A physical information neural network-based arbitrary quadrilateral laminated plate buckling and bending combined evaluation and optimization method

Abstract

This invention discloses a method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on physical information neural networks, relating to the field of composite material structure design technology. The method, for arbitrary quadrilateral laminates, first constructs a method based on classical laminate theory... USA The stiffness matrix explicitly reflects the influence of ply parameters, and a physical domain is transformed to a reference domain through a bitriangular affine mapping to establish a physical information neural network mechanical model. The model is trained with the goal of minimizing the total potential energy or Rayleigh quotient, solving for the buckling factor and layer-by-layer strain limits, and adapting to four typical boundary conditions through a boundary embedding function. Subsequently, using ply angle and thickness as variables, optimization is performed using a genetic algorithm, with the goal of minimizing mass, and the buckling factor and strain limits as constraints. The final output is the optimal ply scheme that satisfies specific boundary conditions. This invention can optimize ply angle and thickness while satisfying buckling stability and strain strength constraints.

Description

Technical Field

[0001] This invention relates to the field of composite material structure design technology, and in particular to a method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on physical information neural networks. Background Technology

[0002] In high-end manufacturing fields such as aerospace, shipbuilding, and energy equipment, composite material structures have been widely used in wing skins, fuselage panels, ship structures, and other critical load-bearing components due to their advantages such as low density, high specific strength, high specific stiffness, and strong structural designability. By rationally designing the layup parameters of composite material structures, such as layup angles and thicknesses, structural performance optimization and weight reduction can be achieved while meeting load-bearing capacity and stability requirements, which is of great significance for improving the overall performance of equipment.

[0003] Composite material structures exhibit significant anisotropy, and their mechanical response is closely related to geometry, material parameters, layup, boundary conditions, and load types, resulting in high variable dimensionality and complex coupling relationships. In current engineering practice, composite material structure design typically relies on finite element analysis (FEM) methods for multiple rounds of modeling and simulation verification to evaluate performance indicators such as bending deflection response, stress-strain distribution, and stability. While this method offers high computational accuracy, its modeling process is cumbersome, computationally time-consuming, and heavily reliant on computational resources, making it difficult to adapt to the demands of rapid design iteration and large-scale optimization in high-dimensional design spaces.

[0004] Therefore, under the premise of ensuring analytical accuracy and result reliability, how to construct an efficient modeling method that can quickly predict the mechanical response of composite material structures, reduce the dependence on high-cost numerical simulation, and is applicable to structural optimization design has become an important technical problem that urgently needs to be solved in this field. Summary of the Invention

[0005] The purpose of this invention is to provide a method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on physical information neural networks, aiming to solve or improve at least one of the above-mentioned technical problems.

[0006] To achieve the above objectives, the present invention provides the following solution: A method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on physical information neural networks includes: Set the relevant parameters of the laminate; the relevant parameters include geometry, ply parameters, material properties, load input, and boundary conditions; the ply parameters include angle, thickness, and mass; Based on the aforementioned parameters and classical laminate theory, laminates are constructed. ABDThe stiffness matrix is ​​used, and the physical domain of the arbitrary quadrilateral laminate is mapped to the reference domain through a bitriangular affine mapping, constructing a physical information neural network mechanical model for arbitrary quadrilateral laminates; wherein, the... ABD The stiffness matrix is ​​obtained by integrating the rotational stiffness matrix of each layer of the laminate and the ply thickness, and is used to explicitly reflect the influence of ply angle and thickness on the overall stiffness. Based on the physical information neural network mechanical model, the minimization of the total potential energy of the bending problem and the minimization of the Rayleigh quotient of the buckling problem are taken as training objectives to solve the buckling factor and the layer-by-layer strain limit. The four types of typical boundary conditions are uniformly adapted by a strong formal boundary embedding function. The four types of typical boundary conditions include simply supported on four sides, fixed on four sides, simply supported on two sides and free on both sides, and fixed on two sides and free on both sides. The ply angle and ply thickness of the laminate under test are used as design variables and optimized using a genetic algorithm. The goal is to minimize the ply mass, with buckling factor and layer-by-layer strain limits as constraints, to output the optimal ply scheme. The optimal ply scheme is a mass-minimizing design obtained by setting one of the four typical boundary conditions, evaluating the buckling factor and strain extreme values ​​through the physical information neural network mechanical model, and using them as constraints in the genetic algorithm for optimization search.

[0007] Optionally, the double-triangular affine mapping specifically includes: The reference domain unit square is divided into two triangular subdomains along the diagonal. An affine transformation relationship between the reference domain and the physical domain is established, and the Jacobian matrix and its determinant are calculated. The equivalent calculation of the energy integral between the physical domain and the reference domain is performed through the Jacobian transformation.

