A diversity-driven topology optimization method for metamaterial structures

Abstract

The application discloses a diversity-driven metamaterial structure topology optimization method, and relates to the technical field of metamaterial design optimization, and comprises the following steps: constructing a design set containing a plurality of initial microstructures, and defining design variables; setting a performance retention coefficient; taking the sum of the minimum shape distance between each microstructure in the design set and the remaining microstructures as an optimization target, taking the performance of each microstructure being not lower than the product of the corresponding optimal target performance and the performance retention coefficient as a performance constraint, and constructing a topology optimization equation; deriving the analytical sensitivity of the performance of the microstructure with respect to the design variable and the analytical sensitivity of the shape distance between the microstructures with respect to the design variable; based on the analytical sensitivity, solving the topology optimization equation by using a gradient-based optimization algorithm, and obtaining a plurality of microstructures which satisfy the performance constraint and have shape differences. The application can automatically and efficiently generate a group of microstructures which satisfy the performance requirements and have significant topological shape differences.

Description

Technical Field

[0001] This invention relates to the field of metamaterial design optimization technology, and in particular to a diversity-driven metamaterial structure topology optimization method. Background Technology

[0002] Metamaterials are a new type of composite material that achieves extraordinary physical properties (such as negative Poisson's ratio, ultra-high specific stiffness, and electromagnetic stealth) that are unattainable by conventional materials in nature by artificially designing their internal microstructure configuration, rather than relying on the material's own chemical composition. The design of the microstructure directly determines the macroscopic properties of metamaterials; therefore, developing advanced structural design methods is the core driving force for advancing metamaterials.

[0003] Topology optimization methods, due to their ability to freely generate, merge, or remove materials within a given design domain, offer extremely high design freedom and have become the most promising tools for microstructure design. Traditional topology optimization methods, such as density-based SIMP (Solid Isotropic Material with Penalization), the Level Set Method, and the Two-Way Asymptotic Structure Optimization (BESO) method, are mathematically constructed as a problem of finding a single optimal solution that satisfies a specific objective function (such as minimum compliance or maximum stiffness) under given boundary conditions and constraints. This characteristic of traditional topology optimization methods leads to the following limitations in practical applications: First, the single-solution dilemma. Traditional topology optimization methods ultimately converge to a definite structural configuration, failing to provide designers with a set of candidate solutions with comparable performance but different topological forms. In practical engineering, in addition to mechanical performance, multiple factors such as structural manufacturability, assembly compatibility, thermal coupling characteristics, acoustic coupling characteristics, and aesthetic requirements often mean that a single optimal solution is not the best feasible solution. Designers urgently need multiple alternative solutions for screening and weighing.

[0004] Second, there is an implicit inhibition of innovation. The characteristic of traditional topology optimization methods converging to a single solution greatly limits the opportunities for designers to gain new design inspiration by observing a large number of potential excellent configurations in the early stages of optimization, which is not conducive to the exploration of materials genome and the discovery of disruptive configurations.

[0005] To address these issues, researchers began exploring topology optimization methods capable of generating diverse structural design schemes. Currently, the main types of topology optimization methods for achieving diverse structural designs are as follows: Topology optimization methods based on external intervention involve generating a large number of diverse initial design configurations manually or through algorithms, or performing multiple independent optimizations by changing optimization problem parameters such as load conditions and boundary conditions, thereby obtaining a batch of different solutions. However, these methods do not actively pursue diversity within a single optimization framework, but rather rely on the differences in initial conditions or external parameters. Their results are random and uncontrollable, and it cannot be guaranteed that the multiple solutions obtained are all at an acceptable high-performance level.

[0006] Topology optimization methods based on objective function embedding directly embed the quantification of structural diversity as an objective function or constraint into the optimization formulation, forcing solution separation during the optimization process. However, existing quantification of structural diversity indices often lack a direct, differentiable mathematical coupling with mechanical performance indices, making sensitivity derivation and efficient gradient basis optimization difficult, and hindering the synergistic optimization of both within a unified optimization framework. Summary of the Invention

[0007] To address some or all of the technical problems existing in the prior art, this invention provides a diversity-driven metamaterial structure topology optimization method that can automatically and efficiently generate a set of microstructures with performance that meets preset performance requirements and with significantly different topological morphologies, thereby achieving a controllable balance between structural mechanical properties and structural diversity.

