An entity element isogeometric analysis method, device and equipment and storage medium for variable-thickness composite materials

Abstract

The embodiment of the application provides a kind of entity unit isogonic analysis method, device, equipment and storage medium for variable-thickness composite material, the method comprises: based on entity geometry model, construct the entity grid model of NURBS, and extract the parameterization information of NURBS;Local coordinate system is established in each gauss point inside geometry entity using parameterization information, determine the first conversion relationship of local coordinate system and global physical coordinate system;Based on entity grid model, construct nonlinear function in parameter space, use L-M algorithm to iteratively solve the intersection of each ply offset surface and thickness direction parameter curve of variable-thickness composite material, determine the integral interval inside each entity unit;Based on geometric volume integral term and the ply trigonometric function of variable-thickness composite material, determine global stiffness matrix;Based on global stiffness matrix, execute isogeometric analysis to determine the laminated structure strain energy, vibration frequency and buckling factor of composite material by loading and defining boundary conditions.

Description

Technical Field

[0001] This application relates to the field of variable thickness composite material analysis technology, specifically to a geometric analysis method, apparatus, equipment, and storage medium for solid elements of variable thickness composite materials. Background Technology

[0002] In related technologies, with the increasing demand for deep-sea resource development and scientific exploration in the marine engineering field, fiber-reinforced polymer (FRP) composites have become the core material for next-generation deep-sea submersibles and high-performance propulsion vehicles due to their advantages such as high specific strength, resistance to seawater corrosion, and designable hydroelastic / acoustic properties. In order to pursue the ultimate mechanical performance and hydrodynamic efficiency in harsh marine environments, modern marine composite structures are showing a trend of high complexity in terms of geometric topology and material distribution. Specifically, the structural forms have evolved from simple plates and shells to three-dimensional solid configurations with variable cross-sections, variable thickness (gradual thickness), and free-form surfaces (e.g., pressure hulls of uniform thickness, hydrofoil sails of varying thickness, and propeller blades with completely free-form surfaces).

[0003] For the mechanical property analysis of such laminated composite structures, the simplified shell theory based on the plane stress assumption is no longer applicable because real marine structures (such as propeller blade roots) are often in a complex three-dimensional stress state, and interlaminar stress is a key driving factor for delamination failure of composite materials. Existing computer-aided engineering (CAE) software and numerical algorithms, whether traditional finite element methods or standard isogeometric analysis methods, cannot simultaneously ensure geometric accuracy, high-fidelity analysis (i.e., solving material discontinuities caused by ply cut-off), and computational efficiency (i.e., avoiding degree-of-freedom explosion and massive repetitive matrix integration calculations) for complex, variable-thickness three-dimensional laminated solid structures. How to perform mechanical property analysis on solids of complex geometrically variable-thickness composite materials has become an urgent technical problem to be solved. Summary of the Invention

[0004] The purpose of this application is to provide a geometric analysis method, apparatus, device, and storage medium for solid elements of composite materials with varying thickness. The specific technical solution adopted is as follows: In a first aspect, a geometric analysis method for solid elements and other components of composite materials with varying thickness is provided, the method comprising: A solid mesh model of three-dimensional non-uniform rational B-splines is constructed based on the solid geometric model, and the parameterization information of the three-dimensional non-uniform rational B-splines is extracted. According to the Gauss-Legendre rule of the Gauss-Legendre integral, a local coordinate system is established at each Gaussian point inside the geometric entity using the parameterized information, and the first transformation relationship between the local coordinate system and the global physical coordinate system is determined. Based on the solid mesh model, a nonlinear function is constructed in the parameter space, and the intersection of the offset surface of each ply of the variable thickness composite material and the thickness direction parameter curve is solved iteratively using the Levenberg-Marquardt LM algorithm to determine the integral interval inside each solid element. The global stiffness matrix is ​​determined based on a preset geometric volume integral term and the ply trigonometric functions of the variable thickness composite material, wherein the preset geometric volume integral term is determined based on the first transformation relationship, the constant matrix of the variable thickness composite material, and the integration interval. Based on the global stiffness matrix, loads are applied and boundary conditions are defined to perform geometric analysis to determine the strain energy, vibration frequency, and buckling factor of the composite laminate structure.

[0005] Secondly, a geometric analysis apparatus for solid elements and other components of variable-thickness composite materials is provided, the apparatus comprising: The construction and extraction module is used to construct a solid mesh model of three-dimensional non-uniform rational B-spline NURBS based on the solid geometric model, and extract the parametric information of the three-dimensional non-uniform rational B-spline. The first determining module is used to establish a local coordinate system at each Gaussian point inside the geometric entity using the parameterized information according to the Gauss-Legendre rule of Gauss-Legendre integral, and to determine the first transformation relationship between the local coordinate system and the global physical coordinate system. The second determining module is used to construct a nonlinear function in the parameter space based on the solid mesh model, and use the Levenberg-Marquardt LM algorithm to iteratively solve the intersection of the offset surface of each ply of the variable thickness composite material with the thickness direction parameter curve, so as to determine the integral interval inside each solid element. The third determining module is used to determine the global stiffness matrix based on a preset geometric volume integral term and the ply trigonometric function of the variable thickness composite material, wherein the preset geometric volume integral term is determined based on the first transformation relationship, the constant matrix of the variable thickness composite material, and the integration interval. The fourth determination module is used to perform geometric analysis, such as applying loads and defining boundary conditions based on the global stiffness matrix, to determine the strain energy, vibration frequency and buckling factor of the composite material laminate structure.

[0006] Thirdly, an electronic device is provided, comprising: a memory and at least one processor, wherein the memory stores instructions; the at least one processor invokes the instructions in the memory to cause the electronic device to perform the above-described method for geometric analysis of solid elements and other components of variable thickness composite materials.

[0007] Fourthly, a computer program product is provided, comprising: computer program code, which, when run on a computer, causes the computer to perform the method described in the first aspect or any possible implementation thereof.

[0008] Fifthly, a computer-readable storage medium is provided that stores computer program code, which, when executed on a computer, causes the computer to perform the methods described in the first aspect or any possible implementation thereof.

[0009] This application offers the following advantages: By applying an intersection algorithm based on LM iteration in the parameter space, the precise integration intervals of each ply cutoff within a variable-thickness solid element are accurately solved. This "layer-by-layer integration" strategy directly encapsulates complex material ply cutoff information within three-dimensional numerical integration points without altering the element mesh structure. It achieves complete decoupling between mesh geometry (degrees of freedom) and physical ply (material), fundamentally avoiding the "degrees of freedom explosion" and element distortion problems in variable-thickness structures. Complex geometric volume integral terms are pre-calculated and stored. When the ply angle changes, the stiffness matrix is ​​updated simply by multiplying the pre-stored matrix with the new angle function, avoiding extremely time-consuming repetitive three-dimensional volume integral calculations. Attached Figure Description

[0010] To more clearly illustrate the technical solutions and advantages in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0011] Figure 1 A schematic diagram illustrating the implementation process of a geometric analysis method for solid elements and other components of composite materials with varying thickness, provided in an embodiment of this application; Figure 2 A schematic diagram illustrating the application of NURBS in parameter space and physical space, provided for an embodiment of this application; Figure 3 A schematic diagram illustrating the geometric mapping and parameter inversion between the physical space and parameter space of a variable thickness laminate structure provided in this application embodiment; Figure 4 A flowchart of the first nonlinear function solving algorithm provided in this application embodiment; Figure 5 A flowchart of the second nonlinear function solving algorithm provided in this application embodiment; Figure 6 This application provides schematic diagrams illustrating the geometric topological evolution of three typical composite material structures in its embodiments. Figure 7 A schematic diagram illustrating the principle and integration interval division of performing a layer-by-layer integration algorithm for structures with different geometric complexities, provided for embodiments of this application; Figure 8 A schematic diagram of a geometric analysis device for solid elements and other components of composite materials of varying thickness provided in this application embodiment; Figure 9 This is a schematic diagram of the structure of a computer device provided in an embodiment of this application. Detailed Implementation

[0012] To further illustrate the technical means and effects adopted by this application to achieve the intended inventive purpose, the following, in conjunction with the accompanying drawings and preferred embodiments, details the specific implementation, structure, features, and effects of a geometric analysis method for solid elements of variable thickness composite materials proposed in this application. In the following description, different "one embodiment" or "another embodiment" do not necessarily refer to the same embodiment. Furthermore, specific features, structures, or characteristics in one or more embodiments can be combined from any suitable form.

