Reconstruction method of deformation shape of plate and shell structure based on inverse finite element

Abstract

The application provides a plate shell structure deformation shape reconstruction method based on inverse finite elements, the core of which is an optimized shear deformation reconstruction algorithm. In the calculation process, a shear correction coefficient does not need to be introduced, the calculation precision is greatly improved, and the application range of the inverse finite element method is effectively expanded.

Description

Technical Field

[0001] This invention belongs to the technical fields of structural health monitoring, structural mechanics analysis, and structural deformation shape reconstruction. Specifically, it relates to an improved inverse finite element method for reconstructing the deformation shape of plate and shell structures, and is an improved inverse finite element method applicable to composite laminates, sandwich panels, and other structures. Background Technology

[0002] Plate and shell structures are widely used in engineering. With the rapid development of aerospace technology, composite plate and shell structures, honeycomb sandwich laminated plate and shell structures, and other high-end equipment have become core structural forms, playing an irreplaceable role. To ensure the service safety and efficiency of these structures, monitoring and evaluating parameters such as structural deformation and stress levels have become crucial technical means. Numerous researchers have conducted extensive studies and proposed a large number of shape reconstruction algorithms, including KO theory, modal transformation methods, and inverse finite element methods (iFEM). Among these, the inverse finite element method is not limited by the geometric complexity of the structure; it can accurately reconstruct the three-dimensional deformation shape of the structure using only discrete strain data measured on the structural surface, attracting widespread attention in areas such as structural deformation shape monitoring and damage identification.

[0003] In plate and shell structures, in-plane membrane strain and flexural curvature can be calculated by simultaneously placing strain sensors on the upper and lower surfaces of the structure. However, transverse shear strain cannot be directly calculated from in-plane strain measurements. For thin-walled structures with relatively small transverse shear deformation, transverse shear strain can be safely ignored. For medium-thickness laminates, sandwich panels, and other structures, ignoring transverse shear deformation leads to a significantly lower reconstructed transverse displacement. According to the equilibrium equations of first-order shear deformation theory, the transverse shear stress on this structure can be calculated using the first derivative of the bending moment. Currently, researchers have solved for the transverse shear strain of the structure based on the above equations, but an additional shear correction factor needs to be introduced during the calculation to ensure accuracy. Although researchers have proposed many formulas to calculate the shear correction factor for specific structures, their solution accuracy is affected by parameters such as the structure's geometry, material properties, laying method, and boundary conditions, further reducing the reconstruction accuracy of the inverse finite element method. Summary of the Invention

[0004] The present invention aims to solve at least one of the above-mentioned technical problems and provide an improved method for reconstructing the shape of a plate and shell structure.

[0005] Some embodiments of the present invention propose a method for reconstructing the deformation shape of a plate and shell structure based on inverse finite element method, specifically including the following steps: selecting appropriate elements to discretize the plate and shell structure; deploying a strain sensing network on the surface of the plate and shell structure to obtain discrete strain data; and calculating the strain at the mid-surface of the discrete structure based on the strain data. and curvature Using Smoothing Element Analysis (SEA) technology, the strain of the mid-surface of the discrete structure is calculated. and curvature Transform into a continuous function e ε κ ε And calculate their partial derivatives respectively. Based on the first-order shear deformation theory and constitutive equation, using the continuous function e ε κ ε , and its partial derivatives Calculate the distribution of in-plane stress σ along the thickness direction; based on the three-dimensional stress equilibrium equation, calculate the actual transverse shear stress using the distribution of in-plane stress along the thickness direction. Distribution along the thickness direction; based on the constitutive equation, using the actual transverse shear stress. Calculation of actual transverse shear strain distribution along the thickness direction Distribution along the thickness direction; based on the energy balance equation, using the actual transverse shear strain. The average transverse shear strain input γ is calculated based on the distribution along the thickness direction. xz γ yz ; and the calculated mid-surface strain e ε , curvature κ ε and transverse shear strain (γ) xz γ yz As the inverse finite element strain input, the shape reconstruction of the plate and shell structure is realized.