[0008] The technical solution of the present invention is as follows: A diversity-driven metamaterial structure topology optimization method is provided, including: Construct a design set containing multiple initial microstructures and define the design variables for the microstructures; A performance retention factor is set, which is used to quantify the maximum performance loss ratio allowed during the optimization process of the microstructure; The optimization objective is to maximize the sum of the minimum morphological distances between each microstructure in the design set and the other microstructures. The performance constraint is that the performance of each microstructure is not lower than the product of its corresponding optimal target performance and the performance preservation coefficient. Based on the optimization objective and the performance constraint, a topology optimization formula is constructed. In the topology optimization formula, a differentiable approximation function is used to continuously approximate the minimum morphological distance. Based on the chain rule, the analytical sensitivity of the performance of microstructures relative to design variables is derived, as is the analytical sensitivity of the morphological distance between microstructures relative to design variables. Based on the analytical sensitivity, the gradient basis optimization algorithm is used to solve the topology optimization formula to update the design variables of all microstructures and obtain multiple microstructures that meet the performance constraints and have morphological differences.

[0009] Furthermore, in some implementations, the design variable is defined as the density field of the microstructure.

[0010] Furthermore, in some implementations, the morphological distance between microstructures is defined based on the characteristics of the density field of the microstructure in the frequency domain.

[0011] Furthermore, in some embodiments, the performance constraints of the microstructure are set as bulk modulus constraints or shear modulus constraints.

[0012] Furthermore, in some implementations, when the performance constraint is a bulk modulus constraint, the topology optimization formulation is expressed as: ; in, This represents the density field of the first microstructure. Indicates the first Density field of microstructures This indicates the total number of microstructures. and Represents a microstructure index. This indicates the preset control parameters. Indicates the first The microstructure and the first The morphological distance between individual microstructures Indicates the first Density field of microstructures Indicates the first The bulk modulus of a microstructure Indicates the performance retention factor. Indicates the first The optimal target bulk modulus corresponding to each microstructure. This represents the number of rows in the discrete grid cells of the density field. The number of columns represents the discrete grid cells of the density field. Indicates the cell index in the direction of the density field row. Indicates the cell index along the density field column direction. Indicates the first The microstructure located in the first Line 1 The volume of the column's cells. Indicates the first The microstructure located in the first Line 1 The density of cells in a column, Represents the volume of the microstructure. This indicates the preset target volume fraction.

[0013] Furthermore, in some implementations, when the performance constraint is a shear modulus constraint, the topology optimization formulation is expressed as: ; in, This represents the density field of the first microstructure. Indicates the first Density field of microstructures This indicates the total number of microstructures. and Represents a microstructure index. This indicates the preset control parameters. Indicates the first The microstructure and the first The morphological distance between individual microstructures Indicates the first Density field of microstructures Indicates the first The shear modulus of a microstructure Indicates the performance retention factor. Indicates the first The optimal target shear modulus corresponding to each microstructure. This represents the number of rows in the discrete grid cells of the density field. The number of columns represents the discrete grid cells of the density field. Indicates the cell index in the direction of the density field row. Indicates the cell index along the density field column direction. Indicates the first The microstructure located in the first Line 1 The volume of the column's cells. Indicates the first The microstructure located in the first Line 1 The density of cells in a column, Represents the volume of the microstructure. This indicates the preset target volume fraction.

[0014] Furthermore, in some embodiments, the analytical sensitivity of the microstructure's performance relative to design variables is expressed as: ; in: ; ; ; ; The bulk modulus of microstructure The first part representing the microstructure Density of each unit This represents the component of the elastic tensor of the microstructure in the X direction when a unit test strain is applied in the X direction. This represents the component of the elastic tensor of the microstructure in the Y direction when a unit test strain is applied in the Y direction. This represents the component of the elastic tensor of the microstructure in the Y direction when a unit test strain is applied in the X direction. This represents the component of the elastic tensor of the microstructure in the X direction when a unit test strain is applied in the Y direction. This represents the preset penalty factor. Young's modulus represents the solid phase. Describes the Young's modulus of the empty phase. Represents the element stiffness matrix. This indicates the first test strain when a unit test strain is applied in the X direction of the microstructure. Displacement response generated by each element This indicates the first test strain when a unit test strain is applied in the Y direction of the microstructure. Displacement response generated by each element, superscript This represents the matrix transpose operation. The symbol represents the partial derivative.

[0015] Furthermore, in some embodiments, the analytical sensitivity of the morphological distance between the derived microstructures relative to the design variables includes: Determine the analytical sensitivity of the morphological distance between microstructures relative to the amplitude spectrum of the microstructures; Determine the analytical sensitivity of the amplitude spectrum of the microstructure relative to the design variables; Based on the chain rule, the analytical sensitivity of the morphological distance between microstructures relative to the amplitude spectrum of the microstructure and the analytical sensitivity of the amplitude spectrum of the microstructure relative to the design variable are combined to obtain the analytical sensitivity of the morphological distance between microstructures relative to the design variable.