[0008] Optionally, the process of solving for the buckling factor and the layer-by-layer strain limit specifically includes: Based on the principle of minimizing total potential energy, a total potential energy functional for the bending problem is constructed as the first loss function, and the physical information neural network mechanical model is used to solve the bending response and layer-by-layer strain limit of the laminate under out-of-plane load. Based on the Rayleigh quotient minimization principle, the ratio of structural strain energy to geometric stiffness energy in the buckling problem is constructed as the second loss function, and the buckling factor of the laminate under in-plane loading is solved using the physical information neural network mechanical model.

[0009] Optionally, the design variables of the genetic algorithm are encoded in the following form: in, x Let be a chromosome vector, representing a candidate layering scheme; For the k-th layer, Let N be the thickness of the k-th layer, where k = 1, 2, ..., N;N This represents the total number of laminate layers.

[0010] Optionally, the genetic algorithm includes selection, crossover, and mutation operators; wherein the angle variable is updated using discrete random sampling or random sampling of the parent generation; and the thickness variable is updated using linear combination or normal distribution perturbation.

[0011] Optionally, the ply angle is a discrete variable selected from the angle set {0°, 45°, -45°, 90°}; the ply thickness is a continuous variable optimized within a preset range.

[0012] The present invention also provides an electronic device, including a memory and a processor, wherein the memory is used to store a computer program, and the processor runs the computer program to enable the electronic device to perform the above-described method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on a physical information neural network.

[0013] The present invention also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on a physical information neural network as described above.

[0014] According to specific embodiments provided by the present invention, the present invention discloses the following technical effects: This invention discloses a method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on a physical information neural network. The method includes: setting relevant parameters of the laminate; the relevant parameters include geometry, ply parameters, material properties, load input, and boundary conditions; the ply parameters include angle, thickness, and mass; and constructing a model of the laminate based on the relevant parameters and classical laminate theory. ABD The stiffness matrix is ​​used, and the physical domain of the arbitrary quadrilateral laminate is mapped to the reference domain through a bitriangular affine mapping, constructing a physical information neural network mechanical model for arbitrary quadrilateral laminates; wherein, the... ABDThe stiffness matrix is ​​obtained by integrating the rotational stiffness matrix of each layer of the laminate and the ply thickness, and is used to explicitly reflect the influence of ply angle and thickness on the overall stiffness. Based on the physical information neural network mechanical model, minimizing the total potential energy of the bending problem and minimizing the Rayleigh quotient of the buckling problem are used as training objectives to solve for the buckling factor and the layer-by-layer strain limit. Four typical boundary conditions are uniformly adapted through a strong formal boundary embedding function. The four typical boundary conditions include simply supported on four sides, fixed on four sides, simply supported on two sides and free on both sides, and fixed on two sides and free on both sides. The ply angle and ply thickness of the laminate under test are used as design variables, and optimization is performed by combining a genetic algorithm. The goal is to minimize the ply mass, and the buckling factor and the layer-by-layer strain limit are used as constraints to output the optimal ply scheme. The optimal ply scheme is a mass-minimizing design obtained by setting one of the four typical boundary conditions, evaluating the buckling factor and strain extrema through the physical information neural network mechanical model, and optimizing it as a constraint in the genetic algorithm.

[0015] This invention enables high-precision prediction of the bending deflection response, stress-strain distribution, and buckling stability of laminated structures under different loads and boundary conditions, without requiring costly and detailed finite element modeling. It also facilitates efficient optimization of ply parameters. The method incorporates the laminate's mechanical governing equations, boundary conditions, and physical constraints into the neural network training process, constructing a physically consistent structural response prediction model. This significantly improves computational efficiency while maintaining accuracy, reducing reliance on large-scale finite element analysis. Furthermore, this invention combines a physically-informed neural network model with intelligent optimization algorithms to optimize the ply angle and ply thickness distribution while satisfying buckling stability and strain strength constraints. Attached Figure Description

[0016] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0017] Figure 1 This is a schematic diagram of the overall process of the present invention. Detailed Implementation

[0018] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0019] The purpose of this invention is to provide a method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on physical information neural networks, aiming to solve or improve at least one of the above-mentioned technical problems.

[0020] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0021] As a first aspect, such as Figure 1 As shown, this invention provides a method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on a physical information neural network, comprising: Set the relevant parameters of the laminate; the relevant parameters include geometry, ply parameters, material properties, load input and boundary conditions; the ply parameters include angle, thickness and mass.

