Inversion Method for Surface Displacement Field of Large Thick-Walled Solid Structures Based on Virtual Plying

Abstract

This invention discloses a method for inverting the displacement field of a thick-walled solid structure based on virtual plying, relating to the field of displacement monitoring and inversion technology for thick-walled solid structures. The steps include: embedding a geometrically attached and mechanically isolated virtual ply on the outer surface of the thick-walled solid structure; collecting surface strain as virtual ply membrane strain by arranging measuring points on one side of the virtual ply neutral plane; constructing a weighted least squares functional containing a membrane strain fitting term and bending and shear strain regularization terms; performing variational decomposition of the functional to obtain the virtual ply nodal displacements; and mapping the virtual ply translational displacements to the corresponding nodes of the solid structure to achieve displacement field inversion. This invention solves the problem of high-precision displacement inversion of thick-walled solid structures with single-sided point arrangement without interfering with structural stress, improves the numerical stability of the inversion, adapts to solid element models, is applicable to structures such as pressure vessels, and can be used for structural health assessment.

Description

Technical Field

[0001] This invention relates to the field of displacement monitoring and inversion technology for thick-walled solid structures, specifically to a method for inverting the displacement field of ply surfaces in thick-walled solid structures based on virtual plying. Background Technology

[0002] In aerospace, energy equipment, and high-end manufacturing, large, thick-walled solid structures endure complex loads such as internal pressure during long-term service. Accurate monitoring of their structural deformation is crucial for safety assessment. The inverse finite element method (IFF), a mainstream displacement field inversion technique, reconstructs the displacement field using strain measurement data from the structural surface. However, traditional IFF methods typically build inversion models based on shell element theory, requiring strain measurement points to be placed on both the upper and lower surfaces of the monitored structure to separate membrane strain and bending strain components. In practical engineering, limitations such as structural enclosure, installation space, and operating environment often limit sensor placement to only one side of the outer wall. This makes traditional methods difficult to apply directly due to the lack of bending strain information. Furthermore, the shell element model differs fundamentally from the volume element model of thick-walled solid structures in terms of degrees of freedom and strain distribution assumptions; forcibly applying the shell element model can easily lead to boundary discontinuities and inversion distortion.

[0003] To overcome the limitations of single-sided measurement point placement, existing technologies have proposed various improvement schemes. For example, some current technologies introduce virtual materials within the unit to optimize design variables. However, these virtual materials are mainly used for design boundary changes in structural shape and topology optimization, and do not involve displacement field inversion based on strain measurement, nor do they solve the problem of displacement field reconstruction of solid structures with single-sided measurement point placement. Furthermore, some current technologies generate solid plies for composite material analysis. These plies are actual solid unit layers participating in structural stress, aiming to accurately simulate the mechanical behavior of composite materials rather than for displacement inversion. They also lack mechanical isolation properties and cannot avoid interfering with the original structural stress characteristics. These existing technologies have not effectively solved the core problem of how to achieve high-precision displacement field inversion in large, thick-walled solid structures with only single-sided measurement point placement, while ensuring that the inversion process does not interfere with the actual structural stress characteristics. Therefore, the core technical problem to be solved by this invention is: how to achieve high-precision displacement field inversion in large, thick-walled solid structures with only single-sided measurement point placement, while not interfering with the actual structural stress characteristics. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of existing technologies and provide a method for inverting the displacement field of the ply surface of a large thick-walled solid structure based on virtual plying. By constructing a virtual ply that is geometrically attached to, mechanically isolated from, and numerically coupled with the outer surface of the solid structure as a dedicated calculation medium for displacement inversion, this method effectively overcomes the dependence of traditional inverse finite element methods on two-sided measurement points and avoids interference with the stress characteristics of the solid structure during the inversion process. This provides a high-precision and high-stability technical means for the health monitoring of large thick-walled solid structures.

[0005] To solve the above-mentioned technical problems, the present invention provides the following technical solution: a method for inverting the displacement field of a ply surface in a large thick-walled solid structure based on virtual plying, comprising the following steps:

[0006] Step 1: Embed virtual plywood on the outer surface of the finite element model of the thick-walled solid structure. The virtual plywood is modeled using shell elements. The nodes of the virtual plywood correspond one-to-one with the nodes of the outer surface of the thick-walled solid structure in terms of geometric position, thereby achieving geometric attachment.

[0007] Step 2: Set the material parameters of the virtual layup so that its material stiffness is much lower than that of the thick-walled solid structure, thereby achieving mechanical isolation;

[0008] Step 3: Arrange strain measurement points on the neutral surface of the virtual layup, collect strain data on the surface of the thick-walled solid structure layup, and use the strain data as the membrane strain of the virtual layup;

[0009] Step 4: Based on the inverse finite element theory, construct a weighted least squares functional, which includes a membrane strain term, a bending strain regularization term, and a shear strain regularization term.