[0016] Furthermore, in some embodiments, the analytical sensitivity of the morphological distance between the microstructures relative to design variables is expressed as: ; in, Indicates the first The microstructure located in the first Line 1 The density of cells in a column, Indicates the first Two-dimensional discrete Fourier transform results of the density field of a microstructure Indicates the first The amplitude spectrum of each microstructure Represents the inverse Fourier transform of a matrix. express The matrix formed by the real parts of each element in the matrix. This represents element-wise matrix multiplication. This indicates that the matrix is ​​divided element by element. The symbol represents the partial derivative.

[0017] Furthermore, in some embodiments, the gradient basis optimization algorithm is an interior-point method.

[0018] The main advantages of the technical solution of this invention are as follows: The diversity-driven metamaterial structure topology optimization method of the present invention maximizes the morphological distance between microstructures by setting a quantifiable performance preservation coefficient and explicitly controlling the upper limit of performance loss of each microstructure. It can automatically and efficiently obtain a set of microstructure design schemes that meet the preset performance requirements and have significantly different topological morphologies through only one gradient basis optimization solution without relying on diverse initial solutions. Thus, it can provide high-performance and highly selective microstructure data for metamaterial design. Attached Figure Description

[0019] The accompanying drawings, which are included to provide a further understanding of embodiments of the invention and constitute a part of this invention, illustrate exemplary embodiments of the invention and, together with their description, serve to explain the invention and do not constitute an undue limitation thereof. In the drawings: Figure 1 A flowchart illustrating a diversity-driven metamaterial structure topology optimization method provided in an embodiment of the present invention; Figure 2 A schematic diagram of a set of initial microstructures provided in an embodiment of the present invention; Figure 3 For based on Figure 2 The diagram shown is a set of microstructures obtained by optimizing the initial microstructure using traditional topology optimization methods. Figure 4 For based on Figure 2 The diagram shown is a set of microstructures obtained by optimizing the initial microstructure using the method provided in the embodiment of the present invention. Figure 5 This is a schematic diagram of another set of initial microstructures provided in an embodiment of the present invention; Figure 6 For based on Figure 5 The diagram shown is a set of microstructures obtained by optimizing the initial microstructure using traditional topology optimization methods. Figure 7 For based on Figure 5The diagram shown is a set of microstructures obtained by optimizing the initial microstructure using the method provided in the embodiments of the present invention. Detailed Implementation

[0020] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below in conjunction with specific embodiments and corresponding drawings. Obviously, the described embodiments are only a part of the embodiments of this invention, and not all of them. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0021] The technical solutions provided by the embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0022] See Figure 1 This invention provides a diversity-driven metamaterial structure topology optimization method, which includes the following steps: Step 1: Construct a design set containing multiple initial microstructures and define the design variables for the microstructures; Step 2: Set the performance retention factor, which is used to quantify the maximum performance loss ratio allowed during the optimization process of the microstructure; Step 3: The optimization objective is to maximize the sum of the minimum morphological distances between each microstructure in the design set and the other microstructures. The performance constraint is that the performance of each microstructure is not lower than the product of its corresponding optimal target performance and performance preservation coefficient. Based on the optimization objective and performance constraints, a topology optimization formula is constructed. In the topology optimization formula, a differentiable approximation function is used to continuously approximate the minimum morphological distance. Step 4: Based on the chain rule, derive the analytical sensitivity of the performance of the microstructure relative to the design variables, and derive the analytical sensitivity of the morphological distance between microstructures relative to the design variables. Step 5: Based on analytical sensitivity, the gradient basis optimization algorithm is used to solve the topology optimization formula to update the design variables of all microstructures and obtain multiple microstructures that meet the performance constraints and have morphological differences.

[0023] In this embodiment of the invention, multiple initial microstructures can be identical, similar, or slightly different. The number of initial microstructures included in the design set is determined based on the actual diversity optimization requirements. For example, it may be necessary to obtain microstructures that meet performance constraints and have morphological differences. If there are 1 microstructure, then the initial number of microstructures in the design set is set to 1. indivual.

[0024] In this embodiment of the invention, an optimization objective function is determined based on the optimization objective, and a performance constraint function is determined based on performance constraints. A topology optimization formula containing both the optimization objective function and the performance constraint function is then constructed. Specifically, in the optimization objective function, a differentiable approximation function is used to continuously approximate the minimum morphological distance to ensure the differentiability of the optimization objective function. The differentiable approximation function is selected according to actual needs, for example, the KS (Kreisselmeier-Steinhauser) function.

