Coupling functional unilateral inverse finite element cylindrical thin-walled structure displacement field monitoring method

Abstract

The invention relates to the field of thin-wall structure displacement field monitoring, and discloses a coupled functional unilateral inverse finite element cylindrical thin-wall structure displacement field monitoring method, which comprises the following steps of: performing discretization by adopting a four-node inverse shell unit, and arranging a sensor at a geometric center position of each unit; based on the basic mathematical model of the inverse finite element, establishing an overall pseudo-stiffness matrix and an overall pseudo-force matrix; and calculating to obtain a displacement field of the structure by solving a linear equation set formed by the overall pseudo-stiffness matrix, the overall pseudo-force matrix and the global displacement vector. According to the method, the structural displacement field is calculated by using the strain information measured at one side, and the global deformation is accurately and reversely deduced under the condition that only the local strain data is obtained by combining the advantages of finite element analysis and inverse problem solution. By optimizing local strain data solution, high cost and low efficiency of global strain measurement in a traditional method are avoided, and the method has remarkable practical advantages.

Description

Technical Field

[0001] The present invention relates to the field of displacement field monitoring of thin-walled structures, and specifically to a method for monitoring the displacement field of a cylindrical thin-walled structure by coupling a functional unilateral inverse finite element method. Background Art

[0002] Due to advantages such as lightweight and high load-bearing efficiency, thin-walled structures are widely used in fields such as aerospace, shipbuilding, and pressure vessels. However, their geometric characteristics make them prone to problems such as buckling and local deformation under complex loads. Real-time monitoring of the displacement field is crucial for structural health assessment. Among current mainstream monitoring technologies, contact sensors (such as displacement gauges and accelerometers) rely on dense layout and are difficult to cover global deformation, especially in complex curved surface scenarios where they are easily interfered by environmental noise, significantly reducing data reliability. Monitoring the displacement or strain response of specific parts using sensors is a commonly used monitoring method for early thin-walled structures. Such methods are suitable for evaluating the structural strength of specific focus points and lack the ability to monitor the overall displacement field or stress field of the structure. Another type of method based on strain reconstruction calculates the displacement field through an inversion algorithm. However, the traditional finite element method requires prior information such as material parameters and boundary conditions, with high modeling complexity and difficulty in meeting real-time requirements, greatly limiting its engineering applicability.

[0003] Traditional monitoring technologies mainly rely on contact sensors (such as displacement gauges and accelerometers) or strain-based reconstruction methods. However, the former has problems of dense layout and difficulty in covering global deformation, while the latter is limited by material parameter dependence and modeling complexity and is difficult to meet real-time requirements. In recent years, the inverse finite element method (iFEM) has gradually become a research hotspot in the field of displacement field reconstruction due to its advantages of not requiring external loads and material constitutive models.

[0004] Inverse finite element realizes the displacement field monitoring of a structure by minimizing the weighted least square functional between the analytical strain and the measured strain, and is especially suitable for real-time monitoring of thin-walled shells. For example, the four-node quadrilateral inverse shell element (iSQ4) developed based on the first-order shear deformation theory has shown good applicability in ocean engineering, and its ability to reconstruct the displacement field from surface strain data has been widely verified. However, traditional inverse finite element methods require bilateral arrangement of strain sensors to obtain complete membrane strain and bending strain data, which is difficult to achieve in thin-walled structures such as wind turbine towers.

[0005] The limitations of the prior art are mainly reflected in that the unilateral layout of sensors leads to incomplete strain data, and the traditional inverse finite element functional does not fully consider the coupling relationship between membrane strain and bending strain, resulting in a significant decrease in the reconstruction accuracy. The limitations of the prior art mainly focus on the engineering problems of sensor layout. First, the traditional inverse finite element method usually requires bilateral symmetric layout of strain sensors to obtain complete membrane strain and bending strain data. However, in the practical application of thin-walled structures such as wind turbine towers, due to factors such as high installation difficulty, complex surface curvature of the structure, or high maintenance cost, unilateral layout becomes an inevitable choice. The unilateral layout leads to serious loss of strain data, and the traditional inverse finite element functional does not fully couple the mapping relationship between unilateral strain and the global displacement field, resulting in a significant decrease in the reconstruction accuracy. Second, the existing inverse finite element methods have too high requirements for the number of measurement points. Taking the four-node quadrilateral inverse shell element (iSQ4) as an example, the traditional bilateral layout requires doubling the number of sensors on the upper and lower surfaces of each unit, significantly increasing the hardware cost and data acquisition complexity. In addition, under the condition of unilateral layout, if the same accuracy as the traditional method is required, it is often necessary to further increase the measurement point density, resulting in a sharp increase in the engineering implementation difficulty and maintenance cost, which becomes the core bottleneck restricting the large-scale application of the inverse finite element technology. Summary of the Invention