[0022] Based on the aforementioned parameters and classical laminate theory, laminates are constructed. ABD The stiffness matrix is ​​used, and the physical domain of the arbitrary quadrilateral laminate is mapped to the reference domain through a bitriangular affine mapping, constructing a physical information neural network mechanical model for arbitrary quadrilateral laminates; wherein, the... ABD The stiffness matrix is ​​obtained by integrating the rotational stiffness matrix of each layer of the laminate and the ply thickness, and is used to explicitly reflect the influence of ply angle and thickness on the overall stiffness. The bitriangular affine mapping specifically includes: The reference domain unit square is divided into two triangular subdomains along the diagonal. An affine transformation relationship between the reference domain and the physical domain is established, and the Jacobian matrix and its determinant are calculated. The equivalent calculation of the energy integral between the physical domain and the reference domain is performed through the Jacobian transformation.

[0023] Based on the aforementioned physical information neural network mechanics model, minimizing the total potential energy of the bending problem and minimizing the Rayleigh quotient of the buckling problem are used as training objectives to solve for the buckling factor and the layer-by-layer strain limit. Specifically, the solution process for the buckling factor and the layer-by-layer strain limit includes: Based on the principle of minimizing total potential energy, a total potential energy functional for the bending problem is constructed as the first loss function, and the bending response and layer-by-layer strain limit of the laminate under out-of-plane load are solved using the physical information neural network mechanical model. Based on the principle of minimizing Rayleigh quotient, the ratio of structural strain energy to geometric stiffness energy for the buckling problem is constructed as the second loss function, and the buckling factor of the laminate under in-plane load is solved using the physical information neural network mechanical model.

[0024] The four types of typical boundary conditions are uniformly adapted by a strong formal boundary embedding function. The four types of typical boundary conditions include simply supported on four sides, fixed on four sides, simply supported on two sides and free on both sides, and fixed on two sides and free on both sides.

[0025] The ply angle and ply thickness of the laminate under test are used as design variables and optimized using a genetic algorithm. The goal is to minimize the ply mass, with buckling factor and layer-by-layer strain limits as constraints, to output the optimal ply scheme. The optimal ply scheme is a mass-minimizing design obtained by setting one of the four typical boundary conditions, evaluating the buckling factor and strain extreme values ​​through the physical information neural network mechanical model, and using them as constraints in the genetic algorithm for optimization search.

[0026] The genetic algorithm includes selection, crossover, and mutation operators; the angle variable is updated using discrete random sampling or random sampling of the parent generation; the thickness variable is updated using linear combination or normal distribution perturbation. The ply angle is a discrete variable selected from the angle set {0°, 45°, -45°, 90°}; the ply thickness is a continuous variable optimized within a preset range.

[0027] Based on the above technical solution, the following embodiments are provided to illustrate the detailed processing procedures of each of the above steps.

[0028] In this embodiment, to efficiently solve the bending response and buckling stability of composite laminates under the combined action of out-of-plane uniform loads and in-plane loads, and to further conduct lightweight optimization design for ply angles (discrete variables) and layer thicknesses (continuous variables), this invention proposes a unified solution for laminate bending-buckling based on Physical Information Neural Network (PINN) and a Genetic Algorithm (GA) optimization method. This method is based on Classical Lamination Theory (CLT) as its mechanical foundation, and calculates the equivalent stiffness matrix of the laminate... A , B , D The construction process of the (Laminate ABD stiffness matrix) and the variational energy principle (minimizing the total potential energy of the bending problem and the Rayleigh quotient of the buckling problem) are explicitly coupled to the neural network training objective. The required displacement field and strain / curvature field derivatives are obtained through automatic differentiation, thus enabling rapid calculation of the bending response and buckling factor of arbitrary quadrilateral laminates under various boundary constraints without the need to establish a finite element mesh and a global stiffness matrix. Based on this, a genetic algorithm optimization process is constructed with the goal of minimizing mass and constrained by buckling factor and layer-by-layer strain limits, automatically searching for ply angle and thickness combinations that satisfy engineering constraints. The specific steps are as follows: Step 1: Problem modeling and input parameter definition.

[0029] This step is used to perform parametric modeling of the mechanical problems of composite laminates. It constructs a parametric input model that can be uniformly set by taking geometric features, material parameters, load conditions and displacement boundary constraints, thereby ensuring that the subsequent PINN solution and optimization module has a unified interface and good transferability under various problem configurations.

[0030] Step 101: Definition of the geometric region and coordinates of the laminate: To facilitate the PINN calculation, a bitriangular affine transformation is needed to transform the physical domain coordinates of the laminate to the reference domain coordinates for the solution. The physical domain of the mid-surface of the laminate is defined as a two-dimensional region. The physical domain coordinates in the middle surface are (x, y), the thickness direction coordinate of the laminate is z (the z coordinate of the middle surface of the laminate is 0), and the z coordinates of the upper and lower surfaces of the laminate are h / 2 and -h / 2 respectively.

