Rapid deformation reconstruction method for large-displacement flexible beam structure

Abstract

The invention discloses a rapid deformation reconstruction method for a large-displacement flexible beam structure, and belongs to the field of structural health monitoring. On the basis of a co-rotation coordinate method, the local displacement is obtained after rigid body displacement is removed from the total displacement of the flexible beam structure. Local displacement obtained through a co-rotation coordinate method is substituted into an iFEM formula based on a small deformation hypothesis, and the deformation reconstruction problem under the geometric nonlinear condition is converted into the problem that the mean square error sum of theoretical strain and actually measured strain is minimum by searching an optimal node displacement parameter to achieve square functional optimization. And solving the optimal solution of the functional by adopting an optimization algorithm. In addition, a simplification strategy which ignores a high-order small quantity is adopted, so that the whole nonlinear deformation reconstruction method remarkably improves the calculation efficiency on the premise that the calculation precision is not sacrificed. According to the method, the node displacement parameter enabling the functional to be minimum can be rapidly solved, and rapid and high-precision deformation reconstruction of the large-displacement flexible beam structure is achieved.

Description

Technical Field

[0001] This invention belongs to the field of structural health monitoring and relates to a rapid deformation reconstruction method for large-displacement flexible beam structures. Specifically, it is an analysis method for rapidly reconstructing the nonlinear deformation displacement field of a beam structure. Background Technology

[0002] Displacement monitoring of beam structures is the process of real-time monitoring of the shape of beam structures. As one of the key damage characteristics in structural health monitoring (SHM) systems, employing efficient and accurate displacement monitoring methods to identify and assess the deformation state and safety of structures in real time has become a core technology for ensuring structural safety and reliability, and also an important early warning means for assessing the risk of structural failure.

[0003] Existing deformation reconstruction methods based on measured strain can be divided into modal methods, Ko displacement theory, and inverse finite element methods (iFEM). Most traditional deformation reconstruction methods focus on small deformation analysis. Among them, the inverse finite element method based on the small deformation assumption is widely used for small deformation reconstruction of various structures due to its excellent performance in terms of accuracy and applicability, and the fact that it does not require elastic or inertial parameters of the material.

[0004] With the continuous advancement of materials science and technology, flexible and complex engineering structures are increasingly widely used in modern engineering. The linear displacement assumption is often no longer applicable to systems subjected to complex loads and nonlinear responses. Existing experimental studies on the nonlinear displacement behavior of beams have found that the deviation between linear analysis predictions and measured values ​​in the nonlinear deformation stage is as high as 12.50%. Therefore, nonlinear analysis must be introduced when reconstructing the nonlinear displacement of beams. However, the introduction of geometric nonlinearity significantly increases the solution time, making it difficult to meet the real-time monitoring requirements for the nonlinear displacement of flexible beam structures in practical engineering applications.

[0005] In the prior art, a Chinese invention patent (application number: 202111469982.5) proposes a classic inverse finite element method framework, which achieves deformation reconstruction by constructing a weighted least squares functional. However, this method is based entirely on linear theory and cannot accurately describe the strain-displacement relationship of the structure under large rotation and large displacement conditions, resulting in a significant decrease in reconstruction accuracy during the nonlinear deformation stage. A Chinese invention patent (application number: 202211569574.1) introduces the corotation coordinate method into the inverse finite element system, improving the ability to describe large rotation problems by separating rigid body motion from pure deformation. Although this method enhances the adaptability to geometric nonlinearity at the theoretical level, its complete corotation formulation and complex Jacobian derivation lead to a sharp increase in computational complexity and low solution efficiency, failing to meet the computational speed requirements of real-time monitoring.

[0006] Therefore, there is an urgent need for a rapid deformation reconstruction method for large-displacement flexible beam structures to predict their deformation and solve the aforementioned technical problems. Summary of the Invention

[0007] To address the problems of existing technologies, this invention proposes a rapid deformation reconstruction method for large-displacement flexible beam structures. This method is based on the corotational coordinate method (CR), obtaining local displacements after removing rigid body displacements from the overall displacement of the flexible beam structure. Substituting the local displacements obtained by the corotational coordinate method into the iFEM formula based on the small deformation assumption transforms the deformation reconstruction problem under geometrically nonlinear conditions into a squared functional optimization problem of finding the optimal nodal displacement parameters that minimize the sum of the mean square errors of theoretical and measured strains. To solve for the optimal solution of this functional, this invention employs a more robust optimization algorithm. This algorithm only requires first-order variational calculation of the global displacement using local displacements and constructing the tangent stiffness matrix, thus avoiding the complex second-order variational derivation of traditional optimization methods. To simplify the calculation of the first-order variation, this invention adopts a simplification strategy of ignoring higher-order small quantities. This simplification significantly improves the computational efficiency of the entire nonlinear deformation reconstruction method without sacrificing computational accuracy. Finally, the nodal displacement parameters that minimize the functional can be quickly solved, achieving rapid and high-precision deformation reconstruction of large-displacement flexible beam structures.

