Structural damage identification method based on random sparse regularization

Abstract

A structural damage identification method based on random sparse regularization comprises the following steps: S1, establishing a finite element model of a to-be-identified damage structure, dispersing the structure into a finite number of units, setting a damage working condition, and simulating the damage working condition; s2, performing modal analysis, calculating a stiffness matrix and a mass matrix of the structure, adding boundary conditions, and solving feature values and feature vectors; s3, constructing an objective function of damage identification; s4, introducing an l1 / 2 regularization optimization method, and establishing a structural damage identification equation; s5, introducing a random class algorithm, optimizing and solving the structural damage identification target equation, and obtaining a solution vector; and S6, determining the damage position and the damage degree in the structure according to the solution vector of the damage identification equation. According to the method, the damage position can be quickly positioned, the damage degree can be accurately quantified, the accuracy of a damage identification result is improved, and the identification time of damage identification is shortened.

Description

Technical Field

[0001] This invention patent belongs to the field of structural health monitoring and relates to structural damage identification technology, specifically to a structural damage identification method based on optimization algorithms. Its application areas include, but are not limited to, machine learning and rapid solution of objective functions. Background Technology

[0002] During their service life, structures inevitably suffer invisible damage due to material degradation, design flaws, environmental erosion, and natural disasters. Accumulated damage can lead to structural collapse. Damage to engineering structures is unavoidable, and accidents not only cause property loss but also seriously endanger human lives. To avoid potentially severe consequences, timely structural health monitoring is essential to determine whether the structure is in a safe service condition and to address any safety issues promptly. This is the core of structural health monitoring technology. As a crucial part of the structural health monitoring system, structural damage identification provides guidance in locating, quantifying, and assessing the extent of damage, ensuring structural safety during operation, and predicting the remaining service life of the structure.

[0003] The process of determining structural damage based on various structural responses is mathematically a typical inverse problem. Structural damage identification is essentially solving an ill-posed inverse problem. It primarily involves inverting the structure's input information through its output response. However, in solving this inverse problem, factors such as noise in the observed data often complicate the model, reducing computational accuracy or preventing accurate identification of structural damage. Summary of the Invention

[0004] The purpose of this invention is to address the aforementioned deficiencies in the prior art and provide a structural damage identification method based on random sparse regularization. Through its implementation steps, this method can quickly locate the damage position and accurately quantify the damage degree, thereby improving the accuracy of damage identification results, reducing the identification time, and exhibiting robustness to environmental noise. Combined with online data reading, it enables health monitoring of actual structures.

[0005] The technical solution adopted by this invention to solve its technical problem is:

[0006] A structural damage identification method based on random sparse regularization includes the following steps:

[0007] Step S1: Establish a finite element model of the damaged structure to be identified. According to the finite element theory, the structure is discretized into a finite number of elements. The damage conditions are set and the damage conditions are simulated by reducing the element stiffness of the finite element model.

[0008] Step S2: Modal analysis, calculate the stiffness matrix and mass matrix of the structure, add boundary conditions, and solve for the eigenvalues ​​and eigenvectors based on the free vibration motion equation of the undamped structure;

[0009] Step S3: Based on the correlation between the modal parameters and physical parameters of the structure, construct the first-order sensitivity equation of the structure and construct the objective function for damage identification;

[0010] Step S4, introduce l 1 / 2 A regularization optimization method is used to establish a structural damage identification equation.

[0011] Step S5: Introduce a randomized (REK) algorithm to optimize the solution of the structural damage identification target equation and obtain the solution vector;

[0012] Step S6: Determine the location and extent of damage in the structure based on the solution vector of the damage identification equation.

[0013] The present invention also has the following additional technical features:

[0014] As a further specific optimization of the technical solution of the present invention: in step S1, a finite element model of a healthy structure is established, the modal parameters of the undamaged complete structure are obtained, a damage condition is set on the undamaged structure, and the modal parameters of the damaged structure are obtained.

[0015] As a further specific optimization of the technical solution of the present invention: In step S2, the modal analysis method includes: for an undamped structure with m elements and N degrees of freedom, its undamped free vibration differential equation is as follows:

[0016] in, x and y are the acceleration and displacement matrices of the structure, respectively, of order N; M and K are the mass and stiffness matrices of the structure, respectively, of order N×N, both real symmetric matrices, with M being a positive definite matrix. The corresponding stiffness matrix K and mass matrix M are as follows:

[0017]

[0018] Assuming the structure undergoes simple harmonic motion, its modal mass (principal mass) and modal stiffness (principal stiffness) can be obtained from its characteristic equation. After normalizing the stiffness matrix and mass matrix of the structure using canonical mode shapes, we can obtain:

[0019]

[0020] In the formula, λ i The i-th eigenvalue is the square of the angular frequency. It is the eigenvector corresponding to the i-th eigenvalue, i.e., the mode shape.