[0013] In the description of the embodiments of this application, unless otherwise stated, " / " means "or". For example, A / B can mean A or B. The "and / or" in the text is merely a description of the relationship between related objects, indicating that there can be three relationships. For example, A and / or B can mean: A exists alone, A and B exist simultaneously, and B exists alone. In addition, in the description of the embodiments of this application, "multiple" means two or more.

[0014] Hereinafter, the terms "first" and "second" are used for descriptive purposes only and should not be construed as implying or suggesting relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature.

[0015] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains.

[0016] To address the challenges of traditional finite element method (FEM) in handling complex variable-thickness composite materials, including difficulties in mesh generation, frequent model trimming errors, and degree-of-freedom explosion, as well as the numerical integration distortion and low computational efficiency of existing isogeometric analysis (IGA) in three-dimensional solid laminates due to internal material discontinuities, this application provides a method for analyzing geometric stress, vibration frequency, and buckling factor of complex geometric variable-thickness composite solids, which can be achieved through the following steps: Step S1: Obtain geometric input and construct geometric analysis models of laminated composite materials, etc. S11: Importing CAD geometric models into computer devices; During implementation, computer equipment acquires initial CAD geometric data for marine composite structures (such as pressure hulls, hydrofoils, or propeller blades). The data source can be analytical geometry generated from standard NURBS coefficients, geometric models generated by CAD software, or NURBS interpolation of spatially discrete data points.

[0017] S12: Extract 3D solid model information from CAD geometric model; During implementation, 3D solid geometry and analysis information can be extracted from the initial geometric model to obtain NURBS parametric information, including node vectors ( U , V , W ), control point coordinates and weights ( P ijk , W ijk ), and the order of the spline polynomial in each direction ( p , q , r ), i , j , k These represent the number of Gaussian points in the three dimensions.

[0018] S13: Without altering the initial precise geometry, improve the order through node insertion or order enhancement (e.g.) k The `-refinement` option performs a mesh refinement operation, constructing a 3D NURBS solid parametric space mesh model for subsequent analysis.

[0019] Step S2: Define the mapping between physical ply and local coordinate system; S21: Utilize NURBS parameterization information, with parameters xUsing the direction as a reference, a local frame field is established as a local coordinate system, and the transformation relationship between the local coordinate system and the global physical coordinate system is determined. .

[0020] S22: Align the material coordinate system with the local curve coordinate system, based on the input composite material thickness and ply angle information (such as ply orientation angle). i ), calculate the fiber orientation transformation matrix T( i ).

[0021] Step S3: Perform layer-by-layer integration based on the Levenberg-Marquardt (LM) algorithm to determine the precise integration interval; To address the issue of internal layer cutoff caused by varying thickness, the 3D integration points are divided into in-plane integration points and thickness direction points based on parametric settings. g (Direction) Integration point. For laminated structures with gradually varying thickness, perform the following intersection algorithm: S31: Determine the integration interval for a two-dimensional constant cross-section variable thickness structure; For structures with two-dimensional variable curvature cross sections (such as hydrofoils), based on the NURBS volumetric CAD model (3D NURBS solid parametric spatial mesh model), along the in-plane parameters ( x , or The integration points are iterated in the direction, and at each integration point, a first nonlinear function of the physical layup offset curve and the thickness direction parameter curve is constructed.

[0022] With initialization parameters (such as [ x =0.5, or =0.5]) and fixed regularization parameter l As the starting point of the first nonlinear function, the intersection point of the planar lines is solved using the LM iterative algorithm.

[0023] The intersection of the resulting plane lines g The coordinates define the precise integration interval boundaries for each ply at that integration location.

[0024] S32: Determine the integration interval for three-dimensional structures with variable cross-section and thickness; For a three-dimensional completely free surface structure (such as a propeller), the second nonlinear function of the three-dimensional ply offset surface and the spatial parameter curve is constructed by traversing the in-plane integration points.

[0025] Set initial iteration values ​​for the second nonlinear function and employ dynamically adjusted regularization parameters. l Perform LM iteration to solve for the intersection points of spatial surfaces and lines, and then determine the integration interval of each intersecting layer within the complex geometry.

[0026] Step S4: Fast calculation of stiffness matrix based on decoupling of integral interval and ply orientation angle within the geometric interior. After establishing the integration interval, the global stiffness matrix is ​​numerically calculated. To avoid repetitive volume integration during the optimization design of laminated structures: S41: Reconstruct the constitutive matrix of the anisotropic material into several constant material matrices and a matrix relating the ply orientation angles. i The product and sum of trigonometric function vectors.

[0027] Based on the equilibrium equation and the principle of virtual work, the expression for the stiffness matrix can be derived as the following formula (1): (1); Where D is usually the constitutive matrix. Represents the strain-displacement matrix. Let B be the transpose of B, and V be the integration interval. Considering the transformation between the global, local, and material coordinate systems in fiber-reinforced laminated structures, the stiffness matrix can be calculated using the following equation (2): (2); Where R is the transformation matrix from the global coordinate system to the local coordinate system, T is the transformation matrix from the local coordinate system to the material coordinate system, and B is the strain-displacement matrix. For the transpose of B, This indicates the first transformation relation. for transpose, This indicates the second transformation relation. express The transpose of , where C is the constitutive matrix of the orthogonal anisotropic material. Let θ denote the constitutive matrix of an orthotropic material, where θ is the ply orientation angle.

[0028] For laminated structures containing multiple layers, the stiffness matrix can be further calculated as follows: (3); in, Represents the strain-displacement matrix. For the transpose of B, This represents the first transformation relationship. for transpose, This represents the second transformation relationship. express The transpose of , where C is the constitutive matrix of the orthogonal anisotropic material. It is the total number of layers (a "layer" refers to a group of consecutive single layers that share the same ply angle). l Indicates a single-level index ( l = 1, 2, ...,n l ),and V k Indicates the first k A single-layer integral volume domain. Specifically, these integrals can be calculated by applying the Gauss-Legendary integral rule; the stiffness matrix can be approximated by performing a three-dimensional Gaussian point integral in the parameter space.

[0029] The constitutive matrix in the stiffness matrix formula can also be expressed as the following formula (4): (4); in, This indicates the second transformation relation. express transpose, A is the constitutive matrix of an orthogonal anisotropic material. i These are some constant matrices related to material properties. f i ( i The trigonometric function vector related to the ply angle θ can be expressed as the following formula (5): (5); in, i This indicates the layup direction angle of the variable thickness composite material.

[0030] Therefore, based on formula (4), the stiffness matrix formula of formula (3) above can be expressed as the following formula (6): (6); Where B is the strain-displacement matrix, For the transpose of B, This indicates the first transformation relation. for transpose, i The ply direction angle, The constant matrix represents the variable thickness composite material. f i,k ( i ) indicates the first k The layup trigonometric function of a single-layer variable thickness composite material n l This indicates the total number of layers with different ply angles. V k Indicates the first k A single-layer integral volume domain i express f The ordinal numbers of the trigonometric functions in the function.

[0031] Furthermore, formula (6) can be further derived into the following formula (7): (7); When the ply angle changes, this formula can be pre-stored. This significantly avoids a large amount of repetitive calculations and has higher computational efficiency. k No. k A single-layer strain-displacement matrix, For B k The transpose operation. k Indicates the first k A single-layer first transformation relationship for k The transpose operation. The constant matrix represents the variable thickness composite material. n l This indicates the total number of layers with different ply angles. V k Indicates the first k A single-layer integral volume domain f i,k ( i ) indicates the first k The layup trigonometric function of a single-layer variable thickness composite material i express f The ordinal numbers of trigonometric functions in the function. i Indicates the ply direction angle.