[0006] Aiming at the deficiencies of the prior art, the present invention provides a coupling functional unilateral inverse finite element displacement field monitoring method for cylindrical thin-walled structures, which solves the problems in the prior art that the unilateral layout of sensors leads to the loss of strain data and affects the reconstruction accuracy, the inverse finite element functional does not fully consider the coupling relationship between membrane strain and bending strain, and the requirements for the number of measurement points are too high and the engineering implementation difficulty is large.

[0007] To achieve the above objectives, the present invention is realized through the following technical solutions: A coupling functional unilateral inverse finite element displacement field monitoring method for cylindrical thin-walled structures, including: discretizing the cylindrical thin-walled structure based on the inverse finite element method, and constructing a least-squares functional including the structural coupling strain and transverse shear strain of the unilateral surface, and solving the least-squares functional through the variational principle to establish the basic mathematical model of the inverse finite element method. The method further includes the following steps: Performing the discretization using four-node inverse shell elements, and arranging sensors at the geometric center position of each unit; Based on the basic mathematical model of the inverse finite element, establishing an overall pseudo-stiffness matrix and an overall pseudo-force matrix, wherein the overall pseudo-stiffness matrix depends only on the geometric shape of the unit and does not depend on the measured values, and the overall pseudo-force matrix is a function of the unilateral measured strain values; Calculating and obtaining the displacement field of the structure by solving the linear equations composed of the overall pseudo-stiffness matrix, the overall pseudo-force matrix, and the global displacement vector.

[0008] Preferably, the discretization includes inverse shell elements with four nodes based on unilateral sensors, and a unilateral inverse finite element mathematical model based on a coupled functional. The specific coupling terms include unilateral measured strain, membrane strain, and bending strain. Specifically, in the traditional bilateral inverse finite element, the relationships between the measured strain values on the upper and lower surfaces of the inverse element and the membrane strain and bending strain are as follows: ; ; Where, and respectively represent the positive strain and negative strain in the x - direction. The positive strain represents the tension of the object, and the negative strain represents compression; and respectively represent the positive strain and negative strain in the y - direction. Similar to the strain in the x - direction, they describe the tension and compression in the y - direction; and respectively represent the positive and negative directions of the shear strain. The shear strain describes the deformation occurring inside the material due to the action of shear force; Indicates that the formula is applicable to the calculation of all nodes or elements. That is to say, the strain calculation in the formula is carried out for all calculation elements or nodes; is the measured strain value on the upper surface of the element, is the curvature strain of the inverse element. Although the shear strain value cannot be directly measured, for most thin - shell structures, the contribution of the shear strain is small and is usually ignored in the inverse finite element calculation. The surface measured strain value is expressed as: ; Where, is the measured strain value on the upper surface of the element, that is, the strain value actually measured on the structure surface; is the membrane strain of the inverse element, representing the average strain component of the element in the plane; is the curvature strain of the inverse element, representing the strain component caused by the bending of the element; is the thickness of the thin - wall structure or a certain scale factor, used to convert the curvature strain into the corresponding surface strain.

[0009] Preferably, the unilateral inverse finite element functional is obtained based on a coupled equation, and the coupled equation includes: ; Where, is the unilateral inverse finite element functional, which measures the consistency between the calculated strain and the measured strain and includes a correction term for stiffness; is the displacement field variable of the inverse finite element unit, which is the unknown quantity that needs to be solved; is the membrane strain of the inverse unit, representing the plane strain part; is the curvature strain of the inverse element, which represents the strain due to bending; is the measured strain on the upper surface of the unit, i.e., the strain value obtained experimentally; is the half-thickness of the shell, which is used to describe the relationship between membrane strain and curvature strain; , and is the weighting coefficient, which is always positive. Its specific value is related to whether the unit is equipped with sensors. It is used to control the consistency between the strain of the analyzed section and the experimental measurement value. and is the influence on the pseudo-stiffness matrix as the pseudo-stiffness correction term for bending and shear.

