; f mi is the measured value of the frequency, f ti is the theoretically calculated value of the frequency; n is the number of influence line coefficients; k is the order of the frequency.

[0169] 4.3, PSO-BP Surrogate Model and Precision Evaluation

[0170] 4.3.1, PSO-BP Surrogate Model

[0171] The initial finite element model parameters are randomly scaled 40 times, and the frequency and deflection influence lines under the corresponding structural parameters are extracted to form 40 neural network sample sets. When training the network, to verify the generalization ability of the network model, the above 40 sample sets are randomly divided into a training set (80%) and a test set (20%). Using the physical parameters of the finite element model structure as the input layer and the constructed frequency and influence line objective functions as the output layer, set the number of training times to 1000, the learning rate to 0.01, and the minimum error of the training target to 1×10 -5 , construct a PSO-BP surrogate model and perform learning training and prediction. Since the parameter orders of magnitude of the input layer and the output layer are different, Min-Max is used to normalize the data:

[0172]

[0173] where x is the normalized parameter value, x max and x min are the maximum and minimum values of the parameter respectively, and x is the original data parameter.

[0174] To determine the number of hidden layer nodes, according to the empirical formula and through comparison of multiple numbers of hidden layer nodes, it is determined that when the number of hidden layer nodes is 4, the training accuracy requirements can be met. The empirical formula is:

[0175]

[0176] where J is the number of hidden layer nodes, p and q are the numbers of nodes in the input and output layers respectively, and i is a random constant in (1, 10).

[0177] Based on the determined number of nodes in each layer, the constructed PSO-BP neural network topology is as shown in Figure 13 . The L-M (Levenberg-Marquardt) backpropagation algorithm is used to train this network, and the result is as shown in Figure 14 . The fitness curve of PSO-BP is shown in Figure 15 . As can be seen from Figure 14 , the PSO-BP neural network reaches the best accuracy in 21 iteration processes, and its training error (MSE) is 5.7971×10 -3 , and the total regression coefficient R = 0.99869. As can be seen from Figure 15 , with the iteration progressing, the fitness value continuously decreases in the initial stage of iteration and performs precise search within the extreme value range in the later stage, and the convergence speed slows down.

[0178] 4.3.2. Agent Model Accuracy Evaluation

[0179] In the constructed PSO - BP agent model, the input and output data are trained to obtain a neural network model between the model parameters and the objective function. When the network training reaches the required accuracy, the training is completed. In this network model, the frequencies and influence line objective functions of different structural parameters are predicted. The relative errors between the predicted values and the sample values are as Figure 16 . As Figure 16 known, there are still errors in the prediction results of the neural network for the frequency objective function. The minimum error is 0.678%, the maximum error is 7.79%, and the error range is controlled within 8%. For the influence line objective function, the minimum error is 1.29%, the maximum error is 6.11%, and the error range is controlled within 7%. The data fitting degree between the predicted values and the sample values is good, indicating that the constructed PSO - BP agent model can reflect the mapping relationship between the model structural parameters and the objective function. The accuracy and generalization ability of the agent model are good, and it can be further used for the correction test analysis of the frequency and influence line double - objective model.

[0180] 4.4. Correction Results and Error Analysis

[0181] 4.4.1. Model Correction Results

[0182] Based on the essence of the correction of the frequency and influence line double - objective model, with the frequency and influence line objective functions as the optimization objectives, within the range of parameters to be corrected, the constructed PSO - BP agent model is called for iterative optimization to search for the coordinated optimal solution of the double - objective optimization problem. After the optimization calculation according to the set parameters, the distribution of non - dominated individuals and the Pareto front of the finite element model correction of the steel - concrete composite continuous beam bridge are obtained, as Figure 17 . The maximum bending angle method is used to find the coordinated optimal solution in the Pareto optimal solution set to realize the optimization of the model parameters. The comparison of the initial values and the corrected values of the parameters to be corrected is shown in Table 3.

[0183] Table 3 Parameters before and after model correction

[0184] Parameter Initial value Corrected value Variation percentage <![CDATA[E2 / Mpa]]> 206000 259000 25.73 <![CDATA[tf1 / mm]]> 20 24.97 24.85 tw / mm 20 23.99 19.95 <![CDATA[tf2 / mm]]> 28 26.58 5.07

[0185] 4.4.2. Model Correction Error Analysis

[0186] The comparison of the modal frequencies before and after model correction is shown in Table 4. The comparison between the corrected value and the measured deflection influence line is shown in Figure 18 . The absolute error of the influence line before and after correction is shown in Figure 19 . Analyzing Table 4, it can be seen that the natural vibration frequencies of the structure after model correction increase compared with those before correction and are closer to the measured frequencies of the bridge. The error between the measured frequency and the calculated frequency of the bridge is reduced from within 20% before correction to within 5% after correction. Among them, the relative error of the first - order frequency is reduced from 17.6% to 3.27%. Analyzing Figure 18It is known that the overall relative error of the deflection influence line before and after correction is reduced from 39.7% to 16%, and the peak relative error of the mid-span control section is reduced from 41.1% to 3.55%. Analysis Figure 19 It is known that the absolute error of the mid-span control section of the deflection influence line is reduced by 0.7546, and the model accuracy is significantly improved. The corrected model is closer to the real bridge structure.