[0032] S42: Extract the complex geometric volume integral terms, which include the strain-displacement matrix and the Jacobian matrix, from the integral formula of the global stiffness matrix. During the initial calculation, perform the calculation of these angle-independent geometric integral terms and store them in the computer memory.

[0033] Here, the complex geometric volume integral term, including the strain-displacement matrix and the Jacobian matrix, is: .

[0034] S33: When a command to change the ply angle is received (e.g., during optimization iteration), the computer directly calls the pre-stored geometric integral term in memory, multiplies it by the trigonometric function vector corresponding to the new angle, and thus quickly generates the updated global stiffness matrix. Here, when the ply angle changes, this formula can be pre-stored... This significantly avoids a large number of repetitive calculations and has higher computational efficiency.

[0035] Step S5: Global matrix assembly, solution, and result reconstruction; S51: Assemble the generated global stiffness matrix, along with the mass matrix and geometric stiffness matrix obtained from the integrals of density and initial stress, into a global matrix system.

[0036] S52: Apply the set external load and boundary conditions, and substitute them into the static equilibrium equation, free vibration equation or linear buckling eigenvalue equation to solve for the nodal displacement, natural frequency or buckling factor.

[0037] S53: Reconstruct the physical field using the solved displacement field coefficients and the original NURBS basis functions, and output the calculated results such as stress and strain for visualization post-processing.

[0038] In this embodiment of the application, the hierarchical integral interval solution algorithm based on the LM algorithm iteration addresses the problem of material discontinuity (ply truncation) within a solid element. The key is that it does not adopt the traditional approach of dividing the mesh, but instead constructs a difference function in the parameter space and uses the LM algorithm to iteratively solve the intersection points of each ply offset surface and the thickness direction parameter curve, thereby accurately determining the boundary of the integral interval within each solid element.

[0039] In this embodiment, the computational decoupling logic of geometric discretization (mesh) and material discretization (layout) is as follows: the key point is to implicitly encapsulate the complex solid layout information (including thickness gradient and angle change) in the numerical Gaussian integration points, so that even coarse meshes can capture the complex internal layout truncation mechanical characteristics, thus eliminating the degree of freedom explosion from the root.

[0040] In this embodiment, the mathematical separation and accelerated pre-calculation of the stiffness matrix integral term and the ply angle function term are achieved as follows: When calculating the global stiffness matrix, the geometric volume integral term, which includes the Jacobian matrix and the strain-displacement matrix, is separated from the ply angle function term. i The trigonometric function terms (1, cos2) i cos4 i sin2 i sin4 i This involves analytical decoupling. The key point is clarifying the accelerated computation logic of pre-stored geometric integral terms and rapid multiplication of angle terms.

[0041] This application provides a geometric analysis method for solid elements and other components in composite materials with varying thicknesses, such as... Figure 1 As shown, this can be achieved through the following steps: Step S110: Construct a solid mesh model of three-dimensional non-uniform rational B-spline NURBS based on the solid geometric model, and extract the parameterization information of the three-dimensional non-uniform rational B-spline. In some embodiments, the geometric model of the entity includes geometrically three-dimensional non-uniform rational B-spline control points, a geometric model generated based on computer-aided design, or a geometric model generated based on discrete points.

[0042] During implementation, computer equipment acquires initial CAD geometric data for marine composite structures (such as pressure hulls, hydrofoils, or propeller blades). The data source can be analytical geometry generated from standard NURBS coefficients, geometric models generated by CAD software, or NURBS interpolation of spatially discrete data points.

[0043] In some embodiments, the characteristics of NURBS curves can be utilized to perform mesh refinement operations on the solid mesh model using node insertion or order boosting methods to obtain a fully subdivided solid mesh model.

[0044] Here, without changing the initial precise geometry, node insertion or order enhancement (such as...) are used to improve the geometry. k The `-refinement` option performs a mesh refinement operation, constructing a 3D NURBS solid parametric space mesh model for subsequent analysis.

[0045] During implementation, 3D solid geometry and analysis information can be extracted from the initial geometric model to obtain NURBS parametric information, including node vectors ( U , V , W ), control point coordinates and weights ( P ijk , W ijk ), and the order of the spline polynomial in each direction ( p , q , r ), i , j , k These represent the number of Gaussian points in the three dimensions.

[0046] Step S120: Based on the Gauss-Legendre rule of Gauss-Legendre integral, establish a local coordinate system at each Gauss point inside the geometric entity using the parameterized information, and determine the first transformation relationship between the local coordinate system and the global physical coordinate system; During implementation, NURBS parameterization information can be utilized, and the Gauss-Legendre integration rule can be applied using parameters. x Using the direction as a reference, a local frame field is established as a local coordinate system, and the first transformation relationship between the local coordinate system and the global physical coordinate system is determined. .

[0047] Figure 2 A schematic diagram illustrating the application of NURBS in parameter space and physical space is provided for embodiments of this application, as shown below. Figure 2As shown, it includes a parametric space 21 and a physical space 22, where...

[0048] Parametric space 21 illustrates a three-dimensional cubic parametric space. The three axes of the cube are labeled as follows: x , or and g Each axis has a value ranging from 0 to 1. Inside the cube, there is a green dot that represents the position of a specific parameter.

[0049] Three vectors are labeled on the three faces of the cube: x Belongs to set U , U =[0 0 0 1 1 1]; or Belongs to set V , V =[0 0 0 1 1 1]; g Belongs to set W , W =[0 0 0 1 1 1]; These vectors represent the weights or parameterization information of NURBS control points.

[0050] Physical space 22 demonstrates the mapping effect of NURBS in physical space, divided into undeformed and undeformed versions. X ) and after deformation (Deformed) x Two states are displayed. The global coordinate system is also shown. o - x.y.z and local coordinate system o'- e 1 e 2 e 3.

[0051] Undeformed state X The graph on the left represents an undeformed NURBS surface, using a global coordinate system. o-xyz The surface is composed of red control points (labeled "NURBS coefficients") and black control grid lines. The control points are connected by curves to form a smooth surface. A local coordinate system is also labeled on the surface. oh-oh 1 e 2 e 3. Used to describe the local geometric properties of curved surfaces.

[0052] Deformed state x The graph on the right represents the deformed NURBS surface, also using the global coordinate system. o - x.y.z and local coordinate system o'- e 1 e 2 e 3. The positions of the control points on the deformed surface change, and the control grid lines are adjusted accordingly, demonstrating the deformation effect of the surface under stress or other factors.

[0053] Figure 3 This application provides a schematic diagram of the geometric mapping and parameter inversion between the physical space and parameter space of a variable thickness laminated structure, as shown in the embodiments. Figure 3 As shown in the diagram, the schematic includes: parameter space 21 and physical space 22, wherein, Physical Space Coordinate System x , y ): Represents the actual geometric coordinates of the structure in the real world. It shows the true form of the structure, which is a multi-layered structure with curvature and potentially varying thickness. Here, the boundaries of the layers are complex curves.

[0054] Parametric Space Coordinate System x , g ξ represents the local (or dimensionless) coordinates used in calculations or modeling. Typically, ξ represents the in-plane or curve direction coordinates (0-1 in the illustration), and ζ represents the thickness or normal direction coordinates. This stretches or maps complex physical geometry onto a relatively regular reference domain (x-axis). x (Normalized to 0 to 1). Although the outer boundary becomes regular, the inner ply boundary curves still exhibit a specific curvature distribution due to the mapping relationship.

[0055] Forward geometric mapping G , indicating from the parameter space ( x , g Mapped to physical space x , y The transformation function of ).

[0056] Inverse Mapping G -1 , indicating from physical space ( x , y Invert back to parameter space ( x , g The transformation function of ).