[0025] In this embodiment of the invention, based on the analytical sensitivity of the performance of the derived microstructures relative to the design variables, and the analytical sensitivity of the morphological distance between microstructures relative to the design variables, the gradient basis optimization algorithm is used to solve the topology optimization formula, thereby updating the design variables of all microstructures in the design set, obtaining multiple microstructures that satisfy the performance constraints and have morphological differences, and the obtained multiple microstructures are the final optimization results.

[0026] The diversity-driven metamaterial structure topology optimization method provided in this invention maximizes the morphological distance between microstructures by setting a quantifiable performance preservation coefficient and explicitly controlling the upper limit of performance loss for each microstructure. It can automatically and efficiently obtain a set of microstructure design schemes that meet preset performance requirements and have significantly different topological morphologies through only one gradient basis optimization solution without relying on diverse initial solutions. This provides high-performance and highly selective microstructure data for metamaterial design.

[0027] Furthermore, in this embodiment of the invention, a density field-based topology optimization framework is adopted, thus defining the design variable of the microstructure as the density field of the microstructure.

[0028] In this embodiment of the invention, when the design variable of the microstructure is defined as the density field of the microstructure, the morphological distance between microstructures is defined based on the characteristics of the density field of the microstructure in the frequency domain.

[0029] Specifically, the morphological distance between two microstructures is defined as the weighted difference of the amplitude spectra corresponding to the two-dimensional discrete Fourier transform results of the density fields of the two microstructures. Specifically, the amplitude spectrum is extracted from the two-dimensional discrete Fourier transform of the density field of the microstructures to obtain the spectrum.

[0030] Furthermore, in this embodiment of the invention, based on the design variables and optimization objectives set above, the optimization objective function can be expressed as: ; in, This represents the density field of the first microstructure. Indicates the first Density field of microstructures This indicates the total number of microstructures. and Represents a microstructure index. Indicates the first The microstructure and the first The morphological distance between individual microstructures.

[0031] To ensure the differentiability of the objective function, this embodiment of the invention uses the KS function to perform a continuous approximation of the minimum morphological distance, resulting in the following objective function: ; in, This represents the density field of the first microstructure. Indicates the first Density field of microstructures This indicates the total number of microstructures. and Represents a microstructure index. This indicates the preset control parameters. Indicates the first The microstructure and the first The morphological distance between individual microstructures This represents an exponential function with the natural constant e as its base.

[0032] Among them, control parameters This parameter is used to control the approximation accuracy; the larger the value, the better the approximation. The specific value of the control parameter is set according to actual needs, for example, 50~60.

[0033] Furthermore, in this embodiment of the invention, the performance and performance constraints of the microstructure are set according to actual needs. For example, the performance of the microstructure can be set as bulk modulus or shear modulus.

[0034] In this embodiment of the invention, when the performance is set to bulk modulus and the performance constraint is set to bulk modulus constraint, based on the above-set design variables, optimization objectives, and performance constraints, the following topology optimization formula is constructed: ; in, This represents the density field of the first microstructure. Indicates the first Density field of microstructures This indicates the total number of microstructures. and Represents a microstructure index. This indicates the preset control parameters. Indicates the first The microstructure and the first The morphological distance between individual microstructures Indicates the first Density field of microstructures Indicates the first The bulk modulus of a microstructure Indicates the performance retention factor. Indicates the first The optimal target bulk modulus corresponding to each microstructure. This represents the number of rows in the discrete grid cells of the density field. The number of columns represents the discrete grid cells of the density field. Indicates the cell index in the direction of the density field row. Indicates the cell index along the density field column direction. Indicates the first The microstructure located in the first Line 1 The volume of the column's cells. Indicates the first The microstructure located in the first Line 1 The density of cells in a column, Represents the volume of the microstructure. This indicates the preset target volume fraction.

[0035] In this embodiment of the invention, the optimal target bulk modulus represents the maximum bulk modulus obtained by structural topology optimization of the microstructure without considering diversity. Specifically, when diversity is not considered, traditional topology optimization methods are used to optimize the microstructure's structural topology.