[0021] As a further specific optimization of the technical solution of the present invention: In step S3, it is considered that when the structure is damaged, the stiffness of its damaged element will be reduced accordingly, ignoring its mass change. The stiffness reduction vector after reduction is represented by {α}. The degree of damage and the location of damage to the structure can be identified based on {α}. {α} is a sparse vector composed of the stiffness reduction coefficients of all elements. The purpose of structural damage identification is to solve for {α}.

[0022] The model correction method based on sensitivity analysis identifies changes in structural parameters by minimizing the difference between analytical predictions and experimental data. Assuming the structure changes linearly before and after damage, the relationship between the stiffness reduction vector {α} and the modal parameter change vector {Δf} can be expressed as:

[0023] [S]{α}={Δf}={V E}-{V 0}

[0024] Among them, {V E} is for measuring modal parameters, {V 0} represents the modal parameters for analysis, and [S] consists of the first-order partial derivatives of the eigenvalues ​​and eigenvectors with respect to {α}, i.e.:

[0025] The stiffness reduction vector {α} can be obtained by solving the following optimization equation:

[0026]

[0027] As a further specific optimization of the technical solution of the present invention: In step S4, a regularization method is usually added during the finite element modeling process to l 1 / 2 Regularization as l p The best regularization method for (0 < p < 1):

[0028] In the formula, β is the regularization parameter.

[0029] As a further specific optimization of the technical solution of the present invention: in step S5, the randomized methods include random coordinate descent algorithm, greedy random coordinate descent algorithm, and relaxed greedy random coordinate descent algorithm (GRCD).

[0030] As a further specific optimization of the technical solution of the present invention: In step S6, the specific unit where the damage occurs is determined according to the solved vector {α}, and the specific value displayed on the unit reflects the degree of damage at the damage location, thereby determining the damage location and degree of damage of the structure.

[0031] Compared with the prior art, the advantages of this invention are:

[0032] Advantage 1: This invention can effectively identify the location and extent of structural damage under different damage conditions and in the presence of noise interference, and it is particularly robust to noise.

[0033] Advantage 2: The l in this invention 1 / 2 Regularization methods can effectively guarantee the sparsity of solutions, reduce misjudgments, and have high computational efficiency.

[0034] Advantage 3: In this invention, because the RCD, GRCD, and RGRCD algorithms solve the equations based on probabilistic random selection, their computational efficiency is significantly improved compared to the least squares method, with the time reduced by an order of magnitude. Generally, damage identification problems involve large amounts of data and are ill-posed; the RCD algorithm exhibits good stability and efficiency.

[0035] Advantage 4: Considering that randomized algorithms can randomly select some equations to solve, this invention can be combined with online data reading methods and applied to practical structural health monitoring problems. It has significant advantages and wide application value, especially for the processing of massive data generated by real-time monitoring of large-scale structures. Attached Figure Description

[0036] Figure 1 This is a flowchart illustrating the damage identification process of the structural damage identification method of the present invention.

[0037] Figure 2 This is a structural model diagram shown in an embodiment of the present invention;

[0038] Figure 3 This is a damage identification result diagram of a single damage condition of the planar truss model in an embodiment of the present invention;

[0039] Figure 4 This is a diagram showing the damage identification results of two damage conditions for the planar truss model in an embodiment of the present invention. Detailed Implementation

[0040] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0041] Example 1

[0042] A structural damage identification method based on random sparse regularization, comprising the following steps:

[0043] Step S1: Establish a finite element model of the damaged structure to be identified. According to the finite element theory, the structure is discretized into a finite number of elements. The damage conditions are set and the damage conditions are simulated by reducing the element stiffness of the finite element model.

[0044] In step S1, a finite element model of the healthy structure is established (the structure to be tested for health is modeled using finite element software; the model is then a finite element model of the healthy structure. Subsequently, by reducing the element stiffness of the finite element model, the desired damaged finite element model can be obtained), the modal parameters of the undamaged complete structure are obtained (the modal parameters are the frequency and mode shape of the structure), a damage condition is set on the undamaged structure, and the modal parameters of the damaged structure are obtained (the modal parameters are the frequency and mode shape of the structure).