[0031] For a quadrilateral laminate of arbitrary shape, the physical domain coordinates of its four corner points can be defined as V1(V1, V2, V3, V4, V5, V6, V7, V8, V9, V1, V1, V2, V1, V2, V3, V4 ... x 1， y 1) V2 ( x 2， y 2) V3 ( x 3， y 3) V4 ( x 4， y 4). A reference domain (a unit square region) is introduced. The coordinates within the reference domain are ( ).in, and These represent the two principal directions of the reference domain coordinates.

[0032] Step 102: Input laminate layup parameters and material properties: Let the total number of laminate layers be... N , No. k The ply angle of each layer (layer numbers increasing from bottom to top) is... , No. k Layer thickness is The total thickness of the laminate is... h Satisfying the relation: And define the coordinates of the interfaces between each layer of the laminate as follows: , , in,z 0 represents the middle surface of the laminate. z coordinate, z k For the first k The coordinates of the upper surface of the layer, z k-1 Let K be the coordinates of the lower surface of the k-th layer. For the first k The z-coordinate of the mid-plane of the layer.

[0033] The engineering parameters of the anisotropic material are as follows (with the principal axis of the material as the first direction of the material coordinate system): tensile modulus in the first direction. E 1- and 2-direction tensile modulus E 2. Shear modulus in 12 directions G 12 12-direction Poisson's ratio 21-direction Poisson's ratio Based on the above anisotropic material parameters, the single-layer stiffness matrix Q (principal axes of the material) under plane stress conditions can be defined as follows: Step 103, Load Input: For bending problems, it is necessary to define out-of-plane load input (uniformly distributed surface load). q ( x,y (Unit: N / m) 2 ): For buckling problems, it is necessary to define in-plane load inputs (uniform wire loads). N x , N y , N xy (Unit: N / m).

[0034] , , Step 104, Boundary Condition Type: This invention supports the following four types of boundary constraint combinations and allows specifying "which two edges are free edges / constrained edges" (implementation is parameterizable, for example, letting...). =0, 1 are constraint edges, let (where =0 and 1 represent free edges), the four types of displacement boundary conditions are as follows: 1) Simply Supported on All Four Edges (SSSS): All four boundary edges are simply supported. 2) Clamped on All Four Edges (CCCC): All four edges are clamped. 3) Simply Supported on Two Opposite Edges and Free on the Other Two Edges (SSFF): The two opposite edges are simply supported, and the other two edges are free. Clamped on Two Opposite Edges and Free on the Other Two Edges (CCFF): Two opposite edges are fixed, while the other two edges are free.

[0035] Step 2: Construction of the physical information neural network mechanical model.

[0036] This step is used to construct a unified physical information neural network mechanical modeling framework for arbitrary quadrilateral laminates. Its core lies in coupling "geometric mapping - construction of laminate theoretical stiffness - expression of displacement field" under the same mathematical system, thereby providing a consistent energy calculation basis for solving subsequent bending problems (minimum total potential energy) and buckling problems (Rayleigh quotient eigenvalues).

[0037] Step 201, Bitriangular Affine Geometric Mapping: First, the unit square region of the reference domain needs to be divided into two triangular regions along the diagonal, resulting in two triangular regions. and The interval expressions for the two triangular regions within the reference domain are as follows: exist Internally defined local affine parameters ( s 1, t 1) Represented using the reference domain coordinate system: Similarly, in Internally defined local affine parameters ( s 2, t 2) Represented using the reference domain coordinate system: Next, the arbitrary shape of the quadrilateral in the physical domain V 1 V 2 V 3 V 4. Divide into two triangles along the diagonal. T 1 and T 2, where triangle T 1. From the vertex V 1 V 2 VComposed of 3, a triangle T 2. From the vertex V 1 V 4 V 3. Composition.

[0038] For the triangular reference domain One point of responsibility , and the physical domain T 1. Establish affine transformation: Similarly, for the triangular reference domain One point of responsibility Establish an affine transformation with the physical domain T2: in, and These are the position vectors of the triangular reference domain, V 1. V 2. V 3. V 4 represents the coordinates of the four vertices of the quadrilateral physical domain, defined as follows: , In the triangular reference domain Inside, define the Jacobi matrix. for: In the triangular reference domain Similarly, internally: Because within the two triangular reference domains, the Jacobian matrix and the local coordinate system... s , t Since it is irrelevant, the Jacobian determinant is a constant within each triangular reference domain.