[0010] Step 5: Perform variational solution on the weighted least squares functional, assemble it into a global linear equation system, and solve for the nodal displacement field of the virtual ply;

[0011] Step 6: Transfer the translational displacement components of the virtual ply nodes to the corresponding nodes on the outer surface of the thick-walled solid structure through the node mapping relationship to realize displacement field inversion.

[0012] Furthermore, the mesh division of the virtual layup is consistent with the mesh division of the outer surface of the large thick-walled solid structure, and each virtual layup node and an outer surface node of the solid structure completely coincide in three-dimensional space coordinates;

[0013] The virtual layup adopts a four-node shell element type, with each shell element having four nodes and each node having six degrees of freedom, including three translational degrees of freedom and three rotational degrees of freedom;

[0014] The thickness of the virtual ply is 1 mm, and the elastic modulus of the material of the virtual ply is 1 times 10 to the power of -11, which is the elastic modulus of the material of the large thick-walled solid structure.

[0015] The Poisson's ratio of the virtual ply is the same as that of the large thick-walled solid structure material;

[0016] The neutral plane of the virtual ply coincides with the outer surface of the large thick-walled solid structure;

[0017] The virtual layup is only used for displacement field inversion calculation and does not participate in the actual stress analysis of the large thick-walled solid structure.

[0018] The virtual layup is configured with low stiffness to ensure that the stiffness matrix and stress distribution of the large thick-walled solid structure are not changed during use.

[0019] The above settings achieve geometric attachment and mechanical isolation between the virtual layup and the solid structure.

[0020] Furthermore, step three, which involves arranging strain measurement points on the virtual layup neutral surface, includes:

[0021] Multiple strain measurement points are arranged on the neutral surface of the virtual ply according to the empirical point layout method. The strain measurement points cover the entire neutral surface of the virtual ply, including the central area, diagonal area and boundary transition area of ​​the plate.

[0022] The number of strain measurement points is one-third of the total number of virtual ply units, and they are densely arranged in the central area, diagonal area and centerline transition area of ​​the plate.

[0023] The arrangement of the strain measurement points utilizes structural symmetry, forming measurement point channels along the centerline of the X direction, the centerline of the Y direction, and the diagonal direction;

[0024] Each strain measurement point is used to collect in-plane strain components along the principal stress direction, including longitudinal strain, transverse strain and shear strain components.

[0025] In step three, the collected strain data is used as the membrane strain of the virtual layup, including assigning the longitudinal strain, transverse strain and shear strain components to the membrane strain components of the corresponding nodes of the virtual layup.

[0026] In inverse finite element inversion, bending strain and shear strain are set to zero at input, and only the membrane strain is used as the measurement input.

[0027] The strain data acquisition is based on the principle of single-sided point placement, eliminating the need to place measurement points on the back of the structure.

[0028] Furthermore, the construction of the weighted least squares functional in step four specifically includes:

[0029] Based on Mindlin shell theory, the relationship between displacement and strain is established, and the following weighted least squares functional is constructed:

[0030]

[0031] in, Let represent the weighted least squares functional. The computational membrane strain representing the virtual layup is obtained through displacement field shape functions. This represents the measured membrane strain, data collected from strain measurement points. The curvature representing the virtual ply is calculated using the displacement field shape function. The calculated shear strain representing the virtual ply is obtained through the displacement field shape function. The weighting coefficients of the membrane strain term, This represents the weighting coefficient of the bending strain regularization term. The weighting coefficients for the shear strain regularization term are represented.

[0032] The membrane strain term of the weighted least squares functional is used to fit the measured membrane strain data, and the bending strain regularization term and the shear strain regularization term are used as regularization constraint terms.

[0033] The calculation of membrane strain, curvature, and shear strain are derived based on the displacement field shape function and geometric relationship of the virtual ply.

[0034] Furthermore, the weighting coefficients , and The signal-to-noise ratio of the measured data is dynamically adjusted, and the adjustment methods include:

[0035] First, estimate the noise variance of the measured strain data, and then set weighting coefficients based on the noise variance. and The value of the weighting coefficient when the noise variance is large. and As the value increases and the noise variance decreases, the weighting factor decreases. and The value decreases, and the weighting coefficient decreases. Set to a fixed value of 1;

[0036] The adjustment of the weighting coefficients is based on the number of measuring points, strain sensitivity, and noise statistical characteristics. and The value ranges from 0.1 to 10;

[0037] By adjusting the weighting coefficients, the contributions between the membrane strain fitting term and the regularization term in the functional are balanced, ensuring the numerical stability of the inversion process.

[0038] Furthermore, step five, which involves performing a variational solution to the weighted least squares functional, includes:

[0039] By taking the variation of the functional and setting the variation to zero, we obtain the pseudo-stiffness matrix and pseudo-load vector of each element in the virtual ply.