[0036] In this embodiment of the invention, when the performance is set to shear modulus and the performance constraint is set to shear modulus constraint, based on the above-set design variables, optimization objectives, and performance constraints, the following topology optimization formula is constructed: ; in, This represents the density field of the first microstructure. Indicates the first Density field of microstructures This indicates the total number of microstructures. and Represents a microstructure index. This indicates the preset control parameters. Indicates the first The microstructure and the first The morphological distance between individual microstructures Indicates the first Density field of microstructures Indicates the first The shear modulus of a microstructure Indicates the performance retention factor. Indicates the first The optimal target shear modulus corresponding to each microstructure. This represents the number of rows in the discrete grid cells of the density field. The number of columns represents the discrete grid cells of the density field. Indicates the cell index in the direction of the density field row. Indicates the cell index along the density field column direction. Indicates the first The microstructure located in the first Line 1 The volume of the column's cells. Indicates the first The microstructure located in the first Line 1 The density of cells in a column, Represents the volume of the microstructure. This indicates the preset target volume fraction.

[0037] In this embodiment of the invention, the optimal target shear modulus represents the maximum shear modulus obtained by structural topology optimization of the microstructure without considering diversity. Specifically, when diversity is not considered, traditional topology optimization methods are used to optimize the microstructure's structural topology.

[0038] In this embodiment of the invention, by solving the constructed topology optimization formula, the design variables of all microstructures can be updated, thereby obtaining multiple microstructures that meet performance constraints and have significantly different morphologies.

[0039] Furthermore, in this embodiment of the invention, in order to facilitate the solution of the topology optimization formula using the gradient basis optimization algorithm and accelerate the optimization solution process, it is necessary to first determine the analytical sensitivity of the performance of the microstructure relative to the design variables, and the analytical sensitivity of the morphological distance between microstructures relative to the design variables.

[0040] In this embodiment of the invention, based on the chain rule, the analytical sensitivity of the performance of the microstructure relative to the design variables is derived, as well as the analytical sensitivity of the morphological distance between microstructures relative to the design variables is derived.

[0041] In this embodiment of the invention, the performance of the microstructure is set as the bulk modulus or shear modulus of the microstructure. Therefore, the analytical sensitivity of the performance of the microstructure relative to the design variables is the analytical sensitivity of the bulk modulus or shear modulus of the microstructure relative to the design variables.

[0042] In this embodiment of the invention, the performance of the microstructure is calculated using a homogenization method. Furthermore, due to the anisotropic nature of the microstructure, when the performance of the microstructure is set to the bulk modulus, the analytical sensitivity of the microstructure's performance relative to the design variables, i.e., the analytical sensitivity of the microstructure's bulk modulus relative to the design variables, can be expressed as: ; When the performance of the microstructure is set to the shear modulus, the analytical sensitivity of the microstructure's performance relative to the design variables, that is, the analytical sensitivity of the microstructure's shear modulus relative to the design variables, can be expressed as: ; in: ; ; ; ; ; The bulk modulus of microstructure The first part representing the microstructure Density of each unit This represents the component of the elastic tensor of the microstructure in the X direction when a unit test strain is applied in the X direction. This represents the component of the elastic tensor of the microstructure in the Y direction when a unit test strain is applied in the Y direction. This represents the component of the elastic tensor of the microstructure in the Y direction when a unit test strain is applied in the X direction. This represents the component of the elastic tensor of the microstructure in the X direction when a unit test strain is applied in the Y direction. This represents the in-plane shear stress generated by the microstructure when a unit tensor shear strain is applied within the XY plane. Represents the volume of the microstructure. This represents the preset penalty factor. Young's modulus represents the stiffness of the solid phase, which is the stiffness of the basic material that constitutes the microstructure. Describes the Young's modulus of the empty phase. Represents the element stiffness matrix. This indicates the first test strain when a unit test strain is applied in the X direction of the microstructure. Displacement response generated by each element This indicates the first test strain when a unit test strain is applied in the Y direction of the microstructure. Displacement response generated by each element This represents the first tensor shear strain applied in the XY plane of the microstructure. Displacement response generated by each element, superscript This represents the matrix transpose operation. The symbol represents the partial derivative.

[0043] Among them, the penalty factor The value of is greater than 1, and its function is to penalize the intermediate density value, so that the unit density approaches 0 or 1.

[0044] To avoid numerical singularities, the Young's modulus of the empty phase is... Instead of setting it to 0, a very small positive value is taken to simulate the extremely low stiffness of the unmaterialized region.

[0045] In this structure, the X-direction corresponds to the row direction of the density field, and the Y-direction corresponds to the column direction of the density field.

