A fast deformation reconstruction method for large displacement flexible beam structure

Abstract

The application discloses a fast deformation reconstruction method for large-displacement flexible beam structure and belongs to the field of structure health monitoring. The application obtains local displacement by removing rigid body displacement from the total displacement of the flexible beam structure based on the co-rotational coordinate method. The local displacement obtained by the co-rotational coordinate method is substituted into the iFEM formula based on the small deformation assumption, and the deformation reconstruction problem under the geometric nonlinear condition is converted into a least square functional optimization problem of finding optimal node displacement parameters to minimize the mean square error of the theoretical strain and the measured strain. An optimization algorithm is adopted to solve the optimal solution of the functional. In addition, a simplification strategy of ignoring high-order small quantities is adopted, so that the whole nonlinear deformation reconstruction method significantly improves the calculation efficiency without sacrificing the calculation precision. The application can quickly solve the node displacement parameters minimizing the functional, and realizes fast and high-precision deformation reconstruction of the large-displacement flexible beam structure.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of structural health monitoring, and relates to a fast deformation reconstruction method for a large-displacement flexible beam structure, in particular to an analysis method for fast reverse reconstruction of a nonlinear deformation displacement field of a beam structure. BACKGROUND

[0002] Displacement monitoring of a beam structure is a process of real-time monitoring of the shape of the beam structure. As one of the key damage features in a structural health monitoring system (SHM), using an efficient and accurate displacement monitoring method to identify and evaluate the deformation state and safety of the structure in real time has become a core technology to ensure the safety and reliability of the structure, and is also an important early warning means to evaluate the failure risk of the structure.

[0003] The existing deformation reconstruction methods based on measured strain can be divided into modal method, Ko displacement theory and inverse finite element method (iFEM). Traditional deformation reconstruction methods mostly focus on small deformation analysis. Among them, the inverse finite element method based on the small deformation assumption is widely used in small deformation reconstruction of various structures due to its excellent precision and applicability and the advantages of not requiring material elastic or inertial parameters.

[0004] With the continuous progress of material science and technology, flexible and complex engineering structures are increasingly widely used in modern engineering. The linear displacement assumption is no longer applicable to systems that bear complex loads and nonlinear responses. Existing experimental studies on the nonlinear displacement behavior of beams show that the deviation between the linear analysis prediction and the measured value in the nonlinear deformation stage is as high as 12.50%, so nonlinear analysis must be introduced when reconstructing the nonlinear displacement of the beam. However, the introduction of geometric nonlinearity greatly increases the solving time, making it difficult to meet the real-time monitoring requirements of the nonlinear displacement of flexible beam structures in actual engineering applications.

[0005] In the prior art, a classic inverse finite element implementation framework is proposed in Chinese invention patent (application number: 202111469982.5), which realizes deformation reconstruction by constructing a weighted least squares functional. However, this method is based on linear theory as a whole, 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 in the nonlinear deformation stage. Chinese invention patent (application number: 202211569574.1) introduces the co-rotation coordinate method into the inverse finite element system, and improves the description ability for large rotation problems by separating rigid body motion and pure deformation. Although this method enhances the geometric nonlinearity adaptability at the theoretical level, the complete co-rotation expression and complex Jacobian derivation lead to a sharp increase in computational complexity, resulting in low solving efficiency, which cannot meet the requirements of real-time monitoring for calculation speed.

[0006] Therefore, there is an urgent need for a large displacement flexible beam structure-oriented fast deformation reconstruction method to predict the deformation of the large displacement flexible beam structure to solve the above technical problems. SUMMARY

[0007] In view of the problems existing in the prior art, the present application provides a large displacement flexible beam structure-oriented fast deformation reconstruction method. This method is based on the co-rotational coordinate method (CR) to obtain the local displacement by removing the rigid body displacement from the overall displacement of the flexible beam structure. The local displacement obtained by the co-rotational coordinate method is substituted into the iFEM formula based on the small deformation assumption, which converts the deformation reconstruction problem under the geometric nonlinear condition into a least square functional optimization problem of finding the optimal node displacement parameter to minimize the mean square error of the theoretical strain and the measured strain. In order to solve the optimal solution of this functional, the present application adopts a more robust optimization algorithm, which only needs to calculate the first-order variation of the local displacement to the global displacement and construct the tangent stiffness matrix, thereby avoiding the complex second-order variation derivation of the traditional optimization method. In order to simplify the calculation of the above first-order variation, the present application adopts the simplification strategy of ignoring high-order small quantities, which significantly improves the calculation efficiency of the entire nonlinear deformation reconstruction method without sacrificing the calculation accuracy. Finally, the node displacement parameter that minimizes the functional can be quickly solved, and the fast and high-precision deformation reconstruction of the large displacement flexible beam structure is realized.