[0040] The Gaussian integral method is used to calculate the integrals of each term in the pseudo-stiffness matrix and pseudo-load vector;

[0041] The pseudo-stiffness matrices and pseudo-load vectors of all elements are assembled into a global linear equation system, which takes the form of: ,in Represents the global pseudo-stiffness matrix. This represents the displacement vector of the virtual ply node. Represents the global pseudo-load vector;

[0042] The global linear equations are a symmetric positive definite system. They are solved using Gaussian elimination or Cholesky decomposition to obtain the displacement field distribution of the virtual ply nodes. The solution process is based on linear algebra theory to ensure computational efficiency and numerical stability.

[0043] Furthermore, step six, which involves transferring the translational displacement components of the virtual plywood nodes to the outer surface nodes of the large, thick-walled solid structure, includes:

[0044] Establish a degree-of-freedom mapping relationship between virtual ply nodes and nodes on the outer surface of the solid structure, wherein the degree-of-freedom mapping relationship is based on the principle of one-to-one correspondence between node coordinates;

[0045] The three translational degrees of freedom displacement components of the virtual ply node are directly assigned to the three translational degrees of freedom of the corresponding node on the outer surface of the solid structure. The displacement transfer process only involves the transfer of displacement information and does not apply any additional load or stiffness.

[0046] The mapping relationship ensures the continuous transfer of the displacement field from the virtual plyshell unit to the solid structural unit;

[0047] The displacement field inversion results are used to reconstruct the displacement distribution and deformation state of the ply surface of a large, thick-walled solid structure.

[0048] By using displacement mapping, consistency between the virtual layup inversion results and the deformation of the solid structure can be achieved.

[0049] Furthermore, the method is applied to large, thick-walled solid structures subjected to internal pressure loads, including pressure vessels, pipelines, and reaction vessels;

[0050] The method is also applicable to deformation monitoring and displacement reconstruction of welded shells, composite material skins, and large equipment;

[0051] The displacement field inversion results are used for strain reconstruction, stress analysis, and structural health assessment.

[0052] The method achieves high-precision displacement identification under single-sided point layout conditions, without the need for measurement points on the back of the structure;

[0053] The introduction of virtual plying enables the inverse finite element method to be directly applied to solid element models, overcoming the limitations of traditional shell element inversion methods;

[0054] The method ensures that the inversion process does not affect the actual stress characteristics of the solid structure through geometric attachment, mechanical isolation, and numerical coupling.

[0055] Compared with existing technologies, this method for inverting the displacement field of ply surfaces in large, thick-walled solid structures based on virtual plying has the following advantages:

[0056] I. This invention embeds a virtual ply with geometrical attachment to the nodes of a large, thick-walled solid structure, and with a material stiffness much lower than that of the solid structure. Combined with the unilateral sampling of the solid surface strain as the virtual ply membrane strain, and based on the inverse finite element theory, a weighted least squares functional containing membrane strain fitting terms and bending and shear strain regularization constraints is constructed. After variational calculation, the virtual ply displacement is obtained and mapped to the solid structure. This enables high-precision displacement field inversion of the surface of a large, thick-walled solid structure under the condition of only unilateral sampling. Moreover, the virtual ply does not participate in the actual stress of the solid structure during the inversion process and does not change the stiffness matrix and stress distribution of the solid structure, effectively meeting the displacement monitoring needs under the constraints of structural enclosure and installation space in engineering.

[0057] Second, this invention introduces bending strain regularization and shear strain regularization terms into the weighted least squares functional, combines a design that dynamically adjusts weight coefficients based on the signal-to-noise ratio of measurement data, and employs an efficient solution method for symmetric positive definite global linear equations. This effectively balances the accuracy and numerical stability of membrane strain fitting, thereby avoiding inversion distortion caused by incomplete strain information due to unilateral point placement. At the same time, the virtual plying shell element modeling design allows the inverse finite element method to be directly adapted to the solid element model of large thick-walled solid structures, thus eliminating the need to simplify the original structural model with shell elements, improving the engineering practicality and adaptability of the method.

[0058] Other advantages, objectives and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination or study, or may be learned from the practice of the invention. Attached Figure Description

[0059] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are merely some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without any creative effort.

[0060] Figure 1 This is a step diagram of the present invention;

[0061] Figure 2 This is a schematic diagram illustrating the relationship between the virtual layering and the solid structure of the present invention;

[0062] Figure 3 This is a flowchart illustrating the operation of the present invention. Detailed Implementation

[0063] To further illustrate the technical means and effects of the present invention in achieving its intended purpose, the following detailed description of the specific implementation methods, structures, features, and effects of the present invention, in conjunction with the accompanying drawings and preferred embodiments, is provided below.

[0064] Example 1

[0065] like Figure 1 and Figure 2 As shown, this embodiment aims to specifically illustrate the implementation process of the displacement field inversion method based on virtual ply layers for large, thick-walled solid structures subjected to internal pressure. This addresses the problems of traditional inverse finite element methods, such as reliance on double-sided point placement, incompatibility with solid element models, and easy interference with the original structural stress. In this embodiment, the pressure vessel is made of Q345 steel, with a designed wall thickness of 50mm, a nominal diameter of 2000mm, and a working pressure of 3MPa. By embedding virtual ply layers on the outer surface of its finite element model, combined with strain measurement point placement, weighted least squares functional construction, and variational solution, high-precision displacement field inversion under single-sided point placement conditions is finally achieved. Furthermore, the inversion process does not affect the actual stress characteristics of the pressure vessel, providing reliable data support for its structural health monitoring.

