Metamaterial adapter optimization design method based on proxy model

Abstract

This invention provides an optimization design method for metamaterial adapters based on a surrogate model, belonging to the field of metamaterial adapter design technology. The method includes: establishing a finite element model of a polyurethane metamaterial adapter; constructing an integrated process for parametric modeling and finite element analysis model simulation; obtaining a high-precision analysis model and a low-precision analysis model through mesh consistency analysis of the finite element model; sampling and generating nested high-precision and low-precision sample points within the constraints of design variables, and obtaining high-precision and low-precision sample data; modeling using the high-precision and low-precision sample data to construct a variable confidence prediction model; establishing an optimization problem for the mechanical properties of the polyurethane metamaterial adapter; and outputting the optimal solution to the optimization problem of the polyurethane metamaterial adapter by combining the variable confidence prediction model with a global optimization algorithm. This invention can balance the prediction accuracy of the model with the efficiency of the optimization design.

Description

Technical Field

[0001] This invention relates to the field of metamaterial adapter design technology, and in particular to an optimization design method for metamaterial adapters based on a surrogate model. Background Technology

[0002] The adapter is a crucial component for connecting the missile to the launch platform and enabling missile launch. Its mechanical structure design directly impacts the stable connection and reliable launch between the missile and the launch platform. Currently, the materials used for missile adapters are primarily rubber, polyurethane elastomers, polyurethane foam, or mixtures of these materials. Adapters made from these materials can have densities ranging from tens to hundreds of kilograms per cubic meter. 3 The strength varies (0.018~0.060W(m·K), is lightweight, high-strength, and has good thermal insulation properties). -1 While suitable for a wide temperature range, traditional adapter structures often cannot simultaneously meet these requirements. For example, some heavy materials have low strength and stiffness, making them unable to withstand vibrations generated during high-speed flight. Furthermore, due to unreasonable structural design, they are prone to deformation and damage, resulting in reduced lifespan, during long-term ballast loading during missile storage, exhibiting poor compressive deformation characteristics and creep resistance. One approach to addressing these issues is to conduct adapter design research using negative Poisson's ratio metamaterials. As a special type of structure, the negative Poisson's ratio structure expands its lateral dimension under tension and shrinks under compression, exhibiting a negative Poisson's ratio. Due to its unique structure, negative Poisson's ratio materials also exhibit special mechanical properties, such as high compressibility, good sound and energy absorption characteristics, high designability, and creep resistance, making them promising for broad research.

[0003] Currently, there are still many challenges in the optimization design process of negative Poisson's ratio metamaterial structures. When a large number of engineering problems involve the optimization design of negative Poisson's ratio metamaterial structures, analysis methods based on physical experiments or numerical simulations are usually adopted. However, due to the complexity of physical experiments and the limited computing resources, the cost of numerical simulation analysis is getting higher and higher, and the nonlinearity and implicitness of the design objective function and constraints are increasing.

[0004] Therefore, how to effectively utilize limited high-precision simulation data to complete the optimized design of negative Poisson's ratio metamaterial adapters while ensuring accuracy is one of the key factors affecting the design efficiency and quality of missile adapters.

[0005] Chinese invention patent application number 202011232458.1 discloses a design method for chiral metamaterial structures with a predetermined negative Poisson's ratio. This method automatically and rapidly designs mechanical metamaterial structures based on specific negative Poisson's ratio requirements, breaking away from traditional experience-based design approaches, improving work efficiency, and saving design time. This prior art only uses a negative Poisson's ratio as a limiting requirement for structural optimization design and does not guarantee data accuracy. Summary of the Invention

[0006] In view of this, the present invention proposes an optimization design method for metamaterial adapters based on a surrogate model. This method makes full use of limited, costly, high-precision analysis data and a large amount of low-cost, low-precision analysis data to construct an adapter performance prediction model, and combines it with a global optimization algorithm for optimization design, which can balance the prediction accuracy of the model and the efficiency of the optimization design.

[0007] The technical solution of this invention is implemented as follows: This invention provides an optimization design method for metamaterial adapters based on a surrogate model, comprising:

[0008] S1 establishes a finite element model of the polyurethane metamaterial adapter and obtains the structural response of the adapter;

[0009] S2 extracts the characteristic geometric design variables of the adapter and determines the constraint range of the design variables. It then performs parametric modeling of the polyurethane metamaterial adapter, using the design variables as input parameters to construct an integrated process of parametric modeling and finite element analysis model simulation.

[0010] S3 obtains high-precision and low-precision analysis models of polyurethane metamaterial adapters through mesh consistency analysis of finite element models;

[0011] S4 samples and generates nested high-precision and low-precision sample points within the constraints of the design variables, and calculates the response values ​​of the high-precision and low-precision sample points under the corresponding high-precision and low-precision analysis models, respectively, to obtain high-precision sample data and low-precision sample data.

[0012] S5 uses high-precision and low-precision sample data to build a variable confidence prediction model between the design variables and mechanical performance response of polyurethane metamaterial adapters.

[0013] S6 establishes an optimization problem for the mechanical properties of a polyurethane metamaterial adapter, and outputs the optimal solution to the optimization problem of the polyurethane metamaterial adapter by combining a variable confidence prediction model with a global optimization algorithm.