[0008] To achieve this purpose, the present application adopts the following technical solutions:

[0009] A large displacement flexible beam structure-oriented fast deformation reconstruction method, comprising 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 discrete units, calculate the length of the th discrete unit.

[0012] The flexible beam structure in the step 1.1 is composed of discrete units, each discrete unit is composed of two nodes; each discrete unit should be independent of each other, i.e. there is no node belonging to different discrete units. In the global coordinate system, the position vector of the discrete unit before initial deformation is , wherein, represents the th discrete unit represents the coordinates of the first node of the th discrete unit of the flexible beam structure before initial deformation in the global coordinate system X, Y axis direction. represents the rotation angle of the first node around the global coordinate Z-axis (following the right-hand screw rule), represents the rotation angle of the second node around the global coordinate Z-axis (following the right-hand screw rule), represents the coordinates of the second node of the first discrete unit in the global coordinate system X, Y axis direction, represents the rotation angle of the second node around the global coordinate Z-axis (following the right-hand screw rule), represents the initial length of the first discrete unit ,

[0013] (1)

[0014] Step 1.2: Set the initial damping parameters of the algorithm, the initial penalty coefficient , , , tolerance , maximum number of iterations , iteration counter . Among them is 1000, is 2, is 1e-7, is 200, and the initial is 1;

[0015] Step 1.3: Input the measured strain of the flexible beam structure surface, calculate the processed discrete unit measured strain and measured curvature , specifically:

[0016] (2)

[0017] (3)

[0018] wherein represents the thickness of the flexible beam structure, and the superscripts “ ” and “ ” respectively represent the upper surface and the lower surface of the flexible beam structure. represents the strain value measured on the surface of the discrete unit;

[0019] Step 1.4: Define the shape function of the discrete unit, derive the displacement field and strain field, and construct the strain-displacement matrix , ; Specifically:

[0020] Through the principle of shape function interpolation, the axial displacement and transverse displacement of any point inside the discrete unit can be represented by shape functions and discrete unit 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 discrete units, and perform a unit-level loop to iterate through the discrete units = 1 to ;

[0031] Step 2.1.1: Describe the first discrete unit of the flexible beam structure using the co-rotational coordinate method, and separate the local displacement of the discrete unit from the global displacement of the flexible beam structure , specifically:

[0032] According to the current global coordinates , calculate the length of the discrete unit using formula (1);

[0033] Calculate the rigid body rotation angle of the first discrete unit in the generalized coordinates : After the deformation of the flexible beam structure, the coordinate difference of the node in the Y-axis direction is , and the displacement of the node in the X-axis direction is ; Since is the rigid body rotation angle of the connecting line of the nodes on the discrete unit after displacement and the X-axis of the global coordinate system, it follows that , and the rigid body rotation angle vector of the first discrete unit in the global coordinates is calculated ; where , and are the axial displacement and transverse displacement of node 1 and node 2 of the first discrete unit in the global coordinate system, respectively; the rigid body rotation angle of the first discrete unit can be obtained by calculating the angle between the connecting line of nodes 2 of the first discrete unit and the X-axis of the global coordinate system;

[0034] After obtaining the rigid body rotation angle of the first discrete unit, substitute it into the general form of the rigid body rotation matrix in the two-dimensional plane to calculate the rotation matrix of the discrete unit as: ;

[0035] The local displacement of the nodes of the discrete unit can be obtained by co-rotational coordinate calculation ; where is the coordinates in the local coordinate system of the initial undeformed configuration; represents the rigid body rotation angle vector of the first discrete unit in the 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; a vector consisting of the residual of the axial strain theoretical value and the axial strain measured value and the residual of the bending curvature theoretical value and the bending curvature measured value; denotes the axial coordinate in the local coordinate system; denotes the equivalent internal force vector of the th discrete unit;

