Inner wall global profile numerical reconstruction method and system based on flexible plate outer wall local displacement field

Abstract

The invention belongs to the technical field of wind tunnel tests, and discloses an inner wall global profile numerical reconstruction method and system based on a flexible plate outer wall local displacement field. According to the method, the finite element model based on the Kirchhoff thin plate theory is established, the local displacement field of the non-reinforcing rib area of the outer wall of the flexible plate is adopted as input, and the engineering obstacle that measuring points are difficult to arrange in the reinforcing rib area connected with the actuating rod is avoided. The result shows that when the input length of the non-reinforcing-rib area in the length direction is not less than 75% of the total length of a single non-reinforcing-rib area, the root-mean-square error of the reconstructed inner wall displacement and the simulation result is less than 0.04 mm, the maximum absolute error is less than 0.13 mm, and the flexible-wall nozzle profile measurement precision requirement is met.

Description

Technical Field

[0001] This invention relates to the field of wind tunnel testing technology, specifically to a method and system for numerical reconstruction of the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate. Background Technology

[0002] Flexible-walled nozzles are key components of hypersonic wind tunnels. Their profile adjustment function utilizes the elastic deformation of flexible walls (hereinafter referred to as flexible plates) to provide aerodynamic profiles at different Mach numbers without replacing the nozzle. Under hypersonic operating conditions, the temperature changes caused by gas compression and expansion, combined with mechanical loads, can cause the flexible plate profile to deviate from the preset value. Therefore, accurately measuring the true full-area profile of the inner wall under operating conditions is a key technical challenge to ensure the quality of the wind tunnel flow field.

[0003] Since the inner wall of the flexible-wall nozzle is in a high-temperature and high-pressure airflow scouring environment, it is difficult to directly deploy sensors. At present, the main method is indirect measurement, that is, the deformation of the inner wall is inverted by measuring the deformation of the outer wall of the flexible-wall nozzle. Existing indirect measurement methods mainly include: (1) Beam model method based on Ko displacement theory: The thin plate is simplified into an Euler beam model. The calculation efficiency is improved by ignoring the deformation in the width direction, shear effect and in-plane stress. It is suitable for wings or beam structures with gradually changing cross sections, but it is difficult to accurately describe the mechanical response under two-dimensional bending deformation, shear deformation and complex boundary conditions. It is prone to significant errors in the application of thin plates or large-span structures. (2) Shape recognition method based on modal method: The first few natural modes of the structure are used as basis functions. The full field displacement is reconstructed by associating discrete strain with strain mode matrix. The reconstruction accuracy of this method is highly dependent on the accuracy of modal basis functions. Moreover, it is difficult to deploy sensors in high-temperature and high-pressure environments, which makes it difficult to meet the actual needs of engineering. (3) Hybrid strategy method based on strain: Local deflection is obtained by integrating strain along the sensing line, and then global expansion is performed with the help of finite element mode. Although it achieves a combination of local accuracy and full-field reconstruction, the calculation process is complex and the reconstruction effect is constrained by the modal basis and sensor layout. (4) Strain-bending moment method: By establishing the differential equation and elliptic integral solution of the nozzle profile, and using plate theory to correct the simplified beam model. The reconstruction accuracy of this method depends heavily on the distribution of strain measurement points, and the applicability of the beam model is limited under complex multi-field coupled loads, making it difficult to achieve fast real-time reconstruction.

[0004] The aforementioned methods generally suffer from common problems such as reliance on simplified mechanical models, high sensitivity to key input parameters, and high computational complexity making real-time reconstruction difficult. Their applicability is limited under the actual working conditions of flexible-walled nozzles with high temperature, high pressure, and large deformation. In particular, existing methods typically require sensors to be placed on the inner wall or in the reinforcing rib area. However, since the reinforcing ribs are directly hinged to the actuators, this area is not only difficult to accommodate measuring devices but also suffers from significant interference from the actuators, resulting in low measurement reliability. Furthermore, existing thermo-mechanical coupling analysis methods usually consider the changes in material properties with temperature, leading to the need to update the element stiffness matrix element by element, resulting in long computation times and making it difficult to meet the real-time measurement and control requirements of wind tunnel tests.

[0005] In view of this, the present invention is proposed. Summary of the Invention

[0006] The present invention aims to solve at least one of the above technical problems, and provides a method and system for numerical reconstruction of the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate.

[0007] To achieve the above objectives, the first technical solution adopted by the present invention is as follows: The method for numerical reconstruction of the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate includes the following steps: The three-dimensional displacement field data of the non-reinforcing rib region on the outer wall of the flexible plate is obtained as a known boundary condition; wherein, the flexible plate is a thin plate structure with multiple transverse reinforcing ribs, and the non-reinforcing rib region is the plate surface region far away from the connection part of the actuator between adjacent reinforcing ribs. Based on Kirchhoff's thin plate theory, the flexible plate is discretized into finite elements to establish an overall stiffness matrix that includes stiffened and unstiffened regions. In this matrix, multiple layers of elements are maintained in the thickness direction, and different element sizes are used to mesh the stiffened and unstiffened regions in the plate surface direction. The displacement boundary conditions are introduced into the overall stiffness matrix, including: assigning zero values ​​to the displacements of all nodes at the fixed end of the flexible plate, and applying the three-dimensional displacement field data of the non-stiffening region as known displacement constraints to the corresponding nodes. Based on the overall stiffness matrix after introducing boundary conditions, the overall equilibrium equation of the structure is solved to obtain the three-dimensional displacement field of all nodes of the inner wall of the flexible plate. The three-dimensional displacement field of all nodes of the inner wall is superimposed with the initial shape of the inner wall to obtain the deformed global shape of the inner wall.