[0010] Preferably, the basic mathematical model is based on the least squares variation principle to minimize the error between the measured strain value and the theoretical strain value. Specifically, according to the functional variation principle, the coupling functional is subjected to displacement differentiation to obtain the basic mathematical model of the unit inverse finite element: ; in, is the one-sided inverse finite element functional; The mapping relationship between the processed element measurement point strain and the node displacement is called the element pseudo-stiffness matrix; is the processed unit measuring point strain, called unit strain vector; is the unit node displacement vector; Denotes the unit variational functional The derivative of .

[0011] Can be simplified to: ; in, The mapping relationship between the processed unit point strain and the node displacement is called the unit pseudo-stiffness matrix. is the processed unit measurement point strain, called the unit strain vector, is the unit node displacement vector.

[0012] Preferably, the coupling functional one-sided inverse finite element inverse unit construction method includes: determining a detailed calculation method of a pseudo-stiffness matrix and a pseudo-force matrix according to the measurement point position, the structural boundary conditions and the displacement-strain relationship, specifically including: The establishment of element pseudo-stiffness matrix: ; in, is the unit area; is the membrane strain matrix, is the bending strain matrix, is the shear strain matrix; Establishment of the element pseudo-force matrix: ; wherein, is the number of internal measurement points of the element; is the measured strain on the upper surface of the element; is the membrane strain matrix; is the bending strain matrix; is the weighting coefficient.

[0013] Preferably, the global pseudo-stiffness matrix and the pseudo-force matrix are assembled based on the element pseudo-stiffness matrix and the element pseudo-force matrix, specifically including: ; ; ; ; wherein, is the coordinate transformation matrix from local coordinates to global coordinates; is the pseudo-stiffness matrix under the coupling functional (a symmetric matrix, independent of the measured values, only requiring inversion once); is the pseudo-force matrix under the coupling functional (a function of the one-sided measured strain values and the coupling matrix of the membrane strain and the bending strain, changing with the measured values); the parameter represents the number of inverse elements; is the global displacement vector, representing the displacement field of the entire structure; is the matrix transpose.

[0014] Preferably, the global pseudo-stiffness matrix contains the rigid body motion modes of the discrete structure and is a singular matrix; after applying the displacement boundary conditions, the global coupling functional one-sided inverse finite element mathematical model can be simplified to: ; wherein, is a positive definite non-singular matrix, which remains unchanged when the sensor layout does not change. Therefore, it only needs to be inverted once during the monitoring of the global displacement field; is the global pseudo-force matrix, representing the equivalent external force generated by the measured strain. This matrix is updated according to the one-sided measured strain values and changes with the measured data, and no cumulative error will be generated. Therefore, this method can achieve high-precision real-time structural health monitoring.

[0015] The present invention provides a method for monitoring the displacement field of a cylindrical thin-walled structure by coupling functional unilateral inverse finite elements, which has the following beneficial effects: 1. The present invention infers the displacement field of the entire structure by using the strain information measured unilaterally. This method combines the advantages of finite element analysis and inverse problem solving, and can accurately infer the global deformation of the structure under the condition of only obtaining local strain data. By optimizing the solution of local strain data, this method can avoid the high cost and low efficiency problems caused by the need for global strain measurement in traditional methods, and has obvious practical advantages.

[0016] 2. The present invention adopts the coupling relationship between unilaterally measured strain, membrane strain and bending strain. This coupling principle stems from the internal relationship between membrane strain and bending strain when the structure is subjected to external loads. Membrane strain reflects the tensile or compressive deformation of the material in the plane, while bending strain represents the stress distribution and deformation of the material under bending. By incorporating the coupling of these two strain forms into the model, the present invention can more comprehensively consider the complex deformation characteristics of the structure, thereby effectively improving the recognition accuracy and accuracy of unilateral displacement field monitoring and reducing the errors caused by single strain data.

[0017] 3. The present invention can effectively handle the uncertainties caused by measurement errors, external environmental changes or local damages through the inverse problem solving process based on a reasonable mathematical model and finite element analysis. Under different loading conditions, this method can still provide stable and reliable displacement field predictions, showing strong adaptability. This robustness makes this method have broad application prospects in practical engineering applications, especially in the fields of structural health monitoring and fault diagnosis. Description of the Drawings

[0018] Figure 1 is a schematic diagram of the method flow of the present invention; Figure 2 is a schematic diagram of an inverse finite element unit; Figure 3 is a schematic diagram of the layout method of unilateral strain measurement points; Figure 4 is a comparison diagram of the displacement field identification of a cylindrical thin-walled structure at a certain moment; Figure 5 is a comparison diagram of the theoretical value and the monitored value of the total displacement of a certain point at 40m height in the time history; Figure 6 is a comparison diagram of the relative errors of a certain point at 40m height under different noises in the time history. Detailed Embodiments

[0019] Next, in combination with the accompanying drawings in the specification of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.

