Structural strain field inversion method based on improved four-node inverse finite element theory

Abstract

This invention discloses a structural strain field inversion method based on an improved four-node inverse finite element theory, belonging to the field of structural health monitoring technology. The method includes the following steps: Step 1: Determine the structural element mesh and sensor network layout; Step 2: Calculate the shape function matrix of each element based on the structural element mesh; Step 3: Solve for the structural surface strain; Step 4: Construct an error function; Step 5: Reconstruct the structural strain field. The method described in this invention improves the four-node element interpolation function. By differentiating the fitted higher-order displacement field function, the strain field of plate and shell structures is obtained. It uses fewer sensors to simultaneously obtain the structural deformation and strain response with high inversion accuracy. This method is simple, convenient, and has strong real-time performance.

Description

Technical Field

[0001] This invention belongs to the field of strain field inversion technology for structural health monitoring, and specifically proposes a strain field inversion method for wall panel structures based on an improved four-node inverse finite element theory. Background Technology

[0002] In recent years, with the rapid development of science and technology in China, the aerospace industry has been continuously advancing. Thin-walled structures such as plate shells have been widely used in the aerospace industry due to their advantages such as light weight and good performance. For example, in the aerospace field, most of the load-bearing components of the airframe are plate shell structures. The processing quality and assembly precision of the plate shell structure determine the mass of the aircraft. Therefore, the stress deformation of plate shell structures is gradually becoming a topic of great interest to researchers.

[0003] During their service life, aircraft often encounter harsh working environments and are subjected to various loads, making them susceptible to varying degrees of unseen damage. Prolonged service can also lead to structural strength failure; for example, composite material plates may experience internal fiber breakage, and metal shell structures may suffer localized deformation. These issues significantly degrade the mechanical properties of typical load-bearing components (fuselage, wings), directly threatening flight safety. Therefore, it is crucial to obtain timely information on the surface strain and deformation of shell structures to provide a reliable basis for structural health assessment and maintenance.

[0004] Currently, there are two main types of methods for reconstructing the strain field of a structure: 1) obtaining the strain value of the entire structural surface through interpolation fitting based on the strain at a finite number of discrete points on the structural surface; 2) conducting theoretical analysis based on the structural stress characteristics and deriving the strain field of the structure according to boundary conditions and mechanical formulas. The modal superposition principle first requires obtaining the strain and displacement modes of the structure through experiments or finite element analysis. Combining the derivative relationship between the strain and displacement modes, the relationship between the strain field and displacement field of the measured structure is established, yielding the strain-displacement transformation matrix. Then, by inputting the strain value measured by the strain sensor, the strain field of the entire structure can be obtained.

[0005] When performing finite element analysis using the modal superposition principle, the inversion accuracy of the strain field is affected by the finite element model and the number of sensors. The modal analysis order selected affects the reconstruction effect. When the number of selected modes is too small, lower-order modes have a significant impact on structural deformation, leading to large errors due to insufficient modal information. Conversely, when the number of selected modes is too large, the calculation results of higher-order modes will produce significant errors, thus affecting the reconstruction effect of the structural strain field. Research has found that as the number of fiber optic FBG sensors increases, more strain information is obtained, and the reconstruction error decreases. Therefore, to address the shortcomings of the modal superposition method, this invention proposes a structural strain field inversion method based on an improved four-node inverse finite element theory, aiming to reduce the use of FBG sensors while maintaining inversion accuracy and simplifying the testing system. Summary of the Invention

[0006] Technical problem: The technical problem to be solved by the present invention is to provide a strain field inversion method for engineering plate and shell structures.

[0007] This method uses fiber optic FBG sensors to measure strain response information at different points on the structure, and uses the acquired strain information as input for inverse finite element method (IFM) calculation to obtain the displacement response of the structure. Since the FBG sensor performs single-point measurements, the obtained displacement information is discrete and discontinuous. Therefore, a higher-order function is used to fit and reconstruct the discrete displacement data to obtain the surface deflection function of the structure. Finally, based on the functional relationship between strain and deflection in Euler-Bernoulli beam theory, the second derivative of the fitted displacement function is calculated to obtain the surface strain function of the structure.