[0187] Table 4 Comparison of modal frequencies before and after model correction

[0188]

[0189] In summary, for the bridge model correction method with the dual-objective function of frequency and deflection influence line in the above embodiments, by using the influence line identification method combining VMD and Tikhonov regularization, the dynamic components of vehicle-induced bridge responses and the interference of multi-axle effects can be effectively eliminated, and the deflection time-history response of the bridge can be restored to the deflection influence line under a unit concentrated load. The effect of identifying the influence line of the variable-section steel-concrete composite continuous beam bridge is good and it has wide applicability; based on the model correction method combining the PSO-BP surrogate model and NSGA-II dual-objective optimization, the complex implicit function relationship between the parameters to be corrected and the objective function can be effectively constructed, and the Pareto coordinated optimal solution under the dual-objective optimization problem can be obtained, and then the correction parameters can be obtained. The optimization range of the model parameters is between 5.07% and 25.73%, which is within the deviation range of actual construction and material properties; after the model is corrected, the relative errors of the first three-order modal frequencies are all reduced to less than 5%; the overall relative error of the deflection influence line is reduced from 39.7% to 16%, and the peak relative error of the mid-span control section is reduced from 41.1% to 3.55%. The corrected bridge model is closer to the real performance of the bridge structure, effectively improving the model analysis accuracy; the model correction method based on the dual-objective optimization of frequency and influence line can significantly improve the description accuracy of the bridge finite element model for the actual operation performance, and has the coordination and applicability of static and dynamic performances. This research method can provide a theoretical basis and method reference for the subsequent prediction of the performance of existing bridges, health diagnosis and safety assessment.

[0190] Although the embodiments of the present invention have been shown and described above, it can be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those of ordinary skill in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of the present invention.

Claims

1. A bridge model correction method with a dual-objective function of frequency and deflection influence line, characterized in that It includes the following steps: S1: Conduct modal testing on the beam bridge based on ambient excitation, analyze it using the EFDD method, and obtain the first three modal frequencies; S2: Use a single heavy vehicle to pass idly for loading, obtain the measured deflection time-history response data, use the VMD method to remove the vehicle-induced response dynamic components, and use the Tikhonov regularization method to remove the multi-axial effect interference to identify the bridge deflection influence line; S3: Construct a combined objective function using the modal frequencies and the deflection influence line and select the parameters to be corrected, use the PSO-BP surrogate model and combine it with the NSGA-II algorithm for bi-objective function optimization to correct the finite element model.

2. A method for correcting a bridge model with a dual-objective function of frequency and deflection influence line according to claim 1, characterized in that In the step S1, the specific processing procedure is as follows: S11: Arrange the measuring points of the continuous beam bridge test section according to the quarter points of the bridge span, arrange multiple measuring points in two measuring lines, upstream and downstream, and collect the velocity signals of each measuring point at the same time; S12: Use the EFDD method to analyze the velocity signals of each measuring point to obtain the first three modal frequencies of the bridge structure.

3. A method for correcting a bridge model with a double-objective function of frequency and deflection influence line according to claim 2, characterized in that In the step S12, the relationship between the input and output signals in the EFDD method is as follows: G yy (jw) = H(jw)G xx (jw)H H (jw) Among them, G yy (jw) is the power spectral density matrix of the output signal, H(jw) is the frequency response function matrix, and G xx (jw) is the power spectral density matrix of the input signal, and the superscript H represents the complex conjugate transpose.

4. A method for correcting a bridge model with a double-objective function of frequency and deflection influence line according to claim 2, characterized in that, In the step S2, the processing procedure of the VMD method is as follows: S201: Under the constraint conditions, the sum of all modes is equal to the original signal, and the bandwidth of each mode is minimized. The constraint conditions are variational constraints, expressed as follows: where, u k is the k-th IMF component, ω k is the central frequency of the k-th IMF component, is the partial derivative with respect to time, δ(t) is the Dirac function, * is the convolution operator, and f(t) is the original input signal, i.e., the measured deflection time history response data obtained; S202: Introduce the alternating multiplicative operator λ(t) and the quadratic penalty factor α, transform the constrained variational problem in step S201 into an unconstrained variational problem, and the expanded unconstrained Lagrange function expression is as follows: By iteratively updating {u k 1}, {ω k 1}, λ 1 to solve until the iterative stop when the allowable error ε is satisfied. The allowable error discriminant is as follows: S203: Starting from K = 2 and gradually taking values, when the main frequency of IMF (K-1) is lower than the structural fundamental frequency, perform signal reconstruction on IMF (K-1 ) and IMF (K) . The obtained signal is the measured quasi-static time history response data of the bridge.