[0066] In this embodiment, the finite element model of the pressure vessel is established using the general-purpose finite element software ANSYS. The solid part uses SOLID185 elements, and the model mesh is structured. The outer surface mesh size is controlled at 20mm×20mm to balance calculation accuracy and efficiency. The total number of nodes in the model is approximately 12,000, and the number of nodes on the outer surface is approximately 3,000.

[0067] Step 1: Embed virtual ply

[0068] Virtual lay-up layers are embedded on the outer surface of the aforementioned pressure vessel finite element model, as follows:

[0069] The virtual layup is modeled using SHELL181 elements. The meshing of the virtual layup is completely consistent with that of the pressure vessel's outer surface. The total number of solid elements on the pressure vessel's outer surface is 900, and the total number of shell elements in the virtual layup is also 900. That is, each shell element in the virtual layup corresponds geometrically to one solid element on the pressure vessel's outer surface. Each node in the virtual layup completely coincides with one solid node on the pressure vessel's outer surface in three-dimensional space (the total number of nodes on the solid outer surface is approximately 3000, and the total number of nodes in the virtual layup is also 3000), achieving geometric attachment.

[0070] The thickness of the virtual layup is set to 1 mm. This thickness is chosen to ensure the effectiveness of shell element modeling without compromising geometric attachment accuracy due to excessive thickness. Geometric attachment ensures that the displacement field of the subsequent virtual layup accurately reflects the displacement state of the pressure vessel's outer surface, laying the foundation for displacement transfer.

[0071] Step 2: Set virtual layup material parameters

[0072] To achieve mechanical isolation between the virtual layup and the pressure vessel, the material properties of the virtual layup are determined based on the material parameters of the pressure vessel:

[0073] The pressure vessel is made of Q345 steel, whose material elastic modulus is... Poisson's ratio The elastic modulus of materials in virtual layups Set as the elastic modulus of the pressure vessel material times, that is

[0074] Poisson's ratio of virtual ply Maintaining the same Poisson's ratio as the pressure vessel, i.e. .

[0075] The mechanical significance of the above parameter settings is as follows: the elastic modulus of the virtual layup is much lower than that of the pressure vessel, making its stiffness negligible. It does not participate in the actual stress on the pressure vessel during the inversion calculation process and will not change the original stiffness matrix and stress distribution of the pressure vessel, thus achieving mechanical isolation. Maintaining a consistent Poisson's ratio ensures that the deformation of the virtual layup and the pressure vessel is coordinated under external forces, avoiding geometric attachment failure due to differences in Poisson's ratio, and ensuring the accuracy of displacement transfer.

[0076] It should also be clarified that virtual layup is only used for displacement field inversion calculations and does not participate in the actual stress analysis of pressure vessels. Its only function is as a dedicated calculation medium for displacement inversion.

[0077] Step 3: Set up strain measurement points and collect strain data

[0078] Strain measurement points were arranged on the neutral surface of the virtual ply, and strain data were collected. The specific process is as follows:

[0079] Measurement point layout principles:

[0080] Strain measurement points are arranged using an empirical point placement method. The measurement points need to cover the entire neutral surface of the virtual ply, including the central area, diagonal area, and boundary transition area of ​​the outer surface of the pressure vessel. The number of measurement points is set to one-third of the total number of virtual ply elements. In this embodiment, the total number of virtual ply elements is 900 (which is exactly the same as the number of solid elements on the outer surface of the pressure vessel), so the number of measurement points is 300.

[0081] Taking advantage of the structural symmetry of the pressure vessel, four measuring point channels are formed along the centerline of the X direction, the centerline of the Y direction, and the diagonal directions. 60 measuring points are arranged along the centerline of the X direction, 60 measuring points along the centerline of the Y direction, and 90 measuring points along each of the two diagonal directions. The measuring points are densified in the central area of ​​the plate and the transition area between the plate and the boundary, while they are arranged at a conventional density in the diagonal areas to improve the strain measurement accuracy in key areas.

[0082] This embodiment adopts the principle of single-sided point layout, and only places measuring points on the outer surface of the pressure vessel, without the need to place measuring points on the inner surface of the vessel, which greatly reduces the difficulty of engineering implementation and adapts to the complex operating environment inside the vessel.

[0083] Additional explanation: In this embodiment, the statements "approximately 3000 nodes on the outer surface of the solid element" and "900 virtual layup elements" are not contradictory. This is because the virtual layup uses four-node shell elements, while the outer surface of the pressure vessel uses four-node solid elements (or eight-node solid elements, but the outer surface mesh is divided into four-node units). Adjacent elements share nodes (e.g., one node belongs to 2-4 elements simultaneously). Therefore, the total number of nodes in the 900 four-node elements is not 900 × 4 = 3600, but approximately 3000 (affected by shared nodes). This is a common phenomenon in finite element mesh generation.