[0014] Based on the above technical solutions, preferably, step S1 includes:

[0015] S11 uses 3D software to establish a 3D geometric model of polyurethane metamaterial according to design and geometric requirements. The overall framework of the component is formed by a centrally symmetric unit cell configuration array with negative Poisson's ratio characteristics.

[0016] S12 meshes the three-dimensional geometric model of the polyurethane metamaterial, performs mesh consistency analysis on the mesh, and evaluates the accuracy and convergence of the mesh.

[0017] S13 imposes constraints on the structure of the polyurethane metamaterial, defines the upper surface of the structure as a rigid displacement plane, and fixes the degree of freedom of the lower surface of the structure;

[0018] S14 applies a pressure load to the finite element model and applies material properties to the polyurethane metamaterial.

[0019] S15 uses a numerical solution method to solve the quasi-static compression problem of polyurethane metamaterials, and obtains the structural response of the polyurethane metamaterial adapter through the solution results.

[0020] Based on the above technical solutions, preferably, the solution to the quasi-static compression problem of the polyurethane metamaterial adapter specifically includes:

[0021] To solve the compression deformation characteristics of a polyurethane metamaterial adapter during compression, the load and deformation data of the polyurethane metamaterial adapter at each time step are extracted by uniformly distributing the simulation time step, and the compression deformation characteristics of the compression process are analyzed.

[0022] The equivalent Poisson's ratio characteristic of the adapter compression process is solved by calculating the equivalent Poisson's ratio value through the difference between the average lateral and longitudinal displacements in the configuration unit cell.

[0023] Based on the above technical solutions, preferably, step S2 includes:

[0024] S21 determines the characteristic geometric parameters of the polyurethane metamaterial adapter, including the fillet radius of the horizontal hole R1, the fillet radius of the vertical hole R2, the center distance of the lower right hole D1, the center distance of the upper right hole D2, the center distance of the upper left hole D3, and the center distance of the lower left hole D4.

[0025] S22 inputs the characteristic geometric parameters as design variables into the 3D software and writes the reconstruction file of the 3D geometric model;

[0026] S23 provides an integrated workflow for parametric modeling and finite element analysis simulation, including model import, mesh generation, load application, and simulation solution.

[0027] Based on the above technical solutions, preferably, step S4 includes:

[0028] S41 sets the number of sampling points according to the constraint range of the design variables;

[0029] S42 uses the Latin hypercube sampling method to generate multiple high-precision sample points and multiple low-precision sample points based on the set number of sampling points. The high-precision sample points and low-precision sample points satisfy a nested relationship.

[0030] S43 inputs high-precision sample points into the high-precision analysis model to calculate high-precision response values;

[0031] S44 inputs low-precision sample points into the low-precision analysis model and calculates the low-precision response value;

[0032] S45 combines high-precision sample points with their corresponding high-precision response values ​​to obtain high-precision sample data, and combines low-precision sample points with their corresponding low-precision response values ​​to obtain low-precision sample data.

[0033] Based on the above technical solutions, preferably, step S5 includes:

[0034] S51 uses low-precision sample data to build a low-precision Kriging proxy model.

[0035] S52 inputs high-precision sample points into a low-precision kriging surrogate model and calculates the predicted response value of the high-precision sample points in the low-precision kriging surrogate model.

[0036] S53 constructs a difference model between the high-precision analysis model and the low-precision analysis model based on the difference between the predicted response value of the high-precision sample point and the high-precision response value of the high-precision sample point.

[0037] S54 constructs a variable confidence prediction model based on the difference model.

[0038] Based on the above technical solutions, preferably, in step S54, the variable confidence prediction model is expressed as:

[0039] f h (x)=k l f l (x)+c h (x)

[0040] In the formula, f h (x) is a high-precision analysis model, k l f is the proportionality coefficient. l (x) represents a low-precision analysis model, c h (x) represents the difference model.

[0041] Based on the above technical solutions, the preferred variable confidence prediction model's predicted response value is calculated using the following formula:

[0042]

[0043] In the formula, Let be the mean of the points to be predicted, c be the covariance matrix between the sample data points X and the predicted point x, C be the covariance matrix between the sample data points X, and Y be the function value of the sample points. Intermediate variables used to obtain the final result.

[0044] Based on the above technical solutions, preferably, the mathematical expression of the mechanical performance optimization problem in step S6 is as follows:

[0045] Find x = (x1, x2, x3, x4, x5, x6)

[0046]

[0047] stg(x)=E s ≥1Mpa

[0048] x1∈[2.0,3.5], x2∈[2.5,3.5],

[0049] x3∈[6.3,7.8], x4∈[9.5,10.5],

[0050] x5∈[17.0,10.0], x6∈[7.0,7.2]

[0051] In the formula, ε X ε Y E s Let x be the characteristic geometric parameter, v(x) be the objective function, and st be the constraint condition.