[0057] Layers 1-4: These represent different material layers or physical regions that make up the structure. These layers correspond to each other in both spaces.

[0058] ( x 0, y 0): Any target point in physical space (marked in layer 2 in the figure).

[0059] ( x 0, g 0): The parameter coordinates of the target point in the parameter space after inverse mapping.

[0060] Coordinate mapping relationship (first transformation relationship): The structure is connected between these two spaces through a mapping function. Given points in the parameter space, it can be transformed through... G Its physical coordinates can be quickly calculated; conversely, if the physical coordinates are known, they can be calculated using... G -1 Solve for its parameter coordinates.

[0061] The red arrows and text ("Parameter Inversion / Material Lookup") in the diagram illustrate the core purpose of the algorithm or process.

[0062] In practical calculations or analyses, it is necessary to know a certain point in physical space ( x 0, g 0) material properties (e.g., which layer it belongs to).

[0063] Due to the complexity of ply boundaries in physical space, direct determination is extremely difficult. Therefore, inverse mapping can be used. G -1 By "inversely" calculating the coordinates of this point back into the parameter space, we can obtain the corresponding parameter coordinates. x 0, g 0).

[0064] In the parameter space, determine the point ( x 0, g 0) Determining which region it falls into (such as layer 2 in this figure) is relatively more standardized and easier. Once the layer is determined, the material parameters corresponding to that layer can be directly accessed or queried.

[0065] Step S130: Based on the solid mesh model, construct a nonlinear function in the parameter space, and use the Levenberg-Marquardt LM algorithm to iteratively solve the intersection of the offset surface of each ply of the variable thickness composite material with the thickness direction parameter curve, so as to determine the integral interval inside each solid unit. Here, to address the issue of internal layer cutoff caused by varying thickness, the 3D integration points can be divided into in-plane integration points and thickness direction points based on parameter settings. g(Integration points in the direction of thickness). For laminated structures with gradually varying thickness, the Levenberg-Marquardt (LM) algorithm is used to iteratively solve for the intersection points of the offset planes of each layup of the variable thickness composite material with the thickness direction parameter curves, in order to determine the integration interval within each solid element.

[0066] Step S140: Determine the global stiffness matrix based on the preset geometric volume integral term and the ply trigonometric function of the variable thickness composite material, wherein the preset geometric volume integral term is determined based on the first transformation relationship, the constant matrix of the variable thickness composite material, and the integration interval; The objective expression for the global stiffness matrix K is: (7); in, ; B represents the preset geometric volume integral term. k Indicates the first k A single-layer strain-displacement matrix, For B k The transpose operation. k Indicates the first k A single-layer first transformation relationship for k The transpose operation. The constant matrix represents the variable thickness composite material. n l This indicates the total number of layers with different ply angles. V k Indicates the first k A single-layer integral volume domain f i,k ( i ) indicates the first k Plyback trigonometric functions for a single-layer variable thickness composite material i express f The ordinal numbers of trigonometric functions in the function. i Indicates the ply direction angle.

[0067] In structural optimization design or variable angle reanalysis, volume integral terms independent of material angles are pre-extracted and stored, and the stiffness matrix is ​​quickly updated by multiplying the constant matrix terms with the new ply angle trigonometric function vector.

[0068] Step S150: Based on the global stiffness matrix, apply loads and define boundary conditions to perform geometric analysis to determine the layer structure strain energy, vibration frequency and buckling factor of the composite material.

[0069] During implementation, the generated global stiffness matrix, along with the mass and geometric stiffness matrices obtained from the density and initial stress integrals, are assembled into a global matrix system. Predefined external loads and boundary conditions are applied, and the solutions are obtained by substituting them into the static equilibrium equation, the free vibration equation, or the linear buckling eigenvalue equation, respectively, to obtain nodal displacements, natural frequencies, or buckling factors. The physical field is reconstructed using the solved displacement field coefficients and the original NURBS basis functions, and the calculated stress, strain, and other results are then visualized and post-processed.

[0070] In this embodiment, the process begins with acquiring the geometric model, extracting 3D NURBS solid parameters, constructing a local coordinate system, performing layer-by-layer integration based on an intersection algorithm to obtain the integration interval, generating the global stiffness matrix, and finally solving for the strain energy, vibration frequency, and buckling factor of the solid element. By applying an intersection algorithm based on LM iteration in the parameter space, the precise integration intervals for the truncation of each ply within the variable-thickness solid element are accurately solved. This "layer-by-layer integration" strategy directly encapsulates complex material ply truncation information within the three-dimensional numerical integration points without altering the element mesh structure. This achieves complete decoupling between mesh geometry (degrees of freedom) and physical ply (material), fundamentally avoiding the "degrees of freedom explosion" and element distortion problems of variable-thickness structures. Complex geometric volume integral terms are pre-calculated and stored. When the ply angle changes, the stiffness matrix is ​​updated simply by multiplying the pre-stored matrix with the new angle function, avoiding extremely time-consuming repetitive three-dimensional volume integral calculations.

[0071] In some embodiments, step S130 above, "constructing a nonlinear function in the parameter space based on the solid mesh model, and using the Levenberg-Marquardt LM algorithm to iteratively solve the intersection points of the offset surfaces of each ply of the variable thickness composite material with the thickness direction parameter curve, so as to determine the integration interval inside each solid element," can be achieved through the following steps: Step A: Determine that the variable thickness composite material is a two-dimensional constant cross-section variable thickness structure; Here, a two-dimensional constant cross-section variable thickness structure refers to a structure that maintains a constant cross-sectional shape along its length (or axial direction), but its thickness varies along the direction of the in-section parameters (e.g., ...). x , or ) Continuously changing. This type of structure is common in composite blades, variable cross-section beams, etc., and needs to be discretized using parametric meshes (such as solid elements).

[0072] Step B, along the plane of the solid mesh model x direction and or The direction parameter traverses the integration points, and at each integration point, constructs the first nonlinear function of the physical layup offset curve and the thickness direction parameter curve; Solid mesh models typically use quadrilateral / hexahedral elements to... x , or (In-plane parameters) and g The thickness direction parameters form a local coordinate system, which is mapped to the global physical space.

[0073] Physical layup offset curves describe the distribution of each layup in the parameter space of a composite material (e.g., fiber orientation of ±45° layup), and may involve nonlinear geometric relationships (e.g., curvature, torsion).

[0074] Thickness direction parameter curve, defining the material thickness as a function of... x , or The variation function (such as a linear, quadratic function, or spline curve) needs to be combined with the ply offset curve to form the first nonlinear function. The following formula (8):

[0075] in, Separation curves for different angles of plywood for Parameter curves for direction.

[0076] In finite element analysis, integration points are located inside the element and are used for numerical integration (such as Gaussian integration). Along x , or By traversing the integration points in different directions, a system of local nonlinear equations can be systematically constructed.

[0077] Step C: Using the initialization parameters and fixed regularization parameters as the starting point of the first nonlinear function, solve for the intersection points of the planar lines using the LM iterative algorithm; Finding the intersection of the physical layup offset curve and the thickness direction parameter curve is a problem involving solving a system of nonlinear equations.

[0078] Fixing the regularization parameter λ can avoid matrix singularity or ill-conditioned problems and ensure iterative stability, especially suitable for ill-conditioned problems (such as when the ply boundaries are close).

[0079] Step D: Define the integration interval boundary of each ply at the corresponding integration position based on the intersection of the plane lines, so as to determine the integration interval inside each solid unit.

[0080] Here, the integral interval of each ply within the unit is divided based on the intersection coordinates to ensure that each sub-interval corresponds to the material properties of a single ply.

[0081] Variable thickness / variable layup structures allow for accurate interpolation of material properties at integration points, avoiding errors from homogenization assumptions. By dividing the integration interval, numerical methods such as Gaussian integration can be used to accurately calculate the element stiffness matrix and stress / strain field, improving the accuracy of finite element analysis.