[0084] Strain data acquisition and processing:

[0085] Resistance strain gauges were attached to each measuring point to collect in-plane strain data of the pressure vessel under rated operating pressure. Three in-plane strain components were collected at each measuring point along the principal stress direction of the structure, representing the longitudinal strain. Lateral strain and shear strain .

[0086] The collected longitudinal strain Lateral strain and shear strain The membrane strain components are directly assigned to the corresponding nodes of the virtual layup, which are the measured membrane strains of the virtual layup. In the subsequent inverse finite element inversion process, since it was impossible to obtain measured data of bending strain and shear strain with only one-sided point layout, the input values ​​of bending strain and shear strain were set to zero, and only the above-mentioned measured membrane strain was used. As the measurement input for inversion calculation.

[0087] Step 4: Construct the weighted least squares functional

[0088] Based on Mindlin shell theory, the displacement-strain relationship of virtual ply is established, and then a weighted least squares functional is constructed, as follows:

[0089] Displacement-strain relationship under Mindlin shell theory:

[0090] Virtual ply nodal displacement vector for:

[0091] ,in , , These represent the translational displacements of the node in the x, y, and z directions, respectively. , These represent the angular displacements of the nodes around the x-axis and y-axis, respectively.

[0092] The strain of the virtual layup includes membrane strain. Bending strain With shear strain All three are achieved through nodal displacement vectors. With unit shape function The calculation yielded:

[0093] Membrane strain : Reflects the tensile and shear deformation within the plane of a shell element, expressed as follows: ,in The membrane strain operator has the following form:

[0094]

[0095] The shape function matrix of a four-node shell element, where a single shape function... The expression is:

[0096] in , These are the local coordinates of the shell element.

[0097] Bending strain : Reflects the bending deformation of the shell element, expressed as: ,in The bending strain operator has the following form:

[0098]

[0099] Shear strain : Reflects the transverse shear deformation of the shell element, expressed as: ,in The shear strain operator has the following form:

[0100]

[0101] (Note: In Mindlin shell theory, shear strain is mainly...) and , here The form has been simplified to an operator corresponding to the transverse shear direction.

[0102] Construction of the weighted least squares functional:

[0103] To achieve displacement field inversion based on measured membrane strain, and to address the lack of bending and shear strain information caused by unilateral point placement, a weighted least squares functional is constructed. The expression is:

[0104]

[0105] The definitions and meanings of each parameter are as follows:

[0106] in, Let represent the weighted least squares functional. The computational membrane strain representing the virtual layup is obtained through displacement field shape functions. This represents the measured membrane strain, data collected from strain measurement points. The curvature representing the virtual ply is calculated using the displacement field shape function. The calculated shear strain representing the virtual ply is obtained through the displacement field shape function. The weighting coefficients of the membrane strain term, This represents the weighting coefficient of the bending strain regularization term. The weighting coefficients for the shear strain regularization term are represented.

[0107] The functional is constructed as follows: the first term is the core fitting term, which ensures the consistency between the inversion results and the measured data; the second and third terms are constraint terms, which solve the problem of incomplete information caused by unilateral point placement and balance the fitting accuracy and numerical stability.

[0108] Step 5: Variational Solution and Global Linear Equation Solution

[0109] For the above weighted least squares functional The variational solution is performed to assemble the global linear equation system and solve for the nodal displacement field of the virtual ply. The specific process is as follows:

[0110] Functional variational derivation and element pseudostiffness matrix and pseudoload vector:

[0111] According to the variational principle, the optimal solution of the displacement field inversion satisfies the functional... The variation is zero, that is .

[0112] right Find the variational function:

[0113]

[0114] Expand the variables:

[0115]

[0116] because Since it is a measured constant, its variation is zero, therefore , It is the variation of the nodal displacement vector.

[0117] Similarly, the variations of the bending strain term and the shear strain term are:

[0118]

[0119] Substituting the above expansion into... After sorting, we get:

[0120]

[0121] make:

[0122] Element pseudo-stiffness matrix :

[0123] Element pseudo-load vector :

[0124] in The area of ​​a single shell unit. For virtual ply thickness, Let Jacobian matrix be the ratio of local coordinates to global coordinates. Let be a small area element in local coordinates.

[0125] Gaussian integral calculation:

[0126] The above calculations were performed using the 2×2 Gaussian integral method. and The reason for this is that the 2×2 Gaussian integral can accurately calculate the integral term of a four-node shell element.

[0127] The coordinates of the integration point of the Gaussian integral are The corresponding integral weight .

[0128] When performing integration, first calculate the shape function at each integration point. strain operators Jacobian matrix and its measured membrane strain Substitute and The expression is used to calculate the integral value, ultimately yielding the value of a single unit. and .