[0008] Technical solution: To solve the above-mentioned technical problems, the technical solution adopted by the present invention includes the following steps:

[0009] 1. A method for inverting the strain field of a wall panel structure based on an improved four-node inverse finite element theory, characterized by comprising the following steps:

[0010] Step 1: Determine the structural unit mesh and sensor layout

[0011] The structure is discretized using four-node inverse shell elements, which are discretized into N four-node inverse shell elements along the length of the structure. Based on the geometric characteristics of the structure, the elements are more densely divided where the curvature is greater. Each element node is numbered 1 to 4 in a counter-clockwise direction. Node 1 is set as the origin O of the element's local coordinate system, with the x-axis along the length of the structure, the y-axis along the width of the structure, and the z-axis perpendicular to the xy plane. n sets of sensors are arranged on the upper and lower surfaces of each inverse shell element, forming a sensor measurement network of N×2n sets on the structural surface. The monitoring area is the entire structural surface. The sensor measurement points are selected according to the actual situation. Each group has three sensors, numbered FBG1, FBG2, and FBG3, which are attached to the structural surface at 0°, 90°, and 45° angles to measure the strain in the x-direction, y-direction, and the direction at a 45° angle to the x-axis at that point.

[0012] Step 2: Calculate the shape function matrix of each element based on the mesh division of the wall panel structure.

[0013] According to classical finite element theory, the displacement of any point within an element can be represented by the linear superposition of the deflection and rotation angle of the four nodes within that element, as expressed in the following expression:

[0014]

[0015]

[0016] In equation (1), u represents the displacement of a point within the element along the x-direction, v represents the displacement of a point within the element along the y-direction, ω represents the deflection of a point within the element, and θ x θ represents the rotation angle of a point within the element along the x-axis. y This represents the rotation angle θ of a point within the element along the y-axis. It is a two-dimensional planar shell element, considering only lateral and longitudinal loads and neglecting the rotation angle θ along the z-axis. z ;u i v i ω i θ xi θ yi These represent the displacement components of each node within each element. i 、L i M i Let i be a four-node element shape function, where i (i = 1, 2, 3, 4) is the number of each element node.

[0017] Each element contains 4 nodes, and the displacement vector of each node can be represented as:

[0018]

[0019] Therefore, the displacement vectors of the four nodes within the element can be expressed as:

[0020]

[0021] In practical applications, the division of structural elements cannot guarantee that all elements are regular rectangular units. Therefore, it is necessary to convert general-shaped quadrilateral elements into regular rectangular units. Introducing a basic coordinate system `st`, the transformation relationship between the `st` coordinate system and the xy coordinate system is as follows:

[0022]

[0023] x2, x3, y3, and y4 are the coordinates of the unit nodes.

[0024] Then the shape function N of the four-node unit i The specific expression is:

[0025]

[0026] L1 = y 14 S4-y 21 S1, M1 = x 41 S4-x 12 S1

[0027] L2 = y 21 S1-y 32 S2, M2 = x 12 S1-x 23 S2

[0028] L3 = y 32 S2-y 43 S3, M3 = x 23 S2-x 34 S3

[0029] L4 = y 43 S3-y 14 S4, M4 = x 34 S3-x 41 S4 (6)

[0030] in,

[0031]

[0032] In formula (6):

[0033] x ij =x i -x j

[0034] y ij =y i -y j (i,j=1,2,3,4) (8)

[0036] In equation (8), x i x j y i y j Let the coordinates be the element node coordinates. Then, the shape function matrix D of each element is shown in the following equation:

[0037] D = [D1 D2 D3 D4] (9)

[0038] Where D i The specific expression is:

[0039]

[0040] Therefore, the shape function of each unit is a 6×24 matrix.