[0082] Figure 4The flowchart of the first nonlinear function solving algorithm provided in the embodiments of this application is as follows: Figure 4 As shown, the solution can be obtained through the following steps: 1. Start: Start the algorithm.

[0083] 2. Traversal x How many iterations are needed to reach the Gaussian point in the direction of the loop? Outer loop (traversal) x (Direction Gaussian point): Determine if the traversal has been completed. x All Gaussian points along the direction (i.e., determining whether the outer loop count has been reached): If so, it means all points have been calculated, and you can proceed to step 10.

[0084] If not, proceed with the calculation process for the current Gaussian point.

[0085] 3. b = 0 .

[0086] Initialize the interval array: Let b = 0 This is used to store the boundary of the integration interval corresponding to the current Gaussian point.

[0087] 4. Traverse the ply separation curves until the specified number of iterations is reached? Inner loop (traversing ply separator curves): Determine if all ply separator curves have been traversed (i.e., determine if the inner loop count has been reached): If so, it means that all curves corresponding to the current Gaussian point have been calculated, and you can proceed to step 9.

[0088] If not, then iteratively solve for the current curve and proceed to step 5.

[0089] 5. Define the objective equation and initial parameters: Define the objective nonlinear equation to be solved: F ( ζ, ζ ) = C k,o ( x ) –Z i ( g ) 。

[0090] Define the initial solution vector for the iteration: x ( ζ, ζ ) = [0.5,0.5] 。

[0091] Set the convergence tolerance and damping coefficient: eps = 10 -3 ? λ = 1000 。

[0092] 6.|norm( F )|> eps ? Iterative solution (convergence determination): Calculate the norm of the objective function value and determine if it is greater than the set tolerance, i.e., |norm( F )|> eps ? If it is (not converged): Perform parameter update step 7: x = x (J T J+ l I) -1 J T F ( x (Where J is the Jacobian matrix and I is the identity matrix). Update x Then return to step 6 to perform convergence judgment again.

[0093] If not (converged): Exit the iteration and proceed to the next step, 8.

[0094] 7. x = x (J T J+ l I) -1 J T F ( x ).

[0095] 8. b = [ b , x (2)].

[0096] Record the results of a single solution: extract the convergent solution x The second component x (2) append it to the array b In, that is, to execute b = [ b , x (2)]. After recording, return to step 4 to continue calculating the next dividing curve.

[0097] 9. b =sort( b ).

[0098] Integral interval sorting: When the inner loop ends, the array accumulated from the current Gaussian point is sorted. b Sort (usually in ascending order), that is b=sort( b After sorting, return to step 2 to calculate the next Gaussian point.

[0099] 10. Obtain all the required integration intervals. b .

[0100] Output: After all loops are completed, extract and summarize all the required integration intervals b.

[0101] 11. End: The algorithm process terminates.

[0102] In this embodiment of the application, for a two-dimensional constant cross-section variable thickness structure (such as a hydrofoil), the offset curve and parameterization are used. Curves are used to construct the difference function, and the intersection points of the planar lines are solved using the LM algorithm. This achieves a closed-loop process from structural parametric modeling and solving of nonlinear equations to defining the integration interval. The fast convergence characteristic of the LM algorithm in nonlinear problems significantly reduces the number of iterations, accelerating the large-scale finite element analysis process. This ensures the accuracy and efficiency of finite element analysis of variable-thickness composite structures.

[0103] In some embodiments, step S130 above, "constructing a nonlinear function in the parameter space based on the solid mesh model, and using the Levenberg-Marquardt LM algorithm to iteratively solve the intersection points of the offset surfaces of each ply of the variable thickness composite material with the thickness direction parameter curve, so as to determine the integration interval inside each solid element," can be achieved through the following steps: Step A: Determine that the variable thickness composite material is a three-dimensional variable cross-section and variable thickness structure; Three-dimensional variable cross-section structures exhibit geometric changes in length, width, and height (such as tapered beams and variable cross-section shells), with thickness varying along spatial parameters (…). x , or , Continuous changes. This type of structure is commonly found in aircraft engine blades, ship propellers, etc., and needs to be discretized using parametric meshes (such as hexahedral elements) to ensure that the mesh changes in sync with the geometry.

[0104] Step B: Traverse the in-plane integration points of the solid mesh model to construct the second nonlinear function of the three-dimensional ply offset surface and spatial parameter curve; In three-dimensional space, ply offset is achieved through a surface offset algorithm (such as offsetting the base plane along the normal to generate a new surface).

[0105] Solving for the intersection of the ply offset surface and the spatial parameter curve (such as the parameter curve in the ξ-η plane) requires constructing a second nonlinear function. The following formula (9): (9); in, Interval curved surfaces between layers at different angles. for Parameter lines in the direction.

[0106] This function may involve: Integral point traversal: within a hexahedral element, along... , , By traversing the Gaussian integration points in the direction, a system of local nonlinear equations is constructed to ensure the accuracy of numerical integration.

[0107] Step C: Set the initial iteration value of the second nonlinear function, and use dynamically adjusted regularization parameters to perform LM iteration to solve for the intersection of spatial surface lines; Solving for the intersection of the ply offset surface and the spatial parameter curve is a three-dimensional nonlinear least squares problem. Gradient descent and the Gauss-Newton method are balanced by adjusting the value of λ.

[0108] Dynamically adjusted regularization parameters l This can avoid matrix singularities, especially when the ply boundaries are close, and ensure iterative stability.

[0109] The initial iteration values ​​should be reasonable (e.g., based on the intersection of the solution from the previous iteration step or the average thickness), and the initialization method (e.g., empirical method, random initialization) should be selected in combination with the physical background of the problem (e.g., material properties, geometric symmetry).

[0110] Step D: Determine the geometric interior of each intersecting ply based on the intersection points of the spatial plane lines. g The integral interval in the direction.

[0111] The integral intervals of each ply within the element are divided based on the coordinates of the intersection points, ensuring that each sub-interval corresponds to a single ply material property. For example, within a hexahedral element, the element is divided into multiple sub-regions by the intersection points, and each sub-region corresponds to a specific ply.

[0112] Figure 5 The flowchart of the second nonlinear function solving algorithm provided in the embodiments of this application is as follows: Figure 5 As shown, the solution can be obtained through the following steps: 1. Start: Start the algorithm.

[0113] 2. Traverse the Gaussian points within the plane ( x , or How many cycles have been reached? Outer loop (traversing Gaussian points within the face): Checks if all points within the face have been traversed. x , or Gaussian point (i.e., determining whether the outer loop iteration count has been reached): If so, it means all points have been calculated, and you can proceed to step 11.

[0114] If not, proceed with the calculation process for the current Gaussian point.

[0115] 3. c = 0 .

[0116] Initialize the interval array: Let c = 0 This is used to store the boundary of the integration interval corresponding to the current Gaussian point.

[0117] 4. Traverse the ply separation curves until the specified number of iterations is reached? Inner loop (traversing ply separator curves): Determine if all ply separator curves have been traversed (i.e., determine if the inner loop count has been reached): If so, it means that all curves corresponding to the current Gaussian point have been calculated, and you can proceed to step 10.

[0118] If not, then iteratively solve for the current curve and proceed to step 5.

[0119] 5. Define the objective equation and initial parameters: Define the objective nonlinear equation to be solved: F ( x , or , g )= S k,o ( x, the ) i,j ( g ).

[0120] Define the initial solution vector for the iteration: x ( x , or , g ) = ( IP i , ip j , 0.5).

[0121] 6. eps = 10 -3 ; l = 1000.

[0122] Set the convergence tolerance and damping coefficient: eps = 10 -3 ; l = 1000.

[0123] 7. |norm (F ) |>eps ? Iterative solution (convergence determination): Calculate the norm of the objective function value and determine if it is greater than the set tolerance. |norm ( F ) |>eps ? If it is (not converged): Perform parameter update step 8: x = x (J T J+ l I) -1 J T F ( x (where J is the Jacobian matrix and I is the identity matrix), update x and return to step 6 to judge again.