[0008] Preferably, the method for establishing the overall stiffness matrix including the stiffened and unstiffened regions includes: The flexible plate is discretized using eight-node hexahedral isoparametric elements. Based on the geometric equations and constitutive relations of elasticity, strain-displacement matrix and stress-strain matrix are established respectively; Based on the principle of generalized virtual work, the element stiffness matrix is ​​obtained by volume integration of the element, and the overall stiffness matrix is ​​formed by superimposing all element stiffness matrices according to the node number.

[0009] Preferably, the displacement of any point within the eight-node hexahedral isoparametric element satisfies a shape function relationship with the node displacement. Based on this shape function relationship, the displacement of any point within the element is calculated from the node displacement, thereby achieving a continuous expression of the displacement field within the element.

[0010] Preferably, the method of meshing the stiffened and unstiffened areas using different element sizes in the plate surface direction includes: The reinforcing rib area adopts a first unit size, and the non-reinforcing rib area adopts a second unit size, wherein the first unit size is smaller than the second unit size in the length direction; In the thickness direction, both the reinforcing rib region and the non-reinforcing rib region adopt two-layer units.

[0011] Preferably, the first unit has a size of 7.5mm×20mm×3mm and the second unit has a size of 15.4mm×20mm×3mm; or the first unit has a size of 5mm×13.3mm×3mm and the second unit has a size of 10.27mm×13.3mm×3mm.

[0012] Preferably, under thermo-mechanical coupling conditions, when establishing the overall stiffness matrix including the stiffened and unstiffened regions, the elastic modulus and Poisson's ratio of the flexible plate material are set as constants, and the influence of the temperature field on the material properties is ignored, so that the element stiffness matrix only needs to be calculated once.

[0013] Preferred options based on Kirchhoff's thin-plate theory include: Assume that there is no displacement parallel to the mid-plane of the flexible plate; Assume that the normal to the mid-plane of the flexible plate remains straight before and after deformation, and that the angle of rotation around the mid-plane is equal to the first partial derivative of the mid-plane deflection. Based on the above assumptions, a mapping relationship is established between the displacement of the outer wall and the displacement of the inner wall about the mid-plane symmetry, wherein the vertical displacements of the outer wall and the inner wall are equal in magnitude and opposite in direction.

[0014] Preferably, methods for solving the overall equilibrium equations of the structure to obtain the three-dimensional displacement field of all nodes on the inner wall of the flexible plate include: The overall equilibrium equation of the structure is solved by Gaussian elimination to obtain the three-dimensional displacement vector containing all nodes of the flexible plate. The nodal displacement components located on the inner wall surface are extracted from the three-dimensional displacement vector to form a three-dimensional displacement field for all nodes of the inner wall.

[0015] Preferably, the method for superimposing the three-dimensional displacement field of all nodes of the inner wall with the initial shape of the inner wall includes: The three-dimensional displacement field of all nodes of the inner wall is vector-superimposed with the corresponding node coordinates of the initial shape of the inner wall. Spatial interpolation of discrete inner wall nodal displacements is performed using finite element shape functions to obtain a continuous global inner wall profile.

[0016] The second technical solution adopted in this invention is: A numerical reconstruction system for the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate includes: The acquisition unit is used to acquire three-dimensional displacement field data of the non-reinforcing rib region of the outer wall of the flexible plate as known boundary conditions; wherein, the flexible plate is a thin plate structure with multiple transverse reinforcing ribs, and the non-reinforcing rib region is the plate surface region far away from the connection part of the actuator between adjacent reinforcing ribs. A finite element is established to discretize the flexible plate into finite elements based on Kirchhoff's thin plate theory, and to establish an overall stiffness matrix including stiffened and unstiffened regions; wherein, multi-layer elements are maintained in the thickness direction, and different element sizes are used to mesh the stiffened and unstiffened regions in the plate surface direction; An introduction unit is used to introduce displacement boundary conditions into the overall stiffness matrix, including: assigning zero values ​​to the displacements of all nodes at the fixed end of the flexible plate, and applying the three-dimensional displacement field data of the non-stiffening region as known displacement constraints to the corresponding nodes. The solver element is used to solve the overall equilibrium equation of the structure based on the overall stiffness matrix after introducing boundary conditions, and obtain the three-dimensional displacement field of all nodes on the inner wall of the flexible plate. The superposition unit is used to superimpose the three-dimensional displacement field of all nodes of the inner wall with the initial shape of the inner wall to obtain the deformed global shape of the inner wall.