[0008] To achieve this objective, the present invention adopts the following technical solution:

[0009] A rapid deformation reconstruction method for flexible beam structures with large displacement includes the following steps:

[0010] Step 1: Finite element discretization and initialization of the flexible beam structure, specifically:

[0011] Step 1.1: Discretize the flexible beam structure into... The nth discrete unit is used to calculate the nth discrete unit. Length of each discrete unit ;

[0012] The flexible beam structure in step 1.1 is composed of The system consists of discrete elements, each composed of two nodes. Each discrete element should be independent of the others; that is, no single node should belong to multiple discrete elements simultaneously. In the global coordinate system, the position vector of a discrete element before initial deformation is... ,in, Indicates the first Discrete unit This indicates the first undeformed section of the flexible beam structure. The coordinates of the first node of a discrete element in the X and Y axes of the global coordinate system; This represents the rotation angle of the first node around the global Z-axis (following the right-hand screw rule). Indicates the first The coordinates of the second node of each discrete element in the X and Y axes of the global coordinate system. This represents the rotation angle of the second node around the global Z-axis (following the right-hand screw rule); via the position vector. The first can be calculated Initial length of each discrete unit :

[0013] (1)

[0014] Step 1.2: Set the initial damping parameters for the algorithm Initial penalty coefficient , tolerance Maximum number of iterations Iteration counter .in For 1000, It is 2. It is 1e-7, The initial value is 200. =1;

[0015] Step 1.3: Input the measured strain of the flexible beam structure surface, and calculate the measured strain of the processed discrete element. With measuring curvature Specifically:

[0016] (2)

[0017] (3)

[0018] in, Indicates the thickness of the flexible beam structure, indicated by the superscript "". "and" "These are used to represent the upper and lower surfaces of a flexible beam structure, respectively." This represents the strain value measured on the surface of the discrete element;

[0019] Step 1.4: Define the shape functions of the discrete element, derive the displacement and strain fields, and construct the strain-displacement matrix. , Specifically:

[0020] Using the principle of shape function interpolation, the axial displacement of any point inside the discrete element can be calculated. and lateral displacement It can be represented by shape functions and discrete element node degrees of freedom:

[0021] (4)

[0022] (5)

[0023] in, These represent the axial displacement, lateral displacement, and rotation angle of node 1 and node 2 in the local coordinate system, respectively. In the subsequent formulas, quantities with horizontal bars above them indicate quantities based on the local coordinate system.

[0024] Axial displacement For local coordinates First variation and lateral displacement Axial coordinates in the local coordinate system The second variation can be expressed as:

[0025] (6)

[0026] (7)

[0027] in, Representing shape functions A first variation; Representing shape functions A first variation; Representing shape functions The second variation; Representing shape functions The second variation; Representing shape functions The second variation; These are the axial coordinates in the local coordinate system.

[0028] The axial strain-displacement matrix is ​​constructed based on formulas (6) and (7). and bending strain-displacement matrix Therefore, for the first Theoretical value of axial strain for a discrete element and the theoretical value of curvature It can be expressed as the product of the discrete element shape function and the discrete element local displacement: ;in, For the first Local displacement of discrete unit;

[0029] Step 2: When the overall equivalent internal force of the flexible beam structure and At that time, the main iteration loop is performed, where Indicates tolerance. Represents an iteration counter. This indicates the maximum number of iterations, specifically:

[0030] Step 2.1: Discretize the flexible beam structure into... For each discrete unit, perform a unit-level loop to traverse the discrete units. =1 to ;

[0031] Step 2.1.1: For the flexible beam structure Each discrete element is described using the co-rotation coordinate method, and the local displacement of the discrete element is obtained from the global displacement of the flexible beam structure. Specifically:

[0032] Based on current global coordinates The length of the discrete element is calculated using formula (1). ;