[0052] Based on the above technical solutions, preferably, in step S6, the optimal solution to the polyurethane metamaterial adapter optimization problem is output by combining a variable confidence prediction model with a global optimization algorithm, including:

[0053] Step 1: Select the improved particle swarm optimization algorithm based on adaptive learning factor as the global optimization algorithm;

[0054] Step 2: Within the range of values ​​for the characteristic geometric parameters of the metamaterial, update the individual variables according to the optimization update rule, which is the iterative formula for the particles;

[0055] Step 3: Calculate the predicted response value under each feature geometric parameter based on the variable confidence prediction model;

[0056] Step 4: Update the learning factor of the algorithm based on the number of iterations;

[0057] Step 5: Determine if the optimization process has converged. If convergence is achieved, proceed to Step 6. If not, add the learning factor obtained in Step 4 to the iterative formula and proceed to Step 2.

[0058] Step 6: Output the optimal solution to the polyurethane metamaterial adapter optimization problem.

[0059] The method of the present invention has the following advantages over the prior art:

[0060] (1) The present invention can fully combine limited, costly, high-precision analysis data and low-cost, low-precision analysis data. The prediction model constructed is more accurate than the single-precision proxy model of the same type. In the optimization design process of negative Poisson's ratio metamaterial adapter, it can take into account both design cost and design efficiency, and ensure optimization accuracy, thereby improving the quality of negative Poisson's ratio metamaterial optimization design.

[0061] (2) The parametric modeling method adopted in this invention can address the issues of repetitiveness and time consumption in sampling during the experimental design process, and achieve the integration of modeling, simulation analysis and optimization calculation, saving designers a lot of effort and time costs, and greatly improving the time efficiency of the optimization process. Attached Figure Description

[0062] To more clearly illustrate the technical solutions in the embodiments of the present invention 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 the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0063] Figure 1 This is a flowchart of a method according to an embodiment of the present invention;

[0064] Figure 2 This is a schematic diagram of the mesh generation of the finite element model according to an embodiment of the present invention;

[0065] Figure 3 This is a schematic diagram illustrating the calculation of the equivalent Poisson's ratio of the polyurethane metamaterial according to an embodiment of the present invention;

[0066] Figure 4 This is a parameterized schematic diagram of the polyurethane metamaterial according to an embodiment of the present invention;

[0067] Figure 5 This is a calculation diagram of the high and low precision analysis model according to an embodiment of the present invention;

[0068] Figure 6 This is a schematic diagram comparing the prediction accuracy of the variable confidence prediction model in an embodiment of the present invention. Detailed Implementation

[0069] The technical solutions of the present invention will be clearly and completely described below with reference to the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0070] This invention focuses on the mechanical structure design of metamaterial adapters, which play a crucial role in the stable connection and reliable launch of missiles and launch platforms. The invention employs a high-molecular-weight polyurethane material with excellent compressive deformation properties as the adapter's substrate. A honeycomb-like macroscopic design is implemented to achieve a negative Poisson's ratio effect. Based on this, an optimization design method is proposed to improve the macroscopic mechanical properties of the metamaterial by optimizing its characteristic geometric dimensions. The optimization design of negative Poisson's ratio metamaterial structures is a CAD-CAE adaptive iterative process, which is costly in numerical simulation analysis and involves strong nonlinearity and implicitness in the design objective function and constraints.

[0071] Please see Figure 1 This invention proposes an optimization design method for metamaterial adapters based on a surrogate model, comprising:

[0072] S1 establishes a finite element model of the polyurethane metamaterial adapter and obtains the structural response of the adapter;

[0073] S2 extracts the characteristic geometric design variables of the adapter and determines the constraint range of the design variables. It then performs parametric modeling of the polyurethane metamaterial adapter, using the design variables as input parameters to construct an integrated process of parametric modeling and finite element analysis model simulation.

[0074] S3 obtains high-precision and low-precision analysis models of polyurethane metamaterial adapters through mesh consistency analysis of finite element models;

[0075] S4 samples and generates nested high-precision and low-precision sample points within the constraints of the design variables, and calculates the response values ​​of the high-precision and low-precision sample points under the corresponding high-precision and low-precision analysis models, respectively, to obtain high-precision sample data and low-precision sample data.

[0076] S5 uses high-precision and low-precision sample data to build a variable confidence prediction model between the design variables and mechanical performance response of polyurethane metamaterial adapters.

[0077] S6 establishes an optimization problem for the mechanical properties of a polyurethane metamaterial adapter, and outputs the optimal solution to the optimization problem of the polyurethane metamaterial adapter by combining a variable confidence prediction model with a global optimization algorithm.

[0078] Specifically, in one embodiment of the present invention, step S1 includes:

[0079] S11 uses 3D software to establish a 3D geometric model of polyurethane metamaterial according to design and geometric requirements. The overall framework of the component is formed by a centrally symmetric unit cell configuration array with negative Poisson's ratio characteristics.

[0080] S12 meshes the three-dimensional geometric model of the polyurethane metamaterial, performs mesh consistency analysis on the mesh, and evaluates the accuracy and convergence of the mesh.

[0081] S13 imposes constraints on the structure of the polyurethane metamaterial, defines the upper surface of the structure as a rigid displacement plane, and fixes the degree of freedom of the lower surface of the structure;

[0082] S14 applies a pressure load to the finite element model and applies material properties to the polyurethane metamaterial.

[0083] S15 uses a numerical solution method to solve the quasi-static compression problem of polyurethane metamaterials, and obtains the structural response of the polyurethane metamaterial adapter through the solution results.