[0129] Assembly and solution of global linear equation systems:

[0130] The pseudo-stiffness matrix of all virtual ply elements Assembled into a global pseudo-stiffness matrix according to node numbers. , all element pseudo-load vectors Assembled into a global pseudo-payload vector according to node numbering. This forms a global system of linear equations:

[0131]

[0132] in for An order matrix, and It is a symmetric positive definite matrix; This is the global node displacement vector; This is the global pseudo-load vector.

[0133] because For a symmetric positive definite matrix, the Cholesky decomposition method is used for solution. This method can... Decompose into a lower triangular matrix Its transpose The product of the products is then solved by forward substitution. Finally, the solution is obtained through backward substitution. The displacement field distribution of all nodes in the virtual ply is obtained. .

[0134] The advantages of the Cholesky decomposition method are its high computational efficiency and good numerical stability, which can effectively avoid rounding errors during the solution process and ensure the accuracy of displacement field inversion.

[0135] Step Six: Realization of Displacement Transfer and Displacement Field Inversion

[0136] The translational displacement components of the virtual ply nodes are transferred to the corresponding nodes on the outer surface of the pressure vessel to achieve displacement field inversion. The specific process is as follows:

[0137] Establishment of node degree of freedom mapping relationship:

[0138] Based on the geometric attachment relationship between the virtual ply and the nodes on the outer surface of the pressure vessel in step one, a degree-of-freedom mapping relationship between the two is established: the three translational degrees of freedom of each node of the virtual ply correspond to the three translational degrees of freedom of the corresponding node on the outer surface of the pressure vessel, and the rotation angle of the virtual ply does not participate in the displacement transfer.

[0139] Displacement transfer operation:

[0140] The translational displacements of the virtual plywood nodes obtained in step five are directly assigned to the translational degrees of freedom of the corresponding nodes on the outer surface of the pressure vessel, i.e.: This displacement transfer process involves only the direct transfer of displacement information and does not apply any additional load or stiffness to the pressure vessel. Therefore, it does not change the original stress state of the pressure vessel and ensures that the inversion results can truly reflect the actual displacement field of the pressure vessel.

[0141] Applications of displacement field inversion results:

[0142] Through the aforementioned displacement transfer, the translational displacements of all nodes on the outer surface of the pressure vessel are obtained, thereby reconstructing the complete displacement field distribution on the outer surface of the pressure vessel. Based on this displacement field distribution, strain reconstruction, stress analysis, and structural health assessment can be further performed.

[0143] In this embodiment, the maximum axial displacement of the outer surface of the pressure vessel is 0.8 mm and the maximum circumferential displacement is 0.5 mm, both of which are less than the design allowable displacement values. This indicates that the deformation of the pressure vessel under the rated working pressure is within a safe range and the structure is in good service condition.

[0144] In summary, this embodiment takes a steel pressure vessel subjected to internal pressure as the object and fully implements a displacement field inversion method based on virtual plying for thick-walled solid structures. By embedding a geometrically attached and mechanically isolated virtual ply, combined with strain acquisition from a single-sided point, weighted least squares functional construction, and variational solution, high-precision displacement field inversion is finally achieved. The implementation process shows that the low stiffness setting of the virtual ply effectively avoids interference with the actual stress characteristics of the pressure vessel; the regularization term of the weighted least squares functional solves the problem of incomplete information caused by single-sided point placement; the Cholesky decomposition method ensures the efficiency and stability of the equation system solution; and the one-to-one correspondence of displacement transfer ensures the accuracy of the inversion results. This embodiment verifies the feasibility and effectiveness of the method in engineering practice and can be widely applied to displacement monitoring and health assessment of thick-walled solid structures such as pressure vessels, pipelines, and reactors.

[0145] Example 2

[0146] like Figure 3 As shown in Example 1, this example elaborates on the specific steps of the method for inverting the displacement field of a ply surface in a large thick-walled solid structure based on virtual plying. The specific steps are as follows:

[0147] 1. Establish a finite element model of the solid structure:

[0148] A three-dimensional solid model of a large, thick-walled solid structure was created using finite element software.

[0149] The outer surface of the structure is meshed to generate nodes and elements, ensuring that the mesh size is appropriate and balancing computational accuracy and efficiency.

[0150] 2. Embedding virtual layers:

[0151] A virtual layer is created on the outer surface of the solid structure, and the shell element type is used for modeling.

[0152] The virtual layer mesh is completely consistent with the outer surface mesh of the solid, realizing a one-to-one correspondence between nodes in three-dimensional spatial coordinates and completing geometric attachment.

[0153] 3. Set virtual layup material properties:

[0154] The elastic modulus of the virtual ply is set to be much lower than that of the solid structure material, while the Poisson's ratio remains consistent with that of the solid structure.

[0155] The virtual ply thickness is set to 1 mm to ensure that it does not participate in the actual stress of the solid structure, thus achieving mechanical isolation.

[0156] 4. Arrange strain measurement points:

[0157] Strain measurement points are arranged on the neutral surface of the virtual layup.