[0006] In some embodiments, for solution domains with complex and irregular geometry, triangular inverse shell elements such as iMiN3 are typically chosen for discretization; while for structures with regular shapes, quadrilateral inverse shell elements such as iQS4 are chosen for discretization.

[0007] In some embodiments, strain sensors are installed on both the upper and lower surfaces of the structure to decouple the neutral membrane strain from the bending curvature.

[0008] In some embodiments, for structures exhibiting symmetrical or antisymmetrical deformation trends, strain sensors are arranged only on one side.

[0009] In some embodiments, the strain of the mid-surface of the structure shown and curvature It is obtained through simple numerical calculations.

[0010] In some embodiments, the SEA method is a method for smoothing scalars, therefore the strain on the mid-surface needs to be considered. and curvature Each component is calculated separately, and the continuous function e of membrane strain and bending curvature is finally obtained. ε κ ε And calculate their partial derivatives respectively.

[0011] In some embodiments, the strain at any point in the structure is first solved, and then the stress at any point in the structure is solved based on the constitutive equation.

[0012] In some embodiments, calculating the distribution of the actual transverse shear strain of a structure along the thickness direction based on the three-dimensional stress balance equation specifically includes the following steps: ignoring body forces and simplifying the first two terms of the three-dimensional stress balance equation;

[0013] The distribution of structural shear stress along the thickness direction of the plate and shell is obtained by piecewise integration of the first two terms of the simplified three-dimensional stress balance equation.

[0014] Based on the distribution of structural shear stress along the thickness direction of the plate and shell, the relationship between transverse shear stress τ(z) and membrane strain and bending curvature of the mid-surface is derived.

[0015] In some embodiments, the transverse shear stress τ is derived based on the distribution of structural shear stress along the thickness direction of the plate or shell, and the transverse shear strain is derived from this.

[0016] In some embodiments, the average transverse shear strain input calculated based on the energy equation includes...

[0017] Derivation of transverse shear strain energy U per unit length α Based on the transverse shear strain energy U per unit length α Solving for the average value of the transverse shear strain in the thickness direction in some sense. That is, the transverse shear strain input required by the inverse finite element method.

[0018] The present invention has the following beneficial effects:

[0019] The present invention proposes a method for reconstructing the deformation shape of plate and shell structures based on inverse finite element method. Some embodiments of this method only require the placement of strain sensors on the upper and lower surfaces of the structure to achieve deformation reconstruction of the plate and shell structure. The calculation process does not require the introduction of shear correction coefficients, which greatly improves the calculation accuracy and effectively expands the application scope of the inverse finite element method.

[0020] The method of this invention can accurately reconstruct the deformation of a structure simply by measuring the strain on the outer surface of the shell structure. It is simple to operate and easy to implement in the field. It provides technical support for condition monitoring of laminated and sandwich panel structures in aerospace, automotive, marine, transportation, petrochemical and other fields, and has the value of wide application.

[0021] The core of the plate and shell structure deformation shape reconstruction method based on inverse finite element method in some embodiments of this application is a shear deformation reconstruction algorithm, and the shape reconstruction of the plate and shell structure is realized based on the inverse finite element method. The method in some embodiments of this application establishes the relationship between transverse shear strain and in-plane strain measurements based on three-dimensional equilibrium equations, and completes the solution of transverse shear strain based on the measured surface strain. Finally, the method in some embodiments of this application uses the obtained transverse shear strain, mid-surface strain, and bending curvature as calculation inputs for the inverse finite element method, providing a general solution for the shape reconstruction of plate and shell structures. Attached Figure Description

[0022] Figure 1 A flowchart of a method for reshaping the shape of a plate-shell structure according to an embodiment of this application;

[0023] Figure 2 This is a schematic diagram of the geometry of a cantilever laminate according to one embodiment of this application.

[0024] Figure 3A This is a schematic diagram of the strain sensor arrangement on a cantilever laminate according to one embodiment of this application.

[0025] Figure 3B Figure 3 is a partial enlarged view showing the angular relationship between multiple strain sensors at the same location.