[0039] Step 202, Energy integral equivalence under Jacobi transform: For the physical domain integral: Replace the integration variable: Therefore, we get: in, For the physical domain The corresponding reference domain, reference domain From triangle and composition.

[0040] Step 203, the derivative chain rule under the Jacobian transform: Define the Jacobian matrix: , Define gradient operator transformation: make: Where a corresponds to the element in the first row and first column of the J-1 matrix, b corresponds to the element in the second row and first column of the J-1 matrix, c corresponds to the element in the first row and second column of the J-1 matrix, and d corresponds to the element in the second row and second column of the J-1 matrix.

[0041] Then for any scalar field w The first derivative can be derived: The second derivative is as follows: Step 204, Derivation of the stiffness matrix of the laminated plate rotation axis: Define the material coordinate system of the laminate Axis around global coordinate system The cosine value c and the sine value s of the rotation angle of the axis, where, Let be the layup angle of the k-th layer of the laminate.

[0042] in, Let be the layup angle of the k-th layer of the laminate.

[0043] Define the first laminate Layer rotational stiffness matrix , The stiffness coefficients of each layer, The rotational stiffness coefficients for each layer are (i,j=1,2,6).

[0044] Step 205, Solving the ABD matrix: According to the constitutive relations of composite laminates, we have: in, N For the resultant external force vector, M For the resultant external force vector, A For the in-plane stiffness matrix, B For the coupling stiffness matrix, D For out-of-plane stiffness matrix, For the mid-surface strain vector, Let be the curvature vector, and its specific expansion is as follows: in, N x for x Components of internal and external forces in the direction plane N y for y Components of internal and external forces in the direction plane N xy For the components of the internal and external forces in the tangential plane, M x External torque x To component, M y External torque y To component, M xy External torque z To component, The middle surface of the laminate x Towards strain, The middle surface of the laminate y Towards strain, Tangential strain on the mid-surface of the laminate, The middle surface of the laminate x Towards curvature, The middle surface of the laminate y Towards curvature, denoted as the mid-surface torsion rate of the laminate. ABD The coefficients of the matrix expansion are as follows: in, For the first laminate k The rotational stiffness matrix of the layer z k For the first k upper surface of the layer z coordinate, z k-1 For the first k The lower surface of the layer z coordinate.

[0045] Step 3: PINN solution method for bending response.

[0046] This step solves for the bending response under a uniformly distributed transverse load and outputs the layer-by-layer strain extrema for constraint determination in subsequent optimization algorithms. To accommodate asymmetric ply layup and bending-tension coupling, this step employs... ABD Solve using the fully coupled matrix energy form.

[0047] Step 301: Physical Domain Displacement Field and Strain-Curvature Relationship: For the quadrilateral region of the physical domain Based on Kirchhoff's thin plate theory, the mid-surface displacement of a laminated plate is defined as follows: in, The displacement of the mid-surface of the laminate along the 1-axis of the material coordinate system. For displacement along the 2-axis direction, w This represents the out-of-plane displacement of the laminate. Additionally, the mid-surface strain of the laminate is defined. and curvature As shown below: , Therefore, the relationship between the strain of the laminate and the thickness direction z can be calculated: Step 302, Strain energy density per unit area in the physical domain: The strain energy density per unit area u within the physical domain is defined as follows: Integrating u over the physical domain, we obtain the total strain energy U in the physical domain, as expressed below: Step 303, External potential energy and total potential energy functional: Uniformly distributed load external potential energy W The definition is as follows: in, q 0 represents the uniformly distributed load acting on the laminate. w This represents the out-of-plane displacement of the laminate.

[0048] Total potential energy functional The definition is as follows: For bending problems, in all displacement fields satisfying the boundary conditions, the real displacement field makes the total potential energy functional The stationary value is usually the minimum value at the stable equilibrium point (small linear elastic deformation).

[0049] Step 304, Energy integral equivalent over the reference domain: Physical domain The corresponding reference domain is Equivalent to the energy integral under the Jacobian transformation in step 202, we obtain: in, and For reference domain coordinates, and For reference domain The two triangular regions are divided along the diagonal. and Let be the determinant of the Jacobian matrix corresponding to the two regions.

[0050] Step 4: Solution method for buckling eigenvalues ​​PINN.

[0051] This step involves applying loads within a given reference plane. N x0 , N y0 , N xy0 Next, the first-order buckling factor is solved. This invention adopts the Rayleigh quotient minimization form and retains... ABD The internal energy term of the fully coupled matrix structure enables the accurate expression of the tension-bending coupling of asymmetric plywood.