[0045] Step S2: Modal analysis, calculate the stiffness matrix and mass matrix of the structure, add boundary conditions, and solve for the eigenvalues ​​and eigenvectors based on the free vibration motion equation of the undamped structure;

[0046] In step S2, the modal analysis method includes: for an undamped structure with m elements and N degrees of freedom, its undamped free vibration differential equation is as follows:

[0047] in, Let x and y be the acceleration and displacement matrices of the structure, respectively, of order N; M and K are the mass and stiffness matrices of the structure, respectively, of order m*n, both real symmetric matrices. M is the mass matrix, and K is the stiffness matrix (which is also a positive definite matrix). The corresponding stiffness matrix K and mass matrix M are as follows:

[0048]

[0049] Suppose the structure undergoes simple harmonic motion, and its characteristic equation (characteristic equation) is used to determine the motion. Modal mass (principal mass) and modal stiffness (principal stiffness) can be obtained. The calculation principle of modal mass (principal mass) and modal stiffness (principal stiffness) is as follows:

[0050] Based on the theory of differential equations, we can assume the general solution is: In the formula, ω is the natural circular frequency of the structure. It is an n×1 column vector.

[0051] Taking the second derivative of equation (2-4) with respect to time t, we get:

[0052] Substituting equations (2-4) and (2-5) into... From the equation, we can obtain its characteristic equation as follows:

[0053] vector It is a non-zero column vector, to ensure the vector If not all n elements in the equation are zero, then the determinant of the coefficient matrix of the characteristic equation (2-6) must be zero, i.e., |K-ω|. 2 M|=0, therefore ω 2 It is the solution when the determinant of the coefficient matrix of the characteristic equation is equal to 0. According to linear algebra theory, if M is a positive definite matrix and K is a positive definite or semi-positive definite matrix, then the characteristic roots are positive or zero, and may have repeated roots.

[0054] Let λ = ω 2 We can obtain: In the formula, λ i Let i be the i-th order eigenvalue. The eigenvector corresponding to the i-th eigenvalue is represented as:

[0055] By using each eigenvalue λ i Substituting back into the characteristic equation yields...

[0056]

[0057] Therefore, it can be seen from the above formulas that they are closely related to the global stiffness matrix K and the global mass matrix M, that is, K and M determine the natural frequency λ and the mode shape. The value of is used to indicate the change in structural stiffness during damage identification, and thus the location and extent of damage.

[0058] Matrix K and matrix M are weighted orthogonal. Suppose we have any two pairs of features: λ i and λ j and Substituting these values ​​into the characteristic equation of equation (2-6), we obtain:

[0059] Let equations (2-10) and (2-11) be multiplied on the left respectively. and We can obtain:

[0060] Transposing both sides of equation (2-12), since matrices K and M are symmetric matrices, they remain the same after transposition. Therefore, we can obtain:

[0061] Subtracting equations (2-13) and (2-14), we get:

[0062] Due to ω i and ω j They are not equal, and it is obvious that: Substituting equation (2-16) back into equation (2-14), we get: The above derivation applies to the case of i(j). If i = j, then:

[0063] because but:

[0064] After normalizing the structural stiffness matrix and mass matrix using regular mode shapes, we can obtain:

[0065]

[0066] In the formula, λ i The i-th eigenvalue is the square of the angular frequency. It is the eigenvector corresponding to the i-th eigenvalue, i.e., the mode shape.

[0067] Step S3: Based on the correlation between the modal parameters and physical parameters of the structure, construct the first-order sensitivity equation of the structure and construct the objective function for damage identification.

[0068] In step S3, it is considered that when the structure is damaged, the stiffness of the damaged element will be reduced accordingly, ignoring the change in mass. The stiffness reduction vector after reduction is represented by {α}. The degree of damage and the location of damage can be identified based on {α}. {α} is a sparse vector composed of the stiffness reduction coefficients of all elements. The purpose of structural damage identification is to solve for {α}.

[0069] The model correction method based on sensitivity analysis identifies changes in structural parameters by minimizing the difference between analytical predictions and experimental data. Assuming the structure changes linearly before and after damage, the relationship between the stiffness reduction vector {α} and the modal parameter change vector {Δf} can be expressed as:

[0070] [S]{α}={Δf}={V E}-{V 0}

[0071] Among them, {V E} is for measuring modal parameters, {V 0} represents the modal parameters for analysis, and [S] consists of the first-order partial derivatives of the eigenvalues ​​and eigenvectors with respect to {α}, i.e.:

[0072] The stiffness reduction vector {α} can be obtained by solving the following optimization equation:

[0073]

[0074] Step S4, introduce l 1 / 2 A regularization optimization method is used to establish a structural damage identification equation.