[0020] To better understand the present invention, the above content will be described in detail below in combination with specific embodiments.

[0021] Please refer to the attached Figures 1-6 , the embodiment of the present invention provides a method for monitoring the displacement field of a coupled functional unilateral inverse finite element cylindrical thin-walled structure. The method is mainly divided into two parts, namely, the part of constructing the basic model and the part of solving the displacement field: The part of constructing the basic model includes: discretizing the cylindrical thin-walled structure based on the inverse finite element method, and constructing a least-squares functional including the structural coupled strain and the transverse shear strain of the unilateral surface, and solving the least-squares functional by the variational principle to establish the basic mathematical model of the inverse finite element method; The part of solving the displacement field includes: Discretize using four-node inverse shell elements, and arrange sensors at the geometric center position of each element; Based on the basic mathematical model of the inverse finite element, establish the global pseudo-stiffness matrix and the global pseudo-force matrix. Among them, the global pseudo-stiffness matrix only depends on the geometric shape of the element and does not depend on the measured values, and the global pseudo-force matrix is a function of the unilateral measured strain values; By solving the linear equations composed of the global pseudo-stiffness matrix, the global pseudo-force matrix and the global displacement vector, the displacement field of the structure is calculated.

[0022] In this embodiment, the cylindrical thin-walled structure is discretized by the traditional inverse finite element method to obtain an inverse finite element model. The traditional four-node inverse finite element is a four-node inverse shell element based on the Mindlin plate theory. On the basis of considering the influence of shear deformation on the structure, the rotational motion in the z direction is introduced at the same time to avoid shear locking of the structure. For plane geometric structures, the general method of adding the local membrane matrix and the bending matrix is used to obtain the element formula.

[0023] Each node of the inverse finite element model has 6 degrees of freedom, including displacement degrees of freedom in three directions and rotational degrees of freedom in three directions. The inverse finite element unit model and node degrees of freedom are as Figure 2 shown. The origin of the local coordinate system is located at the centroid of the element, and the strain value measurement position is also here. The transformation from the element matrix to the global matrix is realized through the coordinate transformation matrix.

[0024] Referring to the finite element method and combining with the Mindlin plate theory, the inverse finite element isoparametric shape functions can be expressed using the following mapping functions: ; ; where, represents the and coordinates at the parametric coordinates ; represents the and coordinates at the parametric coordinates ; represents the shape function; and respectively represent the coordinates of the th node in the x and y directions; According to the kinematic relationship in the assumptions of the first-order shear deformation theory, the three components of the displacement vector of any particle within the element can be expressed as: ;

[0025] ;

[0026] ; where, , , are the displacements along the x, y, and z directions respectively; and are the displacements within the plane element; is the displacement (deflection) across the uniform shell thickness; , are the rotations about the positive x and y axes, and the thickness of the element is 2h.

[0027] The linear strain-displacement relationship of linear elastic theory is as follows: ;

[0028] ;

[0029] ;

[0030] ;

[0031] ; where, , are the rotations about the positive x and y axes; represents the strain in the x direction; Denotes the displacement in the x direction; Denotes the depth or the coordinate perpendicular to the x and y directions in three-dimensional space; Denotes the strain in the y direction; Denotes the displacement in the y direction; Denotes the shear strain in the x and y directions, and are the displacement components along the y and x directions, respectively; Denotes the shear strain in the x and z directions, Denotes the displacement in the z direction; Denotes the displacement component along the z direction; Denotes the shear strain in the y and z directions; According to the Kirchhoff-Love theory, the stress in the z direction , the plate thickness strain .

[0032] The strain-displacement relationship of iSQ4 can be expressed as a strain-displacement relationship expressed in terms of the element nodal displacement vector including membrane strain , bending strain and transverse shear strain : ; ; Where: ; ; Where the matrix is the membrane strain matrix, is the bending strain matrix, is the shear strain matrix, all of which contain the derivatives of the shape functions; , , , Denotes the displacements of the four nodes in the element; is the displacement of the th node in the x direction; The th node's displacement in the y direction; is the displacement of the th node in the z direction; is the rotation angle (angular displacement) of the th node about the x axis; is the The rotation angle (angular displacement) of the th node about the z-axis; represents the transpose of the matrix. These strain matrices are independent of each other.