[0033] Calculate the first in generalized coordinates Rotation angle of a discrete unit rigid body After the flexible beam structure deforms, the coordinate difference of the nodes in the Y-axis direction is: The displacement of the node in the X-axis direction is ;because Let be the rigid body rotation angle between the line connecting the nodes of the discrete element after displacement and the X-axis of the global coordinate system. Calculate the first Rigid body rotation angle vector of a discrete element in global coordinates ;in, ,and The first The axial and lateral displacements of discrete element nodes 1 and 2 in the global coordinate system; Rigid body rotation angle of discrete unit It can be calculated by the first The angle between the line connecting two nodes in a discrete unit and the global coordinate X-axis is obtained;

[0034] Get the first Rigid body rotation angle of discrete unit Then, substitute it into the rigid body rotation matrix in the two-dimensional plane. In the general form, calculate the rotation matrix of the discrete unit. for: ;

[0035] The local displacements of discrete element nodes can be calculated using the co-rotation coordinate method. ;in, These are the coordinates of the initial, undeformed configuration in the local coordinate system. Indicates the first Rigid body rotation angle vectors of discrete elements in global coordinates; Indicates the first after deformation The coordinates of a discrete unit in the global coordinate system;

[0036] Step 2.1.2: Construct the least squares functional Computing the Jacobian matrix ;

[0037] The local displacement of the discrete element is obtained through step 2.1.1. Subsequently, the least squares functional of the discrete unit It can be represented as ;in: , represents the vector composed of the residual between the theoretical value of axial strain and the measured value of axial strain, and the residual between the theoretical value of bending curvature and the measured value of bending curvature; This represents the residual between the theoretical value and the measured value of axial strain. This represents the residual between the theoretical value and the measured value of the bending curvature.

[0038] Through step 2.1.1, the local displacement of the discrete element is determined. The expression is used to calculate the local displacement of the discrete element. For global displacement variation : ,in , representing the rigid body rotation angle vector about the angle of rotation of the rigid body Variations; express unit array; , representing the rigid body rotation angle For global displacement Variations; express When it is 0th Rotation matrix of discrete units The value of is such that the unit Jacobian matrix is ​​expressed as . ;

[0039] Step 2.1.3: Calculate the first [number]th [unit] in the global coordinate system. Equivalent internal forces of discrete elements and tangent stiffness matrix Equivalent internal force of the assembly and the overall tangential stiffness matrix And determine the overall equivalent internal force after constraints based on the boundary conditions. and the constrained global tangent stiffness matrix ;in: Indicates the first The initial length of each discrete unit; This represents a vector consisting of the residual between the theoretical and measured values ​​of axial strain and the residual between the theoretical and measured values ​​of bending curvature; Represents the axial coordinates in the local coordinate system; Indicates the first The equivalent internal force vector of each discrete element;

[0040] Step 2.1.4: End the inner loop of assembling the discrete elements of the flexible beam structure as a whole, and calculate the displacement increment of the flexible beam structure. : ;in, Indicates the current iteration step number is Global coordinates of the flexible beam structure at that time;

[0041] Step 2.2: Step size evaluation and parameter update, calculate gain ratio ,in: Indicates the modulus length; The global coordinates of the flexible beam structure are: Overall equivalent internal force after time constraint The modulus length; The global coordinates of the flexible beam structure are: Overall equivalent internal force after time constraint The modulus length; if This indicates that the step size is accepted, and the node coordinates of the flexible beam structure are... and reduce damping Incrementing step counter ;if This indicates that the step size was rejected, and the node coordinates of the flexible beam structure are... and increase the damping parameter Proceed to step 2.1;

[0042] Step 3: Output the node coordinates of the flexible beam structure ;

[0043] This invention combines the corotational coordinate method (CR) with the iFEM method based on the small deformation assumption. Through this combination, we transform the deformation reconstruction problem under geometrically nonlinear conditions into a square functional optimization problem. The goal of this problem is to find the optimal nodal displacement parameters that minimize the sum of the mean square errors between theoretical and measured strains. This problem is ultimately solved using the LM algorithm. This invention defines the method for reconstructing the deformation of large-displacement flexible beam structures using this method as the large-displacement iFEM method.

[0044] Based on the large-displacement iFEM method, an improved large-displacement iFEM method is proposed to further simplify the calculation and improve the calculation speed. The improved large-displacement iFEM method neglects... High-order, small-scale simplification strategies enable This simplifies the equivalent internal forces of the unit. and tangent stiffness matrix To quickly reconstruct the displacement of a large-displacement flexible beam structure.

[0045] This simplification significantly improves the computational speed of the entire nonlinear deformation reconstruction method without sacrificing computational accuracy, enabling the rapid acquisition of accurate static and dynamic nodal coordinates of flexible beam structures. Analyze the numerical solution.