[0040] Step 2.1.4: end the internal loop of assembling the discrete units of the flexible beam structure into a whole, and calculate the displacement increment of the flexible beam structure ; wherein, denotes the global coordinate of the flexible beam structure at the current iteration step

[0041] Step 2.2: step length evaluation and parameter updating, and calculate the gain ratio , wherein: denotes the module length; denotes the module length of the whole equivalent internal force after constraint when the global coordinate of the flexible beam structure is denotes the module length of the whole equivalent internal force after constraint when the global coordinate of the flexible beam structure is if , it indicates that the step length is accepted, the node coordinate of the flexible beam structure is , and the damping is reduced , and the step counter is incremented ; if , it indicates that the step length is rejected, the node coordinate of the flexible beam structure is , and the damping parameter is increased , and go to step 2.1;

[0042] Step 3: output the node coordinate of the flexible beam structure

[0043] The present application combines the co-rotational coordinate method (CR) and the iFEM method based on the small deformation assumption. Through this combination, we convert the deformation reconstruction problem under the geometric nonlinear condition into a quadratic functional optimization problem. The goal of this problem is to find the optimal node displacement parameter to minimize the sum of squares error between the theoretical strain and the measured strain. This problem is finally solved by the LM algorithm, and the present application will realize the method of large displacement deformation reconstruction of the flexible beam structure, which is defined as the large displacement iFEM method.

[0044] On the basis of the large displacement iFEM method, in order to further simplify the calculation and improve the calculation speed, the improved large displacement iFEM method is proposed. The improved large displacement iFEM method simplifies the calculation by ignoring the high-order small quantity, so that​​​​​​ , and tangent stiffness matrix are simplified to quickly reconstruct the displacement of large displacement flexible beam structure.

[0045] This simplification makes the entire nonlinear deformation reconstruction method significantly improve the calculation speed without sacrificing the calculation accuracy, and can quickly obtain the accurate static and dynamic node coordinates of the flexible beam structure analysis numerical solution.

[0046] The present application has the beneficial effects:

[0047] The present application combines the co-rotation coordinate method with the linear inverse finite element method to construct a weighted least squares functional suitable for large displacement flexible beam structure; in the calculation process of solving the least squares functional to find the optimal node displacement parameter, the Levenberg-Marquardt algorithm is introduced and the calculation of equivalent internal force and tangent stiffness matrix is simplified to realize efficient solution of the optimization problem. When reconstructing the deformation of the large displacement flexible beam structure, the error is less than 2%, the calculation efficiency is improved by 20%, and the calculation time is shortened; it can provide important reference for real-time simulation and digital twin of flexible beam structure which has higher demand for real-time simulation scene. BRIEF DESCRIPTION OF DRAWINGS

[0048] Figure 1 is the shape diagram of the flexible beam structure under dynamic load provided by the embodiment of the present application;

[0049] Figure 2 is a comparison diagram of the deformation reconstruction result of the large displacement flexible beam structure obtained by the large displacement iFEM method and the Ansys simulation result; Figure 2 (a) in (a) of (a) is based on the deformation reconstruction result of the large displacement flexible beam structure obtained by Ansys analysis, and analyzes the accuracy of the large displacement iFEM method in reconstructing the deformation of the large displacement flexible beam structure; Figure 2 (b) in (b) of (a) is Figure 2 (a) A in (a) of (a) is a local enlarged view showing the comparison of the deformation reconstruction result of the large displacement flexible beam structure obtained by the large displacement iFEM method and the Ansys simulation result;

[0050] Figure 3 is a comparison diagram of the deformation reconstruction result; Figure 3 (a) in (a) of (a) is a comparison of the deformation reconstruction result of the large displacement flexible beam structure by the large displacement iFEM method and the improved large displacement iFEM method; Figure 3 (b) in (b) of (a) is Figure 3 (a) A in (a) of (a) is a local enlarged view showing the comparison of the deformation reconstruction result of the large displacement flexible beam structure obtained by the large displacement iFEM method and the improved large displacement iFEM method; ​