[0017] Compared with the prior art, the present invention has the following beneficial effects: This invention establishes a finite element model based on Kirchhoff's thin-plate theory and uses the local displacement field of the non-reinforced region of the flexible plate's outer wall (far from the actuator connection point) as input, thus avoiding the engineering obstacle of arranging measurement points in the reinforced region connected to the actuator. Results show that when the input length of the non-reinforced region along its length is not less than 75% of the total length of a single non-reinforced region, the root mean square error between the reconstructed inner wall displacement and the simulation results is less than 0.04 mm, and the maximum absolute error is less than 0.13 mm, meeting the accuracy requirements for flexible-wall nozzle profile measurement.

[0018] This invention employs a non-uniform mesh partitioning strategy (using different element sizes for stiffened and unstiffened regions) and spatial interpolation based on finite element shape functions. This not only ensures the calculation accuracy of stress concentration areas near stiffeners but also achieves smooth reconstruction from discrete measurement points to a continuous surface across the entire domain, resulting in high computational stability.

[0019] Based on Kirchhoff's assumption of straight normals for thin plates, this invention establishes a displacement mapping relationship between the outer and inner walls about the mid-plane symmetry, proving that the displacement field of the inner wall does not change with different loading forms and is uniquely determined only by the deformation of the outer wall. This provides a theoretical basis for indirect measurement and avoids the cumulative error introduced by the traditional strain integration method. Attached Figure Description

[0020] Figure 1 This is a flowchart illustrating the method for numerical reconstruction of the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate. Figure 2 This is a structural diagram of a flexible plate, where 21 is a reinforcing rib, 22 is a non-reinforcing rib area, and 23 is a fixed end; Figure 3 This is a schematic diagram of a numerical reconstruction system for the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate. Figure 4 This is a diagram showing the relationship between the input area of ​​the outer wall and the reconstruction error in the embodiment. Detailed Implementation

[0021] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be described in detail below with reference to specific embodiments. It should be noted that, unless otherwise specified, the embodiments and features described in these embodiments can be arbitrarily combined with each other. Those skilled in the art should understand that the following descriptions are merely exemplary means of implementing this invention and not limiting conditions; any technical means employed to achieve the same or similar technical effects as this invention should fall within the protection scope of this invention.

[0022] This invention provides a method and system for numerical reconstruction of the entire inner wall profile based on the local displacement field of the outer wall of a flexible plate. It aims to solve the technical challenges of directly measuring the inner wall profile of a flexible-walled nozzle in a hypersonic wind tunnel during operation, as well as the limitations in accuracy and computational time of existing strain inversion methods. This method is based on the principle of elasticity inversion. By measuring the local displacement of easily measurable areas of the outer wall, it reconstructs the three-dimensional profile of the entire inner wall using the finite element method. It is particularly suitable for real-time profile monitoring of thin-plate structures with stiffeners under thermo-mechanical coupling loads.

[0023] Figure 1 This is a flowchart illustrating a method for numerical reconstruction of the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate. (Reference) Figure 1The first embodiment of the present invention provides a method for numerical reconstruction of the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate, including the following steps: S101, Obtain the three-dimensional displacement field data of the non-reinforcing rib region on the outer wall of the flexible plate as a known boundary condition; wherein, the flexible plate is a thin plate structure with multiple transverse reinforcing ribs, and the non-reinforcing rib region is the plate surface region far from the connection part of the actuator between adjacent reinforcing ribs.

[0024] The purpose of this step is to obtain the known boundary conditions as input for finite element analysis. By selecting a specific measurement area on the outer wall (non-reinforcing rib area), the engineering obstacles of difficult sensor or visual marker placement in the reinforcing rib area connected to the actuator are avoided, while ensuring reconstruction accuracy.

[0025] like Figure 2 As shown, the flexible plate structure, from right to left along its length (X direction), includes a fixed end 23, multiple reinforcing ribs 21, and a non-reinforcing rib region 22 located between adjacent reinforcing ribs. The fixed end 23 is connected to a rigid support via high-strength bolts, providing a fixed constraint for the entire flexible plate (displacement is zero at this point). The reinforcing ribs serve as the connection points between the flexible plate and external actuating rods (such as electric actuators). The actuating rods act on the reinforcing ribs via hinges, forming concentrated load application points that drive the deformation of the flexible plate.

[0026] The non-reinforcing rib area is located between two adjacent reinforcing ribs, at the midpoint of the connection point between the two actuators, naturally forming a flat plate surface away from the actuator connection point. Simultaneously, this area is also far from the rigid constraint zone of the fixed end, situated in the middle section where the elastic deformation of the flexible plate is most gradual and continuous.

[0027] This centrally located position gives the non-reinforced area the following technical characteristics: No mechanical obstruction: There are no fixed-end bolts, hinge supports, or other connecting components on the surface, avoiding the dual interference of fixed-end constraints and actuator connecting parts; Moderate deformation: Compared to the fixed end (zero displacement) and the root of the reinforcing rib (point of concentrated force application), the deformation field in this area is smooth and continuous, containing sufficient displacement information while avoiding nonlinear errors caused by local stress concentration; Good maintainability: It does not directly bear the concentrated load of the actuator and is not affected by the clamping force of the fixed end, resulting in high long-term stability of the surface marking points.