[0046] Beneficial effects of this invention:

[0047] This invention constructs a weighted least-squares functional suitable for large-displacement flexible beam structures by combining the co-rotation coordinate method with the linear inverse finite element method. In the calculation process of finding the optimal nodal displacement parameters using the least-squares functional, the Levenberg-Marquardt algorithm is introduced, and the calculation of equivalent internal forces and tangent stiffness matrices is simplified to achieve efficient solution to this optimization problem. When reconstructing the deformation of large-displacement flexible beam structures, the error is less than 2%, and the computational efficiency is improved by 20%, while the computation time is shortened. This invention can provide an important reference for real-time simulation of flexible beam structures and digital twins, and other simulation scenarios with high real-time requirements. Attached Figure Description

[0048] Figure 1 This is a shape diagram of a flexible beam structure under dynamic load provided in an embodiment of the present invention;

[0049] Figure 2 This is a comparison chart of the deformation reconstruction results of the large displacement flexible beam structure obtained by the large displacement iFEM method provided in this embodiment of the invention and the Ansys simulation results; Figure 2 (a) in the figure is based on the deformation reconstruction results of the large displacement flexible beam structure obtained by Ansys analysis, and analyzes the accuracy of the large displacement iFEM method to reconstruct the deformation of this large displacement flexible beam structure. Figure 2 (b) in the middle is Figure 2 The enlarged view in (a)A shows a comparison between the deformation reconstruction results of the large displacement flexible beam structure obtained by the large displacement iFEM method and the Ansys simulation results;

[0050] Figure 3 A comparison chart of the deformation and reconstruction results; Figure 3 (a) in the figure shows a comparison of the deformation reconstruction results of the large displacement iFEM method and the improved large displacement iFEM method for large displacement flexible beam structures; Figure 3 (b) in the middle is Figure 3 The enlarged view in (a)A shows a comparison of the deformation reconstruction results of the large displacement flexible beam structure obtained by the large displacement iFEM method and the improved large displacement iFEM method;

[0051] Figure 4 These are comparison figures of several sets of deformation reconstruction results obtained by the improved large displacement iFEM method provided in this embodiment of the invention and Ansys simulation results;

[0052] Figure 5 This is a comparison chart of the deformation reconstruction results of a flexible beam under dynamic load and the Ansys simulation results using the improved large displacement iFEM method provided in this embodiment of the invention. Figure 5 (a) in the figure is a comparison of the coordinates of the far-end node of the large-displacement flexible beam structure in the X direction reconstructed by this method and the coordinates obtained by Ansys simulation under the time history. Figure 5 (b) in the figure is a comparison between the deformation reconstruction results of the far-end node of the large-displacement flexible beam structure in the Y direction reconstructed by this method and the results obtained by Ansys simulation under the time history.

[0053] Figure 6 This is a comparison chart of convergence speeds provided in an embodiment of the present invention; Figure 6 (a) in the figure compares the number of iteration steps required to reconstruct the deformation of the large displacement flexible beam structure using the improved large displacement iFEM method and the large displacement iFEM method under the LV1 deformation level. Figure 6 (b) in the figure compares the number of iteration steps required to reconstruct the deformation of the large displacement flexible beam structure using the improved large displacement iFEM method and the large displacement iFEM method under the LV2 deformation level. Figure 6 (c) in the figure compares the number of iterations required to reconstruct the deformation of the large displacement flexible beam structure using the improved large displacement iFEM method and the large displacement iFEM method under the LV3 deformation level. Figure 6 (d) in the figure represents a comparison of the number of iteration steps required to reconstruct the deformation of the large displacement flexible beam structure using the improved large displacement iFEM method and the large displacement iFEM method under the LV4 deformation level. Figure 6 (e) in the figure represents a comparison of the number of iteration steps required to reconstruct the deformation of the large displacement flexible beam structure using the improved large displacement iFEM method and the large displacement iFEM method under the LV5 deformation level.

[0054] Figure 7 This is a comparison chart showing the solution speed of reconstructing the deformation of a large-displacement flexible beam structure using the large-displacement iFEM method and the improved large-displacement iFEM method provided in this embodiment of the invention. Detailed Implementation

[0055] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are for illustrative purposes only and are not intended to limit the scope of the invention.