[0084] The solution to the quasi-static compression problem of the polyurethane metamaterial adapter specifically includes:

[0085] To solve the compression deformation characteristics of a polyurethane metamaterial adapter during compression, the load and deformation data of the polyurethane metamaterial adapter at each time step are extracted by uniformly distributing the simulation time step, and the compression deformation characteristics of the compression process are analyzed.

[0086] The equivalent Poisson's ratio characteristic of the adapter compression process is solved by calculating the equivalent Poisson's ratio value through the difference between the average lateral and longitudinal displacements in the configuration unit cell.

[0087] This embodiment uses the finite element method (FEM) software ANSYS 19.2 to perform finite element simulations on the mechanical properties of the metamaterial adapter component. To facilitate an integrated modeling and simulation workflow, the entire simulation process is automated using the software's built-in APDL command flow, including mesh generation, load setting, constraint setting, calculation evaluation, and post-processing.

[0088] like Figure 2As shown, the three-dimensional geometry of the metamaterial adapter is imported into the software. The SOLID186 element has a quadratic displacement mode, which can better simulate the model established by the CAD system. Furthermore, this element supports hyperelasticity, creep, large deformation, and large strain capabilities. Therefore, the SOLID186 element is used to mesh the three-dimensional geometry of the metamaterial adapter. To simulate the actual strain of the metamaterial adapter component under compression, the upper surface of the structure is defined as a rigid displacement plane, coupling the degrees of freedom of all nodes on the upper surface in the vertical direction. The lower surface of the structure is fixed, i.e., the degrees of freedom of the nodes on the lower surface are fully constrained in all directions. A constitutive model of polyurethane material is defined. Through multiple compression experiments at 20% strain, the elastic modulus of polyurethane material is found to remain around 51.5 MPa. Therefore, the elastic modulus of the material in the simulation analysis model is set to 51.5 MPa, and the Poisson's ratio is set to 0.475. To simulate quasi-static compression, a downward vertical force of 4600 N is applied to the upper surface of the structure, uniformly loaded in 20 load steps, and a static analysis of the component structure is performed.

[0089] To evaluate the impact of mesh count on simulation results in the finite element model of the metamaterial adapter, mesh consistency analysis is required for simulations with different mesh counts. A hexahedral swept mesh was used to mesh the geometry. The mesh coarseness was measured by the number of elements in the thickness of the horizontal and vertical beams. Six mesh counts, ranging from 2 to 7 with intervals of 1, were used to simulate the metamaterial adapter component. When the mesh count exceeded 4, the maximum displacement tended to converge. Therefore, based on the mesh consistency analysis results, to reduce the additional computational cost caused by increased mesh count while maintaining the accuracy of the finite element simulation, the high-precision finite element simulation model of the metamaterial adapter component was set to 4 meshes, and the low-precision model was set to 2 meshes, i.e., 4 and 2 elements respectively in the thickness direction of the horizontal and vertical beams within the unit cell.

[0090] like Figure 3 As shown, to facilitate the calculation of Poisson's ratio of the structure, the structural state at the end of the last load step, i.e., when the structural strain is approximately 15%, is extracted for calculating the equivalent Poisson's ratio. The middle unit cell of the structure is extracted for Poisson's ratio analysis. The formula for calculating Poisson's ratio is... In the formula, ε X ε represents the X-direction strain of the compressed structure. Y Let ε be the strain in the Y direction of the structure. Since the Y-direction displacement of the upper surface of the structure is the same at each node, therefore ε Y The displacement data of the upper surface of the structure at 15% compressive strain can be obtained from the structural load-deformation curve. The displacement difference between the upper and lower surfaces of the unit cell is -9.4964 mm. The extracted unit cell X-axis and Y-axis displacement contour maps show the structural X-axis strain ε. XThe average displacement in the X-direction of the left and right boundaries of the unit cell was calculated as follows: Eight symmetrical nodes were selected at the left and right boundaries of the unit cell, and the X-direction displacements at these nodes were extracted. The average displacements at these nodes were averaged to obtain average displacements of 5.5654 mm and -5.5363 mm for the left and right boundaries of the unit cell, respectively. The strain was calculated using the obtained differences in X-direction and Y-direction displacements of the unit cell, and the Poisson's ratio at this moment was obtained using the Poisson's ratio formula.

[0091] Specifically, in one embodiment of the present invention, step S2 includes:

[0092] S21 determines the characteristic geometric parameters of the polyurethane metamaterial adapter, including the fillet radius of the horizontal hole R1, the fillet radius of the vertical hole R2, the center distance of the lower right hole D1, the center distance of the upper right hole D2, the center distance of the upper left hole D3, and the center distance of the lower left hole D4.

[0093] S22 inputs the characteristic geometric parameters as design variables into the 3D software and writes the reconstruction file of the 3D geometric model;

[0094] S23 provides an integrated workflow for parametric modeling and finite element analysis simulation, including model import, mesh generation, load application, and simulation solution.