[0028] In contrast, the area with stiffeners is unsuitable as an input area for displacement measurement because it needs to be hinged to the actuator, resulting in physical obstruction from mechanical connecting parts. Furthermore, the reciprocating motion of the actuator can interfere with the measurement stability of this area. By selecting the non-stiffener area as the input, the engineering obstacles at the actuator connection point are avoided, and the sufficiently large flat area between the two stiffeners is utilized to obtain high-quality displacement data.

[0029] The method for acquiring the three-dimensional displacement field data of the outer wall can be any conventional contact or non-contact measurement method in the art. For example, contact or semi-contact sensors such as resistance strain gauge displacement sensors, laser displacement sensors, and eddy current sensors can be used to measure the displacement of local points; or optical measurement methods such as binocular stereo vision measurement systems, structured light measurement systems, and digital image correlation methods can be used to acquire the displacement data of the entire field. These measurement methods are all conventional techniques in the art, and this invention does not limit them.

[0030] For the acquired discrete displacement data, the displacement values ​​at the finite element nodes can be obtained through interpolation methods. These interpolation methods can be conventional numerical processing methods such as linear interpolation, spline interpolation, or interpolation based on finite element shape functions. The key point of this invention is that it selects the non-stiffening region as the displacement input region, rather than a specific measurement method or interpolation method.

[0031] As an input boundary condition, the input length of the non-reinforced region along the length of the flexible plate is preferably not less than 75% of the total length of a single non-reinforced region. The reconstruction accuracy is optimal when the input length is the full length of a single non-reinforced region. If the input length is too short (e.g., less than 50% of the total length), sufficient constraint information cannot be provided, leading to a significant increase in the inner wall reconstruction error.

[0032] S102, based on Kirchhoff's thin plate theory, the flexible plate is discretized into finite elements to establish an overall stiffness matrix including stiffened and unstiffened regions; wherein, multiple layers of elements are maintained in the thickness direction, and different element sizes are used to mesh the stiffened and unstiffened regions in the plate surface direction.

[0033] The purpose of this step is to establish a mechanical numerical model of the flexible plate structure, form an overall stiffness equation describing the relationship between structural deformation and load, and lay the foundation for subsequent introduction of boundary conditions and solution.

[0034] The Kirchhoff thin plate theory (classical thin plate theory) establishes deformation assumptions based on the mid-plane (the middle plane in the thickness direction of the thin plate, located at 1 / 2 of the plate thickness): (1) No in-plane displacement assumption—no point in the mid-plane of the thin plate has a displacement parallel to the mid-plane; (2) Straight normal assumption—a straight line segment that is perpendicular to the mid-plane before deformation remains straight and perpendicular to the deformed mid-plane after deformation, and the angle of rotation about the mid-plane is equal to the first partial derivative of the mid-plane deflection. Based on the above assumptions, combined with the geometric equations of elasticity, it can be deduced that the displacement of any point in the thin plate can be determined by the mid-plane deflection and its partial derivative. In particular, for a flexible plate with a fixed thickness, the displacement fields of the outer wall and the inner wall are symmetrical about the mid-plane, wherein the vertical displacements of the outer wall and the inner wall are equal in magnitude and opposite in direction, and the horizontal displacements are also antisymmetric.

[0035] The overall stiffness matrix is ​​a matrix in the finite element method that describes the overall stiffness characteristics of a structure. Its dimension is the number of nodal degrees of freedom × the number of nodal degrees of freedom, and it is assembled by superimposing all element stiffness matrices according to the node number.

[0036] Discretizing a flexible plate into finite elements is a conventional technique in this field. Element types can include four-node tetrahedral elements, hexahedral elements, and shell elements. In a specific embodiment of this invention, an eight-node hexahedral isoparametric element is used to discretize the flexible plate. This element has three translational degrees of freedom at each node, and its displacement distribution within the element can be described by shape functions. It offers high accuracy and good computational efficiency, making it a commonly used solid element type in this field.

[0037] The shape function of the eight-node hexahedral element is a conventional three-dimensional linear shape function, and the displacement of any point within the element satisfies the following relationship with the nodal displacements: ,in Let be the shape function matrix, and u, v, w be the displacements of any point within the element along the X, Y, and Z directions, respectively. These represent the three-dimensional displacement components of the element nodes. Based on the geometric equations of elasticity, the relationship between strain and nodal displacement can be established: Among them, for Strain matrix The strain-displacement matrix is... In This is a submatrix of [B]. Establish the relationship between stress and strain: ,in, Here is the stress matrix. This is the stress-strain matrix. Based on the principle of generalized virtual work, the element stiffness matrix is ​​obtained through element volume integration. , Among them, [B] T Let [B] be the transpose matrix, and V be the element volume; and let all element stiffness matrices be superimposed according to node number to form the overall stiffness matrix.

[0038] To improve computational efficiency and ensure accuracy in critical areas, this invention employs a non-uniform mesh generation strategy, using different element sizes for stiffened and unstiffened regions. This is a key technical measure tailored to the characteristics of stiffened flexible plate structures.