[0041] For a two-dimensional planar four-node element, we only consider the transverse and longitudinal loads acting on the structure, that is, we do not consider the rotation angle of the structure around the z-axis. Therefore, all elements in the last row of the element shape function are zero.

[0042] Combining equations (3) and (10), the displacement of any point within each element of the structure can be calculated.

[0043] The displacement is:

[0044]

[0045] Step 3: Solve for the surface strain of the structure

[0046] Structural surface strain ε e It can be expressed as in-plane tensile / compressive strain e(u) e ) and bending strain k(u e Linear combinations of )

[0047] ε e =e(u e )+z0·k(u e (12)

[0048] Where z0 represents the distance between the wall panel surface and the neutral layer.

[0049] The inverse finite element method requires three strain components on the surface of the structure: ε x ε y ε 45° , representing the strain in the x-direction, y-direction, and 45° direction, respectively; the tensile, compressive, and bending strains on the structural surface are expressed using the measured strains in the three directions as follows:

[0050]

[0051] In the formula, "+" represents the surface strain of the structure, and "-" represents the effective surface strain of the structure; γ xy The shear strain in the xy plane can be expressed by the three measured strain components as follows:

[0052] γ xy =ε x +ε y -2·ε 45° (14)

[0053] Equation (13) uses the partial derivative matrix B of the shape function of the four-node element and the nodal displacement u of the element. e Further expressed as:

[0054] ε e =e(u e )+z0·k(u e ) = B m u e +z0·B b u e (15)

[0055] The partial derivative matrix B of the four-node unit shape function is divided into and The specific expression is:

[0056]

[0057] The partial derivative matrix of the shape function of the four-node element can be completely expressed as:

[0058]

[0059] Step 4: Construct the error function

[0060] The strain values ​​measured experimentally on the structural surface are denoted as ε, including tensile and compressive strain e. ε Bending strain k ε and transverse shear strain g ε Then the error function between the actual strain value and the theoretical strain value of the structure is:

[0061] Φ(ε e ,ε)=‖e(u e )-e ε || 2 +‖k(u e )-k ε || 2 +λ‖g(u e )-g ε || 2 (18)

[0063] In the formula, g(u e) represents the theoretical transverse shear strain of the structural surface, which is usually 0; λ represents the penalty parameter (0 < λ < 1) for the correlation between the strain measurement data and the theoretical result.

[0064] Error function Φ on element nodal displacement vector u e By taking the partial derivative and setting it to zero, we can solve the differential equation to obtain the minimum value of the error function, as shown in equation (20).

[0065]

[0066] The matrix equation shown in equation (20) is obtained through calculation:

[0067] k e u e =f e (20)

[0068] In the formula, k e f e It can be calculated using the following formula:

[0069]

[0070] In the above formula, the integration region is the area A of the unit cell. e .

[0071] Substituting equations (21) and (22) into equation (20), the displacement vector u of the structural unit nodes can be obtained. e , will u e Substituting the result back into equation (11), the displacement component at any point within the structure can be obtained.

[0072] Step 5: Reconstruct the structural strain field

[0073] According to Euler-Bernoulli theory, the relationship between surface strain ε and deflection y of a structure is as follows:

[0074]

[0075] In the formula, z0 is the distance from the structural surface to the neutral layer.

[0076] Based on the displacement components obtained in step four, the deflection components at the center points of each element are selected, resulting in N discrete displacements. The relationship between deflection and the x-direction coordinate is then fitted using MATLAB's fitting tool. For a planar four-node element, the displacement function is chosen as follows:

[0077] y = a1x n +a2x n-1 +a3x n-2 +...+a n x+a n+1 (n = 3, 4, 5, ...) (twenty four)

[0079] Substituting equation (24) into equation (23), we can obtain the surface strain function of the wall panel structure as follows:

[0080] ε(x)=z0[n(n-1)a1x n-2 +(n-1)(n-2)a2x n-4 +...+a n-1 ] (25)

[0082] Equation (25) represents the strain distribution law in the x-direction of the structural surface. By substituting the coordinates of different points on the structural surface, the strain field in the x-direction of the entire structural surface can be reconstructed.