[0124] If not (converged): Exit the iteration and proceed to the next step, 8.

[0125] 8. x = x (J T J+ l I) -1 J T F ( x ).

[0126] 9. c = [ c , x (3)].

[0127] Record the results of a single solution: c = [ c , x (3)].

[0128] Extract the third component of the convergent solution x x (3) append it to the array c In, that is, to execute c = [ c , x (3)]. After recording, return to step 4 to continue calculating the next dividing curve.

[0129] 10. c = sort( c ).

[0130] Integral interval sorting: When the inner loop ends, the array accumulated from the current Gaussian point is sorted. c Sort in ascending order, that is c = sort( c).

[0131] After sorting, return to step 2 to proceed with the calculation of the next Gaussian point.

[0132] 11. Obtain all integration intervals c .

[0133] Output: After all loops have finished, extract and summarize all the required integration intervals. c .

[0134] 12. End: The algorithm process terminates.

[0135] In this embodiment of the application, for a three-dimensional variable cross-section and variable thickness freeform surface structure (such as a propeller), protection is requested for the use of offset surfaces and parameterized spaces. The curve is used to construct the interpolation function, and a dynamic regularization parameter is introduced. l The LM algorithm is used to solve for the intersection points of spatial surfaces and lines through iteration. It realizes the entire process from geometric modeling and solving nonlinear equations to defining the integration interval for three-dimensional composite material structures with varying cross-sections and thicknesses. The fast convergence characteristic of the LM algorithm in nonlinear problems can significantly reduce the number of iterations, accelerate large-scale finite element analysis, and ensure the accuracy and efficiency of the analysis.

[0136] In some embodiments, the expression (4) for the constitutive matrix in the formula for the stiffness matrix based on the constant matrix of the variable thickness composite material and the ply trigonometric functions of the variable thickness composite material is: (4); in, This represents the second transformation relationship between the local coordinate system and the material coordinate system of the variable thickness composite material. i This indicates the layup direction angle of the variable thickness composite material. This represents the constitutive matrix of the orthotropic composite material. This represents the constant matrix for the variable thickness composite material. f i ( i ) represents the layup trigonometric function of the variable thickness composite material; The initial expression for the global stiffness matrix is: (3); in, Represents the strain-displacement matrix. For the transpose of B, This represents the first transformation relationship. for transpose, This represents the second transformation relationship. express transpose, This indicates the layup direction angle of the variable thickness composite material. This indicates the total number of layers sharing the same ply angle. l This indicates a single-level index. V k Indicates the first k A single-layer integral volume domain; Substituting the expression for the constitutive matrix into the initial expression for the global stiffness matrix yields the global stiffness matrix. The target expression.

[0137] In this embodiment, the constitutive matrix is ​​characterized based on a constant matrix and ply trigonometric functions, enabling the representation of the geometric volume integral term of the global stiffness matrix and the ply trigonometric functions. Thus, during iterative optimization or reanalysis of composite material ply angles, since the integration domain, NURBS geometric model, and material properties remain unchanged, the system can pre-calculate and store complex geometric volume integral terms. When the ply angle changes, the stiffness matrix can be updated simply by multiplying the pre-stored matrix with the new angle function, avoiding extremely time-consuming repetitive three-dimensional volume integral calculations.

[0138] Based on the above method, this application also provides a solid geometry analysis system for composite materials with complex geometric variations in thickness, including: 1. Model building and initialization module: used to acquire CAD data, extract NURBS parameters, and perform mesh refinement to build the analysis model (3D NURBS solid parameter spatial mesh model).

[0139] 2. Layup and Coordinate Mapping Module: Used to establish transformation matrices between the global, local and material coordinate systems, and to receive layup parameters input by the user.

[0140] 3. Layer-by-layer integration and intersection module for thickness interval: It has built-in nonlinear function (first nonlinear function and second nonlinear function) constructor and Levenberg-Marquardt solver, which is used to calculate the intersection of the ply offset surface and the thickness parameter curve according to the geometric complexity of different dimensions using the above analysis model, transformation matrix and ply parameters, and accurately divide the numerical integration interval.

[0141] 4. Decoupled stiffness acceleration calculation module: It is used to independently calculate and cache the three-dimensional geometric volume integral terms in memory based on the numerical integration interval, and can quickly synthesize the global stiffness matrix according to the real-time changing material ply angle function.

[0142] 5. Equation Assembly and Solver Module: Used to assemble the global system by combining stiffness, mass and geometric stiffness matrices, apply boundary conditions, solve static, vibration and buckling responses and output reconstructed physical field data.

[0143] This application proposes a method for static, vibration, and buckling analysis of solid composite materials with complex geometrical variations in thickness. By introducing a layer-by-layer integration strategy based on the LM algorithm and a fast stiffness matrix calculation algorithm that decouples geometry and material properties, the following significant advantages are achieved: 1. It achieves complete decoupling between mesh geometry (degrees of freedom) and physical layup (material), fundamentally avoiding the "degrees of freedom explosion" and element distortion problems of variable thickness structures.

[0144] Technical Features and Principles: By applying an intersection algorithm based on LM iteration in the parameter space, the precise integration interval of each ply cutoff within a variable-thickness solid element is accurately solved. This "layer-by-layer integration" strategy directly encapsulates complex material ply cutoff information within three-dimensional numerical integration points without altering the element mesh structure.

[0145] Beneficial effects: Compared to traditional finite element method (FEM) which requires mandatory mesh boundary alignment with physical ply boundaries (i.e., mandatory mesh splitting at ply cutoffs), this approach completely avoids the exponential growth in element count caused by increasing ply numbers. When performing high-fidelity analysis on complex structures with numerous plies (such as propellers), only a tiny fraction of the degrees of freedom (DOF) required by traditional finite element method (e.g., thousands vs. hundreds of thousands) is needed to achieve equal or even higher convergence accuracy, significantly reducing hardware memory consumption and preventing computational scale runaway. Simultaneously, it completely avoids the aspect ratio distortion and negative volume elements in the leading / trailing edge mesh caused by the "shell stretching + Boolean trimming" method used in traditional FEM.

[0146] 2. Achieved an order-of-magnitude improvement in the efficiency of structural lamination angle optimization and reanalysis calculation (100-10000 times faster).

[0147] Technical Features and Principles: To address the needs of structural optimization design, a fast algorithm for calculating the stiffness matrix was developed. Mathematically, this algorithm decouples the geometrically related volume integral terms in the integral formula of the stiffness matrix from the trigonometric function terms related to the ply angles.

[0148] Beneficial effects: When performing iterative optimization or reanalysis of composite material ply angles, the system can pre-calculate and store complex geometric volume integral terms because the integration domain, NURBS geometric model, and material properties remain unchanged. When the ply angle changes, the stiffness matrix can be updated simply by multiplying the pre-stored matrix with the new angle function, avoiding the extremely time-consuming repetitive three-dimensional volume integral calculations. Experiments show that as the degrees of freedom and spline order increase, this algorithm can achieve 10... 2 Up to 10 4The speedup ratio is several times that of traditional methods, which greatly breaks through the performance bottleneck of three-dimensional solid elements in iterative design optimization.

[0149] 3. Completely eliminates CAD-CAE geometric discretization errors in complex variable thickness structures, achieving true high-fidelity 3D solid analysis.

[0150] Technical features and principles: A unified three-dimensional solid analysis model is constructed directly using NURBS basis functions of the CAD system, seamlessly covering the geometric configurations of everything from constant-thickness pressure hulls and variable-thickness hydrofoils to completely free-form propeller blades.

[0151] Beneficial effects: It completely eliminates the geometric approximation error caused by the traditional FEM discretization of smooth geometry into low-order polynomial meshes, enabling the analysis model to maintain 100% geometric accuracy even under extremely coarse meshes. In addition, the inherent high-order continuity of NURBS basis functions significantly alleviates the shear locking and volumetric locking problems of low-order solid elements in thin-walled regions. Compared with traditional thin-shell theory, this three-dimensional solid framework can more realistically capture the three-dimensional stress state that may lead to delamination failure at thick sections or ply cutoff points.