[0095] In embodiments of the present invention, a negative Poisson's ratio adapter configuration generated after topology optimization is used as the initial geometric configuration, and an elastic tension-compression structure is formed by horizontal and vertical beams at a certain angle. For example... Figure 4 As shown, the geometric structure consists of an array of "well"-shaped unit cells with rounded corners. This unit cell configuration can be viewed as being formed by mapping a "+" shaped structure with a certain curvature along two central axes. Based on the unit cell configuration, six main geometric parameters are designed, characterizing the size and position of the rounded corners at the nodes. These six parameters are: the radius of the rounded corners of the horizontal holes (R1), the radius of the rounded corners of the vertical holes (R2), the center distance of the lower right hole (D1), the center distance of the upper right hole (D2), the center distance of the upper left hole (D3), and the center distance of the lower left hole (D4).

[0096] Specifically, in this embodiment, to avoid designers manually performing hundreds or even thousands of geometric model reconstructions and numerical simulation analyses, a software-integrated parametric modeling architecture is adopted to achieve the integration of modeling and simulation analysis. The specific implementation steps are as follows:

[0097] ① Determine the design variables for the engineering optimization problem, i.e., the dimensional parameters that need to be changed in SolidWorks software; ② Use VS software to compile a SolidWorks secondary development script in C# and generate a callable dynamic link library file. The script content is to complete the geometric model reconstruction by inputting specific dimensional parameters, including the creation and modification of various features of the part and the saving of the part drawing; ③ Use the classic ANSYS interface to perform a complete numerical simulation analysis process, including model import, mesh generation, load application, and simulation solution; ④ Save the APDL command stream file of the complete simulation process; ⑤ Use MATLAB to write the main program for cyclic simulation: first, call the compiled dynamic link library file to complete the geometric model reconstruction, then call ANSYS to read the APDL command stream file to complete the automatic simulation analysis, obtain the response values ​​of the sample points, and complete one cycle.

[0098] In this embodiment, there are 6 design variables for the metamaterial adapter performance optimization problem. The descriptions and value ranges of each variable are summarized in the table below.

[0099] Table 1. Physical meaning and value range of optimization design variables for metamaterial adapters.

[0100] <![CDATA[x1]]> <![CDATA[Radius R1 of the transverse hole]]> 2.0~3.5mm <![CDATA[x2]]> <![CDATA[Vertical hole fillet radius R2]]> 2.5~3.5mm <![CDATA[x3]]> <![CDATA[Center distance D1 of the lower right hole]]> 6.3~7.8mm <![CDATA[x4]]> <![CDATA[Center distance D2 of the upper right hole]]> 9.5~10.5mm <![CDATA[x5]]> <![CDATA[Distance D3 from the center of the upper left hole]]> 17.0~19.0mm <![CDATA[x6]]> <![CDATA[Center distance D4 of the lower-left hole]]> 7.0~7.2mm

[0101] Specifically, in one embodiment of the present invention, step S3 includes:

[0102] Based on the mesh independence analysis results, a finite element model with 2 mesh parts was set as the low-precision analysis model, and a finite element model with 4 mesh parts was set as the high-precision analysis model. Figure 5 As shown in the table below, the simulation time costs for high- and low-precision sample points are summarized.

[0103] Table 2 Simulation time cost for sample points with different precision

[0104]

[0105] Specifically, in one embodiment of the present invention, step S4 includes:

[0106] S41 sets the number of sampling points according to the constraint range of the design variables;

[0107] S42 uses the Latin hypercube sampling method to generate multiple high-precision sample points and multiple low-precision sample points based on the set number of sampling points. The high-precision sample points and low-precision sample points satisfy a nested relationship.

[0108] S43 inputs high-precision sample points into the high-precision analysis model to calculate high-precision response values;

[0109] S44 inputs low-precision sample points into the low-precision analysis model and calculates the low-precision response value;

[0110] S45 combines high-precision sample points with their corresponding high-precision response values ​​to obtain high-precision sample data, and combines low-precision sample points with their corresponding low-precision response values ​​to obtain low-precision sample data.

[0111] Specifically, in this embodiment, the simulation time cost ratio between low-precision sample points and high-precision sample points is approximately 1:6, meaning the time cost of calculating the response value of one high-precision sample point is equivalent to the time cost of calculating the response values ​​of six low-precision samples. Assume the total budget number of the high-precision analysis model for the metamaterial adapter design optimization problem is... To construct a variable reliability surrogate model describing the performance of the metamaterial adapter, we consider using 40 high-precision sample points and 240 low-precision sample points. The total time cost is approximately equal to that of 80 high-precision sample points. We also use 50 high-precision verification points to evaluate the accuracy of the surrogate model.

[0112] To ensure the uniformity of sample points in the global design space, the Latin hypercube experimental design method is used to obtain high- and low-precision sample point sets for the metamaterial adapter prediction model, and attention is paid to using nested sampling of high and low sample data. The performance prediction of the metamaterial adapter involves establishing a surrogate model by collecting sample point data through physical simulation. For the geometric design of the metamaterial adapter, its equivalent Poisson's ratio and equivalent compressive modulus during compression are important parameters for evaluating its mechanical properties. The equivalent compressive modulus of the structure can be calculated from the Y-direction displacement during compression, and the equivalent Poisson's ratio can be calculated from the X-direction strain and Y-direction strain of the structure. Therefore, in this embodiment, the sample datasets that need to be calculated through time-consuming simulation are the maximum Y-direction displacement of the upper surface and the average X-direction displacement of the central unit cell.