[0083] The advantages of this invention are:

[0084] This invention presents a structural strain field inversion method based on an improved four-node inverse finite element theory. This method uses discrete strain responses measured by a fiber optic FBG sensor network arranged on the upper and lower surfaces of the structure to realize the displacement and strain field inversion. This invention is applicable to engineering applications such as deformation and strain monitoring and inversion of plate and shell structures. Its advantages include: requiring only 4 groups of 12 fiber optic FBG sensors to form the sensor network with minimal sensors, offering advantages such as simple wiring and strong resistance to electromagnetic interference compared to traditional sensing methods. Furthermore, this invention replaces the displacement function of the original quadrilateral element with a higher-order displacement field function, transforming the quadrilateral constant-strain element into an element whose strain value varies with the coordinates of the measurement point. This eliminates the need to consider the material properties of the structure, simplifying the displacement and strain reconstruction process and achieving high inversion accuracy. Attached Figure Description

[0085] Figure 1 It involves structural unit mesh generation and fiber optic FBG sensor layout;

[0086] Figure 2 This is a comparison chart of the reconstructed deflection values ​​and simulation values ​​of the wall panel structure;

[0087] Figure 3 This is a comparison chart of the reconstructed strain values ​​and simulation values ​​of the wall panel structure;

[0088] Figure 4 This is a flowchart of the improved four-node inverse finite element method;

[0089] Figure 5 It is the result of reconstructing the displacement and strain of the structure. Detailed Implementation

[0090] To enable those skilled in the art to better understand the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are merely some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort should fall within the scope of protection of the present invention.

[0091] like Figure 1 As shown, this invention provides a method for monitoring and reconstructing structural deformation and strain based on the inverse finite element method and curve fitting method, comprising the following steps:

[0092] S1. Select appropriate inverse shell elements to discretize the structure.

[0093] S2. Calculate the tensile and compressive strain, bending strain, and shear strain of each element based on the theory of medium-thick plates.

[0094] S3. Select strain measurement points on the upper and lower surfaces of the inverse shell unit, and attach fiber optic FBG sensors to the strain measurement points to measure the strain in real time and obtain strain measurement data.

[0095] S4. Based on the obtained strain measurement data, calculate the membrane strain and curvature of each inverted shell unit.

[0096] S5. Construct an error function based on the least squares method, differentiate the nodal degrees of freedom, obtain the pseudo stiffness matrix and pseudo load array of the inverse shell element, assemble them, and finally calculate the displacement field of the structure by combining the boundary conditions.

[0097] S6. Based on the obtained discrete displacement field of the structure, select an appropriate higher-order function to fit the discrete displacement.

[0098] S7. Take the second derivative of the fitted displacement function and multiply it by 1 / 2 of the wall thickness to obtain the strain function of the structure.

[0099] In a specific implementation, as a preferred embodiment of the present invention, the structure in step S1 includes a complex geometric model composed of all plates and shells, all of which are plates of equal thickness. For example... Figure 1 The diagram shown illustrates the division of aluminum alloy panel units and the layout of the sensor network. In a preferred embodiment of the invention, step S2 is implemented as follows:

[0100] S21. Calculate the tensile and compressive strain of each inverted shell element. The calculation formula is as follows:

[0101] e(u e ) = B m u e (1)

[0102] Among them, u e B represents the displacement vector of each inverse shell element node. m This represents a matrix containing the derivatives of shape functions.

[0103] S22. Calculate the bending strain of each inverse shell element. The calculation formula is as follows:

[0104] k(u e ) = B b u e (2)

[0105] Among them, B b This represents a matrix containing the derivatives of shape functions.