[0033] The inverse finite element method realizes real-time monitoring of the displacement field of the structure by minimizing the weighted least squares functional based on the global discrete node degrees of freedom (DOFs). The inverse finite element unilateral coupling functional established based on the coupling equation has the following specific form: ; where is the unilateral inverse finite element functional, which measures the consistency between the calculated strain and the measured strain and includes a correction term for stiffness; is the displacement field variable of the inverse finite element element, which is the unknown to be solved; is the membrane strain of the inverse element, representing the plane strain part; is the curvature strain of the inverse element, representing the strain due to bending; is the measured strain on the upper surface of the element, that is, the strain value obtained from the experiment; is the half-thickness of the shell, which is used to describe the relationship between the membrane strain and the curvature strain; , and are the weighting coefficients, all of which are positive values. Their specific values are related to whether the element is equipped with sensors and are used to control the consistency between the strain of the analysis section and the experimental measurement value; and are the effects on the pseudo-stiffness matrix and are used as the pseudo-stiffness correction terms for bending and shear.

[0034] Therefore, the squared norm given in this equation can be written in the form of a normalized Euclidean norm: ; ; ; where is the element area; is the half-thickness of the shell.

[0035] Based on the variational principle of the functional, the coupling functional is differentiated with respect to the node displacement to minimize the error between the measured strain and the theoretical strain. The specific form is as follows: ; where is the unilateral inverse finite element functional; The mapping relationship between the processed element measurement point strain and the node displacement is called the element pseudo-stiffness matrix; The processed element measurement point strain is called the element strain vector; is the element node displacement vector; represents the derivative of the element variational functional with respect to.

[0036] It can be simplified to: ; where the mapping relationship between the processed element measurement point strain and the node displacement is called the element pseudo-stiffness matrix, the processed element measurement point strain is called the element strain vector, is the element node displacement vector.

[0037] After performing the least squares functional calculation, the element pseudo-stiffness matrix and the element pseudo-force matrix can be obtained, and their specific forms are as follows: ; ; where is the element area; is the membrane strain matrix, is the bending strain matrix, is the shear strain matrix; is the number of measurement points inside the element; is the measured strain value on the upper surface of the element; , and are the weighting coefficients, all of which are positive values. Their specific values are related to whether sensors are arranged on the element and are used to control the consistency between the analyzed cross-section strain and the experimental measurement values.

[0038] After the element inverse finite element equation is established, the global inverse finite element equation can be assembled in the following way: ; ; ; ; where is the coordinate transformation matrix from local coordinates to global coordinates; is the pseudo-stiffness matrix under the coupling functional (a symmetric matrix, independent of the measurement values, only requires one inversion); is the pseudo-force matrix under the coupling functional (a function of the one-sided measured strain value and the coupling matrix of the membrane strain and the bending strain, which changes with the measurement values); The parameter Indicates the number of inverse units; is the global displacement vector, representing the displacement field of the overall structure; is the matrix transpose.

[0039] To verify the displacement field monitoring effect of the coupled functional unilateral inverse finite element, a certain cylindrical thin-walled structure is used for real-time displacement field monitoring in this example. The layout of strain measurement points is as Figure 3 . The displacement field monitoring results for 40 s are selected and shown in Figure 4 . Figure 5 is the displacement value and reconstructed value of a certain point at a height of 40 m in the time history. Figure 6 is the monitoring error of a certain point at a height of 40 m in the time history under different noises.

[0040] Although the embodiments of the present invention have been shown and described, for those of ordinary skill in the art, it can be understood that various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for monitoring the displacement field of a cylindrical thin-walled structure by using a coupled functional unilateral inverse finite element, comprising: Discretize the cylindrical thin-walled structure based on the inverse finite element method, and construct a least-squares functional that includes the coupled strain and transverse shear strain of the structure with a unilateral surface. Solve the least-squares functional by the variational principle to establish the basic mathematical model of the inverse finite element method, characterized in that the method further comprises the following steps: Perform the discretization using four-node inverse shell elements, and arrange sensors at the geometric center position of each element; Based on the basic mathematical model of the inverse finite element, establish the global pseudo-stiffness matrix and the global pseudo-force matrix, wherein the global pseudo-stiffness matrix depends only on the geometric shape of the element and does not depend on the measured values, and the global pseudo-force matrix is a function of the unilateral measured strain values; Calculate the displacement field of the structure by solving the linear equations composed of the global pseudo-stiffness matrix, the global pseudo-force matrix, and the global displacement vector.