[0113] Specifically, in one embodiment of the present invention, step S5 includes:

[0114] S51 uses low-precision sample data to build a low-precision Kriging surrogate model.

[0115] By utilizing low-precision sample data and the calculated response values ​​of low-precision sample points in a low-precision analysis model with an adapter grid precision of 2, the hyperparameters of the adapter low-precision surrogate model are solved using the likelihood maximization method. A low-precision kriging surrogate model is then established to reflect the general trend of the adapter performance prediction model.

[0116] Kriging interpolation is a statistical interpolation method that predicts values ​​at other locations by spatially interpolating known data points. In low-precision Kriging surrogate models, low-precision sample data is used to train the model, resulting in a coarse model that approximates the system's behavior.

[0117] S52 inputs high-precision sample points into a low-precision kriging surrogate model and calculates the predicted response values ​​of the high-precision sample points in the low-precision kriging surrogate model. Since the low-precision kriging surrogate model is built using low-precision sample data, its prediction results may differ somewhat from the actual high-precision response values.

[0118] S53 constructs a difference model between the high-precision analysis model and the low-precision analysis model based on the difference between the predicted response value of the high-precision sample points and the high-precision response value of the high-precision sample points.

[0119] Based on the difference between the response values ​​of high-precision points in the low-precision model and the response values ​​calculated by the high-precision points using a high-precision simulation analysis model with an adapter mesh precision of 4, a difference model c is constructed. h =f2-k1f1(x2), where f2 represents the function value of the high-precision sample point, k1 represents the scaling factor, and f1(x2) represents the predicted response value of the high-precision sample point x2 on the adapter's low-precision analysis model. Using the high-precision sample points to... h Model the model and solve for the remaining hyperparameters, also using the method of maximizing the likelihood function.

[0120] S54 constructs a variable confidence prediction model based on the difference model.

[0121] A variable confidence prediction model is a model used to estimate the confidence level or reliability of a model's predictions. By analyzing the characteristics and trends of the difference model, a prediction model can be built to predict the confidence level of the model's predictions based on the characteristics of the current sample points. This allows for the correction and adjustment of prediction results from models with different levels of accuracy during the optimization process, thereby improving the accuracy and reliability of the optimization algorithm.

[0122] In this embodiment, the variable confidence prediction model is expressed as:

[0123] f h (x)=k l f l (x)+c h (x)

[0124] In the formula, f h (x) is a high-precision analysis model, k l f is the proportionality coefficient. l (x) represents a low-precision analysis model, c h (x) represents the difference model.

[0125] Specifically, after solving for all hyperparameters, the estimated value of the established variable confidence prediction model can be calculated using the following formula:

[0126]

[0127] In the formula, Let be the mean of the points to be predicted, c be the covariance matrix between the sample data points X and the predicted point x, C be the covariance matrix between the sample data points X, and Y be the function value of the sample points. Intermediate variables used to obtain the final result.

[0128] In this embodiment, three simulation time parameters are used to establish the proxy model: the maximum Y-direction displacement w1 on the upper surface of the metamaterial adapter, the average X-direction displacement (left) w2 of the central unit cell, and the average X-direction displacement (right) w3 of the central unit cell. Meanwhile, in addition to the selected six-dimensional geometric design variables, the performance optimization design of the metamaterial adapter requires all other parameters to be fixed as constraints. These include simulation-fixed variables: the elastic modulus of the polyurethane material E = 51.5 MPa, the Poisson's ratio of the polyurethane material υ = 0.475, and the rated load F = 200 N; and geometric-fixed variables: the tensile thickness t of the structural component. ini =10mm, initial height H of structural component ini =60mm, structural unit cell length L in the X direction ini =50mm, number of structural unit cells n=3.

[0129] To evaluate the predictive performance of the constructed surrogate model, this embodiment compares the prediction models for the three simulation time-consuming parameters using a single-precision kriging surrogate model and the variable confidence surrogate model (co-kriging) in this embodiment. The prediction error indices of the two surrogate models are summarized in the table below. It is clear from the table that the accuracy of predicting the responses of the three targets using the variable confidence surrogate model is generally superior to that of the single-precision surrogate model across all error indices, indicating that the variable confidence surrogate model has higher prediction accuracy. For a comparison of the predicted and actual values ​​of the surrogate model for a specific dimension in the optimization design of the polyurethane metamaterial adapter, please refer to [reference needed]. Figure 6 .

[0130] Table 3 Prediction accuracy of three objective responses in the optimized design of metamaterial adapters

[0131]

[0132]

[0133] Specifically, in one embodiment of the present invention, the mathematical expression of the mechanical performance optimization problem is as follows:

[0134] Find x = (x1, x2, x3, x4, x5, x6)

[0135]

[0136] stg(x)=E s ≥1Mpa

[0137] x1∈[2.0,3.5], x2∈[2.5,3.5],

[0138] x3∈[6.3,7.8], x4∈[9.5,10.5],

[0139] x5∈[17.0,10.0], x6∈[7.0,7.2]

[0140] In the formula, ε X ε Y E s Let x be the characteristic geometric parameter, v(x) be the objective function, and st be the constraint condition.