[0046] Furthermore, based on the morphological distance between microstructures defined above, the analytical sensitivity of the morphological distance between microstructures relative to design variables is derived as follows: Determine the analytical sensitivity of the morphological distance between microstructures relative to the amplitude spectrum of the microstructures; Determine the analytical sensitivity of the amplitude spectrum of the microstructure relative to the design variables; Based on the chain rule, the analytical sensitivity of the morphological distance between microstructures relative to the amplitude spectrum of the microstructure and the analytical sensitivity of the amplitude spectrum of the microstructure relative to the design variable are combined to obtain the analytical sensitivity of the morphological distance between microstructures relative to the design variable.

[0047] In this embodiment of the invention, based on the morphological distance between the two microstructures defined above, the morphological distance between the two microstructures can be expressed as: ; in, Indicates the first The microstructure and the first The morphological distance between individual microstructures and Represents a microstructure index. Represents the spatial index in the frequency domain. Represents the row index in the frequency domain. Column index representing the frequency domain, This represents the number of rows in the discrete grid cells of the density field. The number of columns represents the discrete grid cells of the density field. Represents the preset frequency weighting matrix The Middle Line 1 Column elements, Indicates the first Amplitude spectrum of microstructures The Middle Line 1 Column elements, Indicates the first Amplitude spectrum of microstructures The Middle Line 1 Column elements In this embodiment of the invention, considering that the contributions of each frequency component in the amplitude spectrum to the structural differences of the microstructure are not equal, with low-frequency components dominating the global load-bearing skeleton and overall mechanical behavior, while high-frequency components mainly characterize local details with minimal impact on mechanical performance, treating all frequency components equally could lead to the diversity distance calculation results being dominated by high-frequency noise. Therefore, to avoid interference from high-frequency noise in the diversity distance and to highlight physically meaningful structural differences, a frequency weighting matrix is ​​introduced. Among them, the frequency weighting matrix Determined in the following ways: ; in, Represents the preset frequency weighting matrix The Middle Line 1 Column elements, This represents the corrected spatial index. , , This represents the preset penalty intensity control parameter, which controls the penalty intensity applied to the frequency components of the amplitude spectrum. The specific value of the penalty intensity control parameter is set according to actual needs; the smaller the value, the stronger the penalty applied to the frequency components. For example, set it to 20.

[0048] In this embodiment of the invention, based on the above definition, the analytical sensitivity of the morphological distance between microstructures relative to the amplitude spectrum of the microstructure is expressed as follows: ; in, Indicates the first The microstructure and the first The morphological distance between individual microstructures Indicates the first The amplitude spectrum of each microstructure This represents a preset frequency weighting matrix. Indicates the first The amplitude spectrum of each microstructure This represents element-wise matrix multiplication. The symbol represents the partial derivative.

[0049] In this embodiment of the invention, the analytical sensitivity of the amplitude spectrum of the microstructure relative to the design variables is determined by the following method: set up Indicates the first Two-dimensional discrete Fourier transform results of the density field of a microstructure If is a complex matrix, then the ... Amplitude spectrum of microstructures Based on the properties of complex numbers, we can obtain: ; in, Indicates the first The microstructure located in the first Line 1 The density of cells in a column, express conjugate, This indicates that the matrix is ​​divided element by element.

[0050] because It can be represented as: ; in, express The matrix formed by the real parts of each element in the matrix. express The matrix formed by the imaginary parts of each element in the matrix. It represents the imaginary unit.

[0051] Therefore, we can obtain: ; in, express conjugate, express A matrix formed by the real parts of each element in a matrix.

[0052] Based on the definition of the two-dimensional discrete Fourier transform, we can obtain: ; in, Representation matrix The Middle Line 1 Column elements, Represents the spatial index in the frequency domain. Represents the row index in the frequency domain. Column index representing the frequency domain, Represents the natural constant. This represents the number of rows in the discrete grid cells of the density field. The number of columns represents the discrete grid cells of the density field. Indicates the cell index in the direction of the density field row. Indicates the cell index along the density field column direction. It represents the imaginary unit.

[0053] Define an intermediate matrix And the intermediate matrix The element is set to: ; in, Representing the intermediate matrix The Line 1 The elements of the column.

[0054] Based on the intermediate matrix defined above, we can obtain: ; in, express A matrix formed by the real parts of each element in a matrix.

[0055] Furthermore, in this embodiment of the invention, based on the chain rule, by combining the analytical sensitivity of the morphological distance between microstructures with respect to the amplitude spectrum of the microstructure and the analytical sensitivity of the amplitude spectrum of the microstructure with respect to the design variable, we can obtain: ; in, express transpose, Indicates length is A column vector whose elements are all 1. Indicates length is A column vector whose elements are all 1s.