[0056] This invention proposes a rapid deformation reconstruction method for flexible beam structures with large displacement, comprising the following steps:

[0057] Step 1: Finite element discretization and initialization of the flexible beam structure:

[0058] Step 1.1: Discretize the flexible beam structure into... Calculate the unit. Length of each discrete unit :

[0059] The flexible beam structure in step 1.1 is composed of It consists of discrete elements, each consisting of two nodes; the discrete elements should be independent of each other, that is, no node can belong to different discrete elements at the same time. In the global coordinate system, the position vector of the discrete element in the initial undeformed configuration. Calculate the first Initial length of each discrete unit ;

[0060] Step 1.2: Set the initial damping parameters for the algorithm Initial penalty coefficient , Tolerance Maximum number of iterations Iteration counter .in For 1000, It is 2. It is 1E-7. The initial value is 200. =1;

[0061] Step 1.3: Input the strain data measured on the surface of the flexible beam structure, and calculate the strain of the processed discrete element. With measuring curvature ;

[0062] Step 1.4: Define the shape functions of the discrete element, derive the displacement and strain fields, and construct the strain-displacement matrix. , ;

[0063] like Figure 1 The image shows a reverse beam element. The two-node reverse beam element iBeam3 is constructed from a continuous, homogeneous, and isotropic material. For example... Figure 1 As shown, the beam element contains two nodes, each with three degrees of freedom for displacement. The degrees of freedom of the discrete element nodes are determined by... , ,and composition, =1,2. Where... These represent the translational degrees of freedom along the x-axis and y-axis, respectively. Represents the rotational degree of freedom about the z-axis. Axial displacement at any point within the element. and lateral displacement It can be represented by shape functions and the degrees of freedom of discrete element nodes;

[0064] Axial displacement For local coordinates First variation and lateral displacement For local coordinates The second variation can construct the axial strain-displacement matrix. and bending strain-displacement matrix Therefore, for the first Theoretical value of axial strain for each unit and the theoretical value of curvature It can be expressed as the product of the element shape function and the local displacement of the discrete element: ;in, Representing shape functions A variation, Representing shape functions The second variation, These are the axial coordinates in the local coordinate system.

[0065] Step 2: When the overall equivalent internal force and At that time, the main iteration loop is performed, where Indicates tolerance. Represents an iteration counter. This indicates the maximum number of iterations, specifically:

[0066] Step 2.1: Discretize the flexible beam structure into... Each unit performs a unit-level loop (traversing discrete units). =1 to );

[0067] Step 2.1.1: For the flexible beam structure Each discrete element is described using the co-rotation coordinate method, and the local displacement of the discrete element is obtained from the global displacement of the flexible beam structure. ;

[0068] Based on current global coordinates Calculate the cell length ;

[0069] Calculate the first in generalized coordinates Rotation angle of a discrete unit rigid body After deformation, the coordinate difference of the nodes in the Y-axis direction is: The displacement of the node in the X-axis direction is ;because Let be the rotation angle between the line connecting the deformed nodes on the discrete element and the X-axis of the global coordinate system. This yields the rigid body rotation angle vector in global coordinates. ;in, ,and The first The axial and lateral displacements of discrete element nodes 1 and 2 in the global coordinate system. For the first The angle between the line connecting two nodes of a discrete element and the global X-axis; the local displacement of the discrete element can be obtained by the co-rotation coordinate method. ;

[0070] in: These are the coordinates of the initial, undisplaced configuration in the local coordinate system;

[0071] The local displacements of the discrete element nodes are obtained through step 2.1.1. Subsequently, the least squares functional of the discrete unit It can be represented as ;in: , represents the vector composed of the residual between the theoretical value of axial strain and the measured value of axial strain, and the residual between the theoretical value of bending curvature and the measured value of bending curvature; , representing the residual between the theoretical value of axial strain and the measured value of axial strain; , representing the residual between the theoretical value of the bending curvature and the measured value of the bending curvature;

[0072] Through step 2.1.1, the local displacement of the discrete element is determined. The expression is used to calculate the local displacement of the discrete element. For global displacement variation : ,in , representing the rigid body rotation angle vector about the angle of rotation of the rigid body Variations; express unit array; , representing the rigid body rotation angle For global displacement Variations; express When it is 0th Rotation matrix of discrete units The value, the unit Jacobian matrix is ​​expressed as ;

[0073] Step 2.1.3: Calculate the first [number]th [unit] in the global coordinate system. Equivalent internal forces of discrete elements and tangent stiffness matrix Equivalent internal force of the assembly and the overall tangential stiffness matrix And determine the overall equivalent internal force after constraints based on the boundary conditions. and the constrained global tangent stiffness matrix ;in, Indicates the first The initial length of each discrete unit; This represents a vector consisting of the residual between the theoretical and measured values ​​of axial strain and the residual between the theoretical and measured values ​​of bending curvature; Represents the axial coordinates in the local coordinate system; Indicates the first The equivalent internal force vector of each discrete element;