5. A method for correcting a bridge model with a double-objective function of frequency and deflection influence line according to claim 4, characterized in that In the step S2, the process of identifying the bridge deflection influence line is as follows: S211: Use the Tikhonov regularization method, introduce the error e, and establish a mathematical model for influence line identification: R S = Lφ + e where, Lφ is the measured quasi-static time-history response of the bridge, L is the vehicle information matrix, and φ is the influence line coefficient; S212: Tikhonov regularization uses the L2 norm as a penalty function to limit the least squares expression, and the regularization equation is as follows: where, λ is the regularization coefficient, and T is the regularization matrix: S213: Establish the vehicle information matrix as follows: Among them, A k is the axle load; S214: Substitute the regularization matrix T into the regularization equation and set the derivative function to 0, then the influence line R s The solution expression is as follows: φ = (L T L + λ 2 T T T) -1 L T R s 。 6. A method for correcting a bridge model with a dual objective function of frequency and deflection influence line according to claim 5, characterized in that, In the step S3, the objective function constructed using the modal frequencies and the deflection influence line is as follows: Among them, F1 and F2 are the deflection influence line and the modal frequency objective function; α s , β s are the static and dynamic parameter weight coefficients, with values of 1 / n and 1 / k respectively. u mi is the measured value of the influence line, and u ti is the theoretically calculated value of the influence line; f mi is the measured value of the frequency, and f ti is the theoretically calculated value of the frequency; n is the number of influence line coefficients; k is the frequency order.

7. A method for correcting a bridge model with a dual objective function of frequency and deflection influence line according to claim 1, characterized in that In the step S3, the specific process of selecting the parameters to be corrected is as follows: S301: Initially select the parameters to be corrected as the elastic modulus E1, Poisson's ratio v1, and unit weight r1 of the concrete top slab; the elastic modulus E2, Poisson's ratio v2, unit weight r2, top slab thickness tf1, web thickness tw, and bottom slab thickness tf2 of the rigid box girder; S302: Use the difference method to calculate the relative sensitivity of the modal frequencies and the deflection influence line after regularization of the parameters to be corrected, and the calculation formula is as follows: Among them, S ij is the sensitivity of the modal frequency and the deflection influence line index with respect to each parameter to be corrected, p j is the initial value of the parameter to be corrected, f i represents the structural response, Δp j takes p j / 100, i represents the index of the modal frequency or the deflection influence line index, and j represents the index of the parameter to be corrected; S303: After sensitivity analysis, determine that the final parameters to be corrected are the elastic modulus E2, top slab thickness tf1, web thickness tw, and bottom slab thickness tf2 of the steel box girder, and determine the correction ranges of each parameter to be corrected.

8. A method for correcting a bridge model with a double-objective function of frequency and deflection influence line according to claim 1, characterized in that In the PSO-BP surrogate model in the step S3, use the PSO algorithm to optimize the initial weights and biases of the BP neural network, accelerate convergence and improve the model accuracy at the same time; Among them, in each iteration of the PSO algorithm, the particle velocity and position are updated, and the particles move towards the individual historical optimal center to make the population approach the optimal solution. The specific update formula is as follows: v ij = wv ij + c1r1(p bij - x ij ) + c2r2(g bij - x ij ) x ij = x ij + v ij Among them, v ij represents the velocity of particle i in the j-th dimension, x ij represents the position of particle i in the j-th dimension, w represents the inertia weight, p bij represents the personal best solution of particle i in the j-th dimension, g bij represents the global best solution of the entire population in the j-th dimension, c1 and c2 respectively represent the personal learning factor and the social learning factor, and r1 and r2 are random numbers in (0, 1).

9. A method for correcting a bridge model with a double objective function of frequency and deflection influence line according to claim 1, characterized in that, In the step S3, the specific processing process of the NSGA-II algorithm is as follows: S311: Randomly generate a group of initial solutions, and calculate the fitness value of each solution on the objective function; S312: Perform non-dominated sorting on the solutions in the population, divide them into different levels, and assign better solutions to higher levels; calculate the crowding distance of the solutions within each level to measure the distribution density of the solutions in the objective function space; S313: According to the non-dominated solution level and crowding distance, select high-level solutions as parents, and then generate higher-level solutions; S314: Perform crossover and mutation operations on the selected parent solutions to generate new solutions; S315: Combine the parent solutions and the newly generated solutions into a new population, and truncate it to the size of the original population; S316: Repeat the above steps until the predetermined number of iterations or the termination condition is met; S317: Output the Pareto front.

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