[0106] S23. Calculate the shear strain of each inverse shell element. The calculation formula is as follows:

[0107] g(u e ) = B s u e (3)

[0108] Among them, B s This represents a matrix containing the derivatives of shape functions.

[0109] In a specific implementation, as a preferred embodiment of the present invention, the strains measured in real time by the strain sensor in step S3 are as follows:

[0110]

[0111] Where i represents the i-th inverse shell element, n represents the number of strain measurements within the inverse shell element, and ε x x represents

[0112] Directional strain, ε y ε represents the normal strain in the y-direction. 45 ° indicates strain at 45° on the structural surface; "+"

[0113] "-" indicates the strain on the upper surface of the structure, while "-" indicates the strain on the lower surface of the structure.

[0114] In a specific implementation, as a preferred embodiment of the present invention, the strain sensor is a fiber optic FBG sensor.

[0115] In specific implementation, as a preferred embodiment of the present invention, such as Figure 1 As shown, the fiber optic FBG sensor is arranged along a single axis or in the form of a strain rosette on the upper and lower surfaces of the inverted shell unit.

[0116] In a specific implementation, as a preferred embodiment of the present invention, the specific implementation process of step S4 is as follows:

[0117] S41. Based on the obtained real-time strain measurement data, calculate the tensile and compressive strain of each inverted shell element.

[0118] The formula is as follows:

[0119]

[0120] S42. Based on the obtained real-time strain measurement data, calculate the bending strain of each inverted shell element, and calculate...

[0121] The formula is as follows:

[0122]

[0123] Where z0 represents 1 / 2 of the element thickness.

[0124] In a specific implementation, as a preferred embodiment of the present invention, the specific implementation process of step S5 is as follows:

[0125] S51. For each inverse shell element, the least squares error function is used, and the function expression is:

[0126] Φ(ε e ,ε)=‖e(u e )-e ε || 2 +‖k(u e )-k ε || 2 +λ‖g(u e )-g ε || 2 (7)

[0128] In the formula, e(u e ), k(u e ), g(u e ) represent the theoretical tensile, compressive, bending, and shear strains within each inverse shell element, respectively. ε k ε g ε These represent the tensile, compressive, bending, and shear strains measured in real time for each inverse shell element, respectively. λ represents the penalty parameter (0 < λ < 1) indicating the correlation between the strain measurement data and the theoretical results.

[0129] S52. Differentiate the nodal degrees of freedom of the structure to obtain the pseudo stiffness matrix and pseudo load array of each inverse shell element.

[0130] k e u e =f e (8)

[0131] In the above formula, ke f represents the pseudo-stiffness matrix of the inverse shell element. e Denotes the pseudo-load array of the inverse shell element, where:

[0132]

[0133] In the above formula, the integration region is the area A of the unit cell. e .

[0134] S53. Assemble the pseudo-stiffness matrix and pseudo-load array of the inverse shell element according to the finite element program to obtain the global linear equation system of the discrete structure, as follows:

[0135] K e U e =F e (10)

[0136] In the formula, K e F represents the pseudo-stiffness matrix of the entire inverse finite element method. e This represents the pseudo-load matrix of the entire inverse finite element method. S54. Based on the boundary conditions of the structure, the overall pseudo-stiffness matrix K... e After correction, the displacement field of the structure is obtained as follows:

[0137]

[0138] In the formula, This represents the pseudo-stiffness matrix after applying boundary conditions; it is a positive definite matrix.

[0139] In a specific implementation, as a preferred embodiment of the present invention, the specific implementation process of step S6 is as follows:

[0140] S61. According to the Euler-Bernoulli theory, the relationship between the surface strain ε and the deformation deflection y of a structure is as follows:

[0141]

[0142] In the formula, z0 is the distance from the surface of the wall panel to the neutral layer.