[0051] Figure 4 is a comparison chart of several sets of deformation reconstruction results obtained by the improved large displacement iFEM method and Ansys simulation results provided by the embodiment of the present application;

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

[0053] Figure 6 is a convergence speed comparison chart provided by the embodiment of the present application; Figure 6 (a) in (a) is a comparison of the number of iterations required for the improved large displacement iFEM method and the large displacement iFEM method to reconstruct the deformation of the large displacement flexible beam structure under LV1 deformation degree; Figure 6 (b) in (b) is a comparison of the number of iterations required for the improved large displacement iFEM method and the large displacement iFEM method to reconstruct the deformation of the large displacement flexible beam structure under LV2 deformation degree; Figure 6 (c) in (c) is a comparison of the number of iterations required for the improved large displacement iFEM method and the large displacement iFEM method to reconstruct the deformation of the large displacement flexible beam structure under LV3 deformation degree; Figure 6 (d) in (d) is a comparison of the number of iterations required for the improved large displacement iFEM method and the large displacement iFEM method to reconstruct the deformation of the large displacement flexible beam structure under LV4 deformation degree; Figure 6 (e) in (e) is a comparison of the number of iterations required for the improved large displacement iFEM method and the large displacement iFEM method to reconstruct the deformation of the large displacement flexible beam structure under LV5 deformation degree;

[0054] Figure 7 is a comparison chart of the solving speed of the large displacement iFEM method and the improved large displacement iFEM method for reconstructing the deformation of the large displacement flexible beam structure provided by the embodiment of the present application. DETAILED DESCRIPTION

[0055] The present application will be further described in detail below with reference to the accompanying drawings and embodiments. It can be understood that the specific embodiments described herein are only used to explain the present application, but not to limit the present application.

[0056] The present application proposes a fast deformation reconstruction method for large displacement flexible beam structure, 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 units, and calculate the length of the th discrete unit :

[0059] The flexible beam structure in the step 1.1 is composed of discrete units, each of which is composed of two nodes; each discrete unit should be independent of each other, i.e. there is no node belonging to different discrete units at the same time. In the global coordinate system, the position vector of the discrete unit in the initial undeformed configuration is , and the initial length of the th discrete unit is calculated ;

[0060] Step 1.2: Set the initial damping parameters of the algorithm , the initial penalty coefficient , tolerance , the maximum number of iterations , and the iteration counter . Among them is 1000, is 2, is 1E-7, is 200, and the initial is 1;

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

[0062] Step 1.4: Define the shape function of the discrete unit, derive the displacement field and strain field, and construct the strain-displacement matrix , ;

[0063] As shown in Figure 1 , a reverse beam unit, the continuous, uniform and isotropic material is used to form the two-node reverse beam unit iBeam3. As shown in Figure 1 , the beam unit contains two nodes, each node has 3 degrees of freedom of displacement. The node degrees of freedom of the discrete unit are composed of , , and , =1,2. Among them represent the translation freedom along the x and y axes, 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 unit 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, This indicates that the current iteration step 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] The application provides an improved large-displacement iFEM method. , the calculation complexity of equivalent internal forces and tangent stiffness matrix of discrete units is significantly reduced, so that the large-displacement flexible beam structure static and dynamic deformation is reconstructed efficiently while the reconstruction accuracy is maintained.

[0078] The structural steel with a Young's modulus of 200 GPa and a Poisson's ratio of 0.3 is simulated.

[0079] Figure 2 is a comparison diagram of the Ansys simulation result of the large-displacement flexible beam structure under static load and the deformation result obtained by the method. Figure 2 In (a) of (a), the error of the large-displacement iFEM method in reconstructing the deformation of the large-displacement flexible beam structure is analyzed based on the Ansys analysis result. Figure 2 In (b) of (a), it can be seen that the error of the large-displacement iFEM method in reconstructing the deformation of the large-displacement flexible beam structure is less than 1%.

[0080] Figure 3 is a comparison diagram of the Ansys simulation result of the large-displacement flexible beam structure under static load and the deformation result obtained by the method. Figure 3 In (a) of (a), the accuracy of the large-displacement iFEM method and the improved large-displacement iFEM method in reconstructing the deformation of the large-displacement flexible beam structure is compared. Figure 3 In (b) of (a), it can be seen that the deformation results of the flexible beam structure obtained by the large-displacement iFEM method and the improved large-displacement iFEM method are extremely small and can be ignored.