[0152] Figure 6 Schematic diagrams illustrating the geometric topological evolution of three typical composite material structures provided in the embodiments of this application, such as... Figure 6 As shown in the schematic diagram, the composite pressure hull 31, the composite airfoil sail 32, and the composite propeller 33 are included.

[0153] Figure 6 This study visually demonstrates the progressive evolution of marine laminated composite structures in terms of geometric complexity, successfully covering the following three structures using a unified three-dimensional solid NURBS unit.

[0154] Laminated pressure shell 31: constant cross-section and thickness (such as pressure shell): generated by rotating a two-dimensional generatrix around an axis, having a constant cross-section, thickness and curvature.

[0155] Laminated airfoil sail 32, constant cross section, variable thickness and variable curvature (such as hydrofoil sail): generated by extruding a two-dimensional airfoil cross section along a straight path. Its thickness varies drastically along the chord direction, causing the physical ply to be truncated internally (ply drop-off), where traditional finite element methods are prone to degree-of-freedom explosion.

[0156] 33. Laminated propellers with variable cross-sections, thicknesses, and curvatures (such as propeller blades): These are generated by sweeping and twisting a series of different two-dimensional hydrofoil cross-sections along a complex helical path, representing the most common three-dimensional freeform surface entities.

[0157] The embodiments of this application can directly process structures containing complex twisted ply cuts without splitting the mesh.

[0158] Figure 7 The schematic diagram illustrating the principle and integration interval division of the layer-by-layer integration algorithm for structures with different geometric complexities provided in this application is as follows: Figure 7 As shown, Figure 7 This application illustrates how embodiments of the present application utilize three-dimensional parameter space ( o'-xyz Determine the precise integration interval for each physical layup in ) Figure 7 The solid red dots in the diagram represent the distribution of integration points, and the dashed blue lines represent the interlayer boundaries with the same ply angle, including constant cross-section-constant thickness 41, constant cross-section-variable thickness 42, and variable cross-section-variable thickness 43.

[0159] For structures of uniform thickness ( Figure 7 Upper section with uniform cross-section and uniform thickness 41), thickness direction ( g With the number of plies per layer (in the direction) remaining constant, the system can directly calculate the fixed integration interval and allocate integration points based on the proportion of plies contained in each layer.

[0160] For variable thickness structures (hydrofoils and propellers), Figure 7 In the lower and middle sections (constant cross-section, variable thickness 42 and variable cross-section, variable thickness 43), the total thickness varies along the ξ or η direction, causing the ply to be truncated within the structure (e.g., ...). Figure 7 (The middle section changed from 6 floors to 5 floors, and then to 2 floors). At this time... g The integral intervals in the direction become uneven.

[0161] The intersection algorithm logic is as follows: At each integration point within a surface, the system extracts the ply offset line / surface and... g Parametric curves. For hydrofoils (two-dimensional cross-section), planar line intersection calculations are performed; for propellers (three-dimensional solid), spatial surface line intersection calculations are performed. These intersection points (i.e., the parametric distances in constant cross-section-variable thickness 42 and variable cross-section-variable thickness 43) are accurately determined using the difference function and LM iteration. b i or c i These intersections strictly define the integration limits for each angular layer, enabling subsequent Gaussian numerical integration to maintain extremely high mechanical calculation accuracy at discontinuous material interfaces.

[0162] This application provides a geometric analysis apparatus for solid elements and other components in composite materials with varying thickness. Please refer to [link to relevant documentation]. Figure 8 The device 800 includes: The construction and extraction module 810 is used to construct a solid mesh model of three-dimensional non-uniform rational B-spline NURBS based on the solid geometric model, and extract the parameterization information of the three-dimensional non-uniform rational B-spline. The first determining module 820 is used to establish a local coordinate system at each Gaussian point inside the geometric entity using the parameterized information according to the Gauss-Legendre rule of Gauss-Legendre integral, and to determine the first transformation relationship between the local coordinate system and the global physical coordinate system. The second determining module 830 is used to construct a nonlinear function in the parameter space based on the solid mesh model, and use the Levenberg-Marquardt LM algorithm to iteratively solve the intersection of the offset surface of each ply of the variable thickness composite material with the thickness direction parameter curve, so as to determine the integral interval inside each solid unit. The third determining module 840 is used to determine the global stiffness matrix based on a preset geometric volume integral term and the ply trigonometric function of the variable thickness composite material, wherein the preset geometric volume integral term is determined based on the first transformation relationship, the constant matrix of the variable thickness composite material, and the integration interval. The fourth determination module 850 is used to perform geometric analysis, such as applying loads and defining boundary conditions based on the global stiffness matrix, to determine the strain energy, vibration frequency and buckling factor of the composite material laminate structure.

[0163] Figure 9 This is a schematic diagram of the structure of a computer device provided in an embodiment of this application. For example, as shown... Figure 9 As shown, the computer device 900 includes: a memory 901, a processor 902, and a computer program 903 stored in the memory 901 and running on the processor 902. When the processor 902 executes the computer program 903, the computer device can execute any of the geometric analysis methods for solid elements of variable thickness composite materials described above.

[0164] Furthermore, this application also protects a control device, which may include a memory and a processor. The memory stores executable program code, and the processor is used to call and execute the executable program code to perform a geometric analysis method for solid elements and other components of variable-thickness composite materials provided in this application. This application can divide the control device into functional modules based on the above method examples. For example, each module may correspond to a specific function, or two or more functions may be integrated into a single processing module. The integrated module can be implemented in hardware. It should be noted that the module division in this application is illustrative and only represents a logical functional division; other division methods may exist in actual implementation. It should also be noted that all relevant content of each step involved in the above method embodiments can be referenced to the functional description of the corresponding functional module, and will not be repeated here. It should be understood that the control device provided in this application is used to execute the above-mentioned geometric analysis method for solid elements and other components of variable-thickness composite materials, and therefore can achieve the same effect as the above-described implementation method. When using integrated units, the control device may include a processing module and a storage module. When the control device is applied to a block device, the processing module can be used to control and manage the actions of the block device. The storage module can be used to support block devices in executing mutual program code, etc. The processing module can be a processor or controller, which can implement or execute various exemplary logic blocks, modules, and circuits described in conjunction with the disclosure of this application. The processor can also be a combination of functions that implement computing capabilities, such as a combination of one or more microprocessors, a combination of digital signal processing (DSP) and microprocessors, etc., and the storage module can be a memory.

[0165] Furthermore, the control device provided in the embodiments of this application may specifically be a chip, component, or module. The chip may include a connected processor and a memory. The memory stores instructions, and when the processor calls and executes the instructions, the chip can execute the geometric analysis method for solid elements and other components of variable-thickness composite materials provided in the above embodiments. The embodiments of this application also provide a computer-readable storage medium storing computer program code. When the computer program code is run on a computer, it causes the computer to execute the aforementioned method steps to implement the geometric analysis method for solid elements and other components of variable-thickness composite materials provided in the above embodiments.

[0166] This application also provides a computer program product. When the computer program product is run on a computer, it causes the computer to execute the aforementioned related steps to implement the geometric analysis method for solid elements of variable thickness composite materials provided in the above embodiments. The control device, computer-readable storage medium, computer program product, or chip provided in this application embodiment are all used to execute the corresponding methods provided above. Therefore, the beneficial effects they achieve can be referred to in the beneficial effects of the corresponding methods provided above, and will not be repeated here. Through the description of the above embodiments, those skilled in the art can understand that, for the sake of convenience and brevity, only the division of the above functional modules is used as an example. In practical applications, the above functions can be assigned to different functional modules as needed, that is, the internal structure of the control device can be divided into different functional modules to complete all or part of the functions described above. In the embodiments provided in this application, it should be understood that the disclosed control device and method can be implemented in other ways. For example, the control device embodiments described above are merely illustrative. For example, the division of modules or units is merely a logical functional division. In actual implementation, there may be other division methods. For example, multiple units or components can be combined or integrated into another control device, or some features can be ignored or not executed. Another point is that the mutual coupling or direct coupling or communication connection shown or discussed can be an indirect coupling or communication connection through some interface, control device or unit, and can be electrical, mechanical or other forms.