[0074] Step 2.1.4: End the inner loop of assembling the discrete elements of the flexible beam structure as a whole, and calculate the displacement increment of the flexible beam structure. : ;in, Indicates the current iteration step number is Global coordinates of the flexible beam structure at that time;

[0075] Step 2.2: Step size evaluation and parameter update, calculate gain ratio ,in: Indicates the modulus length; The global coordinates of the flexible beam structure are: Overall equivalent internal force after time constraint The modulus length; The global coordinates of the flexible beam structure are: Overall equivalent internal force after time constraint The modulus length; if This indicates that the step size is accepted, and the node coordinates of the flexible beam structure are... and reduce damping Incrementing step counter ;if This indicates that the step size was rejected, and the node coordinates of the flexible beam structure are... and increase the damping parameter Proceed to step 2.1;

[0076] Step 3: Output the node coordinates of the flexible beam structure ;

[0077] This invention proposes an improved large-displacement iFEM method. This method combines the corotation coordinate method with the linear inverse finite element method, transforming the geometrically nonlinear deformation reconstruction problem into a least-squares optimization problem, which is then solved using the Levenberg-Marquardt algorithm. Its core improvement lies in simplifying the variational relationship between local and global displacements. This significantly reduces the computational complexity of the equivalent internal forces and tangential stiffness matrices of discrete elements, thereby achieving efficient reconstruction of the static and dynamic deformations of large-displacement flexible beam structures while maintaining reconstruction accuracy. The rationality, feasibility, and advantages of this invention can be further illustrated by the following simulations;

[0078] The simulation was performed using structural steel with a Young's modulus of 200 GPa and a Poisson's ratio of 0.3. The CPU used for the simulation was an Intel Core i7-1165G7, and the simulation software was Matlab 2023b.

[0079] Figure 2 This is a comparison chart of the Ansys simulation results of a large displacement flexible beam structure under static load and the deformation results obtained by the method proposed in this invention. Figure 2 (a) uses Ansys analysis results as a benchmark to analyze the error in reconstructing the deformation of a large-displacement flexible beam structure using the large-displacement iFEM method; from the enlarged local view, i.e. Figure 2 (b) in the analysis shows that the error in reconstructing the deformation of a large-displacement flexible beam structure using the large-displacement iFEM method is less than 1%.

[0080] Figure 3 This is a comparison chart of the Ansys simulation results of a large displacement flexible beam structure under static load and the deformation results obtained by the method proposed in this invention. Figure 3 (a) compares the accuracy of the large-displacement iFEM method and the improved large-displacement iFEM method in reconstructing the deformation of a large-displacement flexible beam structure; from the enlarged local view, i.e. Figure 3 As can be seen from (b) in the figure, the difference in deformation results of flexible beam structures obtained by the large displacement iFEM method and the improved large displacement iFEM method is very small and can be ignored;

[0081] To further verify the performance of the improved large-displacement iFEM method proposed in this invention, Figure 4 Ansys simulation results of flexible beam structures under different nonlinear deformation levels (LV1-LV5) were compared with the deformation results of large-displacement flexible beam structures reconstructed by the method proposed in this invention. Practice has proven that the improved large-displacement iFEM method can accurately reconstruct the displacement field of large-displacement flexible beam structures. Based on the Ansys analysis results, the relative displacement error in the X-axis direction is 0.010%-0.605%, and the relative displacement error in the Y-axis direction is 0.061%-0.235%.

[0082] Figure 5This is a comparison chart of the deformation results of the improved large-displacement iFEM method for reconstructing a large-displacement flexible beam structure under dynamic load, provided by the embodiments of the present invention, and the Ansys simulation results. Figure 5 (a) in the figure is a comparison of the coordinates of the far-end node in the X direction of the large-displacement flexible beam structure reconstructed by the improved large-displacement iFEM method and the deformation results obtained by Ansys simulation under the time history. Figure 5 (b) in the figure is a comparison between the deformation results of the far-end node in the Y direction of the large displacement flexible beam structure reconstructed by this method and the results obtained by Ansys simulation under the time history.

[0083] In the X-axis direction, the improved large displacement iFEM method has high calculation accuracy with an average relative error of 0.681%; in the Y-axis direction, the average relative error is 0.963%.

[0084] Figure 6 To improve the computational efficiency of the large displacement iFEM method compared to other methods, Figure 6 The convergence speed comparison shows that the improved large displacement iFEM method significantly reduces the number of iteration steps required to solve the deformation of large displacement flexible beam structures, with the number of iteration steps increased by more than 10%.