[0052] Step 401: Solving the energy functional of the buckling problem: Referring to steps 303 and 304, the reference domain can be obtained. Structural potential energy below U s As shown in the following formula: in, u Strain energy density per unit area and For reference domain coordinates, and For reference domain The two triangular regions are divided along the diagonal. and Let be the determinant of the Jacobian matrix corresponding to the two regions.

[0053] Step 402, Solving for the geometric stiffness-energy term: Define the transverse gradient of the deflection field w ,x and w ,y As shown in the following formula: In the physical domain Internal geometric stiffness U g The expression is as follows: Mapping geometric stiffness to the reference domain Inside, get: Step 403: Solving for Rayleigh quotient and first-order buckling factor: For buckling problems, it is necessary to find a set of non-zero deflection solutions ( u 0, v 0, w ), to satisfy the structural strain energy U s and geometric stiffness energy U g The ratio is the smallest, and this smallest ratio is the first-order buckling factor of the laminate under this load distribution. The definition is as follows: Meanwhile, to avoid calculating trivial solutions (zero displacement fields), it is necessary to introduce a normalized loss term for the buckling modal displacement field, defined as follows: as follows: Therefore, the loss function of the physical information neural network can be obtained. Loss The expression is as follows: in, These are the parameters of the neural network, such as weights and biases. , , The deflection prediction value output by the neural network. β These are the component weights of the loss function.

[0054] Step 5: Unified adaptation method for multi-displacement boundary conditions.

[0055] This step provides mechanical expressions for four types of boundary conditions and offers a template for unified implementation in PINN using "polynomial strong form embedding operators," such that: 1) For different displacement boundary conditions, you only need to switch the different displacement boundary condition operators.

[0056] 2) There is no need to redesign the physical information neural network loss function for the boundary type.

[0057] Step 501, Mechanical expressions for four types of displacement boundary conditions: Define a unit square reference domain The four boundaries are respectively , , and The four displacement boundary condition expressions are as follows: 1) Simply supported on all four edges (SSSS): 2) Clamped on All Four Edges (CCCC): 3) Simply supported on two opposite edges and free on the other two edges (SSFF): 4) Clamped on two opposite edges and free on the other two edges (CCFF): Step 502, Strong Form Polynomial Operator of Neural Networks: Define the neural network displacement prediction field w As shown below: in, The deflection field is the final output of the neural network. For strong-form operators of displacement boundary conditions in neural networks, The four types of strong formal boundary condition operators are shown below for the initial output deflection field of the neural network: Step 6: Layer-based optimization solution method based on physical information neural network mechanical evaluator.

[0058] This step, building upon the "problem modeling—physical information neural network mechanical solution—unified adaptation of multi-displacement boundary conditions" established in steps one through five, introduces a genetic algorithm (GA) to automatically search for and optimize the laminated plate layup parameters. The core objective of this step is to obtain the optimal layup scheme that satisfies buckling and strain constraints in a large-scale discrete-continuous hybrid design space, without relying on gradient information, using a swarm search mechanism.

[0059] Step 601: Design the variable vector and encoding format: For any individual in a genetic algorithm, a candidate layering scheme can be represented as a chromosome vector: in, x Let be a chromosome vector, representing a candidate layering scheme; For the k-th layer, Let N be the thickness of the k-th layer, where k = 1, 2, ..., N; N This represents the total number of laminate layers.

[0060] Step 602, Mathematical expression of objective function and constraint function: (1) Mathematical expression of the objective function: The objective function of a genetic algorithm is defined as minimizing the quality: in, f The objective function (quality function) of the genetic algorithm. ρ For material density, S The area of ​​the region (physical domain). N This refers to the number of laminate layers. t k For the first laminate k Layer thickness.

[0061] (2) Expression of buckling constraint function: The first-order buckling factor of the individual is obtained from the PINN buckling estimator in step four. Set a buckling safety threshold ( Generally between 1 and 1.05), the buckling constraint is defined as... g b As shown in the following formula: (3) Expression of strain constraint function The strain extrema of each layer can be obtained from the PINN bending estimator in step three: in, and These represent the minimum and maximum x-direction strains on the mid-surface of the k-th layer of the laminate, respectively. and These represent the minimum and maximum y-direction strains on the mid-surface of the k-th layer of the laminate, respectively. and These represent the minimum and maximum tangential strains on the mid-surface of the k-th layer of the laminate, respectively (assuming the minimum strain is less than 0, i.e., compressive strain 1, and the maximum strain is greater than 0, i.e., tensile strain). Simultaneously, the allowable tensile strain limit of the material is defined as... The allowable compressive strain limit is The allowable shear strain limit is Therefore, the expression for the three-directional strain constraint function is obtained: in, Let x be the strain constraint value in the x-direction of the k-th layer of the laminate. Let y be the strain constraint value of the k-th layer of the laminate. This represents the shear strain constraint value of the k-th layer of the laminate.