[0039] In the XY plane, the stiffened areas use the first element size, while the unstiffened areas use the second element size, with the first element size being smaller than the second element size in the length direction (X direction). For example, the stiffened areas can use a 7.5mm × 20mm element size, and the unstiffened areas can use a 15.4mm × 20mm element size; or further refinement can be achieved using 5mm × 13.3mm and 10.27mm × 13.3mm element sizes respectively. The denser mesh in the stiffened areas is to accurately capture stress concentration and deformation gradients, while the sparser mesh in the unstiffened areas is to improve computational efficiency.

[0040] In the thickness direction (Z direction), both the stiffened and unstiffened areas preferably use two-layer units. For example, when the thickness of the flexible plate is 6 mm, the unit thickness is 3 mm, so as to accurately simulate the linear distribution of strain caused by bending deformation along the thickness direction.

[0041] Under thermo-mechanical coupling conditions, conventional methods typically consider the changes in material properties (elastic modulus, Poisson's ratio) with temperature, resulting in different stiffness matrices for each element. This necessitates element-by-element calculation and assembly, leading to a large computational burden. This invention adopts a simplified processing strategy: setting the elastic modulus and Poisson's ratio of the flexible plate material as constants (taking values ​​at room temperature), and ignoring the influence of the temperature field on material properties.

[0042] The rationale for this simplification lies in the fact that, for special structural steel, within the temperature range of room temperature to 610K, the variation in elastic modulus does not exceed 7.5% of the room temperature value, and the variation in Poisson's ratio does not exceed 3.6% of the room temperature value. Studies show that the reconstruction error introduced by this simplification strategy (RMS error approximately 0.032mm) is minimal (less than 0.01mm) compared to the strategy considering material property variations (RMS error approximately 0.025mm), but the calculation time is significantly reduced from 215.4 seconds to 4.5 seconds, significantly improving real-time performance. Therefore, in engineering applications, this simplification is a reasonable and more advantageous choice.

[0043] Since the overall stiffness matrix is ​​sparse (non-zero elements are concentrated only near the diagonal), using a sparse matrix storage format is a common technique in this field, which can effectively reduce memory consumption and improve solution efficiency.

[0044] S103, introduce displacement boundary conditions into the overall stiffness matrix, including: assigning zero values ​​to the displacements of all nodes at the fixed end of the flexible plate, and applying the three-dimensional displacement field data of the non-stiffening region as known displacement constraints to the corresponding nodes.

[0045] The purpose of this step is to transform the constraints of the actual structure into mathematical constraint equations, eliminate the singularity of the overall stiffness matrix, and make the system of equations have a unique solution.

[0046] The introduction of displacement boundary conditions is a standard procedure in finite element analysis in this field, and typically includes two types: 1. Fixed end constraint The fixed end of the flexible plate connected to the rigid support has zero displacement in the X, Y, and Z directions. Therefore, all node displacements at the fixed end of the flexible plate are assigned zero values. In the finite element numerical implementation, this constraint can be achieved using the following conventional mathematical processing method: set the rows and columns of the corresponding fixed-end node degrees of freedom in the overall stiffness matrix to 1 (or use the multiplication method or penalty function method), and set the corresponding node force vectors to zero.

[0047] 2. Displacement constraint in non-reinforced regions The three-dimensional displacement field data of the unstiffened region obtained above are used as known displacement constraints and applied to the corresponding nodes. Specifically, for finite element nodes located in the unstiffened region, their corresponding degrees of freedom in the global stiffness matrix are marked as known quantities, the displacement values ​​of these nodes are removed from the solution vector, and the stiffness matrix and load vector are adjusted accordingly. This process ensures that the measured values ​​of the outer wall displacement field are strictly constrained in the numerical model, serving as known conditions for inverting the inner wall displacement.

[0048] S104, based on the overall stiffness matrix after introducing boundary conditions, solve the overall equilibrium equation of the structure to obtain the three-dimensional displacement field of all nodes of the inner wall of the flexible plate.

[0049] The purpose of this step is to solve the deformation response of the structure under known boundary conditions, obtain the displacement vector of all nodes in the field, and then extract the displacement of the inner wall nodes.

[0050] The overall equilibrium equation of the structure is a fundamental equation in the finite element method. ,in, Let [K] be the corrected nodal force vector, and [K'] be the corrected global stiffness matrix. By solving the equilibrium equations using Gaussian elimination, the three-dimensional displacement vector {δ} of all nodes in the flexible plate is obtained, thus realizing the inversion of the global displacement field of the inner wall.

[0051] Solving linear equation systems is a conventional technique in this field, and methods such as Gaussian elimination (direct method), LU decomposition, and conjugate gradient method (iterative method) can be used. In a specific embodiment of the present invention, Gaussian elimination is used to solve the linear equation system composed of the modified nodal force vector and the modified global stiffness matrix, resulting in a three-dimensional displacement vector containing all nodes of the flexible plate. This vector contains the displacement components of all nodes of the flexible plate (including outer wall nodes, inner wall nodes, and internal nodes) in the X, Y, and Z directions.