[0158] The measurement points cover the entire structural area, including the central region, boundary regions, and diagonal directions, and are reasonably densified based on the structural symmetry.

[0159] The principle of single-sided placement is adopted, with sensors placed only on the outer surface, eliminating the need for measurement points on the back.

[0160] 5. Collect strain data:

[0161] Under actual structural service conditions, in-plane strain data at each measuring point are collected using strain sensors, including longitudinal strain, transverse strain, and shear strain components.

[0162] The collected strain data is used as the strain input for the virtual layup membrane, which is then used for subsequent inversion calculations.

[0163] 6. Construct the inversion calculation model:

[0164] Based on the inverse finite element theory, a weighted least squares functional is constructed, which includes a membrane strain fitting term, a bending strain regularization term, and a shear strain regularization term.

[0165] The weighting coefficients are dynamically adjusted based on the signal-to-noise ratio of the strain data to balance fitting accuracy and numerical stability.

[0166] 7. Perform displacement field inversion calculation:

[0167] Variational solutions were obtained for the weighted least squares functional to derive the pseudo-stiffness matrix and pseudo-load vector of each element in the virtual ply.

[0168] Numerical integration methods are used to calculate integral terms in matrices and vectors.

[0169] Assemble the pseudo-stiffness matrix and pseudo-load vector of all elements to form a global linear equation system, and solve the equation system numerically to obtain the displacement field of the virtual ply nodes.

[0170] 8. Transferring displacement to solid structures:

[0171] The three translational displacement components of the virtual ply node are directly assigned to the corresponding nodes on the outer surface of the solid structure through the node mapping relationship.

[0172] This process only transmits displacement information and does not introduce additional loads or stiffness, ensuring that the inversion results truly reflect the deformation state of the solid structure.

[0173] 9. Output and analyze displacement field results:

[0174] Obtaining the complete displacement field distribution on the outer surface of a solid structure can be used for subsequent strain reconstruction, stress analysis, or structural health assessment.

[0175] Verify whether the displacement field meets engineering safety standards and provide data support for structural monitoring.

[0176] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.

Claims

1. A method for inversion of surface displacement field of a thick-walled solid structure based on virtual ply, characterized in that, The method comprises the following steps: Step one, embedding a virtual ply on the outer surface of a finite element model of a large-thickness-wall solid structure, modeling the virtual ply with shell elements, and geometrically attaching the nodes of the virtual ply to the nodes of the outer surface of the large-thickness-wall solid structure one by one; Step two, setting material parameters of the virtual ply to make the material stiffness much lower than that of the large-thickness-wall solid structure, and achieving mechanical isolation; Step three, arranging strain measurement points on the neutral surface of the virtual ply, collecting strain data on the surface of the ply of the large-thickness-wall solid structure, and taking the strain data as the membrane strain of the virtual ply; Step four, based on the inverse finite element theory, constructing a weighted least squares functional, which includes a membrane strain term, a bending strain regularization term, and a shear strain regularization term; Step five, variational solving the weighted least squares functional to assemble a global linear equation system, and solving the node displacement field of the virtual ply; Step six, transferring the translational displacement components of the nodes of the virtual ply to the corresponding nodes on the outer surface of the large-thickness-wall solid structure through node mapping relationship, and achieving displacement field inversion.

2. The virtual ply-based large-thick-wall solid structure ply surface displacement field inversion method according to claim 1, characterized in that, The grid division of the virtual ply is consistent with that of the outer surface of the large-thickness-wall solid structure, and each virtual ply node is completely coincident with a solid structure outer surface node in three-dimensional space coordinates; The virtual ply adopts four-node shell element type, each shell element has four nodes, and each node has six degrees of freedom, including three translational degrees of freedom and three rotational degrees of freedom; The thickness of the virtual ply is 1 millimeter, and the elastic modulus of the virtual ply is 1 times 10 to the power of -11 of the elastic modulus of the material of the large-thickness-wall solid structure; The Poisson's ratio of the virtual ply is the same as that of the material of the large-thickness-wall solid structure; The neutral surface of the virtual ply coincides with the outer surface of the large-thickness-wall solid structure; The virtual ply is only used for displacement field inversion calculation and does not participate in the actual stress analysis of the large-thickness-wall solid structure; The virtual ply is set to be low stiffness, which is used to not change the stiffness matrix and stress distribution of the large-thickness-wall solid structure during use.