[0167] It should be noted that the order of the embodiments described above is merely for descriptive purposes and does not represent the superiority or inferiority of the embodiments. The processes depicted in the accompanying drawings do not necessarily require a specific or sequential order to achieve the desired results. In some embodiments, multiple task processing and parallel processing are possible or may be advantageous. The various embodiments in this specification are described in a progressive manner, and the same or similar parts between the various embodiments can be referred to mutually. Each embodiment focuses on describing the differences from other embodiments. The above content is only a specific implementation of this application, but the protection scope of this application is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the protection scope of this application.

Claims

1. A geometric analysis method for solid elements in composite materials with varying thickness, characterized in that, The method includes: A solid mesh model of three-dimensional non-uniform rational B-spline NURBS is constructed based on the solid geometric model, and the parameterization information of the three-dimensional non-uniform rational B-spline is extracted. According to the Gauss-Legendre rule of the Gauss-Legendre integral, a local coordinate system is established at each Gaussian point inside the geometric entity using the parameterized information, and the first transformation relationship between the local coordinate system and the global physical coordinate system is determined. Based on the solid mesh model, a nonlinear function is constructed in the parameter space, and the intersection of the offset surface of each ply of the variable thickness composite material and the thickness direction parameter curve is solved iteratively using the Levenberg-Marquardt LM algorithm to determine the integral interval inside each solid element. The global stiffness matrix is ​​determined based on a preset geometric volume integral term and the ply trigonometric functions of the variable thickness composite material, wherein the preset geometric volume integral term is determined based on the first transformation relationship, the constant matrix of the variable thickness composite material, and the integration interval. Based on the global stiffness matrix, loads are applied and boundary conditions are defined to perform geometric analysis to determine the strain energy, vibration frequency, and buckling factor of the composite laminate structure.

2. The method as described in claim 1, characterized in that, The method further includes: By utilizing the characteristics of NURBS curves, a mesh refinement operation is performed on the solid mesh model using node insertion or order lifting methods to obtain a fully subdivided solid mesh model.

3. The method as described in claim 1, characterized in that, The process involves constructing a nonlinear function in the parameter space based on the solid mesh model, and using the Levenberg-Marquardt LM algorithm to iteratively solve for the intersection points of the offset planes of each ply of the variable thickness composite material with the thickness direction parameter curves, in order to determine the integration interval within each solid element, including: The variable thickness composite material is determined to be a two-dimensional constant cross-section variable thickness structure; Along the plane of the solid mesh model ξ direction and η The direction parameter traverses the integration points, and at each integration point, constructs the first nonlinear function of the physical layup offset curve and the thickness direction parameter curve; Using the initialization parameters and fixed regularization parameters as the starting point of the first nonlinear function, the intersection points of the planar lines are solved using the LM iterative algorithm; The integration interval boundaries of each ply at the corresponding integration position are defined based on the intersection of the planar lines, so as to determine the integration interval inside each solid unit.

4. The method as described in claim 1, characterized in that, The process involves constructing a nonlinear function in the parameter space based on the solid mesh model, and using the Levenberg-Marquardt LM algorithm to iteratively solve for the intersection points of the offset planes of each ply of the variable thickness composite material with the thickness direction parameter curves, in order to determine the integration interval within each solid element, including: The variable thickness composite material is determined to be a three-dimensional variable cross-section and variable thickness structure; By traversing the in-plane integration points of the solid mesh model, a second nonlinear function of the three-dimensional ply offset surface and spatial parameter curve is constructed. Set the initial iteration value of the second nonlinear function, and use dynamically adjusted regularization parameters to perform LM iteration to solve for the intersection points of spatial surface lines; Based on the intersection points of the spatial planes, the geometric interior of each intersecting ply is determined. ζ The integral interval in the direction.

5. The method according to any one of claims 1 to 4, characterized in that, The objective expression for determining the global stiffness matrix K based on the preset geometric volume integral term and the ply trigonometric functions of the variable thickness composite material is as follows: ； in, ; B represents the preset geometric volume integral term. k Indicates the first k A single-layer strain-displacement matrix, For B k The transpose operation. k Indicates the first k The first transformation relationship of a single layer, for k The transpose operation. The constant matrix represents the variable thickness composite material. n l This indicates the total number of layers with different ply angles. V k Indicates the first k A single-layer integral volume domain f i,k ( θ ) indicates the first k The layup trigonometric function of a single-layer variable thickness composite material i express f The ordinal numbers of trigonometric functions in the function. θ Indicates the ply direction angle.

6. The method as described in claim 5, characterized in that, The method further includes: The expression for the constitutive matrix in the formula for the stiffness matrix based on the constant matrix of the variable thickness composite material and the ply trigonometric functions of the variable thickness composite material is as follows: ； in, This represents the second transformation relationship between the local coordinate system and the material coordinate system of the variable thickness composite material. θ This indicates the layup direction angle of the variable thickness composite material. This represents the constitutive matrix of the orthotropic composite material. This represents the constant matrix for the variable thickness composite material. f i ( θ ) represents the layup trigonometric function of the variable thickness composite material. T for The transpose operation; The initial expression for the global stiffness matrix is: ； in, Represents the strain-displacement matrix. For the transpose of B, This represents the first transformation relationship. for transpose, This represents the second transformation relationship. express transpose, This represents the constitutive matrix of an orthogonal anisotropic material. This indicates the total number of layers sharing the same ply angle. l This indicates a single-level index. V k Indicates the first k A single-layer integral volume domain Indicates the layup direction angle of the variable thickness composite material; Substituting the expression for the constitutive matrix into the initial expression for the global stiffness matrix yields the global stiffness matrix. The target expression.

7. The method according to any one of claims 1 to 4, characterized in that, The geometric model of the entity includes geometric three-dimensional non-uniform rational B-spline control points, a geometric model generated based on computer-aided design, or a geometric model generated based on discrete points.

8. A geometric analysis apparatus for solid elements and other components of composite materials with varying thickness, characterized in that, The device includes: The construction and extraction module is used to construct a solid mesh model of three-dimensional non-uniform rational B-spline NURBS based on the solid geometric model, and extract the parametric information of the three-dimensional non-uniform rational B-spline. The first determining module is used to establish a local coordinate system at each Gaussian point inside the geometric entity using the parameterized information according to the Gauss-Legendre rule of Gauss-Legendre integral, and to determine the first transformation relationship between the local coordinate system and the global physical coordinate system. The second determining module is used to construct a nonlinear function in the parameter space based on the solid mesh model, and use the Levenberg-Marquardt LM algorithm to iteratively solve the intersection of the offset surface of each ply of the variable thickness composite material with the thickness direction parameter curve, so as to determine the integral interval inside each solid element. The third determining module is used to determine the global stiffness matrix based on a preset geometric volume integral term and the ply trigonometric function of the variable thickness composite material, wherein the preset geometric volume integral term is determined based on the first transformation relationship, the constant matrix of the variable thickness composite material, and the integration interval. The fourth determination module is used to perform geometric analysis, such as applying loads and defining boundary conditions based on the global stiffness matrix, to determine the strain energy, vibration frequency and buckling factor of the composite material laminate structure.

9. An electronic device, characterized in that, The electronic device includes: a memory and at least one processor, the memory storing instructions; the at least one processor invokes the instructions in the memory to cause the electronic device to perform geometric analysis methods such as solid elements for variable thickness composite materials as described in any one of claims 1-7.

10. A computer-readable storage medium storing instructions thereon, characterized in that, When the instructions are executed by the processor, they implement geometric analysis methods such as solid elements for variable thickness composite materials as described in any one of claims 1-7.

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