[0085] To further calculate the solution speed, each algorithm was executed 100 times for each deformation level, and the total time required to find the optimal solution was recorded and displayed. Figure 7 As can be seen, the total calculation time for reconstructing the deformation using the improved large displacement iFEM method is less than 0.15 seconds, and the solution speed is reduced by more than 20% in the case of deformation level LV3-LV5.

[0086] Based on the above, the simulation results obtained by the modeling method proposed in this invention are basically consistent with the large displacement iFEM method and the analysis results of Ansys software. At the same time, the solution speed of the improved large displacement iFEM method is significantly improved compared with the large displacement iFEM method, which proves the efficiency and accuracy of the method proposed in this invention.

[0087] The above embodiments are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.

Claims

1. A rapid deformation and reconstruction method for flexible beam structures with large displacement, characterized in that, The rapid deformation and reconstruction method includes the following steps: Step 1: Finite element discretization and initialization of the flexible beam structure; Step 2: When the overall equivalent internal force of the flexible beam structure and At that time, the main iteration loop is performed, where Indicates tolerance. Represents an iteration counter. This indicates the maximum number of iterations, specifically: Step 2.1: Discretize the flexible beam structure into... For each discrete unit, perform a unit-level loop to traverse the discrete units. =1 to ; Step 2.2: Step size evaluation and parameter update; Step 3: Output the coordinates of the nodes of the flexible beam structure ; Combining the corotation coordinate method (CR) with the iFEM method based on the small deformation assumption, the deformation reconstruction problem under geometric nonlinear conditions is transformed into a square functional optimization problem. The goal is to find the optimal nodal displacement parameters to ensure that the mean square error between theoretical strain and measured strain is minimized. The square functional optimization problem is solved by the LM algorithm. The method for realizing the deformation reconstruction of large displacement flexible beam structure is defined as the large displacement iFEM method, and the improved large displacement iFEM method is obtained by simplification. The improved large-displacement iFEM method ignores... High-order, small-scale simplification strategies enable This simplifies the equivalent internal forces of the unit. and tangent stiffness matrix Rapidly reconstruct the displacement of a large-displacement flexible beam structure to obtain accurate static and dynamic nodal coordinates of the flexible beam structure. Analyze the numerical solution.

2. The rapid deformation and reconstruction method for large-displacement flexible beam structures according to claim 1, characterized in that, Step 1 shown is specifically as follows: Step 1.1: Discretize the flexible beam structure into... The nth discrete unit is used to calculate the nth discrete unit. Length of each discrete unit ; The flexible beam structure in step 1.1 is composed of It consists of discrete elements, each consisting of two nodes; the discrete elements are independent of each other; in the global coordinate system, the position vector of the discrete element before initial deformation is... ,in, Indicates the first Discrete units This indicates the first undeformed section of the flexible beam structure. The coordinates of the first node of a discrete element in the X and Y axes of the global coordinate system; This represents the rotation angle of the first node around the global Z-axis. Indicates the first The coordinates of the second node of each discrete element in the X and Y axes of the global coordinate system. This represents the rotation angle of the second node around the global Z-axis; via position vector Calculate the first Initial length of each discrete unit : （1） Step 1.2: Set the initial damping parameters for the algorithm Initial penalty coefficient , tolerance Maximum number of iterations Iteration counter ,initial Set to 1; Step 1.3: Input the measured strain of the flexible beam structure surface, and calculate the measured strain of the processed discrete element. With measuring curvature , Step 1.4: Define the shape functions of the discrete element, derive the displacement and strain fields, and construct the strain-displacement matrix. , Specifically: Using the principle of shape function interpolation, the axial displacement of any point inside the discrete element can be calculated. and lateral displacement Represented by shape functions and discrete element node degrees of freedom: （4） （5） in, These represent the axial displacement, lateral displacement, and rotation angle of node 1 and node 2 in the local coordinate system, respectively. Quantities with horizontal bars above them indicate quantities based on the local coordinate system. Axial displacement For local coordinates First variation and lateral displacement Axial coordinates in the local coordinate system The second variational expressions are respectively expressed as: （6） （7） in, Representing shape functions A first variation; Representing shape functions A first variation; Representing shape functions The second variation; Representing shape functions The second variation; Representing shape functions The second variation; These are the axial coordinates in the local coordinate system. The axial strain-displacement matrix is ​​constructed based on formulas (6) and (7). and bending strain-displacement matrix ; For the Theoretical value of axial strain for a discrete element and the theoretical value of curvature It can be expressed as the product of the discrete element shape function and the discrete element local displacement: ;in, For the first Local displacement of discrete unit.