[0081] To further verify the performance of the improved large-displacement iFEM method proposed in the application, Figure 4 The Ansys simulation results of the flexible beam structure under different nonlinear deformation degrees (LV1-LV5) are compared with the deformation results of the large-displacement flexible beam structure reconstructed by the method.

[0082] Figure 5is a comparison chart of the deformation results of the large displacement flexible beam structure reconstructed by the improved large displacement iFEM method and the Ansys simulation results under dynamic load according to the method provided in the embodiments of the present application; Figure 5 (a) in the comparison chart is a comparison chart of the coordinates of the far-end node of the large displacement flexible beam structure in the X direction reconstructed by the improved large displacement iFEM method and the deformation results obtained by the Ansys simulation under the time history; Figure 5 (b) in the comparison chart is a comparison chart of the deformation results of the far-end node of the large displacement flexible beam structure in the Y direction reconstructed by the method and the results obtained by the Ansys simulation under the time history;

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

[0084] Figure 6 For the calculation efficiency of the improved large displacement iFEM method relative to the large displacement iFEM method, Figure 6 The convergence speed comparison chart shows that the improved large displacement iFEM method significantly reduces the number of iteration steps required to solve the deformation of the large displacement flexible beam structure, and the number of iteration steps is increased by more than 10%.

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

[0086] According to the above, the simulation results obtained by the modeling method proposed in the present application are basically consistent with the analysis results of the large displacement iFEM method and the Ansys software, and the solving speed of the improved large displacement iFEM method is obviously improved relative to the large displacement iFEM method, which proves the high efficiency and accuracy of the method proposed in the present application.

[0087] The above-described embodiments only express the implementation of the present application, but cannot be interpreted as a limitation on the scope of the present application, and it should be pointed out that, for those skilled in the art, without departing from the concept of the present application, a number of modifications and improvements can be made, which all belong to the protection scope of the present application.

Claims

1. A fast deformation reconstruction method for large displacement flexible beam structure, characterized in that, The rapid deformation reconstruction method comprises the following steps: Step 1: finite element discretization and initialization of the flexible beam structure; Step 2: When the flexible beam structure is in overall equivalent internal force and a main iteration cycle is performed, wherein represents a tolerance, represents an iteration counter, represents the maximum number of iterations, in particular: Step 2.1: Discretize the flexible beam structure into discrete units, perform a unit-level loop, iterate through the discrete units = 1 to ; Step 2.2: step evaluation and parameter updating; Step 3: Output flexible beam structure node coordinates ; In combination with the co-rotational coordinate method CR and the iFEM method based on the small deformation assumption; the deformation reconstruction problem under the geometric nonlinear condition is converted into a quadratic functional optimization problem, and the optimal node displacement parameter is sought to ensure that the sum of squares errors between the theoretical strain and the measured strain reaches the minimum; The quadratic functional optimization problem is solved by the LM algorithm, and the method for realizing the deformation reconstruction of the 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 a high-order small quantity simplification strategy, further simplifies the cell equivalent internal force and tangent stiffness matrix quickly reconstructs the displacement of the large-displacement flexible beam structure, and obtains the accurate static and dynamic node coordinates of the flexible beam structure analyzes the numerical solution.