[0062] (4) Individual fitness expression: No. i The total fitness of an individual V i The final optimization objective is obtained by summing the objective function, buckling constraint fitness, and three-directional strain constraint fitness, as shown in the following equation: in, w 1. w 2. w 3 represents the weights of the objective function, buckling constraint, and strain constraint, respectively.

[0063] Step 603, Construction of Genetic Algorithm Operators: (1) Selection operator: Setting the first g The population vector is The expression is as follows: in, For the first g Generation 1 i Individual design variable vector, N The total number of individuals in the population. This is determined by a rotation operator based on sorting or tournament rules. S (.), construct the parent set .

[0064] (2) Crossover operator: for g Two parent individuals in the population and The offspring generation expression is as follows. For the ply angle design variable, we have: in, For the first g Any offspring angle variable in the population, For the first g The parental perspective variable of this individual in the population. For the first g The other parent angle variable for this individual in the population. The offspring angle variable is set to a random floating-point number between 0 and 1.r The variables are randomly selected from the perspectives of the two parents.

[0065] For the ply thickness variable, we have: in, Let α be the ply thickness sequence of a certain offspring of the g-th generation population, and α be the crossover weight (between 0 and 1). and The two parent layer thickness sequences of this individual.

[0066] (3) Mutation operator: For the angle variable, random resampling is performed in the angle sequence {0°, 45°, -45°, 90°} to obtain a new individual angle sequence.

[0067] For the thickness variable, let the difference between the upper and lower limits of the ply thickness variable be denoted as . The mutation expression is as follows: in, This represents the layup thickness sequence of a certain offspring individual in the g-th generation population. The layup thickness sequence of its parent individuals, where N(0,1) is a standard normal distribution. α A parameter that controls the magnitude of variation (between 0 and 1).

[0068] Step 604: Optimize termination conditions: The optimization stops when any one of the following conditions is met: 1) Achieve the maximum number of optimization algebras; 2) Or the improvement in the optimal value over several consecutive generations is less than the threshold; Once the optimization termination condition is met, the optimal individual design variables are output. x * .

[0069] In summary, this invention, based on a mechanical modeling method combining classical laminate theory and physical information neural networks, constructs a unified computational framework for evaluating and optimizing the bending and buckling performance of composite laminates, offering the following technical effects and advantages: 1. It can achieve high-precision and rapid prediction of the mechanical response of laminates while ensuring physical consistency.

[0070] This invention directly incorporates the energy functional of the bending and buckling problem of laminates into the neural network training process, ensuring that the network's solution process strictly aligns with real physical laws. This avoids the shortcomings of traditional data-driven surrogate models, which "only fit the data and do not reflect the mechanism." Stable and reliable mechanical response results can be obtained without requiring a large amount of finite element sample data, significantly improving computational efficiency and result reliability.

[0071] 2. It can fully reflect the true impact of composite laminate layup parameters on structural performance.

[0072] This invention explicitly introduces laminates during the modeling process. ABD The stiffness matrix accurately reflects the effects of ply angle and thickness variations on bending stiffness, in-plane stiffness, and tension-bending coupling effects. Compared to a simplified model that only uses equivalent bending stiffness, this invention can more realistically reflect the structural mechanical behavior in complex engineering scenarios such as asymmetric plying and multi-material lamination.

[0073] 3. It can uniformly adapt to various boundary conditions, improving the versatility of the method and its applicability in engineering.

[0074] This invention achieves unified support for various typical boundary conditions, such as simply supported four sides, fixed four sides, simply supported two sides with both sides free, and fixed two sides with both sides free, by constructing a polynomial-form boundary embedding function. Under different boundary conditions, there is no need to reconstruct the solution model or algorithm flow; adaptation can be achieved simply by adjusting the boundary expression, significantly improving the applicability of the method to different structural configurations and engineering problems.

[0075] 4. It can directly output the key performance indicators required for engineering design, which facilitates integration with the optimization design process.

[0076] This invention can directly output the first-order buckling factor and the strain extreme values ​​of each ply, with the index format highly consistent with the composite material structure design specifications, and can be directly used for strength and stability constraint determination. This feature allows the method to be naturally embedded into the structural optimization process, providing a standardized performance evaluation interface for the automatic optimization of ply parameters.

[0077] 5. It can significantly reduce the dependence on finite element modeling in composite material structure design and improve design efficiency.