[0052] According to Kirchhoff's thin plate theory, the vertical displacements of the inner and outer walls are equal in magnitude and opposite in direction, while the horizontal displacements are symmetrical about the mid-plane. Therefore, the correctness of the extraction results can also be verified through this theoretical relationship.

[0053] S105, superimpose the three-dimensional displacement field of all nodes of the inner wall with the initial shape of the inner wall to obtain the deformed global shape of the inner wall.

[0054] The purpose of this step is to combine the obtained displacement increment with the original geometry to obtain the absolute surface coordinates after deformation, thus realizing the final transformation from displacement field to geometric surface.

[0055] The initial inner wall profile refers to the three-dimensional geometric coordinates of the inner wall surface when the flexible plate is unloaded (zero position); the deformed inner wall global profile refers to the three-dimensional coordinates that reflect the true shape of the flexible plate under actual working conditions after superimposed displacement.

[0056] This step transforms the displacement field into a geometric surface through vector superposition. For each node on the inner wall surface, the three-dimensional displacement components obtained in the above steps are algebraically added to the initial coordinates of the corresponding node. The mathematical expression is: In the formula, This represents the three-dimensional surface of all nodes on the inner wall after deformation. Let be the initial coordinates of the global nodes on the inner wall. The displacement field of all nodes on the inner wall can be obtained through the vector {δ}.

[0057] Since finite element method (FEM) calculations yield discrete nodal coordinates, to obtain a continuous surface description for visualization or aerodynamic analysis, spatial interpolation can be performed on the discrete inner wall nodal displacements based on the finite element shape functions to obtain a continuous global inner wall surface. Specifically, within each quadrilateral element, a bilinear shape function is used to calculate the displacement of any point inside the element using the known displacements of the four vertices, thereby reconstructing a smooth and continuous surface.

[0058] Through the above steps, complete surface data containing the coordinates of all nodes on the inner wall of the flexible plate after deformation is finally obtained, completing the entire process from measuring the local displacement of the outer wall to reconstructing the surface of the entire inner wall.

[0059] Figure 3 This is a schematic diagram of a numerical reconstruction system for the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate. (Reference) Figure 3 The second embodiment of the present invention provides a numerical reconstruction system 300 for the global surface profile of the inner wall based on the local displacement field of the outer wall of the flexible plate, including: acquisition unit 301, establishment unit 302, introduction unit 303, solution unit 304, and superposition unit 305. The detailed functions of each unit are described below.

[0060] The acquisition unit 301 is used to acquire three-dimensional displacement field data of the non-reinforcing rib region of the outer wall of the flexible plate as known boundary conditions; wherein, the flexible plate is a thin plate structure with multiple transverse reinforcing ribs, and the non-reinforcing rib region is the plate surface region far away from the connection part of the actuator between adjacent reinforcing ribs. Element 302 is established to discretize the flexible plate into finite elements based on Kirchhoff's thin plate theory and establish an overall stiffness matrix including stiffened and unstiffened regions; wherein, multi-layer elements are maintained in the thickness direction, and different element sizes are used to mesh the stiffened and unstiffened regions in the plate surface direction. The unit 303 is used to introduce displacement boundary conditions into the overall stiffness matrix, including: assigning zero values ​​to the displacements of all nodes at the fixed end of the flexible plate, and applying the three-dimensional displacement field data of the non-stiffening region as known displacement constraints to the corresponding nodes. Solver element 304 is used to solve the overall equilibrium equation of the structure based on the overall stiffness matrix after introducing boundary conditions, and obtain the three-dimensional displacement field of all nodes of the inner wall of the flexible plate. The superposition unit 305 is used to superimpose the three-dimensional displacement field of all nodes of the inner wall with the initial shape of the inner wall to obtain the deformed global shape of the inner wall.

[0061] The following section, using software simulation data, details the specific implementation process and verification results of the method of this invention.

[0062] Example The flexible plate is made of special structural steel with geometric dimensions of 2050 mm × 580 mm × 6 mm (length × width × thickness) and an elastic modulus of 2 × 10⁻⁶. 11 Pa, Poisson's ratio 0.3, yield strength 1000 MPa, coefficient of thermal expansion 1.2 × 10⁻⁶ -5 K -1 The thermal conductivity is 60.5 W / (m·°C). The 50 mm wide area on the right is the fixed end. A 15 mm wide reinforcing rib is set every 400 mm along the length direction, for a total of 5 ribs, forming 5 non-reinforcing rib areas. Each non-reinforcing rib area is 385 mm long.

[0063] The mesh was created using eight-node hexahedral elements. The element size for the stiffened areas was 7.5mm × 20mm × 3mm, while the element size for the non-stiffened areas was 15.4mm × 20mm × 3mm. Both areas had two layers of elements in the thickness direction. The overall model consisted of 51,157 nodes and 9,358 elements.

[0064] To verify the convergence of the numerical solution, three grid density models were constructed: Type 1 unit: ribbed area 15×30×3mm, non-ribbed area 22.6×30×3mm; The second type of unit: the area with reinforcing ribs is 7.5×20×3mm, and the area without reinforcing ribs is 15.4×20×3mm; The third type of unit: the area with reinforcing ribs is 5×13.3×3mm, and the area without reinforcing ribs is 10.27×13.3×3mm; A spatial interpolation strategy based on finite element shape functions was adopted to map the global displacement field of the inner wall obtained by the three types of elements onto a unified node. The root mean square error and maximum absolute error between the models were calculated with the Z-direction displacement as the key index. The results are shown in Table 1.