[0141] In this embodiment, the performance optimization design problem of polyurethane metamaterial adapter can be simply summarized as follows: based on the topological configuration of the metamaterial adapter, using six selected geometric characteristic parameters as design variables, a predictive model describing the mechanical properties of the metamaterial adapter is constructed, and the structural equivalent Poisson's ratio is minimized using the compression modulus as a constraint.

[0142] This optimization problem has 6 design variables, 1 objective function, and 1 constraint. The objective function and constraint cannot be explicitly expressed and must be calculated using the established surrogate model, as shown in the table below.

[0143] Table 4. Calculation Methods for Optimization Problem Variables

[0144]

[0145]

[0146] In this embodiment, obtaining the optimal solution to the optimization problem by combining the metamaterial adapter prediction model with a global optimization algorithm specifically includes the following steps:

[0147] Step 1: Select the improved particle swarm optimization algorithm based on adaptive learning factor as the global optimization algorithm.

[0148] An improved particle swarm optimization (PSO) algorithm based on an adaptive learning factor was selected as the global optimization algorithm. PSO is a swarm intelligence-based optimization algorithm that searches for optimal solutions by simulating the behavior of flocks of birds or schools of fish. The adaptive learning factor is an improvement on the traditional PSO algorithm; it automatically adjusts the size of the learning factor to improve the algorithm's convergence speed and global search capability. This global optimization algorithm can effectively search for optimal solutions for metamaterials and provide good initial solutions for subsequent optimization processes.

[0149] Step 2: Within the range of characteristic geometric parameters of the metamaterial, update the individual variables according to the optimization update rule, which is the iterative formula of the particles.

[0150] By iteratively updating individual variables, the particle swarm optimization algorithm can search the entire design space to find the optimal solution. This update rule helps the algorithm gradually converge to the vicinity of the optimal solution during the search process.

[0151] Step 3: Calculate the predicted response value under each feature geometric parameter based on the variable confidence prediction model.

[0152] Based on the previously constructed variable confidence prediction model, the confidence of the response value can be predicted according to the current values ​​of the feature geometric parameters. This allows for the correction and adjustment of the predicted response value for each feature geometric parameter during the optimization process, improving the accuracy and reliability of the optimization algorithm.

[0153] Step 4: Update the learning factor of the algorithm based on the number of iterations.

[0154] The learning factor is one of the key parameters controlling the update of individual particle positions in particle swarm optimization (PSO). By dynamically adjusting the learning factor based on the number of iterations, the algorithm can focus more on global search in the early stages and more on local search in the later stages. This improves the algorithm's global search capability and local search accuracy, thus accelerating its convergence speed.

[0155] Step 5: Determine if the optimization process has converged. If convergence is achieved, proceed to Step 6. If not, add the learning factor obtained in Step 4 to the iterative formula and proceed to Step 2.

[0156] The algorithm determines whether the optimization process has converged. If convergence is achieved, proceed to step six; otherwise, add the learning factor obtained in step four to the iterative formula and proceed to step two. By determining whether the optimization process has converged, it can be determined whether the algorithm has found the optimal solution. If the convergence condition is met, the algorithm stops iterating and outputs the optimal solution; if the convergence condition is not met, the iterative formula is updated based on the learning factor obtained in step four, and the next round of iteration continues. This ensures that the algorithm fully utilizes the information from the learning factor during the search process, improving the algorithm's search efficiency and accuracy.

[0157] Step 6: Output the optimal solution to the polyurethane metamaterial adapter optimization problem.

[0158] After the algorithm converges, the optimal solution to the polyurethane metamaterial adapter optimization problem is output. This optimal solution is obtained through a global optimization algorithm search and, after considering the influence of the variable confidence prediction model and the learning factor, exhibits high reliability and accuracy. This can provide effective guidance and reference for the design and optimization of metamaterials.

[0159] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for optimizing the design of metamaterial adapters based on a surrogate model, characterized in that, Includes the following steps: S1. Establish a finite element model of the polyurethane metamaterial adapter and obtain the structural response of the adapter; S2 extracts the characteristic geometric design variables of the adapter and determines the constraint range of the design variables. It then performs parametric modeling of the polyurethane metamaterial adapter, using the design variables as input parameters to construct an integrated process of parametric modeling and finite element analysis model simulation. S3 obtains high-precision and low-precision analysis models of polyurethane metamaterial adapters through mesh consistency analysis of finite element models; S4 generates nested high-precision and low-precision sample points within the constraints of the design variables, and calculates the response values ​​of the high-precision and low-precision sample points under the corresponding high-precision and low-precision analysis models respectively, thus obtaining high-precision sample data and low-precision sample data. S5 uses high-precision and low-precision sample data to build a variable confidence prediction model between the design variables and mechanical performance response of polyurethane metamaterial adapters. Step S5 includes: S51 uses low-precision sample data to build a low-precision Kriging proxy model; S52 inputs high-precision sample points into a low-precision kriging surrogate model and calculates the predicted response value of the high-precision sample points in the low-precision kriging surrogate model. S53 Based on the difference between the predicted response value of high-precision sample points and the high-precision response value of high-precision sample points, construct a difference model between the high-precision analysis model and the low-precision analysis model; S54 Construct a variable confidence prediction model based on the difference model; S6 defines the optimization problem of the mechanical properties of the polyurethane metamaterial adapter, and outputs the optimal solution of the optimization problem of the polyurethane metamaterial adapter by combining a variable confidence prediction model with a global optimization algorithm.