[0075] In step S4, generally speaking, the least squares method is used to solve linear inverse problems. However, in general practical problems, the resulting matrix [S] is often ill-conditioned. Under certain perturbations, such as measurement noise and data loss, the final result will be greatly affected, which may lead to inaccurate results. Therefore, the solution obtained by the least squares method is unstable. To solve this problem, a regularization method is usually added to the finite element model process, that is, a regularization term is added to the objective equation. In addition, since structural damage often has sparsity, {α} is a sparse vector. In order to better solve the sparsity problem, Xu Zongben et al. improved l p Regularization was studied in depth; the results show that, compared with l1 regularization, l p (0 < p < 1) Regularization yields sparser and more accurate solutions; therefore, l 1 / 2 Regularization as l p The best regularization method for (0 < p < 1):

[0076] In the formula, β is the regularization parameter.

[0077] Step S5: Introduce a randomized (REK) algorithm to optimize the solution of the structural damage identification target equation and obtain the solution vector.

[0078] In step S5, the effectiveness of the algorithm for solving the regularization model is closely related to the solution of the linear algebraic equation system Ax = b, especially the calculation method of the least squares solution of the incompatible equation system. The traditional calculation method is to solve the equation system by using the singular value decomposition of the matrix to calculate the least squares solution with the smallest 2-norm and generalized inverse. This type of algorithm involves matrix decomposition and has a large workload, and the efficiency of the algorithm is greatly affected by matrix A. For the sensitivity matrix [S], which is large in size and has error perturbation in the data, the equation system is incompatible and the data is highly correlated, which brings great difficulties to the calculation. Considering the data construction conditions of the objective equation, the randomized algorithm can more effectively solve the optimal solution of the regularization model in this paper because it selects some equations of the linear algebraic equation system to solve according to a certain probability.

[0079] The random class methods include the following:

[0080] (1) Random Coordinate Descent (RCD) algorithm;

[0081] (2) Greedy Random Coordinate Descent Algorithm (GRCD);

[0082] (3) Relaxed Greedy Random Coordinate Descent Algorithm (GRCD).

[0083] Step S6: Determine the location and extent of damage in the structure based on the solution vector of the damage identification equation.

[0084] In step S6, the specific unit where the damage occurred is determined based on the solved vector {α}. The specific value displayed on the unit reflects the degree of damage at that location (the specific value obtained in step 6 is expressed as a percentage, reflecting what percentage of damage occurred at that location). Thus, the location and degree of damage to the structure can be determined.

[0085] Example 2

[0086] A structural damage identification method based on random sparse regularization is established based on the finite element theory of truss structures, as follows: Figure 2 The model shown is a 21-member simply supported steel truss, where n = 21. The members have circular cross-sections with a radius of 0.1m and a cross-sectional area of ​​0.03142m². 2 The moment of inertia is 4.91 × 10⁻⁶ m. 4 The elastic modulus is 206 GPa, and the density of the rod is 7850 kg / m³. 3 The degree of damage to truss structural units is quantified by the stiffness reduction rate of the corresponding units. Three damage conditions are set: an undamaged condition (T0) and two damaged conditions (T1 and T2), as detailed in Table 1.

[0087] Table 1 Damage Conditions of Planar Trusses

[0088] Damage conditions Damage type Damage unit Degree of damage T1 Single loss 9 30％ T2 Double loss 9,12 30％，20％

[0089] The following algorithms were used respectively: Random Coordinate Descent (RCD), Greedy Random Coordinate Descent (GRCD), and Relaxed Greedy Random Coordinate Descent (RGRCD). 1 / 2 The regularization method is used to identify damage in a planar truss structure model. The damage identification results for each damage condition are as follows: Figure 3 , Figure 4 As shown, in Figure 3 , Figure 4 In this context, the three methods mentioned above are abbreviated as IT1 / 2-RCD, IT1 / 2-GRCD, and IT1 / 2-RGRCD, respectively.

[0090] The damage identification results show that the IT1 / 2-RCD, IT1 / 2-GRCD, and IT1 / 2-RGRCD methods can effectively identify damage in planar truss structures. They demonstrate accurate location and high identification accuracy in both single-damage and multi-damage scenarios, with no false positives. Furthermore, the computation times for single-damage scenarios using the IT1 / 2-REK, IT1 / 2-PREK, and IT1 / 2-FMDEK methods are 0.0077s, 0.0069s, and 0.0063s, respectively, showing a significant improvement in computational efficiency.