[0056] Furthermore, the formula and formula Substitute into the formula It can then be found that, This can be represented using the inverse Fourier transform, therefore... This can be further expressed as: ; in, Represents the inverse Fourier transform of a matrix. express A matrix formed by the real parts of each element in a matrix.

[0057] Furthermore, in this embodiment of the invention, the gradient basis optimization algorithm employs the interior-point method. Specifically, based on the analytical sensitivity derived above, the interior-point method is used to solve the topology optimization formulation to update the design variables of all microstructures, thereby obtaining multiple microstructures that satisfy performance constraints and have morphological differences. Since the interior-point method is common knowledge in the art, it will not be elaborated upon here.

[0058] Furthermore, in this embodiment of the invention, when solving the topology optimization formula using the gradient basis optimization algorithm, in order to prevent the sensitivity of the optimization objective function to the design variables, i.e. the analytical sensitivity of the morphological distance between microstructures to the design variables, from being too small in the first iteration, a random perturbation is applied to the analytical sensitivity of the morphological distance between microstructures to the design variables in the first iteration.

[0059] refer to Figures 2-4 , Figure 2 This is a schematic diagram of a set of initial microstructures provided in an embodiment of the present invention. Figure 3 For based on Figure 2 The diagram shown illustrates a set of microstructures obtained by optimizing the initial microstructure using traditional topology optimization methods. Figure 4 For based on Figure 2 The diagram shown illustrates a set of microstructures obtained by optimizing the initial microstructure using the method provided in this embodiment of the invention. When optimizing using the method provided in this embodiment of the invention, the performance is set to bulk modulus, the performance retention factor is set to 0.9, the target volume fraction is set to 0.6, and the penalty factor is set to 7.5. According to... Figures 2-4 It can be seen that when using the traditional topology optimization method, the topological shapes of the three microstructure optimization results are basically the same. However, when using the method provided in this embodiment of the invention, three microstructure optimization results with different topological shapes can be obtained, and the performance of all three can meet the requirements.

[0060] refer to Figures 5-7 , Figure 5 This is a schematic diagram of another set of initial microstructures provided in an embodiment of the present invention. Figure 6 For based on Figure 5 The diagram shown illustrates a set of microstructures obtained by optimizing the initial microstructure using traditional topology optimization methods. Figure 7 For based on Figure 5 The diagram shown illustrates a set of microstructures obtained by optimizing the initial microstructure using the method provided in this embodiment of the invention. When optimizing using the method provided in this embodiment of the invention, the performance is set to shear modulus, the performance retention coefficient is set to 0.9, the target volume fraction is set to 0.6, and the penalty factor is set to 8.0. According to... Figures 5-7It can be seen that when using the traditional topology optimization method, the topological shapes of the three microstructure optimization results are basically the same. However, when using the method provided in this embodiment of the invention, three microstructure optimization results with different topological shapes can be obtained, and the performance of all three can meet the requirements.

[0061] It should be noted that, in this document, relational terms such as “first” and “second” are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms “comprising,” “including,” or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus.

[0062] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A diversity-driven metamaterial structure topology optimization method, characterized in that, include: Construct a design set containing multiple initial microstructures and define the design variables for the microstructures; A performance retention factor is set, which is used to quantify the maximum performance loss ratio allowed during the optimization process of the microstructure; The optimization objective is to maximize the sum of the minimum morphological distances between each microstructure in the design set and the other microstructures. The performance constraint is that the performance of each microstructure is not lower than the product of its corresponding optimal target performance and the performance preservation coefficient. Based on the optimization objective and the performance constraint, a topology optimization formula is constructed. In the topology optimization formula, a differentiable approximation function is used to continuously approximate the minimum morphological distance. Based on the chain rule, the analytical sensitivity of the performance of microstructures relative to design variables is derived, as is the analytical sensitivity of the morphological distance between microstructures relative to design variables. Based on the analytical sensitivity, the gradient basis optimization algorithm is used to solve the topology optimization formula to update the design variables of all microstructures and obtain multiple microstructures that meet the performance constraints and have morphological differences.

2. The diversity-driven metamaterial structure topology optimization method according to claim 1, characterized in that, The design variable is defined as the density field of the microstructure.

3. The diversity-driven metamaterial structure topology optimization method according to claim 2, characterized in that, The morphological distance between microstructures is defined based on the characteristics of the density field of the microstructure in the frequency domain.

4. A diversity-driven metamaterial structure topology optimization method according to claim 2 or 3, characterized in that, The performance constraints of the microstructure are set as either bulk modulus constraints or shear modulus constraints.