2. The displacement field monitoring method of the coupled functional unilateral inverse finite element cylindrical thin-walled structure according to claim 1, characterized in that The discretization includes arranging four-node inverse shell elements based on unilateral sensors, and using the unilateral inverse finite element mathematical model based on the coupled functional. The specific coupling terms include unilateral measured strain, membrane strain, and bending strain, expressed as: ； Among them, is the measured strain on the upper surface of the element, that is, the strain value actually measured on the structure surface; is the membrane strain of the inverse element, representing the average strain component of the element in the plane; is the curvature strain of the inverse element, representing the strain component caused by the bending of the element; is the thickness of the thin-walled structure or a certain scale factor, used to convert the curvature strain into the corresponding surface strain.

3. The displacement field monitoring method of the coupled functional unilateral inverse finite element cylindrical thin-walled structure according to claim 2, characterized in that, The unilateral inverse finite element functional is obtained based on the coupled equations, and the coupled equations include: ； Among them, is a unilateral inverse finite element functional, which measures the consistency between the calculated strain and the measured strain and includes a correction term for stiffness; is the displacement field variable of the inverse finite element element and is the unknown to be solved; is the membrane strain of the inverse element, representing the plane strain part; is the curvature strain of the inverse element, representing the strain due to bending; is the measured strain on the upper surface of the element, that is, the strain value obtained from the experiment; is the half thickness of the shell, which is used to describe the relationship between the membrane strain and the curvature strain; , and are weighting coefficients, all of which are positive values. Their specific values are related to whether the element is equipped with sensors and are used to control the consistency between the strain of the analysis section and the experimental measurement value; and are the influences on the pseudo stiffness matrix and are used as the pseudo stiffness correction terms for bending and shear.

4. The displacement field monitoring method of the coupled functional unilateral inverse finite element cylindrical thin-walled structure according to claim 3, characterized in that The basic mathematical model is based on the least-squares variational principle to minimize the error between the measured strain values and the theoretical strain values, including ； Among them, is the mapping relationship between the processed unit measurement point strain and the nodal displacement, which is called the unit pseudo stiffness matrix, is the processed unit measurement point strain, which is called the unit strain vector, is the unit nodal displacement vector.

5. The displacement field monitoring method of the coupled functional unilateral inverse finite element cylindrical thin-walled structure according to claim 4, characterized in that The construction method of the coupled functional unilateral inverse finite element inverse element includes: determining the detailed calculation methods of the pseudo-stiffness matrix and the pseudo-force matrix according to the measuring point positions, the structural boundary conditions, and the displacement-strain relationship, specifically including: Establishment of the element pseudo-stiffness matrix: ； wherein, is the unit area; is the membrane strain matrix, is the bending strain matrix, is the shear strain matrix; Establishment of the element pseudo-force matrix: ； Among them, is the number of internal measurement points of the unit; is the measured strain on the upper surface of the unit; is the membrane strain matrix; is the bending strain matrix; is the weighting coefficient.

6. The displacement field monitoring method for the coupled functional unilateral inverse finite element cylindrical thin-walled structure according to claim 5, characterized in that The global pseudo-stiffness matrix and the pseudo-force matrix are assembled based on the element pseudo-stiffness matrix and the element pseudo-force matrix, specifically including: ； ； ； ； Among them, is the coordinate transformation matrix from local coordinates to global coordinates; is the pseudo-stiffness matrix under the coupling functional; is the pseudo-force matrix under the coupling functional; The parameter represents the number of inverse elements; is the global displacement vector, representing the displacement field of the overall structure; is the matrix transpose of.

7. The displacement field monitoring method of the coupled functional unilateral inverse finite element cylindrical thin-walled structure according to claim 6, characterized in that, The global pseudo-stiffness matrix contains the rigid body motion modes of the discrete structure and is a singular matrix; after applying the displacement boundary conditions, the global coupled functional unilateral inverse finite element mathematical model can be simplified to: ； Among them, is a positive definite non-singular matrix, which remains unchanged when the sensor layout does not change. Therefore, only one inversion is required during the overall displacement field monitoring process; is the global pseudo-force matrix, representing the equivalent external force generated by the measured strain. This matrix is updated according to the unilateral measured strain value as the measurement data changes and does not produce cumulative errors. Therefore, this method can achieve high-precision real-time structural health monitoring; The displacement DOFs results of non-boundary nodes obtained by reconstructing after the coupled functional unilateral inverse finite element calculation, based on which the displacement field monitoring of the structure can be realized.

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