3. The rapid deformation and reconstruction method for large-displacement flexible beam structures according to claim 2, characterized in that, In step 1.3, the discrete element measures the strain. With measuring curvature The calculation formula is: （2） （3） in, Indicates the thickness of the flexible beam structure, indicated by the superscript "". "and" "These are used to represent the upper and lower surfaces of a flexible beam structure, respectively." This represents the strain value measured on the surface of the discrete element.

4. The rapid deformation and reconstruction method for large-displacement flexible beam structures according to claim 3, characterized in that, Step 2.1 specifically involves: Step 2.1.1: For the flexible beam structure Each discrete element is described using the co-rotation coordinate method, and the local displacement of the discrete element is obtained from the global displacement of the flexible beam structure. Specifically: Based on current global coordinates The length of the discrete element is calculated using formula (1). ; Calculate the first in generalized coordinates Rotation angle of a discrete unit rigid body ; Get the first Rigid body rotation angle of discrete unit Then, substitute it into the rigid body rotation matrix in the two-dimensional plane. In the general form, calculate the rotation matrix of the discrete unit. for: ; The local displacements of the discrete element nodes are calculated using the co-rotation coordinate method. ;in, These are the coordinates of the initial, undeformed configuration in the local coordinate system. Indicates the first Rigid body rotation angle vectors of discrete elements in global coordinates; Indicates the first after deformation The coordinates of a discrete unit in the global coordinate system; Step 2.1.2: Construct the least squares functional Computing the Jacobian matrix ; The local displacement of the discrete element is obtained through step 2.1.

1. Subsequently, the least squares functional of the discrete unit Represented as in: , represents the vector composed of the residual between the theoretical value of axial strain and the measured value of axial strain, and the residual between the theoretical value of bending curvature and the measured value of bending curvature; This represents the residual between the theoretical value and the measured value of axial strain. This represents the residual between the theoretical value and the measured value of the bending curvature. Through step 2.1.1, the local displacement of the discrete element is determined. The expression is used to calculate the local displacement of the discrete element. For global displacement variation : ,in , representing the rigid body rotation angle vector about the angle of rotation of the rigid body Variations; express unit array; , representing the rigid body rotation angle For global displacement Variations; express When it is 0th Rotation matrix of discrete units The value of is such that the unit Jacobian matrix is ​​expressed as . ; Step 2.1.3: Calculate the first [number]th [unit] in the global coordinate system. Equivalent internal forces of discrete elements and tangent stiffness matrix Equivalent internal force of the assembly and the overall tangential stiffness matrix And determine the overall equivalent internal force after constraints based on the boundary conditions. and the constrained global tangent stiffness matrix ;in: Indicates the first The initial length of each discrete unit; This represents a vector consisting of the residual between the theoretical and measured values ​​of axial strain and the residual between the theoretical and measured values ​​of bending curvature; Represents the axial coordinates in the local coordinate system; Indicates the first The equivalent internal force vector of each discrete element; Step 2.1.4: End the inner loop of assembling the discrete elements of the flexible beam structure as a whole, and calculate the displacement increment of the flexible beam structure. : ;in, Indicates the current iteration step number is The global coordinates of the flexible beam structure at that time.

5. The rapid deformation and reconstruction method for large-displacement flexible beam structures according to claim 4, characterized in that, In step 2.1.1, the first Rotation angle of a discrete unit rigid body The calculation method is as follows: After deformation, the coordinate difference of the nodes in the Y-axis direction of the flexible beam structure is: The displacement of the node in the X-axis direction is ;because Let be the rigid body rotation angle between the line connecting the nodes of the discrete element after displacement and the X-axis of the global coordinate system. Calculate the first Rigid body rotation angle vector of a discrete element in global coordinates ;in, ,and The first The axial and lateral displacements of discrete element nodes 1 and 2 in the global coordinate system; Rigid body rotation angle of discrete unit It can be calculated by the first The angle between the line connecting the two nodes of each discrete unit and the global coordinate X-axis is obtained.

6. The rapid deformation and reconstruction method for large-displacement flexible beam structures according to claim 5, characterized in that, Step 2.2 specifically involves: Calculate the gain ratio ,in: Indicates the modulus length; The global coordinates of the flexible beam structure are: Overall equivalent internal forces after constraints The modulus length; The global coordinates of the flexible beam structure are: Overall equivalent internal force after time constraint The modulus length; if This indicates that the step size is accepted, and the node coordinates of the flexible beam structure are... and reduce damping Incrementing step counter ; if This indicates that the step size was rejected, and the node coordinates of the flexible beam structure are... and increase the damping parameter Proceed to step 2.1.

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