[0026] Figure 4 The graph shows a comparison of the lateral displacement results of the AB midline position reconstructed by the improved inverse finite element method of this example, with the curves of the lateral displacement results of the AB midline position reconstructed by the traditional inverse finite element method and the finite element calculation results. Detailed Implementation

[0027] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0028] The method in this application is particularly applicable to structures where transverse shear deformation is not negligible, such as non-thin-walled shell structures.

[0029] The following section uses a cantilever laminate as an example to describe in detail the inverse finite element method for reconstructing the deformation shape of a plate and shell structure. One such cantilever laminate in the embodiments of this application is... Figure 2 As shown, its length a and width b are 250mm and 125mm respectively. The width-to-thickness ratio of the cantilever laminate is b / t = 5. The ply angle of each single layer is [0 / 90 / 90 / 0], and the normalized single layer thickness is [0.125 / 0.125 / 0.125 / 0.125]. The material parameters of the single layer are shown in Table 1.

[0030] Table 1. Material properties of composite single-layer plates

[0031]

[0032] like Figure 1 As shown, the plate and shell structure deformation shape reconstruction method based on inverse finite element method provided by the embodiments of the present invention includes the following steps:

[0033] Step 1: Based on the above geometry and material properties of the cantilever laminate, a quadrilateral inverse shell element such as iQS4 can be selected for discretization.

[0034] Step 2: Deploy a strain sensing network on the surface of the structure to obtain discrete strain data;

[0035] In specific implementation, the arrangement of strain sensors on the cantilever laminate structure is as follows: Figure 3A , 3B As shown, multiple strain sensors are arranged at multiple strain measurement points on the upper and lower surfaces of the cantilever laminate structure. Each group of strain sensors includes three strain sensors arranged at an angle, such as three strain sensors spaced at 45-degree intervals, to acquire discrete strain data.

[0036] Step 3: Calculate the strain on the mid-surface of the discrete structure based on the discrete strain data obtained in Step 2. and curvature

[0037] In a specific implementation, as a preferred embodiment of the present invention, in step 3, the strain of the mid-surface of the structure shown is... and curvature It can be obtained through simple numerical calculations:

[0038]

[0039]

[0040] In the formula, the superscripts "+" and "-" represent the upper and lower surfaces of the structure, respectively; the superscript ε indicates that there is a measurement error in the obtained strain data; the subscripts i and n represent the location of the strain measurement point and the total number of sensors deployed within the unit, respectively. x and ε y These represent linear strains along the x and y directions, respectively, with γ representing the shear strain in the xy plane. xy express;

[0041] Step 4: Smoothing Element Analysis (SEA) technique is used to analyze the strain of the discrete mid-surface obtained in Step 3. and curvature Transform into a continuous function e ε κ ε And calculate their partial derivatives respectively.

[0042] In a specific implementation, as a preferred embodiment of the present invention, step 4 requires checking the strain of the mid-surface. and curvature Each component is calculated separately, ultimately yielding the first-order partial derivatives of the membrane strain and bending curvature. (Continuous function e) ε κ ε And calculate their partial derivatives respectively.

[0043] Step 5: Based on the first-order shear deformation theory and constitutive equation, calculate the distribution of stress σ along the thickness direction within the structural plane.

[0044] For example, step 5 may specifically include the following steps:

[0045] Step 5.1, according to the first-order shear deformation theory, the strain at any point in the structure can be expressed as:

[0046] ε=e ε +zκ ε , ε=[ε x ε y γ xy ] T (3)

[0047] Step 5.2, based on the constitutive equation, the stress at any point of the structure can be expressed as:

[0048] σ = Qε, σ = [σ x σ y τ xy (4)

[0049] Where Q is the stiffness matrix of the plate shell.