3. The virtual ply-based large-thick-wall solid structure ply surface displacement field inversion method according to claim 1, characterized in that, The step three of arranging strain measurement points on the neutral surface of the virtual ply comprises: Arranging a plurality of strain measurement points on the neutral surface of the virtual ply according to an empirical point arrangement method, the strain measurement points covering the whole domain of the neutral surface of the virtual ply, including the center area, the diagonal area, and the boundary transition area; The number of strain measurement points is one-third of the total number of virtual ply units, and the strain measurement points are densely arranged in the center area, the diagonal area, and the midline transition area; The arrangement of the strain measurement points utilizes the structural symmetry to form measurement point channels along the X-direction midline, the Y-direction midline, and the diagonal direction; Each strain measurement point is used to collect in-plane strain components along the principal stress direction, including longitudinal strain, transverse strain, and shear strain components; The step three of taking the collected strain data as the membrane strain of the virtual ply comprises assigning the longitudinal strain, transverse strain, and shear strain components to the membrane strain components of the corresponding nodes of the virtual ply. In the inverse finite element inversion, the bending strain and the shear strain are set to zero at the input, and only the membrane strain is used as the measured input; The strain data acquisition is based on the principle of unilateral point distribution, without the need to arrange measuring points on the back of the structure.

4. The virtual ply-based large-thick-wall solid structure ply surface displacement field inversion method according to claim 1, characterized in that, The step four of constructing the weighted least squares functional specifically includes: Based on the Mindlin shell theory, the displacement and strain relationship is established, and the following weighted least squares functional is constructed: wherein, represents the weighted least squares functional, represents the calculated membrane strain of the virtual ply, calculated from the displacement field shape function, represents the measured membrane strain, data collected from the strain gages, represents the calculated curvature of the virtual ply, calculated from the displacement field shape function, represents the calculated shear strain of the virtual ply, calculated from the displacement field shape function, represents the weight coefficient of the membrane strain term, represents the weight coefficient of the bending strain regularization term, represents the weight coefficient of the shear strain regularization term; The membrane strain term of the weighted least squares functional is used to fit the measured membrane strain data, and the bending strain regularization term and the shear strain regularization term are used as regularization constraint terms; The calculation of the membrane strain, the calculation of the curvature and the calculation of the shear strain are derived based on the displacement field shape function of the virtual layup and the geometric relationship.

5. The virtual ply-based large-thick-wall solid structure ply surface displacement field inversion method according to claim 4, characterized in that, The weight coefficient , and According to the signal-to-noise ratio of the measurement data, the adjustment method comprises: First, the noise variance of the measured strain data is estimated, and then the weight coefficient is set based on the noise variance and The value of the weight coefficient increases as the noise variance is large, and the value of the weight coefficient decreases as the noise variance is small and The value of the weight coefficient increases as the noise variance is large, and the value of the weight coefficient decreases as the noise variance is small and The value of the weight coefficient increases as the noise variance is large, and the value of the weight coefficient decreases as the noise variance is small is set to a fixed value of 1. The adjustment of the weight coefficient is based on the number of measuring points, strain sensitivity and noise statistical characteristics, and the weight coefficient and is in the range of 0.1 to 10.

6. The virtual ply-based large-thick-wall solid structure ply surface displacement field inversion method of claim 1, wherein, The step five of variational solving of the weighted least squares functional includes: Taking the variation of the functional and setting the variation to zero, the pseudo stiffness matrix and the pseudo load vector of each element of the virtual layup are obtained; The Gaussian integral method is used to calculate the integral of each term in the pseudo stiffness matrix and the pseudo load vector; The pseudo stiffness matrix and the pseudo load vector of all elements are assembled into a global linear equation group in the form of wherein denotes a global pseudo stiffness matrix, denotes a virtual ply node displacement vector, denotes a global pseudo load vector; The global linear equation system is a symmetric positive definite system, which is solved by using the Gaussian elimination method or the Cholesky decomposition method to obtain the displacement field distribution of the virtual layup nodes, and the solving process is based on the linear algebra theory.

7. The virtual ply-based large-thick-wall solid structure ply surface displacement field inversion method of claim 1, wherein, The step six of transmitting the translational displacement components of the virtual layup nodes to the outer surface nodes of the large thick-walled solid structure includes: The freedom mapping relationship between the virtual layup nodes and the outer surface nodes of the solid structure is established, and the freedom mapping relationship is based on the principle of one-to-one correspondence of node coordinates; The three translational freedom displacement components of the virtual layup nodes are directly assigned to the three translational freedoms of the corresponding nodes of the outer surface of the solid structure, and the displacement transmission process only involves displacement information transmission without applying any additional load or stiffness; The mapping relationship ensures the continuous transmission of the displacement field from the virtual layup shell element to the solid structure element; The displacement field inversion result is used to reconstruct the displacement distribution and deformation state of the layup surface of the large thick-walled solid structure.

8. The virtual ply-based large-thick-wall solid structure ply surface displacement field inversion method of claim 1, wherein, The method is applied to large thick-walled solid structures under internal pressure load, including pressure vessels, pipelines and reaction kettles; The method is also applicable to deformation monitoring and displacement reconstruction of welded shells, composite skins and large equipment; The displacement field inversion result is used for strain reconstruction, stress analysis and structure health assessment; The method realizes high-precision displacement identification under the condition of unilateral point distribution without the need for back measuring points of the structure; The introduction of the virtual layup makes the inverse finite element method directly applicable to solid element models; The method realizes geometric attachment, mechanical isolation and numerical coupling.

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