2. The surrogate model-based metamaterial adapter optimization design method as described in claim 1, characterized in that, Step S1 includes: S11 uses 3D software to establish a 3D geometric model of polyurethane metamaterial according to design and geometric requirements. The overall framework of the component is formed by a centrally symmetric unit cell configuration array with negative Poisson's ratio characteristics. S12 meshes the three-dimensional geometric model of the polyurethane metamaterial, performs mesh consistency analysis on the mesh, and evaluates the accuracy and convergence of the mesh. S13 imposes constraints on the structure of the polyurethane metamaterial, defines the upper surface of the structure as a rigid displacement plane, and fixes the degree of freedom of the lower surface of the structure; S14 applies a pressure load to the finite element model and applies material properties to the polyurethane metamaterial. S15 uses a numerical solution method to solve the quasi-static compression problem of polyurethane metamaterials, and obtains the structural response of the polyurethane metamaterial adapter through the solution results.

3. The surrogate model-based metamaterial adapter optimization design method as described in claim 2, characterized in that, The solution to the quasi-static compression problem of the polyurethane metamaterial adapter specifically includes: To solve the compression deformation characteristics of a polyurethane metamaterial adapter during compression, the load and deformation data of the polyurethane metamaterial adapter at each time step are extracted by uniformly distributing the simulation time step, and the compression deformation characteristics of the compression process are analyzed. The equivalent Poisson's ratio characteristic of the adapter compression process is solved by calculating the equivalent Poisson's ratio value through the difference between the average lateral and longitudinal displacements in the configuration unit cell.

4. The surrogate model-based metamaterial adapter optimization design method as described in claim 1, characterized in that, Step S2 includes: S21 Determine the characteristic geometric parameters of the polyurethane metamaterial adapter, including the fillet radius of the transverse holes. Vertical hole fillet radius Distance between the center of the lower right hole Distance between the centers of the upper right holes Distance between the center of the upper left hole Distance between the center of the lower left hole ; S22 Input the characteristic geometric parameters as design variables into the 3D software and write the reconstruction file of the 3D geometric model; S23 provides an integrated workflow for constructing parametric modeling and finite element analysis simulation models, including model import, mesh generation, load application, and simulation solution.

5. The surrogate model-based metamaterial adapter optimization design method as described in claim 1, characterized in that, Step S4 includes: S41 Set the number of sampling points according to the constraint range of the design variables; S42 uses the Latin hypercube sampling method to generate multiple high-precision sample points and multiple low-precision sample points based on the set number of sampling points. The high-precision sample points and low-precision sample points satisfy a nested relationship. S43 Input high-precision sample points into the high-precision analysis model to calculate high-precision response values; S44 Inputs low-precision sample points into the low-precision analysis model to calculate low-precision response values; S45 combines high-precision sample points with their corresponding high-precision response values ​​to obtain high-precision sample data, and combines low-precision sample points with their corresponding low-precision response values ​​to obtain low-precision sample data.

6. The surrogate model-based metamaterial adapter optimization design method as described in claim 1, characterized in that, In step S54, the variable confidence prediction model is expressed as: ； In the formula, For high-precision analysis models, This is the proportionality coefficient. This is a low-precision analysis model. This is a difference model.

7. The surrogate model-based metamaterial adapter optimization design method as described in claim 6, characterized in that, The predicted response value of the variable confidence prediction model is calculated using the following formula: ； In the formula, The mean of the points to be predicted. Let x be the covariance matrix between the sample data point X and the predicted point x. Let X be the covariance matrix among the sample data points, and Y be the function values ​​of the sample points. Intermediate variables used to obtain the final result.

8. The surrogate model-based metamaterial adapter optimization design method as described in claim 1, characterized in that, In step S6, the mathematical expression of the mechanical property optimization problem is: ； ； In the formula, , , Let x be the characteristic geometric parameter, and let x be the variable. Let be the objective function. These are constraints.

9. The surrogate model-based metamaterial adapter optimization design method as described in claim 8, characterized in that, In step S6, the optimal solution to the polyurethane metamaterial adapter optimization problem is output by combining a variable confidence prediction model with a global optimization algorithm, including: Step 1: Select the improved particle swarm optimization algorithm based on adaptive learning factor as the global optimization algorithm; Step 2: Within the range of values ​​for the characteristic geometric parameters of the metamaterial, update the individual variables according to the optimization update rule, which is the iterative formula for the particles; Step 3: Calculate the predicted response value under each feature geometric parameter based on the variable confidence prediction model; Step 4: Update the learning factor of the algorithm based on the number of iterations; Step 5: Determine if the optimization process has converged. If convergence is achieved, proceed to Step 6. If not, add the learning factor obtained in Step 4 to the iterative formula and proceed to Step 2. Step 6: Output the optimal solution to the polyurethane metamaterial adapter optimization problem.

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