[0065] Table 1 Comparison of displacement field errors among different mesh models .

[0066] The results show that the root mean square error of the displacement in the Z direction for the second and third element types is only 0.049 mm, and the maximum absolute error is 0.236 mm, indicating comparable error levels. From the perspective of balancing accuracy and efficiency, the second element type is preferred as the standard mesh scheme.

[0067] The influence of different input lengths on the reconstruction accuracy in the non-stiffening region was studied, and four input lengths were selected: 107.8 mm (28%), 200.2 mm (52%), 292.6 mm (76%), and 385 mm (100%). These different outer wall displacement fields were input into the inner wall reconstruction algorithm of this invention to obtain the corresponding global displacement field of the flexible plate inner wall. The result was compared with the inner wall deformation obtained from simulation software. The root mean square error was as follows: Figure 4 As shown.

[0068] The results show that the reconstruction errors of the in-plane deformation of the flexible plate in the X and Y directions are similar in magnitude, both smaller than the surface deformation error in the Z direction. As the input area along the length of the flexible plate increases from 107.8 mm to 385 mm, the reconstruction error of the inner wall shows a significant decreasing trend, indicating that the area of ​​the input displacement field on the outer wall of the flexible plate has a direct impact on the reconstruction accuracy. When x = 385 mm, i.e., using the global displacement data of the non-reinforcing rib region as input, the reconstruction result is closest to the simulation result, with the errors in all directions reaching their minimum, and the root mean square error of the Z-direction displacement being only 0.032 mm. In practical engineering applications, since the size of the reinforcing rib is only x = 15 mm and it is tightly connected to the actuator, directly obtaining the displacement field of the reinforcing rib is challenging. This demonstrates that using the global displacement data of the non-reinforcing rib region as input can effectively reconstruct the inner wall profile.

[0069] A heat source is set above the inner wall of the flexible plate corresponding to the No. 2 actuator to form a non-uniform temperature field (maximum temperature 610K) inside the flexible plate through radiation. At the same time, the actuator is subjected to stroke loads (20mm, 90mm, 77mm, 28mm, 6mm) to form thermo-mechanical coupling deformation.

[0070] Two strategies were used for reconstruction: Strategy A: Consider the changes in material properties with temperature (elastic modulus and Poisson's ratio are functions of temperature). Strategy B: Ignore the effect of temperature on material properties (material properties are constant).

[0071] The global displacement fields of the inner wall obtained by the two reconstruction strategies were compared and analyzed with the simulation baseline results to evaluate the reconstruction error of the two strategies under the action of a non-uniform temperature field. At the same time, the computation time of the two reconstruction strategies was compared to analyze their computational efficiency. The results are shown in Table 2.

[0072] Table 2. Whether the impact of material property variations on reconstruction error and computation time is considered. .

[0073] The results show that strategy A takes 215.4 seconds, with a root mean square error of 0.025 mm in the Z direction and a maximum absolute error of 0.101 mm; strategy B takes 4.5 seconds, with a root mean square error of 0.032 mm in the Z direction and a maximum absolute error of 0.124 mm. The difference in accuracy between the two strategies is less than 0.01 mm, but strategy B improves efficiency by 48 times, which verifies the rationality and advantages of using the assumption of constant material properties in engineering applications.

[0074] Through the above embodiments, the present invention achieves rapid and high-precision reconstruction of the inner wall profile of the flexible plate, providing reliable technical support for the precise control of the flexible wall nozzle. Those skilled in the art should understand that the above specific embodiments are only for illustrating the present invention and not for limiting the scope of protection of the present invention. Any modifications, equivalent substitutions, or improvements made to the technical solution of the present invention without departing from the technical principles of the present invention should fall within the scope of protection of the present invention.

Claims

1. A method for numerical reconstruction of the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate, characterized in that, Includes the following steps: The three-dimensional displacement field data of the non-reinforcing rib region on the outer wall of the flexible plate is obtained as a known boundary condition; wherein, the flexible plate is a thin plate structure with multiple transverse reinforcing ribs, and the non-reinforcing rib region is the plate surface region far away from the connection part of the actuator between adjacent reinforcing ribs. Based on Kirchhoff's thin plate theory, the flexible plate is discretized into finite elements to establish an overall stiffness matrix that includes stiffened and unstiffened regions. In this matrix, multiple layers of elements are maintained in the thickness direction, and different element sizes are used to mesh the stiffened and unstiffened regions in the plate surface direction. The displacement boundary conditions are introduced into the overall stiffness matrix, including: assigning zero values ​​to the displacements of all nodes at the fixed end of the flexible plate, and applying the three-dimensional displacement field data of the non-stiffening region as known displacement constraints to the corresponding nodes. Based on the overall stiffness matrix after introducing boundary conditions, the overall equilibrium equation of the structure is solved to obtain the three-dimensional displacement field of all nodes of the inner wall of the flexible plate. The three-dimensional displacement field of all nodes of the inner wall is superimposed with the initial shape of the inner wall to obtain the deformed global shape of the inner wall.