5. The diversity-driven metamaterial structure topology optimization method according to claim 4, characterized in that, When the performance constraint is a bulk modulus constraint, the topology optimization formula is expressed as: ； in, This represents the density field of the first microstructure. Indicates the first Density field of microstructures This indicates the total number of microstructures. and Represents a microstructure index. This indicates the preset control parameters. Indicates the first The microstructure and the first The morphological distance between individual microstructures Indicates the first Density field of microstructures Indicates the first The bulk modulus of a microstructure Indicates the performance retention factor. Indicates the first The optimal target bulk modulus corresponding to each microstructure. This represents the number of rows in the discrete grid cells of the density field. The number of columns represents the discrete grid cells of the density field. Indicates the cell index in the direction of the density field row. Indicates the cell index along the density field column direction. Indicates the first The microstructure located in the first Line 1 The volume of the column's cells. Indicates the first The microstructure located in the first Line 1 The density of cells in a column, Represents the volume of the microstructure. This indicates the preset target volume fraction.

6. The diversity-driven metamaterial structure topology optimization method according to claim 4, characterized in that, When the performance constraint is a shear modulus constraint, the topology optimization formula is expressed as: ； in, This represents the density field of the first microstructure. Indicates the first Density field of microstructures This indicates the total number of microstructures. and Represents a microstructure index. This indicates the preset control parameters. Indicates the first The microstructure and the first The morphological distance between individual microstructures Indicates the first Density field of microstructures Indicates the first The shear modulus of a microstructure Indicates the performance retention factor. Indicates the first The optimal target shear modulus corresponding to each microstructure. This represents the number of rows in the discrete grid cells of the density field. The number of columns represents the discrete grid cells of the density field. Indicates the cell index in the direction of the density field row. Indicates the cell index along the density field column direction. Indicates the first The microstructure located in the first Line 1 The volume of the column's cells. Indicates the first The microstructure located in the first Line 1 The density of cells in a column, Represents the volume of the microstructure. This indicates the preset target volume fraction.

7. The diversity-driven metamaterial structure topology optimization method according to claim 5, characterized in that, The analytical sensitivity of the performance of the microstructure relative to the design variables is expressed as: ； in: ； ； ； ； The bulk modulus of microstructure The first part representing the microstructure Density of each unit This represents the component of the elastic tensor of the microstructure in the X direction when a unit test strain is applied in the X direction. This represents the component of the elastic tensor of the microstructure in the Y direction when a unit test strain is applied in the Y direction. This represents the component of the elastic tensor of the microstructure in the Y direction when a unit test strain is applied in the X direction. This represents the component of the elastic tensor of the microstructure in the X direction when a unit test strain is applied in the Y direction. This represents the preset penalty factor. Young's modulus represents the solid phase. Describes the Young's modulus of the empty phase. Represents the element stiffness matrix. This indicates the first test strain when a unit test strain is applied in the X direction of the microstructure. Displacement response generated by each element This indicates the first test strain when a unit test strain is applied in the Y direction of the microstructure. Displacement response generated by each element, superscript This represents the matrix transpose operation. The symbol represents the partial derivative.

8. The diversity-driven metamaterial structure topology optimization method according to claim 4, characterized in that, The analytical sensitivity of the morphological distance between the derived microstructures relative to the design variables includes: Determine the analytical sensitivity of the morphological distance between microstructures relative to the amplitude spectrum of the microstructures; Determine the analytical sensitivity of the amplitude spectrum of the microstructure relative to the design variables; Based on the chain rule, the analytical sensitivity of the morphological distance between microstructures relative to the amplitude spectrum of the microstructure and the analytical sensitivity of the amplitude spectrum of the microstructure relative to the design variable are combined to obtain the analytical sensitivity of the morphological distance between microstructures relative to the design variable.

9. The diversity-driven metamaterial structure topology optimization method according to claim 8, characterized in that, The analytical sensitivity of the morphological distance between the microstructures relative to the design variables is expressed as follows: ； in, Indicates the first The microstructure located in the first Line 1 The density of cells in a column, Indicates the first Two-dimensional discrete Fourier transform results of the density field of a microstructure Indicates the first The amplitude spectrum of each microstructure Represents the inverse Fourier transform of a matrix. express The matrix formed by the real parts of each element in the matrix. This represents element-wise matrix multiplication. This indicates that the matrix is ​​divided element by element. The symbol represents the partial derivative.

10. The diversity-driven metamaterial structure topology optimization method according to claim 1, characterized in that, The gradient basis optimization algorithm is the interior point method.

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