[0050] Subsequently, in step 6, the actual transverse shear stress is calculated based on the three-dimensional stress equilibrium equation. Distribution along the thickness direction;

[0051] Step 6 may specifically include the following steps:

[0052] Step 6.1, neglecting body forces, the first two terms of the three-dimensional stress equilibrium equation can be expressed as:

[0053] σ x，x +τ xy,y +τ xz，z =0; σ y,y +τ xy,x +τ yz,z =0 (5)

[0054] Step 6.2, the distribution of structural shear stress along the thickness of the plate and shell can be obtained by piecewise integration of equation (5):

[0055]

[0056] Where c1 and c2 are integration constants, which can be uniquely determined by the boundary conditions of the upper and lower surfaces; τ xz τ is the shear stress acting in the plane perpendicular to the x-axis and parallel to the z-axis. yz This is the shear stress acting on the plane perpendicular to the x-axis and parallel to the z-axis.

[0057] Step 6.3, the relationship between the transverse shear stress τ(z) and the membrane strain and curvature of the mid-surface in the comprehensive equations (3)-(6) can be expressed as:

[0058]

[0059] In the formula, Represents a Boolean matrix, (), α This represents the derivative with respect to x and y.

[0060]

[0061] Step 7: Calculate the actual transverse shear strain based on the constitutive equation. Distribution along the thickness direction;

[0062] In step 7, the transverse shear strain can be expressed as:

[0063] γ αz =Cτ αz (9)

[0064] In the formula, C is the compliance matrix of the plate shell.

[0065] Step 8: Calculate the average transverse shear strain input γ based on the energy balance equation. xz γ yz ;

[0066] Step 8 may specifically include the following steps:

[0067] Step 8.1, derive the transverse shear strain energy U per unit length according to formula (9). α Transverse shear strain energy U α (U xz U yz This can be expressed as:

[0068]

[0069]

[0070] in, This represents the average value of the transverse shear strain in the thickness direction, which is the transverse shear strain input required for inverse finite element method.

[0071] Step 8.2, calculate the average transverse shear strain according to formula (10):

[0072]

[0073] Finally, following step 9, the calculated mid-surface strain e is... ε , curvature κ ε and transverse shear strain As an inverse finite element strain input, the shape reconstruction of the plate and shell structure is realized.

[0074] In Figure 3, the solid line represents the reference solution for the lateral displacement, i.e., the deflection, of the centerline of the cantilever laminate AB obtained by the finite element method. The dotted line away from the solid line represents the deflection solution obtained by the applicant using the original iFEM method. The dashed line near the solid line represents the deflection solution obtained by the applicant using the iFEM method in this application. It can be seen that the method proposed in this application obtains a more accurate deflection.

[0075] The prior description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the invention. Various modifications to these embodiments will be apparent to those skilled in the art, and the general principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not intended to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

[0076] Of course, the above description is not intended to limit the present invention, and the present invention is not limited to the examples given above. Any changes, modifications, additions or substitutions made by those skilled in the art within the scope of the present invention should also fall within the protection scope of the present invention.

Claims

1. A method for reconstructing the deformation shape of a plate and shell structure based on inverse finite element method, characterized in that: Including the following steps: Based on the properties of the plate and shell structure, select appropriate elements to discretize the plate and shell structure; A strain sensing network is deployed on the surface of the plate shell structure to obtain discrete strain data; The strain on the mid-surface of the discrete structure is calculated based on the measured discrete strain data. and curvature ; The strain of the mid-surface of the discrete structure is obtained using the smooth element analysis method. and curvature Transform into a continuous function , And calculate their partial derivatives respectively. , ,in ; Based on the first-order shear deformation theory and constitutive equations, using the aforementioned continuous function , , and its partial derivatives , Calculate the internal stress of the structural plane Distribution along the thickness direction; Based on the three-dimensional stress equilibrium equation, the actual transverse shear stress is calculated using the distribution of in-plane stress along the thickness direction. , Distribution along the thickness direction; Based on the constitutive equation, the actual transverse shear stress is used. , Calculation of actual transverse shear strain distribution along the thickness direction , Distribution along the thickness direction; Based on the energy balance equation, the actual transverse shear strain is used. , The average transverse shear strain input is calculated based on the distribution along the thickness direction. , ; as well as The calculated mid-surface strain , curvature and transverse shear strain ( , () is used as the inverse finite element strain input to reconstruct the deformed shape of the plate and shell structure.