[0143] S62. Based on step S54, the discrete deflection components are obtained, and the relationship between deflection and x-direction coordinates is fitted using MATLAB fitting tools. For this embodiment of the invention, the displacement function is selected as:

[0144] y = a1x 3 +a2x 2 +a3x+a4 (13)

[0145] S63. Substituting equation (13) into equation (12), with the thickness of the example being 5mm, the strain field function of the wall panel structure surface can be obtained as follows:

[0146] ε(x)=z0(6a1x+a2x) (14)

[0147] The reconstruction yields the displacement and strain results of the structure, such as Figure 5 As shown.

[0148] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit them. Those skilled in the art can still modify the technical solutions described in the foregoing embodiments.

Claims

1. A structural strain field inversion method based on an improved four-node inverse finite element theory, characterized in that, Includes the following steps: Step 1: Determine the structural unit mesh and sensor layout Four-node inverse shell elements are used to discretize the structure, which is then discretized into N four-node inverse shell elements along the length of the structure. Based on the geometric characteristics of the structure, the elements are more densely divided where the curvature is greater. Each element node is numbered 1 to 4 in a counter-clockwise direction. Node 1 is set as the origin O of the local coordinate system, with the x-axis along the length of the structure, the y-axis along the width, and the z-axis perpendicular to the xy plane. n sets of sensors are arranged on the upper and lower surfaces of each inverse shell element, forming a sensor network of N×2n sensors on the structural surface, monitoring the entire structural surface. The sensor measurement points are selected according to the actual situation. Each group has three sensors, numbered FBG1, FBG2, and FBG3, which are attached to the structural surface at 0°, 90°, and 45° angles to measure the strain in the x-direction, y-direction, and the direction at a 45° angle to the x-axis at that point. Step 2: Calculate the shape function matrix of each element based on the structural element mesh. According to classical finite element theory, the displacement of any point within an element can be represented by the linear superposition of the deflection and rotation angle of the four nodes within that element, as expressed in the following expression: In equation (1), u represents the displacement of a point within the element along the x-direction, v represents the displacement of a point within the element along the y-direction, ω represents the deflection of a point within the element, and θ represents the displacement of a point within the element. x θ represents the rotation angle of a point within the element along the x-axis. y This represents the rotation angle θ of a point within the element along the y-axis. It is a two-dimensional planar shell element, considering only lateral and longitudinal loads and neglecting the rotation angle θ along the z-axis. z ;u i v i ω i ,θ xi ,θ yi ,θ zi N represents the displacement components of each node within each element; i L i 、M i For the shape function of a four-node unit, the specific calculation method is shown in the following equation (6), where i (i = 1, 2, 3, 4) is the number of each unit node; Each element contains 4 nodes, and the displacement vector of each node can be represented as: Therefore, the displacement vectors of the four nodes within the element can be expressed as: In practical applications, the division of structural elements cannot guarantee that all elements are regular rectangular units. Therefore, it is necessary to convert general-shaped quadrilateral elements into regular rectangular units. A basic coordinate system, st, is introduced. The transformation relationship between the st coordinate system and the xy coordinate system is as follows: x2, x3, y3, and y4 are the coordinates of the unit nodes; Then the shape function N of the four-node unit i The specific expression is: L1=y 14 S4-y 21 S1,M1=x 41 S4-x 12 S1 L2=y 21 S1-y 32 S2,M2=x 12 S1-x 23 S2 L3=y 32 S2-y 43 S3, M3=x 23 S2-x 34 S3 L4=y 43 S3-y 14 S4, M4=x 34 S3-x 41 S4 (6) in, In equation (6): x ij =x i -x j y ij =y i -y j (i,j=1,2,3,4) (8) In equation (8), x i x j y i y j Let the coordinates be the element node coordinates. Then, the shape function matrix D of each element is as follows: D = [D1 D2 D3 D4] (9) Where D i The specific expression is: Therefore, the shape function of each unit is a 6×24 matrix; For a two-dimensional planar four-node element, we only consider the transverse and longitudinal loads acting on the structure, that is, we do not consider the rotation angle of the structure about the