2. The fast reconfiguration method for large displacement compliant beam structure according to claim 1, wherein, The specific step 1 is: Step 1.1: Discretize the flexible beam structure into N discrete units, calculate the length of the Nth discrete unit Step 1.2: Calculate the length of the Nth discrete unit Step 1.3: Calculate the length of the Nth 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; By position vector Computing the initial length of the discrete element : （1）； Step 1.2: Set initial damping parameter of 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, calculate the processed discrete element measured strain With the measured curvature , Step 1.4: Define the shape function of the discrete element, derive the displacement field and strain field, and construct the strain-displacement matrix , , in particular: The axial displacement and lateral displacement of any point inside the discrete element are expressed by the shape function and the freedom of the node of the discrete element and lateral displacement of any point inside the discrete element are expressed by the shape function and the freedom of the node of the discrete element （4）； （5）； wherein, are the axial displacement, lateral displacement and rotation angle of node 1 and node 2 in the local coordinate system, respectively, the quantities with horizontal bars represent quantities based on the local coordinate system; axial displacement the first variation of the axial coordinate with respect to the local coordinate system the second variation of the axial coordinate with respect to the local coordinate system （6）； （7）； wherein represents a first variation of the shape function ; represents a first variation of the shape function ; represents a second variation of the shape function ; represents a second variation of the shape function ; represents a second variation of the shape function ; is an axial coordinate in a local coordinate system; Based on equation (6) and equation (7), the axial strain-displacement matrix and the 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 fast reconfiguration method for large displacement compliant beam structure according to claim 2, wherein, In step 1.3, the discrete unit measures the strain With the measured curvature The calculation formula is: （2）； （3）； wherein denotes the thickness of the flexible beam structure, the superscript " and " are used to denote the upper and lower surfaces of the flexible beam structure, respectively; denotes the strain value measured at the surface of the discrete unit.

4. The fast reconfiguration method for large displacement compliant beam structure according to claim 3, wherein, The specific step 2.1 is: Step 2.1.1: The local displacement of the discrete unit of the flexible beam structure is separated from the global displacement of the flexible beam structure by using the co-rotation coordinate method , which is specifically: ​ According to the current global coordinates , the discrete unit length is calculated by formula (1) ; calculating the first discrete unit rigid body rotation angle of the generalized coordinate under the first discrete unit ; The rigid body rotation angle of the discrete element is obtained The rigid body rotation matrix of the discrete element is calculated The rigid body rotation matrix of the discrete element is calculated The rigid body rotation matrix of the discrete element is calculated is: ; Local displacements of the discrete element nodes are calculated from the corotational method ; wherein are the coordinates of the initial undeformed configuration in the local coordinate system; denotes the rigid body rotation angle vector of the th discrete element in the global coordinate system; denotes the coordinates of the deformed th discrete element in the global coordinate system; Step 2.1.2: Constructing the least squares functional with the Jacobian matrix ; Local displacement of discrete elements by step 2.1.1 Afterwards, the least-squares functional of the discrete elements is represented as where: represents the vector of the residuals of the axial strain theoretical values and the axial strain measured values and the residuals of the bending curvature theoretical values and the bending curvature measured values; represents the residuals of the axial strain theoretical values and the axial strain measured values; represents the residuals of the bending curvature theoretical values and the bending curvature measured values; 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 rotation angle 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 units into a whole for the flexible beam structure, calculate the displacement increment of the flexible beam structure : ; wherein, denotes the global coordinates of the flexible beam structure at the current iteration step .

5. The fast reconfiguration method for large displacement compliant beam structure according to claim 4, wherein, The calculation of the rotation angle of the discrete unit rigid body in step 2.1.1 is as follows: The calculation of the rotation angle of the discrete unit rigid body in step 2.1.1 is as follows:​ The coordinate difference of the node in the Y-axis direction after the deformation of the flexible beam structure is The displacement of the node in the X-axis direction is ; since is the rigid body rotation angle of the connecting line between the nodes after displacement on the discrete unit and the X-axis of the global coordinate system, therefore , the rigid body rotation angle vector of the first discrete unit in the global coordinate is calculated ; wherein, , and are the axial displacement and the transverse displacement of the first discrete unit node 1 and node 2 in the global coordinate system; the rigid body rotation angle of the first discrete unit is obtained by calculating the angle between the connecting line between the nodes of the first discrete unit 2 and the X-axis of the global coordinate.

6. The fast reconfiguration method for large displacement compliant beam structure according to claim 5, wherein, The specific step 2.2 is: Computing the gain ratio wherein: denotes the module length; denotes the global coordinate of the flexible beam structure the overall equivalent internal force after constraint the module length of the overall equivalent internal force after constraint denotes the global coordinate of the flexible beam structure the overall equivalent internal force after constraint the module length of the overall equivalent internal force after constraint If , the node coordinates of the flexible beam structure are updated , and the damping is reduced ; If , the node coordinates of the flexible beam structure are updated , and the damping parameter is increased, and go to step 2.1.

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