[0078] This invention eliminates the need to construct complex finite element models and mesh generation. It only requires input of geometry, material, load, boundary, and layup parameters to complete analysis and optimization. Compared with the traditional "modeling-solving-post-processing" finite element process, it significantly reduces modeling complexity and labor costs, making it particularly suitable for applications such as parameter sensitivity analysis, scheme comparison, and intelligent optimization design.

[0079] As a second aspect, the present invention also provides an electronic device, including a memory and a processor, the memory for storing a computer program, the processor for running the computer program to enable the electronic device to perform the above-described method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on a physical information neural network.

[0080] As a third aspect, the present invention also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on a physical information neural network as described above.

[0081] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. The same or similar parts between the various embodiments can be referred to each other.

[0082] This document uses specific examples to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the core ideas of the present invention. Furthermore, those skilled in the art will recognize that, based on the ideas of the present invention, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on physical information neural networks, characterized in that, include: Set the relevant parameters for the laminate; The relevant parameters include geometry, ply parameters, material properties, load input, and boundary conditions; the ply parameters include angle, thickness, and mass. Based on the aforementioned parameters and classical laminate theory, laminates are constructed. ABD The stiffness matrix is ​​used, and the physical domain of the arbitrary quadrilateral laminate is mapped to the reference domain through a bitriangular affine mapping, constructing a physical information neural network mechanical model for arbitrary quadrilateral laminates; wherein, the... ABD The stiffness matrix is ​​obtained by integrating the rotational stiffness matrix of each layer of the laminate and the ply thickness, and is used to explicitly reflect the influence of ply angle and thickness on the overall stiffness. Based on the physical information neural network mechanical model, the minimization of the total potential energy of the bending problem and the minimization of the Rayleigh quotient of the buckling problem are taken as training objectives to solve the buckling factor and the layer-by-layer strain limit. The four types of typical boundary conditions are uniformly adapted by a strong formal boundary embedding function. The four types of typical boundary conditions include simply supported on four sides, fixed on four sides, simply supported on two sides and free on both sides, and fixed on two sides and free on both sides. The ply angle and ply thickness of the laminate under test are used as design variables and optimized using a genetic algorithm. The goal is to minimize the ply mass, with buckling factor and layer-by-layer strain limits as constraints, to output the optimal ply scheme. The optimal ply scheme is a mass-minimizing design obtained by setting one of the four typical boundary conditions, evaluating the buckling factor and strain extreme values ​​through the physical information neural network mechanical model, and using them as constraints in the genetic algorithm for optimization search.

2. The method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on physical information neural networks according to claim 1, characterized in that, The bi-triangular affine mapping specifically includes: The reference domain unit square is divided into two triangular subdomains along the diagonal. An affine transformation relationship between the reference domain and the physical domain is established, and the Jacobian matrix and its determinant are calculated. The equivalent calculation of the energy integral between the physical domain and the reference domain is performed through the Jacobian transformation.

3. The method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on physical information neural networks according to claim 1, characterized in that, The specific process for solving the buckling factor and the layer-by-layer strain limit includes: Based on the principle of minimizing total potential energy, a total potential energy functional for the bending problem is constructed as the first loss function, and the physical information neural network mechanical model is used to solve the bending response and layer-by-layer strain limit of the laminate under out-of-plane load. Based on the Rayleigh quotient minimization principle, the ratio of structural strain energy to geometric stiffness energy in the buckling problem is constructed as the second loss function, and the buckling factor of the laminate under in-plane loading is solved using the physical information neural network mechanical model.

4. The method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on physical information neural networks according to claim 1, characterized in that, The design variable encoding form of the genetic algorithm is as follows: in, x Let be a chromosome vector, representing a candidate layering scheme; For the k-th layer, Let N be the thickness of the k-th layer, where k = 1, 2, ..., N; N This represents the total number of laminate layers.

5. The method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on physical information neural networks according to claim 1, characterized in that, The genetic algorithm includes selection, crossover, and mutation operators; the angle variable is updated using discrete random sampling or random sampling of the parent generation; the thickness variable is updated using linear combination or normal distribution perturbation.

6. The method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on physical information neural networks according to claim 1, characterized in that, The ply angle is a discrete variable selected from the set of angles {0°, 45°, -45°, 90°}; the ply thickness is a continuous variable and is optimized within a preset range.

7. An electronic device, characterized in that, The device includes a memory and a processor, the memory being used to store a computer program, and the processor running the computer program to enable the electronic device to perform the method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on a physical information neural network according to any one of claims 1-6.

8. A computer-readable storage medium, characterized in that, It stores a computer program, which, when executed by a processor, implements the method for joint evaluation and optimization of buckling and bending of arbitrary quadrilateral laminates based on a physical information neural network as described in any one of claims 1-6.

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