[0091] As illustrated by the above embodiments, the present invention, through its specific implementation steps, can quickly locate the damage position and accurately quantify the damage degree, with high computational efficiency. The above embodiments are preferred embodiments of the present invention, but the implementation of the present invention is not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention should be considered equivalent substitutions and are included within the protection scope of the present invention.

Claims

1. A structural damage identification method based on random sparse regularization, characterized in that: The structural damage identification method comprises the following steps: Step S1, a finite element model of a structure to be identified is established, the structure is discretized into a finite number of units according to the finite element theory, a damage condition is set, and the unit stiffness of the finite element model is reduced to simulate the damage condition; Step S2, modal analysis is performed, the stiffness matrix and the mass matrix of the structure are calculated, boundary conditions are added, and the eigenvalues and eigenvectors are solved according to the undamped structure free vibration motion equation; Step S3, a first-order sensitivity equation of the structure is constructed according to the correlation between the modal parameters and the physical parameters of the structure, and a target function for damage identification is constructed; Step S4, introducing l 1 / 2 The regularization optimization method establishes a structural damage identification equation. Step S5, a random class (REK class) algorithm is introduced, and the structural damage identification target equation is optimized and solved to obtain a solution vector; Step S6, the damage position and the damage degree in the structure are determined according to the solution vector of the damage identification equation.

2. The structural damage identification method based on random sparse regularization according to claim 1, characterized in that: In step S1, a finite element model of a healthy structure is established, the modal parameters of the undamaged complete structure are obtained, and the modal parameters of the structure after damage are obtained by setting a damage condition on the undamaged structure.

3. The structural damage identification method based on random sparse regularization according to claim 1, characterized in that: In the step S2, the modal analysis method comprises: for a non-damped structure with m units and N degrees of freedom, the non-damped free vibration differential equation of the structure is as follows: wherein, and x are acceleration and displacement arrays of the structure, respectively, of order N; M and K are mass and stiffness matrices of the structure, respectively, of order m*n, both being real symmetric matrices, M being the mass matrix and K being the stiffness matrix, the corresponding stiffness matrix K and mass matrix M being: When the structure is subjected to simple harmonic vibration, the modal mass (main mass) and the modal stiffness (main stiffness) can be obtained through the characteristic equation, and the stiffness matrix and the mass matrix of the structure are normalized by using the regular mode to obtain: where λi i is the i-th eigenvalue, i.e. the square of the circular frequency, is the eigenvector corresponding to the i-th eigenvalue, i.e. the mode shape.

4. The structural damage identification method based on random sparse regularization according to claim 1, characterized in that: In step S3, when the structure is damaged, the stiffness of the damaged unit will be reduced accordingly, and the mass change is ignored. The stiffness reduction vector after reduction is represented by {alpha}, which can be used to identify the damage degree and the damage position of the structure. The stiffness reduction vector {alpha} is a sparse vector composed of unit stiffness reduction coefficients of all units. The purpose of structural damage identification is to solve the stiffness reduction vector {alpha}; The model updating method based on sensitivity analysis identifies the change of the structure parameters by minimizing the difference between the analysis prediction and the experimental data. When the structure changes linearly before and after damage, the relationship between the stiffness reduction vector {alpha} and the modal parameter change vector {Delta f} can be represented as: [S] {a} = {Af} = {V E} - {V 0}; where {V E} is the measured modal parameter, {V 0} is the analytical modal parameter, and [S] is the first-order derivative matrix of {α} with respect to the eigenvalue and eigenvector, i.e.: The stiffness reduction vector {alpha} can be obtained by solving the following optimization equation:

5. The structural damage identification method based on random sparse regularization according to claim 1, characterized in that: In the step S4, a regularization method is usually added in the finite element model, that is, a regularization term is added in the target equation; the l 1 / 2 The regularization is the best regularization method of the l p (0 < p < 1) regularization Where beta is a regularization parameter.

6. The structural damage identification method based on random sparse regularization according to claim 1, characterized in that: In step S5, the random class method includes a random coordinate descent algorithm, a greedy random coordinate descent algorithm, and a relaxation greedy random coordinate descent algorithm.

7. The structural damage identification method based on random sparse regularization according to claim 1, characterized in that: In step S6, according to the solved vector {alpha}, the specific unit where the damage occurs is determined, and the specific numerical value displayed on the unit reflects the damage degree of the damage position, so that the damage position and the damage degree of the structure can be determined.

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