z-axis. Therefore, all elements in the last row of the element shape function are zero. Combining equations (3) and (9), the displacement of any point within each element of the structure can be calculated. The displacement of any point within an element is: Step 3: Solve for the surface strain of the structure Structural surface strain ε e It can be expressed as in-plane tensile / compressive strain e(u) e ) and bending strain k(u e Linear combinations of ) ε e =e(u e )+z0·k(u e ) (12) Where z0 represents the distance between the wall panel surface and the neutral layer; The inverse finite element method requires three strain components on the surface of the structure: ε x ε y ε 45° , representing the strain in the x-direction, y-direction, and 45° direction, respectively; the tensile, compressive, and bending strains on the structural surface are expressed using the measured strains in the three directions as follows: In the formula, "+" represents the strain on the upper surface of the structure, and "-" represents the strain on the lower surface of the structure; γ xy The shear strain in the xy plane can be expressed by the three measured strain components as follows: c xy =e x +e y -2·e 45° (14) Equation (12) uses the partial derivative matrix B of the shape function of the four-node element and the nodal displacement u of the element. e Further expressed as: ε e =e(u e )+z0·k(u e )=B m u e +z0·B b u e (15) The partial derivative matrix B of the four-node unit shape function is divided into and The specific expression is: The partial derivative matrix of the shape function of the four-node element can be completely expressed as: Step 4: Construct the error function The strain values ​​measured experimentally on the structural surface are denoted as ε, including tensile and compressive strain e. ε Bending strain k ε and transverse shear strain g ε Then the error function between the actual strain value and the theoretical strain value of the structure is: F(e e ,ε)=‖e(u e )-e ε ‖ 2 +‖k(u e )-k ε ‖ 2 +λ‖g(u e )-g ε ‖ 2 (18) In the formula, g(u e ) represents the theoretical transverse shear strain of the structural surface, which is usually 0; λ represents the penalty parameter (0 < λ < 1) indicating the correlation between the strain measurement data and the theoretical results; Error function Φ on element nodal displacement vector u e By taking the partial derivative and setting it to zero, we can solve the differential equation to obtain the minimum value of the error function, as shown in equation (19): k e f represents the pseudo-stiffness matrix of the structure. e This represents the pseudo-load array of the structure; The matrix equation shown in equation (20) is obtained through calculation: k e u e =f e (20) In the formula, k e f e It can be calculated using the following formula: In the above formula, the integration region is the area A of the unit cell. e ; Substituting equations (21) and (22) into equation (20), the displacement vector u of the structural unit nodes can be obtained. e , will u e Substituting the result back into equation (11), the displacement component at any point within the structure can be obtained. Step 5: Reconstruct the structural strain field According to Euler-Bernoulli theory, the relationship between surface strain ε and deflection y of a structure is as follows: In the formula, z0 is the distance from the structural surface to the neutral layer; Based on the displacement component obtained in step four, the deflection component at the center point of each element is selected. There are N discrete displacements. The relationship between deflection and the x-direction coordinate is fitted using MATLAB's fitting tool. For a planar four-node element, the displacement function is chosen as follows: y=a1x n +a2x n-1 +a3x n-2 +...+a n x+a n+1 (n=3,4,5,...) (24) Substituting equation (24) into equation (23), we can obtain the strain function of the structural surface as follows: ε(x)=z0[n(n-1)a1x n-2 +(n-1)(n-2)a2x n-4 +...+a n-1 ] (25) Equation (25) represents the strain distribution law in the x-direction of the structural surface. By substituting the coordinates of different points on the structural surface, the strain field in the x-direction of the entire structural surface can be reconstructed.

2. The structural strain field inversion method based on the improved four-node inverse finite element theory according to claim 1, characterized in that: The sensor is an FBG fiber optic sensor.

3. The structural strain field inversion method based on the improved four-node inverse finite element theory according to claim 1, characterized in that: The sensors are arranged in a strain rose pattern at the same location on the upper and lower surfaces of the structure.

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