2. The numerical reconstruction method for the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate as described in claim 1, characterized in that, Methods for establishing the global stiffness matrix, including stiffened and unstiffened regions, include: The flexible plate is discretized using eight-node hexahedral isoparametric elements. Based on the geometric equations and constitutive relations of elasticity, strain-displacement matrix and stress-strain matrix are established respectively; Based on the principle of generalized virtual work, the element stiffness matrix is ​​obtained by volume integration of the element, and the overall stiffness matrix is ​​formed by superimposing all element stiffness matrices according to the node number.

3. The numerical reconstruction method for the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate as described in claim 2, characterized in that, The displacement of any point within the eight-node hexahedral isoparametric element satisfies the shape function relationship with the nodal displacement. Based on this shape function relationship, the displacement of any point within the element is calculated from the nodal displacement, thereby achieving a continuous expression of the displacement field within the element.

4. The numerical reconstruction method for the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate as described in claim 1, characterized in that, Methods for meshing stiffened and unstiffened regions using different element sizes in the plate surface direction include: The reinforcing rib area adopts a first unit size, and the non-reinforcing rib area adopts a second unit size, wherein the first unit size is smaller than the second unit size in the length direction; In the thickness direction, both the reinforcing rib region and the non-reinforcing rib region adopt two-layer units.

5. The numerical reconstruction method for the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate as described in claim 4, characterized in that, The first unit has a size of 7.5mm×20mm×3mm, and the second unit has a size of 15.4mm×20mm×3mm; or the first unit has a size of 5mm×13.3mm×3mm, and the second unit has a size of 10.27mm×13.3mm×3mm.

6. The numerical reconstruction method for the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate as described in claim 1, characterized in that, Under thermo-mechanical coupling conditions, when establishing the overall stiffness matrix including the stiffened and unstiffened regions, the elastic modulus and Poisson's ratio of the flexible plate material are set as constants, and the influence of the temperature field on the material properties is ignored, so that the element stiffness matrix only needs to be calculated once.

7. The numerical reconstruction method for the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate as described in claim 1, characterized in that, Kirchhoff's thin-plate theory includes: Assume that there is no displacement parallel to the mid-plane of the flexible plate; Assume that the normal to the mid-plane of the flexible plate remains straight before and after deformation, and that the angle of rotation around the mid-plane is equal to the first partial derivative of the mid-plane deflection. Based on the above assumptions, a mapping relationship is established between the displacement of the outer wall and the displacement of the inner wall about the mid-plane symmetry, wherein the vertical displacements of the outer wall and the inner wall are equal in magnitude and opposite in direction.

8. The numerical reconstruction method for the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate as described in claim 1, characterized in that, Methods for solving the overall equilibrium equations of a structure to obtain the three-dimensional displacement field of all nodes on the inner wall of a flexible plate include: The overall equilibrium equation of the structure is solved by Gaussian elimination to obtain the three-dimensional displacement vector containing all nodes of the flexible plate. The nodal displacement components located on the inner wall surface are extracted from the three-dimensional displacement vector to form a three-dimensional displacement field for all nodes of the inner wall.

9. The numerical reconstruction method for the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate as described in claim 1, characterized in that, The method for superimposing the three-dimensional displacement field of all nodes of the inner wall with the initial shape of the inner wall includes: The three-dimensional displacement field of all nodes of the inner wall is vector-superimposed with the corresponding node coordinates of the initial shape of the inner wall. Spatial interpolation of discrete inner wall nodal displacements is performed using finite element shape functions to obtain a continuous global inner wall profile.

10. A numerical reconstruction system for the global surface profile of the inner wall based on the local displacement field of the outer wall of a flexible plate, characterized in that, include: The acquisition unit is used to acquire three-dimensional displacement field data of the non-reinforcing rib region of the outer wall of the flexible plate as known boundary conditions; wherein, the flexible plate is a thin plate structure with multiple transverse reinforcing ribs, and the non-reinforcing rib region is the plate surface region far away from the connection part of the actuator between adjacent reinforcing ribs. A finite element is established to discretize the flexible plate into finite elements based on Kirchhoff's thin plate theory, and to establish an overall stiffness matrix including stiffened and unstiffened regions; wherein, multiple layers of elements are maintained in the thickness direction, and different element sizes are used to mesh the stiffened and unstiffened regions in the plate surface direction; An introduction unit is used to introduce displacement boundary conditions into the overall stiffness matrix, including: assigning zero values ​​to the displacements of all nodes at the fixed end of the flexible plate, and applying the three-dimensional displacement field data of the non-stiffening region as known displacement constraints to the corresponding nodes. The solver element is used to solve the overall equilibrium equation of the structure based on the overall stiffness matrix after introducing boundary conditions, and obtain the three-dimensional displacement field of all nodes on the inner wall of the flexible plate. The superposition unit is used to superimpose the three-dimensional displacement field of all nodes of the inner wall with the initial shape of the inner wall to obtain the deformed global shape of the inner wall.

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