2. The method for reconstructing the deformation shape of a plate and shell structure based on inverse finite element method according to claim 1, characterized in that: For solution domains with complex and irregular geometry, triangular inverse shell elements iMiN3 are selected to discretize the plate and shell structure; while for plate and shell structures with regular shapes, quadrilateral inverse shell elements are selected for discretization.

3. The method for reconstructing the deformation shape of a plate and shell structure based on inverse finite element method according to claim 2, characterized in that: The triangular inverse shell unit is iMiN3; the quadrilateral inverse shell unit is iQS4.

4. The method for reconstructing the deformation shape of a plate and shell structure based on inverse finite element method according to claim 1, characterized in that: The method of deploying a strain sensing network on the surface of the plate and shell structure to obtain discrete strain data includes: simultaneously installing strain sensors on the upper and lower surfaces of the plate and shell structure to decouple the neutral membrane strain from the bending curvature; or for structures exhibiting symmetrical or antisymmetrical deformation trends, arranging strain sensors only on one side.

5. The method for reconstructing the deformation shape of a plate and shell structure based on inverse finite element method according to claim 1, characterized in that: Strain on the mid-surface of the plate shell structure and curvature We obtained this through simple numerical calculations: ； ； In this context, the superscripts "+" and "-" represent the upper and lower surfaces of the structure, respectively; the superscript... This indicates that the strain data obtained from the measurement contains measurement errors; the subscripts i and n represent the location of the strain measurement point and the total number of sensors deployed within the unit, respectively. and These represent linear strain along the x and y directions, respectively. Used to represent shear strain in the xy plane.

6. The method for reconstructing the deformation shape of a plate and shell structure based on inverse finite element method according to claim 1, characterized in that: The smoothing element analysis method includes strain analysis of the mid-surface. and curvature Each component is calculated separately, ultimately yielding a continuous function of membrane strain and bending curvature. , And calculate their partial derivatives respectively. , , .

7. The method for reconstructing the deformation shape of a plate and shell structure based on inverse finite element method according to claim 1, characterized in that: Calculate the internal stress of the structural plane The distribution along the thickness direction includes: Derivation of strain at any point in the structure ; And based on the constitutive equation, derive the stress at any point in the structure. : ; in, Let be the stiffness matrix of the plate and shell structure.

8. The method for reconstructing the deformation shape of a plate and shell structure based on inverse finite element method according to claim 1, characterized in that: The calculation of actual transverse shear stress , The distribution along the thickness direction includes the following steps: Ignoring body forces, the first two terms of the three-dimensional stress balance equations are expressed as follows: ； The distribution of shear stress along the thickness direction of the plate and shell structure is obtained by piecewise integration of the first two terms of the three-dimensional stress balance equation: ; in, and It is the integration constant, which can be uniquely determined by the boundary conditions of the upper and lower surfaces; The shear stress acts on the plane perpendicular to the x-axis and parallel to the z-axis. The shear stress acts on the plane perpendicular to the x-axis and parallel to the z-axis; The transverse shear stress is obtained based on the stress at any point, the strain at any point, the first two terms of the stress balance equation, and the distribution of shear stress along the thickness direction of the plate and shell structure. The relationship between the membrane strain and the curvature of the mid-surface is as follows: ； In the formula, Represents a Boolean matrix. This represents the derivative with respect to x and y; 。 9. The method for reconstructing the deformation shape of a plate and shell structure based on inverse finite element method according to claim 1, characterized in that: The transverse shear strain is expressed as: ; In the formula, C is the flexibility matrix of the plate shell.

10. The method for reconstructing the deformation shape of a plate and shell structure based on inverse finite element method according to claim 1, characterized in that: Calculate the average transverse shear strain input , The specific steps include: Derivation of transverse shear strain energy per unit length , ,in: ； ； in, , The transverse shear strain input required for inverse finite element method represents the average value of the transverse shear strain in the thickness direction in some sense. The average transverse shear strain is obtained by transformation: ； 。

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